ICRF Antenna Coupling and Wave Propagation in a Tokamak Plasma - CaltechTHESIS

ICRF Antenna Coupling and Wave Propagation in a Tokamak Plasma - CaltechTHESIS
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ICRF Antenna Coupling and Wave Propagation in a Tokamak Plasma
Citation
Greene, Glenn Joel
(1984)
ICRF Antenna Coupling and Wave Propagation in a Tokamak Plasma.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/ppan-gp26.
Abstract
A variety of experiments are reported pertaining to the excitation, propagation, and damping of waves in the ion cyclotron range of frequencies (ICRF) in the Caltech Research Tokamak.
Complex impedance studies on five different RF antennas addressed the nature of the anomalous density-dependent background loading observed previously in several laboratories. A model was proposed which successfully explained many of the observed impedance characteristics solely in terms of particle collection and rectification through the plasma sheath surrounding the antenna electrode. Peaks were observed on the input resistance of the shielded antennas and were coincident with toroidal eigenmode production; their magnitude was explained by a simple coupling theory.
The toroidal eigenmodes were studied in detail with magnetic field probes. The mode dispersion curves in density-frequency space were mapped out and the results compared with various theoretical models. A surprising result was that all of the antennas, both magnetic and electric in nature, coupled to the eigenmodes with comparable efficiency with respect to the antenna excitation current. Wave damping was investigated and found to be considerably higher than predicted by a variety of physical mechanisms. A numerical model of the wave equations permitting an arbitrary radial density profile was developed, and a possible mechanism for enhanced cyclotron damping due to density perturbations was proposed. Toroidal modes were identified using phase measurements between pairs of magnetic probes; they were found to have
= 1 poloidal character and low integral toroidal mode numbers, in accordance with theoretical predictions.
A new approach to the study of ICRF wave propagation was investigated: wave-packets were launched and their propagation was followed around the tokamak using magnetic probes. This technique avoided the dominant effect of the eigenmode resonances because it observed propagation on a time scale short compared to the formation time for the modes. The transit time of the packets around the machine yielded the toroidal group velocity, and the results of the experiments were compared with several theoretical models. The inclusion of a vacuum layer at the plasma edge was useful in explaining some of the observations.
Finally, a plasma-compatible Rogowski current probe was developed and used to observe, for the first time, RF particle current in a tokamak plasma. The diagnostic permitted investigation of the spatial form of the RF current driven in the edge plasma by the electric field antennas. The results dramatically showed that the current from these antennas flows largely along the toroidal field lines. This highly localized current distribution suggests a mechanism for the good coupling to the eigenmode fields observed with these antennas.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics)
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Gould, Roy Walter
Thesis Committee:
Gould, Roy Walter (chair)
Bellan, Paul Murray
Corngold, Noel Robert
Bridges, William B.
Phillips, Thomas G.
Defense Date:
9 February 1984
Funders:
Funding Agency
Grant Number
Westinghouse Electric
UNSPECIFIED
Earle C. Anthony Foundation
UNSPECIFIED
Rockwell
UNSPECIFIED
Department of Energy (DOE)
DE-AS03-76SF-00767
Record Number:
CaltechETD:etd-11102005-144119
Persistent URL:
DOI:
10.7907/ppan-gp26
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No commercial reproduction, distribution, display or performance rights in this work are provided.
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4494
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10 Nov 2005
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17 Jul 2023 23:35
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ICWF ANTENNA COUPLING AND WAVE PROPAGATION
IN A TOKAMAK PLASMA

Thesis by
Glenn Joel Greene

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California
1984
(Submitted February 9, 1984)

@ 1984
Glenn Joel Greene

This thesis is dedicated
to the memory of m y mother, Sylvia,
and to m y father, Franklin

I would like to thank my thesis advisor, Dr. Roy Gould, for his guidance
during the course of this work, and for the benefit of his clear insight into
many aspects of physics.

I enjoyed discussions with Eric Fredrickson, Larry Sverdrup, and Dr. Mark
Hedemann on a variety of topics, including, occasionally, plasma physics. I am
indebted to Dr. Mark Dolson for a clear introduction to digital sgnal processing, as well as for providing company during many late evenings on the second
floor of Steele laboratories.
The facilities and staff of C.E.S. (Central Engineering Services) proved
invaluable during the experimental phase of t h s work. In particular, I extend
my sincere thanks to Louis Johnson, Asst. Manager of C.E.S., for services which
went far beyond his official duties. On many occasions when an experiment
was halted for lack of a part (particularly before a long holiday), Louis
somehow found a way to have it produced, I have also benefited from his
tutorials on materials technology and production technique. I am indebted to
Norman Keidel, Manager of C.E.S., for h s help on many projects, and to staff
members Michael Gerfen, Marty Gould, Ralph Ortega, and Ricardo Paniagua for
their skillfull work. I thank Herb Adarns for demonstrating that function can
indeed still have form.

I would also like to thank Frank Cosso for h s elegant work on the power
and diagnostic systems of the tokamak, on the data acquisition system, and on
the laboratory computers. His expertise in electronics was a valuable asset to
the laboratory.

To all of my friends from the Keck House Colony, particularly Dr.'s Mark
Dolson, Neil Gehrels, Jack Kaye, Ken McCue, Andy Pesthy, Satwindar Sadhal,
Betty Vermeire, Vatchc! VorpGrian, and Ellen Williams, my appreciation for contributing to an enjoyable community. I am also grateful to Mrs. Marilyn
Chandler for her generous hospitality during my stay in her home.

I genuinely enjoyed my interactions with Dr. James Hudspeth, who
catalyzed my interest in neurobiology and provided many stimulating &scussions. I extend my appreciation as well to Edith Huang, who was proficient in
eliminating many problems associated with the typesetting program on which
this thesis was produced.
Organizations which provided support for this research include the Westinghouse Corporation, the Earle C. Anthony Foundation, Rockwell International
Corporation, and the U.S. Department of Energy.

I am grateful to my family and relatives for their continued support and
encouragement daring my stay at Caltech.
Finally, I thank Wendy All for apprehending me thirteen years ago a t the
San Diego Science Fair, Her return to my life has made these last difficult
years more enjoyable. I am also grateful to her for the many excellent illustrations whch enhance t h s thesis.

ABSTRACT
A variety of experiments are reported pertaining to the excitation,

propagation, and damping of waves in the ion cyclotron range of frequencies
(ICRF) in the Caltech Research Tokamak.
Complex impedance studies on five different RF antennas addressed the
nature of the anomalous density-dependent background loading observed previously in several laboratories. A model was proposed which successfully
explained many of the observed impedance characteristics solely in terms of
particle collection and rectification through the plasma sheath surrounding
the antenna electrode. Peaks were observed on the input resistance of the
shielded antennas and were coincident with toroidal eigenmode production;
their magnitude was explained by a simple coupling theory.
The t,nrnidal eigenmndes were skudied in

kt_hmagnetic field probes,

The mode dispersion curves in density-frequency space were mapped out and
the results compared with various theoretical models. A surprising result was
that all of the antennas, both magnetic and electric in nature, coupled to the
eigenmodes with comparable efficiency with respect to the antenna excitation
current. Wave damping was investigated and found to be considerably higher
than predicted by a variety of physical mechanisms. A numerical model of the
wave equations permitting an arbitrary radial density profile was developed,
and a possible mechanism for enhanced cyclotron damping due to density perturbations was proposed. Toroidal modes were identified using phase measurements between pairs of magnetic probes; they were found to have m = 1
poloidal character and low integral toroidal mode numbers, in accordance with
theoretical predictions.

A new approach to the study of ICRF wave propagation was investigated:

wave-packets were launched and their propagation was followed around the
tokamak using magnetic probes. This technique avoided the dominant effect
of the eigenmode resonances because it observed propagation on a time scale

short compared to the formation time for the modes. The transit time of the
packets around the machine ylelded the toroidal group velocity, and the
results of the experiments were compared with several theoretical models. The
inclusion of a vacuum layer a t the plasma edge was useful in explaining some
of the observations.
Finally, a plasma-compatible Rogowsla current probe was developed and
used to observe, for the first time, RF particle current in a tokamak plasma.
The &agnostic permitted investigation of the spatial form of the RF current
driven in the edge plasma by the electric field antennas. The results dramatically showed that the current from these antennas flows largely along the
toroidal field lines. T h s hghly localized current distribution suggests a
mechanism for the good coupling to the eigenrnode fields observed with these
antennas.

3.4 Data Acquisition ......................................................................................
97
3.5 Diagnostics ..............................................................................................100

3.6 Discharge Cleaning ................................................................................
102
3.7 Gas Puffing..............................................................................................

104

3.8 Plasma Characteristics ......................................................................

105

3.9 Charge Exchange Diagnostic .............................................................. 108
3.9.1 Introduction ................................................................................
108
3.9.2 Design and Construction ...........................................................110
3.9.3 Operation .................................................................................... 116

TV. RF Apparatus

.............................................................................................

121

4.1 RF System Design ..................................................................................
121
4.1.1 RF Exciters ..................................................................................
121
4.1.2 System Grounding ......................................................................
123
4.1.3 RF Shielding ................................................................................. 124

4.2 High Power Amplifier ............................................................................ 128
4.2.1 Introduction ................................................................................ 128
4.2.2 RF Design ................................................................................... 129
4.2.3 Construction ............................................................................. 131
4.2.4 Operation .....................................................................................
141
4.3 Directional Coupler ...............................................................................142
4.4 Impedance Matching Network ......................................................... 148
4.4.1 Introduction ................................................................................
148
4.4.2 Design .........................
.........................................................
151
4.4.3 Construction .............................
. ...........................................156
4.5 bnear RF Detectors ..........................................................................

160

4.6 RF Phase Detectors ............................................................................. 162
4.6.1 Introduction ...............................
.......................................... 162
4.6.2 DBM Phase Detector ........................ .
................................. 163
..................................165
4.6.3 Digital Phase Detector .......................
4.6.4 Harmonic Distortion .................................................................. 168
4.6.5 Dynamic Range............................................................................ 169

V . RF Antennas and Plasma Probes........................................................

172

5.1 RF Antennas ........................................................................................... 173

.....................................
5.1.1 Antenna Feedthru Assembly ........... .
173
5.1.2 Bare h o p Antenna .....................................................................
180
5.1.3 Ceramic-Insulated Loop Antenna .............................................
182
5.1.4 Faraday-Shielded Loop Antenna ..............................................
184

7.2 Wave-Packet Experiment ................................................................. 365
7.2.1 Introduction ............................................................................
365

7.2.2 Experimental Method ................................................................
366
7.2.3 Experimental Data .................................................................. 369
7.2.4 Theoretical Model ......................................................................
301

7.3 RF Current Probe Experiment ..........................................................395
7.3.1 Experimental Method .................................................................395
7.3.2 Experimental Results .........................
.................................
398

VIII. Summary and Conclusions..................................................................
405

Appendix A: Coupling Efficiency to ICRF Toroidal Eigenrnodes
and Transmission between Two Identical Antennas ..............411
Appendix B: Transient RF Heating of a Conducting Cylinder.....420
References .................................................................................................................
437

CHAPTER I
Introduction

Controlled thermonuclear fusion promises to provide a virtually unlimited
source of energy, provided the enormous technical challenges can be overcome. Among the current schemes to achieve a demonstration of scientific
feasibility, the tokamak approach has been the most promising.
In order to achieve the desired goal of a net energy yleld from a controlled
fusion reaction, the plasma must be both c o n k e d for a sufficient period of
time and heated to a sufficient temperature. In the tokamak configuration of
magnetic confinement, the poloidal field whch provides the rotational
transform is generated by driving a large current toroidally around the
machne. Besides providing one component of the necessary confinement field,
this current provides initial heating of the plasma through collisional losses.
The conductivity of a high-temperature plasma increases as the temperature rises, and the efficiency of this ohmic-heating consequently decreases.
The maximum permissible toroidal current is limited by the onset of MHD
instabilities, and it has long been recognized that some form of supplemental
heating will be necessary to achieve ignition temperatures [Artsimovich, 19721.
Neutral-beam heating (injection of high-energy neutralized particle beams
across the toroidal field lines) has been used successfully [Hawryluk, 1981;
Fonck, 19031, and it is projected that it d l be sufficient to reach breakeven in
the TFTR tokamak a t Princeton within a few years. Nevertheless, it is possible
that neutral-beam heating will not be easily or economically scalable to reac-

t o r regimes [Furth, 19831, and alternate plasma heating schemes are being
actively pursued.
One of the most promising supplemental heating mechanisms involves the
u s e of electromagnetic waves near t h e ion cyclotron frequency. To first order,
t h e plasma particles execute helical orbits about the magnetic field lines, precessing a t their respective cyclotron frequencies. If a circularly polarized electric field with the appropriate phase and frequency could be somehow imposed
in the plasma volume, the particles would be accelerated by the wave field and
would gain energy. This approach is particularly attractive since energy can be
deposited directly in the ion population. The plasma, however, is a complex,
anisotropic m e h a , and one cannot in general impose a prescribed field.
Rather, a wave is launched from some point on the plasma boundary, and it
propagates into the plasma, being damped as it travels. Careful consideration

of the properties of the waves being excited is necessary to insure that energy
is deposited in the desired region. Note that because the toroidal magnetic
field falls off as 1 / R , where R is t h e major radius, resonant cyclotron effects
take place on a cylindrical surface of constant major radius where the cyclotron frequency (which is now a function of major ra&us) becomes equal to the
excitation frequency.
In the ion-cyclotron range of frequencies (ICRF), cold plasma theory
predicts that, in a hydrogenic plasma, two waves can propagate: the so-called
fast and slow waves (see Chapter 2). The slow wave, also called the ion cyclotron wave, has a resonance (wavenumber -, -) a t the ion cyclotron frequency
and does not propagate immediately above it. For propagation below the ion
cyclotron frequency and along the static magnetic field &rection, however, the
electric field of the slow wave is completely left-hand circularly polarized, i.e.,
polarized in the same sense as the ion precession. The earliest attempts to use
ICRF waves to heat a plasma utilized the slow wave in a stellerator device [Stix

and Pallahno, 19601. The waves were launched in a region where the excitation frequency ( w) was less than the cyclotron frequency ( w~ ). The slow wave
was excited and propagated into a magnetic "beach region where the
confinement field decreased until w > w~ ; wave energy was absorbed by the
plasma at the resonant ( w = wd ) layer. In a tokamak geometry, however, the
static magnetic field gradient is in the radial hrection, and calculations suggest that the slow wave does not propagate to the plasma core but is rather
absorbed in the outer plasma region [Klima, 1975; Colestock, 19831.
The other propagating wave is called the fast or magnetosonic wave; at
h g h e r frequencies it becomes the whstler or electron cyclotron wave. This
wave does not have a resonance a t w = w ~and easily propagates throughout
t h e plasma. At the fundamental cyclotron frequency, however, the electric
field of the fast wave is completely right-hand circularly polarized with respect
to the static magnetic field; hence no fundamental cyclotron absorption might
be expected. Inclusion of warm-plasma effects prehcts a small left-hand component of the wave electric field a t w = wd ; the absorption, however, is calculated to be very small (for a hydrogenic plasma). Nevertheless, a variety of
other damping mechanisms exist by which the plasma particles can absorb
energy from the fast wave (see section 2.3), and the use of ICRF fast waves to
heat tokamak plasmas has been successful in some laboratories.
Introduction of a second ion species results in new resonances (e.g., the
ion-ion hybrid resonance) and further wave damping mechanisms. For
instance, when a small concentration of a light ion is added to a plasma consisting of heavier ions, the wave propagation and field structure is still dominated by the majority species, Thus a fast wave can propagate which has a
substantial left-hand circularly polarized electric field component at the fundamental resonance of the lighter minority species. Minority heating and ionion hybrid heating have been studied on several tokamaks [Adam et al., 1974;

Takahashi et al., 1977; Josea et al., 19791. The Caltech Research Tokamak,
however, has only been operated using pure hydrogen, so there are no minority
ions with fundamental resonant frequencies above that of the protons. Wave
damping due to high harmonic cyclotron absorption by relatively cold, heavy
impurity ions is probably negligible.
Including thermal effects in the wave propagation theory shows the possibility of a mode conversion process in which energy from the fast wave is converted, via a gradient in some plasma parameter, to a n electrostatic thermal
wave (e.g., an ion Bernstein wave) whch may be heavily damped. A detailed
analysis of some of these effects is given by Colestock [1982]; these effects are
not considered in t h s thesis.
Heating of a tokamak plasma a t the second harmonic of the cyclotron frequency ( 2w& ) has also been effective in several tokamaks. This mechanism is
attractive because it allows heating a t a higher frequency which, in a reactor
configuration, could allow the use of waveguide coupling to the plasma. Initial
results from the T.F.R. tokamak [Adam, 19741 showed significantly hgher wave
damping than was predicted theoretically. This was later found to be due to
the presence of a small proton impurity in a predominantly deuterium plasma;
ion-ion hybrid effects substantially enhanced the damping. More recent experiments on the PLT tokamak used a hydrogen plasma to avoid ion-ion hybrid
effects and still succeeded in coupling -3MW of power to the plasma, producing an effective plasma temperature of -4.8 keV [Hwang et al., 19831.

Efforts to heat plasmas using ICRF fast waves have, however, been unsuccessful in some tokamaks. Experiments a t power levels of 100 kW on the
Alcator A h g h density tokamak a t M.I.T. showed evidence of antenna loading
and wave generation but no evidence of bulk heating; the destination of the
power dissipated by the antenna was unclear [Gaudreau, 19811. Efforts on the
Microtor tokamak a t U.C.L.A. also failed to demonstrate plasma bulk heating.

Heating experiments on the larger Macrotor tokamak showed a dramatic and
deleterious decrease in particle confinement time [Taylor et al., 1981]. The
reasons for these diverse results are not currently understood.
In some regimes, such as that in which the Caltech Research Tokamak
operates, the wave damping per tokamak circumference is small, so waves can
propagate around the machine many times and interfere with each other,
forming cavity resonances. In other regimes (for instance, when an ion-ion
hybrid resonance layer is present), wave damping can be sufficiently large that
eigenmodes are not seen.
Fast wave cavity resonances, or toroidal eigenmodes as they are often
called, were first observed in a tokamak plasma in the TM-1-Vch and TO-1
t ~ k a m a k sa t the Kurchatov Institute in Moscow [Ivanov et al., 1971; Vdovin et
al., 19711 and have been studied in a number of other tokamaks (see
Chapter 7). Eigenmodes are potentially useful from a plasma heating point of
view for two reasons. First, the resonant buildup of the electric field a t the
antenna causes an increase in the loading resistance of the antenna. Since the
antenna loading due to excitation and damping of waves in the plasma is often
small ( & 1 n), an increase in antenna loading can increase the efficiency of
power transfer to the plasma. Second, the eigenmodes are global modes, so
energy can be deposited uniformly around the tokamak rather than in a localized region near the antenna.
One problem associated with the use of fast wave eigenmodes for heating a
reactor-sized plasma involves mode separation. As is shown in Chapter 2, for a
fixed plasma density and magnetic field, there exist an infinite set of modes a t
discrete frequencies. The modes have a finite width due to wave damping, and
the number or density of modes per unit frequency interval increases as the
size of the tokamak and its density increases [Stix, 19751. For fusion reactor
parameters and excitation frequencies near the ion cyclotron frequency, there

is expected to be a virtual continuum of overlapping modes. Unfortunately, as

Stix points out [1983], simultaneous excitation of multiple modes is undesirable as it leads to very strong fields in the local vicinity of the antenna which
may cause substantial and deleterious plasma edge heating. There has
recently, however, been renewed interest in utilizing the advantages of eigenmode heating for future reactor-grade plasmas [Stix, 19831. Mode separation
c a n be retained by lowering the excitation frequency. Because the fast wave
h a s no resonance a t the fundamental cyclotron frequency ( od),the excitation
frequency w can be lower than w c i . Although cyclotron absorption would no
longer be present, it is speculated that other damping mechanisms could still
lead to efficient wave absorption and plasma heating.
It was the point of this thesis to study various features of fast wave propagation, damping, and antenna-plasma coupling in a tokamak plasma, with an
emphasis on the physics of the processes rather than on bulk heating. Small
tokamaks such as the Caltech Research Tokamak provide a unique opportunity
t o make use of probes in the outer plasma; larger machines cannot tolerate
such experiments due to the much higher plasma temperatures involved.
Motivation for some of this work came from unexplained observations in
several other laboratories. One effect was the so-called "anomalous ICRF
antenna loading" which was seen with certain antennas in some tokamaks;
this was a density-dependent background resistive loading which was
apparently unconnected with wave propagation and was not predicted theoretically. Simple theory predicts a series of peaks on the input antenna resistance
coincident with excitation of toroidal eigenrnodes in the plasma volume.
This anomalous antenna loading (sometimes called parasitic loading) was
seen in the TO-4tokamak a t the Kurchatov Institute [Buzankin et al., 19761.
There, the fast wave was excited in the frequency range wci < o < 3od with a
loop antenna a t the plasma edge which was uninsulated but protected with

side diaphragms. The background loading depended greatly on the particular
discharge characteristics and did not exhibit any peaks associated with eigenmode generation, The authors speculated that the loadmg was due to heating
and/or expulsion of plasma entering the gap between the loop and the protective diaphragm.
The Erasmus tokamak at the Ecole Royale Militaire in Brussels was also
was used for studies of ICRF antenna loading. One antenna investigated consisted of a full loop which completely encircled the plasma column. The
antenna was simply a formed rod of copper encased in a glass tube and placed

in the shadow of a lirniter. With this antenna, a large background loa&ng was
observed ( 1 - 10 R ) which did not vary during the eigenmode resonances which
were observed elsewhere in the plasma with magnetic probes. The loading was
explained as arising from the excitation of some form of electrostatic waves in
the plasma periphery [Bhatnagar e t al., 1978a]. It was noted that with a
smaller electrostatically shielded side loop antenna, the background loading
was greatly reduced and peaks on the loading due to eigenmodes were
observed.
Parasitic or anomalous ICRF antenna loading was also observed in both
the Microtor and Macrotor tokamaks a t U.C.L.A. [Taylor and Morales, 19781.
The loading was reported to increase with plasma density and excitation frequency and was observed with a wide variety of antennas, both shielded and
unshielded ( l e a l n g the authors to refer to the phenomenon as "universal
plasma loading"). In Microtor, the parasitic loading swamped out resonant
loading due to eigenmodes under all plasma conditions; in Macrotor, the eigenmode loading could be seen, but only a t low density when the parasitic loading
was small.
The f i s t observations of fast wave eigenmodes on the Caltech Research
Tokamak were reported by Hwang [1978a, 19791, who also investigated the

complex antenna impedance. Using a small two-turn glass-covered loop
antenna, only resonant peaks on the antenna loading resistance were found, as
expected from theory. No evidence of background anomalous loading was
found.
Identification of the origin of the anomalous ICRF antenna loa&ng is of
some importance, since energy deposited in that impedance is probably not
being deposited in the plasma bulk and is thus wasted. The above observations
led this author to an extensive impedance study with a series of antennas
specifically designed to enhance the anomalous loadmg. On the basis of that
study, a model is proposed which accounts for many of the observed antenna
impedance features solely in terms of particle collection through the plasma
sheath surrounchng the antenna.
Another surprising observation in early ICRF experiments was that the
observed wave damping in the tokamak was much too great to be explained in
terms of classical damping mechanisms (i.e., cyclotron, Landau, 'ITMP, collisional, and resistive). Ths result was observed both in pure hydrogen plasmas
such as are used in the Caltech tokamak [T.F.R.Group, 1977; Bhatnagar et al.,
197Ba], and in two-ion mixtures [Takahashi, 19771. The results for the two-ion
case have been reasonably well accounted for in terms of mode conversion a t
the ion-ion hybrid layer, but the anomalously high damping observed in a pure
hydrogen plasma remains unexplained. These observations led to a series of
investigations of wave propagation and damping in the Caltech machine. In
addition to measurements of the eigenmode fields using magnetic probes, a
new approach involving the propagation of wave-packets was utilized to study
the waves unencumbered by the toroidal resonance effects.
Unexpected results obtained with simple electric field antennas led to the
design of a new &agnostic - a high frequency Rogowski current probe which
was compatible with the tokamak edge plasma. This allowed direct observation

for the first time of RF plasma particle current. The current probe was used to
investigate the spatial distribution of current in the tokamak plasma driven by
a n electric field antenna.

Thesis Outline
This thesis begins in Chapter 2 with a review of cold-plasma wave theory
appropriate to the conditions encountered in the Caltech Research Tokamak.
The torus is modeled as a cylinder with periodic boundary conditions, and the
fast-wave toroidal eigenmodes which result are discussed. A review of some
classical wave damping mechanisms (fundamental cyclotron, second-harmonic
cyclotron, electron Landau and transit-time magnetic pumping, collisional, and
resistive wall) is presented, and expected cavity Qs for various eigenmodes are
calculated. A previously overlooked wave damping mechanism due to dielectric
losses in the toroidal insulating gap is similarly evaluated for the eigenmodes

of interest.
The wave model is then extended to allow an arbitrary rad-ia! density
profile (within some mild constraints). Quadratic density profiles are considered, and the resulting effects on the eigenmode dispersion curves and on
t h e wave fields are discussed. Radially-oscillating density profiles are investigated to simulate coherent plasma density fluctuations, and a dramatic effect
o n the left-hand circularly polarized component of the wave magnetic field is
discovered. I t is speculated that t h s effect could lead to enhanced cyclotron
wave damping.

A description of the Caltech Research Tokamak is presented in Chapter 3,
including some features of its design and construction.'

Operation of the

tokamak and its associated diagnostics and data acquisition system is dis1. Some material in Chapter 3 has been previously reported [Greene and Hedemann, 19781.

cussed, and data from typical tokamak shots are shown. A section is also
included detailing the construction and operation of the new charge-exchange
diagnostic w l c h provided an ion temperature measurement for the first time
o n the Caltech tokamak.
Chapter 4 provides a detailed description of the RF systems and components which were constructed for these experiments, including the high
power ( 100 kW) RF amplfier, directional coupler, impedance-matcbng network, and amplitude and phase detectors. Phase detectors were required both
for antenna impedance measurements and for investigations of the mode
structure of the wave magnetic fields in the tokamak. Because the antenna
reactance was typically much larger than its resistance, the phase between the
antenna voltage and current was nearly 9 0 h n d changed very little during the
plasma shot. The impedance measurements thus required a phase detector
whrch could respond to changes of a fraction of a degree. The wave measurements, on the other hand, required a detector with a range of 360" which was
insensitive to the large amplitude variations (up to -35 db) occurring a t the
eigenmode resonances. The design of a novel and simple phase detector which
satisfied the above requirements is presented.
Chapter 5 describes the five RF antennas which were used in these investigations: the bare loop, insulated loop, Faraday-shielded loop, bare plate, and

"T" antenna^.^ The antennas were designed to be easily interchangable and
could be removed from the tokamak through an airlock chamber. The construction of the magnetic field probes and the Langmuir probes is also discussed. Magnetic field measurements near the loop antennas, made in a
laboratory test stand, are presented and some effects of the Faraday shield are
noted. Finally, the design and construction of a plasma-compatible Rogowski
2. Some material in Chapters 4 and 5 has been previously presented [Greene and
Gould, 19011.

RF current monitor is described, along with a novel mounting fixture which
allowed investigation of the spatial dependence of the RF current in the edge
plasma near a small electric field antenna.
Impedance measurements on the five RF antennas are presented in detail
i n Chapter 6. The impedance was investigated as a function of the excitation
frequency, the toroidal magnetic field, the plasma density, the antenna radial
position, and the antenna current levek3 Langmuir probe studies are also discussed here.

The shielded antennas display loading resistance peaks

corresponding to the excitation of eigenmodes, while the uninsulated antennas
all show the large anomalous density-dependent background loading. A simple
theoretical model is described, based on particle collection by the antenna and
making use of the Langmuir probe measurements, which successfully explains
many of the observed features of the background loabng.
Wave experiments are discussed in Chapter 7. Magnetic field probes are
used to observe the eigenmode wave fields, and the locations of the resonances
in frequency-density space are mapped out and compared to theoretical predictions. Eigenmode excitation efficiencies are examined and seen to be comparable for all of the antennas. Investigations of the wave field mode structure
using pairs of magnetic probes are presented and found to be in accordance
with theoretical expectations. Further experiments measured the eigenmode
radial magnetic field profiles and the eigenmode resonance Qs and wave damping lengths.
The wave-packet experiments are then described and the observed variation of the packet velocity with plasma and RF parameters is displayed. A wave
propagation model whch includes a vacuum layer at the plasma edge is
applied to the theoretical calculation of the group velocity, and comparison
3. Some results presented i n Chapters 6 and 7 have been previously reported [Greene and
Gould, 19831.

with the experimental results ylelds surprisingly good agreement.
Finally, results from the measurements with the RF Rogowski current
monitor are presented. The spatial distribution of the RF plasma current
driven by a small electric field antenna is investigated and found to be highly
localized along the toroidal field direction. T h s result suggests a mechanism
for the good coupling to the eigenmodes observed with the electric field antennas.
Chapter 8 concludes the thesis with a brief summary of the major experimental results.
Two appendices are included which are not directly related to the rest of
this work. Appendix A presents observations of efficient power coupling
between two identical ICRF antennas in the Caltech t ~ k a m a k ,an
~ experiment
which ties in well with a simple circuit model of the eigenmode resonances.
Appendix B discusses the effect of a transient RF pulse on surface heating in a
conducting cylinder, a subject which may be important for the design of future
very b g h power RF systems.

4. This paper has been previously presented [Greene and Could, 19791.

Wave Theory

This chapter begins with a review of cold plasma theory appropriate to
study the propagation of waves near the ion cyclotron frequency. The tokamak
is then modeled as a cylinder with periodic boundary conditions, and solutions
for the normal modes of the system are found for parameters appropriate to
t h e Caltech Research Tokamak. Retaining the electron mass terms reveals the
form of the small E, component of the waves. The approximation me4 0 is
then considered and the solutions are compared with the previous results. A
variety of wave damping mechanisms are reviewed and the resulting cavity Q s
a r e calculated for appropriate parameters and modes; the dependence of Q

on k,,is noted. A numerical solution of Maxwell's equations is then developed
for the cold plasma model which allows an arbitrary radial density profile to be
included. Several profiles are considered, including quadratic and oscillatory
functions, and the effect of the profiles on the eigenmode dispersion curves
and on the wave fields is investigated. The left-hand circularly polarized component of the electric field is found to be very sensitive to radial density perturbations, and the possibility of enhanced cyclotron damping arising from
this source is discussed. Finally, previous work on the effects of the poloidal
magnetic field and of toroidal geometry are briefly reviewed.

2.1 Review of Cold Plasma Wave Theory
2.1.1 Cold Plasma Dielectric Tensor
The well-known cold plasma effective &electric tensor can be derived from
t h e two-fluid momentum conservation equation together with Maxwell's equations. Neglecting pressure and collision terms, the equations can be written as
[Krall and Trivelpiece, 19731:

where a refers to a species of charged particle (electrons or ions), n,, ma,
and q , are the particle density, mass, and charge, respectively, and Va is the
fluid velocity of species a . B and E are the magnetic induction and electric
field, respective!y, ~ n cd is the ve!ocitl7J of hnht.
Assamire a steady state back-

ground magnetic field, &, and linearizing the momentum equation for small
perturbations with harmonic time dependence ( e-iut ) ylelds

where the tilde in&cates a perturbed variable. The above equation can be
solved for the mobility tensor, pa,defined by Fa =pa.

E. The result, in Carte-

sian ( x ,y , z ) or cylindrical ( p , 8,z ) coordinates, is

where wca = qaRO and loca is the cyclotron frequency for charge species a
ma c

a n d F a = l - - W c a . The particle current in the plasma is
u2

related to the electric field in the plasma through

?=z

nuqaVa and is

T=z- 2 ,where

Note that since the conductivity tensor is anti-Hermitian, energy absorption is
not described in t h s model [Ichimaru, 19731. Equation 2 , l b can be written as

t)

where I is the unit dyad. The effects of the particle current

can thus be

4?Tincluded in an effective dielectric tensor &+
& = I- -a
. The cold plasma dielec++

ZW

tric tensor, y , is found from equations 2.3 and 2.4; in cartesian (or cylindri-

caij coorciinates, the resuit is

where

is the plasma frequency for species a.

For a hydrogenic plasma such as is used in the Caltech tokamak, the general forms of E l , E X , and Ell, as functions of frequency, are shown in Figure
2-1,

has a pole a t

o =a, r [oiO+ oii]

w=O

and a zero a t the plasma frequency

'. EL has poles a t the electron and ion cyclotron frequen-

cies, and the solutions for its two zeroes at w = W I ~ ,

are called the lower

and upper hybrid frequencies. C, has poles also a t the cyclotron frequencies
and has zeroes a t o = 0,w. For the Caltech tokamak, the ion cyclotron frequency is small compared to the other characteristic frequencies (except at
the plasma-wall boundary) which are ordered approximately as

2.1.2 Plane Waves
Maxwell's equations can now be written in the form

where the subscripts in&cating perturbed quantities have been dropped. The
wave equations for the fields are then

Considering infinite boundary plane wave solutions of the form e i k - x - i w t ,
where k is the wavevector, the equation for E becomes

nx(nxE) + g - E = O ,

(2.10)

where n = -k and n = In1 is called the refractive index. The magnitude of

FIG. 2-1. Plots of the elements of the cold plasma dielectric tensor, ell, €1,c x ,
and of ( ~- fE ~ ) / c as
~ functions of frequency, for a hydrogenic plasma. Summations are over electrons and ions. w,, and w,, are the cyclotron and plasma
frequencies for species a ; w h and w* are the lower and upper hybrid frequencies; € A = 1 + 4 7 ~ n q c ~f / . Plots are not to scale.

t h e phase velocity of the wave is then u,, = g = and the direction is normal
k:
t o surfaces of constant phase, i.e., in the direction of k . Tahng the steady
state background magnetic field to be & = B o % ,where 2 is a unit vector in the
z -direction, and taking n to lie in the x - z

plane and e to be the angle

between n and &, the wave equation becomes

For E, = E, = 0 , the only non-trivial solutions are e = 0 or n, qi= 0 (i.e.,
?T
3n
+ o:i =oz,the plasma oscillation) and e= or - n 2=ell (i.e.,
2 '

02= oie

02= o+$ k 2 c 2 ,the ordinary transverse electromagnetic wave),

For E, or &!I

not equal to zero, the determinant of the matrix in equation

2.11 must be set equal to zero, resulting in the well known unbounded cold
plasma dispersion relation:

where

Equation 2.12 gives n as a function of the plasma parameters (i.e., plasma
density, static magnetic field, and o) and of the angle of propagation, e. The
dispersion relation may be rearranged to yield 8 as a function of the plasma
parameters and of n :

where R = €1+ zx and L = EL - cx [Boyd and Sanderson, 19691
In general, equation 2.12 admits two distinct solutions, although one or
both may be evanescent ( n Z< 0 ) for a particular set of plasma parameters and
angle of propagation 8. For propagation in the direction of the principal axes

( e= 0, l[_)
the
, &spersion relation may be greatly simplified. For propagation
along the static magnetic field ( 8 = 0 ) , and for w << I w,, 1, up,, the dispersion

relation may be written in the form

where v A = Bo/

is the Alfv8n velocity. From equation 8.11, the polar-

ization of the two waves can be found:

so the waves are circularly polarized, the electric field vector rotating in the
same sense or in the opposite sense as the direction of ion precession (this Tnll
be called left-hand and right-hand circular polarization, respectively, by convention). The wave associated with the "+" sign in equation 2.12, the righthand circularly polarized wave, is variously called the fast or compressional
Alfvgn wave; a t higher frequencies, it is termed the whistler or electron cyclotron wave. The wave associated with the "-" sign is left-hand circularly polarized and is called the ion cyclotron wave, or the slow or torsional Alfven wave.
(Note that its phase velocity, a t a given frequency, is sloww than that of the
compressional wave). I t is also apparent that the slow wave does not propagate
immediately above w C i , since k is imaginary there.
For propagation across the static magnetic field ( e = E ) ,the dispersion

relation reduces to

f o r the "+" sign in equation 2.12; the other solution is evanescent ( n Zpropagating wave is found to be elliptically polarized in the x-y plane:

Note that a t the fundamental ion cyclotron frequency (wCi),the wave associated with the "+" sign (the fast wave) is right-hand circularly polarized for all
directions of propagation.

2.1.3 Phase and Group Velocity Surfaces
The phase velocity surface (sometimes called the wave-normal surface)
may be defined by a polar plot of up, as a function of 6 ,for fixed plasma
parameters and fixed w . ("Surface" here refers to the surface of revolution
wlxch would be generated by rotation of the plot of vph(@)about the z-axis.
The three-dimensional surface is rarely plakted.) The CMA &agram (aftsr Clemmow, Mullaly, and Allis [Stix, 19621) provides a useful classification of cold
plasma waves based on the topology of the phase velocity surface as the
plasma parameters are varied. Fqure 2-2 is a CMA diagram for a two component

(hydrogenic)

plasma

and

shows

parameter

space

(i.e.,

density-magnetic field space) divided into 13 regions with &stinct phase velocity surfaces. In regions where both waves propagate, one phase velocity surface is always (except for some possible singular points) contained within the
other. The wave corresponding to the outer surface is called the fast wave; the
other one, the slow wave.
For

waves

of

interest

in

these

experiments,

al-3
uci

and

u << w,,, w&, up . These waves then occur in region 10 on the CMA diagram. For

4/a2OR

DENSITY

FIG. 2-2. The CMA diagram for a hydrogenic plasma, showing the topology
of the phase-normal surfaces as functions of the plasma density and the
static magnetic field (after Chen [1974]). The experiments described in
this thesis were confined to region 10.

w

wave does not propagate for all angles 8). As o increases and crosses the ion
cyclotron resonance (at w~ ), the slow wave becomes evanescent (for all e)
whle the fast wave is largely unaffected by the transition. Hence the mode in
region 10 is also called the fast wave.
The vector phase velocity is vph = L k . The velocity of a local msturbance
aw
o r wave packet, however, is the group velocity vg = - (provided the hsper-

ak

sion and spread of wavenumbers are not too great) and is, in general, not in
t h e same direction as the phase velocity. The group velocity can be written as

where f;= - and ^e=kx(kx^z) is a unit vector in the e-mrection. Hence vg
and vph will lie in the same direction only when a0 = 0 . From the symmetry

ae

of the dispersion relation, this occurs a t least a t 8 = 0, 2 '

A group velocity surface may be defined, in similar fashon to the phase
velocity surface, as a polar plot of vg versus eg,where 6, is the angle between

vg and the static magnetic field Bo2. From equation 2.18,

and

Plots of the phase and group velocity surfaces are shown in Figure 2-3 for
several mfferent frequencies; the Alfvdn velocity ( v A )is also shown for comparison. It should be noted that, in general, e$h # e9, except along the princi-

FTG. 2-3. Polar plots of ug(es) and v~~(€+,~)
(right column) and plots of e9 vs.
15,~(left column) for different frequencies. Units for the plots of eB and f$h
are degrees. The plasma density is n = 1013~ m - ~
the; static magnetic field is
Bo =4.0 kG. The dashed circle in each polar plot represents the Alfvdn velocity
~ ~ = 2 . 8 cm/sec.
~ 1 0 ~ a) Q = C I / C I ~ ~ =
b)O R=1.0
.~

em

FIG. 2-3, continued. c) R =3.0 d) Q = 10.0

pal axes. At low frequencies (near w & ) , the two surfaces do not differ greatly.

37T, while at
The phase and group velocities become equal a t Bph = Bg = 2. +h

= eg = O , T ,the group velocity exceeds the phase velocity by a factor of

[D. ] [&+ .

approximately 2 -+

21-I

At high frequencies ( >> o n ) , the group

velocity becomes substantially larger than the phase veloeity for 8, near 0 or
7 ~ ,.and the group velocity surf ace may become multi-valued.

2.1-4 Wave Ehergy
The energy associated with a wave propagating through a lossless plasma
has three components: the magnetic field energy, the electric field energy, and
the particle kinetic energy. The energy density associated with the magnetic
field, averaged over the wave period, is just

where

"*" indicates the complex conjugate, whle that associated with the

vacuum electric field is

Since the effects of the particle motion are being included in the effective
dielectric tensor, the energy density associated with the particle motions may
be expressed in terms of this tensor. Brillouin [1960] considers a process in
whch the wave is built up slowly from E = 0 to E=Z e T i o t and integrates the
incremental energy 6 W = E*-6D, where D = H(o) - $. The result. valid pro47r

aF is not too great, is
vided the medium is lossless and aw

The ratios of the electric field energy to the magnetic field energy, and of
the particle kinetic energy to the magnetic field energy, are plotted as functions of the angle of propagation ($h) in Figures 2-4a and 2-4b, for parameters appropriate to the experiments on the Caltech tokamak. The ratio of electric to magnetic field energies is peaked for propagation perpendicular to the
static magnetic field (Figure 2-4a), and this directional asymmetry as well as
the magnitude of the ratio increases as w increases. Even a t w = 6 ad,however, the peak of the ratio reaches only 3.1 x

hence the vacuum electric

field energy is a negligible fraction of the total wave energy. Although not
shown in this plot, the ratio of electric to magnetic field energies increases
linearly with

density in the

range

of

parameters

beq

considered

( n = 1012-1014

The ratio of particle ktnetic energy to magnetic field energy (Figure 2-4b)
is larger, reaching a peak of -5 for propagation across the static magnetic
field. The directional asymmetry also increases as the frequency increases, but
the magnitude of the ratio decreases. For propagation nearly perpendicular to
the static field, the ratio of energies depends little on frequency; also, the
ratio is nearly independent of density.
Using the form of the dielectric tensor

z , the expression 2.22 for the par-

ticle energy can be separated into terms arising from electron and ion motion.
The ratio of the electron to ion kinetic energy associated with the wave is plotted in Figure 2-4c as a function of

%, for several frequencies. The electron

energy exceeds the ion energy by a moderate factor (4-16) and the ratio is
peaked for propagation along the static magnetic field. As the frequency
decreases, the ion portion of the ktnetic energy becomes more important
(although for frequencies below w & , the ion component decreases as the frequency decreases). The values of the ratio are also nearly independent of density over the range 1012- lo1*emS.

FIG. 2-4. Polar plots of the ratio of the vacuum electric field energy to the magnetic
field energy (a), and the ratio of the particle kinetic energy to the magnetic field energy (b), for a plane wave, as a function of the angle of propagation ( 8)with respect
to the static magnetic field. In each case, the curves are shown lor three frequencies: R=u/ud = 1.5, 3.0, 6.0. The plasma density is n = 1 0 ' ~cmS and the static
magnetic field is B o = 4 kG. The dashed circles are drawn a t the radius corresponding to the maximum ratio of the energies for each plot; that maximum is 3.2 x lo9
for the figure in a), 5.0 for thc figure in b).

FIG. 2-4, continued. c) Ratio of electron to ion kinetic energy
associated with a plasma wave, as a function of angle of propagation ( 6 ) with respect to the static magnetic field. The curves
are shown for three frequencies: hd = 1.5, 3.0, 6.0. The plasma
density is n = 1013 cmS and the static magnetic field is Bo =4.0
kG. The dashed circle is drawn a t a radius corresponding to the
maximum ratio ( 16.3) for the hd =6.0 curve.

Hence, over the range of parameters appropriate to the Caltech tokamak,
t h e energy associated with the fast wave is largely shared between the magnetic field and the particle motions; the vacuum electric field energy is very
small. This, then, is why traditional approaches to the coupling of energy to the
f a s t wave generally employ magnetic field antennas.

2.2 Toroidal Eigenmodes
2.2.1 Introduction
The torus which forms the tokamak wall can be regarded as a resonant
cavity which may support a variety of electromagnetic modes. In the absence
of plasma, there are no propagating modes because the lowest resonant frequency of the vacuum-filled torus is well above the frequencies of interest here.
This follows because the free-space wavelengths involved are much larger than
the dimensions of the tokamak. With the addition of plasma, however, the
wavelengths may become much smaller (since the elements of the effective
plasma dielectric tensor can be very large), and, if the damping of the waves is
sufficiently small, cavity modes may then exist.
Analysis of the resonant frequencies of the plasma-filled torus is complicated by the toroidal geometry and by the anisotropic nature of the plasma
dielectric tensor. Several simplifying assumptions are made to make the problem tractable. The toroidal geometry is replaced with a cylinder (Figure 2-5);
the length of the cylinder (L) is taken to be the "unfolded" length of the
tokamak (2nR) and the r a h u s of the cylinder ( p o ) is equal to the minor radius
of the tokamak. The closed nature of the torus is taken into account by
imposing periodic boundary conditions: all field components ( 9 ) must satisfy
@ ( z )= @(z+ NL) ,

N = 0,rtl, rt2, . - .

(2.23)

The external magnetic field Bo is assumed to be constant and parallel to the

FIG.2-5. Geometry of the periodic cylinder model of the tokamak. The radius of
the cylinder ( p o ) is equal to the minor ra&us of the torus, while the length of
the cylinder ( L ) is equal to the major circumference of the torus (2nR).
Periodic boundary conditions are imposed at the ends of the cylinder to simulate the closed nature of the torus.

cylinder axis. Thus the 1/ R variation of the actual toroidal magnetic field in
t h e tokamak is neglected in t h s model as well as the effect of the poloidal magnetic field generated by the plasma current.
The case of a homogeneous plasma completely filling a conducting
cylinder is considered first. The wave equation and the general relations
between field components are derived, following the approach of Allis [1963].
Boundary conditions are then applied to the solutions of the wave equation,
yielding an eigenvalue equation which determines the frequencies of the fundamental modes of the system. The equation is solved numerically for parameters appropriate to the Caltech tokamak, and the ra&al dependence of the
wave fields associated with these eigenfrequencies is obtained.
The approximation m, -,0 (which requires E, +O) is then considered and
the effects on the wave fields and on the normal mode dispersion relation is
noted.

I t is useful in the following discussion to introduce a separation of the
wave fields into components parallel and perpendicular to the external static
magnetic field %=Bog. The subscript "_L" refers to a component perpendicular to

, while subscripts " /I" or " z " refer to components parallel to & .

Then a vector field A can be written as A = & + A , 2, where A, = A - 2 and
qi = ('ix A) x 2 . We look for solutions where all field quantities vary in the 2
direction as ei*llz, where kll is called the parallel (or toroidal) wavenumber.
The gradient operation can then be written as V = VL + i k l 1 2 ; the divergence as

V - A= Vl. & + i ill
A, ; the Laplacian as v2A= Vf A - klf A; and the curl as

V x A = V'X A + i k l l %x A , Some useful relations are easily derived:

and

Maxwell's equations may then be written as

where ko = - 1s the free-space wavenumber and an e-iwt time dependence has
W .

been assumed. Talang the curl of equation 2.26a yelds a wave equation for the
electric field:

It will be shown subsequently that a knowledge of E, and B, is sufficient
to completely determine all components of the wave fields, hence we consider
the z -component of equation 2.27:

Since there are no external charges in this system (those comprising the
plasma having already been taken into account by the effective dielectric tensor), the Poisson equation is just:

V - D = V-[F-E]= o .

(2.29)

Using the form of the cold-plasma dielectric tensor discussed previously,
C)

c - E = clEl+ i ~ ~ ^ z x cllEZhX.
Q +

Expanding Poisson's equation, we have:

(2.30)

Substituting equation 2.31 into equation 2.28 yields the wave equation for Ez :

The wave equation for B is obtained by taking the curl of equation 2.26b:

V ~ B = ~ ~ ~ V X [ F C - E ] .

(2.33)

The z -component of the above is, using equations 2.24 and 2.30,

Since

V'x El=i koBz and V'x ( 2 x El) = ^z(vLEl), the above equation can be

recast, using equation 2.31 , as

Equations 2.32 and 2.35 are a pair of coupled second-order differential
equations for the field components Ez and Bz . These equations can be solved
for E;, or B z , resulting in a single fourth-order equation satisfied by both Ez
and Bz. Introducing new constants a , b , f , and g , the wave equations can be
written as

where

Solving for E, or B, then yields

Since a homogeneous plasma is being considered a t this point, the
coefficients a , b , f , and g are not functions of position, and the equation can
be rewritten in biquadratic form:

Expandmg equation 2.38 and comparing with equation 2.37 pelds an equation
f o r p , a n d p a interms of a , b , f , and g :

where p

, is associated with the "+" sign and p 2 is associated with the "-" sign.

The four linearly independent solutions which are associated with equation
2.37 are then given by the two pairs of solutions of the second-order equation:

for pi = p l o r p z .
To proceed further with the solution of the wave equation requires
specification of the transverse geometry. The periodic cylinder model is used
and equation 2.40 becomes, in cylindrical coordinates,

where $ = E, or B, . Separation of variables, $(p, 8)= R ( p ) p ( ~, )yields

where m is a constant called the poloidal mode number. Solutions of the

above are simpIy p(e) = ei m 8 and R(p)= Jm(;Pip) , Y m ( p i p ) , where J , and Ym
are the Bessel functions of order m . Physically, the solutions must be periodic
functions of 8 with period 2 n , and rn must therefore be an integer. Solutions
for Ez and Bz whch are finite as p + 0 can then be written explicitly:

where m=O,k1, k2 ..., a and @ are arbitrary constants, and equations 2.36
have been used.
The above results are not valid, however, when kii= 0 , for then b and g
vanish and equations 2.36 are uncoupled. In that case the solutions are given
by

Bz(p,
e, 2 ) = S J m ( q p ) e i m e,
where y and 6 are arbitrary constants.

2.2.3 Relations Between Field Components
As mentioned earlier, all field components can be derived from & and B,

and their spatial derivatives. The explicit relations between the field components are derived from Maxwell's equations together with the form of the
cold plasma dielectric tensor. Taking the cross-product of
2.26a and using equation 2.25 ylelds
i k o 2 x B l = 2 x (VxE) = VLE' - i k l l E l

The cross-product of ^z with equation 2.26b is given by
^ z x ( V x B )= - i k o 5 i x E - l $ )

2 with equation

where equations 2.25 and 2.30 have been used. Now the cross-product of
$ with equation 2.45 is
% X (VIEz>= i k i i ^ X x E- ~
ikQB~,

(2.48)

whle the cross-product of 2 with equation 2.47 is

%x(VIB,) = i k I l 2 x B.5~Z ~ Q E-k0cXCixEl.
~ E ~

(2.49)

Equations 2.45, 2.47, 2.48, and 2.49 can now be combined into a matrix expression relating VlE, and V i B , to E, and B, and their respective cross-products
with 2:

What is actually desired, however, is the relation between the transverse components El, Bl and the gradents of E, , B, . The above matrix must therefore
be inverted. The algebra is straightforward [ ~ l l i es t al., 19631, yielding

where

rsnd A = k f -2kg klf cl+k,j ( c f - ~ ; ) ,and the inverse exists provided A # 0 .
For cylindrical geometry, it is convenient to derive explicitly the relations
between Ep, Ee, Bpr Be, and Ez, B,.

For azimuthal variation e i m e , the

im
transverse gradient is V'= i j - + 6 -.
Writing out the p^ and 8 components
8P

of the first two terms in equation 2.51 yields the desired result, which can also
be expressed in matrix form:

2.2.4 Boundary Conditions
Thus far we have determined acceptable solutions to Maxwell's equations
for a periodic-cylinder geometry filled with a homogeneous plasma magnetized
in the B direction. Appropriate boundary conditions must be applied which
will restrict the set of allowable solutions.
The tokamak wall is made of stainless steel much thicker than the skin
depth a t the frequencies of interest ( > w d ) , and it is continuous except for
small access ports and a toroidal insulating gap. The effect of the gap is

ignored in this discussion and the boundary for the model is taken to be a perfectly conducting cylinder a t radius p o . The tangential components of the
electric field ( E , and EB) and the normal component of the magnetic field

(3,)
must therefore vanish a t p = p ~ .
Periodicity in the azimuthal ( G ) direction has already been taken into
account by restricting m to be an integer. Periodicity in the 2 direction
requires that k =

5.

where N = 0 . 1 , 2 , . . , will be termed the toroidal mode

number, and R is the major ra&us of the tokamak.

Case 1: N # 0
Imposing the boundary conchtion & (PO)= 0 on equation 2.43a yields
aJrn(P1~0)

8 Jrnb2~0>
=0

(2.53)

where N # 0 . The P components of the fields are then determined to within an
arbitrary constant a :

Using equation 2.52, the boundary concbtion Ee(po)= 0 can be written as

Using equation 2.54a to evaluate the derivatives yields, after some algebra and
using equation 2.53,

For Axed toroidal and poloidal mode numbers N and m , and fixed plasma
density and static magnetic field, the left-hand side of equation 2.56 is a function only of frequency. The roots of the function therefore determine distinct
frequencies for which the given mode satisfies the boundary condition. These
eigenfrequencies correspond to different radial propagation constants p l and

p,, and each solution is labeled a different "radial mode", 2 . By convention,
that solution corresponding to the lowest frequency (if there is one) is called

the first ra&al mode (I = I), and so on. Alternatively, for fixed frequency (typical of actual experiments), the left-hand side of equation 2.56 is a function of
density, and its roots give the distinct densities at which the modes occur.

When N = 0 (corresponding to a mode with no toroidal variation), the solutions for the fields are given by equations 2.44. Application of the boundary
condition on Ez results in two possibilities: either y = 0 or 6 p o= Cnzn,where

Cm,

is the nthzero of the Bessel function of order rn. However, a < 0 for

w

are all real [Abramowitz

and Stegun, 19721, it follows that y = 0. This implies that Ez is zero everywhere
and so the only allowable solution in this case is a TE (transverse electric)
mode.
The boundary condition on E, is, from equation 2.52,

Using equation 2.44b and the definitions of GI and G2,this becomes

where .f0=*8[

'1

&f EL
- Ex

and f 0 > 0 for W < U ~F o r k e d m ,u , m d B C then,

this equation determines the allowable plasma densities for each mode. The
fields in t h s case are just

E' = 0

(2.6Oa)

where a is an arbitrary constant.

It is useful to investigate the form of p

and pz for parameters appropri-

ate to the Caltech tokamak. py is plotted in Figure 2-6a, for a density of
1013emS, as a function of frequency for various parallel wavenumbers
( k i l= N / R ) ; p r

as a function of frequency is plotted for N = 1 and different

densities in Figure 2-6b. Similar plots are shown in Figures 2-7a and 2-7b for

p f . It is clear that for parameters appropriate to the experiments performed
(R

w 1 -3;

n a 1012- 1013 cmS),

p f is less than zero and p$ is greater

oci

than zero. Hence p , is imaginary while p 2 is real. The Bessel functions
J,(p ,p) then become (i)mim(
jp, lp) . From Figure 2-6a, lp lpo > 90 for
n = 1013 cmW3and R > 1. Since, for large [ , I, (f)

-dl%$-'
el:

it follows that the

contributions from the I, functions will be important only near the wall (i.e.,
near p =po),
Equations 2.56 and 2.59 were solved numerically to find the locations of
the eigenmodes in density-frequency space. All of the modes occurring in the

FIG. 2-6. a) Plots of P? as a function of Q for different values of the toroidal
mode number N = k l Ro
l . The plasma density is n = 1013 cmS ; the static magnetic field is Bo = 4 kG, b) Plots of ~f as a function of Q for different densities
n and fixed toroidal mode number N = 1 . Curve A: n = l o i 4 cmS , vertical scale
vertical
x 10. Curve B: n = 1013 cmS, vertical scale x 1, Curve C: n = 1012
scale x lo-' . Curve D: n = 1011 cmg , vertical scale x 1o - ~

FIG. 2-7. a) Plots of P: as a function of R for different values of the toroidal
mode number N = k i lRo. The plasma density is n = 10" cm-'; the static magnetic field is Bo = 4 kG. b) Plots of P$ as a function of R for different densities
n and fixed toroidal mode number N = 1 , Curve A: n = 1014 cmg , vertical scale
x 10. Curve B: n = 10" cmS , vertical scale x 1. Curve C: n = 1012ern-', vertical
scale x lo-'. Curve D: n = 1011cmS, vertical scale x

region bounded by 0.55R53.0 and 0 < n 5 1.5 x 10'' ~ r n -are
~ shown in Figure

2-8; more than 75 modes are present. The modes become very dense as either

R or n are increased.
The modes are conveniently categorized by their poloidal ( m ) , toroidal (N)
and radial (1) mode numbers. Plus and minus N modes are degenerate, yield-

ing the same solutions in equation 2.56 ( t h s follows from the symmetry of the
geometry). Plus and minus m modes, however, are not degenerate and yleld
different solutions. This arises physically from the anisotropy introduced by

the static magnetic field: wave field patterns rotating in the direction of electron precession see a different environment from those rotating in the opposite sense. T h s means that the eigenfunctions of e are necessarily complex;
sin(m@)and cos(m@)functions will not satisfy the equations.
Figure 2-9a shows the solutions to the eigenvalue equation for the m = 1
modes; the solid lines are I = 1 modes while the dotted lines, occurring at
higher frequencies and densities, are the 1 = 2 modes. There are no higher
radial modes present within the region plotted. As the toroidal mode number
increases, the frequencies and densities a t which the modes occur also
increase; furthermore, there are no modes whch occur below the N = 0 mode
(the mode "cutoff), as below this mode the parallel wavenumber becomes
complex. Solutions for m = 0 and m = 2 modes are shown in Figures 2-9b and
2-9c, respectively. Cutoff curves are shown in Figure 2-9d for all of the 1 = 1
modes occurring within the region; the m = 1 cutoff mode is the first mode
encountered as either the frequency or the density is increased.
In an actual experiment, the frequency of the exciter is fixed while the
density is increased from

1012 cmS to

1013 cm-a through gas puffing. We

then expect to see a succession of peaks in the amplitude of the wave fields in
the tokamak as the density sweeps through successive eigenmode resonances.
For frequencies R s 2 , for instance, the first five modes encountered as the

eter space bounded b y ~ = 3 3and
n = 1.5 x 1013 cmL3. The static magnetic field is Bo= 4.0 kG.

FIG. 2-9. Eigenmode curves for various modes. Solid lines are the first radial
mode solutions ( 1 = 1 ) and are labeled by their toroidal mode number ( N ) ;
dashed lines are the second radial modes ( 1 =2). The static magnetic field is
Bo= 4.0 kG. a) Eigenrnode curves for m = 1 poloidal modes. b) Eigenmode
curves for m = 0 poloidal modes.
Note that throughout this thesis, the suffur " X l o L " on an axis label means that
the numbers labeling that axis are to be multiplied by l o L .

FIG. 2-9, continued. c) Eigenmode curves for m = 2 poloidal modes.
d) Eigenmode cutoff curves ( N = 0 ) for the first radial mode and for various
poloidal mode numbers.

density is ramped up are all 1 = 1, m = 1 modes; the toroidal mode numbers
a r e N = 0, 1, 2, 3, and 4,
The wave fields may be found from equations 2.43, 2.44, and 2.52 once the
frequency and density have been chosen to satisfy the eigenvalue equation for
a particular mode. Note that if E, is taken to be real (by choosing a
appropriately), then Ee and Bp are also purely real, while Ep, B, and Be are
purely imaginary (i.e., out of phase by n/2 radians with respect to the real
field components).
Another quantity of interest is the particle current J associated with the
wave. Although the effects of the particle current have been lumped into the
effective dielectric tensor, it is the particle motions that are fundamentally
responsible for wave propagation. The particle current is given by J = g - E, and
the components can be written explicitly as

The ra&al distributions of the components of the electric field, the magnetic field, and the particle current are shown in Figure 2-10 for the 1 = 1,

m = 1, N = 0 , 1, 4 modes and the 1 = 1, m = 0 , N = 0, 1, 4 modes. The plots are
normalized so that the peak magnetic field is one gauss; the units for the plots
of the electric field are volts/cm, while the current density is plotted in units
of amperes/cm2.
In all cases, B, is the largest component of the magnetic field near the
wall; for N=O modes, B, is the only non-zero component. As the toroidal
mode number increases, the p and z components of the magnetic field

FIG. 2-10. Ra&al distributions of the components of the wave magnetic field (top),
electric field (middle) and particle current density (bottom) for the N=O, m = 1,
1 = 1 mode (left-hand column) and the N = 1, m = 1, 1 = 1 mode (right-hand
column). The horizontal axis in each plot is the radius p ; the limits are 0 (left-hand
end) and po (right-hand end). Also R = 2.0 and Bo= 4.0 kG.

FIG. 2-10, continued. Radial distributions of the wave magnetic field, electric field,
and particle current density for the N = 4, m = 1, 1 = 1 mode (left-hand column)
and the N = 0 , m = 0 , 1 = 1 mode (rght-hand column).

FIG. 2-10, continued. Radial distributions of the wave magnetic field, electric field,
and particle current density for the N = 1, rn =0, 1 = 1 mode (left-hand column)
and the N = 4, m = 0, 1 = 1 mode (right-hand column).

become relatively larger. For the m = 1 modes, 8, + 0 as p + 0 , while the magnitudes of Bp and Bg become equal a t p = O . For m = 0 modes, B, is finite a t
p = 0 while Bp and Be vanish there. As is necessary to satisfy the boundary

conditions, B, always vanishes a t the wall ( p = P O ) .
The magnitudes of Ep and Eg also become equal a t p = O ; they vanish
there for m = O modes and are finite for m $0 modes. For N $0, E, is nonzero although, as is evident from the plots, its amplitude is very small. The
magnitude of E, changes very abruptly near the wall; this is the due to the
influence of the I m ( l p l lp) function. Ez and .Ee vanish a t the wall, while E,
and EB vanish a t p = O for m # 0.
Although the magnitude of E, is several thousand times smaller than the
other components, the magnitude of J, is comparable to the other components because the parallel conductivity is large. The relative size of J,
increases as the toroidal wavenumber N increases, and in some regions Jz
may be the largest component of the current.

2.2.6 Zero Electron Mass Approximation
An approximation often found in the literature dealing with fast wave
propagation is to set Ez equal to zero. In this section, the approximation is
seen to follow from ignoring the electron mass in the wave equations.
In the limit me + 0 , the elements of the cold plasma dielectric tensor
become

Ex

-= Ex, ,

hence €1and &, remain finite while €11 4 -".

The coefficients b and f of the

wave equation (2.36) also remain finite as m, + 0 , but a and b are proportional to m;' .
The ra&al propagation constant pz remains finite as me -,0,

whilep, doesnot:

where lim (p 12) < 0 a t least for w,i < w < wu,.
me+O

The solutions for the fields Bg and E, are still given by equation 2.54.
Noting that lim (p 12) = lim ( a ) and that lim [J, (p l p ) ]= (i)m
%+O
%-to
ng+O
42-

IPIIP

it

follows that

and

= -if?,,

lim -b.

me+o a

(2.65b)

Since lirn - .c - = me + 0 , it follows that Ego= 0. The above expressions are
ng+o a
EII
valid for k l l= 0 as well, and p 2 = f

in that case. The other field components

are obtained from equation 2.52, where it is noted that the factors in the
matrix are all k i t e as me 40. The eigenmode equation (2.56) similarly
reduces, as m, + 0, to

which can be shown to be equivalent to the result derived by Gould [1960]. The

p and 6 components of the particle current can be obtained from equations

2.6ia and 2.61b, but the expression for the z component of the current given
in 2 . 6 1 is
~ indeterminate as m, -+ 0. The current in t h s case can be obtained

more easily from the Maxwell equation:

Numerical evaluation reveals that the zero-m, approximation is quite
accurate for most purposes. In particular, calculation of the eigenrnode
dispersion curves yielded the same results as the finite-me model within the
accuracy of the numerical routine. The only signiflcant differences in evaluat-

ing the field and current components involved Ez, J,, and Be. Ez vanishes in
the zero electron mass model, while the full calculation shows the actual magnitude of E, and the form, which has a rapid transition near the wall. The
current Jz calculated from the zero-m, model also agrees with the full calculation except near the wall, where a significant difference occurs. Jz calculated
from the finite-me model goes to zero at the wall, while the zero-me model
yields a finite value there. An example for R = 2 is shown in Figure 2-1 l a . The

Be components calculated from the two models agree well except in a small
region near the wall (Figure 2-1 ib); even there the difference is small.

2.3 Wave Damping
2.3.1 Introduction

The wave model discussed thus far includes no mechanism for energy dissipation and thus allows waves excited a t a toroidal resonance to have
unbounded amplitude. In fact, energy can be absorbed within the plasma by

ZERO-me MODEL

FIG. 2-1 1. Comparison of the zero-m, and finite-% models applied
to the calculation of a) j, , and b) Be. This example is for an
N = l , m=1, 1 = 1 mode; h2=2.O and B0=4.0kG.

both collisional and collisionless processes, and in the bounding structure
which surrounds the plasma. Both the calculated and the observed damping
are small in the experiments described in this thesis in the sense that a wave
can propagate many times around the tokamak before losing an appreciable
fraction of its energy; hence the concept of a toroidal eigenmode is still
appropriate.
Energy dissipation in a weakly damped resonant cavity is often discussed
in terms of the cavity " Q ", defined as the time-averaged stored energy associated with the wave divided by the energy dissipated in one wave period. The Q

of each toroidal eigenmode can be estimated experimentally by observing the
width of the resonance in the wave field amplitude as the density increase
sweeps through successive modes (see Chapter 7). Another approach is to consider the attenuation that a wave would undergo if it propagated from its point
of excitation down an infinitely long cylinder, experiencing constant absorption

--z
per unit length. The wave fields would then decay as e

L~

. where z is the dis-

tance along the cylinder axis and LD is defined as the characteristic damping
length. In the actuai tokamak geometry, a useful model is to view tne antenna

as exciting waves whch propagate toroidally in both directions [Stix, 19751.
Since the damping length is n o t small compared to the circumference of the
tokamak, the waves interfere with each other and the wave amplitude a t a
given point is an infinite sum with contributions from each pass of the wave
around the torus. In this view, the toroidal eigenmode resonances arise from
the constructive interference which occurs when some integral multiple of the
wavelength in the toroidal direction (which depends, of course, on the plasma
density) becomes equal to the circumference of the torus. As pointed out by
Takahashi [1979], the damping length Lo for a mode can be estimated experimentally from measurements of the wave field amplitude at a resonance and
a t an adjacent anti-resonance (see Chapter 7).

The cavity Q and the damping length, LD, are both measures of the wave
absorption, and are related [Takahashi, 19771 by the parallel (i.e., toroidal)
group velocity of the wave:

The group velocity, ug I,, can be independently measured by a wave-packet technique which is also described in Chapter 7.
In the following sections, a variety of well-known damping mechanisms are
briefly described, and expected cavity Qs are calculated for each for parameters appropriate to the Caltech Research Tokamak.

2.3.2 Cyclotron Damping
The perpendicular component of the orbit of an ion in a magnetic field
consists, to first order, of simple Larmor precession a t the cyclotron frequency, and it is natural to try to couple wave energy to this motion. If a circularly polarized electric field were imposed which rotated in the same sense
and with the same frequency aiid phase as the ion motion, the ion would feel a
constant acceleration and energy would be transferred from the wave to the
particle. The velocity of the ion would increase linearly with time, and the ion
Larmor radius would similarly increase until it suffered a collision either with
another particle or with the vacuum chamber wall.
A prescribed electric field cannot, however, be simply imposed in the

plasma volume. Rather, sources a t the plasma edge excite waves whch then
propagate through the plasma according to the wave equations discussed earlier. Below the ion cyclotron frequency, both the slow and fast waves propagate.
The slow wave has a resonance ( / k / -r m ) a t w~

( lkj2 < 0 ) immediately above w,,;

and does not propagate

the fast wave propagates above and below

wGi and is unaffected by the fundamental cyclotron resonance.

Absorption of the slow wave a t the fundamental cyclotron resonance was
used for heating in early stellerator experiments [Stix, 19601. The waves were
launched in a region of high magnetic field ( w < o , )

and propagated into a

"magnetic beach" of lower field where a cyclotron resonant layer ( w = w , )

was

present a t whlch the wave was damped. In a tokamak geometry, however, the
static magnetic field gradient is perpendicular to the static magnetic field
direction, and the cyclotron resonance surface is a cylinder of constant r a l u s .
To take advantage of the fundamental resonance, the slow wave would probably have to be launched from the high field side of the torus, i.e., from the
inside. Calculations suggest, however, that the slow wave does not propagate to
the plasma core but is absorbed earlier in the plasma periphery by electron
Landau damping [Klima, 1975; Colestock, 19031.
The fast wave, on the other hand, penetrates the plasma easily. Cold
plasma theory predicts that the wave is completely right-hand circularly polarized a t w = w, , the electric field vector rotating in the sense opposite to that of
the ion precession. Hence, this simplest picture predicts no absorption of the
fast wave a t the fundamental cyclotron resonance. Hot plasma effects, however, prevent the wave from becoming completely right-hand polarized at
w = w G i , and the small left-hand component can then contribute to wave damp-

ing.
Calculation of the cyclotron damping in a tokamak configuration is complicated by several factors. The resonance surface is a cylinder of constant
radius, and ions following the helical static magnetic field lines can pass
through the resonant region a t most twice during each poloidal revolution. If
the wave field has a toroidal variation f k I I$ 0 ) then the resonance layer has
effectively a finite width due to the velocity distribution of the ions and the
variation of w, with major radius. Taking into account the Doppler shift, an
ion with a parallel (i.e., toroidal) velocity vll experiences the fundamental

resonance layer when the condition w - k i l v l=l w& is satisfied, where w is the
excitation frequency. For a toroidal magnetic field which varies as 1 / R and a
Maxwellian ion velocity distribution, the effective width of the resonant layer is
given by

where ti =

2kBG

z 1s.the ion thermal velocity. On each pass through the

resonance layer, an ion ( j ) receives a " k i c k Avlj in its perpendicular velocity.
Because the phase of the ion Larmor precession as it approaches the resonance region is assumed to be randomly &stributed with respect to the phase

of the wave field, the distribution of Auk is symmetric about zero: the ion
perpendicular velocity is as likely to be increased after passing through the
resonance zone as it is to be decreased. Net energy is still transferred to the
ion population, however, since the average energy increment is given by

< AE, > =

~ m (nuLj
)2

> . which is positive.

A simple calculation of fast wave cyclotron damping in tokaPmak geometry

is given by Stix [1975], who averages over randomly-distributed initial orbit
phases and assumes that collisions randomize the orbit phases between successive passes through the resonance zone. The analysis also assumes that the
wave electric field (calculated from cold plasma theory) is constant along and
across the resonance surface, that the parallel wavenumber is small enough
that

= w >> V A , and that the cyclotron resonance layer passes through
ll

the center of the plasma. The result for the cavity Q is

where kB is Boltzmann's constant and

is the ion temperature; a similar

result was also obtained by Perkins [1972]. The Qs predicted from equation
2.70 are plotted in Figure 2-12 for the N = 1-5, m = 1, I = 1 modes, as functions of the ion temperature. Note that the Q decreases with increasing ion
temperature and also with increasing kli. The cyclotron frequency on axis was
taken to be 8 MHz for this calculation, whch corresponds to the lowest frequency used in the experiments described in Chapters 6 and 7. This implies
that the static toroidal magnetic field on axis is 5.3 kG, whch is nearly the
upper limit to the field which can be obtained in the Caltech tokamak. For
most of the wave propagation and coupling experiments, the excitation frequency was high enough that the fundamental cyclotron layer was not inside
the tokamak vessel and fundamental absorption was not important.
Absorption can also take place a t harmonics of the fundamental cyclotron
frequency where the fast wave electric field has a finite left-hand component,
even in cold plasma theory. T h s mechanism requires both a finite Larmor
radius (i.e., non-zero temperature) and a spatial gradient in the wave field.
Consider an ion precessing at frequency wCi in a magnetic field. If a uniform
left-hand circularly polarized electric field a t frequency 2 w& were imposed,
the ion would be accelerated during one half of its orbit and decelerated
equally during the other half; the time-averaged energy change would be zero.
If, however, the magnitude of the electric field had a gradient in the direction
perpendicular to the static magnetic field, so that its magnitude varied over
the orbit of the ion, then the acceleration and deceleration during the orbit
would be unequal and a net transfer of energy from the wave to the ion could
take place. The wave damping in this simple model is proportional to
pf / v ~
EL/ I

IZ,

where p~ is the Larmor radius and

is the left-hand com-

ponent of the electric field, so harmonic damping favors high temperatures
(large Larmor radii) and fields which vary rapidly in the transverse direction
[Takahashi, 19771.

FUNDAMENTAL CYCLOTRON DAMPING

LOG

--

ION TEMPERATURE (EV)

F'IG. 2-12. Plots of the Qs of the N = 1-5, m = 1, L = 1 toroidal eigenmodes,
as functions of the ion temperature, for fundamental cyclotron damping.
Here w/2n = 8 MHz, B0 = 5.26 kG, and the ion cyclotron resonance layer
passes through the center of the plasma.

SECOND HARMONIC DAMPING

LOG (GI)

0.

50.

100.

150.

200.

ION TEMPERATURE (EV)

F'IG. 2-13. Plots of the Qs of the N = 0 -5, m = 1, L = 1 toroidal eigenmodes,
as functions of the ion temperature, for second-harmonic cyclotron damping. Here W / 2~ = 12 MHz, Bo= 3.95 kG, and the second-harmonic resonance
layer passes through the center of the plasma.

An analysis of fast wave second-harmonic cyclotron damping in tokamak
geometry is also given by Stix [I9751 and uses the same approximations made
for the fundamental damping calculation. The result for the cavity Q , assuming that the resonance layer is at the center of the plasma, is simply

where E = p o / R is the inverse aspect ratio, and Pi1= B~Z / ( 8 7 r q kB Td) is the
perpendicular ion beta.

A more detailed calculation by Paoloni [1975a] includes the form of the
toroidal eigenmode wave fields (using the model described in section 2.2.6) and
does not restrict k i p The result, assuming again that the second-harmonic
resonance layer passes through the center of the plasma colurnn, is

where

2%i
Ai = Z(p
pL)Z, kA = -- , D = 2

v4

+P'
k j -k,f '

and p

is the radial

wavenumber defined by equation 2.63. The Qs caiculated from the above formula are plotted f o r several low order eigenmodes, as functions of the ion temperature, in Figure 2-13. Note that the Q decreases with ion temperature but
increases with k t , .

2.3.3 Landau Damping and TTMP
When there exists a population of particles in the plasma with thermal
velocities parallel to the static magnetic field comparable to the parallel wave
phase velocity, the wave can transfer energy to the particles via Landau damping [Stix, 19621. For toroidal eigenmodes, the wave phase velocity is simply
0 vphll
=-W R , end its amplitude decreases for successive toroidal modes.
kll

For an N = 4 mode, for example, a t a frequency of 12 MHz,the phase velocity is
vphll = 8.5 x 10' cm/sec. For a plasma in thermal equilibrium a t a tempera-

ture of 100 eV, the electron thermal velocity is vt, = 4 . 2 10'
~ cm/sec, while
~ cm/sec. Because the expression for the
the ion velocity is vti = 9 . 8 lo6
power absorbed due to Landau damping involves terms proportional to

_wz

, where vt, is the electron or ion thermal velocity and a Maxwellian

distribution is assumed, the damping by the ions is negligible compared to that
due to the electrons. Further, the electron Landau damping decreases as the
toroidal mode number decreases (because of the consequential increase in

An analogous damping process for an electromagnetic wave propagating
parallel to the static magnetic field is transit-time magnetic pumping (TTMP).
Whereas the mechanism of Landau damping involves the interaction between
the parallel component of the wave electric field and the electric charge of the
particles (F=qJ$), damping due to TTMP involves the interaction between the
effective magnetic moment of the particle (due to its gyro-orbit) and a gradient
in the wave magnetic field amplitude ( F= -,u -VB).Stix [I9751 has shown that
TTMP and electron Landau damping are coherent processes and must be calculated together; the result is that the net damping is equal to half of that due
to TTMP alone, Paoloni [1975a] has calculated the expected toroidal eigenmode Q due to these processes, and finds

where R = w / w~ . The Qs calculated for toroidal modes appropriate to the Caltech experiments are plotted as functions of electron temperature in Figure
2-14. The Qs decrease as the electron temperature increases (since there are
then more particles with thermal velocity near v*~,) and also decrease as the

LOG

0.

50.

100.

150.

200.

ELECTRON TEMPERATURE (EV)

FIG. 2-14. Plots of the Qs of the N = 1-5, m = 1, 1 = 1 toroidal eigenmodes,

as functions of the electron temperature, for electron Landau damping and
TTMP. Note that Q = m for the N=O mode. Here ~ / 2 n =12 MHz and
Bo= 3.95 kG.

TOKAMAK WALL

LOG (Q)

0.5

1.0

1. 5

2. 0

2. 5

3. 0

FIG. 2-15. Plots of the Qs of the N = 0 -4, m = 0,1, 1 = 1 toroidal eigenmodes, as functions of 0 , for resistive wall damping. Bo= 3.95 kG.

toroidal mode number (and hence k i l )increases. The Q for the N = O (cutoff)
mode is infinite, since the parallel phase velocity is infinite and there is then
no Landau damping.

2.3.4 Resistive Wall Damping

Wave damping can also occur due to ohmic losses in the conducting
boundary surrounchng the plasma. The wave model described earlier in this
chapter assumed that the conductivity of the boundary was infinite, so that
the RF fields vanish inside the wall. This requires that the tangential magnetic
fields (B,, B,)

a t the inner wall of the cylinder be balanced by surface

currents flowing on the wall:

where j, is the surface current, j3 is a unit vector in the radial direction, and

.& is the tangential component of the magnetic field a t the surface of the conductor. In actuality, the conductivity of the tokamak vacuum vessel is finite,
and the surface currents flow largely in a layer of thckness 6 :

where [ is the perpenchcular distance into the conductor, measured from the

surface, and 6 , called the skin depth, is defined by 6 = (u,uw)-',

where u and p

are the conductivity and permeability of the conductor, respectively. Then the
total power lost in the conductor, per unit surface area, is given by

where jso = 1 j s o / , whle the total current flowing in the conductor, per unit
length perpendicular to the &rection of current flow, is

Hence the power lost in the conducting wall, per unit area, is

and it is apparent that R, = - can be viewed as the effective surface resist-

06

ance of the conducting boundary, Note that the surface resistance of stainless
steel (of whch the tokamak wall is made) a t a frequency of 12 MHz is

fZ, = 5 . 2 x

Q/square (mks units).

In calculating the power dissipated in the boundary, the usual approach
[Jackson, 19621 is to assume that the fields in the plasma are, to first order,
unchanged by the finite resistivity of the wall. The total currents flowing in the
wall are then calculated from the tangential components of the wave magnetic
field a t the boundary, and the distribution of current in the metal is assumed
to obey equation 2.75. The net power lost is then found by integrating equation 2.78 over the entire boundary surface. For tokamak geometry, the result
is

In order to calculate the cavity Q for an egenmode, the total stored energy
associated with the wave, E t O t ,is needed. This is obtained by integrating the
expressions for the wave electric, magnetic, and kinetic energy densities (equations 2.20 - 2.22) over the plasma volume, using the field distributions
appropriate
Qwrrll

for

particular

eigenmode;

the

is

then

given by

2n EtOt . Calculated Qs for various eigenmodes appropriate to the
= -0 Pwll

Caltech tokamak are shown in Figure 2-15 as functions of Q. Note that the
dependence of the Q on the toroidal mode number (i.e., k l l )is relatively small,

particularly near fl = 2.

2.3.5 Insulating Gap Damping
The tokamak must have an insulating gap a t a particular toroidal location
in order to prevent shorting out the electric field induced by the ohmic heating
transformer (see Chapter 3). In the Caltech tokamak, this gap consists of an
insulating spacer sandwiched between two stainless steel flanges welded to the
vacuum vessel. The gap represents a perturbation to the boundary conditions
of the wave-propagation problem and has apparently not been included in any
theoretical models thus far. Besides potentially altering the wave fields within
the plasma, the gap presents an additional mechanism for energy dissipation,

viz., through dielectric losses. The insulating spacer on many tokamaks is
made of

alumina, but the Caltech tokamak utilizes Bakelite (phenol-

formaldehyde). The dissipation factor (the reciprocal of the Q of the material,
sometimes called the loss tangent) for Bakelite is relatively high: D m0.03 at a
frequency of 10 MHz (the value can change somewhat with ambient humidity).

The i n s ~ u l a t ~spacer
an& the two adjacent flanges form a capacitor whose
value is Cc = 850 pF in the Caltech tokamak. Surface currents induced on the
inside of the tokarnak wall due to the RF fields can flow through the spacer as
displacement current, giving rise to dielectric losses; some of the current may
also flow out the gap and around the outside of the tokamak vessel, causing
energy loss through radiation. Only the former mechanism is considered here.
As an estimate of an upper bound of the energy loss in the insulating gap,
we make the approximation that all of the current flowing in the toroidal
direction on the inner wall flows through the gap as displacement current. The
surface current is calculated, as before, from the periodic cylinder model, and
the power dissipated in the dielectric is then

where I, is the net current traversing the gap,

and Rc = -is the effective resistance of the gap. The stored energy associQ

CG

ated with the wave is calculated as described in section 2.3.4; the energy
stored in the electric field in the dielectric gap is found to be small in comparison and is neglected,
The Qs for various eigenrnodes resulting from &ssipation in the gap are
plotted as functions of Q in Figure 2-16. Note that m = 0 as well as N=O
modes have no component of Be, hence there is no z-component of surface
current and no energy loss in the dielectric; these modes have intinite Q. The
Qs of the rn = 1 modes decrease as N (or k l i ) increases; this is a result of the
relative increase of Be as compared to B,, a t the plasma edge, as N is
increased (cf. Figure 2-10). A n increase in Be(po) causes an increase in j,,
which results in more cu-rrent passing through the dielectric gap and more

energy dissipation.

2.3.6 Collisional Damping
Wave energy can also be damped by collisions between particles in the
plasma. Ohmic or joule damping arises from collisions between electrons and
ions, described by "Spitzer resistivity" [Spitzer, 19621,while collisions with neutral particles can also lead to wave damping [Frank-Kamenetskii, 19601.
Paoloni [1975a] calculates the wave damping due to electron-ion collisions
for the fast wave in a cylindrical geometry. Only fluid motion along the static
magnetic field is considered, and & is approximated from the Maxwell equa-

BAKELITE GAP

4-

LOG (Q>

N= 1

3-

2.
0. 5

1.0

1. 5

2. 0

2.5

3. 0

FIG. 2-16. Plots of the Qs of the N = 1 -4, m = 1 , L = 1 toroidal eigenmodes,
as fxnctions of 0 , for dsmping due to dielectric losses in the insulating gap.
Note that Q = for N = 0 or m = 0 modes. Bo = 3.95 kG.

10

LOG(Q>

COLLISIONAL DAMPING

0.

50.

100.

150.

200.

ELECTRON TEMPERATURE (EV)

FIG. 2-17. Plots of the Qs of the N = 1 - 5 , m = 1, L = 1 toroidal eigenmodes,
as functions of the electron temperature, for electron-ion collisional damping. Note that Q = m for the N = 0 mode. Here w/2rr = 12 MHz and
Bo= 3.95 kG.

tion for V x B:

where B is calculated from the zero-m, model and qil is the parallel Spitzer
resistivity (in units of statohrn-cm). The ohmic power loss due to collisions is
then just

where the integration is over the volume of the tokamak. Paoloni calculates
the stored wave energy as described in earlier sections, and finds the following
expression for the Q for fast wave eigenmodes due to collisions:

For parameters appropriate to the Caltech tokamak, the Qs for the N=1-5,

m = 1, 1 = 1 eigenmodes are plotted as functions of the electron temperature
in Figure 2-17. Note that the Qs decrease as N increases, but that even for
the N = 5 mode, for T, = 100 eV, the Q is greater than lo7.

2.3.7 Comparison
In the preceding section, six &fferent wave damping mechanisms were
considered:

cyclotron fundamental, second harmonic cyclotron, electron

Landau and TTMP, resistive wall, dielectric gap, and collisional damping. These
mechanisms are compared, for some specific plasma parameters and frequencies, in Tables 2- 1 and 2-2.
Table 2-1 compares the Qs for the different damping mechanisms for the
case where w = wd a t the tokamak minor axis, i.e., the fundamental cyclotron
resonance layer passes through the center of the plasma column. The excitation frequency is taken to be 8 MHz, so the toroidal field on axis is 5.26 kG. The

T (eV)

Fundamental

EL + lTMP

Collisional

Wall

3.58

4.79

14.7

7.77

3.59

4.18

5.23

7.21

3.63

3.83

3.59

6.93

3.68

3.58

3.07

6.76

3.73

3.39

2.85

6.63

3.78

3.58

4.49

8.04

8.22

3.59

3.88

3.45

7.67

3.63

3.53

2.71

7.38

3.68

3.28

2.51

7.21

3.73

3.09

2.44

7.08

3.78

Total

50

100

TABLE 2-1. Values of log(Q) predicted from various damping mechanisms (fundamental ion cyclotron, electron Landau + TTMP, electron-ion collisional, wall
resistivity, and dielectric losses in the insulating gap) and the total Q , for the
N = 1 - 5, m = 1, I = 1 toroidal eigenmode in the Caltech tokamak. Here
G)/27~
= 8 MHz, Bo= 5.26 kG,and the fundamental cyclotron layer passes through
the center of the plasma; the second-harmonic resonance layer is not in the
tokamak. Results of the calculations are shown for T, = !& = 50 and 100 eV.

T(eV)

50

EL+ ZTMP

Collisional

Wall

Gap

Total
3.22

3.30

DD

DD

3.98

3.38

>20

7.95

3.98

4.19

3.23,

3.57

9.61

7.41

3.98

3.63

3.21

3.81

5.56

7.13

3.98

3.34

3.14

4.05

4.19

6.96

4.00

3.17

3.03

4.27

3.60

6.85

4.02

3.06

2.90

3.00

3.98

3.08

16.75

8.41

3.98

4.19

3.00

3.27

5.67

7.86

3.98

3.63

3.06

3.51

3.72

7.58

3.98

3.34

2.97

3.75

3.10

7.41

4.00

3.17

2.76

3.97

2.84

7.30

, 4.02 , 3.06

2.60

100

2.96

TABLE 2-2. Values of log(Q) prehcted from various damping mechanisms
(second-harmonic ion cyclotron, electron Landau + TTMP, electron-ion collisional, wall resistivity, and dielectric losses in the insulating gap) and the total
Q , for the N = 1 - 5, rn = 1, I = 1 toroidal elgentnode in the Caltech Tokamak.
Here o/2n = 12 MHz, B0 = 3.95 kG, and the second-harmonic resonance layer
passes through the center of the plasma; the fundamental resonance layer is
not in the tokamak. Results of the calculations a r e shown for T, = 1?, = 50 and
100 eV.

second harmonic resonance layer is then outside the tokamak and does not
contribute to the damping. The Qs are shown for the N = 1 -5, m = 1, 2 = 1
toroidal eigenmodes, for temperatures of 50 and 100 eV (the electron and ion
wall resistivity and
temperatures are taken to be equal). For small N (or k,,),
dielectric losses in the gap are the major energy loss mechanisms; as N
increases, fundamental cyclotron and electron Landau damping becomes more
important. As the temperature increases, both fundamental and Landau
darnping increase; for T = 100 eV and N = 5, the most important mechanism is
electron Landau damping. Collisional damping is negligible for all the cases
shown (it is only of consequence for very low temperatures).
The total Q is also shown for each toroidal mode number and is obtained
by summing the damping rates for each separate mechanism, assuming they
are independent. The total Q is observed to decrease as N increases and is
smaller for the kugher-temperature case. As N varies from 0 to 5 (for T = 100
eV), the total Q decreases from 3800 to 155.
An example where w = 2w& on axis is shown in Table 2-2. The excitation
frequency is taken to be 12 MHz and the taroida! magnetic field is 3.95 kG. The
fundamental cyclotron resonance layer is now inside the tokamak "hole" and
does not contribute to the damping. For small N the most important dissipation mechanism is seen to be second-harmonic cyclotron damping; for larger

N , electron Landau damping and dissipation in the dielectric gap become the
dominant mechanisms. For large N, the total Q decreases, and for IVa 1-2
the total Q reaches a maximum. For T = 100 eV, the total Q varies from 1150
( N = Z ) to 398 ( N = 5 ) .

2.4 Radial Density Profile
2.4.1 Numerical Model
The plasma density in the actual tokamak is not constant over the crosssection of the torus; it is generally peaked in the center. When the plasma
density varies as a function of radius, the elements of the dielectric tensor 2
are no longer constants. The spatial derivatives of 2 must then be included in
the derivation of the wave equations and simple analytic solutions are not
feasible for arbitrary profiles. In this section a numerical approach is taken,
solving for E and B by directly integrating Maxwell's equations. The periodiccylinder model is again used, and the plasma density is allowed to be an arbitrary function of radius witlun some mild constraints.
We

assume that

i ( m 6 t -z)

-iwt

&=O

and that

all field components vary as

, where, as before, m and N are the poloidal and toroidal mode

numbers and k i i= N/R . Writing out the components of the Maxwell equation

VXE=---

c at

then yields:

p- component:

-k,, Be = kOBp

(2.85a)

0- component:

kilEp= koBe

(2.85b)

whle the p and 8 components of the Maxwell equation V X B= -i k o F - E are
p-component :

--B,

+ k i l B e= k O € l E p- iFcocxEe

(2.86a)

The above constitute five equations in five unknowns, and it is easy to solve for
the derivatives of Eg and B, in terms of Eo and B, alone:

Note that E, and EL are functions of density and that k f - clkf > O for

wCi< o < oh . We assume that the plasma density n (p) satisfies

=0

and that n ( p ) is everywhere sufficiently large that uLh
fp)> o .
Expanding the above equations near p = O ylelds the same solutions as for
the constant density case:

where a is a constant and p is defined in equation 2.63. Equations 2.87a and
2.87b are then integrated by a simple numerical shooting code. First, the
static magnetic field Bo, the frequency w , and the functional form of n (p) are
chosen.

Po

An initial mean (i.e., line-averaged) density E = -Jn(p)dp
Po 0

is

selected, and initial values of the fields to start the integration a t p =6po are
found from equations 2.88a and 2.88b. The value of 6 is typically chosen to be
the solution is not sensitive to this choice provided 6 is sufficiently
small. The equations are then integrated from p =dpo to p=po using a standard numerical routine, and the value of Ge at p =po is recorded. The mean
density is then incremented and the procedure is repeated. The roots of the
function E b o ; E )thus generated are the &stinct densities a t which the boundary condition EB(po)= O is satisfied for the specified frequency w . The frequency is then incremented and the process is continued, mapping out the

solutions within a specified area in density-frequency space. For a given frequency, once the eigendensity is found, the wave field components as functions
of radius are found by integrating the equations for Ee and Bz and by using
equation 2.52 to find E,, Be and B,.
Several density profiles were investigated with this model. A quadratic
profile, often used in theoretical models, is given by

where a is a parameter related to the steepness of the profile. Another profile,
designed to simulate coherent radial density perturbations, was composed by
adding a Jo Bessel function to a constant background density:

where h = 1 +

Cm o

d q is the appropriate normalization factor and fm is

the mULzero of the Bessel function Jo. Here a is related to the magnitude of
the perturbation and rn , to the period.

2.4.2 Effects on Eigenmode Dispersion Curves
Dispersion curves for modes using the quadratic density profiles are
shown in Figures 2-18a and 2-1Bb for the first radial mode with N=O, 4 and

rn =O,1. Figure 2-18a shows the results for the profile nI(p; 0.5), corresponding to a profile with a peak (center) density twice the edge density; Figure
2-18b is for the profile n1(p; 0.9),corresponding to a profile with a peak density of ten times the edge density. Results from the constant density profile
model are also plotted (dashed lines) for comparison. The overall effect is that
the curves are displaced downward (i.e., to smaller densities and frequencies),

FIG. 2-18. Eigenmode curves for N = 0, 4, m = 0, 1, L = 1 modes. Solid lines
are for the quadratic density profile shown in the inset; dashed lines are the
solutions for a constant density. a) nI(p; 0.5) density profile, b) nI(p; 0.9)
density profile.

FIG. 2-19. Eigenmode curves for N = O ,4, m =0,1, 1 = 1 modes. Solid lines
are for the oscillatory density profile shown in the inset; dashed lines are the
solutions for a constant density. a) n g ( p ; 0.4; 5) density profile. b)
nn(p; 0.4; 20) density profile.

the displacement being greater for the profile which is steeper. T h s means
that, for a given frequency and mean density, the parallel wavenumber kll is
increased by changing the density profile from a constant to a quadratic form.
Note that the effect is considerably larger for the m = 1 modes than for the
m = O modes and that the relative displacement for the m = 1 curves is sub-

stantial.
Eigenmode dispersion curves for two density profiles with oscillatory perturbations are shown in Figures 2-1Qa and 2-19b. Figure 2-19a pertains to the
profile nrdp; 0.4; 5 ) which has 2.25 periods of oscillation radially; the density
a t the center of the plasma ( p = 0) is

1.4 times the mean density. The eigen-

mode curves are displaced slightly up in the frequency-density plane relative to
their positions for the constant-density case. For a given frequency and density, then, the imposition of the oscillatory profile decreases k l l ; the effect,
however, is rather small. The sign of the displacement is independent of the
phase of the density perturbation: inverting the profile (i.e., changing its
phase by n radians by changing the sign of a ) still ylelds an upward displacement of the curves.
Figure 2-19b shows the results for the nrdp; 0.4; 20) density profile. The
density a t p = 0 is still -1.4 times the mean density, but there are now

10

periods of oscillation in the radial profile. The eigenmode curves are still displaced upwards (i.e., lower klI for fixed 62 and n ) but the relative change is
less than 1%. Thus, as the wavelength of the radial density perturbation
decreases, the resulting effect on the location of the toroidal eigenmodes in
density-frequency space diminishes.

2.4.3 Effects on Field Profiles

The radial profiles of the components of B and E for the N = 4, m = 1,

I = 1 eigenmodes, with a constant density profile, are shown in Figure 2-20a.

Here the electric field is plotted in terms of the right- and left-hand circularly
polarized components:

E' = E' - i E8 and EL = Ep + i EB. The field profiles for

the n J ( p ;0.5) quadratic density function are shown in Figure 2-20b; the relative change in the fields is quite small.
Figures 2-20c and 2-20d show the field profiles using the nIdp; 0.4; 5 ) and
nIAp; 0.4; 20) density functions. The effect on the field components is now

quite noticeable: the components are perturbed with the same wavelength as
the density perturbation. It is of particular interest that the left-hand component of the electric field, E L , is altered markedly from its constant-density
form. This is significant because it is the left-hand component of the field that
is responsible for fundamental cyclotron damping, and second-harmonic
damping is related to the transverse (radial) gradient of this field component.
Figure 2-21 shows the ramal form of

/ V L4
~ 1

1'

for the density profile

nIAp; a; 20) as a is increased from 0.0 to 0.4. The transverse gradient of the

field is seen to be extremely sensitive to the ramal density perturbations: for
the a = 0 . 4 case, the peak of

~V'I 1 /1
EL

is more than two-thousand times

greater than the peak for the constant density ( a = 0 ) case.
The wave damping, of course, is dependent not only on the peak of the
transverse gradient, but on some form of integral of the transverse gradient
over the cyclotron resonance layer. A full calculation of the Qs expected for
the eigenmodes using these density perturbations remains to be carried out.
It is likely, however, that the dramatic increase of the transverse gradient of

EL will result in substantially increased damping and hence lower values for
the Qs.
The preceding model of ra&al density perturbations is not presented as a
realistic emulation of the tokamak environment; it is unlikely that such oscillatory density profiles as described actually exist. The model is intended only

FIG. 2-20. Components of B and E as functions of radius for the N = 4 , m = 1, 1 = 1
toroidal eigenmode, for different density profiles. Here w / 2n = 12 MHz and Bo= 4 kG. The
horizontal axis in each plot is the radius p; the limits of the axis are 0 (left end) and po
(right end). The top plot in each column shows the density profile. a) Left-hand column constant density profile. b) Right-hand column - quadratic density profile n I ( p ; 0.5).

FIG.2-20, continued. c) Left-hand column - oscillatory density profile nn(p;0.4; 5).
d) Right-hand column - oscillatory density profile nn(p;0.4; 20).

FIG. 2-21. Plots of the radial dependence of / VL 1 ELI 1 for the N =4, m = 1, 1 = 1
toroidal eigenmode and the n n ( p ; a; 20) oscillatory density profile, as a is
increased from 0 to 0.4. Horizontal axis in each plot is the radius p; the axis limits
are 0 (left end) and po (right end). Here w/2n = 12 MHz and Bo= 4 kG. Field distributions have been normalized so that B, (po)= 1 G in each case.

FIG. 2-21,continued.

to suggest that spatial density perturbations may be important to consider in
calculations of cyclotron wave damping.
There is evidence that global density perturbations do exist in tokamaks.
Low frequency ( <have been observed in many tokamaks. The "sawtooth oscillation, for
instance, has been seen with an m = 0 (poloidal) and N = 0 (toroidal) structure; superimposed on t h s is a lower amplitude, higher rn-number, hlgher frequency perturbation [Von Goeler, 1974; Jahns et al.,19781. Many of the initial
measurements were made with soft-X-ray detectors which respond to both
electron temperature and density fluctuations. The T.F.R. group investigated
the sawtooth fluctuations with a variety of external diagnostics in order to
separate the density and temperature effects [T.F.R. Group, 19761. They

F - 17%; for

estimated typical values for the temperature fluctuations to be T
the density fluctuations,

X g 3 %. Unfortunately, there is little information

available about the radial profile of the density during sawtooth or other MHD
oscillations.
Another kind of spatial density fluctuation occurring on a time scale

2n

much longer than - has been observed in a number of tokamaks [Surko and
uci

Slusher, 1900; Mazzucato, 1982; Zweben and Gould, 19831. These fluctuations,
referred to as "microturbulence", typically have broad frequency spectra
!E
(below -200 kHz), significant amplitudes ( - 0,01-0.5) which depend on

the ra&al position, and short correlation lengths (

1 cm). On the time scale

of the RF period, then, the plasma density appears "grainy". It is conceivable
that this form of density perturbation could also induce short-range fluctuations in the left-hand component of the wave electric field, resulting in
enhanced cyclotron damping. Clearly, more work needs to be done to assess

this possibility.

2.5 Other Effects
2.5.1 Poloidal Magnetic Field
The &scussion of toroidal eigenmodes has thus far neglected the poloidal
component of the confinement magnetic field whch is generated by the ohmicheating plasma current. To first order, this field varies as l / p , where p is the
minor radius, and its magnitude a t the tokamak wall is typically 10% of the
toroidal field.
The poloidal field breaks the degeneracy of the -irN egenmodes: waves
traveling in the direction of the plasma current see a different environment
than waves traveling in the opposite &rection. The net result is to change
slightly the densities a t whch the + N and -N modes become resonant, leading to mode "splitting" [Takahashi, 19771. The effect has been considered
theoretically by several authors with similar results [Adam and Jacquinot,
1977; Messiaen, 19781. Messiaen considers a uniform plasma with a poloidal
field arising from a uniform plasma current density:

BPo1
= yB,, where y = P

qR '

R is the major ra&us, and q is the usual safety factor. Keeping only terms to
*st

order in y and neglecting the electron mass, a wave equation is derived

for B,:

where p has been defined previously (equation 2.63) and /I

kt! E X
. Note
k8 €1- k,?

that for q -, m, correspondmg to zero poloidal field, or if m = 0 or k,,
= 0, the
wave equation reduces to the form found earlier (equation 2.40). Imposition of
the appropriate boundary condition ( E,(po) = 0 ) yields, as before, an equation

which specifies the density for a mode given the frequency, i.e., equation 2.66
where p is replaced by
For parameters typical of the Caltech tokamak, the relative shift in density of the resonances associated with a plus and minus N toroidal mode
number pair is small (typically < 5%). A s will be mentioned in Chapter 7, resonance peak splitting which may be due t o the poloidal field is sometimes, but
not consistently, observed in actual experiments. Resonances due to a wave
traveling in a single direction (i.e.,a plus or minus N mode alone) are traveling wave resonances and the magnitude of the wave fields should not vary as a
function of toroidal angle. If, however, the resonances due to the k N modes
substantially overlap (the resonances have finite width due to damping), then
the modes become standing wave resonances and the amplitude of the fields
should have nodes and peaks as a function of the toroidal angle.

2.5.2 Toroidd Geometry
The effects of toroiiia! geometry on fast wave propagation and elgemodes
have been considered by only a few authors [Pridmore-Brown, 1966; Gould,
1975; Swanson, 19751. The most important effect of toroidal geometry is the

inhomogeneous toroidal magnetic field which varies as 1/ R. Physically, this
means that the absorption which may occur a t the cyclotron fundamental and
harmonic frequencies is confined to a cylindrical surface of constant major
radius. Although t h s is clearly of great importance in calculating the wave
damping, in cold plasma theory the fast wave has no dramatic transition a t
these frequencies and so this effect may not be important in determining gross
propagation characteristics. Unfortunately, the wave equation in toroidal
geometry cannot be solved by separation of variables; numerical methods are
possible, but the problem is formidable.

Swanson considered a plasma-filled toroidal cavity of rectangular crosssection with the appropriate toroidal magnetic field. He calculated approximate fast wave egenmode frequencies and found them to be within about 10%
of the frequencies calculated from a straight rectangular model, for low order
modes. It was noted, however, that there were in general no single mode solutions whch simultaneously satisfied all of the boundary conditions.
Numerous questions remain about the theory of fast wave resonances in a
truly toroidal cavity; no satisfactory analysis has yet emerged.

The Caltech Research Tokarnakt

The Caltech Research Tokamak is a small circular cross-section device
with an air-core ohmic heating transformer. I t has been used primarily for
investigations of RF wave propagation near the ion cyclotron frequency
[Hwang, 19791 and for studies of electric [Kubena, 19781 and magnetic [Levine,
1980; Hedemann, 19821 field fluctuations within the plasma outer edge. A
recent focus of research has irzvolved turbulence in the plasma edge and its
relation to energy transport [Zweben et al., 19831. The tokamak is in regular
operation with toroidal fields of 3 - 6 kG, plasma currents of 15 - 35 kA, and
plasma densities of 1012-1013 cmS. Pulse lengths are typically 10 - 20 msec,
and ion and electron temperatures are 50 - 100 eV.

3.1 Vacuum Vessel
The Caltech Research Tokamak was constructed during 1975 and 1976.
The vacuum chamber was formed from four 7.5 mm thick stainless steel (316L)
90 degree elbows whch were welded together to form two 180 degree flanged
sections. Azimuthal copper water cooling channels were attached directly to
the chamber (Figure 3-1). The completed torus had a major radius ( R o ) of
45.7 cm (i.d.) and a minor radius (a)of 16.2 cm (id.); hence the aspect ratio
was 2.8. The chamber, which had no plasma limiter, was evacuated by a
t This chapter is a n updated version of the reference by Greene and Hedemann [1978].The
work described, except for the section on the charge-exhange diagnostic, represents the
combined efforts of a number of individuals.

Half of torus showing azimuthal water cooling channels and ports.

'/

(CHARGE EXCHANGE) \

FIG. 3-2. Top view of tokamak, showing locations of ports used in these experiments. Ports #1-4 are 2.2 cm (i.d.1, ports #5 and #7 are 9.8 cm tall x 2.2
cm wide, port #6 (the RF antenna port) is 9.8 cm tall x 4.7 cm wide, port #8
(one of the microwave interferometer ports) is rectangular, 2.2 cm x 3.5 cm.
All ports are 15 cm long and end in high vacuum flanges.

Leybold-Heraeus turbomolecular pump, with a speed of 450 liters/sec, which
permitted base pressures of 2 x lo-? Torr to be attained without the use of a
cold trap. Viton O-rings and copper gaskets were used throughout the vacuum
system. The two halves of the torus were separated by a Viton O-ring backed
by insulating Bakelite spacers in order to prevent azimuthal current flow.

A variety of ports were welded into the vacuum chamber. The largest,
whose cross-section measured 9.8 cm tall x 4.7 cm wide, was connected, via a
14 cm i.d, elbow, to the high vacuum pump. A 10 cm i.d, port was later welded
into the elbow to allow the use of the pumping port for the RF antennas. Five
ports were available which measured 9.8 cm tall x 2.2 cm wide; they were
located, like the pumping port, a t the midplane around the outer radius of the
chamber (R = Ro+ a). A number of pairs of 2.2 cm i.d, cylindrical ports were
located along the top and bottom of the chamber, and a pair of rectangular
ports, 2.2 cm x 3.5 cm, contained the horns for the microwave interferometer.
All of the ports were about 15 cm long and ended in high vacuum flanges. The
positions and orientations of the ports used in the experiments described in
this thesis are shown in Figure 3-2.

3.2 Coil Design
The toroidal field coil was wound directly on the vacuum liner. It consisted of 480 turns of #1/0 AWG wire (nominally rated at 600 V!) wound in a
double toroidal layer. Small inhomogeneities in the toroidal field coil winding
(from ports projecting through the coil, etc.) resulted in horizontal and vertical field components (referred to as error field components) which
significantly affected the plasma confinement time and the peak value of the
plasma current,
Rather than measure the error fields inside the plasma volume, an

indirect procedure was used to infer these fields from measurements made
outside the chamber. The toroidal field winding (minor radius = 17 cm) was
modeled as a perfect current sheet with the addition of a number of azimuthal
(i.e., toroidal) filamentary currents on the winding surface which were responsible for the error fields. To find the positions and amplitudes of these
assumed currents, twelve one-turn azimuthal loops, spaced 30 degrees
poloidally, were placed a t a minor radius of 30 cm (i.e., outside the windings).
The toroidal field was pulsed and the voltage induced in each loop was measured. From these data, the flux through each loop was obtained; t h s flux was
attributed to the error currents, since a n ideal winding would produce no flux
external to the winding surface. By trial and error calculations, a set of five
azimuthal ring currents was found which reproduced the measured fluxes (Figure 3-3). The error fields inside the vacGum chamber due to these currents
were then computed. To cancel these fields, a set of azimuthal correction coils
were designed whch were located outside the toroidal field coil winding surface
and which produced a field distribution inside the torus approximately equal
and opposite to the error fields inferred by the above procedure (Figure 3-4).
These error fields turned out to be largely horizontal in direction, hence the
correction coils are referred to as horizontal field coils, Proper correction for
all values of the toroidal field was insured by connecting the correction coil in
series with the toroidal field windmg. A second set of horizontal field correction coils was added to the tokamak several years after initial operations
began in order to further increase the plasma lifetime. This set of coils was
energized independently of the toroidal field by a separate DC supply. Correcting for the toroidal field errors had a large effect on the performance of the
tokamak. Prior to field correction, the plasma pulse length was about 2 msec
and the maximum plasma current was 10 kA; after field correction the maximum pulse length and plasma current rose to 20 msec and 35 kA, respectively.

0 EXPERIMENTAL ERROR FLUX

ASSUMED ERROR CURRENT POSITIONS

- FLUX FROM ASSUMED CURRENTS

-120°

r- ,t

-,

- 1

-60"

FIG. 3-3. Measured values of the flux versus poloidal angle and the
Aux due to the five assumed currents.

MAJOR
AX1 s

ONE-TURN LOOPS
0 ASSUMED CURRENTS

CORRECTION COILS

FIG. 3-4. Cross-section of the torus, showing locations of one-turn
loop measurement coils, assumed currents and correction coils.

The design of the ohmic heating coil required little or no magnetic field
inside the plasma due to the current in the coil (to avoid disturbing the plasma
equilibrium), whle it was desirable to have all windings lie on a single toroidal
surface for ease of construction. To insure that the magnetic field inside the
plasma was zero, the ohmic heating coil was designed to have a flux line coincide with the winding surface'

(major radius = 45.7 cm, minor ra&us =

27.0 cm). The positions of the coil windings on this toroidal surface were cal-

culated with the following simple model. A nearly circular flux line at the
minor radius of the winding may be produced using only a few symmetrically
placed azimuthal ring currents (Figure 3-5). The ring currents can be replaced
by an assumed azimuthal sheet current on the winding surface, where the
current at any point is taken to be proportional to the magnitude of the field
from the ring currents at that point. The field outside the winding surface is
then the same as that produced by the ring currents, while the field inside is
zero. Breaking this sheet current into finite wires introduces only small error
fields into the plasma volume.
The ohmic heating coil was wound on eight wooden frames clamped to the
vacuum chamber, using #1 AWC cable. The error fields produced withm the
vacuum chamber due to this coil were calculated to be less than 1%of the
ohmic heating field amplitude at the center of the torus (along the major axis).
The vertical field coils (used to control plasma radial position) were wound
on the same frames which supported the ohmic heating coil and were designed
to produce a slghtly curved field within the plasma volume. The magnetic field

a& where B, is the vertical field magnitude and R is the
index, q = -B, aR '
major radius, is a measure of the curvature of the field and was chosen to be
1. Then, from Ampere's law, it follows that the field must vanish inside the winding surface
since there are no current sources there.

FLUX LINE AT MINOR
RADIUS OF WINDING

CURRENTS

FIG. 3-5. Several flux lines of the ohmic heating field are shown, including
the one a t the minor radius of the winding. The flux lines are produced
by a model consisting of two azimuthal filamentary ring currents of
different magnitudes, flowing in the same direction, one a t 0.63 R and the
other a t 0.75 R, where R is the major radius of the toms.

FPG. 3-6. The Caltech Research Tokamak, showing
confinement and ohmic heating coils.

about -0.5 on the minor axis of the torus. The coils consisted of 40 azimuthal
turns of #1 AWG cable, wound on the same toroidal surface as the ohmic heating coil. The turns were uniformly spaced poloidally, but the 20 turns on the
inside (R < 45.1 cm) were wired so that the current flow through them was in
the opposite dwection to that in the 20 turns on the outside (R > 45.7 cm).

A photograph of the tokamak, showing the toroidal field, horizontal
correction field, ohmic heating, and vertical field coils is shown in Figure 3-6.

3.3 Power Supplies
A tokamak shot involved the firing of a number of power supplies at

specified times, A bank of digital timing modules in the power supply room
provided easy adjustment of the relative timing. The repetition rate of the
tokamak was limited by the toroidal field capacitor bank charging time and
was generally set to a one minute cycle.
The sequence of events involved in firing the tokamak began with the
switching on of a tungsten filament located in one of the bottom tokamak
ports. The filament, biased 10-20 V negative with respect to the tokamak
chamber, provided a supply of free electrons and promoted consistent breakdown of the hydrogen filling gas. Several seconds after the filament was turned
on, the toroidal field bank was switched into the toroidal field winding.
Power for the toroidal field coil was provided by a 240 kJ, 10 kV capacitor
bank which was capable of providing a maximum field of about 13 kG on axis.
Mechanical stresses in the coils became rather large a t high fields, however, so
in practice the toroidal field was rarely used above 5 kG. The bank was
switched by ignitrons, and in order to prevent L-C ringing during the plasma
shot, a crowbar circuit was used. An ignitron shorted the toroidal field coil
when the current in it reached a maximum, i.e., when the voltage across the

capacitor bank crossed zero after the first quarter-cycle of the L-C ringing
(the quarter-cycle period was typically -9 msec). Thereafter, the dominant
parameters of the circuit were the toroidal field coil inductance (kc= 7 mH)
and resistance (Rtc =0.17 0). The toroidal field amplitude then decayed
k c

exponentially with a time constant T = - = 42 msec.
4 c

The ohmic heating coil was energized by a power supply which consisted of
two capacitor banks, referred to as the "fast" and "slow" banks, which were
switched into the ohmic heating coil through ignitrons. The two banks, which
had different capacitances and could be charged to different voltages, provided
some degree of flexibility in shaping the ohmic heating current waveform.
It was desirable for the ohmic heating field to have a fast initial rise time
in order to generate a large electric field in the tokamak chamber to promote
complete breakdown of the plasma. Since the rise time of the current was

e-.

where LOH and CoH are the inductance of the ohmic heating

coil and the capacitance of the bank, respectively, the fast bank had a smaller
capacitance and was operated at htgher voltages than the corresponding slow
bank. The fast bank was fired first (about 2 msec before the peak of the
toroidal field current), and the slow bank fired when the voltage on the fast
bank had decayed to the point where the voltages on the two banks were equal.
The fast bank capacitance was 650 mF and could be charged to a maximum of
2.5 kV, while the slow bank consisted of two halves each of which had a capacitance of 0.22 F at up to 450 V. The two halves of the slow bank could be connected in series or in parallel to provide further flexibility.
Transient supressors were used a t each end of the coaxial transmission
lines (RG-215) which connected the ohmic heating and toroidal field coils to
the power supplies, located in an adjacent room. These simple R-C combinations reduced the voltage spikes which occurred due to reflections from the

impedance mismatch at the ends of the cables.
The vertical field coils were also energized by a power supply consisting of
"fast" and "slow" capacitor banks. The fast bank consisted of 2 pF of capacitance, was typically charged to 100 -200 V, and was discharged into the vertical field coil through an SCR. The slow bank, consisting of 400 fiF of capacitance, was charged to 40 -60 V and was switched into the coil when the voltage
on the fast bank had decayed to the level of the voltage on the slow bank. The
vertical field fast bank was fired about 100 psec after the ohmic heating fast
bank.
To provide for reliable initial breakdown of the plasma, a 1.5 mF capacitor, charged to 6 - 9 kV, was discharged into the ohmic heating coil about

2 msec before the firing of the main ohmic heating bank. This "preionization"
supply resulted in an oscillating current in the ohmic heating coil with a frequency of about 13 kHz and a duration of about 1 msec. The oscillating magnetlc flux produced a large toroidal electric field in the tokamak, causing initial breakdown of the gas by accelerating the electrons provided by the
tungsten filament.

3.4 D a t a Acquisition
The data acquisition system was housed in the control cabinet which also
included the RF &agnostics electronics (Figure 3-7). The data system evolved
through many incarnations, but only the most recent (and that used for most
of the experiments described in this thesis) will be described.
The four signals which were generally most useful in diagnosing the
plasma global behavior (the one-turn loop voltage signal, the plasma current
signal, the soft-UV monitor output, and the microwave interferometer detector
output) were recorded on a four-channel waveform recorder (Biomation model

FIG. 3-7. Data acquisition system (center) and RF electronics (left-most
cabinet). The dedicated LSI-11 computer is visible at the right.

1016). Each channel had a memory of 1024 bytes and a resolution of 8 bits.
The sampling rate of the Biomation recorder was usually set to 50 kHz so that
the record length, 20 msec, was sufficient to cover all but the longest plasma
shots. The time variation of the above-mentioned signals was quite slow, so
that anti-aliasing filters were in general not required. The four channels were
displayed on a CRT monitor irnme&ately after each tokamak shot, and an
interface allowed transfer of the data to a DEC LSI-11/03 computer whch was
dedicated to data collection. Data transferred to the computer were stored on
floppy disks for later analysis with an LSI-11/23 computer.

A CAMAC crate containing a number of LeCroy A-D modules was used for
recording other experimental signals. Two LeCroy model 2264 transient recorders provided up to sixteen channels of 8-bit data recording capability with a
record length for each channel of 4 kilobytes and a maximum sampling rate
(in the sixteen channel mode) of 500 kHz. A plug-in module for the CAMAC
crate2 allowed display of the sixteen data channels on CRT monitors automatically after the tokamak shot and also permitted transfer of the data to the
computer. Because the LeCroy modules had only fixed full-scale inputs of
k256 mV, a series of sixteen buffer-amplifiers with attenuators were constructed which provided calibrated full-scale inputs from 0.2 V to 20.0 V and
also included offset adjustments. Twelve of the buffers had -3 db bandwidths of
250 kHz; the other four had bandwidths of 1 MHz. Since the sampling rate
used for most of the experiments was either 50 or 100 kHz, low-pass filters
(with cut-off frequencies of about 20 and 40 kHz, respectively) were sometimes
used to eliminate the possibility of aliasing from higher-frequency signal components.

2. Designed and built by F.Cosso, E.E.(Caltech).

3.5 Diagnostics
A variety of standard tokamak diagnostics for measuring average macro-

scopic plasma parameters were available; they are described here only briefly
(for further information, see DeMichelis [1978] or Cross [1977]).
Plasma current was measured with a Rogowski coil which encircled the
chamber poloidally. The signal from the Rogowski coil was proportional to

s,
where 4 is the net plasma current, and was integrated electronically to
at
provide an output of 0.5 V/kA.
Plasma position (in an average sense) was monitored by "up-down" and
"in-out" coils. These coils were similar to Rogowski coils, but the number of
turns per unit length was made proportional to cos(6) or sin(@ (where B is the
poloidal angle) with the directican of winding reversing appropriately as the
cosine or sine changed s g n . The integrated voltage from the coils was divided
by the plasma current electronically, and the resulting voltage was roughly
proportional to the vertical or horizontal plasma mean-center displacement.
The in-out coil was used primarily as a n aid in adjusting the charging voltages
for the vertical field capacitor banks.
The plasma loop voltage 5 (the "one-turn voltage") is the integral of the
electric field in the plasma around the tokamak toroidally. Coupled with the
plasma current Ip , tlvs velds a value for the plasma resistance, 4 , from
wlvch the electron temperature can be deduced. The loop voltage was measured indirectly using a single loop of wire placed on top of the torus. The voltage V measured by this loop is not the same as the plasma loop voltage, % ,
due to the poloidal flux between the plasma and the wire. The measurement,
however, may be used to calculate the plasma resistance from

where L2, is the effective inductance of the toroidal plasma [Dimock et al.,
19731, The electron temperature is then related to J$ by

where Zeff =

(x22 (x

Zana) is the effective atomic nurnber of the plasma

nu)/

(n, is the density of ionic charge species a , in emp3, and Z, is its atomic

number), Te is the electron temperature in eV, and A = 1.54X 101° ~ ; ~ e/ ~ ' n - ~ / ~
Solution for the electron temperature is carried out by numerical integration.
A microwave interferometer operating a t 75 GHz measured the average

electron density along a vertical chord through the center of the tokamak.
The additional phase shift of the microwave signal passing through the
tokamak chamber, due to the presence of the plasma, is approximately

provided U$~(C)<< w2 for -po s (. S r O . Defining the line-averaged electron density % , it follows that

so the electron density is linearly proportional to the phase shift or to the
number of fringes observed on the output of the interferometer's crystal
detector (one fringe corresponds to a phase shift of
W =75GHz,
2lT

the

constant

of

proportionality

2rr ra&ans). For
is

approximately

1 . 8 lo1'
~ ~m-~/fringe.

The microwave source was a Hughes Impatt oscillator (model 41343H-001)
which provided about 75 mW of power and had to be well shielded (see section
4.1.3). The interferometer was constructed from WR- 15 waveguide plumbing;

the detector was a IN53 &ode in a Baytron 1V-30 mount. The detector functioned as a homodyne mixer, and the output was proportional to sin(A9 + ao),
where Go is a constant. One fringe shift therefore corresponded to a change in

A@ of 2n radians. The sign. of the phase s h f t could not, however, be directly
determined, and this sometimes led to ambiguity in determining the density
evolution during part of a tokamak shot. The art of reconstructing % ( t ) from
the interferometer output is discussed by Levine [1980].
Two light detectors were used to provide general indications of impurity
levels in the plasma. A visible light detector utilized a phototransistor (Fairchild F'PT 120A) with a spectral response from 500 to 1000 nm. This detector
was generally mounted on an optical port on the RF antenna feedthru flange
and was used to register signs of arcing from the RF antennas. A soft-UV
vacuum photodetector [Zweben et al., 19791 was very useful in determining
global relative impurity levels. The detector consisted simply of a bare plate of
tungsten, located in one of the bottom tokamak ports and appropriately
shielded from the plasma. The photoelectric current from the plate was measured and was proportional to the incident flux of UV photons; the response of
the detector was roughly from 20 to 120 nm (the "soft" UV region). The detector response during a tokamak shot was apparently largely dominated by radiation from oxygen impurity lines. The magnitude of the signal correlated
inversely with the peak plasma current and plasma shot duration: low level
signals in&cated clean plasmas and vice versa. This diagnostic was used routinely to estimate the relative "cleanliness" of the machne after various
probes were introduced.

3.6 Discharge Cleaning
The inside wall of the tokamak was regularly cleaned by procedures

recommended by R. J. Taylor [Oren and Taylor, 19771. The ohmic heating coils
were pulsed with a 3 kW, 30 kHz oscillator for 30 -50 msec, a t a repetition rate
of about 2 Hz. During the discharge cleaning, a current of 70 -80 A was passed
through the toroidal field coil, producing a magnetic field on axis of about
150 G and lengthening the confinement time of the plasma. The gas fill pres-

sure was 5 - 10 x

Torr of hydrogen which was admitted to the tokamak, as

during regular plasma shots, through a leak valve on the forepump side of the
turbopump3.
Through processes that are poorly understood, the low density plasma
knocks low-Z impurities off of the walls of the vacuum vessel. The primary
impurities, carbon and oxygen, then combine in the plasma to form methane
and water vapor which are pumped out through the turbopump. Although the
dynamics of the chemical processes occurring in the plasma have not been well
characterized, the overall effect on the tokamak performance is dramatic.
Without &scharge cleaning, the tokamak plasma shots are limited to a duration of only a few msec and to plasma currents of about 10 kA; after sufficient
discharge cleaning, the shots last 10 - 20 msec with currents of up to 35 kA.
Evolution of impurity levels in the tokamak neutral gas during discharge
cleaning was monitored with a quadrupole residual gas analyzer (Varian
VGA-100); relative impurity levels required for clean operation of the tokamak
were established through experience. Daily discharge cleaning of 4-6 hours
was generally sufficient, but after opening the tokamak to air (to change a
probe, for instance), cleaning for 15 - 25 hours was usually necessary.

3. Since the pumping speed of the turbopump was considerably smaller for hydrogen than
for higher mass impurities, admitting the gas on the forepump side served t o increase
the purity of the gas reachmg the tokamak.

3.7 G a s Puffing
When the tokamak was fired, the plasma density rose rapidly ( < 1 msec)
to a peak value of 5 - 10 X 10'' em-'.
about 1 -2 msec to a level of

The density then decayed over a period of

10'' em-' where it remained for the rest of the

shot. The peak density was smaller by a factor of 2 - 3 than what would be
expected if all of the hydrogen gas in the chamber were ionized: a typical
filling pressure of

3.5 x l o 4

Torr4 would yleld an electron density of

2.5 x 1013

if fully dissociated and ionized (the volume of the tokamak was

237 liters).

Since optical measurements in&cated a nearly fully ionized

plasma, the discrepancy was probably due to hydrogen being driven into the
tokamak walls or into the ports. There was some evidence that the background neutral gas pressure in the ports rose during the plasma shot.
The density fall following the initial rise was presumably due to the loading
of the inner wall with hydrogen. In order to increase the plasma density again,
a fast-opening ( - 1 msec) piezoelectric gas valve (Veeco model PV-10) was
installed. The valve was located on one of the bottom tokamak ports, and was
connected t o 8 hydrcgen reservoir which was typically kept a t a pressure of
500 -800 Torr. The duration and temporal position of the voltage pulse which
opened the valve were controlled by two digital timing modules. The valve was
typically pulsed for 2 -8 msec, beginning 2 -5 msec after the start of the
tokamak shot.
The additional gas introduced by the piezoelectric valve was quickly ionized by the plasma and caused the plasma density to rise. A short pulse of gas
typically resulted in a n approximately linear rise in density, over a period of
3 -6 msec, up to a peak of 1.0 -1.5 x 1013em-'.

While the density rise after gas

4. The ionization gauge used to measure gas pressure in the tokamak was calibrated for ni-

trogen; its sensitivity for hydrogen gas was smaller by a factor of about 0.43. This prep
sure, and all other pressures reported in this thesis, are corrected ion gauge pressures.

puffing was fairly reproducible from shot to shot, the subsequent fall in density
sometimes was not. With a clean machine, the density decay would occasionally virtually mirror the rise; more often, however, the fall was much more
abrupt. Gas puffing also shortened the overall plasma lifetime, Without gas
puffing, shot lengths of 20 msec were easy to obtain; with gas p u f i g to a density of about l 0 I 3

the maximum shot lengths were 10 - 15 msec.

3.8 Plasma Characteristics
Records of the one-turn voltage, the plasma current, the soft-W monitor
output, and the plasma density for a typical clean tokamak shot (with gas
puffing) are shown in Figure 3-8. For comparison, a shot in which a contarninated probe was in the plasma is shown in Figure 3-9.
The one-turn loop voltage is a useful monitor as it provides confirmation
of the proper firing of the ohmic-heating fields. The first oscillation observed,
at around t = 1.5 msec, is due to the preionization capacitor being discharged
into the ohmic heating winding. About one msec later, the loop voltage signal
goes off scale; this results from the fast bank being fired. After an interval of
another msec, the fast bank voltage has decayed sufficiently that the slow
bank fires; this transition is evident on the trace. Misfirings due, for instance,
to insufficient preionization voltage or a faulty ohmic heating supply ignitron
are easily diagnosed using the loop voltage signal.
The plasma current increases abruptly (within about 0.5 msec) to a level
of 10-15 kA itnmeckately after the ohmic heating fast bank is fired. The
current then increases slowly over a period of 3 -5 msec, peaks a t a current of
20 -30 kA, and begins to decrease. In a clean tokamak, the current decay is
smooth; in a dirty machne the decay is much more abrupt and the plasma
shot length is decreased.

TIME (MSEC)
FIG. 3-8. Typical "clean" plasma discharge. a) One-turn loop voltage. b) Plasma current. c) Soft-W signal. d) Plasma density,
derived from the interferometer output. Squares indicate halffringe increments.

TIME (MSEC)
FIG. 3-9. Typical "&rty" plasma discharge. a). One-turn loop voltage. b) Plasma current. c) Soft-W signal (units are the same as
in Figure 3-Bc), d) Plasma density.

The soft-UV signal is the most useful indicator of the impurity level of the
machme. When the tokamak is clean, the signal is small and varies slowly during the plasma shot, roughly following the plasma density evolution. In a contaminated machne, however, the W signal is much larger and may increase
dramatically throughout the abbreviated plasma shot. The behavior shown in
Figure 3-9 is often characteristic of a probe "burning up" in the plasma.
The density evolution shown in Rgure 3-8 is characteristic of a clean
plasma discharge; the gas puff causes a linear increase in density up to a peak
of around 1013

followed by a similar density decay. The density evolution

of the dirty discharge shows a fast increase and an abrupt fall. Occasionally
dirty discharges yleld very lvgh peak densities but the shots are rarely repeatable.
Reproducibility of even a clean tokamak discharge was sometimes a problem. Typically, about 10% of the shots during a run simply did not form a
plasma due to failure of one of the ignitrons to fire properly. About 15%of the
remaining shots were usually unsatisfactory because their density or current
evolution were abnormal.

3.9 Charge-Exchange Diagnostic
3.9.1 Introduction

An estimate of the ion temperature in the Caltech Research Tokamak was
obtained using a charge-exchange diagnostic. Neutral atoms of hydrogen
within the plasma undergo charge-exchange collisions with energetic ions

( H + Hf -+ Hf + H) resulting in a population of neutral atoms with an energy
distribution similar to that of the plasma ions. The energetic neutrals are not
confined by the magnetic field and their spectrum can be measured outside
the tokamak.

Although the tokamak plasma is nearly fully ionized, there is always a
small background of neutral hydrogen, even a t the plasma center [Goldston,
19821. These neutrals arise from desorption of gas from the tokarnak wall and

from recombination. Neutrals diffusing into the plasma from the wall region
are generally cold ( T < 5 ev) and have short mean free paths for electron
impact ionization. Before being ionized, however, they can undergo chargeexchange collisions with the hotter plasma ions (since the rate coefficient for
charge-exchange is larger than that for impact ionization a t energies less that

10 keV). The charge-exchange reactions produce energetic neutrals which

can penetrate farther into the plasma and undergo further charge-exchange
reactions. Through t h s cascade of charge-exchange reactions, cold neutrals
from the wall can reach the hot plasma core.
The neutral particle energy spectrum, observed with a diagnostic outside
the plasma, is related to the ion energy distribution, but interpretation is complicated by a number of factors. The spectrometer sees a chord through the
plasma and hence receives neutral particles which were "born" throughout the
plasma, The plasma density and temperature are peaked a t the center, while
the neutral atom density is largest a t the periphery. Furthermore, neutrals
produced through charge-exchange collisions in the plasma may be attenuated
on their way out by impact ionization or further charge-exchange.
Most of the charge-exchange neutral flux received by the &agnostic arises
from the low temperature region near the plasma edge. Hence, to observe the
charge-exchange energy distribution characteristic of the core plasma, it is
necessary to look a t the hgh-energy part of the spectrum, ire.,energies greater
than that correspondmg to the plasma core temperature [Eubank, 19791.

If the neutral particle distribution was uniform and homogeneous in space
and Maxwellian in energy, the energy distribution of the flux entering the
charge-exchange spectrometer, F ( E ~, ) would be simply proportional to

--El
k4 , where

f i e

& is the ion temperature, k is Boltzmann's constant, and

El is the kinetic energy component along the chord which the spectrometer
views. A plot of

from
then yields a straight line of slope -2
k$

whch the temperature is found. In practice, t h s is what is generally plotted
from the charge-exchange data [Afrosimov et al., 1961; Eubank, 19791
although the resulting curve is, not surprisingly, generally not a straight line.
In fact, observations on tokamaks often yield a "two-temperature" distribution
resembling a broken line with two slopes, the higher slope (i.e.,lower temperature) part of the curve occurring at lower energies. This form of spectrum
apparently results from the inhomogeneous density and temperature profiles
present in the actual tokamak [Parsons and Medley, 19741. In particular, the
large number of low-energy atoms which undergo their final charge exchange
reaction (i.e., the last before emerging from the tokamak and being detected)
in the outer edge of the plasma (where the neutral density is highest) are
responsible for the large flux observed a t low energies. The high-energy part of
the spectrum is thought to be representative of the central ion energy hstribution.

3.9.2 Design and Construction
A neutral particle spectrometer was constructed in order to measure the

energy spectrum. of the neutral flux from the Caltech Research Tokamak. In
general, these devices function by converting the neutral atoms back into ions
(once they are outside the magnetic fields of the tokamak) and performing
energy analysis on the ions. The neutrals are ionized by passing them through
a "stripping cell" containing a relatively hlgh pressure of some gas such as
hydrogen, nitrogen or helium, or through an oven containing an alkalai metal
vapor such as cesium. Inelastic scattering results in ionization of some

fraction of the neutrals (through a reaction such as H + N2 -, Hf + N2 + e-)
and the ions are then dispersed in energy by any of a variety of schemes.
Energy drspersion has been accomplished using electrostatic deflection (parallel plate analyzers, cylindrical analyzers, etc.), using combined electric and
magnetic fields (when mass-sensitive energy dispersion is required), and using
time of flight measurements.
The ions must be detected after energy selection, and this is usually
accomplished with a channeltron [Barnett and Ray, 19721 or via secondary
electron emission from a metal target [Fleischmann and Tuckfield, 19681. The
channeltron [Goodrich and Wiley, 19621 is a high gain ( - 10') tubular continuous electron multiplier whch has a high efficiency for counting a wide variety
of particles (e.g., electrons, ions, and photons). The mouth and inside of the
tube are coated with a material which has a high secondary electron emission
coefficient; a potential of about 2.8 kV is applied between the ends of the
device, with the mouth of the channeltron biased negative with respect to the
other end. Incident particles strlking the funnel-shaped mouth of the tube
generate secondary electrons which are then accelerated through the tube,
making multiple collisions with the wall. A single incident particle results in an
output pulse containing about 10' electrons, whch are detected with a collector cup near the end of the tube.
The neutral atom spectrometer constructed for the Caltech Research
Tokamaks followed a design first described by Barnett and Ray [1972]. A
schematic of the spectrometer is shown in Figure 3-10. The spectrometer was
attached to port #7 of the tokamak (Figure 3-2). A turbomolecular pump (with

a speed of 450 liters/sec) and a series of baffles maintained the pressure in the
charge-exhange spectrometer chamber during operation a t about 5% of the
5. Part of the neutral particle spectrometer system was manufactured by 2. Lucky Enterprises, Los Angeles, Ca.

pressure in the tokamak. Neutral atoms from the tokamak were collimated
and entered the nitrogen stripping cell which was fed continuously from a
reservoir through a leak valve. The pressure in the stripping cell was monitored approximately by a Schulz-Phelps ionization gauge; the cell was usually
operated at a pressure of about 2 x l o 9 Torr.
Ions emerging from the stripping cell then passed into a 45" parallel plate
electrostatic energy analyzer. This type of analyzer, first described by Harrower [1955], utilizes a uniform electric field created by applylng a potential
between the two plates. The plate closer to the stripping cell was grounded to
the chamber wall, as was the stripping cell itself. Ions entering the input aperture travel in parabolic paths, and those with energies within a specific range
emerge through the exit aperture. The energy of those ions that exit the
analyzer is linearly proportional to the potential difference between the plates;

the relation is Uo= qi Vb -,

2d

where Uo is the kinetic energy of the ion entering

the analyzer, Vb is the bias potential, d is the separation between the plates,
s is the linear distance between the input and exit apertures, and qi is the ion

charge. For the analyzer constructed here, the relation is Uo = 1.5e J$, . The
resolution of the analyzer depends on the size of the apertures, on the spacing
between them, and on the range of angles the incident particles make with
respect to the entrance aperture. For the analyzer described here, the relative
energy resolution was calculated to be better than 12%. A channeltron (Galileo
model 4039) faced the exit aperture; it was mounted within a shielded box to
reduce background noise from W photons.
The energy spectrum was obtained by varying the analyzer bias from shot
to shot. (Some experiments using a fast ( = 1 msec) h g h voltage sweeper were
also performed, but the total particle flux was too small to provide adequate
counting statistics.) In order to relate the signal from the channeltron to the
actual energy spectrum of the incoming neutrals, the energy dependence of

each component of the system had to be examined. The cross section for ionization in the nitrogen stripping cell was strongly dependent on the neutral particle energy, decreasing rapidly below about 1 keV. Calibration curves for the
stripping efficiency between 10' and lo4 eV were presented by Barnett and Ray
(19721 and were used here in unfolding the spectrum. The energy dependence
of the parallel plate analyzer has already been discussed, but that of the channeltron must be considered. For a channeltron operating with its entrance
aperture a t ground potential, the counting efficiency drops off rapidly below
about 800 eV [Iglesias and McGarity, 19711. The technique of preacceleration
[Burrous et al., 19671 greatly reduced t h s energy dependence. The channeltron was simply biased so that the mouth of the multiplier was a t a potential
of -2.8 kV with respect to the grounded plate of the analyzer. Then any ions
emerging from the exit aperture received an additional 2.8 keV of energy
before str~kingthe channeltron. Burrous et al. [1967] have measured the relative efficiency as a function of incident proton energy for a preaccelerated
channeltron detector and found a variation of less than 10% between 200 eV
and 18 key.
The exit end of the channeltron was grounded; the collector was a small
copper cap suspended about 1 mm from the end of the channeltron tube.
Because of decreased gain at high input fluxes, the channeltron was used in a
pulse counting mode. A discriminator followed by a one-shot multivibrator and
a low-pass filter was used to insure that each detected ion contributed equally
to the output signal. The electronics limited the maximum counting rate to
about 1 MHz.
A photograph of the tokamak lab whch shows the installed charge-

exchange spectrometer is shown in figure 3-11.

FIG, 3-11, The tokamak lab, showing the charge-exchange spectrometer
(the cylindrical chamber a t lower right) attached to the tokamak. The RF
antenna port is on the right side of the tokamak.

3.9.3 Operation
No problems were observed with contamination of the tokamak from
nitrogen from the stripping cell. In fact, the nitrogen peak on the residual gas
analyzer did not change noticeably when the charge exchange system was
opened to the tokamak, indicating that the differential pumping of the
chamber was adequate.

A series of tokamak shots was recorded whle changing the analyzer bias
between 100 V and 600 V in increments of 50 V; about 4-6 shots were recorded
a t each voltage. Typical shots a t analyzer voltages of

150 and 400V

(corresponding to particle energies of 225 and 600 eV, respectively) are shown
in Figures 3-12 and 3-13. Consider the charge-exchange signal in Figure 3-12.
The peak a t around 4.5 msec corresponds to the density minimum following
the initial gas breakdown. As the gas puffing begins to increase the plasma
density, the charge-exchange signal decreases initially but reaches a minirnum
a t around 6.5 msec (while the plasma density is still rising) and increases
again. Figure 3-13 shows almost the same initial rise and fall of the chargeexchange signal, but the magnitude of the signal after the gas puff begins is
substantially lower; this implies cooling of the plasma due to the gas puff.
The full charge-exchange spectrum was plotted out for two points in the
discharge: a t the density minimum occurring a t around 4.5 msec, and at the
density peak occurring a t around 8.5 msec. The results are shown in Figure
3-14; the characteristic "two-temperature" spectrum is evident. The energy
spectra coincide for low energies but &verge for energies above about 350 eV.
Fitting a straight line (by least squares) to the higher-energy parts of the two
curves yields an estimated ion temperature of 80 eV for the h g h density case
and 130 eV for the low density case. Hence the gas puffing cools the plasma
substantially. Measurements by Levine [19BO] of the electron temperature
(from the plasma resistivity) also indicate a decrease in temperature as the

T I M E (MSEC)
FIG. 3-12. Typical tokamak shot; analyzer bias = 150 V. a) 1-turn
voltage. b) Plasma current. c) Charge exchange spectrometer output signal (from discriminator). d) Plasma density (from microwave interferometer).

T I M E (MSEC)
FIG. 3-13. Typical tokamak shot; analyzer bias = 400 V. a) 1-turn
voltage. b) Plasma current. c) Charge exchange spectrometer output s~gnal(from discriminator). d) Plasma density (from microwave interferometer).

ENERGY (EV)
FIG. 3-14. Plots of the charge exchange spectrum at low density (0)and high
density ( A ) points during the discharge. Vertical axis is In[ F ( E ) / m]+ C ,
where F ( E ) is proportional to the counting rate of the spectrometer (corrected
for the stripping cell efficiency) when the analyzer bias is set for energy E, and
C is an arbitrary const,ant. Least-squares fits to the high energy part of the
curves yield temperatures of 130 eV (low density) and 80 eV ( h g h density).

density increases during gas puffing under similar circumstances:

the

decrease noted was 22%, a somewhat smaller change than that observed for
the ions.

CHAPTER 4
Apparatus

A large quantity of RF equipment was constructed by the author during
the course of these experiments because the required devices were generally
not commercially available. A number of changes also had to be made in the
tokarnak and its electrical systems and &agnostics in order to permit operation of the RF experiments. This chapter describes the design, construction
and installation of the RF system, includmg the kgh-power amplifier, directional coupler, impedance-matching system, and amplitude and phase detectors.

4.1 RF System Design
4.1.1 RF Exciters
Several different RF amplfiers were used in these experiments. The signal
source, gating circuitry, and first stage of amplification were the same in all
cases, but the output stages varied depending on the application.
The RF signal generator was a Hewlett-Packard 8601A which provided an
output of up to 20 dbm (referenced to 50 0). The signal was gated by an RF
switch consisting of a double-balanced mixer (Minicircuits model ZAY-3) whose

IF port was driven by a square pulse from a pulse generator (Systron-Donner
model 101). The generator, which had easily adjustable pulse width and delay
controls, could be triggered by a low-frequency signal generator (0.001 - 10 Hz)
for testing purposes. During an experiment, it was triggered, via a fiber-optic

link, from a timing module in the tokamak power supply room.
The output from the RF switch passed through a series of three 50 Q RF
attenuators which provided 0 - 100 db attenuation in 0.1 db steps. The signal
was then amplified by a broadband (0.15 - 300 MHz), 10 W, 40 db gain amplifier

(EN1 model 411LA) and was passed through a 30 MHz low-pass filter (Miller
model C-511-T) to attenuate harmonics generated by the RF switch, At this
point, the signal could be amplified by one of three systems, or it could be used
directly.
For low-power wave propagation and impedance measurement experiments, the RF pulse was amplified by a 300 W broadband (0.35 - 30 MHz) power
amplifier (EN1 model A-300). This amplfier was very convenient to use as it
required no tuning and was unconditionally stable (i.e., stable for arbitrary
load impedance).
For operation a t higher power levels (up to 1 kW), an amateur-radio linear
amplifier was employed (Dentron Clipperton-L). The signal was amplified to an
appropriate driving level (-60 W) by a 100 W broadband (7 - 30 MHz) linear
amplifier ( ~ a r k omodel loo), passed through another harmonic attenuation
network, and was fed to the Dentron input. The Dentron amplifier utilized four
572B tubes in parallel in a grounded-grid design, with an untuned input and a
n-network output tank circuit. The output networks for the 40 and 20 meter
bands were modified to allow operation from 8 - 12 and 14 - 16 MHz. In addition, power supply regulation was improved substantially to reduce 60 Hz
modulation of the output.
For operation a t power levels up to 100 kW, a high power class C amplifier
was constructed (section 4.2). The 1 kW RF source described above was used to
drive the larger amplifier. Both the high power amplifier and the driver system
components were housed in a large cabinet in a room adjoining the tokamak
lab. Approximately 20 m of RG-17 50 Q coaxial cable connected the h g h power

amplifier to the impedance-matching network which was mounted on the end
of the antenna feeder tube; RG-8 was used to connect the lower-power systems.
Directional couplers were used to monitor forward and reflected power,
both at the output of the amplifier system being used and a t the input of the
impedance-matchng box. For power levels lower than 1 kW, Bird model 4266
dual broadband couplers (-30 db, 2 - 32 MHz) were used; for the hgh-power
amplifier, a special dual coupler was built into the RG-17 transmission line
itself (section 4.3).

4.1.2 System Grounding
Identification and elimination of ground loops throughout the RF generator, tokamak, and data acquisition systems was a tedious but essential task.
Large transient magnetic fields are present throughout the laboratory during a
tokamak shot, largely from the preionization and ohmic heating power supplies. Topological loops formed by multiple grounding connections or by the
i ~ t e r c ~ n n e c t i nshields
of coaxial cables can lead to pickup of spurious signals. The changing magnetic flux intercepted by such a loop generates an EMF
around the loop whch can be falsely interpreted as a genuine signal.
The

pickup

problem

was

serious

enough

to

require

complete

reconfiguration of the tokamak laboratory grounchng system. The tokamak
vacuum vessel was connected directly to the building earth ground in the
power supply room via a 2.5 cm diameter copper tube. T h s was done in order
to reduce the danger from a potential catastrophc failure of the toroidal field
winding insulation which could result in a n arc to the vacuum chamber. This
connection formed the only path from the chamber to the building ground.
The power supplies of equipment which had to be connected to the vacuum
chamber (i.e., the tokamak turbopump, the data acquisition system console,

and the RF source cabinet) were isolated from the building ground using
electrostatically-shielded isolation transformers. The chassis ground of the
data aquisition console was connected to the tokamak system ground by a
3.8 cm diameter copper tube wbch attached to the RF impedance-matching

box. The RF source cabinet ground was connected to the tokamak system
ground only through the sheath of the RF output transmission line which also
attached to the impedance-matching box. A schematic of the grounhng system is shown in Figure 4-1.
The only signal cables whose sheaths were required to make contact with
the tokamak vessel (and thus form ground loops) were those running from the
data acquisition system console to the impedance-matching box (i.e., cables
for forward and reflected voltage, and RF antenna current and voltage signals).
The potential ground loops that these cables could form was avoided by routing them through the copper tube that provided the chassis ground connection, and by bundling them tightly together.
All other signal cables leading to tokamak diagnostics were isolated from
making electrical contact with the tokamak vessel. The plasma current, loop
voltage, and visible light monitors &d not require physical connection to the
tokamak vessel; mylar washers at waveguide junctions isolated the microwave
interferometer. The vacuum flange that housed the UV light monitor was isolated from the tokamak chamber by a teflon spacer; the magnetic probes were
also isolated from the chamber.

4.1.3 RF Shielding

Interference and pickup from the RF generator also caused significant
problems, particularly a t high power levels. Initial operation with the high
power amplifier generally saturated most of the tokamak diagnostics. Two
approaches were taken to eliminate these problems: reduction of the RF

TOKAMAK LAB

POWER SUPPLY
ROOM
ADJOINING
LAB

-1

AC

ISO

RF EXCITER
AND
HIGH POWER
AMPLl FlER
CABINET

FIG. 4-1. Grounding configuration of tokamak and RF system. IS0 = electrostatically-shelded isolation transformer
with RF bypass filter.

leakage, and shielding of the diagnostics and data acquisition system.
RF leakage can occur at joints in the amplifier cabinet or impedancematching box, at the terminating connectors of the transmission line connecting the two, from the transmission line itself, or from the antenna-tokamak
assembly. Radiation from imperfect electrical contact between the frames of
the cabinets and their covers was minimized by using wide mating surfaces
(generally 5 cm) and fastening them with closely spaced screws. Critical joints
in the amplifier cabinet were sealed with a conductive RF caulk.'
The vacuum O-ring seal assembly through which the antenna tube passed
into the tokamak was found to be a path through which RF currents were
flowing, giving rise to substantial radiation. Nylon bushings had been installed
in the O-ring holders to prevent scoring of the antenna tube; these actually
insulated the tube from the tokamak chamber and resulted in the antenna
"floating" with respect to the tokamak wall. This problem was solved by installing a spring-finger grounding assembly, at the end of the tube, which provided
a very good RF seal to the t o k m a k port.
RF pickup through the data acquisition system, the tokamak diagnostics,
and the RF electronics cabinet (containing amplifiers, amplitude and phase
detectors, and physically connected to the data acquisition cabinet) was investigated by exciting, a t low power, a dipole antenna which was strung across the
lab. This flooded the lab with RF radiation and its effect on the various electronic systems could be easily observed. EM1 filters (Corcom model 10R1) were
added to the power supplies of all of the electronic systems to eliminate RF
pickup from the power lines.
The plasma current integrator box was rebuilt to afford better shielding
and was moved from its location overhanging the tokamak (whch required a
1, Eccoshield VY-C, manufactured by Emerson & Curning, Canton, MA.

long power supply cable) to the data aquisition cabinet. The plasma current
Rogowski coil was connected by a 50 R coaxial cable to the integrator box
where it was terminated in 50 R. A 100 kHz, 50 R low-pass filter preceded the
termination and eliminated RF pickup from the Rogowski coil. The loop voltage
monitor was similarly terminated at the data acquisition cabinet with a 50 R
low-pass filter.
The visible light detector was found to be easily saturated by low levels of
RF radiation, and it was completely rebuilt. The phototransistor and a
preamplifier were housed in a small aluminum box which was mounted (via an
insulating, light-tight tube) to an optical window on the antenna feedthru
flange. The hole in the box that allowed light to enter was shielded with a 200mesh stainless steel screen, affuced with a conductive silver epoxy (Emerson &
Cuming Eccobond 66C). The signal and power lines from the detector box were
brought back to an amplifier in the data acquistion cabinet via a shielded
multi-conductor cable. The cable shield (aluminum-coated polyester) was
sealed to both the detector box and the amplifier box using the silver epoxy.
The microwave interferometer system also required some modification to
eliminate severe interference from the RF generator. The Impatt oscillator was
enclosed in a steel box to shield it electrically and magnetically (the oscillator
was sensitive to DC magnetic fields as well as RF interference). Power lines
leading to the oscillator were heavily filtered and the oscillator was electrically
isolated from the waveguide forming the interferometer with a mylar disc. The
output of the detector diode was connected to a 50 R coaxial cable which led
back to the data acquisition system cabinet where it was terminated with
another 100 kHz, 50 R low-pass filter. The &ode detector mount was also
electrically isolated from the rest of the microwave system.
During the course of testing the microwave system, it was discovered that
the detector output was quite sensitive to physical vibration of the detector

mount. Vibrations of the waveguide, caused by the jump of the tokamak
chamber that occurs during a shot, were causing a large signal in the 1 10 kHz range on the detector output. This vibration-induced noise, which had
plagued earlier operation of the microwave system [Levine, 19001, made
interpretation of the interference fringes dficult. The noise was greatly
reduced by clamping lead weights to various parts of the waveguide in order to
damp out the unwanted vibrations.

4.2 High Power Amplifier
4.2.1 Introduction
A high-power RF amplifier was built in order to have a wide range of power
available for investigating coupling and m-tenna-plasma interactions, A class C
amplifier was designed around an Eimac 4CW100,OOOE water-cooled ceramic
envelope power tetrode and provided output power of up to 100 kW for pulses
of 5 - 10 msec duration. The amplifier could be tuned over a frequency range
of 5 - 20 MHz using different tank coils.
In a class C design, the tube is biased to conduct for only a small fraction
of the total RF cycle. The tube applies current pulses to the output tank circuit, wlvch therefore must be of sufficiently high Q to keep the tank current
flowing when the tube is cut-off. This non-linear behavior of the tube means
that the amplifier efficiency can be high, exceeding the 50% limit of a linear
amplifier [Martin, 19551.

A tetrode tube was chosen for several reasons. The control grid - anode
feedback capacitance is very small for tetrodes (-0.3 pF in this case) due to
shielding by the screen grid, so neutralization is not generally required.
Tetrode amplifiers also can be driven with considerably less power than a
triode, minimizing the requirements of the driver section [Sutherland, 19671.

4.2.2 RF Design
A simplified schematic of the amplifier is shown in Fgure 4-2; it will be

referred to in the following discussion of the amplifier design and construction.

A grounded cathode design was employed for simplicity, since the filament
transformer was then a t RF ground potential and did not require large, highcurrent RF chokes. Because the Eimac tetrode was physically large (N 20 cm
0.d. x 29 cm long), its input (control grid - cathode) and output (anode cathode) capacitances were correspondingly high (430 pF and 100 pF, respectively). The input and output RF tuned circuits had to be designed to incorporate these added reactances.
The grid tuned circuit was a parallel resonant L-C

network in which the

variable capacitor was, a t RF frequencies, effectively in parallel with the tube
input capacitance and thus simply added to it. A 1500 R non-inductive resistor2 fixed the real part of the grid input impedance in order to swamp out variations in impedance a t high power due to secondary grid emission. An
impedance step-up was necessary to match the impedance of the driver
amplifier (50 R) to that of the grid circuit (1500 R). This was accomplished
using two biflar-wound broadband RF transformers in series [Ruthroff, 1959;
Sevick, 19761.
Grid bias was applied through the same 1500 R resistor through an RF
bypass filter; a 0.01 pF capacitor blocked the DC voltage from the input tuned
circuit. RF grid drive was obtained from the 1 kW Dentron amplifier described
in section 4.1.1.
The output tank circuit consisted of

a C-L-C

n-network which

transformed the resonant plate impedance of the tube ( 4-600 R for typical
operating con&tions3 ) to the desired output impedance ( RL = 50 R ) . Note that
2. Most of the non-inductive resistors used in this amplifier were provided by the Carborundurn Company, Niagara Falls, N.Y.
3. Calculated from Eimac constant-current tube curves.

Output RF

I RF Plate Voltage/2000

Capacitive
Voltage
Divider

Transformer1

Eimac

Amplifier

ELECTRONICS

Signal
Generator

FIG.4-2. Schematic of high-power RF amplifier.

the output capacitance of the tube was effectively in parallel with the first
capacitor of the tank, C1, since the impedance of the 0.004 ,uF DC-blocking
capacitor was negligible a t the frequencies being considered (5 - 20 MHz).
The loaded Q of the output tank circuit, QL, is of some importance. If QL
is too small, the harmonic content content of the output will be high; if

QL

is

too large, power losses from the circulating current in the tank will be excessive, reducing the amplifier efficiency. A loaded Q between 10 and 20 is generally chosen in order to balance the two effects [DeMaw, 19791; a design value
of 12 was chosen for this amplifier. Once the resonant plate impedance Rp,
the load impedance RL, the operating frequency w/2n, and the loaded Q are
selected, the component values for the tank circuit ( CT1, CT2, LT) may be
found using the impedance transformation relations of a n-network [DeMaw,
19761:

(4.lb)

and

Plate voltage was applied through a hlgh voltage RF choke (see below), and
a 25 R series resistance limited plate current in the event of an internal tube
arc.

4.2.3 Construction
The amplifier was built withn an aluminum frame (2.0 m high x 1.2 m
wide x 0.7 m deep) which was covered with aluminum sheet panels (Figure 4-3).

FIG. 4-3. Front view of RF amplifier. Exciter compartment is at upper right,
water cooling connections are a t lower left. Rule at bottom is 30 cm long.

FIG. 4-4. a) Back view of RF amplifier (cover panels removed).

FIG. 4-4. b) Schematic of FIG. 4-4a. A - RF exciter and gating electronics compartment. B - Grid
bias power supply and control logic compartment. C - Plate and screen bias supplies and crowbar
compartment. D - Capacitor bank compartment. E Plate tank circuit compartment. F - Grid
tuned circuit compartment. 1 - Power supply for Marko 100 W amplifier. 2 - EN1 10 W amplifier. 3 Screen supply bleeder resistors. 4 - Ignitron crowbar. 5 - 20 kV plate power supply. 6 - Plate supply capacitors. 7 - Water system coils. 6 - Plate supply RF choke. 9 - Ehmac 4CW100000E tetrode.
10 - DC-blocking capacitor. 11 - Tank circuit vacuum capacitor (Crl). 12 - Tank circuit inductor
( L T ) . 13 - Water cooling lines. 14 - Capacitor adjustment mechanism. 15 - Filament transformer.
16 - Tube base cooling blower. 17 - Water system connections. 18 - RF output connector.

The chassis was divided into six shielded and isolated sections: the output
tank circuit compartment, the input grid tuned circuit compartment, the plate
and screen power supplies and crowbar compartment, the plate supply capacitor bank compartment, a compartment for the grid bias power supply and control electronics, and one for the exciter and RF gating electronics (Figures
4-4a,b). Good shieI&ng between the compartments was essential as the overall
gain of the whole system (including the exciter) was greater than 60 db.
The tube was mounted in an Eimac SK-2000 air-cooled socket on an aluminum partition whch separated the input and output RF compartments. A
blower mounted below the socket provided a forced-air flow of 5 ms/min. The
air intake, in the grid compartment, and the exhaust, in the plate compartment, used RF shielded air filters4 to prevent RF leakage through the large
apertures. The tube socket contained an integral bypass capacitor for the
screen grid electrode, so external RF bypassing of the screen power supply was
not necessary.
The layout of the RF components was very important because of their
large physical size. Stray inductances and capacitances associated with the
various elements could be substantial and had to be taken into account in the
design.
The control-grid tuned circuit was mounted directly below the tube socket
and consisted of a 0 - 1000 pF, 3 kV vacuum variable capacitor and an inductor
wound from copper strap. A capacitive voltage-divider allowed monitoring of
the grid RF voltage.
The plate tank circuit was constructed on the same partition that held the
tube socket. The n-network consisted of two 0 - 1000 pF, 15 kV vacuum capacitors5 and a plug-in inductor wound from 0.95 cm 0.d. silver-plated copper
4. Teckcell filters, manufactured by Tecknit, Santa Barbara, CA.

5. Manufactured by the Jennings Gorp., Palo Alto, CA.

tubing. Since the inductance L T pre&cted from equation 4 . 1 ~was small

( < 1 pH) for the operating parameters of interest, it was necessary to mount
the tube and the two capacitors close together in order to minimize the stray
inductances associated with the capacitor bodies. Teflon sheet insulation was
used when necessary to prevent RF breakdown.
The plate supply choke was wound on a phenolic tube 5 cm 0.d. x 50 cm
long. I t consisted of 400 turns of #I8 AWG magnet wire, wound in six sections
of decreasing width (thls served to increase the resonant frequencies of the
choke which arose from capacitive coupling between adjacent turns). The
measured inductance was 0.65 mH, the resistance, 1.2 R. A high voltage cable
led from the bottom of the RF choke, through a shlelded tube, to the plate supply capacitor bank compartment.

An RF plate voltage monitor was constructed using a shlelded vacuum
voltage avider capacitor (Jennings model VDF-2.8-60s) whose input capacitance was -2.8 pF; the division ratio was chosen to be 2000. The output RF
current was monitored using a high frequency current transformer (Ion
Physics Go. model CM-1-S).
The filament power supply consisted of a step-down transformer which
provided 14 V (rms) a t 200 A. The transformere was designed with a n air gap
in the core so that the leakage inductance was high. This limited the shortcircuit current to insure that the filament specifications were not exceeded
during the filament warm-up period (the resistance of the cold filament was
much lower than the resistance a t operating temperature). The transformer
was mounted directly below the tube socket, in the grid tuned-circuit compartment, so that the connecting straps could be kept very short. Connections on
the secondary side were silver plated t o minimize contact resistance. The pri6.Built by Teledyne - Crittenden Transformer Works, Los Angeles, CA.

mary power (115 V, 30 A) was passed through an RF filter a t the point where it
entered the grid compartment (as were all other power lines entering the
chamber) in order to prevent RF energy from being conducted onto the AC
power lines.
The grid bias supply used a sealed transformer/rectLfier unit (Plastic
Capacitor Co. model HV50-502M) to provide up to 1 kV at 5 rnA. The DC grid
current drawn during an RF pulse was usually negligible.
The screen bias supply used a transformer and diode bridge to provide up
to 2.5 kV a t 400 mA. Secondary electron emission from the screen grid can
lead to reverse screen current under some operating conditions. It was therefore important to provide a low-impedance path for this current in order to
keep the screen voltage constant during the RF pulse [Sutherland, 19671. This
was accomplished by bleeding the supply with a resistor between the output
and ground; a bleeder current of 80 mA was found to be sufficient. The screen
grid supply voltage was maintained during the RF pulse by the 80 pF capacitor
across its output.
The plate bias power supply was composed of a 15 kV (rms), 6Ci mA
current-limited transformer (originally used to light a neon sign!), a rectifier
stack, and a capacitor bank. Full wave bridge rectification was employed, using
four pairs of 10 kV, 300 mA &odes (International Rectifier #30AV100). The case
of the transformer had to be insulated since it was connected to the center-tap
of the secondary. The capacitor bank consisted of three low-inductance energy
storage capacitors in parallel, totaling 42 pF and rated a t 20 kV. This was
sufficient to keep the plate voltage sag during an RF pulse to less than 10%.
Voltage adjustment and regulation were accomplished in the same fashion
for the grid, screen, and plate ~ u p p l i e s .A~ triac was used to modulate the
7. The voltage regulator circuitry was designed and built by F. Cosso, E.E. (Caltech).

power delivered to the transformer primary; the triac was in turn triggered
from a feedback loop which sensed the high voltage DC output through a resistive voltage divider. T h s enabled full charging current to be applied to the
capacitors immediately after an RF pulse, thus increasing the maximum
repetition rate (which was helpful during tuning of the amplifier). Regulation
of the bias supply voltages was typically held to 1%.
The Eimac tube could be damaged by excessive anode current, by internal
arcing to one of the grids, or by heat from the filament without suffLcient cooling. As the grids could be damaged by an arc carrying an energy of only
-50 joules,' it was necessary to provide adequate protection circuitry. The

screen and plate currents were monitored, and, if they rose above preset limits
(15 A for the plate, 1.2 A for the screen), crowbars were triggered which

discharged the capacitors withn - 2 psec. The plate supply crowbar consisted
of an ignitron (General Electric model GL7703) in series with a 0.2 Q noninductive resistor connected across the capacitor bank. The resistor was
chosen to critically damp the resulting L-C-R

circuit (where L arose from

the inductance of connecting leads). The screen supply crowbar consisted simply of an SCR and a damping resistor.
Three other interlocks were provided which discharged the capacitors and
shut off the filament power: an air flow switch (for the blower cooling the tube
base), a water flow switch (for the anode cooling system described below), and
a water temperature switch. Control circuitry was arranged so that the cooling
systems continued to operate for several minutes after the amplifier was
turned off.
The Eimac tube required both forced air and anode water cooling, even a t
the low duty cycle used here, due to the large continuous filament power being
8. Conversation w i t h R. Artego, Eimac Division of Varian, San Carlos,CA.

dissipated (-3 kW) and the fact that 20 - 30 % of that power ended up in the
anode. The water jacket which covered the tube anode was a t a DC potential of
15 - 20 kV during operation of the amplifier. It was therefore necessary to

build a deionized water system which could maintain the water specific resistivity a t a sufficiently high value ( > 0 . 5MR-em), both to prevent current from
flowing through the water channels and to prevent electrolysis which could
damage the tube anode.'

A schematic diagram and a photograph of the cooling system are shown in
Flgures 4-5a,b. A 360 liter polypropylene tank was used for the water reservoir,
All plumbing and components were made of plastic or stainless steel to elirninate corrosion. One water pump provided 20 liter/min for anode cooling and
another provided 2 liter/min for the purifying loop, which included a 1 micron
particle filter, a mixed-bed deionizing cartridge, and an oxygen-removal cartridge.''
The deionized water system was cooled by a water-to-water heat exchanger
consisting of 25 m of 2.5 cm i.d. copper tubing, welded into a grid, nickelplated, and suspended withn the polypropylene tank. The heat exchanger was
connected to a recirculating water cooling system whch serviced the building
and required a flow of 8 liter/min during operation of the amplfier.
In order to provide a water path of sufficient resistivity and yet allow adequate water flow, it was necessary to coil 6 m of 1.9 cm i.d. plastic tubing inside
the amplifier tank circuit compartment. The water was grounded to the
cabinet through conductive feedthrus, and under operating conditions the
current flow through the water channel was less than 0.5 mA.

9. Eimac Application Bulletin 16, "Water Purity Requirements in Liquid Cooling Systems".
10. The filters were manufactured by the Barnstead Co., Boston, MA.

Nickel-Plated

@ TEMPERATURE GAUGE

9PRESSURE GAUGE
9CONDUCTIVITY METER
Amplifier Cabinet

FIG. 4-5. a) Schematic diagram of high-purity water cooling system b) Photograph of water cooling system. Rule a t bottom is 30 cm long.

4.2.4 Operation
The amplifier had to be tuned before the Eimac tube could be energized.
Grid tuning was adjusted using the directional coupler in the driver line. The
driver system was operated at low power and the grid tuning capacitor was
adjusted to minimize the reflected power. During operation, the resonant grid
impedance was considerably higher than the 1500 R which paralleled it, so little readjustment was usually necessary.
Plate tuning was accomplished by connecting a 50 Q signal generator and
directional coupler to the 50 R a t p u t cable of the amplifier. Now, however, the
resonant plate impedance of the tube had to be simulated. The impedance was
calculated at the desired operating point using the tube constant-current
curves, and a non-inductive resistor of approximately the same value was connected between the anode and cathode of the tube. The two n-network capacitors were adjusted (without energizing the tube) to minimize the reflected signal from the directional coupler. The resistor and the signal generator were
then removed and the filament power and high voltages were applied.
The 1 kW Dentron amplifier had some residual 60 H z output ripple, so it
was found useful to synchronize the amplifier pulses to the line voltage. When

a trigger pulse was received, an output pulse was generated a t the next zerocrossing of a line voltage reference, The tokamak timing cycle was also
modified so that it fired at a zero-crossing of the line reference.
Initial testing of the amplifier was performed by connecting a noninductive, air-cooled 50 R resistor to the output cable. The amplifier was
pulsed a t a repetition rate of

- Hz
0.5

with a pulse width of 2 - 5 msec. RF plate

and grid voltages were monitored during operation as well as output RF
current and DC plate current. The DC plate voltage and current gave the tube
input power, while the output RF current together with the load impedance
ylelded the output power.

Some low frequency ( 50 kHz) parasitic oscillations were observed during
early operation of the amplifier. They were traced to a resonant circuit composed of the grid bypass capacitor together with a n RF choke in the grid bias
power supply. The problem was solved by replacing the choke with an R-C
network. It was also necessary to shape the input RF gating pulse to slow its
rise time to -0.2 msec. No high frequency parasitics were observed.
The tube was typically operated with a plate potential of +15 kV, a screen
potential of + 1.5 kV, and a grid potential of -540 V. Under these conditions, a n
output power of 160 kW was achieved (at 14 MHz). The DC input power was
230 kW, yielding an efficiency of 70%. Lower powers (down to - 4 kW) were also
easy to deliver.

4.3 Directional Coupler
The solutions to Maxwell's equations applied to a coaxial transmission line
are traveling waves proceeding in both &rections. A directional coupler is a
device which generates an output signal proportional t o t h e amplitude of the
wave traveling in a single direction along the line. This was a necessary instrument for tuning the impedance matching network (section 4.4) which
transformed the amplifier output impedance to the impedance presented by
the RF antenna. A perfect match was indicated by a vanishing reflected wave
amplitude.
Much of the developmental work on directional couplers (sometimes
referred to as reflectometers) was done in the 1940's; a n excellent bibliography of the work was given by Schwartz [1954]. Many of the coupler schemes
utilized a t microwave frequencies (e.g., multiple hole or slot couplers) are not
practical to implement a t lower RF frequencies ( < 100 MHz). In this range,
several authors have presented designs which involve a loop coupled both

electrically and magnetically to a coaxial line section [Pistolkors and Neumann, 1941; Parzen, 1949; Boff,1951]. The basic idea is that the voltage
across the loop is the result of contributions from capacitive coupling to the
center element of the coaxial line and from magnetic coupling to the current
carried by the center element. The contributions are in phase or out of phase
with each other, depending on the direction of the wave on the line. If the coupling coefficients are adjusted appropriately, the two contributions will cancel
when out of phase, leakng to zero output for a wave traveling in one krection,

If the dimensions of the coupler are very small compared to a wavelength,
a lumped-element equivalent circuit model may be justfied. A schematic draw-

ing and a simple circuit for a loop-type coupler are shown in Figures 4-6a,b.
Here L represents the mutual inductance between the center conductor and
the loop, C1 and C2 represent the capacitances between the loop and the
center and outer conductor, respectively, RT is the resistive termination on
the sensing side of the loop (usually a terminated 50 R coaxial cable), and Z is
the impedance terminating the other side of the loop. Vl is the potential
between the inner and outer conductors and Vz is the voltage induced across
the loop due to the current in the center conductor. Solutions for the current

( I )and voltage ( V ) waves on a coaxial line [Chpman, 19681 are given by

where the subscripts "f" and "r" refer to waves propagating from left to right,
and from right to left, respectively (as in Figure 4-6a). Ro is the characteristic
impedance of the transmission line and k is the wave number.
Taking the loop position to be a t position x = 0 , it follows that V1 = V' + Ir,
and

iwL
V = -(
V -)
Ro

. Using Grchhoff's equations, the solution for the

wI

- -J
L - - - ,

-------------------I

TRANSMISSION LINE
CHARACTERISTIC IMPEDANCE R,

I,? "r
REFLECTED

FORWARD
If9

"f

FIG. 4-6. a) Schematic of loop-type dwectional coupler, b) Equivalent circuit
model of loop-type coupler. The "+" symbols i n b c a t e relative phasing of the
two voltage sources.

output voltage is

where Z ' =

1 +iwCzZ

is the impedance of the parallel combination of Z and

C2. When there is only a forward traveling wave on the line, the numerator of

(4.3)vanishes provided

If the coupling parameters can be arranged so that relation 4.4 is satisfied,
then a forward-traveling wave will produce no signal a t Vo,while a wave traveling in the reverse direction wilt yreld a non-zero output. The loop then functions as a &rectional coupler.

If the area enclosed by the loop is sufficiently small that wL << Ro is a valid

cz .

C1

c1 Ro

approximation, and if the ratio - 1s not too large, so that --<<

uL

1 is still

valid, then C2 can be neglected, 2'-, Z , and relation 4.4 becomes simply

z =-, L
Cl Ro

f 4.5)

a result first derived by Parzen [1947]. The coupling factor in the reverse
direction is then found from equation 4.3,

where we have set RT=Ro . The pre&cted coupling therefore increases linearly
with frequency.
Some loop-type directional couplers are available commercially, but those
capable of handling high power levels (100 kW) are expensive and designed for

large-diameter rigid coaxial air-line. Since the duty cycle of the RF amplifier
was very low (always < I%),a relatively small (2.2 cm 0.d.) flexible 50 R coaxial
cable (RG-17A/U) was used to connect it to the impedance matching network
in the tokamak lab. An extremely simple dual directional coupler was built
into the cable itself.

A diagram of the dual coupler is shown in Figure 4-7a. The coaxial line
section for the coupler was just a 10 cm length of the RG-17 cable. The loop
was formed by simply inserting an insulated wire between the outer conductor
(copper braid) and the dielectric insulation of the cable. It was found, however, that the resistive termination Z specified by equation 4.5 was not adequate; much higher directivity was obtained by adding a small trimmer capacitor in parallel with the resistance." The necessity for t h s complex impedance
termination probably arose from stray reactances due to the physical size of
the elements (not included in the model) or from the approximations used in
the derivation of equation 4.5.
One end of the loop was terminated with a miniature non-inductive variable resistor (100 n) in parallel with a trimmer capacitor ( 5 -80 pF) and soldered to the copper braid; the other end was soldered to a small 50 R cable
whch led to a BNC connector. Two couplers were built into the same section of
coaxial line and were oriented in opposite senses in order to measure both the
forward and reflected waves. The entire assembly was enclosed in a copper box
to eliminate external interference.
The couplers had to be adjusted initially to optimize their rhrectivity. The
procedure for tuning the reflected wave coupler will be described; the other
coupler was adjusted in a similar fashion. The ends of the RG-17 cable were
attached to low-VSWR connectors. The left-hand end of the cable (referring to
11. Similar couplers built using the much smaller RG58 for the coaxial line did not require

the additional parallel capacitance.

REFLECTED
OUTPUT

.. ... .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. ..

FORWARD
OUTPUT

----A
10 cm LENGTH OF #20AWG
INSULATED WIRE

FREQUENCY (MHz)
FIG. 4-7. a) Diagram of dual directional coupler built into RG-17
cable (not to scale). b) Forward and reflected coupler calibrations.

Figure 4-7a) was connected to a 50 Q signal generator (capable of several watts
of output power) which was swept from 1 - 30 MHz,while the right-hand end
was connected to a 50 0 termination. There were then only forward traveling
waves propagating in the coax. The sensing side of the reflected wave coupler
was connected to an oscilloscope via a 50 0 terminated line, and the variable
resistor and capacitor were adjusted simultaneously to minimize the output
signal over the entire frequency range.
The coupling coefficient is plotted as a function of frequency for the two
couplers in Figure 4-7b and the form agrees well with the linear frequency
prediction of equation 4.6. The directivity of the couplers was typically greater
than 30 db, while the power handling capability ( > 2 MW) was limited only by
the breakdown voltage of the coaxial line. Impedance measurements on the
coaxial line in&cated that the perturbation of the characteristic impedance of
the coupler section by the wire beneath the braid was negligible.
A photograph of the completed coupler (without the copper shield) is

shown in Figure 4-8.

4.4 Impedance Patching Network
4.4.1 Introduction

In order to efficiently transfer R F energy to the plasma, it is necessary to
match the impedance of the generator (50 R) to that presented by the antenna
in the plasma. This is in general accomplished with some sort of passive reactive network placed between the generator and the antenna.
If the antenna impedance were constant, it would be a relatively easy
matter to design a network to transform the impedances appropriately. However, since the plasma waves launched by the antenna are not highly damped,
resonant toroidal eigenmodes can develop which cause peaks to appear in the

real part of the antenna input impedance (for shelded loop antennas) a t particular values of the plasma density. These peaks can be 5 - 10 times the offresonance loading resistance (see Chapter 6 ) . Unshelded antennas do not
exhibit eigenmode peaks in the input resistance (probably because it is
swamped out by the increased background loading) but do exhibit a loading
resistance which increases (for magnetic antennas) or decreases (for electric
antennas) as the plasma density increases, typically changing by a factor of

2 - 5 during the tokamak shot. The reactive part of the antenna impedance
generally decreases slightly as the plasma density increases (for all of the
antennas) but rarely by more than

10%.

Ideally, the impedance-matchng network would be active, its elements
changing value as the load impedance changed in order to maintain a perfect
match. in practice, this has not been feasible because sf the time scale over
which the antenna impedance changes (half-widths of eigenrnode peaks are
typically 50 - 500 gsec in these experiments) and the mechanical dficulties
associated with automatically adjusting large inductors and vacuum capacitors.
Most of the work directed towards plasma heating with ICRF waves has
been done using shielded loop antennas which, in one-component plasmas, give
rise to toroidal eigenmodes and corresponding peaks in the antenna input
resistance. It is desirable, from the point of view of heating, to stay locked to a
single resonance peak, as the increased loading resistance leads to better coupling efficiency. Since the occurrence of a particular eigenmode depends on
the plasma density and the RF frequency, several approaches have been tried
to keep the system locked to a resonance. The PLT group has used precise
control of plasma density as well as active frequency feedback to stay tuned to
a mode [Hwang et al., 1978b1. The matchng network, whch utilized stub
tuners, was fixed. Several other groups [Bhatnagar et al., 1978b; Biddle, 19801

have attempted, with mixed results, to solve both the impedance matchng and
mode trachng problems by incorporating the antenna impedance into the
tank circuit of a high power RF oscillator.
Since the major thrust of the work presented here was to study wave
propagation and coupling, and not heating, no attempt was made to track a
particular eigenmode. The RF frequency was fixed during the plasma shot, as
was the matching network.

4.4.2 Design
Some of the simple reactive impedance-matching networks which were
considered were r-networks, T-networks and quarter-wave transformers. An
L-C-L

T-network was chosen (Figure 4-9), as it let to component values which

were easily implemented ( t h s was an important factor, since the available high
voltage vacuurn capacitors were limited to values & 1000 pF) and exhibited reasonable insensitivity to changes in the load impedance (see below).
A typical value for the loop antenna impedance (with plasma), used for

preliminary design of the matching network, was Z L O M 0.5 + 5.0i R f a t 12 MHz).
The impedance seen by the matching network, however, was the antenna
impedance ZLo as transformed by the antenna feeder tube. This frequencydependent transformation is &scussed in detail in Chapter 6; here we simply
call the resulting impedance ZP = R' + C X . Once the source impedance

f li$=SOR) and the load impedance ( Z P ) for the matching network are
specified, there are still an infinite number of ways to select L1, L2, and C to
achieve a perfect match a t a particular frequency. The families of component
values which provide an impedance match are usually specified in terms of a
parameter QL (which is called the loaded Q of the network). Solutions for the
components [DeMaw, 19701 are then given by

ANTENNA

ANTENNA
FEEDER

IMPEDANCE-MATCHING
- - - - - - - -BOX

F'IG. 4-9. Schematic of L-C-L

impedance-matching network.

10.

20.

LOADED Q
F'IG. 4-10. a) Solutions a t three frequencies for impedance-matching network
inductance L 1 as a function of QL. Matching-network load impedance is
ZAO= 0.5 + 0.42f i ohms, where f is the frequency in MHz. Network transforms
the load impedance to 50 ohms.

10.

LOADED Q

FIG. 4-10, continued. b) Solution for impedance-matching network inductance
Lz as a function of QL. c) Solution for impedance-matching network capacitance
C as a function of QL.

L2 = R ' B -X'

[ j, '

where A =Rg(l+Q~') and B = -- I] . Solutions for L , , i 2
and C as functions of QL , are shown in Figure 4-10 for the three operating frequencies typically used (0, 12 and 16 MHz). In order to keep component values withn practical limits, a QL of 5 - 10 was generally selected.
Once having chosen an assumed antenna impedance ( ZLO= RLO+ iXLO) and
Q L , the values of

the components of the matching network are determined.

With these values fixed, the fraction P of the power available from the generator which is deposited in the load decreases as the antenna resistance RL or
reactance XL varies. Plots of P as functions of RL and XL are shown in Figures 4-1l a , b for several different values of the design antenna impedance ZLo.
Note that for each curve, the factor P is unity only when ZL = ZLO,since that is
the impedance for whch the network is tuned.
For a typical ZLO ( 0 . 5f 5 . 0 i R), Figure 4 - l l a shows that greater than 70%
of the available power is delivered to the load provided 0.14 R < RL< 1.70R (for

X L = X L O ) Hence the matching network is not extremely sensitive to the
changes in RL which we expect to see during the actual plasma shot. If RL is
held fixed a t RLO,and XL is varied, Figure 4-1 l b shows that greater than 70%
of the available power is delivered to the load if 4.3R much smaller (14%) relative allowable change than was found for the case
where RL was varied, but fortunately during a plasma shot XL changes by less
than 10%. Hence a simple T-network, appropriately tuned, can provide sub-

FIG. 4-11. Calculated sensitivity of impedance-matching network to changes in
load impedance. a) Fraction P of power available from generator which is deposited in load for fixed matchng network parameters, as a function of RL, The
frequency is 12 MHz and the load resistance is 0.5 0. Four drfferent curves are
shown, correspondrng to setting the impedance-matchng network for a perfect
match a t ZL = & + 5 . i R, where & = 0.2, 0.5, 1.0 and 2.0 R. b) Fraction P as a
function of XL, for fixed load resistances &. For each curve, the matching network has been set to yeld a perfect match a t ZL = & + 5 . i R.

stantial amounts of power to the antenna-plasma load even with the
impedance changes that occur during the shot.
It should be pointed out that the above discussion assumed that the generator impedance, Rg,remained constant as the load impedance changed.
T h s was true for some of the RF exciters (e.g. the EN1 A-300 amplifier), but the
effective output impedance of the h g h power tetrode amplfier did depend on
the power level and the load mismatch. Although the impedance matching network design may not have been optimal for the high-power amplifier, large
amounts of power were still successfully coupled to the plasma.

4.4.3 Construction

The T-network matching system (Figures 4-12 and 4-13) was built within
an alurninum box which measured 55 x 65 x 42 cm, and was, during operation,
mounted adjacent to the RF antenna on a table next to the tokamak. The RF
input cable and the antenna feeder tube attached to the box via specially constructed RF-tight fittings.
The input cable center conductor was connected to the middle of a twoposition high voltage air-powered switch. The switch consisted of two 2.2 cm
o.d, brass cone-shaped plugs connected to flexible copper braid and facing
opposite sides of a central cylindrical brass electrode whch had conical
depressions machined in each end. The fixed center electrode was mounted in
a hole bored perpendicularly through a 7.6 cm 0.d. plexiglass pillar which was
bolted to the bottom of the aluminum box. The two movable electrodes were
mounted, via plexiglass insulators, to the ends of air cylinders whch were in
turn mounted on two facing sides of the box. The air cylinders had strokes of
about 10 cm and were pneumatically connected via solenoid valves to an air
source and arranged so that when one was extended, the other was withdrawn.
The large separation was necessary to prevent RF arcs when operating at high

FIG. 4-13. Schematic of Figure 4-12. 1 - Aluminum housing (cover removed). 2 - KF input cable (RG-17 A / U ) whlch plugs into center electrode of high voltage switch. 3 - Plexlglass pillar
housmg center electrode of high voltage switch wh~chconnects input either to a d m r n y load
or to inductor L,. Tbe switch itself is hidden beneath the bracket supporting the pillar. 4 k r cylinders whtch drive the high voltage switch. 5 - 50 R dummy load resistor. 6 - Inductor
( L , ) . 7 - Piexiglass insulating tube. 8 - S11d;mg brass cylinder to vary L I . 5 - Adjustment rod
f ~ brass
cylinder. 10 - Vacuum capacitor (C). 11 - Adjustment knob for vacuum capacitor.
12 - Inductor (L2).13- Vacuum voltage divider capacitor. 14 - HF current monitor. 55 - Plexglass housing to insulate current monitor. 16 - Ceramic insulator of antenna feeder tube.
17 - Antenna feeder tube.

power levels. The electrodes, whch had a contact area of about 3.5 cm2, were
silver-plated to minimize contact resistance. One side of the switch was connected to an air-cooled 50 Q non-inductive resistor which served as a dummy
load for testing purposes; the other side led to the inductor L1 of the
impedance-matchng network. The switch could be activated remotely from
the amplifier cabinet so that the high power amplifier could be easily tuned
using the durnmy load before attempting to tune the matching network.
Capacitor C was a 0 - 1000 pF, 15 kV vacuum capacitor, one flange of
which was mounted to the aluminum box wall. The inductor L 1 was wound
from 1.3 cm 0.d. soft copper tubing, insulated with heat-shrink tubing, and
mounted on a 10 cm &ameter plexiglass tube. The coil used for most of the
experiments consisted of nine turns over a length of 18 cm and had an inductance of 5 pH (at 12 MHz). The inductance was varied (i.e., decreased) by sliding a copper tube, 8.9 cm in diameter, inside the plexiglass tube underneath
the coil windings. Since the tube was much thicker than a skin depth, flux was
excluded from its interior (through the induction of appropriate surface
currents) and the net flux linking the coil was reduced. The tube also
increased the stray capacitance between turns of the coil, changing the selfresonant frequency. In practice, the coil reactance could be lowered by up to
-50% using this method, and although the Q of the coil itself was reduced,

there was no significant decrease in the overall matching network Q (indicating that power losses in the sli&ng tube were not important). Inductor Lz was
not adjustable and consisted simply of the strap connecting the vacuum capacitor to the center conductor of the antenna feeder tube.
RF antenna current was measured with a current monitor (Ion Physics Co.
model CM-100-L) mounted in a plexiglass shield directly over the ceramic insulator a t the end of the antenna feeder tube; the sensitivity of the monitor,
connected to a 50 R terminated cable, was 0.1 V/A. RF voltage was measured

with a capacitive voltage divider (similar to that described in section 4.2.3)
which was connected to the antenna tube inner electrode a t the point where it
emerged from the O-ring seal at the end of the ceramic insulator. The output
of the divider, also connected to a 50 R terminated cable, was 0.55 mV/V a t
12 MHz. The divider had a nearly flat frequency response over the range of

interest but was calibrated a t each frequency used.

4.5 Linear RF' Detectors
In order to record the amplitudes of RF signals (forward and reflected
voltages, antenna voltage and current, magnetic probe outputs) on the lowfrequency ( <1 MHz bandwidth) &gital transient recorders, amplitude or
envelope detectors were required. Square-law diode detectors were originally
used but resulted in compressing the 48 db dynamic range of the 8-bit transient recorders to only 24 db. Since the magnetic probe signals may vary by

10 db more than t b s , a Linear detector was desirable. The availability of a
linear detector -video amplifier integrated circuit (designed for radar signalprocessing applications) provided a simple solution.
A ten-channel RF detector system was constructed; the schematic for a

single channel appears in Figure 4-14a. A broadband 1:4 RF transformer coupled the input signal (terminated in 50 R) to the detector's long-tailed pair
input. Some R-C

filtering was required a t the video amplifier output to

prevent spurious oscillations and was followed by a low frequency 20 db gain
amplifier. The time response of the detector was determined by the R-C filter
and was typically set to -2 psec. The frequency response was quite flat, varying by less than 0.5 db from 1 to 100 MHz (Figure 4-14b). The input-output
voltage calibration curve for a typical detector is shown in Figure 4-14c; it is
reasonably linear over a 40 db range.

RF DETECTOR

(a)

V out

- ' 5 v 0 ~ ~ 1~5" 0 +
y fI.

-.

U1 - Plessey SL510C
U2- LF 356
T 1 - 1 4 broadband RF transformer
(Minicircuits T 1 - 1T)

.O1

FREQUENCY (MHz)

.1
1.
RF INPUT (PEAK VOLTS)

10.

FIG. 4-14. a) Schematic of linear RF detector (one channel). b) Frequency
response of detector: DC output voltage versus frequency for fixed input amplitude. c) Linearity of detector: DC output voltage versus RF input peak voltage
for a fixed frequency.

The actual calibration curves for the detectors were stored on disk and
used during data analysis to accurately unfold the RF amplitude signals.

4.6 RF Phase Detectors
4.6.1 Introduction

RF phase detectors were required both for measuring the phase between
the antenna current and voltage (for impedance measurements) and for
measuring the phase of the wave fields detected by various probes around the
tokamak. The different applications imposed different constraints on the
phase detectors.
When an eigenmode came into resonance, the magnetic probe signal magnitude could increase by a factor of up to 25 - 35 db (see Chapters 6 and 7).
The temporal width of the resonance was determined by the & of the mode
and by the rate of change of plasma density. The half-width of the resonance
peaks on the probe signals was typically 50 - 500 psec in these experiments,
during which the phase of the signal could undergo transitions of up to about
27~radians.
The characteristics of the RF signals from the antenna voltage and
current monitors were somewhat different. Sharp peaks associated with eigenmodes were seen with some antennas, but the change in signal magnitude during the plasma shot was generally less than 10 db. However, the change in

phase between the voltage and current signals was very small - less than 1
degree in some experiments. In this case, therefore, phase noise and dependence on input amplitude had to be minimized.

4.6.2 DBM Phase Detector
Several approaches to the problem of phase detection were considered.
Double-balanced mixers (DBMs) are commonly used as phase detectors at RF
frequencies [Mini-Circuits Laboratory, 19801 and were used here in early
impedance-measuring experiments. When two signals of the same frequency
are applied to the RF and LO (local oscillator) ports of a DBM, the DC component of the output a t the IF (interme&ate frequency) port is a function of
the phase difference between the signals. The approximate phase characteristic of t h s kind of detector is Vo,, =AIAzcosp,where A l and A2 are the amplitudes of the input signals, p is the relative phase &fference between them, and
provided A l and Az are well below the compression level of the mixer. In order
to unambiguously determine the phase over a 2n radian range, a quadrature
system using two DBMs must be used (Figure 4-15). The phase difference
between the input signals may then be found from

where tan-' is the principal branch and --<

7r

p < -.

This approach has the

advantage of simplicity, good harmonic rejection, and lack of active components but suffers from several drawbacks. The phase characteristic function

( cosp ) is only an approximation; its actual form must be found by careful
calibration and depends somewhat on input power level. The dynamic range of
the system (important for magnetic probe measurements) is limited because
of the proportionality of the output voltage to the input signal amplitude (the
data recor&ng system has a fixed dynamic range of 48 db). Finally, a single
phase signal requires two data channels, which are often a scarce commodity.

0"

A, sin(w t ) o

A,sin(~t t +) 0

- H1

-LO

oO-

1 LRF

90" -

DBM1

- LO

IF
I,

0 V,aA ,A2cos+

Low PASS

DBM2

ti2
0" -

LOW PASS
FILTER AND
AMPLIFIER

RF

FILTER AND
AMPLIFIER

IF

oV2aA, ApsinO

H 1 - Wide bandwidth quadrature hybrid (Anzac J H-6-4)
H2- 0° Power splitter (Minicircuits ZSC - 2 - 1)
DBM 1,2 - Matched pair, high compression level
double - balanced mixers (Minicircuits ZAY - 3 - I )

FIG. 4-15. Quadrature phase detector using two double-balanced mixers.

'-t

rP

4.6.3 Digital Phase Detector

A simple &gital phase detector has been devised which circumvents the
above problems. The relative phase between two RF sinusoids is proportional
to the time interval T ~ ,between successive positive (or negative) - going zerocrossings of the signals: p = ~ T T , , / T ~ where
~ d is the RF period. An ideal
phase detector could therefore consist of a two-state "black box" whch would
be set high whenever one input signal had a positive-going zero-crossing and
would be set low whenever the other signal had a positive-going crossing. The
output of this device would then consist of a train of pulses whose duty-cycle
would be proportional to the input phase difference. The phase information
could be extracted by low-pass filtering the output.
The design of a four-channel, ZT radian range phase detector for use with
the magnetic probe signals is shown in Figure 4-16. Because T~ could be as
short as 50 nsec (at a frequency of 20 MHz), circuitry with very fast switching
times was required. ECL (emitter-coupled logic), with transition times on the
order of 1 nsec, was found adequate. The function of the above-mentioned
"black box" and the subsequent low-pass filtering was realized using an ECL
comparator, two ECL D-type flip-flops, and a low-frequency differential
amplifier. The sinusoidal input signals were converted to square waves of constant amplitude by the comparators. Each flip-flop had only one input which
was positive-edge triggered: the clock (Ck) input. The other inputs (R (reset),
and S (set)) were state-sensitive,
Consider the function of fhp-flop U3a in Figure 4-16. As long as the
positive-going transition a t the Ck: input occurs while R is low, the output at
Q is a pulse-width modulated square wave whose duty-cycle is linearly propor-

tional to the phase difference between the inputs a t Ck and R . During .rr radians of the phase characteristic, however, the transition a t Cjcc occurs whle R
is h g h , and so the output Q remains low. The operation of flip-flop U3b is

INPUT I

r~#
""

-5v

ID

70k

U3a Q

PHASE
OUTPUT I

-5v

-5v

MATCHED LENGTHS OF
5 0 n COAX
REFERENCE
INPUT

TO PHASE
'DETECTORS

m,
~ 1U,2 - Plessey S P 9 6 8 5 (ECU
U3a,b - Motorola 10H 131 ( E N )

U 4 Motorola L F 3 5 6
U5o,b,c,d - Motorola 10189 (ECL)

FIG. 4-16. Schematic diagram of 2n radian range digital phase detector (one
channel shown). Compact layout of ECL circuitry was crucial to optimize performance.

WAVEFORM - I
-1,8-6
AT TP1
(VOLTS) -2.6-

2-l~

37

3r

-I-

WAVEFORM -1.8
AT TP2

27T

PHASE
OUTPUT
(TPI-T P2)

FIG. 4-17. Waveforms a t TP1 and TP2 (in FIG. 4-16) as functions of the phase
&fference between the input signals. Subtracting the two waveforms provides a
phase characteristic with a 2 n radian range.

identical except that its CK: and R inputs are interchanged with respect to
U3a. Therefore, output Q of U3b provides a pulse-width modulated output
during the n radian portion of the characteristic when Q of lJ3a remains low,
and conversely it remains low during the n radian portion when U3a is functional. The ffip-flop outputs are low-pass filtered by an R-C combination and
are subtracted by a hfferential amplifier ( U4). The variation of filtered outputs (at T P 1 , TP2 in Figure 4-16) with input phase difference is shown in Figure 4-17, and it is clear that subtracting the two provides a linear phase output
characteristic with a 27r radian range.
The time response of the phase detectors to a step change in phase was
determined by the time constant of the low-pass filter and was generally set to
- 5 psec. The frequency range over which the phase detector operated was

limited on the upper end by the rise and fall times of the ECL logic. Distortion
of the phase characteristic begins to occur when the rise and fall times
become a substantial fraction of the RF period. Typically these phase detectors functioned at frequencies of up to about 70 MHz.

4.6.4 Harmonic Distortion

The dgital phase detector is, however, sensitive to the presence of harmonics on the input signals. To obtain an estimate of the magnitude of this
effect, consider an idealized comparator, with no input offset or hysteresis,
whose output changes state as the input voltage passes through zero:

where VH and VL are the two output voltage states of the comparator. If a
sinusoid Aosin(wot) is applied to the input, the output will change states at
times t = n n / wo, where n is an integer. If a harmonic of the fundamental frequency is added to the input, the transition times will be shfted. For an input

Vh = A o sin(oot) +An sin(n wet + p,) , the positive-going zero-crossing which had

occurred a t t = 0 before the introduction of the harmonic now occurs approximately at time

t,, =

-& sinp,

The shift in transition time is a function of the relative phase of the harmonic,
cp, , and its maximum occurs a t p,

= T F / ~
f o,r nA,

<< A O , leading to a max-

An
imum possible shift in zero-crossing time of At,, = -. This corresponds to
Aowo
the introduction of a phase error of & / A o

ra&ans. Thus, to keep the error

introduced by a harmonic to a level less than 0.1 degree (which is necessary in
some of the impedance-measuring experiments), the amplitude of the harmonic should be a t least 55 db below the fundamental. Experimental simulations agree reasonably with this estimate.
By observing the RF antenna current and voltage signals during plasma
shots with a spectrum analyzer, it was found that the second (Zw) and third
( 3 0 ) harmonic amplitudes were generally lower than 30 db below the fundamental. This was sufficient, however, to require additional filtering preceding
each input to the phase detector. A pair of 50 R, six-pole bandpass filters were
constructed for each fundamental operating frequency using image-parameter
design. A typical 12 MHz filter had a 3 db bandwidth of 2 MHz and an attenuation of 60 db above 23 MHz, w l c h was more than sufficient to reduce the effect
of input signal harmonics on the phase detector output to a negligible level.

4.6.5 Dynamic Range

The dynamic range of the phase detection system can be defined as the
ratio of maximum to minimum input signal amplitude (at constant input

phase) over which the detector output varies by less than a given amount.
Variation of the detector output as the input amplitude is decreased is primarily a consequence of the input offset voltage and finite gain of the comparators. The upper limit (A,,)

to the allowable input amplitude is typically

slightly less than half of the comparator power supply voltage swing. As the
input signal envelope decreases, the point during each cycle a t which the comparator triggers the subsequent ECL logic changes. This introduces a phase
error whose magnitude can be simply estimated.
The voltage transfer function for a comparator may be modeled as

where

II"

G . v for V L / G < V < T / x / G

\ k ( V ) = VH

for V > V H / G
for V < & / G

Here G is the comparator voltage gain and Voffis the input offset voltage. Let
the input voltage be V , = A sin(&) and the positive-going voltage at whch the
subsequent ECL switches, T/, . Then the time t, at which the transition occurs
is given by the solution of

The apparent phase shift Ap introduced by changing the input signal amplitude from A,,

to A-

is then

where $ = 'Ir, -( VH+ VL)/2 . Parameters appropriate to the Plessey SP9685
(one of the fastest comparators available) are:
T/', = 0.15 V, A,,,

G =650, Vofl = 3.0 mV,

= 4.0 v.lZ For a phase error of less than 10 degrees, then,

equation 4.13 yelds a dynamic range of 48 db. For an error of less than
0.1 degree, the dynamic range is reduced to 11 db. The phase detector system

described was quite adequate for the magnetic probe measurements where a

2n radlan range was used and the signals varied by a t most 35 db. For the
impedance measurements, however, where a 1 degree full scale range was often
used, the dynamic range was marginal and had to be increased.
Improvements in dynamic range can be made by increasing the gain of the
input stage and by reducing the input offset voltage. A phase detector was constructed for the impedance measurements which incorporated input offset
adjustments and another stage of RF amplification for each comparator. The
Plessey SL532 low phase-shft limiting amplifier was used, provi&ng an additional 12 db of gain and increasing the estimated dynamic range to 23 db,
whch was adequate for the impedance measurements.
In laboratory simulations, the observed dynamic ranges of the phase
detectors were somewhat larger than predicted. In addition, the magnitudes of
the current and voltage signals for the impedance measurements changed in
close synchrony during the plasma shot (since

/ Zmt 1 did not change very

much) which tended to reduce phase errors.

12. Radar and Radio Communications IC Handbook, 1981, published by Plessey Semiconductors, Irvine, CA.

CHAPTER 5
RF Antennas and Plasrna Probes
The structure which couples energy to the plasma, the RF antenna, is of
crucial importance in the deslgn of any ICRF system whch transfers energy to
plasma waves, yet the interactions between the antenna and the plasma
remain poorly understood. A n area of current controversy, for example, is the
role of the Faraday sheld for traditional magnetic loop antennas [Faulconer,
1981; Messiaen and Weynants, 19821; another is the nature of the anomalous
density-dependent antenna loading resistance which is seen in some tokamaks
[Buzankin et al,, 1976; Bhatnagar et al., 1978a; Taylor and Morales, 19701.
This chapter begins with a description of a series of RF antennas which
were built in order to illuminate various features of the antenna-plasma coupling problem. The antennas were of small poloidal extent because of the limited size of the tokamak ports and were easily changed through an air-lock.
Three magnetic loop antennas (bare, insulated, and Faraday-shielded) and two
electric field antennas (bare plate, "T") are described. The vacuum fields associated with the magnetic loop antennas are &splayed, showing some of the
effects of the Faraday shield. Magnetic probes are then discussed in some
detail as they were the major source of information regarding wave propagation in the plasma. A brief mention is made of several conventional Langmuir
probes which were used both to measure radial density profiles and to investigate the modification of the probe I - V characteristic in the plasma near an
antenna due due to application of RF power. The chapter then concludes with

a section on the design and construction of a RF plasma current probe and the
rotatable antenna with which it was used.

5.1 RF Antennas
5.1.1 Antenna Feedthru Assembly
The tokamak port used for the introduction of RF antennas had the large s t cross-section of any port on the machne (9.8 cm high x 4.7 cm wide) and
also functioned as the pumping port (Figure 5-1). The antennas had to be
about 1 m long in order to pass t h r ~ u g hall of the vacuum plumbing into the
tokamak chamber. The portion of the antenna which provided this length was
a 3.8 cm 0.d. coaxial line section which will be referred to as the antenna
feeder tube. The RF coupling structure was mounted on one end of this tube;
t h e other end consisted of a n alumina high voltage insulator with an O-ring
vacuum seal through which the center conductor emerged (Figure 5-2). The
end with the insulator plugged into the impedance-matching box (Figure 5-3)
where it was electrically connected to the T-network (section 4.4).
The center element of the feeder tube was removable and was contiguous
with the radiating element a t the end of the tube. A close-fitting Pyrex tube
was slid into the feeder tube to provide high voltage insulation between the
coaxial elements. The loop antennas required a low impedance feeder tube
(see Chapter 6) and the center element of these antennas consisted of a

2.54 cm 0.d. copper tube, yielding a characteristic impedance of 16 R. The
electric field antennas, however, used a higher impedance section (99 R) for
which the center element of the feeder tube was a 6.4 mm 0.d. copper rod.

A vacuum seal was made on the outside of the antenna feeder tube
through double, differentially-pumped O-ring seals which allowed sliding
motion and rotation of the tube. The drfferential pumping was found to be

necessary in order to prevent large bursts air from leaking in past the seal
when inserting the antenna. The surface of the copper feeder tube was soft
and easily scored by the O-ring fittings, causing small leaks to occur. This
problem was solved by plating the outside of the tube with a 25 micron layer of
hard chrome; nylon bushings were also added to the insides of the O-ring
fittings to decrease the lateral play in the seals. The double O-ring seal assembly was part of a 15.2 cm o.d, vacuum flange which also had five small O-ring
vacuum seals for viewing ports and electrical feedthrus (Figure 5-4). This
flange was mounted on a bellows assembly which allowed the antenna support
structure (described below) to be aligned with the tokamak port as it was being
inserted, thus preventing binding. The bellows assembly was, in turn, bolted to

an 18 cm long, 10 cm i.d. air-lock chamber which had a gate valve at its end.
The air-lock chamber allowed antennas to be easily changed -without opening
the tokamak to air; it was pumped down to the 10" Torr level with an
auxiliary vacuum system before opening it to the tokamak. The antenna
assembly, attached to the tokamak and the impedance-matching box, is shown
in Figure 5-5.
The antenna feeder tube was grounded to the tokamak port by means of a
clamp-on support structure (visible in Figure 5-4) which had spring-finger contact strips1 along its edges. The support was machined to conform to the contours of the tokamak port, and large areas were milled through its face to
maximize the pumping speed through it. An effective RF seal was made when
the structure was slid into the port, compressing the silver-plated ftngers. The
support structure also served as a base for attachment of the Faraday shield
and the Langmuir probes.

1. Manufactured by the Instrument Specialties Co.,Little Falls, N.J.

5.1.2 Bare Loop Antenna
The simplest magnetic loop antenna consisted of a bare loop of 6.4 mm
0.d. OFHC copper rod. One side of the loop was welded to the 2.54 cm 0.d.
center conductor of the antenna feeder tube; the other side was clamped in a
split copper block which was welded to the outer conductor of the feeder tube.
The hole through the block, and the end of the loop which fit into it, were
silver-plated to reduce contact resistance. A photograph and diagram of the
antenna are shown in Figure 5-6.
The antenna could be inserted about 5 cm into the plasma (measured
from the inner wall of the tokamak to the leading edge of the loop) with virtually no effect on the global plasma parameters. This distance corresponded to
the end of the feeder tube being flush with the tokamak inner wall, The loop
was usually left in the tokamak during the &scharge-cleaning procedure and
no outgassing problems were observed. Note that the entire antenna feeder
tube was under vacuum, the seal being a t the end which was attached to the
im-pedance-matching box.
The Q of the antenna (the ratio of its reactance to its resistance), including the antenna feeder tube, was measured using the method described by
DeMaw [1976]. A high Q (-800) variable vacuum capacitor was connected
with copper strap directly across the input to the feeder tube, and a 50 R signal generator and a 50 R oscilloscope were loosely coupled to the resonant circuit through 1.0 pF capacitors. The vacuum capacitor was adjusted to achieve
resonance a t the desired frequency, and the frequency was then varied to find

the -3 db points from which the Q was simply calculated ( Q = -----).

A value

b ~ s d b

of 157 was obtained at 12 MHz with the antenna front edge 5 cm into the
tokamak; this decreased to 148 with the loop withdrawn into the tokamak
port (the decrease was due to induction of eddy currents in the stainless steel

SPRING FINGER
GROUNDING CLIPS

SPLIT-BLOCK
, CLAMP

FIG. 5-6. Bare loop antenna.

OFHC COPPER
ROB (.63 cm O.D.)

port which increased the resistive part and decreased the reactive part of the
loop impedance). Since the input reactance was approximately 18 0, the Q
measurement implied a n antenna resistance of about 0.1 R.
Calculations of the antenna resistance a t 12 MHz based on its geometry,
however, yielded a figure of about 0.02 R, a factor of 5 lower than observed.
T h ~ simplied that the major power losses in the antenna occurred not from distributed resistance, but from connections (i.e., between the feeder tube and
the matching box, and a t the clamp for the loop rod a t the tokamak end of the
tube). No effort was made to increase the antenna Q , however, since the losses
in the antenna were sufficiently small for the experiments being performed.

5.1.3 Ceramic-Insulated b o p Antenna
The ceramic-insulated loop antenna was a modification of the bare loop
antenna which prevented plasma from reaching the antenna element. A fivesided hollow box was constructed from 3 mm thick Macor2 sheet; the outside
x were B.9 c m high :
: 5.4 cm wide x 1.9 cm thick. The
dimensions of the b

edges of the Macor pieces were lapped to a smooth finish and they were assembled in a jig. The box was then glued together from the inside using a thin
bead of low vapor pressure epoxy wbch was applied though a hypodermic
syringe. The very close fit of the pieces of the box insured that no plasma was
able to reach the epoxy. The box was then slid over the bare loop and epoxied
in place, also from the inside. The completed assembly is shown in Flgure 5-7.
The Macor was con&tioned by leaving it in the &scharge-cleaning plasma.
After such cleaning, it could typically be inserted about 4 cm past the wall
without adversely affecting the tokamak plasma. After several months of use,
the side of the box which faced the electron runaways was noticeably
2. Hacor, a machineable glass-ceramic, is manufactured by Corning Glass Works, Corning,
N.Y.

MACOR BOX
(1.9 em DEEP)

FIG. 5-7. Ceramic-insulated loop antenna, showing
probes attached to the side of the antenna.

blackened. The dscoloration affected only the outer 20 - 30 microns of the
Macor and did not seem to affect the plasma discharge.
Several small probes were built which could be attached to the side of the
Macor box ( a magnetic and a Langmuir probe are shown in Figure 5-7). The
bodies of the probes were 2.18 m m 0.d. 50 62 semi-rigid coax whch could easily
be bent to the desired contour. The last 10 cm of the probe was sheathed in a n
alumina tube to protect the teflon insulation of the cable and to minimize
current flow to the outer conductor of the cable. The coaxial cables were
clamped to the antenna support structure where they ended in SMA connectors. Lengths of a small, flexible 50 62 cable (RG-l78), whose insulating jackets
had been stripped off (to decrease outgassing), connected the probes to
vacuum feedthrus on the antenna feedthru flange (Figure 5-4).

5.1.4 Faraday-Shielded Loop Antenna

A Faraday shield was built for the bare loop antenna to reduce the electric
field components produced by the antenna. There couid be a large potentiai
dBerence across the loop antenna due t o its finite inductance: an RF potential
of 1000 V applied to the input of the antenna feeder tube ( a t 12 MHz) produced
a potential difference of about 140 V between the top and bottom of the loop.
The large toroidal electric fields which could thus be generated are thought to
couple to unwanted electrostatic modes [Messiaen and Weynants, 19821.
The Faraday s h e l d consisted simply of strips of metal covering the loop
and effectively shorted out the z-component of the electric field. The shield,
shown in Figure 5-8, was welded from 0.8 mm thick copper sheet. Four strips,
2.0 cm tall, were bent to form a 2.1 cm wide box covering the loop. The strips
were separated by 2.0 mm wide gaps which extended 3.8 cm along the sides of
the shield. The slots were baffled in order to prevent plasma streaming along
the magnetic field lines from reaching the inner loop through the gaps. The

SHIELD FORMED FROM 0.8 mm
THICK COPPER SHEET

I------3.8cm-I

2 mm

FIG. 5-8. Loop antenna with Faraday shield. The shaded areas in
the scale drawing represent the slots in the shield which are hidden in the photograph by the baffles.

shield assembly was bolted to the antenna support structure, and a gap of
4.0 mm separated the front of the loop from the inside front of the shield.

The shield decreased the & of the antenna to about 126 a t a frequency of
12 MHz,due to the decrease in inductance of the loop (by about 12%) and the
increase in resistance (by about 9%) because of eddy current losses. The Faraday shield also had the unfortunate effect of almost completely blocking the
port, vastly reducing the pumping speed of the tokamak turbopump. It was
necessary to use the turbopump from the charge-exchange diagnostic, pumping on another port of the tokamak, to acbeve adequate pumping speed. The

shielded antenna could be inserted about 4 cm past the tokamak wall before
the global plasma properties were affected.

5.1.5 Bare Plate Antenna
In order to investigate the effect of the electric fields generated by the
loop, several antennas were designed to maximize these fields. The first such
structure investigated consisted of a bare rectangle of copper (5.1 ern long x
3.8 cm wide x 1.5 mm thick) connected to the inner element of the antenna

feeder tube (figure 5-9). For tlvs antenna, the inner element consisted of a
6.3 mm 0.d. copper rod which was centered within the feeder tube with a single

teflon spacer at the midpoint of the tube. In operation, then, the plate was
simply being driven at an RF potential with respect to the tokamak wall.
This type of antenna is referred to here as an electric field antenna
because, in the absence of plasma, the antenna produces essentially no magnetic field. It is recognized, of course, that in the plasma environment the
antenna may drive currents (through the plasma) which then produce magnetic fields; nevertheless the label "electric field antenna" will be used to distinguish this design from the loop antennas.

OFHC COPPER PLATE

FIG.5-9. Bare plate antenna.

OFHC COPPER
ROD (6.3mm O.D.)

FIG. 5-10. "T" antenna,

5.1.6 "T"Antenna
The possibility that the bare plate antenna was being "shorted out" by
plasma in the mouth of the tokamak port led to the construction of a mo&fied
electric antenna. This antenna consisted of a rod of copper ( 6 . 3 mm o.d, x
5.1 cm long), attached a t its midpoint to another rod which in turn was connected to the inner conductor of the 99 R antenna feeder tube (Figure 5-10).
The middle rod was insulated with a Macor tube which insured that any
current flow driven in the plasma by the antenna came from the bar a t its end
(displacement current through the Macor was negligible). The inner element of
the feeder tube could be rotated to align the cross bar of the "T" parallel or
perpendicular to the toroidal magnetic field.
Both electric field antennas could be inserted 5-6 cm past the tokamak
wall without any adverse effects. Both of the antennas could also be operated
either "floating" (i.e., with a series capacitor between the matchng network
and the RF input) or at DC ground with respect to the tokamak wall (using an

RF choke between the matching network input and ground). When isolated, the
antennas assumed the plasma floating potential, while when a t DC ground they
generally drew a net electron current from the plasma.

5.2 Vacuum Fields
Measurements were made in a test jig to examine the effect of the Faraday
shield on the loop antenna fields in vacuum (or air, to be precise).
The antenna feeder tube was mounted through a large sheet of brass
whch served as a ground plane. The sheet was located 12 cm behind the tip of
the loop (Figure 5-11), and the end of the feeder tube was connected coaxially
to a signal generator. The fields produced in this test set-up were, of course,
different from those produced in the tokamak chamber because of the

FIG. 5-11. Diagram of the test configuration for measurement of the
antenna vacuum magnetic flelds. The x -axis is collinear with the
center element of the feeder tube; the y-axis lies in the plane of the
antenna loop. The front edge of the loop is located at x = -2.5 cm.

FIG. 5-12. Plots of normalized 1 B, (x, 0,O) 1 for sluelded and unsluelded
loop antennas. The dots are the experimental points. Vertical scale units
are milligauss per ampere of antenna current, where the current is measured at the input to the antenna feeder tube. Data points for the
shielded antenna have been multiplied by 10.

FIG. 5-13. Plots of normalized 1 BZ(0,
y,0)1 for shielded and unshielded
loop antennas.

FIG. 5-14. Plots of normalized I Bz( 0 ' 0 ,z ) 1 for shielded and unshielded
loop antennas.

different geometry, yet useful information was still obtainable from measure-

ments near the antenna where boundary effects were less important.
Measurements of the z-component of the magnetic field (analogous to the
toroidal direction in the tokamak) were made along each of the three axes
shown in the figure. The x-axis of the coordinate system was colinear with the
axis of the antenna feeder tube, and the tip of the loop was located in this system a t x = -2.5 cm. When the Faraday shield was added, the loop position
remained fixed and the front edge of the shield was then located a t
x = -2.0 cm. The magnetic field probes described in section 5.3 were used for
these measurements, which were made a t a frequency of 12 MHz.
The amplitude of Bz is shown for the shielded and bare loop antennas, as
a function of position along the three axes, in Figures 5-12, 5-13 and 5-14. The
field falls off rapidly in the x-direction (roughly as l / r , where r is the distance to the current-carrylng element) and &splays a localized peak around
the origin in the y - and z-drections. The most notable feature of these measurements was that the introduction of the shield reduced the magnetic field
amplitude by a factor of about 3 -4 and had little effect on its spatial dependence. The fact that the loop was about 5.0 mm b e h n d the front edge of the
Faraday shield meant that the maximum magnetic field seen by the plasma
(that at the front edge of the antenna) was even lower for the shielded antenna

- about 10%of the field seen a t the front edge of the bare loop antenna, for a
given excitation current. This reduction of magnetic flux available for coupling
to the ICRF waves was responsible for the very low loading resistances observed
with the shielded antenna (see Chapter 6).
One of the supposed advantages of the Faraday shield for ICRF work is the
reduction of Ez produced by the potential difference between the loop and the
wall. By covering the loop with metal strips running in the z-direction, which
present a low-impedance path to the wall, E, near the antenna surface is

greatly reduced. The Faraday sheld allows B, to pass through by the induction of surface currents in the strips. To first order, the current flows up the
outside surface of the strips in the y-direction, through the gaps between the
strips, and back down the inside surface. The current distribution on the
inside of the strips is just what is needed to cancel the field produced by the
loop so that there is no field inside the conducting strips themselves. The
currents running along the outer surfaces of the strips therefore generate the
magnetic field seen outside the antenna so that, in a sense, the low-inductance
shield becomes the radiating element of the antenna.
Although the inductance associated with the current path on the outside
of the sheld is small, there still must exist a potential difference between the
top and bottom of the sheld. To first order, the electric fields produced by the
shield are in the x- and y-directions (corresponding to the radial and poloidal
directions in the tokamak geometry), but the fringing fields associated with
any antenna of k i t e toroidal width produce components of

E in the

z -direction as well.

The potential &fferences across the gaps between strips of the sheld were
measured experimentally. A vector voltmeter with a 10:l divider probe was
used to measure the potentials across the five gaps a t 1.0 cm intervals along
the gaps (Figure 5-15). The probe ground clip was bent to be parallel and very
close to the probe tip so that the induced EMF from magnetic flux threading
the probe part of the circuit was negligible; the RF generator was also floated
to minimize any RF ground loop currents. The results are shown in Figure 5-16
where S represents the path length along the gap (the front center of a 2 crn
wide strip corresponds to S = 0 , whle S = k 1 cm corresponds to the corners of
the sheld, etc.). The potentials across the gaps are peaked along the front of
the sheld ( I S / & 1 cm) and the largest potential jumps are across the central
gap (i.e., the gap along the midline of the antenna). Summing the potentials

FIG. 5-15. Diagram of the test configuration for measurement of
poten- tials across the Faraday shield gaps. S is the path length along
each gap and S = 0 corresponds to the front center of the shield. The five
gaps are labeled A - E. The baffles which cover each gap are not shown.

FIG. 5-16. Measured RF potentials across the five gaps of the Faraday
shield, as functions of S . Vertical scale units are millivolts per ampere of
antenna current, where the current is measured a t the input to the antenna feeder tube.

across the five gaps a t S = O yields a n estimate of the potential difference
between the top and bottom of the front of the shield: 87 mV (per arnpere of
antenna current, measured a t the input to the feeder tube). This should be
compared to the potential difference of about 2.5 V/A between the top and bottom of the bare loop, Hence the electrostatic potentials seen by the plasma
are substantially smaller with the Faraday sheld in place, even after normalization to the same magnetic flux at the front edges of the two antennas. This
is actually just a result of the lowered inductance of the effective current path
due to the introduction of the shield. The same result could presumably be
obtained by simply using a very wide antenna loop conductor.

5.3 Magnetic Probes
5.3.1 Introduction
Magnetic probes are a routine diagnostic for investigating the field structure of plasmas [Lovberg, 19651. The most common type of magnetic probe,
and the type to be discussed here, consists of a small loop or coil of wire
encased in an insulating jacket which is inserted into the plasma. Other
approaches are possible, such as charged particle beam probes or optical
methods, but are generally much more complex experimentally,

If the dimensions of the coil are small compared to the characteristic
lengths over whch the magnetic field changes, then the voltage output of the
coil gives an indication of the time rate of change of a component of the magnetic field a t the location of the coil. It is necessary, however, to consider how
the field that the probe senses is related to the field that would have been a t
that point in the plasma in the absence of the probe. The probe interacts with
the plasma in a number of ways: local plasma which intercepts the probe is
cooled; the presence of the probe perturbs the flow of plasma and hence the

particle currents which generate the wave magnetic fields; impurities from
boil-off or erosion of the surface of the probe envelope may diffuse through the
plasma bulk and cool it substantially through bremsstrahlung radiation,
Other problems include destruction of the probe due to the large heat flux
incident on it and frequency response limitations due to the self-inductance of
t h e coil.
The effect of the probe on the bulk plasma can be monitored by observing
t h e global plasma diagnostics: the visible light and soft-UV detectors, the
plasma current and loop voltage monitors, and the microwave interferometer.

If these signals change negligibly as the probe is pushed into the plasma, it
may be assumed that the probe is not interacting greatly with the bulk plasma.
If the probe is pushed too far into the plasma, or if it is contaminated, the
soft-UV and visible light signals increase substantially during the plasma shot,
the plasma current may be reduced, and the density evolution is typically substantially changed. A test of the effect of a probe on the large-scale propagation of the waves being observed was performed by using two probes separated
toroidally by a large fraction of the tokamak circumference and exciting one of
the RF antennas. One probe was fixed in the plasma and the wave fields were
measured. The other probe was then inserted into the plasma and any resulting effect on the signals from the first probe was observed. In fact, no effect
was found until the second probe was inserted far enough into the plasma to
affect the global plasma &agnostic signals by itself (4 - 6 em).
The effect of the probe on local plasma and the resulting effect on the
magnetic fields at the probe are &fficult to analyze [Ecker et al., 1962; Malmberg, 1964; Lovberg, 19651. These effects were likely to be small in t h s experiment because the fields being measured arose from global plasma properties
and the scale length over which the fields and eigenmodes varied was large
compared to the dimensions of the probe.

Experimental confirmation,

however, was not available
Physical damage to probes inserted into the plasma was a common problem. The Pyrex envelopes used to house magnetic probes withstood the plasma
environment moderately well, generally lasting many months ( > 1000 plasma
shots) before requiring replacement. Damage was almost entirely confined to
the side of the probe which faced electron runaways. Deterioration of the
probe began with discoloration from clear to amber, followed by progressively
worse fracturing and flaking of the glass surface. Other probe envelope
materials were tried, with mixed results. An alumina tube3 inserted 2 cm into
the plasma was severely eroded after only 100 plasma shots. Boron nitride4
and Macor withstood a similar test much better, showing only some discoloration and very slight erosion. Pyrex was chosen because it was easy to repair
and vacuum-tight glass-to-metal seals were readily available.

5.3.2 Description of Probes
The magnetic probes used in tirese e~pe~ii-nents
consisted of small coils
wound on insulating forms and protected from the plasma by a Pyrex tube.
Two types of probes were constructed: single-coil and triple-coil probes. The
single-coil probes were oriented with the magnetic axis of the coil perpendicular to the probe shaft. These probes could measure Be or B, by simply rotating the probe shaft by 90 degrees (where 8 corresponds to the poloidal angle
and z corresponds to the toroidal direction in the tokamak). The triple-coil
probe consisted of three coils whose magnetic axis were mutually orthogonal;
these probes measured all three components of the magnetic field a t a single
point. Construction of the triple-coil probes is described here; the single-coil
3. Alumina tubes were obtained from the Coors Co., Boulder, Col.
4. Boron nitride, grade HPC, was obtained from Union Carbide Corp., Carbon Products Divi-

sion, New York, N.Y.

probes were built in a similar fashon.
The probe shaft was a 1.27 cm 0.d. x 0.83 cm i.d. stainless steel tube whose
surface was polished. One end of this tube was welded to a 0.95 cm 0.d.
stainless-to-Pyrex transition, and the end of the Pyrex tube was closed forming
a vacuum-tight envelope. The probe was introduced into the plasma through a
single O-ring seal which allowed the probe to slide and rotate (Figure 5-17), A
well-polished surface was found to aid in preventing bursts of air from leaking
into the tokamak as the probe shaft was inserted. Since the Pyrex could be
damaged by extended discharge cleaning, the probe tips were generally left in
the ports 3 -5 cm from the inner wall of the tokamak and were pushed in past
the wall prior to a tokamak run.
The three orthogonal coils were wound on a 3 rnm 0.d. phenolic form with
seven turns each of #34 AWG magnet wire (Figure 5-18). The two leads from
each coil were twisted to prevent pickup and were epoxied to the side of the
form. At the end of the form each pair of wires was soldered to a 2.18 mm 0.d.
semi-rigid 50 R ceaxia! cable (which had a solid copper outer conductor), taking care to keep the two wires extremely close to each other to minimize any

enclosed area. The three coaxial cables were soldered together along their
length and were centered in, and epoxied to, a 1.15 cm 0.d. phenolic tube which
could then be slid into the stainless steel probe shaft. The coils, and the rod
they were wound on, were completely covered with a thin coating of a hightemperature epoxy. The epoxy also covered the gap between the rod and the
semi-rigid coax, and the end of the coaxial cable bundle itself. The epoxycoated end of the probe, along with about 1 cm of the adjoining coaxial cable
bundle, was given three coats of a colloidal silver lacquer (Emerson & Cuming
Eccocoat CC-2). The surface resistance of the resulting film was about
1 O/square and provided an effective electrostatic shield. The silver coating,

however, was easily scratched and was protected by a high-temperature teflon-

- 199 BNC CONNECTORS
SEMI - RIGID COAX
PHENOLIC TUBE
(INSULATES COAX)

LINEAR POSITION
SCALE
ANGULAR ORIENTATION
SCALE

/ NYLON-LINED SPLIT-RING
CLAMP (LOCKS PROBE POSITION

VACUUM O-RING SEAL

CONFLAT VACUUM
FLANGE (7 cm O.D.)

POLISHED STAINLESS
STEEL TUBE (127mrn 0 D )

STAINLESS TO PYREX.\TRANSITION (9.5mm OD)

COIL ASSEMBLY

F7G. 5-17. Magnetic probe holder.

STEEL PIN
(KEEPS COIL ASSEMBLY ALIGNED)

(a)

THREE

SEMI-RIGID 5 0 0
COAXIAL CABLES (2.18 m m O.D.)
SOLDERED TOGETHER

MAGNET WIRE

SILVER LACQUER LAYER (-10 /sq.)
L k L o N PRo-rEcTvE COATING

FIG. 5-18. Magnetic probe construction. a) Bare phenolic form fixed in place
with stainless steel pin. b) Coils wound on mutually orthogonal axes. c) Electrostatic shield applied.

FIG. 5-19. a) Magnetic probe before slniel&ng. b) Probe
after coating with epoxy and silver lacquer.

based ~ o a t i n g Application
.~
of the electrostatic shield is shown in Figure 5-19.
Output ends of the three semi-rigid cables were soldered to BNC jacks, and the
completed assembly could withstand temperatures of 300 " C (probe tip temperatures greater than 100 " C have been observed in some experiments).

5.3.3 Circuit Model of Magnetic Probe
The wavelengths of the fields that the probe senses are generally much
greater than the &mensions of the probe. Therefore, a lumped-element circuit
model is used to investigate the response of the probe [Ramo and Whinnery,
19.141.

A simple equivalent circuit used by a number of authors [Segre and Allen,

1960; Ashby et al,, 1963; Decker and Honea, 19721 is shown in Figure 5-20.
Here 4 represents the self-inductance of the coil, 4 is the resistance of the
coil (which is frequency-dependent due to the skin effect), Cp is a n effective
capacitance representing the stray capacitance of the coil and its leads, and

RL is tne ioad impedance. i.;, is the EMF generated by t h e external magnetic
fields; it is proportional to the number of turns N ,to the coil area A , and to
the time rate of change of flux linking the coil. It is assumed that the cot1 area

A is small compared to the scale length over which the magnetic field changes.
The probes are used to detect waves with a sinusoidal time variation, and
I$
is then given by

I$ = -iioNAB, where B is the magnitude of the magnetic

field component parallel to the axis of the coil a t the location of the coil, and

B and I$
vary as e + i w t . The output voltage Vo is then related to the magnetic
field B by

5. TFE Coating #6065, manufactured by Crown Industrial Products Co.,Hebron, El.

I<

MAGNETIC PROBE COIL

TERMINATED

I TRANSMISSION
LINE

FIG. 5-20, Magnetic probe equivalent circuit model.

C\]

MAGNETIC PROBE W3

FREQUENCY (MHz)

FIG. 5-21. Typical magnetic probe calibration as a function of frequency. Sensitivity is defined as the rms voltage output of the
probe (terminated in 50 R) divided by the rms magnetic field amplitude (in gauss). Circles are the measured points; the solid line
is the result of equation 5-2a with NA = 0.82 cm2 and .& = 0.28 FH.

where

04

WRPCP+L

8 = tan-'

1-W"L,

+ -4 '
RL

f 5. l b )

The RF resistance of the coil was generally small and could be neglected
provided Rp << RL, -. The radius of the wire used to wind the coils ( a ) was
CP

80 microns. Since this was still large compared to the skin depth ( 6 ) at the
frequencies of interest (5 - 20 MHz), the resistance per unit length was given by
p = q / (2na6), where q is the DC resistivity. A t a frequency of 12 MHz, this

yields a resistance of p =0.018 R/cm. The total length of wire used for each
coil was about 12 cm, so the RF resistance of a coil at 12 Mhz was R, = 0.2 R. It
was convenient to use standard terminated 50 R transmission line to connect
the magnetic probe output to the RF electronics; hence RL = 5 0 R and the condition I$ <<& was satisfied.
The coil capacitance C;, was at, T Z ? L ? Sa~ few picofarads, and its effect was
negligible provided Cp << ( ~ ~ b )(u&)-l.
- ~ ,

The coil self-inductance 4 was on

the order of 0.3 p H and therefore both & and Cp could be neglected in equation 5. l , ylelding

where

The wave fields detected by these probes were often small, and to improve
the signal-to-noise ratio it was desirable to maximize the sensitivity of the
probe. The output of the probe for a given field amplitude could be increased

by enlarging the area of the coil. This had to compete, however, with the
necessity of minimizing the perturbation of the plasma by the probe, which
required small probe dimensions.
For a fixed frequency and coil diameter, the sensitivity of the probe,
defined as S = Vo/ B volts per gauss (V/G), is a function of N and can be maximized by choosing N appropriately. The self-inductance of the coil is given
approximately by Lp =N'LO, where Lo is the inductance of one turn ( -6 nH
for these probes). The magnitude of the voltage source Vp increases as N , but
the coil inductance forms a voltage divider with the load impedance RL,and
the voltage division ratio increases as A@ for large N . It is easy to show that
the sensitivity of the probe (at a given frequency) is maximized for

and the maximum value is just

where the units of A are cm2. For a frequency of 12 MHz, then, N,
S,,,

= 8 and

=0.7 V/G. The sensitivity can be maximized as a function of the number

of turns only for a single frequency, so in practice a compromise value of N = 7
was chosen.

5.3.4 Experimental Probe Response

A small Helmholtz coil was built in order to calibrate the magnetic probes.
It was necessary to keep the length of the current-carrylng conductors much
smaller than a wavelength a t the frequencies being used. Each coil of the
Helmholtz pair was a one-turn loop wire with a &ameter of 7.6 cm; the pair
yelded a field of 0.24 G / A (gauss per ampere of current flowing through the
coil) in the central region between the coils. The Helrnholtz coil current was

monitored with a hlgh-frequency current transformer a t the input to a coil.
Magnetic probes could be inserted through guides either along the axis or
between the Helmholtz coils, allowing calibration of all three coils in a triplecoil probe. A typical frequency response curve for a probe is shown in Figure
5-21. The expected response from equation 5.2 is also plotted and agrees well

with the experimental results.
Isolation between the coils of a triple-coil probe is a measure of the actual
orthogonality of the coil axes as well as intra-coil capacitive coupling. Isolation
between two coils can be defined as the ratio, expressed in db, between the output of one coil when the external field is parallel to its axis and the output of
the other coil (which ideally would be zero). Isolation between the B, and Be
coils was typically 45 db; between the Bz and 3, coils, 30 db.
Large electrostatic fields in the plasma may couple to the magnetic probe
coils (whch are not balanced) and generate spurious signals. The silver coating described earlier encloses t h e coils i n a continuous grounded electrostatic
sLLIG
~-laU,
; greatly reducing the coil-plasma capacitance which can iead. to pick-up.
However, the effect of the silver lining on the magnetic signals must be considered. A perfect conductor encasing the magnetic probe coils would shield
out external fields entirely. If, however, the conductivity and thckness of the
silver layer are chosen so that the s h n depth in the layer is much larger than
its thickness, then the layer will have a negligible effect on the magnetic fields
being measured.
The effectiveness of the shiel&ng was tested in the laboratory. Copper foil
was wrapped tightly around the last few centimeters of the Pyrex envelope of
the probe shaft. An RF potential was applied between the foil and the probe
coaxial cable shield emerging from the end of the stainless shaft. The electrostatic coupling factor was defined as t h e ratio, expressed in db, between the
potential applied to the foil cap and the output from a magnetic probe coil

(terminated, as always, in 50 R). An unshielded probe exhibited a coupling factor, a t 12 Mhz, of 64 db. Shelding the probe with a silver coating with a conductivity of about 1 R/square improved the coupling factor to 90 db. The frequency response of the probe was unchanged by the shelding, as expected,
since the skin depth in the silver layer ( - 1 rnm) was much larger than the
thickness of the layer (-0.03 mm).

5.4 Langmuir Probes
Langmuir probes [Langmuir and Mott-Smith, 19241 have long been used to
measure electron temperature, plasma density, and floating and space potentials [Chen, 19651. In its simplest form, the probe consists of a n insulated wire,
whose tip is exposed, which is inserted into the plasma and biased with respect
to the walls of the confinement chamber. The bias is varied (either swept or
from shot to shot) and the I - V characteristic is recorded, from which the
various plasma properties are inferred.
Although the measurement is very simple to perform, the results must be
interpreted with considerable caution. The presence of the probe perturbs the
plasma (as discussed in section 5.3.1) and there is some evidence that currentdrawing probes may generate plasma fluctuations [Lecioni, 1968; Schmidt,
19681. The probe theory which is used to interpret the I - V characteristic is
very complicated in the presence of a magnetic field [Chen, 19651. Some
observed effects of a strong magnetic field are a decrease in the ratio of electron to ion saturation currents and a disappearance of a sharp knee in the

I - V characteristic w h c h ordinarily occurs a t the plasma space potential
[Sato, 19721. Probes have nevertheless been used even in the edge plasma of
large tokamaks [Budny, 19821. The usual approach is to use field-free probe
theory, provided pi <

(where pi is the ion Larmor radius and pp is the

cylindrical probe radius), and to correct for the area of the probe (since the
plasma tends to stream along the magnetic field lines, the effective area of the
probe is reduced). It should be recognized, however, that interpretation of the
probe characteristic requires the assumption of a dwtribution function, and
the usual simple Maxwellian may not be adequate in a tokamak environment
(due to mirroring, for instance).
Probe results in this thesis include measurements of entire I- V characteristics as functions of radial position (section 6.3.1) and, for the probe
mounted on the side of the ceramic-insulated loop antenna, as functions of RF
antenna current (section 6.3.2). Also investigated were the effects of inserting
an RF antenna on the ion saturation current of a &stant probe and the
dependence of the antenna probe floating potential on various parameters.
The probe tips were made from 1.0 mm 0.d. tungsten rod and had an
exposed length of 3.8 rnm. The tungsten rod (about 5 cm long) was crimped to
the inner conductor of a 2.18 rnm 0.d. semi-rigid coaxial cable. The probes
used for radial profile scans were insulated with an a l l ~ ~ i tube
n a which was in

turn inserted inside a stainless steel tube (3.0 mm 0.d.) as shown in Figure
5-22. The alumina insulator was recessed about 1 mrn from the end of the
stainless tube, and the tungsten was well centered in order to prevent arcing to
the stainless tube when drawing large probe currents. The Langmuir probe
whch attached to the side of the ceramic-insulated loop antenna (section
5.1.3) was built in the same way except that an additional ceramic tube

covered the stainless steel jacket to prevent RF currents from flowing along it.
The probes were connected via coaxial cable to a 0.2 F capacitor which was
charged to the desired bias voltage. Care was taken in the routing of the signal
cables to insure that ground-loop signals were negligible. Probe current was
measured either by the voltage drop across a 1.0 hZ resistor in the output line
or with a current monitor which had a n I - t

saturation product of about

1 A-sec. The probe I - V characteristic was mapped out by varyng the bias
f r o m shot to shot.

5-5 RF Plasma Current Monitar
5.5.1 Introduction
A current probe was constructed in order to investigate the dstribution of

R F current flowing in the tokamak plasma near an RF-excited electric field
antenna. The probe had to be capable of operating a t RF frequencies, be
sufficiently sensitive, and be compatible with the plasma plasma environment.
T b s section describes the design and construction of such a probe, based on
t h e familiar Rogowski coil [Rogowsk, 19121.
The Rogowski current monitor consists of a uniform toroidally-wound coil
which measures some function of the total current passing through its apert u r e and has the advantage that no physical contact with the current-carryng
medium is necessary. The Rogowski coil, in various forms, has been used
extensively in plasma physics research, primarily to measure the large
currents flowing through various magnetic field coils. It is also routinely
wrapped around tokamaks, in the poloidal direction, and used as a diagnostic
of the total plasma current [DeMichelis, 19781.
There has, however, been very little use of the Rogowski coil to directly
measure current distributions inside a plasma; indeed only four references in
the literature have been found. The earliest was a Russian group [Golovin et
al., 19581 which used three fixed concentric Rogowski coils in a cyhndrical
Z-pinch experiment to obtain information on the plasma current radial &stribution. Two groups have discussed the use of a Rogowski monitor in mapping
the plasma current distribution inside an MPD (magnetoplasmadynamic) arc
jet [Schneiderman and Patrick, 1967; Belinski, 19731. finally, an Italian group

[Piperno and Solaini, 19751 has also used a small Rogowsla coil to probe the
current density profile in a 2-pinch device. In all of the above cases, the
Rogowski coil was used to investigate the background (i.e., low-frequency or
DC) component of the current; no attempt was made to observe plasma

currents associated with high-frequency waves.
h k e all physical probes introduced into a plasma, the Rogowsla coil does,
to some extent, perturb the plasma and thus the current it is trylng to measure. As with other probes, the extent and effect of this perturbation are
difficult to assess. The c o m e n t s made earlier in relation to magnetic probes

(section 5.3) apply here as well. As a minimum. condition, it is necessary for
the global plasma properties (density, current, etc.) to remain largely
unaffected by the insertion of the probe. Also, since the Rogowski coil measures current passing through its aperture, it is desirable to maximize the
aspect ratio of the coil while still keeping the major diameter small compared
to the plasma characteristic size. In a tokamak, the plasma flows largely along
the magnetic field lines, and so to minimize the perturbation of the plasma
flowing through the aperture of the coil, the normal to the plane of the aperture should be oriented parallel to the background magnetic field. If the normal to the plane of the aperture was oriented perpendicular to the field, the
only plasma passing through it would be that due to cross-field diffusion
(although that in itself could suggest a useful experiment).

5.5.2 Design of the Current Monitor
A schematic diagram of a simple Rogowski coil is shown in Figure 5-23.

The toroidal solenoid consists of N turns, uniformly spaced, and the return
lead is brought back through the winding (in order to prevent a net single loop
whch would respond to -).

at

From Faraday's law, the EMF induced around

FIG.5-23. Schematic of a simple Rogowsh coil.

FIG. 5-24. Sketch of the diffusion of magnetic field lines through the
Rogowski coil shield. a) Flux is excluded from the aperture for t < < T ~ .
b) Flux penetrates the aperture for t >> T ~ .

any one turn is approximately

where B is the magnetic field, Ei is a unit vector arected along the minor axis
of the torus a t the position of the turn, A is the area of the loop and mks
units are being used. Then the total voltage induced around the Rogowski coil
is just the sum over all of the turns,

For large N ,the sum can be approximated by an integral:

where S is a closed surface bounding the closed curve C , and L = NAl is the
major circumference of the toroidal coil. The second term in equation 5.7 is
due to displacement current and is ordinarily neglected, although in some special circumstances it can be important [Klein, 19751. The displacement
current due to a charged particle beam traveling towards a Rogowski monitor
has been investigated by Stygar and Gerdin [1902], who show that the major
effect is an increase in the rise time of the output signal (on the order of
1 nsec for a 25 keV beam). The effect is negligible for conditions llkely to be

encountered in the Caltech tokamak. The induced voltage around the coil then
becomes

where I is the total current passing through the aperture of the torus.
Hence, in its simplest form, the Rogowski coil may be modeled as a voltage
source, given by VR, in series with the self-inductance of the coil, LR, and terminated by a load impedance impedance RT. The resistive load RT used here
was always a 50 R terminated coaxial cable. The time response of the Rogowski

r,
coil is then, with t h s model, limited only by the inductive time constant, -.
RT

Since the coil is designed to look for RF currents, we assume that all fields and
currents vary as e+iwt. The output voltage, Vo , is then given by

hence the sensitivity of the current monitor is a linear function of frequency if
the inductance is sufficiently small ( wLR << RT).
In practice, several other effects must be considered in the design of the
coil. Large currents are often accompanied by large electrostatic potentials,
and capacitive coupling to the coil windings can lead to spurious output signals, particularly at high frequencies. To protect against this problem, high
frequency current monitors are usually encased in a toroidal conducting shell
whch is then grounded. A perfect shield, however, excludes all time-varying
magnetic flux from the interior of the torus where the coil is situated; the
shield is therefore slotted toroidally in order to allow Be to enter. The shield
also performs another useful function: it prevents any z-component of the RF
magnetic field from penetrating to the inside of the coil where it could otherwise generate spurious signals due to inhomogeneities in the win&ng [Anderson, 19711. A grounded shield, however, could substantially alter the current
distribution within the plasma by providing an alternate route for current flow.
For t h s reason, the shielded Rogowski coil built for these experiments was

encased in an insulating shell; displacement current through the shell was
negligible.
The high frequency behavior of the coil departs from that given by equation 5.9 when resonances due to the distributed capacitance between the turns
of the coil, and between the coil and the sheld, become important. A more
realistic model of the device, that of a distributed delay line, has been analyzed
by several authors [Cooper, 1963; Nassisi and Luches, 19791 who also &scuss
methods of increasing the coil self-resonant frequencies. The resonant frequencies of the coil built here, however, turned out to be much hgher than the
operating frequencies of interest even without t a h n g special precautions.

A more important consideration, from the point of view of the Rogowsh
coil as a tokamak diagnostic, is the effect of the coil and its shield on the
confining magnetic fields. Although it is desirable for the shield to prevent the
penetration of any RF components of Bz, it is crucial to allow the penetration
of the low frequency (essentially static) external confining field (Bzo,3,)

into

the coil aperture. If the conductivity of the shell were infinite, even the low frequency confining fields would be excluded by the surface currents which would
be set up in the sheld. The result would be that the net confining field would
bend around the coil and none of it would thread the aperture (Figure 5-24);
consequently, plasma would be excluded as well. It is therefore important to
make the diffusion time of the confining magnetic field lines through the shield
short compared to the characteristic duration of the tokamak shot. In the
present design, this was accomplished by making the shield from sufficiently
for diffusion of
thin copper (0.25 mm). The time constant ( T ~appropriate
field lines through a conducting cylindrical shell whose axis is parallel to the
field is given by Lovberg [1965]: +rd = p r f j o f 2 , where ,u and o are the permittivity and DC conductivity of the material, respectively, r is the radius of the
cylinder, 6 is the wall thickness, and mks units are being employed. Using the

dimensions of the Rogowski coil which was built for the Caltech tokamak, the
above yields rd =0.13 msec. Perhaps a more appropriate method for calculation of the diffusion time of the field lines through the shield, given its
geometry, is to calculate the inductive time constant L / R for a solid ring
whose cross-sectional area is the same as the total cross-sectional area of the
shield, and whose 0.d. is the same as that of the shield. T k s approach yelds a
similar value for the diffusion time:

-rd

= 0.15 msec. Hence the diffusion time

of the field lines through the shell is short compared to the plasma lifetime of
10 -20 msec.

5.5.3 Construction of the Current Monitor
The mechanical design of the current monitor had to take into account
some special constraints: it had to be compatible with the high vacuum
environment (i.e., not outgas) and had to be capable of withstanding the
plasma and the heat deposited by it. The construction of a monitor which
satisfied these zritzria is shorn inFigures 5-25 and 5-26.
The coil was wound with 100 equally spaced turns of #34 AWC tefloninsulated wire on a ceramic (Macor) ring of square cross section (31 mm 0.d. x
1.6 rnm x 1.6 mm). The return lead was brought back underneath the winding

through a groove in the outer face of the ring. The electrostatic shield,
machined from copper bar, was made in two pieces (a body and a cover) and
had a wall thickness of 0.25 rnm. The pieces of the shield, when joined together

by a pressure fit, left a 0.5 mm gap between them a t the inner wall (see Figure
5-25). The coil lay within the body of the shield, and one of its output leads
was joined to the inner conductor of a 2 m long, 3.58 mm o.d., semi-rigid 50 R
coaxial cable which fit through a hole in the outer edge of the sheld. The
other output lead, and the outer conductor of the coaxial cable, were joined to
the copper shield body. The three connections were made with minute

/MACOR

RlNG

MACOR RlNG WOUND
WITH #30 AWG
TEFLON- INSULATED

SEMI-RIGID COAX
(3.6 mm O.D.9
FIG. 5-25. Schematic hagram of RF plasma current monitor

FIG. 5-26. Construction of RF plasma current monitor. a) Macor ring wound
with 100 turns of #34 AWG teflon-insulated wire. Return lead is brought back
around coil through groove in ring. Coil is installed in one half of electrostatic shield. b) Split electrostatic shield in place. Gap (0.5 mm) between two
halves of s h e l d is visible.

FaG. 5-26, continued. c) One half of ceramic shell attached to electrostatic
shield. d) Completed ceramic insulation of current monitor.

4.

5.

6.

7.

LOG (FREQUENCY (HZ) 1

FIG. 5-27. Frequency response of current monitor.

amounts of a hgh-melting point solder, and the assembly was throughly
cleaned and degreased. The electrostatic shield containing the coil was then
insulated with a close-fitting two-piece ceramic shell, both to limit the temperature rise of the teflon insulation of the wire inside and to minimize the
effect of the grounded sheld on the RF current drstribution withn the plasma.
The shell was machined from Macor and had a wall thickness of 1.6 mm. The
overall dimensions of the finished current monitor were 38 mm 0.d. x 25 mm
i.d. x 7.5 mm t h c k . The cover of the ceramic shell was affixed to the body of
the copper shield with several small dots of high-vacuum epoxy. The epoxy was
kept centered between the i.d. and the 0.d. of the ring, well away from the
mating surfaces of the Macor pieces which had been lapped to a smooth finish
(this insured that the plasma was kept well away from the epoxy).
The sensitivity of the coil was calibrated from 1 kHz to 100 MHz with a
coaxial fixture similar to that described by Vignone [1968]. The result is shown
in Figure 5-27; no serious resonances are present within t h s range. The coil
response begins to depart from linearity with frequency a t about 20 MHz,
whch is consistent with calculations using equation 5.9 (the inductance of the
coil in its shield was about 0.3 pH).

5.5.4 Rotatable Antenna Fixture
The current monitor probe was built in order to try to investigate the
current distribution near an electric field type RF antenna. This was mechanically difficult, however, due to the awkward geometry and small ports of the
tokamak. One approach would have been to simply rotate the Rogowski coil
around the antenna, but t h s would not have allowed the Rogowski coil major
axis to remain parallel to the background confinement field (which was largely
in the toroidal direction, since the poloidal magnetic field generated by the
plasma ohmic-heating current was less than 10% of the toroidal field a t the

outer wall). Instead, a small antenna was designed to rotate about a fixed
current probe.

A schematic and photographs of the assembly are shown in Figures 5-28
and 5-29. The transmitting antenna was simply a short cylindrical plug of
copper (1.0 cm 0.d. x 1.0 cm long) which was connected to the center conductor of a 50 R semi-rlgid coaxial cable. The coaxial line passed through a
vacuum flange which was attached to the end of a 1 m long, 3.81 cm 0.d. stainless steel tube that supported both the transmitting antenna and the current
monitor. The support tube passed through the same &fferentially-pumped
double O-ring seal flange that was used for the other RF antennas; the tube
could thus be rotated and linearly translated. The semi-rigid coax was bent as
shown in the diagram to offset it from the central axis of the support tube.
About 7 cm of the coax adjacent to the copper antenna was encased in a Macor
sheath to prevent the shield of the coax from exciting waves or driving current
in the plasma (since the shield of the cable was grounded to the tokamak wall
some 12 cm away, the shield a t the antenna end of the cable was not necessarily at wall potential), The offset bends were made appropriately to maximize the diameter of the circle that the copper plug traced out when the support tube was rotated, while still allowing the assembly to fit through the
tokamak port when the copper plug was oriented vertically above the current
monitor ( t h s circle diameter turned out to be -8.9 cm).
The semi-rigid cable from the current monitor passed through a vacuum
O-ring seal which was welded to the flange on the end of the support tube;
hence it could be rotated or translated independently of the transmitting
antenna. The copper plug antenna could therefore be rotated around the
Rogowski coil while keeping the orientation of the coil fixed. Angular scales
attached to the support tube and to the end of the semi-rigid coax from the
current monitor

allowed accurate setting of

the orientations of

the

FIG. 5-29. Rotatable antenna assembly, showing two positions of
the transmitting probe as it rotates around the current monitor.

transmitting antenna and the current monitor. The transmitting probe connector was attached, via a short length of flexible coax, to the output of the
impedance matching box; the output of the Rogowski monitor led to an
amplifier in the R F electronics cabinet.
In operation, the transmitting probe was fixed radially a t a position
corresponding to the probe tip being located 3.0 cm past the inner wall of the
tokamak. No outgassing or contamination problems were observed from the
transmitting probe. The Rogowski coil, however, required substantial conditioning through discharge-cleaning and tokamak shots. Eventually it could be
inserted approximately 3.8 cm past the wall without affecting the global properties of the plasma (corresponding to the entire coil being just past the inner
wall of the tokamak).

CWTER6
Impedance Measurements

A fundamental diagnostic of the interaction between the plasma and the

RF antenna is the antenna impedance. The antenna may launch waves and
excite cavity resonances in the tokarnak plasma and may interact with local
plasma through particle collection; these effects in turn change the input
impedance of the antenna. Knowledge of the antenna input impedance and its
variation with plasma parameters is necessary for the design of impedancetransformation networks which efficiently couple energy to the plasma and can
also illuminate the physics of the coupling process.
This chapter presents the results of impedance stucfies made on the series
of RF antennas described in the preceding chapter. The experimental
approach to the measurements is described in detail, and the transformation
necessary to relate the measured impedance to the impedance a t the input to
the coupling structure is developed, Typical tokamak shots and associated raw
impedance data are &splayed for each of the five antennas.

Density-

dependent background loading (apparently unconnected with wave excitation)
is clearly seen with all of the uninsulated antennas; the insulated and shielded
antennas show peaks on the input loading resistance associated with eigenmode production. Studies of the effects of various parameters (plasma density, antenna insertion, excitation frequency, toroidal magnetic field, and
antenna current) on the complex input impedance are presented.
Langmuir probe 1-V curves (at DC) taken with and without high power RF

antenna excitation are discussed, and the dramatic change of the floating
potential near an insulated RF antenna is noted. Finally, a simple model is
presented which explains some features of the observed loading of the uninsulated antennas solely in terms of particle collection.

6.1 Experimental Method
6.1.1 General Considerations
The antenna impedance 2, is defined here as 2, = Va/ I,, where Va and I,
are the RF voltage across and current through the antenna a t a specified point
(an e f i W t time dependence is assumed, and all quantities are complex). A convenient point a t which to measure the impedance for comparison with theoretical analysis is a t the input to the antenna coupling structure, although in
practice this may not be feasible.
RF excitation systems generally consist of the same three elements,
whether they be for h g h power heating or low power wave-launching experiments: a coupling structure (referred to here as the RF antenna), an
impedance-matchng system, and an RF generator (figure 6-1). In order to
prevent unwanted radiation and power loss, the three sections are connected
by transmission lines of various kinds. The RF antenna is usually close to or in
the plasma, and power must be conducted to it through a transmission line
wbch includes a vacuum-air transition.
The impedance which the antenna presents to the matching system is, of
course, a function of the plasma interaction with the coupling structure and in
general changes with plasma parameters. The RF generator, on the other
hand, is almost always designed to operate into a specified load impedance,
equal to the internal impedance of the generator Zg (usually 50 Q). To optimize power transfer between the generator and the antenna (crucial for

TOKAMAK
TRANSMISSION
LINE

ANTENNA
FEEDER
IMPEDANCE

GENERATOR

MATCH ING
NETWORK

zm

ANTENNA

FIG. 6-1. General configuration for RF heating or wave propagation experiments, showing the generator, impedancematching network, antenna feeder tube, and antenna. The
, is measured a t the input to the antenna
impedance 2
feeder tube; the impedance is later transformed to the
antenna-end of the feeder tube.

plasma heating experiments), the matclung system must transform the
impedance of the antenna so that Z, = Zg*, where the " *" represents the comp l e x conjugate.
The antenna input impedance in the absence of a plasma, Zo =Ro+iXo,is
largely reactive for magnetic loop antennas but inevitable losses from the
finite conductivity of the metals used in the antenna, from connecting joints,
a n d from eddy currents induced in the stainless steel tokamak ports all contrib u t e to Ro. It was difficult in practice to keep the effective antenna background resistance much below 0.1 0.
The change in antenna impedance due to the presence of a plasma, AZ,,
generally has both real and imaginary (resistive and reactive) components:

AZ, =AR, ti&.

P, =

The power

I, 1 (Ra+A&), where I

dmsipated in the impedance

Z,

is then

is the peak RF antenna current. The- term

-211 I, / R~ represents energy lost in joule heating of the antenna, while the
t e r m -1 I, / A&

represents additional power dissipated due to the presence of

t h e plasma. This adhtional power will occasionally be referred to here as
power dissipated "by the plasma", although the destination of t h s power is not
always clear. The power may be deposited in the plasma bulk via damping of
waves launched from the antenna; it may heat the plasma surface; it may,
through some interaction with local plasma, simply be heating the antenna
itself.
The ratio of the power dissipated by the plasma to the power input to the

R F antenna, a measure of power transfer efficiency, is just A&/(Ro+AI&).
Thus to reduce losses in the antenna structure to, say, 25% of the total dissipated power, the ratio

Ro
must be kept less than0.33. ICRF experiments
ARa

are usually characterized by small values for AR,, typically ranging from a few

tenths of an ohm to a few ohms. Efficient coupling of power to the plasma is
possible, then, provided some care is taken to minimize losses in the antenna.
Similarly, losses in the transmission line section connecting the antenna with
the matchng system, and in the matchng network itself, must be kept small.
Ohmic losses in the antenna, transmission line, and matching network
arise from joule heating of the conductors. Both the RF resistance of the
current-carrylng conductors (governed by the skin effect) and contact resistance a t mechanical connections contribute to these losses. The distribution of
--

high frequency current density in a conductor is given by j

, where x is

the distance into the conductor measured from and perpendicular to its surface. 6, called the skin depth, is defined by 8=l/-

where ,u and a are

the permeability and conductivity of the material, respectively, and it has been
assumed that 6 <conditions, the particular geometry of the cross section of the conductor is
unimportant; the current flows largely in a layer of thickness 6 a t the surface
of the conductor. It is the surface area that determines the high-freqiieiicy
resistance of a conductor, hence physically wide straps are used to interconnect elements in the impedance-matctung network. Note that the surface
resistance of copper a t a frequency of 16 MHz is about 1 mQ/square.

6.1-2 Experimental Approach

A simple and direct approach was used in these experiments to measure
the complex antenna impedance. The magnitudes of the RF antenna current

(I,) and voltage ( T/,) and the relative phase between the two (a,)

were

measured at the input to the short transmission line section (the antenna
feeder tube) which connected the impedance-matching system to the antenna;
the experimental arrangement is shown in Figure 6-2. The current and voltage

IMPEDANCE

FEEDER TUBE

HIGH POWER
FIBER OPTIC

DETECTOR

LINEAR RF
DETECTORS

PHASE (I ,V)

liml

TOKAMAK
TRIGGER
MODULE

ivml

FIG. 6-2. Schematic of the impedance measurement experiment. The R F current and voltage and the phase between
the two are measured on the antenna-side of the matching
network, a t the input to the feeder tube.

signals were first passed through variable attenuators to reduce the signal levels, if necessary, and were then led to buffer amplifiers which served to present
a constant (50 R) impedance to the voltage and current monitors. The outputs
of the buffers were passed through bandpass filters (with typical bandwidths of
1 MHz) to reduce any harmonics on the signals, then to power splitters which

led to the amplitude and phase detectors. The characteristic impedance of all
of the system components was 50 Q. For low-power experiments, the buffer
amplifiers were replaced with +30 db wide-band RF amplifiers (TRW CA3020).
The impedance looking into the transmission line section was then given
vm

by Z, = -ei'",

and the power dissipated by the antenna and its feeder tube

rrn

was just P, = -VmI,cos(prn).

The primary advantage of this approach was

that the measurements were made on the antenna side of the matchng network and were thus independent of the tuning of the network. The measurements were simple to interpret since there was no need to transform the measured impedance across a high-ratio impedance-matchng network in order to
find the antenna impedance.
The reflection-coefficient method used by Hwang [1980] measures the
complex reflection coefficient ( p ) a t the input (i.e.,the generator side) of the
matching network; a dwectional coupler is used to measure the forward and
reflected wave amplitudes on the transmission Line and the phase between
them. The input impedance a t the location of the coupler can then be found:

Z* =Zc &,
1 -p

where 2'' is the characteristic impedance of the transmission

line (in which the directional coupler is located) leadmg from the generator to
the matchng network. This method requires the matchng network to be reasonably well tuned to the RF antenna in order for p to be determined with
sufficient accuracy. The method also requires accurate characterization of the
impedance- transformation properties of the matching network for each new

setting of the network. Since the transformation ratios across the network for
the components of the impedance are large (the resistive part of the generator
impedance is 50 R, while that of the antenna impedance is typically < 1 R; the
reactive part of the generator impedance is typically < 1 hl, while that of the
antenna impedance may be 15-30 R), small errors in determining the
transformation properties of the matching network can lead to significant
errors when mathematically relating the measured impedance to the
impedance on the antenna-side of the network. On the other hand, the
reflection-coefficient method does have greater sensitivity to the relatively
small changes in the imaginary part of the antenna impedance that occur during the plasma shot.
The independence of the impedance measurements from the particular
setting of the matching network in the experiments described here means that
the only effect of the matchng network tuning is on the level of power
delivered to the plasma.

6.1.3 Antenna Feeder Tube Transformations

The short transmission-line section which connects the RF antenna structure with the matching network does transform the impedance of the antenna
somewhat, and it is desirable to unfold this transformation. The feeder tube is
a coaxial section which also serves as the vacuum feedthru. The voltage and
current measurements are made in air, at the end of the feeder tube which
terminates in the matching network housing. The geometry of the feeder section is precisely known; hence the impedance transformation between the
point at which the voltage and current monitors are located and the other end
of the feeder tube (where the antenna coupling structure is located) can be
derived from transmission-line theory.

The feeder transmission line consists of

concentric copper tubes

separated by a glass insulating tube (section 5.1); the geometry of a crosssection of the line is shown in Figure 6-3. The characteristic impedance of
such a line is easily calculated. The distributed capacitance is found to be
C=

27-r~~

Farad/ meter,

and the distributed inductance is

where EO and po are the permittivity and permeability of free space, respectively, and

is the permittivity of the glass tube. The characteristic

impedance is then Z, =-,
The line was
vph = 1/m.

and the phase velocity of a wave on the line is
uniform except near the end of the tube a t the

vacuum break. Two &fferent feeder transmission lines were actually used in
the experiments: a low-impedance section (2,-- 15 0) used with the magnetic
loop antennas, and a hgher impedance section ( 2, -99 0) used with the electric field antennas. The dimensions of the two sections are given in Flgure 6-3.
The effect of the electrically very short transition region at the vacuum
break can be modeled by a series of transmission-line sections of different
characteristic impedances. Consider the main transmission-line section, with
characteristic impedance Z,,, and let it be terminated a t its end by the
antenna coupling structure with impedance Z,,.

Then Z,,

is related to the

input impedance of t h s section ( Z,,) by

where 1 , is the length of the section and

is the wavelength in the section

F'IG. 6-3. Geometry of the cross-section of the main section
of the antenna feeder tube. For the low-impedance feeder
tube, a = 1 . 2 7 c m , b=1.40cm, c=1.59cm, d = 1 . 7 5 c m .
For the high-impedance feeder tube, a = 0 . 3 2 cm,
b = 1.40 cm, c = 1.59 em, d = 1.75 cm. The dielectric constant of the glass was -4.8.

[Chipman, 19681. The transformation of impedance across any section may be
written as

where

The impedance a t the input to any section j

may then be related to the

impedance at the antenna-end of the line by

The low-impedance line, for example, is modeled by four serial sections, shown
in Figure 6-4, and the antenna impedance is given by Z,, = F1(F2(F3(F,(Z,,)))),

where Z,, is the actual quantity measured in the experiment.
The transformation can be simplified considerably because the lengths of
sections 2 , 3, and 4 are very small compared to a wavelength. For section j , we
have

provided pi9jlj << 1, and the inverse transformation is then given by

Thus ad&ng a short section of transmission line to the input of the line is
equivalent to adding a small amount of inductive or capacitive reactance in
series with the input, depending on whether the input impedance is less than
or greater than the characteristic impedance of the added section.

IMPEDANCE MATCHING BOX
RF CURRENT
MONITOR

<- - -

SECTION 4

SECTION 3

SECTION 2

SECTION 1

FIG. 6-4. Geometry of the antenna feeder tube used to calculate
the impedance transformation relations, showing the separation
into four sections. The area marked "A" is the 2.54 cm o.d, copper
tube used for the low-impedance line; this tube is absent for the
high-impedance line. Za, represents the measured impedance,
Za, , the antenna impedance.

'\
ANTENNA COUPLING
STRUCTURE

The low-impedance line was used only with the loop antennas and, for all
experiments with these. the relation

/ Zai/ ZGiI << 1 was satisfied ( j=2.3.4).

Therefore, the transformation for the short sections can be approximated as

- iZci pj Lj , and, for the composite transform,

The above equation is then used with equation 6.4 to relate the measured
impedance (Za4)to the antenna impedance (Z,,). The small sections each act
as a series imaginary impedance proportional to frequency, i.e., as an inductance with value ZCfLj d m .
Some examples of the specific transformations between the measured
impedance ( Z,

= Z,,) and the antenna impedance (Zao)are shown for the high

and low impedance lines in Figures 6-5 and 6-6. Note that the ratio of the real
part of the antenna impedance to the real part of the measured impedance
depends somewhat on the imaginary part of the measured impedance and is
generally between one and two for the range of parameters appropriate to the
experiments described here. The transformation of the imaginary part of the
measured impedance depends slightly on the real part of the measured
impedance; the overall effect is largely to add a constant reactance to the
antenna impedance.
The measured impedance during the plasma shot, for the low-impedance
line and a loop antenna, typically varied over the range: Re(Z,)

0.1 -2.0 Q,

Im(Zm)%15.-20. 0. Therefore the phase measured between the voltage and
current. pm = tan-'[lrn(~,)/~e(Z,)]

. was quite close to 90 degrees and varied

by only a small amount ( 0.1 -4.0 degrees) during the plasma shot. A sensitive
and noise-free phase detection system was needed for this measurement; the
detector is discussed in detail in section 4.6.

Ra
(OHMS)

0.

5.

10.
R,

15.

20.

(OHMS)

Xa

(OHMS)

30.

50.
X,

(OHMS)

F'IG. 6-5. Examples of the transformation between the measured
impedance ( & =R, +iX,) and the impedance a t the input to the
antenna coupling structure ( Z,,= 4 +i&) for the hgh-impedance feeder
tube. a) R, vs. & for various values of &, b) & vs. ITC, for various
values of R, .

LOW IMPEDANCE FEEDER

Ra

(OHMS)

R,

(OHMS)

X,

(OHMS)

Xa
(OHMS)

F'IG. 6-6. Examples of the transformation between the measured
impedance ( Z m = R m + i X , ) and the impedance a t the input to the
antenna coupling structure ( Za, = Ra +t& ) for the low-impedance feeder
tube. a) R, vs. IZ, for various values of &. b) A& vs. & for various
values of R, .

In summary, the voltage-current-phase measurement of impedance close
to the RF antenna eliminates the problem of accurately characterizing the
hgh-ratio impedance-matching network.

The remaining

transformation

across the feeder tube from the measurement point to the antenna coupling
structure is independent of the tuning of the matching network and leads to
small transformation ratios.

6.1.4 Data Collection and Analysis
Data collection was accomplished with the sixteen-channel LeCroy A-D system and the four-channel Biomation waveform recorder as described in section 3.4. During impedance measurements, the following signals were always
recorded: one-turn voltage, plasma current, soft-UV signal, microwave interferometer output, forward and reflected RF voltages (from the dwectional
coupler), and the RF voltage, current, and phase, measured a t the input to the
antenna feeder tube, Additional channels often recorded included the visible
light (from a photodetector) and X-ray (from a PIN diode) sigaals, the outputs
from various magnetic probes, and the gas-puff valve voltage waveform. In
addition, the hydrogen gas pressure and the settings of the toroidal field, vertical field, horizontal field, and ohmic-heating field power supplies were
recorded (manually) for each shot.
Each recorded channel generally consisted of 1024 equally-spaced samples
of eight-bit resolution. The data from each plasma shot were transferred,
between shots, from the transient recorder memory, through an BI-11/03
computer, to storage on 8-inch floppy disks. The data archtved from all of the
experiments described in t h s thesis comprise some 3000 tokamak shots and
occupy

40 megabytes of disk space,

A typical tokamak run began with extensive discharge cleaning (section
3.6) to lower the level of impurities adsorbed on the tokamak walls. After

sufficient cleaning, as judged from the quadrupole residual gas analyzer,
tokamak shots were started. The tokamak usually required several shots to
settle into a repeatable discharge mode. The piezoelectric gas-puff valve voltage waveform was then adjusted, together with the voltage and timing of the
tokamak power supplies, to try to achieve the desired plasma current and density evolution. After considerable experience, combinations of the various
parameters were found which usually led to fairly reproducibly discharges with
t h e desired plasma properties. Three factors which acted to make the
discharge less reproducible were identified: problems with the gas-puff valve
(due to the piezoelectric diaphragm "sticking"), irregular firing of the ohmicheating or preionization power supply ignitrons, and insufficiently clean walls.
Data collection was begun once the tokamak was firing in a mode which
was both reproducible and had similar properties to previously recorded
discharges with which the data run was to be compared. Generally three to
seven shots were recorded for each setting of the experimental parameters
(e.g.,RF antenna position, excitation frequency, etc.). The tokamak could typically be run for two to five hours before impurity adsorption on the walls
increased to the point where the plasma shots became noticeably shortened
and less reproducible. The collected raw data sets were then plotted out in
compact form (eight data channels per page) to provide for rapid review of the
experiment.
Calibration curves for the directional coupler, the RF detectors, and the
current and voltage monitors were used in the computer programs which were
written to analyze the data; the curves were interpolated as necessary using a
cubic-spline fit. The signal attenuation due to the connecting coaxial cables,
buffer amplifiers, bandpass filters, and power splitters was also measured a t
each frequency used and entered into the programs.

Analysis of a series of similar plasma shots for the dependence of the
impedance on a particular parameter yelds a set of data points for each setting of the experimental parameter. The mean of the set is usually plotted as
the experimental point associated with the particular experimental parameter;
the scatter in the data is indicated with "error bars" of amplitude kg,where rr
is the standard deviation of the data set. In general, the scatter in the data
from shot to shot was greater than the actual "error" due to, for instance,
instrumental background noise, calibration error, or quantization error (from
the eight-bit resolution of the data system). In cases where the scatter in the
data did not vary much as the experimental parameter was changed, the vertical lines denoting the scatter may be omitted from some of the plotted
points. High-frequency noise was also present on some of the signals associated with the impedance measurement; this appeared to be due to density
fluctuations in the plasma intercepting the antenna. T h s noise was sometimes
reduced using some simple &gital filters [ H a m i n g , 19771.

6.2 Impedance Measurement R d t s
This section presents the results of impedance measurements on the five
antennas described earlier. For each antenna, an impedance analysis for a
typical shot is first shown. Eight of the raw data channels are displayed, followed by the results from the analysis program: the antenna current and voltage, the power dissipated by the antenna, and the real and imaginary parts of
the complex antenna impedance (all transformed from the point of measurement to the antenna-end of the feeder tube). Following t h s display of a
characteristic plasma shot, the dependence of the real and imaginary parts of
the antenna impedance on various parameters is shown. The curves drawn
through the experimental points in these plots are generally fits to low-order
polynomials and are meant to "guide the eye". Comparison of some of the data

- 244 with theoretical models is discussed in section 6.4,

6.2.1 Bare h o p Antenna
Raw data channels for a typical tokamak shot using the bare loop antenna
are shown in Flgure 6-7. The excitation frequency was 12 MHz, the toroidal
magnetic field on axis was 4.0 kG, and the front edge of the antenna loop was
2.5 cm past the tokamak wall. The matchng network was tuned to match the
generator impedance to the antenna impedance; from the forward and
reflected voltage signals it is apparent that, during the plasma shot, most of
the incident RF power is being absorbed by the antenna-plasma system. The
line-averaged density measured by the microwave interferometer ramps up
almost linearly after the gas puff begins (at t -4 msec); the density fall is also
fairly linear. The phase signal decreases by

4 degrees as the plasma density

increases; the antenna current and voltage vary by a factor of almost two during the plasma shot. Although the amplitude variations of the RF antenna
current

voltage Were considerably less than the dynamic range of the

phase detector (whch was greater than 20 db for a change in output
corresponding to 0.1 degree), the independence of the phase detector output
on the input signal amplitude was checked in each experiment under actual
operating conditions and signal levels.
The raw phase signal provides a useful in&cation of the resistive part of
the antenna loadmg impedance. Because the antenna current and voltage
track each other fairly closely, it follows that / Z, / does not change very much
during the plasma shot. Also, the phase a(,
during the shot. Writing a(,

where terms of order

= n / 2 -&(a,

remains close to n/2 radians

it follows that

and higher have been dropped. The antenna

II

I-TURN VOLTAGE (20 V F.S.)

I1

PLASMA CURRENT (40 KA F.S. )

PHASE (I ,V)

(DEGREES)

FORWARD VOLTAGE (200 V F.S.)

REFLECTED VOLTAGE (200 V F.S.)

TIME (MSEC)

TIME (MSEC)

FIG. 6-7. Raw data from a tokamak shot using t h e bare loop antenna. Here
w / 2 7 ~ = 1 2 MHz, B0=4.0kG, and the leading edge of t h e loop was 2.5 cm past the
tokamak wall. Note the good Impedance match indicated by the small reflected voltage
signal during the discharge. The density curve is plotted from analysis of the
microweve interferometer signal.

TENNA CURRENT
(PEAK AMPS)

ANTENNA VOLTAGE
(PEAK VOLTS)

POWER D I S S I P A T E D
(WATTS)

TIME (MSEC)
FIG. 6-8. Results for the antenna current, voltage, and dissipated power from
the computer analysis of the raw data for the tokamak shot shown in Figure 6-7.
The results have been transformed from the measurement location to the
antenna-end of the feeder tube. Note the dissipation of -30 watts in the background antenna resistance after the discharge has ended.

RE( Z 1
(OHMS)

IM( Z 1
(OHMS)

DENSITY X 1012

( c M -> ~

TIME (MSEC)

FIG. 6-9. Real and imaginary components of the bare loop antenna impedance,
and the plasma density, for the tokamak shot shown in Figure 6-7. The analysis
program has tranformed the impedance to the antenna-end of the feeder tube..

RE( Z >
(OHMS)

10.

IM( Z >
(OHMS)

5. -

o.,

12.

8. -

4. -

DENSITY X l o 1 *
( c M - >~

0.

TIME (MSEC)
FIG. 6-10. Results of the impedance analysis for another tokarnak shot where
the l e a l n g edge of the loop was 5.0 cm past the tokamak wall; again
W / ZTT= 12 MHz and Bo= 4.0 kG. The increase in the real part and the decrease
in the reactive part of the impedance are both larger than for the previous shot.

loading resistance, measured a t the input to the antenna feeder tube, is then
just &

/ 2, / Cip, . Hence, the raw phase signal in Figure 6-7, viewed upside-

down, is approximately proportional to the antenna loa&ng resistance.
The RF antenna current and voltage, transformed to the input of the
antenna coupling structure using the computer analysis program, are shown
in Fgure 6-8; also shown is the net power chssipated by the antenna and
plasma. Note that the transformed antenna current is somewhat larger than
the raw current signal, and the transformed voltage is substantially smaller
than the raw signal (because of the large inductance added to the antenna
loop by the feeder tube).
The real and imaginary parts of the antenna impedance (transformed to
the end of the antenna feeder tube), together with the plasma mean density,
are plotted in Figure 6-9. Note that the real part of the impedance does in fact
follow the form suggested by the raw phase signal. The background loading
resistance (due to ohmic losses in the system) is -80 mR. The loading resistznce increzses with density hut appears to saturate a t % - 5 - 7

x 1012

somewhat before the density reaches its peak. The peak of the loading resistance is nearly 1.0 R.
The imaginary (reactive) part of the antenna impedance is

-R

in the

absence of plasma; this merely reflects the inductance of the antenna loop.
The reactive part of the impedance decreases during the plasma shot, the
decrease being roughly proportional to the increase in plasma density. The
relative noise on the plot of Im(2) in Figure 6-9 is large, i.e., comparable to the
change in the mean of the signal. T h s is typical of the reactive impedance
measurements, and the source of the noise appears to be related to plasma
density fluctuations. As with fluctuations associated with Langmuir probe ion
saturation current measurements, the relative level of the fluctuations in the
imaginary part of the antenna impedance decreases as the antenna is moved

farther into the tokamak plasma.
The results of the impedance analysis for another shot where the front
edge of the antenna loop was 5.0 cm past the tokamak wall are shown in figure
6-10. Although the density evolution is comparable to that of the previous shot

(Fgure 6-7), the density seen by the antenna is now higher because it is
further into the plasma; the plasma density profile is peaked in the center of
the tokamak. The real part of the antenna loading impedance is now larger,
reaching a peak of

1.4 R . The evolution of the antenna resistance with den-

sity, however, shows a new feature. After the initiation of the gas puff a t

t -4.5 msec, the density begins an approximately linear increase, The
antenna resistance also begins to increase, but reaches a peak a t t -6 msec,
corresponding to a mean plasma density of n -5.5 x 10" em-'.

After this

point, the antenna resistance slowly decreases until it reaches a minimum of

1 . 2 R at a time corresponding to t h e density maximum. This behavior is

approximately repeated during the density fall, with the antenna resistance
first increasing and then decreasing as the density decays.
The imaginary part of the antenna impedance in the absence of the
plasma is now -7.5 R . The slight increase in this background reactance, as
compared with the shot shown in Figure 6-9, is due to the loop being slightly
less shielded by the tokamak port and walls. The antenna reactance again
decreases with the appearance of plasma, the decrease still being roughly proportional to the plasma density increase. The magnitude of the change is
somewhat larger for this shot, the reactance reaching a minimum of -4.5 Q a t
a time corresponding to the density peak. Note that the relative fluctuation
level of the antenna reactance is smaller than that of the previous shot.
The variation of the complex antenna impedance with different parameters is shown in Figures 6-11 to 6-15. In each case, the data points shown in
the plots represent the mean of the experimental points accumulated from a

DENSITY X 10 l2 (CM -3)

FIG. 6-11. Variation of the complex impedance of the bare loop antenna with
plasma density, as measured with the microwave interferometer.
w/27~=12MHz, Bo=3.5kG, and the antenna was inserted 5.1 cm past the wall.
The dashed line represents the background impedance measured without any
plasma,

i. 5

WITH PLASMA
NO PLASMA

ANTENNA DISTANCE PAST WALL (CM)

ANTENNA DISTANCE PAST WALL (CM)

FIG. 6-12. Variation of the complex impedance of the bare loop antenna with
insertion of the antenna past the tokamak wall. The horizontal axis is the &stance from the front edge of the antenna loop to the tokamak wall. Here
W / Zx = 12 MHz, Bo = 4.0 kG, and the data set was taken a t a plasma density of
n - 6 . 0 ~1012 ern-'. The background impedance (without plasma) is also plotted.

1.5-

12.

0 WITH PLASMA
NO PLASMA
, -?

Ln

1.0-

,-.

ar: 0.5w

0. 0
5.

10.
15.
FREQUENCY (MHz)

0 WITH PLASMA
NO PLASMA

20.

84.*--

0.
5.

10.
15.
FREQUENCY (MHz)

20.

FIG. 6-13. Variation of the complex impedance of the bare loop antenna with
excitation frequency. The front edge of the antenna was 3.8 cm past the
tokamak wall, the data set was taken a t a plasma density of n -8.0 x 1012
and Bo = 4.0 kG. The background impedance (without plasma) is also plotted.

r 1.0I
5!
0 WITH PLASMA

0 WITH PLASMA
NO PLASMA

2.
6.

0. 0
0.

4.

TOROIDAL FIELD (KG)

TOROIDAL FIELD (KG)

FIG. 6-14. Variation of the complex impedance of the bare loop antenna with
toroidal magnetic field. The front edge of the antenna was 5.1 cm past the
tokamak wall, the data set was taken a t a plasma density of n ~ 8 .x11012cm-',
and W / ~ T T =12 MHz. The background impedance (without plasma) is also plotted.

1.5

10.

r 1.0-

Gj

tl

F?

C1

;;i

C1

(21

0 WITH PLASMA
0.5NO PLASMA

0. 0 ,

5. -

0 WITH PLASMA
NO PLASMA

..

10

100

0.
1000

RF ANTENNA CURRENT (PEAK AMPS:

10

100

1000

RF ANTENNA CURRENT

FIG. 6-15. Variation of the complex impedance of the bare loop antenna with RF
excitation current. The front edge of the antenna was 3.8 cm past the tokamak
wall, the data set was taken a t a plasma density of n - 8 . 1 x 10'' cm-',
W / 271 = 12 MHz,and Bo = 4.0 kG. The background impedance (without plasma) is
also plotted.

series of similar tokamak shots; vertical bars denoting the scatter of the data
are also shown for some points. The lines marked "no plasma" are the results
of the loadmg measurements when no plasma was present. In all cases, the
impedance results plotted have been transformed to the antenna-end of the
feeder tube, using the expressions derived in section 6.1.3.
The variation of the complex antenna impedance with mean plasma density (as measured with the microwave interferometer) is shown in Figure 6-1 1.
For this experiment, the front edge of the antenna loop was 5.1 cm past the
tokamak wall, the excitation frequency was 12 MHz,and the toroidal magnetic
field was 3.5 kG. The variation of antenna impedance with plasma density was
measured during the density rise following the initiation of the gas puff. The
real part of the impedance initially rises as the density increases, then reaches
a peak and decreases slightly. The imaginary part of the antenna impedance
decreases monotonically as the plasma density rises.
Variation of the antenna impedance with insertion of the antenna past the
tokamak wall is p!otteO in Figure 8-12. Eere the excitation freqilencjr was
12 MHz,the toroidal magnetic field was 4.0 kG, and the data set was taken during the density rise following the gas puff, when the density reached a value of
n - 6 . 0 ~i012

The real part of the antenna impedance increases mono-

tonically as the antenna is inserted into the plasma, although its rate of rise
decreases as it moves farther in. In the absence of plasma, the imaginary part
of the impedance increases slightly as the antenna moves into the tokamak
chamber due to decreased shelding of the antenna loop. In the presence of
plasma, the imaginary part of the itnpedance decreases somewhat as the
antenna is inserted.
The effect of the excitation frequency on the complex antenna impedance
is shown in Figure 6-13. The front edge of the antenna loop was now 3.8 cm
past the tokamak wall, the toroidal magnetic field on axis was 4.0 kG, and the

data set was taken a t a plasma density of n -8.0 x 1012cm-' . Because of the
extensive calibrations that had to be carried out at each new excitation frequency, in general only three &fferent frequencies were used in these investigations: 8, 12, and 16 MHz. Even with only three points on the plots, some
trends are evident. The real part of the antenna impedance clearly increases
with frequency, changing by a factor of

1.7 as the frequency doubles. The

background resistance does not change significantly over this frequency range
(it increases slightly, due to the skin effect). Both the antenna background
reactance and the reactance in the presence of plasma increase as the frequency is raised, the increase being approximately linear within the accuracy
of the measurement.
The variation of the antenna impedance with the toroidal magnetic field is
displayed in Figure 6-14. For this experiment, the front edge of the antenna
loop was 5.1 cm past the tokamak wall, the excitation frequency was 12 MHz,
and the data set was taken at a mean plasma density of n ~ 8 .x1l~l%m-'.
Within the scatter of the data points, there is no significant change in either
the real or imaginary parts of the impedance as the toroidal magnetic field
varies from 3.0 to 5.0 kG.
Finally, the variation of the antenna impedance with the level of RF
current exciting the antenna is shown in Figure 6- 15. The excitation frequency
for this experiment was 12 MHz, the toroidal magnetic field was 4.0 kG, and the
data set was taken at a point during the plasma shots corresponding to a
mean density of n -8.1 x 1012cm-'.

The RF antenna current (transformed, as

always, to the end of the antenna feeder tube) was varied from

1 A (peak) to

-300 A (peak) using a variety of amplifiers ranging from the 10 watt EN1

amplifier to the 100 hlowatt tetrode amplifier. Over this range of excitation
current, no significant change is observed in either the real or imaginary part
of the complex antenna impedance.

6.2.2 Bare Plate Antenna
Typical raw signals from a tokamak shot using the bare plate antenna are
shown in Figure 6-16. For this shot, the excitation frequency was 12 MHz,the
toroidal magnetic field on axis was 4.0 kG, and the leading edge of the copper
antenna plate was 3.8 cm past the tokamak wall. Also, the antenna was
oriented so that the plane of the surface of the plate was normal to the
toroidal magnetic field lines. The antenna was operated with an R F bypass
choke to ground (inside the impedance-matching box) so that the plate was a t
DC ground potential (or more accurately, a t tokamak wall potential). The

impedance-matching network was tuned for a fairly good match: the reflected
wave voltage (from the directional coupler) is a small fraction of the forward
wave voltage during most of the plasma shot. The phase signal now has a
rather different form from that observed with the bare loop antenna. As the
plasma density increases, the phase also increases, i.e., moves closer to
90 degrees. This means that the real part of the impedance decreases as the

density rises, in contrast to the increase observed with the bare loop antenna.
Results from the computer analysis of the RF antenna current and voltage
and the hssipated power (transformed to the end of the feeder tube) are
shown in Figure 6-17. Again, the voltage a t the input to the antenna coupling
structure is substantially smaller than that measured by the voltage divider
because of the inductance associated with the antenna feeder tube. The
antenna current is largely unchanged by the transformation.
The complex impedance a t the antenna-end of the feeder tube is plotted
along with the density evolution for this shot in Figure 6-18. The real part of
the impedance scales approximately as the reciprocal of the mean plasma density, and its peak value (at low density) is -5 R. At low density, the imaginary
part of the antenna impedance is quite noisy; these fluctuations hminish as
the density increases during the shot. The mean value of the imaginary part of

1-TURN VOLTAGE ( 2 0 V F.S.)

PLASMA CURRENT (40 KA F.S. )

FORWARD VOLTAGE' ( 1 5 0

v F'.
S. )

REFLECTED VOLTAGE ( 1 5 0 V F. S. )

10
T I M E (MSEC)

ANTENNA VOLTAGE ( 7 5 0 V F.S. )

20

10
T I M E (MSEC)

20

F?G. 6-16. Raw data from a tokamak shot using the bare plate antenna. Here
o/ 2n = 12 MHz, Bo= 4.0 kG, and the leading edge of the plate was 3.8 cm past the
tokamak wall. The plane of the surface of the plate was normal to the toroidal field
lines, and the antenna was a t DC ground potential.

15.

10.-

5. -

ANTENNA CURRENT
(PEAK AMPS)

0.

150.

100. -

50. -

ANTENNA VOLTAGE
(PEAK VOLTS)

0.

5.

10.

POWER DISSIPATED
(WATTS)

0.

15.

TIME (MSEC)
FIG. 6-17. Results for the antenna current, voltage, and dissipated power from
the computer analysis of the raw data for the tokamak shot shown in Figure 716. Note that results can only be plotted for the period during the plasma shot
when the raw phase signal is on-scale (i.e.,80" < p, < 90").

8.

6. -

RE( Z )

4. -

(OHMS)
2. -

IM( Z 1
(OHMS)

DENSITY X 1 0 "

( c M - ~>

T I M E (MSEC)

FIG. 6-18. Real and imaginary components of the bare plate antenna
impedance, and the plasma density, for the tokamak shot shown in Figure 7-16.
The analysis program has transformed the impedance to the antenna-end of the
feeder tube.

the impedance decreases slightly as the density increases.
Several effects are noticed when the antenna is floated by removing the DC
grounding choke. With no application of RF power and without changing any of
the tokamak settings, if the antenna is changed from a grounded to a floating
configuration, the plasma density and current evolution change markedly. Figure 6-19 shows the density and current evolution for two successive tokamak
shots during an otherwise reproducible run: for the first shot, the antenna
was grounded; for the second, it was floating. In both cases, the plane of the
antenna plate was normal to the toroidal magnetic field lines, and the leading
edge of the plate was 3.8 cm past the tokamak wall. The density during the
second shot (antenna floating) reaches a peak value of only -60% of that
observed during the first shot (antenna grounded). The plasma current is
somewhat larger for the shot with the antenna floating, and the discharge
duration is considerably longer. Increasing the gas puff with the antenna floating can simulate conditions that occur with the antenna grounded, i.e., adding
more gas externally with all other parameters &xed increases the peak density
during the shot and decreases the discharge duration.
The origin of this effect is not understood. One explanation involves
absorption of hydrogen by the copper plate. When the antenna is floating,
there is no net current to it. Since the floating potential is, in these experiments, always negative with respect to t h e wall potential, a grounded antenna
draws a net DC electron current (typically 20-30 amps). It is possible that
surface heating due to this current, or some other effect of the increased electron flux strilang the surface, causes absorbed hydrogen in the metal to evolve
more rapidly during the discharge. This would then increase the density, for a
constant gas puff, with respect to the case with the antenna floating.
The effects of grounding or floating the bare plate antenna, and of changing its orientation with respect to the toroidal magnetic field, on the plasma

TIME (MSEC)

FIG. 6-19. Evolution of the plasma density (from the microwave interferometer
signal) and plasma current f o r two successive tokamak shots: one with the bare
plate antenna grounded, the other with the antenna floating. The leading edge
of the plate was 3.8 cm past the tokamak wall and the plane of the surface of
the plate was normal to the toroidal axis; also, Bo = 4.0 kG.

ANTENNA FLOATING

ANTENNA GROUNDED

TIME (MSEC)

PERPENDICULAR

TIME (MSEC)

FIG. 6-20. Effects of grounding and floating the bare plate antenna on the real part of the
antenna impedance and on the plasma density evolution, with the plane of the surface of the
plate oriented perpendicular and parallel to the toroidal axis. Here w / 27r = 12 MHz,
Bo=4.0 kG, and the leading edge of the plate was 2.5 cm past the tokarnak wall. a) Antenna
grounded. b) Antenna floating.

density evolution and the real part of the antenna impedance are shown in Figures 6-20a and 6-20b; no significant effect was observed on the imaginary part
of the impedance. These shots were taken with an excitation frequency of
12 MHz and a toroidal magnetic field on axis of 4.0 kG, and the leading edge of
the antenna plate was 2.5 cm past the tokamak wall. In these figures, the
curves labeled "perpen&cularU refer t o the antenna orientation where the
toroidal magnetic field lines are normal to the plane of the surface of the
antenna plate; the label "parallel" refers to the orientation where the toroidal
field lines are parallel to the surface of the plate.
The antenna was at DC ground potential in Figure 6-20a; on the two successive shots shown, the orientation of the antenna was changed from perpendicular to parallel leaving all other parameters unchanged. Thus, for the
grounded antenna, changing the orientation from perpendicular to parallel
increases the plasma density at a given time and also increases the antenna
resistance a t a given density. When the antenna is floating (Figure 6-20b),
changing the orientation from perpendicular to parallel decreases the density
but still increases the antenna resistance, by a somewhat larger ratio than
that observed for the grounded antenna. Note that the gas puff was increased
between the shots with the antenna grounded and those with it floating in
order to counteract the density decrease discussed previously. For all of the
following plots of impedance data from the bare plate antenna (Figures 6-21 to
6-25), the orientation of the plate was perpendicular to the toroidal field direction.
The variation of the complex antenna impedance (transformed to the
antenna end of the feeder tube) with plasma density is shown in Figure 6-21.
The excitation frequency was 12 MHz, the toroidal field on axis was 4.0 kG, the
leading edge of the antenna plate was 3.8 cm past the tokamak wall, and the
antenna was a t DC ground potential. The real part of the impedance decreases

6.

ANTENNA GROUNDED

ANTENNA GROUNDED

4. -

2. -

0.
0.0

0.5

1. 5

1. 0

DENSITY X 10 '"CM

-3)

FZG. 6-21.Variation of the complex impedance of the bare plate antenna with
plasma density. Here o/2n= 12 MHz, Bo= 4.0kG, the leading edge of the plate
was 3.8 cm past the tokamak wall, and the antenna was a t DC ground potential.

ANTENNA DISTANCE PAST WALL (CM)

ANTENNA GROUNDED

ANTENNA DISTANCE PAST WALL (CM)

FIG. 6-22.Variation of the complex impedance of the bare plate antenna with
insertion of the antenna past the tokamak wall. The horizontal axis is the &stance between the leading edge of the plate and the tokarnak wall. Here
W / 2n = 12 MHz, Bo= 4.0kG, and the data set was taken a t a plasma density of
n -8.1 x 1012cmP3; also, the antenna was at DC ground potential.

3.

cn

2.-

r\

CY

10.

1.-

0 ANTENNA

FLOATING

A ANTENNA GROUNOED

0.
5.

10.

j5-:/

0 ANTENNA FLOATING
V ANTENNA GROUNDED

0.
5.

15.

20.

10.

15.

20.

FREQUENCY (MHz)

FREQUENCY (MHz)

FIG. 6-23. Variation of the complex impedance of the bare plate antenna with

excitation frequency, for both grounded and floating configurations. The leading edge of the plate was 3.8 cm ast the wall, the data set was taken a t a
plasma density of n 8.1x 1 0 ' ~cm$. and Bo = 4.0 kG.

3.

12.

cn

2.-

8. -

CY

1.-

0 ANTENNA FLOATING

4. -

o ANTENNA FLOATING

C-l

V ANTENNA GROUNDED

0.
0.

V ANTENNA GROUNDED

2.

4.

TOROIDAL FIELD (KG)

6.

0.
0.

2.

4.

6.

TOROIDAL FIELD (KG)

FIG. 6-24. Variation of the complex impedance of the bare plate antenna with
toroidal magnetic field, for both grounded and floating configurations. The leading edge of the plate was 3.8 cm ast the wall, the data set was taken a t a
plasma density of n B 1x 10" cmg, and o/2a = 12 MHz.

8.-

0 ANTENNA

cn

6.-

12.

V ANTENNA GROUNDED

C;

5!

FLOATING

4. -

8.-

E?

4i!L

F T" $YI 0

P.

a 2. -

0.
10

-'

10

lo2

RF ANTENNA CURRENT (PEAK AMPS)

4.0 ANTENNA

FLOATING
V ANTENNA GROUNDED

0.
lo-'

10

lo2

RF ANTENNA CURRENT (PEAK AMPS)

FIG. 6-25. Variation of the complex impedance of the bare plate antenna with
RF excitation current, for both grounded and floating configurations. The leading edge of the plate was 3.8 cm past the tokamak wall, and the data set was
taken a t a plasma density of n li8.l x 10'' cm-3 ; also w / 2n = 12 MHz and
Bo=4.0 k c . Note that the measured antenna current has, as in all of the plots,
been transformed to the current at the antenna-end of the feeder tube.

substantially and monotonically as the plasma density increases; the imaginary part of the impedance exhibits a small decrease.
Variation of the antenna impedance with insertion of the antenna past the
tokamak wall is plotted in Figure 6-22. The excitation frequency was 12 MHz,
the toroidal field was 4.0 kG, and the data was taken a t a mean plasma density
of n 28.1 x 10" em-';

also, the antenna was at DC ground potential. The real

part of the antenna impedance decreases abruptly as the antenna enters the
plasma; the rate of this fall decreases as the antenna moves farther into the
tokamak. The reactive part of the impedance also decreases as the antenna is
inserted, but the relative decrease is much smaller.
The dependence of the complex impedance of the bare plate antenna on
excitation frequency is shown in Figure 6-23. The toroidal magnetic field was
4,O kG, the leading edge of the antenna plate was 3.0 cm past the tokamak wall,
~ .with
and the data set was taken a t a plasma density of n ~ 8 .x11012~ m - As
the bare loop antenna, both the real and imaginary parts of the antenna
impedance increase as the frequency is raised. The results are presented for
the antenna both floating and grounded. The imaginary part of the impedance
for the two cases is almost identical; however, the real part of the impedance,
and the rate of rise with frequency, is greater for the floating antenna than for
the grounded antenna.
Variation of the complex antenna impedance with toroidal magnetic field
is shown in Figure 6-24. Here the excitation frequency was 12 MHz,the leading
edge of the antenna plate was 3.0 cm past the tokamak wall, and the plasma
density was n -0.1 x 10'' emT3. Again, the results are presented both for floating and grounded antennas. The real part of the impedance for the floating
antenna shows a small decrease as the toroidal magnetic field is raised from
3.0 to 5.0 kG; none of the other plots show signdieant changes within the limits of the measurement.

Finally, the change in antenna impedance with antenna excitation current
is presented in Figure 6-25, for both floating and grounded antennas. The excitation frequency for this experiment was again 12 MHz,the toroidal magnetic
field was 4.0 kG, the antenna insertion was 3.8 cm, and the data set was taken
a t a plasma density of

n = 8.1 x 1012 cm-'.

The imaginary part of the

impedance for both the floating and grounded antennas shows no significant
change with excitation current; the real part of the impedance of the
grounded antenna also shows little change. The real part of the floating
antenna impedance, however, shows a dramatic increase as the current
increases past

10 amps. The maximum current that was driven with the

floating antenna was -30 amps; as the current increased further, the antenna
resistance increased beyond the range over which the impedance matching
network was useful.

6.2.3 "T" Antenna
Rciw signals from E typical tokamak shot using the "T" antenna are S~OP.TI
in Figure 6-26. For this shot, the excitation frequency was 12 MHz,the toroidal
magnetic field was 4.0 kG, and the leading edge of the "T" was 3.8 cm past the
tokamak wall. The general features of the RF signals are similar to those of
the bare plate antenna (Figure 6-16). In particular, note that the phase signal
has the same general form with respect to the density evolution. In all experiments using the "T" antenna, the cross-bar of the "T" was oriented vertically,
i.e., with its axis perpendicular to the direction of the toroidal field; also, the

" T was grounded via an RF choke in the impedance-matching box.
Results of the computer analysis of this shot for the antenna current and
voltage and the dissipated power are shown in Figure 6-27. The large increase
in power dissipation during the density rise is due to the increase in antenna
current as the antenna impedance becomes better matched to the generator.

1-TURN VOLTAGE ( 2 0 V F.S.)

90

PLASMA CURRENT (40 KA F.S. )

- .-

PHASE (1,V)

(DEGREES)

ANTENNA CURRENT ( 2 2 A F.S. )

FORWARD VOLTAGE ( 3 5 0 V F.S.)

ANTENNA VOLTAGE ( 8 8 0 V F.S. )

REFLECTED VOLTAGE ( 3 5 0 V F.S. )

- -

- -

TIME

(MSEC)

TIME

(MSEC)

FIG. 6-26. Raw data from a tokamak shot using the "T" antenna. Here o / 2 n = 12 MHz,
Bo= 4.0 kG, and the front edge of the cross-bar of the " T was 3.8 cm past the wall. The
cross-bar was oriented perpendicular to the toroidal field &rection, and the antenna
was a t DC ground potential.

ANTENNA CURRENT
(PEAK AMPS)

ANTENNA VOLTAGE
(PEAK VOLTS)

POWER DISSIPATED
(WATTS)

TIME (MSEC)
FIG. 6-27. Results for the antenna current, voltage, and dissipated power from
the computer analysis of the raw data for the tokamak shot shown in Figure
6-26. The results are plotted for the period during the plasma shot when the
raw phase signal is on-scale (i.e.,80' < (om < 90" ).

6. -

RE( Z >

4. -

(OHMS)

2. -

0.

15.

10.-

IM( Z >
(OHMS)
5. -

0.t

5.

10.

15.

T I M E (MSEC)

FIG. 6-28. Real and imaginary components of the "T" antenna impedance, and

the plasma density, for the tokamak shot shown in Figure 6-26. The analysis
program has transformed the impedance to the antenna-end of the feeder tube.

Note that the power incident on the matchng network, related to the square of
the forward voltage signal shown in Figure 6-26, also increases during the density rise. The power outputs of most of the RF generators were dependent on
the load impedance which they saw and thus changed in a complicated fashion
as the antenna load impedance changed during the plasma shot; only the EN1
broadband power amplifiers were capable of delivering a constant incident (or
forward) power to an arbitrary load.
The complex impedance analysis for t h s characteristic shot is shown in
Fgure 6-28; the results are quite sirnilar to those shown for the bare plate
antenna (Figure 6-17). Again, the real part of the impedance decreases substantially as the density rises; the relative decrease of the imaginary part of
the impedance is somewhat smaller.
The variation of the antenna impedance with plasma mean density is
shown in Figure 6-29. For this experiment, the excitation frequency was
8.5 MHz,the toroidal magnetic field was 3.6 kG, and the front edge of the "T"
was 3.5 ern past the tokama! wall. Note that both the magnitude ant! the func-

tional form of the real part of the impedance are quite similar to that found
for the bare plate antenna (Figure 6-18), with the antenna resistance varymg
approximately as the reciprocal of the plasma density. The imaginary part of
the impedance also decreases somewhat as the density increases; the magnitude of the reactance, for a given plasma density, is slightly smaller than that
found for the bare plate antenna. It should be recalled, however, that the data
set for the bare plate antenna was taken a t a frequency of 12 MHz and a
toroidal field of 4.0 kG.
The variation of antenna impedance with the insertion of the antenna
past the tokamak wall is shown in Figure 6-30. For these data points, the excitation frequency was 12 MHz,the toroidal field was 4.0 kG, and the plasma density was n -8.1 x 10" emp3. As with the bare plate antenna, the real part of

0.0

0.5

1. 0

1.5

0. 0

0. 5

1.0

1.5

DENSITY X l o i 3 ( c M - ~ )

DENSITY X 1013 ( c M ' ~ )

FIG. 6-29. Variation of the complex impedance of the "T" antenna with plasma
density. Here o/2n = 8.5 MHz, Bo = 3.6 kG, and the front edge of the cross-bar of
the " T was 3.8 cm past the tokamak wall.

9.

2.

4.

ANTENNA DISTANCE PAST WALL (CM)

6.

0.l
0.

2.

4.

6.

ANTENNA DISTANCE PAST WALL (CM)

FIG. 6-30. Variation of the complex impedance of the "T" antenna with insertion
of the "T" past the tokamak wall. The horizontal axis is the distance between
the leading edge of the cross-bar of the "T" and the tokamak wall. Here
W / 27~
= 12 MHz, Bo = 4.0 kG, and the data set was taken at a plasma density of
n~8.1~10'~

5.

15.
FREQUENCY (MHz)
10.

20.

5.

15.
FREQUENCY (MHz)
10.

20.

FIG. 6-31. Variation of the complex impedance of the "T" antenna with excitation frequency. The leading edge of the "T" was 3.8 cm past the tokamak wall,
the data set was taken a t a plasma density of n - 8 . 1 ~10'' emS, and
Bo= 4.0 kG.

TOROIDAL FIELD ( K G )

TOROIDAL FIELD (KG)

FIG. 6-32. Variation of the complex impedance of the "T" antenna with toroidal
magnetic field. The leadmg edge of the " T was 3.8 cm past the tokamak wall,
~ ~ the
, excitathe data set was taken a t a plasma density of n -6.1 x 1 0 ' ~ c mand
tion frequency was w / 2n = 8.5 MHz.

10

100

1000

RF ANTENNA CURRENT (PEAK AMPS)

FIG. 6-33. Variation of the complex impedance of the " T antenna with antenna
excitation current. Here o/2n = 12 MHz, Bo= 4.0 kG, the leading edge of the "T"
was 3.8 cm past the tokamak wall, and the data set was taken at a plasma density of n N 8.1 x 10" ~ m - ~ .

the impedance decreases abruptly as the antenna leading edge moves past the
tokamak wall, but does not decrease below -2 R. The dependence of the imaginary part of the impedance on the antenna insertion is somewhat different
from that found for the bare plate antenna. For the "T" antenna, the reactive
part of the impedance increases only slightly (from -7.3 CI to -8.6 0) as the
antenna leadng edge is inserted from 0.5 to 5.6 cm past the wall. In contrast,
the reactive part of the impedance of the bare plate antenna decreases by
-30% over a similar range of antenna insertion.

The effect of excitation frequency on the grounded "T" antenna is shown
in Figure 6-31. Here, again, the toroidal field was 4.0 kG, the leading edge of
the "T" was 3.8 cm past the tokamak wall, and the data was taken at a plasma
density of n -8.1 x 1012 emS.

The general trend is again evident: both the

real and imaginary parts of the antenna impedance increase as the frequency
increases.
The variation of antenna impedance with toroidal magnetic field is plotted
in Ykure 8-32. The zxcitation frequency was 8.5 MHz, the leading e6ge of the
antenna was 3.8 cm past the tokamak wall, and the data set was taken at a
~ . is virtually no change in the imagplasma density of n -8.1 x 1012 ~ m - There
inary part of the antenna impedance as the toroidal field is varied from 2.5 to
5.0 kG; the real part of the impedance, however, decreases by about 30% over
the same range.
Lastly, the variation of antenna impedance with antenna excitation
current is shown in Figure 6-33. For this experiment, the excitation frequency

was 12 Mhz, the toroidal field was 4.0 kG, the antenna insertion was 3.8 cm, and
the plasma density was again n

8.1x lo1* cmS . No significant variation of

the imaginary part of the impedance is observed as the excitation current is
varied from

2 A to

100 A; the real part of the impedance increases slightly,

but the increase is not much greater than the scatter in the data.

6.2.4 Ceramic-Insulated Loop Antenna
Raw signals from a typical tokamak shot utilizing the ceramic-insulated
loop antenna are shown in Figure 6-34. The excitation frequency for t k s shot
was 16 MHz,the toroidal magnetic field on axis was 4.0 kG, and the front edge
of the Macor box surrounding the loop was 2.5 cm past the tokamak wall. The
phase signal has a rather different form from those dzscussed previously. The
overall phase change during the plasma discharge is small: the full-scale range
of the phase plot in the figure is 2.0 degrees. As the density rises, a series of
sharp spikes in the phase signal (in the direction of decreasing phase) are
encountered. From the discussion of section 6.2.1, this implies that real part
of the loading resistance has a series of sharp peaks as the density increases;
this will be seen more clearly when the computer analysis of the data is discussed. Coincident with the negative-going peaks in the phase signal, the
antenna current and voltage also have sharp minima, decreasing by up to
-40%. The impedance-matching network was tuned to match the antenna

impedance to the generator a t the peaks in the loading resistance. The sharp
minima in the reflected voltage signal coincide with the peaks in the phase slgnal (i.e.,the peaks in the antenna resistance) and show that a t the peaks, the
reflection coefficient can be made very small. Detuning the impedancematching network so that the reflected and incident voltage signals are nearly
equal eliminates the effect of the antenna loading on the impedance which the
generator sees. In this case, the antenna current and voltage signals do not
show sharp minima, yet the phase signal is largely unchanged, demonstrating
that in fact the phase detector is not sensitive to input amplitude variations.
Results of the computer analysis of the raw data for the antenna current,
voltage, and dzssipated power, transformed to the end of the antenna feeder
tube, are shown in Figure 6-35. The sharp minima in the current and voltage
signals are evident, as are some peaks in the dissipated power. The peaks in

N S I T Y X 1012

(C M - ~ )

1-TURN VOLTAGE ( 2 0 V F.S. )

TIME

(MSEC)

TIME

(MSEC)

FIG. 6-34. Raw data from a tokamak shot using the ceramic-insulated loop antenna.
Here w/2n= 16 MHz, Bo=4.0 kG, and the front edge of the Macor box surrounding
the loop was 2.5 cm past the tokamak wall. Note that the full-scale range of the
phase plot is 2.0 degrees.

ANTENNA CURRENT
(PEAK AMPS)

ANTENNA VOLTAGE 300m
(PEAK VOLTS)

POWER DISSIPATED
(WATTS)

0.

5.

10.

15.

T I M E (MSEC)
FIG. 6-35. Results for the antenna current, voltage, and dissipated power from
the computer analysis of the raw data for the tokamak shot shown in Figure
6-34.

- 280 0.4

0.3

RE( Z 1

0.2

(OHMS)
0. 1

0. 0

.. .

15.

10.-

IM( Z >
(OHMS)
5. -

15.

10.-

DENSITY X

loi2

(cM-~
5. -

0.
0.

5.

10.

15.

TIME (MSEC)

FIG. 6-36. Real and imaginary components of the ceramic-insulated loop
antenna impedance, and the plasma density, for the tokamak shot shown in Figure 6-34. The sharp peaks in the real component of the impedance are coincident with the excitation of toroidal eigenmode resonances.

the power arise because, even though the antenna current and voltage may
have minima a t that point, the real part of the antenna impedance has a maximum.
Results of the analysis for the complex impedance are plotted in Figure
6-36. The distinct sharp peaks on the real part of the impedance (corresponding to the minima of the phase signal) are clearly seen. The background
antenna resistance is
measurement is

80 mR, and the instrumental noise associated with the

15 mR. The largest peak in the antenna resistance has an

amplitude of -0.32 R, whch is approximately four times the background loading resistance. The imaginary part of the antenna impedance does not change
significantly during the plasma &scharge, although two or three small peaks,
just above the noise level, whch coincide with peaks on the real part of the
impedance can be discerned.
The sharp peaks in the RF signals are, of course, related to the excitation
of toroidal eigenmode resonances in the tokamak cavity. Details of the wave
fie!ds associated with the eigenmodes \re presented in Chapter 7. Here we
merely note that, in general, there is a one-to-one correspondence between the
peaks seen on the real part of the antenna impedance and the peaks in the
wave fields around the tokamak as detected with magnetic probes. The magnitude of the peaks in the antenna resistance seen in Figure 6-36 can vary from
shot to shot by up to -50 %. Lowering the excitation frequency or plasma density, or raising the toroidal magnetic field, reduces the number of modes which
appear during the plasma shot (which is in agreement with the discussion of
Chapter 2) and also broadens the modes and enhances their reproducibility.
Figures 6-37 and 6-38 present the variation of the amplitudes of the peaks
of the real part of the antenna impedance with antenna insertion and frequency, respectively. For both of these plots, the density of modes during the
discharge was sufficiently high that an indwidual mode could not be

RE( Z )
(OHMS)

0.

2.

6.

4.

ANTENNA DISTANCE PAST WALL (CM)

FIG. 6-37'. Variation of the amplitudes of the peaks of the real part of the
ceramic-insulated loop antenna impedance as a function of the insertion of the
antenna. The horizontal axis is the distance between the front edge of the
ceramic box surrounding the antenna loop and the tokamak wall. The points
plotted represent the averages of the second, t h r d , and fourth peaks observed
after the initiation of the gas puff. Here w / 2 w = 12 MHz and Bo= 4.0 kG. The
antenna background resistance (without plasma) is also plotted.

0. 3 -

0. 2 RE( Z )
(OHMS)

0. 1 ANTENNA
BACKGROUND
RESISTANCE

0. 0.
5.

10.

15.

20.

FREQUENCY (MHz)

FIG. 6-38. Variation of the amplitudes of the peaks of the real part of the
ceramic-insulated loop antenna impedance as a function of the excitation frequency. The points plotted represent the averages of the second, third, and
fourth peaks observed after the initiation of the gas puff. The leading edge of
the insulating box was 3.2 cm past the tokamak wall and Bo=4.0 kG. The
antenna background resistance (without plasma) is also plotted.

confidently identfied from shot to shot. Consequently, the points plotted in
these graphs are the averages of second, third, and fourth modes observed
after the start of the density rise associated with the gas puff. Each experimental point plotted represents data from five to ten tokamak shots. The vertical bars are the standard deviation of the data set composed of the average,
for each shot, of the peak amplitudes of the three modes.
The variation of the amplitudes of the peaks with antenna insertion is
shown in Figure 6-37. The excitation frequency was 12 MHz and the toroidal
magnetic field on axis was 4.0 kG. Although the scatter in the data is large, the
peaks clearly increase in amplitude as the antenna moves into the tokamak.
Similarly, Figure 6-38 presents the data for the variation of the peak amplitudes as a function of the excitation frequency. For these shots, the leading
edge of the antenna was 3.2 cm past the tokamak wall and the toroidal magnetic field was 4.0 kG. The amplitude of the peaks is seen to increase
significantly as the frequency is raised.
Carefully csntrollin~the plasma density evolukinn by adjusting the gas
puff valve waveform to slow the rate of gas injection permits identification of
particular eigenmodes from shot to shot. The particular density a t which a
mode occurs is a function of the toroidal field and the excitation frequency,
and the eigenmode dispersion curves can be mapped out through careful
experiments as described in Chapter 7. Figure 6-39 presents the amplitudes of
the peaks on the real part of the antenna impedance, as functions of the
toroidal magnetic field, for the first five modes which occur after the gas puff.
For this experiment, the excitation frequency was fixed a t 12 MHz and the leading edge of the antenna was 3.2 cm past the tokamak wall. The scatter in the
data is large, but the amplitudes of the peaks in the loading resistance clearly
increase substantially for the modes occurring at higher density. The variation
of peak amplitude on the toroidal field strength is not significant, except

0.25

0.20 -

0. 15 -

RE( Z 1

(OHMS)
0. 10 -

------------------------------------------------------------------------

O5 -

0.002.

ANTENNA
BACKGROUND
RES l STANCE

3.

4.

5.

6.

TOROIDAL FIELD (KG)
FIG. 6-39. Variation of the amplitudes of the peaks of the real part of the
ceramic-insulated loop antenna impedance as a function of the toroidal magnetic field, for the first five modes which appear after the initiation of the gas
puff. The leahng edge of the insulating box surroundmg the antenna loop was
3.2 cm past the tokamak wall and w / 27r = 12 MHz. The curves for each mode are
labeled from 1 to 5 , in order of their appearance during the density rise. The
antenna background resistance (without plasma) is also plotted.

perhaps for the fifth mode, where a decrease in peak amplitude is noticed as
the toroidal field increases.

6.2.5 Faraday-Shielded Loop Antenna
Raw signals from a typical tokamak shot using the Faraday-shelded loop
antenna are shown in Figure 6-40. The excitation frequency for this shot was

12 MHz,the toroidal magnetic field was 4.0 kG, and the front edge of the shield
was 5.1 cm past the tokamak wall. In order to see any effect of the plasma on
the antenna impedance, the antenna had to extend fairly far into the plasma

( 2 3 -4 cm). With the antenna 5 cm into the plasma, the impurity levels began
to increase and the plasma discharge duration began to decrease. Inserting
the antenna any further than this resulted in abrupt increases in impurities,
as measured by the soft-W detector, and early disruptions of the plasma
current.
The change of the phase signal due to the appearance of plasma is now
very small: the full-scale range of the phase piot is oniy one degree. The phase
signal after the plasma terminates, at t =12 msec, shows the instrumental
noise associated with the phase detector - the rms phase noise is less than
0.04 degree. Note that the amplitudes of the raw antenna current and voltage
signals vary during the plasma shot by less than 2 db, a factor small enough to
ensure that the phase signal was not being affected by the input amplitude
variations, even a t this sensitivity. The phase signal during the discharge exhibits a small decrease which appears to be proportional to the mean density
rise; superimposed on this are several negative-going peaks which are coincident with the excitation of eigenmodes as seen on magnetic probes (and discussed in Chapter 7).
Figure 6-41 shows the results of the computer analysis for the antenna
current and voltage and the dissipated power, transformed to the antenna end

20

DENSITY X 1012
(cM'~)

1-TURN VOLTAGE ( 2 0 V F.S.)

PLASMA CURRENT ( 4 0 KA F .S )

PHASE ( I ,V)

i-r

(DEGREES)

ANTENNA CURRENT (80 A F.S .)

REFLECTED VOLTAGE ( 1 0 0 V F.S. )

ANTENNA VOLTAGE ( 1 4 0 0 V F .S. )

10

TIME

(MSEC)

10

TIME

20

(MSEC)

FIG. 6-40. Raw data from a tokamak shot using the Faraday-shielded loop antenna.
The leading edge of the ceramic box surrounding the antenna loop was 5.1 cm past
the tokamak wall, also w / 2 n = 12 MHz and Bo=4.0 kG. Note that the full-scale
range of the phase plot is 1.0 degree.

60.

40. ANTENNA CURRENT
(PEAK AMPS)

20. -

0.
300.

200. ANTENNA VOLTAGE
(PEAK VOLTS)

100. -

0.
120.

80.
POWER DISSIPATED
(WATTS)

-1

40. -

0.
0.

5.

10.

15.

TIME (MSEC)
FIG. 6-41. Results for the antenna current, voltage, and dissipated power from
the computer analysis of the raw data for the tokamak shot shown in Figure
8-40.

RE( Z 1
(OHMS)

IM( Z >
(OHMS)

DENSITY X 1012
(cM-~

TIME (MSEC)
FIG. 6-42. Real and imaginary components of the ceramic-insulated loop
antenna impedance, and the plasma density, for the tokarnak shot shown in Figure 8-40.

of the feeder tube. The plots are unremarkable; the increase of the dissipated
power with the appearance of plasma is only just above the noise level. This is
a reflection of the very small increase in the real part of the antenna
impedance in the presence of plasma. Figure 6-42 shows the actual complex
impedance resulting from the computer analysis. The real part of the
impedance follows the general form suggested by the phase signal of Figure
6-40: a small density-dependent background loading with a few small peaks
discernible from the noise. Note that the background antenna resistance is
-70 mR and that the largest loading observed during the shot is only

mR

larger than t b s . The imaginary part of the impedance shows no change above
the noise level during the tokamak discharge.
Because the loading was so small, and the reproducibility was poor, the
data sets from the Faraday-shielded antenna were not analyzed further.

6.3 Iangrnuir Probe Studies
Langmuir probes were used both to study edge plasma properties with and
without RF power applied to an antenna, and to provide the I - V curve data
for the impedance model to be chscussed in section 6.4. Construction of the
probes is described in section 5.4; the probes were operated at a fixed bias and
probe curves were mapped out by varymg the bias from shot to shot.
The basic features of an idealized Langmuir probe curve have been discussed by many authors [hngmuir and Mott-Smith, 1924; Chen, 19651. As the
probe is biased negative with respect to the actual plasma space potential a t
the probe location, virtually all of the ions whose paths intersect the probe are
collected, but fewer and fewer of the electrons have sufficient energy to reach
the probe. In this region, the net current collected by the probe is approximately given by

where n is the plasma density, e is the electronic charge, A is the probe area,

5 is the probe potential, T/, is the space potential, T, is the electron temperature, kg is Boltzmann's constant, U, is the mean electron thermal speed
(a Maxwellian distribution is assumed), and 4 is the ion sound speed (roughly
equal to the ion thermal speed if the ion and electron temperatures are equal)
[Budny, 19821. A logarithmic plot of this electron-repelling part of the probe
curve (after subtracting off the ion saturation current) then ylelds a straight

line of slope -from which the electron temperature is found. As the
k3 Te
probe bias becomes more positive than I/, , almost all of the electrons but
fewer and fewer of the ions are collected by the probe. The ion-repelling region
of the probe curve ( Vp > j / s ) is difficult to analyze theoretically, but the following simple form is sometimes useful in unmagnetized plasmas:

The ion saturation current (for I$
<<, ) is then given by I~ rt? --neA ui

whle the electron saturation current (for Vp >> I/,) is I, N neA U, . Thus. the

saturation currents are proportional to the plasma density and to the square
root of the temperature. In the presence of a magnetic field, however, the electron saturation current is usually significantly smaller than predicted by this
simple expression.
When the probe floats with respect to the plasma, the net current to it
must vanish. Because the flux of electrons striking a probe a t the space potential is much larger than the flux of ions (due to their higher thermal velocity) a
floating probe charges up to a sufficiently negative potential (with respect to

the space potential) to retard the flux of electrons and render the electron and
ion currents equal. The floating potential can be found from equation 6.1 1:

where a hydrogen plasma of equal electron and ion temperatures has been
assumed. Hence, the difference between the space potential (if it can be
identified) and the floating potential can yleld information on the electron
temperature. Unfortunately, in the presence of a magnetic field, the space
potential (usually identifled as a "knee" in the curve) often becomes obscured
[Sato, 19721, and generally only the electron-repelling part of the probe curve
is useful.

6.3.1 Probe Results Away From RF'Antenna
A typical tokamak shot showing the ion-saturation probe current is shown
in Figure 6-43. The probe tip was 4.0 cm past the tokamak wall and the probe
bias was -200 volts (-tith respect to the tokamak chamber). Note that there is
substantial noise on the signal (and this trace has already been filtered somewhat); these fluctuations are characteristic of Langmuir probe traces. The
magnitude of the current roughly follows the mean density inferred from the
microwave interferometer, except that the probe current reaches its peak
shortly before the mean density peaks. This is probably because the plasma
moves slightly inward toward the end of the discharge (as seen with the in-out
coils [section 3.51) and the Langmuir probe was located on the outside of the
torus. With proper adjustment of the vertical field power supplies, the mean
density and the probe saturation current roughly track each other during
most of the discharge. Note that the ratio of the peak mean density to the
density just before the gas puff begins is somewhat larger than the ratio of the
saturation currents a t the corresponding times. Part of the explanation for

1-TURN VOLTAGE
(VOLTS)

30.

20.

PLASMA CURRENT
(KA)
10.
0.

DENS I TY X lo1* ( c M - ~)
(FROM INTERFEROMETER)

2.
1.

PROBE CURRENT
(AMPSICM~ 1
0.

- 1.0.

5.

10.

15.

TIME (MSEC)
FIG. 6-43. A typical tokamak shot, showing the one-turn voltage, the plasma
current, the plasma density (from the microwave interferometer signal) and the
ion-saturation Langmuir probe current. The probe tip was 4.0 cm past the tokamak
wall and the bias was -200 V with respect to the tokamak chamber. The probe was
located in tokamak port #1 (cf. Figure 3-I), and the RF antenna was withdrawn
from the tokamak.

this observation may lie in the dependence of the saturation current on both
the density and the temperature. As mentioned in section 3.9, both the electron and ion temperatures of the bulk plasma are observed to decrease during
the gas puff. The edge plasma where the probe is located is probably cooled a t
least as much as the bulk plasma, leading to reduced saturation current.
A Langmuir probe I-V

curve is shown in Fgure 6-44a. Here the probe tip

was 1.0 cm past the tokamak wall, and the data points were taken when the
mean plasma density reached n

7.0 x 10'' em-'.

Fitting a n exponential to

the electron-repelling part of the curve yields an electron temperature of

9.5 eV; together with the ion saturation current, t h s implies a density of

-0.51x 1012 em-'.

Similar plots of the I-V

curve for probe insertions of

2.0 cm and 4.0 cm past the tokamak wall are shown in Figures 6-44b and 6-44c.

As the probe moves farther into the plasma, the slope of the transition region
of the curves decreases, which means that the electron temperature increases.
The ion saturation current also increases, by a larger factor than would be
expected from the temperature increase alone; hence the density must also be
increasing as the probe moves in. Plots of the electron temperature and density, derived from the I-V curves using the simple theory discussed above, are
shown in Flgure 6-45 as functions of the probe insertion. Thus, the temperature and density profiles appear to be almost linear functions of radius near
the plasma edge. At the most interior point investigated (4.0 cm past the wall),
the electron temperature was -32 eV and the density was

4.1 x 1012ern-', or

some 60% of the line-averaged plasma density as measured by the microwave
intsrf erometer.
Langmuir probe curves are perhaps most useful in establishing general
trends in plasma density and temperature. Due to the significant complexities
and uncertainties of probe theory in the presence of a magnetic field, the absolute values of plasma parameters derived from the probe traces should be

LANGMUIR PROBE ON PORT #1
PROBE 1 CM PAST WALL

- 1.
-200.

-100.

0.
100.
PROBE BIAS (VOLTS)

200.

LANGMUIR PROBE ON PORT # 1

PROBE BIAS (VOLTS)
F'IG. 6-44. Langmuir probe I -V curves. The probe was located in tokamak
port #l, and the data sets were taken a t a mean plasma density of
n -7.0 x 1012~ m - The
~ . exposed area of the probe tip was 0.12 cm2. For plots a),
b), and c), the RF antenna was withdrawn from the plasma. a) Probe tip 1.0 cm
past the tokamak wall, b) Probe tip 2.0 cm past the tokamak wall.

LANGMUIR PROBE ON PORT #1
RF ANTENNA CURRENT = 0

PROBE 4 CM PAST WALL

-200.

- 100.

0.

100.

200.

PROBE BIAS (VOLTS)

LANGMUIR PROBE ON PORT # 1
RF ANTENNA CURRENT = 4 0 0 r 5 0 AMPS (PEAK)
PROBE 4 CM PAST W A i i

-200.

- 100.

0.

100.

200.

PROBE BIAS (VOLTS)

FIG. 6-44, continued. c) Probe tip 4.0 cm past the tokamak wall. d) Probe tip
4.0 cm past the tokamak wall, and the ceramic-insulated loop antenna was inserted
3.2 cm past the wall and excited with -400 A of RF current at w / 27r = 12 MHz.

0.

5.

1.
2.
3.
4.
PROBE DISTANCE PAST WALL (CM)

4. -

3.-

5.

I-

2.-

1.-

r-(

Ln

0.
0.

1.
2.
3.
4.
PROBE DISTANCE PAST WALL (CM)

5.

FIG. 6-45. The electron temperature and density, derived from the
Langmuir probe plots using simple theory, as a function of radial
position.

regarded with a t least moderate skepticism. In this spirit, then, an investigation of the effect of insertion of the Macor-covered loop antenna on the plasma
edge density is presented in Figure 6-46. The question being addressed was
whether or not the the Macor box acted as a limiter, depressing the edge density globally around the tokamak. A Langmuir probe was mounted on tokamak
port #1 (Figure 3-2), which was 180 degrees toroidally away from the RF
antenna port, and was biased at -200 V with respect to the tokamak chamber
in order to draw ion saturation current. For each of three positions of the
Macor box (the leading edge of the box being 0.0, 2.5, or 5.0 cm past the
tokamak wall), the Langmuir probe was inserted from 0.0 to 6.0 cm past the
wall in 1.0 cm steps, and at each probe position, 3 -7 plasma shots were
recorded. It should be noted that at a probe insertion of -7 cm or greater,
the impurity levels in the plasma (observed with the soft-UV detector) began to
increase markedly, indicating contamination of the plasma by the probe. For
each plasma shot, the probe current was recorded at a mean plasma density of
n -6.0 x 1012 cm".
The results in Figure 6-46 show, surprisingly, that the ion saturation
current at a particular radial position actually increased when the Macorshielded antenna was moved into the plasma. Since it was unlikely that the
plasma temperature increased upon insertion of the Macor, this implies that
the plasma density increased, However, the differences observed between the
saturation currents when the Macor box was 2.5 or 5.0 cm past the wall were
not significant. Since the mean (line-averaged) plasma density measured by
the microwave interferometer was the same for all of the data points, t h s plot
suggests that the plasma density profile (normally peaked in the center) was
being flattened or broadened by the insertion of the Macor. However, this supposition cannot be substantiated without a reliable radially-resolved density
diagnostic (such as a multi-channel microwave interferometer). The defensible

2.

4.

6.

PROBE DISTANCE PAST WALL (CM)

FIG. 6-46. Effect of the insertion of the ceramic-insulated loop
antenna on the radial dependence of the ion saturation current of
a Langmuir probe. The probe was located in tokamak port #1
( 180" toroidally away from the RF antenna) and was biased a t
-200V with respect to the tokarnak chamber. The leading edge of
the Macor box was positioned at 0.0, 2.5, and 5.0 cm past the
tokamak wall; there was no RF excitation.

8.

conclusion of this experiment is simply that the insertion of the XiTacorinsulated antenna does n o t diminish the plasma edge density.
The effect of hgh-power RF excitation on the Langrnuir probe I-V

curve

was investigated using the probe on port # l . For t h s experiment, the probe
was inserted 4.0 cm past the tokamak wall and the bias was again varied from
shot to shot. The RF antenna used was the Macor-insulated loop antenna,
inserted 3.2 cm past the tokamak wall. The excitation frequency was 12 MHz,
the toroidal magnetic field was 4.0 kG, the RF antenna current (transformed to
the antenna end of the feeder tube) was approximately 400 A (peak), and the
data points were taken a t a mean plasma density of n -7.0 x 1012em-'.
resulting I-V

The

curve is plotted in Figure 6-44d; comparison with the curve of

Fgure 6-44c (taken with the same experimental parameters but with no RF
current) shows that RF excitation with the insulated antenna produces no
noticeable effect on the I-V curve far away from the antenna.

6-32Probe Results H e a r IS Antenna
This section presents the results of investigations with a Langmuir probe
attached to the side of the Macor-insulated loop antenna (section 5.1.3). The
tip of the probe was 1.0 cm back from the front edge of the Macor box and the
perpendicular distance from the probe axis to the Macor surface was 2 mm.
An I-V

curve taken with t b s probe, without any RF excitation, is shown

in ngure 6-47a. The leading edge of the Macor box was 3.2 cm past the
tokamak wall; hence the tip of the Langmuir probe on the side of the box was
2.2 cm past the wall. The toroidal magnetic field on axis was 4.4 kG, and the
data points were taken when the microwave interferometer in&cated a plasma
mean density of n -7.2 x lo1*

The part of the curve below the floating

potential is similar to that of Figure 6-44b (which was taken with a Langmuir
probe 2.0 cm past the wall, located 180 degrees toroidally away from the RF

.200.

0.
PROBE BIAS (VOLTS)

LANGMUIR PROBE ON RF ANTENNA
RF ANTENNA CURRENT = 4 0 0 r 5 0 AMPS (PEAK)

PROBE BIAS (VOLTS)
FIG, 6-47. I - V characteristics of a Langmuir probe mounted on the side of the
ceramic-insulated loop antenna. The probe tip was located 1.0 cm back from the
front edge of the ceramic box, which in turn was located 3.2 cm past the tokamak
wall. The data sets were taken a t a mean plasma density of n - 7 . 2 ~10'' em-',
and B o = 4 . 4kC. a) Langmuir probe curve with no RF excitation, b) Langmuir
probe curve with RF antenna current of -400 A (peak) at a frequency of 12 MHz.

antenna). The part of the curve above the floating potential, however, exhibits
some differences. The current does not appear to have saturated, even at
+200V bias, and the "knee" which appears in the other I - V

curves is no

longer clear. The ion saturation currents for the two curves are almost equal.
Neglecting possible temperature changes, this implies that the density 1.0 cm
back from the leading edge of the ceramic box is comparable to the density at
the same location in the absence of the antenna, for the same mean plasma
density.
Excitation of the RF antenna with a large current changes the Langmuir
probe characteristic markedly. Figure 6-47b shows the I-V

curve obtained

with an RF current of -400 A (peak) at a frequency of 12 MHz; the other
plasma and tokamak parameters were unchanged from those of ngure 6-47a.
The entire f-V curve appears to have been shifted to the left; the floating
potential now occurs a t
potentials below

-240 V. The ion current still appears to saturate for

-350 V, and the value of the saturation current is about the

same as for the curve taken without R F current (Figure 6-47a). The slope of
the curve in the transition region, however, is significantly smaller than for the
curve without RF current. Also, the electron current does not appear to
saturate, increasing almost linearly with the probe bias above

-100 V.

The dramatic change of the floating potential with the application of RF
current to the insulated loop antenna led to further explorations of this effect.

A typical plasma shot is shown in Figure 6-48; the antenna and tokamak
parameters were unchanged from those given above. The antenna probe floating potential remains fairly constant throughout the shot. Note that the RF

pulse begins slightly before the ohmic-heating banks fire (which is coincident
with the sudden rise of plasma current). The floating potential also decreases
immediately with the onset of the RF current which suggests the possibility
that the effect is simply electrostatic pickup by the probe. In fact, this is not

DENS I TY X lo1* ( c M - ~ >
(FROM I NTERFEROMETER)

RF ANTENNA CURRENT
(PEAK AMPS)

400*1
200.

o.,

0.
ANTENNA PROBE
FLOAT I NG POTENT I AL

0°- -

(VOLTS)

0.

5.

10.

15.

TIME (MSEC)
FIG. 6-48. Typical raw data for a tokamak shot, showing the plasma current, the
plasma density, the RF antenna current, and the floating potential of the Langmuir
probe on the side of the ceramic-insulated loop antenna. The leading edge of the
ceramic box was 3.2 cm past the tokarnak wall, w / 2 ~ =12 MHz, and Bo=4.4 kG.

ANTENNA PROBE
FLOATING POTENTIAL
(VOLTS)

- 100.

-200.

TIME (MSEC)
100.

(b)
0.

ANTENNA PROBE
FLOAT I NG POTENT I AL
(VOLTS)

- 100.
-200.
-300.

TIME (MSEC)

(c)
ANTENNA PROBE
FLOATING POTENT I AL
(VOLTS)

100.
0.
- 100.

-200.
-300.

5.

0.

10.

15.

TIME (MSEC)
FIG. 6-49. Floating potential of the Langmuir probe on the side of the ceramicinsulated loop antenna. The leading edge of the ceramic box was 3.2 cm past the
tokarnak wall. a) Floating potential with no RF current applied to the loop.
b) Floating potential with a loop current of 400 A (peak) a t a frequency of 12 MHz.
The horizontal bar shows the duration of the RF pulse. c) Floating potential with RF
excitation and with all tokamak fields firing, but with no gas and hence no plasma.

the case. Figures 4-49a, 4-49b, and 4-49c show the antenna probe floating
potential for three different tokamak shots. In the first plot (Figure 6-49a), no
RF current is applied, and the floating potential varies during the shot by less
than 50 V. Note that these potentials are, of course, the potentials with
respect to the tokamak wall, and not with respect to the space potential. Figure 6-47b shows the antenna probe potential for a similar discharge with

400 A of RF current applied during only part of the shot. The floating poten-

tial change upon application of the RF current has a very short rise-time; the
decay-time when the RF pulse ends (at whch point the plasma is still present)
is equally short. Finally, Figure 6-47c shows the floating potential of the
antenna probe with the application of the RF current and with all tokamak
fields firing, but without any gas in the machine and consequently no plasma.
There is no discernible pickup by the probe.
The question then remains as to why the probe floating potential drops
suddenly upon the application of RF current in the shot shown in Figure 6-46,
apparently before the formation of the plasma. The answer is that there actually is a low-density plasma in the tokamak for some time before the ohmicheating bank begins to drive the large plasma current. The preionization
power supply discharges into the ohmic-heating coils several milliseconds
before the ohmic-heating bank is fired, in order to initially break down the gas
(section 3.3). This preionization can, in fact, be observed with a visible light
detector. The preionization plasma density has not been measured directly,
but it is estimated to be in the range of 1011 - 1012cm-'.

This suggests, then,

that the change in the antenna Langmuir probe floating potential with application of RF current to the antenna is rather insensitive to the plasma density.
Thrs argument is further supported by the observation that the floating potential changes little during the tokamak shot shown in Figure 6-48, while the
plasma mean density varies by a large factor during the gas puff.

The relation between the change in floating potential seen by the antenna
Langmuir probe and the magnitude of the RF current exciting the antenna
loop was also investigated. Figure 6-50 shows the floating potential of the
probe as a function of the peak RF current for two excitation frequencies: 12
and 16 MHz. The data set was taken a t a plasma density of n - 7 . 2 ~10'' cm4,
although, as mentioned before, the signal was not very sensitive to the density.
Here the leading edge of the Macor was again 3.2 cm past the wall of the
tokamak, and the toroidal magnetic field on axis was 4.5 kG. The floating
potential drops monotonically and almost linearly as the antenna RF current
increases, with the change for the excitation frequency of 16 MHz being slightly
larger than for the 12 MHz case.
Finally, the relation of the floating potential during the RF pulse to the
toroidal magnetic field is &splayed in Figure 6-51. Here the excitation frequency was 12 MHz, the RF antenna current was approximately 185 A (peak),
and the toroidal field was varied from 3.6 to 5.2 kG. Each point plotted in this
graph represents data from three to six tokamak shots. As the toroidal field
increases, for fixed RF excitation current, the change in floating potential is
observed to decrease slightly. The decrease is on the order of 25% over the
range of toroidal field used.

6.4 Discussion
6.4.1 General Impedance Characteristics

The impedance measurements on the five RF antennas present a variety of
dflerent features. Qualitatively, the results from the bare plate and "T" antennas were similar to each other, as were results with the ceramic-insulated and
Faraday-shielded loop antennas. These two sets of antennas, however, behaved
quite &fferently from each other and from the bare loop antenna. Those

LANGMUIR PROBE ON RF ANTENNA

RF ANTENNA CURRENT (PEAK AMPS)
FIG. 6-50.Floating potential of the Langmuir probe on the side of
the ceramic-insulated loop antenna, as a function of the RF
antenna current, for two frequencies: 12 and 16 MHz. The leading
edge of the ceramic box was 3.2 cm past the tokamak wall, the
data set was taken at a plasma mean density of n -7.2 x 1012
and Bo = 4.5kG.

0.

LANGMUIR PROBE ON RF ANTENNA

FREQUENCY = 12 MHz
ANTENNA CURRENT = 185 + 10 AMPS (PEAK)

-50.-

-100.-

- 150.
3. 0

4.
TOROIDAL
FIELD (KG)
5. 0

6. 0

FIG. 6-51. Floating potential of the Langmuir probe on the side of
the ceramic-insulated loop antenna, as a function of the toroidal
magnetic field. The excitation frequency was 12 MHz, and the RF
antenna current was -185 A (peak). The leading edge of the
ceramic box was 3.2cm past the tokamak wall, and the data set
was taken a t a plasma mean density of n -7.2 x 1012 cmS .

antennas with the center current-carrylng element exposed directly to the
plasma (i.e., the bare loop, bare plate and "T" antennas) exhibited a large
density-dependent loading loading resistance. The loading increased with
plasma density for the bare loop antenna but decreased with density for the
bare plate and "T" antennas; the magnitude of the resistance was typically

2 -8 0. In contrast, the dominant feature of the characteristic loading resistance of the ceramic-insulated loop antenna was a series of sharp peaks, as the
density rose, whch were typically 0.1 -0.3 h2 in amplitude, or 2 - 3 times the
background resistance due to ohmic antenna losses. There was also a small
continuous component of the loading which was roughly proportional to density, typically 0.05 -0.10 R in amplitude. The input resistance of the Faradayshielded loop antenna was similar in that the density-dependent loading was
present, although somewhat smaller in magnitude (typically <0.04 0 ) . The
peaks on top of the background loading observed with the insulated antenna
were also seen with the Faraday-shielded antenna but were of considerably
smaller amplitude ( < 0.05 0).
The effect on the loading resistance of moving the antennas farther into
the plasma (with the measurements made a t constant mean density) was comparable to the effect of a density increase at a fixed antenna location. Thus
moving the bare loop antenna farther past the tokamak wall increased the
measured resistance; moving the bare plate or "T" antennas farther in
decreased the resistance. For the ceramic-insulated and Faraday-shielded
antennas, the magnitudes of both the loading resistance peaks and of the
background continuous loadmg increased with density.
All of the antennas exhibited an increase in loading resistance as the excitation frequency increased; the increase was generally smaller than a factor of
two as the frequency was doubled.

The loadmg resistance of the "T" antenna decreased by -30% as the
toroidal magnetic field was raised by a factor of two. None of the other antennas showed any significant change in impedance as the toroidal field was
varied.
The bare loop and ceramic-insulated loop antennas showed no significant
variation of loa&ng resistance with antenna current. The bare plate and " T
antennas, however, exhibited a substantial increase in resistance as the
current increased above a certain level.
The parametric dependence of the imaginary part of the antenna
impedance was similar for the bare loop, bare plate, and "T" antennas. The
antenna reactance decreased as the density increased or as the antenna was
moved farther into the plasma; the decrease was typically 10-30% over the
range of parameters measured. Similarly, these antennas all exhbited an
increase in reactance as the excitation frequency was raised; the increase was
typically 30 - 100 % as the frequency was doubled. None of these three antennas showed any significant change In reactance as either the excitatio~
current or the toroidal magnetic field were varied. The insulated and Faradayshielded antennas showed no significant change in reactance at all, in the presence of the plasma.
The sharp, discrete peaks observed on the loading resistance of the insulated and shielded antennas have a &fferent physical origin from the continuous density-dependent loading seen with the bare exposed antennas. As mentioned in Chapter 7, the sharp peaks coincide with the appearance of peaks in
the RF magnetic fields observed globally around the tokamak (the toroidal
eigenmodes). As discussed below, some features of the observed peaks in the
loading can then be understood in terms of wave generation and damping.
Although the bare antennas also excite discrete eigenmodes, there is no
continuous wave excitation corresponding to the density-dependent loading. It

is likely that the continuous density-dependent loading results not from wave
excitation and subsequent damping but rather from particle collection effects
in the immediate vicinity of the antenna.

6.4.2 Particle Collection Model
When a conductor is placed in the tokamak and biased with respect to the
chamber wall, a space charge sheath forms around the electrode and, if the
fluxes of electrons and ions collected by the electrode are unequal, a net
current wdl flow. Simple Langmuir probe theory whch describes the relation
between the applied DC potential and the DC current drawn by the electrode
was discussed earlier. T h s section models the bare plate and "T" antennas as
Langmuir probes driven with periodic excitation.
At

an electron temperature

of

30 eV and

a plasma

density of

2 x 10'' emp3, the Debye length, a measure of the thickness of the space

charge sheath surrounding the electrode, is very small (-0.03 mrn). The thermal velocities are sufficient for even ions to traverse the sheath in a very smaii
fraction of an RF period (TRF-100 nsec). We then assume the steady state
(DC) I - V characteristic to be valid instantaneously during the periodic excitation. In addition to conduction (particle) current, displacement current can
flow through the capacitance arising from charge separation across the sheath
region. Simple estimates of the plasma sheath capacitance suggest that the
resulting shunt reactance seen by the antenna is large compared to the experimentally observed impedance values; hence this sheath capacitance is probably not important in the frequency and plasma parameter region of interest.
For the purposes of this model, an analytical function was needed to fit
the experimentally observed Langmuir probe I - V
developed was

curves. The function

where V, and Vf are the probe potential and the floating potential, respectively, measured with reference to the tokamak wall, and

4 is the probe

current (other symbols are defined in section 6.3). Here a is related to the
ratio of electron (I,,) to ion ( I & )saturation currents by

and the experimental value for a is typically -10. Once Vf and cx are fixed
(from the experimental probe traces), this functional form scales appropriately with temperature, density, and probe area and models well the curves of
Fzgure 6-44.

If a sinusoidal voltage, Vmcos(ot), is applied to the antenna electrode,
the resulting current depends on the fashion in which the electrode is connected to the generator. Two cases are ascussed here: the electrode is either
grounded with respect to the chamber wall (by means of a low-pass filter) or is
left floating (via an isolation capacitor). These cases correspond to the two
operating modes of the bare plate and "T" antennas.
Figure 6-52 shows a typical Langmuir probe I - V curve plotted from
equation 6-14, with

Te=30eV, n = 4 x 1 0 l ~ c m - ~A, = 1 2 c m \

a=10

and

Vf = -20 V. Also plotted is an example of the applied voltage for a grounded
antenna, $ ( t ) = VTRFcos(ot)and the resulting antenna current. The antenna
current response is clearly non-sinusoidal, and there is a net DC time-averaged
current flow.

If the electrode is allowed to float by removing all DC current paths to it, a

1 (AMPS)
DOC

1 (AMPS)

FIG. 6-52. Periodic excitation of a Langmuir probe. The static I - V curve is
shown together with one period of the imposed sinusoidal probe voltage (100 V
peak) and the resulting current drawn by the probe. The I - V curve is from
equation 6-14, with T, = 30 eV, n = 4 x 1012cm", A = 12 e m 2 , and Vf = -20 V.
a) Probe a t DC ground potential. Note that the probe draws a net DC current.
b) Probe floating, The probe charges to potential Vo such that the DC component of the resulting current vanishes. Note that for the same excitation voltage, the peak current is reduced for the floating probe.

FIG. 6-53. The antenna bias potential, Vo,plotted as
a function of the RF excitation amplitude, Vm, for
the I - V curve shown in Figure 6-52.

charge will build up which shifts the operating point on the I - V curve. The
condition that the antenna float requires the integral of the current over one
RF period to vanish. Figure 6-52b graphically illustrates this condition. The
excitation voltage is the same as in figure 6-52a, but the floating antenna
charges up to a sufficiently negative potential that the areas under the positive
and negative parts of the current waveform are equal. Note that this results in
a substantially lower peak current as well. Mathematically, the condition
describing a floating antenna can be written as

where Vo is the bias potential resulting from charge buildup on the antenna,
and T is the RF period. The above integral equation can be solved numerically
for the self-charging bias potential, V o , as a function of the excitation amplitude, Vm. As an example, for the I- V curve in Figure 6-52, a plot of Vo vs.
VM is shown in Figure 6-53. Note that when the excitation voltage vanishes,

the bias voltage is simply the DC floating potential, y..
The DC bias potential of the bare plate or "T" antenna during RF excitation was not directly measured experimentally. A related experiment, however,
was performed with the ceramic-insulated loop antenna, as described in section 6.3.2. In the absence of RF excitation, the surface of an insulator will also
charge up to the DC plasma floating potential, since the time-averaged current
to it must vanish. When the loop is excited with an RF current, an RF potential
exists between the loop and the tokamak wall due to the finite inductance of
the loop; the magnitude of the potential varies with location around the loop.
Although the Macor ceramic was 3 rnrn thick, displacement current could still
flow through the resulting capacitance and into the plasma, resulting in the
potential of the surface of the ceramic being driven a t the RF frequency. Thus,
in some respects, this situation is the same as a floating bare plate or "T"

antenna.
The potential near the surface of the ceramic on the side of the loop
antenna was investigated with a floating Langmuir probe. The relation between
the measured probe potential and the RF excitation current in the loop is
shown in Figure 6-50. Note the similarity between the form of the curve and
the theoretical calculation shown in Figure 6-53. A strict comparison cannot
be made, however, for several reasons. The Langmuir probe axis was 2 mm
from the Macor, so the potential seen by the probe was not necessarily the
same as that a t the surface of the ceramic. Both capacitive coupling between
the loop element and the probe, and particle currents between the probe and
the plasma and between the probe and the ceramic surface contribute to the
potential measured by the probe. Further, the relation between the antenna
excitation current and the potential developed on the outside of the ceramic is
not linear. Since the impedance presented by the capacitance due to the
ceramic is substantially higher than that presented by the plasma or by the
loop itself, the insulator acts as a voltage divider, and the potential on the
plasma-side of the ceramic should be proportional to the plasma impedance
(whch changes with excitation level).

Bare Plate a n d " T' A n t e n n a Impedance
The simple model of the bare plate and "Ti' antennas as driven Langmuir
probes can be used to find the expected antenna RF impedance. The actual
impedance experiments measured the fundamental components of the RF
voltage and current, and the phase between them. The current response,

4,

to a periodic voltage excitation, V'cos(ot), will, of course, have harmonic
components; these are eliminated in the experiment using bandpass filters.
The fundamental component of the impedance due to particle collection by the
antenna is then simply the excitation vottage divided by the fundamental com-

ponent of the current:

where Vo is given as a function of V w by the solution of equation 6.16 for a
floating antenna, or Vo= 0 for a grounded antenna. Note that this impedance
is purely real, i.e., a resistance. An inductive component of the impedance in
series with t h s real part arises due to the geometry of the return current path
between the electrode and the tokamak wall. This will not be considered
theoretically, but evidence that the current leaving these antennas is spatially
localized along the toroidal field lines is presented in Chapter 7.
Since the current

4 is linearly proportional to plasma density, the RF

impedance given by equation 6.17 should vary inversely with density. The variation of the real part of the impedance of the "T" antenna with plasma lineaveraged density is shown in Figure 6-54. The solid line is a least-squares fit to
a. l / n density dependence. The agreement is reasonable, considering that the

average density measured by the microwave interferometer does not strictly
track the density in the outer plasma where the antenna is located. Note also
that this model neglects possible temperature effects due to cooling of the
plasma during the gas puff or due to heating of the plasma by the RF fields.
The qualitative behavior of the impedance as the bare plate antenna is
rotated from a parallel to a perpendicular orientation can be understood in
terms of this model. From equation 6.14, the current $ is proportional to the
antenna electrode area. In a strong magnetic field the plasma particles, to
first order, stream along the field lines. Then, since the ion gyro-rachus
( p i W l m ) is much smaller than the antenna &mensions, the area that

enters into equation 6.14 is the projected area perpendicular to the field direction. Reducing t b s area by rotating the antenna then has the same effect as

T ANTENNA

RE( Z
(OHMS)

MEAN DENS1TY X 10 l2

( c M - 1~

FIG. 6-54. Variation of the real part of the impedance of the "T"
antenna as a function of the mean (line-averaged) plasma density.
The experimental points are from the data of figure 6-29; the
solid line is a least-squares fit to a l/n density dependence.

BARE PLATE ANTENNA

o ANTENNA FLOATING

v ANTENNA GROUNDED

RE( Z
(OHMS)

0. 1

1.

10.

RF ANTENNA CURRENT (PEAK AMPS)

FIG. 6-55. Variation of the real part of the bare plate antenna
impedance as a function of the antenna excitation current, for
both floating and grounded c o ~ g u r a t i o n s . The experimental
points are from Figure 6-25; the solid lines are the results calculated from equation 6.17, with n = 4 x 10" cm",
T, = 30 eV,
A =12 cm2, and VI =-5V.

100.

lowering the density: the antenna current decreases and hence the impedance
increases. This effect was confirmed by the experiment illustrated in Figure
6-20. The impedance of both the floating and grounded antennas increases
substantially for the orientation where the projected antenna area is much
smaller.

A more severe test of the model involves the predicted dependence of the
impedance on the excitation amplitude. The real part of the impedance of the
bare plate antenna is plotted as a function of the fundamental component of

the antenna excitation current in Figure 6-55 for both grounded and floating
antennas. The leading edge of the plate was 3.8 cm past the tokamak wall. A
reasonable density to assume, based on kngmuir probe profiles, was

4 x 1012~ r n ;- ~the temperature a t this location was typically -30 eV.

Using these values and an effective antenna area of 12 cm2, the impedance
was calulated from equation 6-17 and is also plotted in the figure. Note that
the horizontal axis of the plot is the fundamental component of the excitation
current rather than the excitation voltage. The agreement between the
theoretical curves and the experimental results is rather good, especially considering the simplicity of the model and the uncertainties in establishng the
various parameters. It should be noted that the lower curve, for the grounded
antenna, eventually turns up sharply as the current is raised further. This follows intuitively, since for very large excitation voltages, the current
approaches saturation both on the positive and negative excursions and hence
the impedance must increase as the voltage is raised further. Note again that
this model does not consider possible heating of the local or bulk plasma at
h g h excitation current levels. Some evidence that the plasma actually is
heated is seen in Figure 6-47b, where the slope of the transition region of the

I - V curve is considerably smaller than for the case without RF power.

Bare Loop A n t e n n a Impedance
The simple particle collection model appears to account reasonably well
for some of the observed impedance characteristics of the bare plate and "T"
antennas. The impedance of the bare loop antenna, however, has the opposite
behavior with respect to plasma density and antenna insertion. Because of the
loop inductance, the potential of the antenna varies around the loop. Each
point on the loop which is in contact with the plasma will draw a particle
current whch depends on the density a t that point and the potential of that
point with respect to the wall. A very simple model of the loop antenna which
nevertheless proves useful is shown in Figure 6-56. Rather than considering
the more d~fficultdistributed-impedance problem, the effect of the plasma is
lurnped into a real impedance $ which shunts the loop at its midpoint. LA is
the loop inductance and Lf is the stray inductance representing that part of
the loop not in the plasma. The input impedance of the antenna is then given
by

It is clear from the circuit model that the real part of the antenna impedance
has a maximum for a particular value of the plasma shunt impedance, Z p .
When the shunt impedance is very small (at high plasma density), most of the
current flows through it and the input impedance is small. When the shunt
impedance is very large (at low plasma density), most of the current flows
through the inductance whch is in parallel with Zp , and since the inductor is
lossless, the input impedance is still small. The maximum of Re(Zmt) occurs
when Zp = oLA/ 2 and has the value uLA/ 4 .
The model was applied to the experimental impedance data for the bare
loop antenna. The antenna loop inductance was approximated, from its

geometry, as 80 nH. The plasma shunt impedance, from the previous dscussion, was taken to be inversely proportional to the plasma density:

Z' = q / n ,

where 7 is a constant. The shunt impedance arises from the parallel contributions of the distributed impedance along the loop and is dominated by the
impedance of that part of the loop in the region of highest density plasma, i.e.,
the front or leading part of the loop. Since the height of the loop and the
length of the cross-bar of the "T" antenna were comparable, a reasonable value
for the constant 7 was taken from the impedance data of the "T" antenna:

1.4x 1013h l - ~ m - ~ . Taking the value of the stray inductance to be

Lf = 75 nH, the results from equation 6-18 for the real and imaginary parts of
the antenna impedance are plotted in Figure 6-57; the experimental points are
from the data of Figure 6-11.
The reasonable agreement between the theoretical curves and the experimental data supports the hypothesis that the impedance of bare loop antenna
is dominated by a shunting of the loop reactance due to plasma particle collection.

6.4.3 Impedance Due to Eigenmodes

The input loading resistance of the ceramic-insulated loop antenna was
dominated by a series of sharp peaks associated with eigenmode excitation.
Such peaks were first observed on the TO-1 tokarnak in Moscow [Ivanov, 19731
and on the ST tokamak a t Princeton [Adam et al., 19741 and have since been
seen in a number of other experiments.
The peaks on the impedance are a direct result of the damping of waves in
the hgh-Q toroidal cavity. As discussed in Chapter 2 , the fast waves excited in
the frequency and plasma parameter range of concern are only weakly
damped; interference effects of waves traveling around the torus in both
directions leads to a spectrum of discrete cavity resonances a t whch the wave

BARE LOOP ANTENNA

RE( Z >
(OHMS)

MEAN DENS ITY X 10

'* ( c M - ~)

IM( Z >
(OHMS)

MEAN DENSITY X 1 0 '"(CM-~

FIG. 6-57. Real and imaginary components of the impedance of the bare loop
antenna, as a function of the mean plasma density. The experimental points
are from the data of Figure 6-11; the solid h e s are the results of the theoretical model described by equation 6.18.

fields can build up to large amplitudes. The input impedance of the antenna
can be written as the ratio of the power dissipated by the antenna to the
square of the antenna current:

where rant is the antenna current, E, and 6, are the electric field and current
on the surface of the antenna loop element, and the integral is taken over the
loop. For a given antenna current, the lower the damping in the cavity, the
hgher will be the amplitude of the wave electric field throughout the cavity (at
a resonance), and, in particular, the higher will be the back-EMF at the
antenna loop itself. Thus, lower damping in the cavity leads to higher input
loading resistance a t the resonances. Stix [I9751 has shown that the loading
resistance is 2/0( times higher in the periodic-cylinder model than it would be
for an infinite cylinder with no interference effects, where e-a is the attenuation the wave suffers in traveling a length equal to the circumference of the
tokamak.
The discussion in Chapter 2 treated the boundary value problem without
external sources and solved for the normal modes of the system. To solve the
problem including the R F antenna requires solutions of Maxwell's equations
with the appropriate source terms. In order for the fields to remain finite at
the resonances, appropriate wave damping must be included. The fields in the
cavity can be written as a summation over all of the eigenmode solutions of the
corresponding source-free problem. The coefficients of the individual terms
can then be found in principal by imposing the boundary con&tions on the
conducting wall and on the antenna element. For hgh-&,well separated eigenmodes, the situation can be simplified somewhat. When the excitation frequency is near a resonant frequency for a particular eigenmode, the amplitude
of the resonant term dominates the response, and contributions from the

other modes can be neglected. Paoloni [1975a] has carried out a perturbation
expansion about the eigenmodes and arrives a t a simple expression for the
input impedance of a loop antenna when the excitation frequency is equal to
the resonant frequency for an eigenmode:

where W is the total time-averaged wave energy for the mode and Ipl is the
eigenmode flux linked by the loop. Here W and r9 are evaluated using the
source-free dissipationless field solutions, and Q is defined, as before, as the
ratio of the tirne-averaged wave energy to the energy dissipated in one wave
period for the particular mode. Thus the magnitude of the antenna resistance
a t an eigenmode is directly proportional to the Q of that mode. Note that the
G2
term - does not depend on the magnitude of the field, but only on the radial

2W

form of the field and on the antenna geometry.
The reason for the increase in loading a t the peaks of the modes as the
antenna :s xoved into the plasma (5g1x-e 6-37) is now apparent. As the
antenna moves in, r9 increases because more of the loop is past the wall and
because the magnitude of the component of the wave field which links the loop,

3,, is a decreasing function of minor radius in the outer plasma for the modes
of interest fcf. Figure 2-10).
The variation of the antenna loading at the eigenmode peaks as a function
of the excitation frequency, toroidal magnetic field, or mode number is more
difficult to analyze because Q and W are no longer constant. The term

(P
2W

was calculated for the N = O -5, m = 1, 1 = 1 toroidal eigenrnodes as a function
of R = w / o~ using the periodic-cylinder, constant-density, zero-electron mass
model described in Chapter 2 . The flux r9 was calculated by integrating the z component of the wave magnetic field over the area inclosed by a polygon

approximation to the loop boundary; the front edge of the loop was taken to
be 2.5 cm past the tokamak wall. The results are plotted in Figure 6-58. For
92
R 6 2 . 3 , the value of decreases with increasing toroidal mode number; the

zw

term decreases with 0 for N = 0, 1, 2 and varies little with h2 for N = 3, 4, 5.
In order to calculate the input loacbng resistance, the Qs of the modes
must be known. As is discussed in the following chapter, the theoretical calculations for the mode Qs which were presented in Chapter 2 do not correctly
predict the observed values; the theoretical values are generally too high.
Estimates of the mode Qs based on wave field measurements are presented in
the following chapter. Drawing on those results in advance, the expected
antenna impedance can be calculated as a function of the mode number.
Table 6-1 presents the theoretical and experimental data for the first five
modes encountered as the density rises, for a toroidal field of 4.0 kG.
G2
Specifically, the experimental value of Q and the theoretical evaluation of 2W

are used in equation 6.20 to calculate the predicted antenna eigenrnode
impedance. Except for the first mode, the results agree within a factor of
about 50%, with the predicted value being consistently lower than the observed
result. The agreement, except for the first mode, is reasonable considering the
large scatter in the data for the Qs and for the antenna impedance. Phase
measurements also presented in Chapter 7 confirm, as expected from the discussion of Chapter 2, that the observed eigenmodes are m = 1 modes of successively increasing toroidal mode number. The toroidal mode numbers were
not identified absolutely, however, so the assumption in the table that the first
observed mode was an N = 0 mode may be incorrect. However, even if the first
G2
observed mode were an N = 1 or N = 2 mode, the change in the values for 2W

and hence the change in the predicted impedance would be relatively small

( x 10-'~OHM -

FIG.6-58. Calculation of the quantity @'/ (2 W) as a function of R = w / w d for the
N = 0 - 5 , m = 1, 1 = 1 toroidal eigenmodes. The wave fields and wave energy W
are found from the zero-electron mass, constant-density model described in
Chapter 2; the flux 4? is found by integrating the z-component of the magnetic
field over the area inclosed by the loop antenna. The front edge of the loop is
2.5 cm past the tokamak wall and its boundary is approximated by a polygon.

Antenna Loading due to Eigenrnodes
(p2/ (2 W )
Re(Zani)

Mode
Order
(observed) (assumed) (observed)
27
155
309
537
091

(X 10-l2 ohm-see)

(calculated)
2.17
2.13
1.99
1.77
1.51

Re(Zani)
(rnilliohms) (milliohms)
(calculated) (observed)
19.
4.
43.
25.
70.
46.
91.
72.
134.
101.

TABLE 6-1. Comparison of expected values (from equation 6.20) and observed
values (from Figure 6-39) of the real part of the insulated loop antenna
impedance at the eigenmode peaks, Here Q is taken from the data presented in
Chapter 7 and Q2/(2W) is calculated from the wave model of Chapter 2. The
column labeled "Mode" lists the order of the observed eigenmodes, and N is the
assumed toroidal mode number associated with each mode. Also,
0/(27r) = 12 MHz, Bo =4.0 kG, and the front edge of the loop is taken to be 2.5
cm past the tokamak wall. Note that the background loading due to ohmic
losses in the antenna has been subtracted from the observed values for
Ret Zani) .

Wave Experiments

Waves associated with toroidal eigenmodes in a tokarnak plasma were first
observed in the TM-1-Vch and TO-1 tokamaks a t the Kurchatov Institute
[Ivanov et al., 1971; Vdovin et al., 19711 and have since been seen in a variety
of other machines. The most detailed investigations were carried out on the ST
and ATC tokamaks a t Princeton [Adam et al., 1974; Takahashi, 19791, on the
T.F.R. tokamak a t Fountenay-aux-Roses [T.F.R. Group, 19771 and on the
Erasmus tokamak in Brussels [Messiaen et al., 19781. In several experiments,
the spectrum of the modes in density-frequency space was mapped out, and
mode identification was carried out in some cases using phase measurements
of the wave magnetic fields a t the plasma edge.
Toroidal elgenmodes were first observed in the Caltech Research Tokamak
by Hwang [1979] who characterized the general location of the modes in
density-frequency space. Mode identification was not carried out, however, as
the major emphasis of the work involved complex antenna impedance measurements.
This chapter presents the results of a series of experiments deslgned to
investigate the waves launched in the Caltech tokamak, using the RF antennas
and magnetic probes previously described. The chapter is divided into three
main sections. The first reports observations of the magnetic fields associated
with the toroidal eigenrnodes. The most surprising result is that the all of the
antennas, including the bare plate, bare loop, and "T" antennas, excited the

eigenmodes with comparable efficiency in terms of wave amplitude normalized
to the antenna current. Eigenmode dispersion curves were mapped out by
tracking the density a t which various modes occurred as the magnetic field
was incrementea. Probes which were separated toroidally and poloidally were
used to estimate the mode structure of the waves, and radial profiles of the
fields were obtained as well. Finally, the wave field amplitude evolution was
analyzed for the various modes to obtain estimates of the cavity dE) and the
damping length for each mode as a function of the toroidal magnetic field.
The second section of this chapter describes a new approach to the investigation of ICRF waves propagating in a tokamak. The antennas were excited
with very short bursts (4-5 cycles) of RF current, generating waves on a time
scale short compared to the formation time for the eigenmodes. The wave
packets thus launched by the antenna were observed with the magnetic field
probes and could be seen to propagate several times around the machine. The
time delay between the peaks of the excitation and received signal envelopes
gave direct information about the group velocity of the waves in the toroidal
direction. The group velocity was investigated as a function of a variety of
plasma parameters for comparison with theoretical models. Both the ceramicinsulated loop antenna and the "T" electric field antenna were used in these
experiments and yelded similar results.
The last section of this chapter presents an investigation of the RF
current density distribution in the outer tokamak plasma near a very small
electric field antenna, using the RF current probe discussed earlier (section
5.5). The experiment apparently represents the first direct measurement of

RF particle current in a plasma. The current leaving the antenna was found to
be quite localized along the toroidal field direction; the result suggests a
mechanism for the good coupling to eigenmodes observed with the electric
field antennas.

7.1 Eigenmode Observations
The experimental configuration of the tokamak and RF system was basically the same as that described for the impedance measurement experiments
(section 6.1). The wave fields were detected with the magnetic probes detailed
in section 5.3. The probes were mounted on various tokamak ports as necessary for the particular experiment; up to four probes could be used simultaneously. Each probe was connected to a 50 R attenuator and a +30 db
broadband RF amplifier. The output of each amplifier led to a power splitter,
one output of which led to a linear RF detector and the other to an input of the
four-channel phase detector. The reference input of the phase detector could
be connected to any of the probe signals but was usually used with the antenna
current signal. The outputs of the amplitude and phase detectors were
recorded with the data system described previously (section 3.4). In many
experiments, the antenna impedance was monitored simultaneously, as
described in section 6.1, both as an aid in tuning the matching system and to
observe the relation between the eigenmode wave field peaks and the peaks on
the antenna loading resistance. Impedance information was also required for
the Q measurements in order to convert the measured (loaded) Q to the
unloaded cavity Q .
The loop antennas were always positioned so that the plane of the loop
was normal to the toroidal direction. This orientation was chosen in part
because the toroidal component of the magnetic field associated with the
eigenmodes was the largest component of the field near the wall where the
antenna had to be positioned, and it was desirable to maximize the eigenmode
flux linked by the loop in order to increase the coupling efficiency. The bare
plate and "T" antennas, however, could be rotated throughout a full 360"
range.

7.1.1 Typical Probe Data

An example of the magnitude of the signal from a magnetic probe is plotted in Figure 7-1, along with the plasma density derived from the microwave
interferometer signal. The cerarnic-insulated loop antenna was used, the front
edge of the antenna was 3.2 cm past the tokarnak wall, the excitation frequency was 8.5 MHz,and the toroidal magnetic field on axis was 3.8 kG; hence
u/uCiN 1.5 a t the plasma center. The coil axis of the probe was located 2.0 cm

past the wall, in tokamak port #1 ( 180" toroidally away from the RF antenna),
and the orientation of the coil was such as to measure the toroidal component
of the wave magnetic field. Four eigenmodes are clearly visible during the density rise; the same modes can also be seen during the density fall. Several
qualitative features can be seen from this plot. The etgenmodes occur a t
approximately the same density during the rising and falling portions of the
density curve; this implies that they are in fact the same modes. The amplitudes of the modes increase as the density increases; also the amplitudes of
the modes during the density fall are smaller than those occurring during the
density rise. The density rises and falls approximately linearly with time, and
it is clear that the temporal width of the resonances decreases as the density
rises. From shot to shot there was considerable variation in the amplitude of
each mode, but the general features described above were characteristic. In
many cases, the density fall was much more abrupt than the rise, and the
modes occurring during the fall were not easily identifiable.
A surprising result of t h s investigation was that all of the RF antennas

were capable of exciting eigenmodes, including the bare plate and "T" antennas. In previous work at other laboratories, only loop antennas of various
sorts have been used to excite ICRF waves.
Typical probe signals for the various RF antennas are shown in Figures
7-2, 7-3, and 7-4, for excitation frequencies of 8.5, 12, and 16 MHz, respectively.

T I M E (MSEC)
FIG. 7-1. Magnitude of the toroidal component of the wave magnetic field for a
typical tokamak shot, as measured with a magnetic probe, and the density evolution following the gas puff. The ceramic-insulated loop antenna was excited a t
a frequency of 8.5 MHz and the static toroidal magnetic field on axis was 3.8 kG.
The magnetic probe was located 180" toroidally away from the RF antenna and
the axis of the probe coil was 2.0 cm past the wall. The four modes observed
during the density rise occur at approximately the same densities as those seen
during the density fall.

BARE LOOP
ANTENNA

CERAMICINSULATED
LOOP
ANTENNA

BARE PLATE
ANTENNA
Jn,

5.

10.

15.

TIME (MSEC)

FIG. 7-2. Typical magnetic probe signal amplitudes for the bare loop, ceramicinsulated loop, bare plate, and "T" antennas. The excitation frequency was
8.5 MHz and the static magnetic field on axis was 4.0 kG (hence o / w d 1.4 a t
the plasma center). The probe, oriented to measure the toroidal component of
the wave magnetic field, was located in tokamak port #2, 180" toroidally and
90" poloidally away from the RF antenna. The density evolution and peak density were different for each shot.

BARE LOOP
ANTENNA

CERAMICINSULATED
LOOP
ANTENNA
FARADAYSH I ELDED
LOOP
ANTENNA

IR

TIME (MSEC)

FIG. 7-3. Typical magnetic probe signal amplitudes for the bare loop, ceramicinsulated loop, Faraday-shielded loop, bare plate, and " T antennas. The excitation frequency was 12. MHz and the static magnetic field on axis was 4.0 kG
(hence o/wCi-2.0
a t the plasma center). The probe, oriented to measure the
toroidal component of the wave magnetic field, was located in tokamak port #2,
180" toroidally and 90' poloidally away from the RF antenna. The density evolution and peak density were &fferent for each shot.

CERAMICINSULATED
LOOP
ANTENNA

TIME (MSEC)

FIG. 7-4. Typical magnetic probe signal amplitudes for the bare loop, ceramicinsulated loop, bare plate, and "T" antennas. The excitation frequency was
16. MHz and the static magnetic field on axis was 4.0 kG (hence w / w & -2.6 a t
the plasma center). The probe, oriented to measure the toroidal component of
the wave magnetic field, was located in tokamak port #2, 180" toroidally and
90" poloidally away from the RF antenna. The density evolution and peak density were different for each shot.

TIME (MSEC)
FIG. 7-5. Effect of the static toroidal magnetic field on the magnetic probe signal
amplitudes, for a fixed excitation frequency of 8.5 MHz. The ceramic-insulated
loop antenna was used and the magnetic probe was located in tokamak port #1
( 180" toroidally away from the antenna). The density evolution was different
for each shot; the arrows indicate the time a t which the plasma density reached
its maximum. value, The peak densities were similar for all of the shots:
n ~ ( 8 & 2 ) x 1cm".
0~~

The toroidal magnetic field was 4.0 kG in all cases, hence o/o& = 1.4, 2.0,and
2.6 at the plasma center for the three frequencies. The leading edge of each

antenna was 3.8 cm past the tokamak wall, and the density evolution was
different for each shot. Several effects are worthy of comment. As the fre-

quency is raised, the modes become more densely spaced in time, more modes
are observed, and the temporal width of the modes decreases. For each frequency, the character of the modes for each antenna is similar. The modes a t
each frequency do not occur a t identical times or even in exactly the same
numbers because the density evolution and the peak density associated with
each curve are different; the plasma density profiles may differ as well.
Typical behavior of the eigenmodes as the toroidal magnetic field was
varied is shown in Figure '7-5. The ceramic-insulated loop antenna was used,
and the excitation frequency was 8.5 MHz; the ratio w/wd

a t the plasma

center was thus varied from 1.1 to 1.9. The plasma density evolution could
change significantly as the confinement field was varied, and the time a t which
the density reached its peak is in&cated for each curve with an arrow. The
peak densities acheved in each shot were the same withn -20%. The density
decay for these shots was much faster and more abrupt than the density rise;
thus the eigenmodes appearing to the right of the arrows cannot in general be
matched with those occurring during the density rise. The plot shows the
dramatic effect of changing the magnetic field: as the field is raised, fewer and
fewer modes come into resonance during the density rise. In addition, the
modes that do appear become broader and broader. We shall see subsequently
that t h s is an indication of increased wave damping.
The effect of raising the toroidal magnetic field on the characteristics of
the modes is thus qualitatively the same as the effect of lowering the excitation
frequency. T h s is consistent with the simple theoretical models described in
Chapter 2: in the frequency and plasma parameter range of interest, the wave

dispersion relation depends only on the ratio w/w& rather than on w or Bo
separately.
The average wave excitation efficiencies for the five RF antennas are
shown in Figure 7-6, as functions of the peak antenna current. For these
measurements, the excitation frequency was 12 MHz, the toroidal magnetic
field was 4.0 kG (hence o / w d - 2 . 0

on axis) and the leading edge of each

antenna was 3.8 cm past the tokarnak wall. The toroidal component of the
wave magnetic field was measured with a probe in tokamak port #2 ( 180"
toroidally and 90" poloidally away from the antenna); the center of the coil
was 2.0 cm past the wall. The excitation efficiency is defined here as the peak
eigenmode field amplitude measured by the probe, divided by the antenna
current. The amplitudes of the second, third, and fourth modes to appear
after the initiation of the gas puff were divided by the currents a t the
corresponding times, and the results were averaged. Each point on the plot
represents the average of 5 - 10 tokamak shots analyzed in this fashion.
The ceramic-insulated loop antenna was the most eficient coupler; the
Faraday-shielded antenna was the least efficient. The bare loop, bare plate,
and "T" antennas all showed comparable efficiencies, ranging from 5 0 -70% of
the ceramic-insulated antenna efficiency. The efficiency of the Faradayshielded loop antenna was only

30 % that of the ceramic-insulated antenna.

This was expected, since the vacuum field measurements (section 5.2) showed
that the magnetic flux available for coupling to the eigenmodes was substantially reduced by the imposition of the sbeld. None of the antennas showed
significant variation in coupling efficiency as the antenna current changed over
a wide range. Note that even a t the highest antenna current, using the most
efficient antenna, the wave field amplitude measured by the probe was only

0.7 gauss, a small fraction of the toroidal or poloidal confinement field mag-

nitude. Hence the linearization of the fluid momentum equation (2.1a) for

WAVE EXCITATION EFFICIENCIES

EIGENMODE AMPLITUDE
ANTENNA CURRENT

0 MACOR-COVERED LOOP ANTENNA
0 BARE LOOP ANTENNA
0 T- ANTENNA
v BARE PLATE ANTENNA
x FARADAY-SHIELDED LOOP ANTENNA

ANTENNA CURRENT (PEAK AMPS)
FIG. 7-6. Eigenmode excitation efficiences as functions of the antenna current, for the bare
loop, ceramic-insulated loop, Faraday-shielded loop, bare plate, and "T" antennas. The magnetic
probe, located in tokamak port #2,was oriented to measure the toroidal component of the wave
magnetic field; the axis of the probe coil was 2.0 cm past the wall. The excitation frequency was
12 MHz,and the front edge of each of the antennas was located 3.2 cm past the wall. The experimental points represent the averages of the excitation efficiencies for the second, third, and
fourth modes to appear after the initiation of the gas p a ; each point is the average of 5 - 10
tokamak shots analyzed in this fashion.

small wave amplitudes is experimentally justified.

7.1.2 Eigenmode Dispersion Curves
At a given excitation frequency and toroidal magnetic field, the specific
densities at which eigenmode resonances appear can be found in principle
from the dispersion characteristics of the waves in the bounded system. As
discussed in Chapter 2, the form of the radial density profile can substantially
affect these egendensities. It was found that, for the same line-averaged density (which is what the microwave interferometer measures), the more peaked
the profile, the lower the eigendensity. Since the actual radial density profile
in the Caltech tokamak plasma has not been measured except at the plasma
edge, direct comparison with theory is not possible; nevertheless, comparison
of the general features of the eigenmodes is useful.
Eigenmode curves (i.e., plots of the density at which a specfic mode
appears as a function of w/ wCi ) were mapped out experimentally. In the range

of parameters appropriate to the experiments considered here, the theoretical
curves depend only on the ratio o / o & . For convenience in the experiments,
o/w,; was changed by varylng the toroidal magnetic field rather than the RF

frequency. The experiments required considerable care in adjusting the density evolution via the gas puff valve voltage waveform. A slowly rising, smooth
density was necessary in order to have clean, well-separated modes for
identification from shot to shot. Probe signals such as the one shown in Figure
7-1 were analyzed, recording the density corresponding to each of the eigen-

mode peaks occurring after the initiation of the gas puff. The toroidal magnetic field was then adjusted slightly and the procedure repeated. Typically
4 -6 shots were recorded a t each field strength; the eigendensities for each

mode were averaged.

The resulting eigenmode dispersion curves for the ceramic-insulated loop
antenna are shown in Figure 7-7a. The excitation frequency was 12 Mhz, and
the leading edge of the ceramic box shielding the antenna was 3.2 cm past the
tokamak wall. The probe used to measure the wave fields was located in
tokamak port ff2, and the probe coil axis, oriented to measure the toroidal
component of the field, was 2.0 cm past the wall. Generally only the first 5 - 6
modes occurring during the density rise were sufficiently reproducible and
clearly separable; the curves for the f i s t five modes are plotted in the figure.
As expected from the discussion in Chapter 2, the eigendensities decrease as
w/ud increases. The labels 1-5 identifying the modes represent the observed

order in which they appear after the initiation of the gas puff.
Also plotted in the figure is a curve calculated from theory, using the
quadratic density prome n l ( p ; 0.9) discussed in section 2.4, for the N = O ,
m = 1,L = 1 toroidal eigenmode (dashed line). The theory of Chapter 2 predicts

that this is the lowest cutoff mode, i.e., that there are no eigenmodes occurring
a t densities below the curve. The density profile used in the calculation is a
quadratic function of minor radius, with a peak density a t the center ten times
that at the edge and consequently a line-averaged density of seven times the
edge density. This curve is plotted only as an indication that the observed
modes do in fact occur within the region of density-frequency space predicted
by simple cold-plasma theory. Because the exact location and spacing of the
modes depends strongly on the radial density profile chosen, mode
identification cannot be carried out by comparison of the theoretical and
experimental curves. Note that the theoretical curves for a constant density
model (plotted in Fgure 2-9) are more closely spaced near the cutoff; t h s is
true for the quadratic density profile model as well. In contrast, the spacing of
the observed modes in Figure 7-7a is relatively uniform. One explanation for
this effect is that the N = 0 or 1 modes are simply not excited sufficiently to

CERAMIC-INSULATED LOOP ANTENNA

"T" ANTENNA

FIG. 7-7. Eigenmode dispersion curves for the ceramic-insulated loop antenna and
the "T" antenna. The mean density (from the microwave interferometer) at which
the modes apear is plotted as a function of R w / wd . Each experimental point
(circles) represents the average of the eigendensities for that particular mode
from 4-6 tokamak shots. For comparison, the theoretical result for the cutoff
mode ( N =O, m = 1, L = 1) using the n I ( p , 0.9) quadratic density profile is plotted
(dashed line). a) Ceramic-insulated loop antenna, excited at a frequency of
12 MHz. b) "T" antenna, excited at a frequency of 8,5 MHz.

observe and that the first modes seen are N = 2 or hgher order modes.
Another possibility involves the presence of a very low density region a t the
tokamak wall - plasma interface. As d l be seen later in this chapter when the
group velocity results are discussed, inclusion of a vacuum layer a t the plasma
edge can substantially modify the modes near cutoff.
Similar results with the "T" antenna are displayed in Figure 7-7b. In this
experiment, the excitation frequency was 8.5 MHz and the front edge of the
cross-bar of the "T" was 2.8 cm past the wall; the magnetic probe used was the
same as described above. The spacing of the modes and their general location
in density-frequency space are similar to the results obtained with the loop
antenna, except that the first observed mode occurs a t a somewhat higher
density.

7.1.3 Global Characteristics: Amplitude and Phase
The amplitude evolution of the wave magnetic field associated with the
eigenmode was simjlar at all locations investigated, Figure 7-8 displays signals from four magnetic probes, at different locations, taken during a single
tokamak shot. The ceramic-insulated loop antenna was used to excite the
waves a t a frequency of 12 MHz, and the toroidal magnetic field on axis was
4.0 kG. The probes were located in tokamak ports #1, 3, 4, and 5; their loca-

tion with respect to the RF antenna is shown in Figure 3-1. The probe coils
were all oriented to observe the toroidal component of the wave magnetic field
and the axes of the probe coils were 2.0 cm past the tokamak wall. The gas

puff began a t t -3 msec and the plasma density reached its peak a t

12 msec. The first five modes are clear and well separated, and the signals

from all four probes are quite similar. The relative amplitudes of the successive modes vary somewhat from one probe to another and from shot to shot.
The signal amplitudes from the B,,and Be probe coils were quite similar to

PROBE #5
AMPLITUDE

PROBE #1
AMPLITUDE

PROBE #3
AMPLITUDE

PROBE #4
AMPLITUDE

0.

5.

10.

15.

20.

TIME (MSEC)
FIG. 7-8. Magnetic probe signal amplitudes from four probes during a typical
single tokamak shot. The probes were located in tokamak ports #1, 3, 4 and 5
and were oriented to measure the toroidal component of the wave magnetic
field. The ceramic-insulated loop antenna was used to excite the modes at a frequency of 12 MHz; also Bo= 4.0 kG on axis.

those from the Bz coil, except for an overall change in magnitude.
The eigenmodes are thus seen to be global modes, which implies that the
wave damping length is, as expected, large compared to the tokamak circumf erence.
Phase measurements were carried out in order to further identify the
mode structure. The phase detector used in t h s experiment was described in
section 4.6; it had an unambiguous range of Zn radians and sufficient
dynamic range to track the phase during the large wave amplitude peaks a t
the eigenmode resonances. The phase detector reference input signal was
taken from the antenna RF current monitor signal using a 0" power splitter. A
signal from one of the magnetic probes could also be used as the reference
input.
A typical tokamak shot showing the phase detector output signals from

two magnetic probes is shown in Flgure 7-9; the phase reference signal was the
RF antenna current. Also shown for comparison is the amplitude of the signal
from one of the probes. The probes were located in tokamak port #1 ( 180"
toroidally and 90" poloidally away from the RF antenna) and in port #4 (90"
poloidally away from the RF antenna and at the same toroidal location as the
antenna). The probes were positioned 2.0 cm past the tokamak wall and were
oriented to measure the toroidal component of the wave magnetic field. The
ceramic-insulated loop antenna was used for the measurements, with an excitation frequency of 12 MHz,and the toroidal magnetic field on axis was 4.0 kG.
The first four ergenmodes after the initiation of the gas puff (at

t -4.5 msec) are clearly visible on the trace of the signal amplitude. The very
abrupt jumps in the phase signals from - 1 8 0 V o +180° represent the mapping of the actual continuous phase signal into a 360" range by the phase
detector electronics. Due to the fast response of the detector ( 7 - 5 psec),
these phase jumps generally occur within one sample period of the data

PROBE #1
PHASE
(DEGREES)

PROBE #4
PHASE
(DEGREES)

TIME (MSEC)
FIG. 7-9. Phase detector output signals for the magnetic probes in tokamak
ports #1 and #4, and the amplitude of the signal from probe #1, for a typical
tokamak shot, The phase detector reference signal was the RF antenna current.
The ceramic-insulated loop antenna was excited a t a frequency of 12 MHz, the
toroidal magnetic field on axis as 4.0 kG, and the magnetic probes were oriented
to measure the toroidal component of the wave magnetic field,

PROBES #1,
PHASE
(DEGREES)

TIME (MSEC)

1 8 0 - - - - - - - - - - - - - - - - - - -II - - - - - - - - - - - - - - - - -

PROBES #I. #5
PHASE
(DEGREES)

--

----------I

-180

4.

- - - - - - -I- - - - - - - t - - - - - - i - - - - - - - -

8.

6.

10.

TIME (MSEC)

FIG. 7-10. Relative phase between pairs of magnetic probes. The vertical dashed
lines mark the times at whch the eigenmode peaks in the wave amplitude occur;
the lines are labeled in order of appearance of the modes after the gas puff. The
waves were excited with the ceramic-insulated loop antenna a t a frequency of
8.5 MHz; the toroidal magnetic field on axis was 3.5 kG, a) Phase between two
probes separated by 180" in the toroidal &rection. b) Phase between two
probes separated by 90" in the poloidal direction.

recording system and can be easily separated from the actual phase variation
of the signals. Note that these 360" phase jumps are not coincident with the
eigenmode amplitude peaks.
The phase between the signal from magnetic probe #4 and the antenna
current changes by -360" from one mode to the next. The phase undergoes a
fairly rapid transition of

180" during the eigenmode amplitude resonance

and a slower variation between mode peaks. The phase of the signal from
probe #/I shows similar behavior except that the rate of change of phase is
smaller by a factor of -2.
The phase signals were generally quite clean and reproducible from shot
to shot. Note that because the magnetic probes were all constructed identically and the connecting cables were matched (within -0.5 cm), the relative
phase difference between the signal from probes fil and#4 therefore
represents the actual phase difference between the wave magnetic fields at the
probe locations.
A more useful represe~tztionis to display the relative phase between pairs

of probes. A computer program was written wlvch removes the 360" phase
detector jumps and then subtracts the phase signals from each other. Figure
7-10a shows the resulting phase difference between two probes separated by
180Voroidally during the part of the tokamak shot following the gas puff.
Note that the phase a t the beginning of the plot was chosen to lie in the
0" -360" interval; this choice was arbitrary to an additive constant of an
integral multiple of 360". The vertical dashed lines mark the times at which
the eigenmode resonances occurred on the probe signal amplitude; for this
particular shot, five modes were clearly identifiable. The labels 1-5 at the top
of the plot simply identify the order of appearance of the modes after the initiation of the gas puff.

The phase difference between the two probes continuously increases as
the density rises. At each eigenmode resonance (the intersection of the
dashed vertical line with the phase curve), the phase difference is an almost
exact multiple of 180°, with the phase increasing by 180" between successive
modes. T h s result is consistent with the wave theory described in Chapter 2,
whch pre&cted that the first 4-5 modes accessible in the Caltech tokamak
would have successively increasing toroidal mode numbers. If the wave fields
a t toroidal eigenmodes vary as eWv , where N is the toroidal mode number and
p , the toroidal angle (note that p = z / Ro in the periodic cylinder model), then

the predicted phase difference between two locations separated toroidally by
180' is precisely that whch was observed: successive increases of 180".
Because only two probes were used, the toroidal mode number can only be
identified absolutely as even or odd. According to Figure 7-10a, then, the first
observed mode is odd, probably an N = 1 or N = 3 mode. Physically, the parallel (i.e.,toroidal) wavelength of the fields in the plasma decreases as the density rises; eigenmodes occur when the wavelength is such that an integral
number of waves can fit around the tokamak. The cold plasma theory does not
describe the behavior of the wave fields between eigenmode resonances as wave
damping must then be considered.
The phase difference between two probes a t the same toroidal location,
but separated by 90" poloidally, is shown in Figure ?-lob for another tokamak
shot. The experimental arrangement was as described above but the probes
were located in tokamak ports #1 and #5, 180" toroidally away from the RF
antenna. The vertical dashed lines again denote the times at which the eigenmode resonances occur,
The phase difference is very close to 90" a t each of the eigenmodes, and it
varies little between modes. If the wave fields vary, as assumed in the periodic
cylinder model, as eime, where m is the poloidal mode number and 6 is the

poloidal angle, then this result implies that m = 1 + 4 1 , where L is some
integer. Thus, the measurement is consistent with values for m of 1, 5, -3,
9, -7, etc. The simple cold-plasma theory predicts that the plasma densities

required to reach even the m = 5 or m =-3 modes are beyond the limits of
the Caltech tokamak, for the frequency range of interest (cf. Figure 2-9d), and
that other lower-order m modes would be excited first. Therefore, it is reasonable to conclude that the observed waves have an m = +1 poloidal mode
structure.
In conclusion, phase measurements of the wave magnetic fields support
the predictions of the simple periodic-cylinder, cold-plasma model that the
first modes encountered are m = +1 modes of low and successively increasing
toroidal mode numbers.

7.1-4 Wave Amplitude Radial Profiles
The radial profiles of the amplitudes of the magnetic field components
associated with the eigenmodes were investigated experimentally. A three-coil
magnetic probe was located in tokamak port #1 to provide a measurement of
the toroidal, poloidal, and radial components of the magnetic field simultaneously. The cerarnic-insulated loop antenna was used to excite the ergenmodes

a t a frequency of 8.5 MHz and the toroidal static magnetic field was 3.5 kG on
axis; hence w / w d

1.6 a t the plasma center. After conditioning the probe

with discharge cleaning and tokamak shots, it could be inserted almost 4 cm
past the wall without seriously affecting the plasma discharge. During the wave
measurements, the probe was inserted into the plasma in increments of 0.5 or
1.0 cm. The plasma density evolution was adjusted carefully, via the gas puff
valve voltage waveform, so that the first 4 -5 eigenmodes were distinct and
clearly separated, and the tokamak discharge was fairly reproducible. At each
position of the magnetic probe, 3 -6 shots with similar plasma current and

density evolution were recorded. At each eigenmode peak, the components of
the magnetic field were normalized to the antenna current at that time; the
data for each field component from the tokamak shots a t each probe location
were averaged.
The results of this field component survey are displayed in Figures
7 - l l a , b , c ,and d for each of first four eigenmodes to for each of the first four
eigenmodes to appear after the initiation of the gas puff. Notice that the radial
form of the field components is quite similar for each of the modes, apart from
an overall scale factor. Over most of the radial range, B, is the largest component of the wave field, and it increases monotonically as the probe is
inserted radially into the plasma. The p-component of the field also increases
as the probe moves into the plasma, but its magnitude is considerably less
than that of the z-component. Note that as the probe moves in, the rate of
rise of Bp increases, while that of Bz decreases; for the first mode at the maximum probe insertion, B,, becomes larger than B, . The 6-component of the
field is approximately constant over the range of the probe position for all four
of the modes. The overall coupling efficiency to the modes (the field amplitude
divided by the antenna current) increases for each successive mode.
The experimental results can be compared with the theoretical field
profiles whch are shown in Figure 2-10. These profiles are calculated from the
zero-electron mass, constant-density, periodic-cylinder model. For the m = l
poloidal mode and the first radial mode, the profiles for the N = 0, 1, 4 toroidal
modes are plotted. From the &scussion of the phase measurements in the
previous section, the poloidal mode number of the observed modes in the
tokamak is m = 1 , and the toroidal mode number of the first observed mode is
probably N = l or M = 3 , with successive modes having integrally increasing
toroidal numbers,

MAGNETIC
AMPLITUDE

MAGNETIC

FIELD

(~G/A)

FIELD

AMPLITUDE

FIG. 7-11. Radial profiles of the magnetic field components of the elgemnode
fields, for the first four modes observed after the initiation of the gas puff. The
ceramic-insulated loop antenna was used to excite the waves a t a frequency of
8.5 MHz and the toroidal magnetic field on axis was 3.5 kG. a) Field profiles for
the first mode. b) Field profiles for the second mode.

MAGNETIC
AMPLITUDE

MAGNETIC
AMPLITUDE

F lELD
(~G/A)

FIELD

(~G/A)

FIG. 7-11, continued. c) Field profiles f o r the third mode. d) Field profiles for
the fourth mode.

The general form of the p- and 2-components of the field over the range
of the measurement (0.75 < p/po < 1.00) agrees roughly with the calculated
form: Bz and Bp both increase as p decreases, B, is considerably larger than

B p , and the rate of increase of Bp is larger than that of Bz as p decreases.
The major difference in the form of the calculated profiles as the toroidal mode
number increases is a relative decrease in the magnitude of B, compared to
the other components. T h s effect is not observed experimentally: the
observed relative magnitude of B, with respect to Bp is approximately constant for the four modes. The calculated curves exhibit several other features
which do not agree with the measurements. The theory predicts that Bp
should vanish at p = p o in order to satisfy the boundary conditions of a perfectly conducting wall. The measurements show a small but finite value for Bp
when the probe coil is at minor radius p o . This is not surprising, however,
because the probe does not actually measure the field component a t the surface of the conducting boundary. The probe is located in the center of a
2.2 cm i.d. cylindrical port; this perturbation to the boundary will certainly
alter the radial component of the field near it. Another point is that the wall,
being made of stainless steel, is not a perfect conductor; in fact, the conductivity of the wall is not too &Berent from the parallel conductivity of the
plasma at the temperatures typically encountered in the tokamak. The effect
of the finite wall conductivity on the field profiles near the wall has not yet
been considered theoretically. Finally, the low plasma density at the plasmawall interface in the actual tokamak also affects the eigenmode fields near the
wall.
Another point of disagreement involves the 8-component of the wave
field. The experimental results show that Be is fairly constant as the probe is
moved in as far as 4.0 cm past the wall. The theoretical curves in Figure 2-10,
for the m = 1, N = 1, 4 modes, show Be monotonically decreasing as the minor

radius decreases, passing through zero a t some point. The change in sign of

B, simply means that the phase has changed by 180"; the appropriate quantity for comparison with the experiment is the absolute value of the Be curve.
It is not clear why the experimental and theoretical results are distinctly
drfferent. Besides some of the effects mentioned above which can alter the
wave fields near the plasma edge, a possible explanation involves the poloidal
component of the confinement magnetic field. The total static magnetic field
in the tokamak is the sum of the toroidal and poloidal components (the component from the vertical field coils is negligible). At typical plasma currents of
25 kA and toroidal magnetic fields of 3.5 kG, the angle between the toroidal
direction and the direction of the total confinement field at the plasma edge is
-5".

If we assume, as a first approximation, that the wave fields are tied to

the total static magnetic field [Messiaen, 19781, then the parallel component of
the wave field will also contribute to the signal measured with a probe oriented
to measure the @-component of the field (in the tokamak toroidal coordinate
frame). Simple misalignment of the probe would yeld the same effect. The
three probe coils were orthogonal within

5", and the alignment of the B, coil

with the toroidal axis of the tokamak was also accurate within - 5 ' . Nevertheless, even 5 h f error due to misalignment or due to the poloidal field effect
would be sufficient to substantially alter the +component of the field measured by the probe, particularly if the actual ratio of B, to Be was large.
In conclusion, some of the features of the eigenrnode magnetic field radial
profile measurements are in agreement with predictions of the simple theoretical model presented in Chapter 2 for m = 1 modes. The z - and p-components
of the field have roughly the expected form, although the @-component does
not. The overall agreement, however, is much better than for modes with
m # 1. The four successive eigenmodes investigated all have similar field

profiles, further supporting the conclusions of the section 7.1.3 that the field

structure is that of m = 1 modes of successively increasing toroidal mode
number.

7.1.5 Wave Damping Measurements
The disposition of the energy lost in t h e antenna-plasma system is of considerable importance, both for understanding the physics of the wave coupling
and propagation, and for the more practical motivation of optimizing the
plasma heating. For the fast wave near the ion cyclotron frequency in the Caltech tokamak, the wave attenuation is small so that cavity resonances dominate the field response. Measurements involving the form of these resonances
can yleld information about the damping processes in the plasma.

Eigenmode Q M e a s u r e m e n t s
The cavity Qs expected for a variety of damping mechanisms were calculated for different eigenmodes in section 2.3. In a simple cavity resonator, the
Q of a mode may be found by observing the response as the frequency is swept
urns

through the resonance: Q = -, where om, is the resonant frequency for a
AGJ

particular mode and Ao is the difference in frequency between the points a t
whch the field amplitudes are reduced from their peak value by a factor of

I/*.

Although the driving frequency could in principle be swept, it is easier

to utilize the changing plasma density. It can be shown that, a t a fixed fre2n,
, where n
, is
quency, the Q of a mode is approximately given by Q N An.

the density a t which the resonance peak occurs, and An is the difference in
density between points a t which the field amplitudes are reduced by I/*
[Takahashi, 19771. The cavity Qs for each mode can then be estimated by
careful analysis of the density and magnetic field amplitude traces. Because
the cavity is loaded by the RF antenna, however, the Q estimated in this

fashion is the loaded Q ; the theoretical calculations are for the unloaded Q .

If the RF antenna were coupled very weakly to the eigenmode, then there would
be no difference between the two. In many experiments, however, the antenna
is tuned to maximize the coupling to the etgenmodes. If the antenna were perfectly matched to the mode, so that a t resonance there was no power reflected
back to the generator from the antenna, it can be shown that the loaded Q
would be smaller than the unloaded Q by a factor of two. Hwang [l979] has
shown that the loaded and unloaded cavity Qs can be related by measurement
of the antenna impedance:

where QL and Qx are the loaded and unloaded Qs, 1.2, is the antenna resistance at the eigenmode resonance, Ro is the background antenna resistance
is the antenna resistance off-resonance,
(in the absence of plasma), and ROfl
i.e.,between the resonance peaks on the loading.
Measurements of the cavity Qs were carried out using the ceramicinsulated loop antenna to excite the waves. The excitation frequency was
12 MHz, and the toroidal magnetic field on axis was varied from 3.25 kG to
5.50 kG in increments of 0.25 kG. The antenna input impedance was simultaneously monitored, as described in. Chapter 6. For each of the first five
modes occurring during the density rise after the gas puff, the loaded Q was
calculated as described above; the impedance measurements were then used
to convert this value to the unloaded Q .
The results are shown in Rgure 1-12, where the unloaded Q is plotted on
a logarithmic scale. The curves for each mode are labeled in order of their
appearance during the density rise. The Qs of the modes increase dramatically for successive modes; they range, a t a toroidal field of 4 kG, from -25
for the first mode to -900

for the fifth mode, with the largest increase

2.

3.

4.

5.

6.

TOROIDAL FIELD (KG)
FIG. 7-12. The unloaded Qs for the first five eigenmodes t o appear during the
density rise after the gas puff, as functions of the toroidal magnetic field. The
ceramic-insulated loop antenna was used to excite the waves a t a frequency of
12 MHz. The measured (loaded) Qs were converted to the unloaded Qs using
the simultaneous impedance measurements. Note that the second-harmonic
( 2w& ) layer enters the tokamak when B0-2.5 kG and exits when Bo-5.3 kG;
the fundamental ( w k ) layer enters the tokamak when Bo-5.1 kG.

occurring between the first and second modes. Each experimental point on
the plot represents the average of data from a nwnber of tokamak shots; note
that the indicated scatter in the data is substantial. The first four modes exhibit a substantial decrease in Q as the toroidal magnetic field is raised above
- 5 kG (the density is not sufficient to reach the Wth mode at these high
fields). Noting that the fundamental ion cyclotron layer enters the tokamak
when Bo-5.1 kG, the decrease in Q a t this point suggests fundamental cyclotron damping as the mechanism.
The observed Q values are not in agreement with any of the theoretical Q
calculations described in section 2.3. Referring to Figures 2- 12 through 2-1 7 ,
the only damping mechanism which predicts eigenrnode Qs which increase
with the toroidal mode number is second harmonic cyclotron damping.
Although the second harmonic cyclotron layer is within the plasma over most
of the range of toroidal field used in the experiment (it enters the tokamak
when B O w 2 . 5kG and exits when B o N 5 . 3kG), the results are not in quantitative agreement. For an ion temperature of 100 eV, the most heavily damped
mode (the cutoff or N = 0 mode) has a pre&cted Q due to second harmonic
damping of

1000. The actual first observed mode in these experiments is

likely an N = 1 or N = 3 mode; the predicted Qs for these modes are

1200

and -3200, respectively, much higher than the observed Q of - 2 5 . Adding
the effects of

the various theoretical damping mechanisms, assuming

independence, lowers the predicted Q as shown in Tables 2-1 and 2-2. The
total Q , however, still decreases dramatically as the toroidal mode number
increases, in contrast to the opposite experimental result. Thus it appears
that some physical damping mechanism other than those considered in
Chapter 2 must be responsible for the wave damping observed in the Caltech
tokamak. One possibility [Thompson, 19821 involves mode conversion of the
fast wave, near the second-harmonic resonant layer, to an electrostatic

Bernstein wave (which appears only when thermal effects are considered).

Damping Length Measurements
Another view of wave damping in the tokamak is to consider the attenuation that a wave suffers in traveling around the torus. If the wave were propagating in an infinite cylinder rather than in a torus, the damping could be
measured by simply observing the difference in wave amplitudes with two
probes spaced along the cylinder axis. In the tokamak geometry, however,
where the wave attenuation length is long compared to the toroidal circumference and interference effects are important, the situation is more complicated.
A simple model can yield a relation between wave magnetic field quantities

and the wave damping per unit length. Assume a periodic cylinder model of
the tokamak, where z is the axial length along the cylinder corresponding to
the toroidal coordinate in the tokamak. The RF antenna is located at z = O
and the ends of the cylinder (at whch the periodic boundary conditions apply)
are at z = rtrR0,where Ro corresponds to the major ra&us of the tokamak. If
the effect of the poloidal field is ignored, the antenna launches waves symmetrically in both the + z and -z

directions. A wave, upon reaching one of the

boundaries, is assumed to appear with the same phase at the other boundary.
For a given antenna excitation current, the field at a point in the cylinder can
be found by summing the contributions from each pass of the wave. Constant
wave absorption per unit axial length is assumed, and the wave propagates in
the z -&rection as e

*(lz-alz/

, where kll is the parallel wavenumber, a = l / L n is

the inverse wave damping length, and a harmonic time dependence is assumed.
hmiting the discussion to positions 0 < z s n R o , the contribution to a field
component from waves traveling in the +z direction is

and the contribution from the waves traveling in the -z direction is

where A+ and A - are constants related to the coupling efficiency of the
antenna to the waves and proportional to the antenna current. Assuming
equal coupling coefficients, the total field a t location z is just
cosh ( z - nRo)(ik - a ) ]

B=B++B-=A

s i n h [ n ~ k~l(l-i a ) ]

(7.4)

The parallel wavenumber, k l l ,is a function of the plasma density. In an
actual experiment, the excitation frequency is fixed, and k l l increases as the
density rises. Eigenmodes, or peaks in the wave amplitude, occur when

k l l = N / R o , where N , an integer, is the toroidal mode number, Antiresonances, or points where the wave amplitude is a minimum, occur when
k i i= ( N +%)/Ro.A particularly useful relation can be found by considering the

fields a t location z =rrRo,corresponding, in the tokamak geometry, to a position 100" toroidally away from the RF antenna. At that location, the magnit N/ R o ) to that a t an adjacent
tude of the ratio of the field a t a resonance ( k i =

anti-resonance ( kil= ( N +%)/R o ) can be easily shown to be

where ES, is the field component at the resonance, Badi is the component a t
the anti-resonance, and the antenna coupling efficiency is assumed to not
change significantly over the plasma density range between the resonance and
anti-resonance. The above equation can be solved for the damping length:

a result first derived by Takahashi 119771. Thus the wave attenuation length in

an infinite cylindrical plasma, L D , can be estimated by measurements in the
tokamak of the eigenmode wave field amplitude at a resonance and an adjacent anti-resonance.
The set of tokamak shots which provided the eigenmode Q data was
analyzed in the above fashon for the damping length of each mode; the
results are plotted in Figure 7-13 as functions of the toroidal magnetic field.
The curves for each of the five eigenmodes are labeled in the order of their
appearance during the density rise after the gas puff. The damping length
increases for each successive mode, and for the hgher three modes, the damping length is considerably longer than tokamak major circumference

(2nRo= 2.9 m). Note also that the damping length decreases for all of the
modes as the toroidal field increases. These results show the same features as
were suggested by the Q measurements: the plasma wave damping decreases
for successive toroidal modes and increases as the fundamental cyclotron
layer enters the plasma.
The eigenmode Q measurements and the wave damping length measurements are different views of the same physical absorption process. Takahash
[1977] shows that the two are related by the wave group velocity:

where vg is the group velocity of the wave in the toroidal direction. Thus the
independent measurements of LD and Q can be used to estimate the group
velocity. Table 7-1 shows the calculated group velocity for the five modes a t a
toroidal magnetic field of 4.0 kG. The modes occur at successively higher densities; thus the table gives information about the variation of the group velocity with density. Note the data predict that the group velocity decreases as the
density increases.

Lo
(METERS)

2.

3.

4.

5.

6.

TOROIDAL FIELD (KG)

FIG.7-13. The axial damping length, Lo, for the first five modes to appear during the density rise after the gas puff, as functions of the toroidal magnetic field,
The ceramic-insulated loop antenna was used to excite the waves a t a frequency
of 12 MHz. The data set analyzed was the same as for Figure 7-12. Note that the
second-harmonic ( 2wci) layer enters the tokamak when BoN 2.5 kG and exits
when Bo-5.3 kG; the fundamental (wci) layer enters the tokamak when
Bo25.1 kG.

Mode
Order

Density

(~m-~)

(cm /set)

TABLE: 7-1. Calculation of the parallel group velocity, u o , from
measurements of the wave Q and the damping length Lo, using
equation 7.7. The excitation frequency was 12 MHz and the
toroidal magnetic field on axis was 4.0 kG.

7.2 Wave-Packet Experiments
7.2.1 Introduction
The wave response in the tokamak plasma is dominated by interference
leading to toroidal eigenmodes a t specific plasma densities. The large amplitude peaks a t the resonances complicate wave measurements, and some are
only feasible a t the &screte eigendensities. For instance, the wave damping
and cavity Q measurements (and consequently, the group velocity calculation), described in section 7.1.5 yleld information only a t the discrete set of
densities corresponding to modes. It would be useful, from the point of view of
understanding the physics of the wave propagation and damping, to be able to
observe the waves without interference effects, as would be the case if the
waves were propagating in a n infinite cylinder geometry. Since the waves
travel with finite group velocity, one approach is t o examine the waves on a
time scale short compared with the characteristic time for eigenmode formation, i.e., short compared with the transit time of the wave around the torus.
Launching wave-packets from the antenna with very short bursts of RF current
was found to be a useful approach t o study wave propagation without interference. The time-of-flight of the wave disturbance around the tokamak gives
direct information about the parallel (toroidal) group velocity.
Wave-packet techniques have been used previously in plasma physics, primarily to investigate the propagation of electron plasma surface waves in
cylindrical geometry [Demokan e t al., 1971; Anicin et al., 1972; Landt et al.,
19741. A good review of this work is given by Moisan [1982]. In tokarnaks, the
only previous use of the technique appears to be a n experiment which
launched lower hybrid waves in the Doublet IIa tokamak [Luxon et al., 1980].
The antenna in that case was excited with 100 nsec bursts of RF current a t
800 MHz. Use of the technique in the ion cyclotron range of frequencies is
d f i c u l t in the Caltech Research Tokamak because the tokamak dimensions

are small and the group velocity is relatively h g h : the wave can propagate a
substantial fraction of the toroidal circumference in one wave period. In order
to localize the excited wave-packet in frequency space, long RF pulses consisting of many oscillations are desirable. T h s must compete, however, with the
need to make the spatial extent of the wave-packet small compared to the
toroidal circumference of the tokamak in order to maximize the sensitivity of
the time-delay measurement.

7.2.2 Experimental Method
The RF system required some modification to permit excitation of the
antenna with very short wavetrains. A typical antenna current waveform in
these experiments consisted of about 4 - 5 periods of a sine wave; at a typical
frequency of 12 MHz, the duration of this RF burst was then -300 nsec. The
apparatus used to generate such a pulse is shown in Figure 7-14. A doublebalanced mixer (Minicircuits model ZAY-3-1) was used as a very fast RF switch
(switching time

- 1 nsec), gating a continuous signal generator. The output of

the mixer was amplified by a 10 watt broadband amplifier (EN1 411LA); for
lower frequencies a specially modified Marko solid state 100 watt amplifier
(originally deslgned for amateur radio use) could be used as well.
The hgh-Q impedance-matching network which was used in all of the previously described experiments to match the antenna impedance to that of the
generator was removed for these experiments; t h s was necessary in order to
prevent the network from ringing for a long period of time when excited with
the short RF burst. The output of the broadband amplfier led, through a long
50 R coaxial cable, directly to the ceramic insulator on the end of the antenna
feeder tube (a coaxial connector was constructed for t h s arrangement). Since
the impedance rnismatch between the cable from the generator and the
antenna was now very large, the RF pulse incident on the antenna gave rise to

a substantial reflected pulse. This reflected pulse traversed the cable back to
the generator a t whch point another reflected pulse was generated, traveling
again toward the antenna. The object of t h s experiment was to excite the RF
antenna with a short, isolated burst of RF current. Therefore, the cable
between the generator and the antenna was made sufficiently long (300 m)
that the round-trip propagation time for the reflected wave (-3 psec) was long
compared to the time scales of interest. In order to prevent excessive power
loss in the incident pulse, the delay line was made from a large-diameter cable
(RG-17 A/U).
The antenna current waveform for the ceramic-insulated loop antenna
was observed with an RF current monitor (Ion Physics Go. model CM-100-L)
mounted a t the end of the antenna feeder tube. The "T" antenna was also used
to excite wave-packets, and it was more convenient in this case to use a broadband directional coupler, mounted at the end of the antenna feeder tube, to
monitor the incident waveform. The wave fields in the tokamak were detected
with the magnetic probes described previously; the signal from the probe was
sometimes amplified with a broadband ( 1 -520 MHz) +30 db gain amplifier

(TRW model CA2020). The only available data recorhng system capable of handling the sub-microsecond time scales of these experiments was a Tektronix
model 466 storage oscilloscope with a bandwidth of 100 MHz. The scope trace
was recorded with a standard Polaroid camera. In order to avoid the need for
corrections, the cables leading from the magnetic probe and from the antenna
current monitor (or directional coupler) to the RF electronics cabinet were
matched in length. Also, the time delays associated with the antenna current
monitor and the directional coupler were investigated and found to be negligible.

A timing module in the tokamak control rack provided the initial signal to
start the experiment, The module provided a pulse which could be delayed to

any portion of the tokamak shot. The slgnal triggered a pulse generator
(Systron-Donner model 101) with two outputs. One of the outputs gated the RF
switch; the other trggered the fast oscilloscope. High-speed optical links (H.P.
model HBFR-0500) with very low triggering jitter were used for isolation
between the tokamak timing module and the pulse generator, and between the
pulse generator and the oscilloscope. These electrical breaks eliminated all
ground loops from the system. A schematic of the entire experiment is shown
in Figure 7-14.
Since only one hgh-speed storage channel was available, it had to be
switched between the exciting signal (the RF antenna current) and the received
signal (from the magnetic probe). In order to ascertain the propagation time
between the launched wave and the received signal, it was crucial that the
oscilloscope trigger at precisely the same time from shot to shot. This was
investigated a t some length during actual tokamak shots; the triggering was
found to be completely independent of interference from the tokamak stray
magnetic fields and the triggering jitter was negligible.

7.2.3 Experimental Data
Some typical raw probe signals are shown in Figure 7-15. The diagram a t
the top of the figure depicts the experimental geometry; in this case the magnetic probe was located in tokamak port #1, 180" toroidally and 90" poloidally
away from the RF antenna. The antenna was the ceramic-insulated loop, the
fundamental frequency of excitation was 12 MHz, the toroidal magnetic field on
axis was 4.0 kG, and the magnetic probe was 3.0 cm past the tokamak wall,
oriented to measure the toroidal component of the wave field. The RF pulse
was triggered a t a time during the plasma shot when the density was
- 0 x 1012

In each plot of raw signals, the top trace is the RF current (or

incident voltage) which excites the antenna and the bottom trace is the

- 370 TOKAMAK

ANTENNA
CURRENT

B PROBE
SIGNAL

B PROBE
SIGNAL

FIG. 7-15. Typical magnetic probe and antenna current signals for
two different time scales, using the ceramic-insulated loop
antenna. The probe was located in tokamak port # l .

TOKAMAK

TOKAMAK
PROBE
TRANSMITTING
LOOP ANTENNA

B PROBE
SIGNAL

FIG. 7-16. Typical magnetic probe and antenna
current signals for the ceramic-insulated loop
antenna; the probe was located in tokamak port #3.

FIG. 7-17. Typical magnetic probe and antenna forward voltage signals for the " T antenna; the probe
was located in tokamak port #1.

received signal from the magnetic probe; the time axis is the same for both.
Note that the antenna current and magnetic probe signals were taken from
successive tokamak shots (because only one data channel was available). The
antenna excitation current signal varied negligibly from shot to shot and was,
in fact, unaffected by the tokamak firing (i.e.,the same trace was obtained by
triggering the RF system alone).
The antenna current pulse shown in the center plot of Figure 7-15 is about
four periods long. The received signal has a distinct structure. It consists of a
series of envelopes of diministung amplitude, with the first envelope somewhat
delayed with respect to the antenna current pulse. As depicted in the diagram
a t the top, these signals apparently result from wave-packets traveling syrnmetrically in opposite &rections around the tokamak. The first envelope
results when the packets have traveled half way around the torus; the second
envelope results from the packets traveling another full circuit around the
torus. Note that the time delay between the centers of the antenna current
envelope and the first received signal envelope is roughly half of the time delay
between the centers of the first and second received signal envelopes; this is
consistent with the idea that the &sturbance travels toroidally with constant
velocity,
The lower plot in Figure 7-15 shows the signals from another tokamak
shot on a longer time scale; all other parameters were approximately the
same. The small signal appearing a t t -3.5 psec on the antenna current trace
is the first reflection from the incident pulse, having traveled through the
delay line and back to the antenna again. Note that the received probe signal
displays at least four distinct envelopes of diminishing amplitude. The successive envelopes appear to become broader and eventually the sqnal loses its
coherent structure. The envelopes which are clear are approximately equally
spaced in time, supporting the hypothesis that they result from wave-packets

making nlultiple transits around the tokamak.
Figure 7-16 shows typical raw signals when the magnetic probe was located
in tokamak port #3, 105" toroidally and 90" poloidally away from the RF
antenna. Other experimental parameters were the same as for the previous
figure. Now only the first two received signals envelopes are a s t i n c t . The peak
of the second envelope occurs about 300 nsec after the peak of the first
envelope, whereas in Figure 7-15 the peaks of the first two envelopes are
separated by -600 nsec. T h s is not unreasonable, if we postulate that two
wave-packets are indeed propagating in opposite directions around the
tokarnak a t constant: velocity ug . From Figure 7-15, the packet velocity is
approximately 5 x 10' ern/ sec . Then for the geometry of Figure 7-16, the
expected time delay between the arrivals of
counterclochse-going

wave-packets

at

the

the clochse-going and
magnetic

probe

is

240 nsec, in reasonable agreement with the obser-

vation.
The wave-packets propagating around the tokamak have a finite spatial
extent, and this Limits the precision of the time-of-aght measurement. The
toroidal length of the wave-packet could be shortened by reducing the length
of the current excitation pulse. T h s does not in general help to increase the
clarity of the received signal envelopes, however, because the shorter the
pulse is in time, the broader it is in frequency-space. Due to non-linear I s p e r -

a2u7
sion (
# O), the wave-packet envelope broadens as it travels; the shorter
%I

the excitation pulse, the greater the broadening. Experimentally, the optimum
antenna current pulse typically consisted of 4 - 5 cycles. Locating the magnetic probe 180" away from the RF antenna in the toroidal direction also
optimized the received signal envelope separation, as that position allowed the
greatest time between passes of the wave-packet.

The "T" antenna was also used to excite wave-packets. An example of the
raw signals using this antenna is shown in Figure 7-17. The excitation frequency was 12 MHz, and the toroidal magnetic field on axis was 5.5 kG. The
magnetic probe was positioned, as shown in the hagram, 180" toroidally away
from the R F antenna. The probe coil was 2.0 cm past the tokamak wall and
was oriented to measure the toroidal component of the wave magnetic field.
The first received signal envelope is clearly visible, although the second is less
distinct. The amplitude of the second envelope was typically less with the "T"
antenna than the corresponding amplitude using the loop antenna. Note that
the time delay of the received signal with respect to the excitation current is
longer than for the previously displayed shots. The reason for t h s , as will be
seen in more detail later, is that the group velocity decreases as the toroidal
msgnetic field increases; t h s shot used a substantially hgher toroidal field.

Wave- Packet P a r a m e t e r Scans
Approximately 500 tokamak shots were recorded (using the digital data

m for the wavesystem for the standard tokamak &agnostics and Polaroid f
packet signals) while changing various tokamak and RF parameters. The rnagnetic probe used for these investigations was located in tokamak port #1 and
was oriented to respond to the toroidal component of the wave magnetic field.
The resulting data set was analyzed to yield the packet velocity as a function of
the RF frequency, the toroidal magnetic field, the plasma density, and the
probe insertion past the wall; in addition, the amplitude of the packet signal
was investigated as a function of the probe insertion. The packet velocity was
defined as the major toroidal circumference (2.92 m) divided by the time delay
between the centers of the first and second received signal envelopes. In cases
where the second signal envelope was not clear (as with some of the data taken
with the "T" antenna), half of the major circumference was divided by the time

delay between the centers of the antenna current envelope and the first
received signal envelope. Because the signal records were photograpkc in
nature, the digital data analysis techniques used for all of the other experiments were unavailable. Tracings were (laborously) made of the signal on each
photograph, and the envelopes were sketched in; measurements were then
made on the tracing using a scale subdivided to 0.25 m. T h s procedure was,
of course, subject to some arbitrary judgements. Nevertheless, although the
precision of the experiment was not great, useful trends in the wave-packet
velocity were clearly established.
The variation of the wave-packet velocity with the RF excitation frequency
is shown in Figure 7-10 for both the ceramic-insulated loop antenna and the
"To antenna. The toroidal magnetic field on axis was 4.0 kG, and the RF pulse
was triggered a t a point during the density rise after the gas puff when the density was approximately 8 x lo1*

The results for the two antennas were

similar, with the packet velocity increasing almost linearly with frequency.
The variation of the wave-packet velocity with the toroidal magnetic field
is shown in Figure 7-19 for the loop and "T" antennas. The excitation frequency was 12 MHz and the density at the time at which the RF pulse was triggered was again -0x 1012cmS.

The packet velocities for the two antennas

were similar in amplitude and decreased as the toroidal field was raised.
The dependence of the packet velocity on the plasma density is shown in
F ~ u r 7-20
for the loop antenna; for this experiment the excitation frequency
was 12 MHz and the toroidal field on axis was 4.0 kG. The density was
increased a t the time of the RF pulse by keeping the gas puff constant and
varying the time during the discharge density rise a t which the RF pulse was
triggered. The results show that the packet velocity decreased somewhat as the
density increased.

MAGNETIC ANTENNA

PACKET VELOCITY
X 10' CM/SEC

FREQUENCY (MHz)
ELECTRIC ANTENNA

PACKET VELOCITY
X 10' CM/SEC

0.

20.
FREQUENCY (MHz)
10.

30.

FIG. 7-18. Variation of the wave-packet velocity with excitation frequency, for the ceramic-insulated loop antenna (top)
and the "T" antenna (bottom).

M A G N E T I C ANTENNA

PACKET VELOCITY

X 10' CM/SEC

0.

2.

4.

6.

T O R O I D A L F I E L D (KG)

E L E C T R I C ANTENNA

PACKET VELOCITY

X 10' CM/SEC

0.

2.

4.

6.

T O R O I D A L F I E L D (KG)

FIG. 7-19. Variation of the wave-packet velocity with
toroidal magnetic field, for the ceramic-insulated loop
antenna (top) and the "T" antenna (bottom).

MAGNETIC ANTENNA
8.

PACKET VELOCITY
X 10' CM/SEC

6. -

4. -

2. -

f = 12 MHz

B = 4 k G

o.,
0. 0

0. 5
1.0
DENSITY X 1013 ( c M - ~

1. 5

FIG. 7-20. Variation of the wave-packet velocity with plasma
density, for the ceramic-insulated loop antenna.

MAGNETIC ANTENNA

PACKET AMPLITUDE
(ARB UNITS)

-2.

0.

2.

4.

6.

8.

PROBE DISTANCE PAST WALL (CM)

PACKET AMPLITUDE
(ARB UNITS)

-2.

0.

2.

4.

6.

8.

PROBE DISTANCE PAST WALL (CM)
FIG. 7-21. Variation of the wave-packet peak magnetic
probe signal with probe insertion past the wall, for the
ceramic-insulated loop antenna (top) and the "T" antenna
(bottom).

MAGNETIC ANTENNA
8.

PACKET VELOCITY
X 10' CM/SEC

[ + o0 0

4. -

2.-

f = 12 MHz

B = 4 k G

n = 8 x 10'' cm-3

0.
-2.

0.
2.
4.
6.
PROBE DISTANCE PAST WALL (CM)

8.

ELECTRIC ANTENNA
8.

6. -

PACKET VELOCITY
X 10' CM/SEC

4.

2' -

:jH?&-i

f = 12 MHz

B = 4 k G
n = 8 x

0.
-2.

lo1* cm-3

2.
4.
6.
PROBE DISTANCE PAST WALL (CM)
0.

8.

FIG. 7-22. Variation of the wave-packet velocity with insertion of the probe past t h e wall, for the ceramic-insulated
loop antenna (top) and the "T" antenna (bottom).

The wave-packet peak magnetic field amplitude is plotted as a function of
the probe insertion in Figure 7-21 for both the loop and "T" antennas. The
excitation frequency was 12 MHz, the toroidal magnetic field on axis was
4.0 kG, and the plasma density a t the time of the RF pulse was

1x1013 ~ r n - ~ .

The packet amplitude increases substantially as the probe moves into the
plasma. This result shows that the observed wave-packet was not simply a
surface-wave phenomena; the fields propagated throughout the plasma.
Finally, the variation of the wave-packet velocity as the magnetic field
probe was inserted into the plasma is shown in Figure 7-22 for both the loop
and " T antennas. Again, the excitation frequency was 12 MHz, the toroidal
magnetic field on axis was 4.0 kG, and the plasma density at the time of the RF
pulse was

0 x 10'' cmW3. There was no significant change in the velocity as

the probe moved up to 6 cm into the plasma. Thus, the wave-packet propagated toroidally with a constant velocity, independent of radius.

7.2.4 Theoretical Model
A disturbance localized in space can be represented as

where B is some field component and k is the wavevector and is related to w
through the dispersion relation. For a localized wave-packet, the coefficient
Afk) is peaked around some specific wavenumber k, . If the dispersion is not
too great, then wfk) can be expanded:

where w, = w ( b ) . Using this expansion i n expression (7.8) ylelds [Jackson,
19601:

so the initial spatial &sturbance propagates without change in form (apart
from an overall phase factor) a t the group velocity vg =

aw
a.
Including higher-

order terms in the expansion 7.9 shows that the initial spatial form of the
packet spreads out or broadens as it propagates.
The theoretical models discussed in Chapter 2 describe the propagation of
waves in a cylindrical model of the tokamak. The fields away from the excitation source can be written as a sum over all of the normal modes:

where m and 1 are the poloidal and radial mode numbers, and the cylinder is
assumed to be of infinite extent so that k i l is not discretized. For each mode,
k i i is related to w through the dispersion relation, which, for the cold plasma,
zero-electron mass model, is given by equation 2.66. The observed modes were
propagating in the z-ihrection and were not evanescent in either the radial or
poloidal directions; hence k l i , m , and 1 must all be real. From Figure 2-9d, it
is expected that over most of the density and frequency parameter range of
the experiments, the only toroidally-propagating ( kil> 0) mode is the m = 1,
1 = 1 mode. This presumption is also supported by the results of section 7.1.3

whch show that the first 4-5 eigenmodes all have m = 1 poloidal character.

If only a single mode is propagating, the velocity at which a disturbance travels
can be easily calculated.
The group velocity in the toroidal direction for the m = 1 modes can be
found by calculating

*.

Making use of the dispersion relation for the zero11

electron mass model (equation2.66), the group velocity is predicted to
decrease sharply as the density decreases and approaches the cutoff density;

this can be seen intuitively from Figure 2-9a. The eigenmode curves can be
viewed as curves of constant and equally-spaced k l l in density-frequency space;
these curves are more closely spaced near the cutoff ( N = 0 ) curve. Thus, for a
given A w , the resulting change in k i l is greater for densities near the cutoff

A" is correspondingly smaller. The
curve than it is for higher densities, and Ak 11

fact that this dependence of group velocity on density was contradictory to the
experimental result (Figure 7-20) led to the consideration of further models.
The effect of a radial plasma density profile on the dispersion relation was
considered in section 2.4. Using the numerically computed dispersion relation
to calculate the group velocity did not change the form of the group velocity
dependence on density near the cutoff; the group velocity was still found to
decrease as the density decreased. The ra&al density profiles considered in
section 2.4 were restricted in that the density had to be sufficiently large that
kt: - zlk8 > O

everywhere, where

dielectric tensor and k o = -.

is the component of the cold-plasma

For the cutoff mode, this requires the density to

be large enough that the lower hybrid frequency ( o h ) is everywhere greater
than the excitation frequency ( w). The effect of a region of density low enough
that q,< w can be modeled with a vacuum layer a t the plasma edge. The most
dramatic effect of a vacuum layer, first &scovered by Paoloni [1975b], is that
the waves then have no cutoff, i.e., a t a given frequency they can propagate
down to very low densities.
Consider a radial plasma density profile given by
Region I:

n ( P ) = no

( 0 %(7.12)

Region11

+I

Assuming that the fields vary as e

=0
i ( k i j z+ m e - w t )

(PV P S PO)
, and neglecting the electron

mass so that E, = 0 , the solution for the z -component of the magnetic field in

region I is, from the discussion of section 2.2.6,

where p is the radial wavenumber given by equation 2.63 and the field has
been normalized so that EIz(po)= 1 . In region 11, the wave equations (2.36)
become uncoupled:

V? B% - T

~ =B

(7.14a)

where T 2 = k i f - u2 . The solutions for B, and E, in region I1 are then given

by

where I,

and Km are the modified Bessel functions and a, P, y, and 6 are

arbitrary constants. The boundary a t p=po is assumed to be perfectly conducting, hence E' and ED, vanish there. Continuity of the tangential component of the electric field a t the interface between the two regions requires

E% (pv) = 0 , hence it follows that y = 6 = 0 and Ez vanishes everywhere.
Using the relation between field components given by equation 2.52, the
remaining components can be written as:

in region I, where the factors GI, G2, G3, and G4 are defined in equation 2.51.
In region 11, the fields are given simply by

Region 11:

Since the boundary is perfectly conducting, it follows that the normal

= 0 . Using the
component of the magnetic field must vanish there: BBP(po)
above relations, this boundary condition yields a relation between a and 8 :

Continuity of the z-component

of the magnetic field a t the plasma-

vacuum interface, BIz(p,) = BDz(p,) gives another relation between a and @ :

which, together with equation 7.18, allows solution for a and /3 in terms of the
plasma parameters. Note that the 8-component of the magnetic field will, in
general, not be continuous across the vacuum-plasma interface, because the
infinite plasma conductivity in the z -direction arising from neglect of the
electron mass allows a surface current to flow in the z -direction.
The @-component of the electric field must also be continuous at the
plasma-vacuum interface, i.e., E',(p,) = Eir,(pv), which yields the following relation:

Since a and p are determined from the other boundary conditions, the above
equation represents the dispersion relation, relating w to k i t , the plasma
parameters, and the geometry.
Numerical evaluation of the dispersion relation for parameters appropriate to the Caltech tokamak demonstrates the lack of a cutoff when the vacuum
layer is present. Figure 7-23 shows a plot of k i t as a function of the plasma
density, for fixed frequency (12 MHz) and toroidal magnetic field (4.0 k c ) .
Evaluations for the m = 1, first radial mode, are shown for vacuum layers
0.1 cm and 1.0 cm t h c k , as well as the result for a constant density profile

without a vacuum layer. The density plotted is the line-averaged density for
each curve, i,e., no&. Note that the curve evaluated without a vacuum layer
Po
shows a cutoff at a density of -3.6 x 1012

while the curves for the cases

with vacuum layers do not display cutoffs. For a given (line-averaged) density,
the parallel wavenumber increases as the vacuum layer thickness increases.

0. 0

0. 5

1. 0

DENSITY X 1013 (CM -3 >

FIG. 7-23, Theoretical evaluation of kll as a function of the line-

averaged plasma density, for vacuum layers 0.1 cm and 1.0 cm
thick; the result without a vacuurn layer is also shown. Here
w / w , ~= 12 MHz, Bo= 4.0 kG, po = 16.2 cm, and the calculations are
for the m = 1, L = 1 mode. Note the absence of a cutoff when the
vacuurn layer is present.

1. 5

1 . 0 CM VACUUM L A Y E R

FIG. 7-24. a) Effect of various vacuum layers on the eigenmode &spersion curves
for the N = 1, 5 , m = 1, L = 1 modes. Here Bo=4.0 kG and pa= 16.2 cm; also
fl r w / o ~ . b) Eigenmode curves for the N = 1 -10, m = 1, L = 1 modes, with a
1.0 cm vacuum layer. Again Bo = 4.0 kG and pa = 16.2 cm. These curves should be
compared with those of Figure 2-9a, calculated for the same parameters but
without a vacuum layer.

Having obtained t h s result, it is easy to e x m i n e the effect of a vacuum
layer on the eigenmode curves in density-frequency space. Referring to Figure
7-23,

the

eigenmodes

occur

at

the

densities

for

which

k,,= -r?i (0.022)N emh1,where Ro is t h e the major toroidal radius and N is

Ro

the toroidal mode number. F g u r e 7-24a shows the effect of various vacuum
layers on the eigenmode curves for the rn = 1, 1 = 1, N = 1,5 modes. For a
given excitation frequency, the larger the vacuum layer, the lower the
corresponding density a t which the eigenmode occurs. For a given vacuum
layer thickness, the relative displacement of the eigenmode curve decreases
for higher toroidal mode numbers (larger k,,);this is also evident from Rgure
7-23. For a 1.0 cm thick vacuum layer, the first ten toroidal eigenmode curves
are shown in Figure 7-24b. Note that these results show the same qualitative
effect as was observed with the quadratic density profile considered in section 2.4.2. In that case, however, the waves still suffered a cutoff a t sufficiently
low densities.
Returning to the discussion of the group velocity for wave packets, the
dispersion relation in a n infinite cylinder inciudmg a vacuum layer (equa-

aw , for the
tion 7.20) can be used to calculate the group velocity, vg = ak 11
desired poloidal and radial mode. In keeping with the previous &scussion, we
consider only the m = 1, 1 = 1 mode; all other modes are assumed to be
evanescent in the toroidal direction. Numerical evaluations of the group velocity as functions of the density, frequency, and toroidal field, for various
vacuum layers, are displayed in Figures 7-25, 7-26, and 7-27; also plotted are
the data from the wave-packet experiments.
Figure 7-25 shows the group, or wave-packet, velocity as a function of the
plasma line-averaged density; the experimental points shown are for the
ceramic-insulated loop antenna, taken from Figure 7-20. For these experi-

10.

PACKET
VELOCITY 5.
X 10' CM/SEC

A CERAMIC-INSULATED
LOOP ANTENNA

0.
0. 0

0. 5
1.0
DENSITY X 1013 ( c M - ~ >

1. 5

FIG, 7-25. Wave-packet velocity as a function of the line-averaged
plasma density. The theoretical results (solid lines) are plotted for
vacuum layers of 0.0, 0.1 and 1.0 cm; the experimental points are
from Figure 7-20. Also w / ( 2 7 ~ =
) 12 MHz and Bo =4.0 kG.

PACKET
VELOCITY
X 10' CM/SEC

0.

10.

20.

30.

FREQUENCY (MHz)

FIG. 7-26. Wave-packet velocity as a function of the excitation frequency, The theoretical results (solid lines) are plotted for
vacuum layers of 0.0, 0.1, and 1.0 em; the experimental points are
from Figure 7-18. Also n -8 x 10'' cm-9 and Bo =4.0 kG.

PACKET
VELOCITY
X 10' CM/SEC

8.

2.

CERAMIC-INSULATED
LOOP ANTENNA

"T" ANTENNA

4.

6.

TOROIDAL FIELD (KG)

FIG. 7-27. Wave-packet velocity as a function of the toroidal magnetic field. The theoretical results (solid lines) are plotted for
vacuum layers of 0.0, 0.1, and 1.0 cm; the experimental points are
from Figure 7-19. Also w/(2n) = 12 MHz and n -8 x 1012 cm-'.

ments and calculations, w = 12 MHz and Bo= 4.0 kG. The most stribng feature
of the results is that the inclusion of a vacuum layer in the theoretical model
pelds a group velocity which increases as the density decreases, a t low densities. This is in contrast to the result shown for the case without a vacuum
layer which predicts the opposite dependence. Note that the curve for the
0.1 cm thlck vacuum layer shows reasonable agreement with the experimental
points; the other two curves show significant disagreement with each other
and with the experimental data a t low densities.
The theoretical and experimental variation of the group velocity with excitation frequency is shown in Figure 7-26. Here the density was 8 x 10" cmS
and the toroidal magnetic field was 4.0 kG; the experimental points are from
Flgure 7-18. The theoretical curves for the various vacuum layers do not &ffer
greatly above a frequency of -8 MHz. Below t'nis frequency, the curves for the
models with vacuurn layers show increasing group velocity as the frequency
decreases, whle the curve for the case without a vacuum layer continues to
decrease. The lowest frequency investigated experimentally, however, was
8.5 MHz, so the experiment cannot resolve the difference between the vacuum

layer and constant density models a t low frequencies. The agreement between
the experimental points and the theoretical curves is reasonable, although a t
higher frequencies the experimental points occur a t somewhat lower velocities
than the theoretical curves. Part of the discrepancy a t higher frequencies may
be due to the excitation of other poloidal modes (most likely the m = O mode).
Variation of the group velocity with toroidal magnetic field is shown in Figure 7-27. Here the frequency was 12 MHz, the density was 8 x 1012 cmS, and
the experimental points are from Figure 7-19. The agreement between the
theoretical curves and the experimental points is again reasonable, except for
the curve associated with the 1.0 cm vacuum layer; a t high fields that curve
predicts increasing group velocity with toroidal field, in contrast to the

experimental observation.
In summary, the agreement between the experimentally observed wave
packet velocities and the theoretically calculated group velocity is surprisingly
good. Inclusion of a vacuum layer a t the plasma edge in the theoretical model
pelds the experimentally observed increase in group velocity as the density
decreases, in contrast to the opposite result obtained without a vacuum layer.
The best agreement is found for a vacuum layer

0.1 cm thick. The depend-

ence of group velocity on frequency is similar for all of the theoretical models
over the frequency range pertinent to the experimental results; substantial
dflerences are predicted, however, a t points below the lowest experimental frequency. The experimental data and the theoretical curves show a similar
increase as the frequency is raised, but the agreement becomes worse a t high
frequencies; this may be the result of excitation of other poloidal modes.
Finally, reasonable agreement between the experiment and the theory is
observed for the variation of group velocity with toroidal magnetic field, except
for part of the curve calculated with the 1.0 cm t h c k vacuum layer.
The results of this study are not intended to suggest t h a t a vacuum layer
such as described actually exists, but rather that the very low density plasma
region near the tokamak wall can significantly affect the wave propagation. It
should be noted that the drscontinuous plasma density function assumed in
the vacuum layer model neglects the lower hybrid resonance which occurs
when kif - zLkg = 0 ; the singularities which appear in the wave equations a t
this point may be important. Proper treatment of this effect requires consideration of warm-plasma effects and is beyond the scope of this thesis.

7.3 RF Current Probe Experiment
A surprising result of the impedance invesigations reported in Chapter 6

was that the exposed electric field antennas (the bare plate and "T" antennas)
exhibited a large density-dependent loading whch was independent of wave
excitation. Nevertheless, the eigenmode excitation efficiency, defined as the
wave amplitude divided by the antenna current, was comparable for both the
electric field antennas and the more conventional loop antennas. The mechanism of coupling of the bare plate and " T antennas to the fast wave eigenmodes, however, is not clear. One possibility is that the electric fields associated with the antennas couple directly to the wave electric fields; another is
that the antennas drive currents in the plasma which then interact with the
magnetic component of the wave fields. Because of the large continuous
antenna loading, these antennas drive substantial RF currents throughout the
tokamak shot.
The RF current leaving an electric field antenna must return via some
path to the tokamak wall. The current within the plasma is carried largely by
the charged particles; displacement current is usually negligible. This section
presents the first direct observation of RF particle current in a tokamak
plasma. Using a plasma-compatable Rogowski current monitor with a small,
rotatable electric field antenna, the spatial distribution of this RF plasma
current was studied.

7.3.1 Experimental Method
The design and construction of the RF plasma current monitor was
described in detail in section 5.5, as was the antenna fixture with whch it was
used. The orientation of the antenna fixture in the tokamak chamber is shown
in Figure 7-28. As mentioned in section 5.5, in order to minimize perturbation
of the plasma by the Rogowski monitor, it was necessary to keep the major axis

ROTATABLE TUBE
(HOUSES COAXIAL CABLES)

COPPER TRANSMITTING

RF CURRENT PROBE

MAJOR A X I S

FIG. 7-28. Orientation of the rotatable antenna fixture with the R F
plasma current monitor in the tokamak port. Both the transrnitting antenna electrode and the current probe can be translated
and rotated independently.

of the monitor parallel to the static toroidal magnetic field. Therefore, in
order to investigate the distribution of current leaving the small electric field
antenna, the antenna element was moved rather than the current monitor.
Although the ability to move the electric field antenna in orthogonal directions with respect to the monitor would have been useful, the mechanical
dficulties associated with such a design, considering the rather limited access
within the tokamak, were prohibitive. The simplest degrees of freedom were
utilized, that is, rotation and translation of the antenna probe with respect to
the current monitor. As seen in the diagram and described in section 5.5, the
output semi-rigid coaxial cable of the current monitor passed through a
vacuum O-ring seal which allowed translation and rotation of the monitor; this
section of coax was colinear with the axis of the stainless steel tube into which
it passed. The semi-rigid coax lea&ng to the transmitting probe also made a
vacuum seal a t the end of the stainless steel tube but was rigidly fixed to the
tube. The stainless steel tube itself passed through a &fferentially-pumped
double O-ring seal which allowed it to translate and rotate. Thus, during an
experiment, the current monitor could be held fixed in position w l l e the
transmitting antenna rotated around it. Similarly, the rotatable antenna
could be fixed in a position and the current monitor could be independently
translated into or out of the plasma. Thus, even with the limited access to the
tokamak, several degrees of freedom were available for the investigations.
The transmitting antenna was fed from the impedance-matching box, as
were all of the other RF antennas. The only difference in operation was a special coaxial fitting which connected the output of the matching box to the BNC
connector on the end of the semi-rigid coaxial cable section. The one-meter
length of coaxial cable had a characteristic impedance of 50 R , and there was
no diEculty in tuning the matching network to the antenna, even though the
antenna impedance was rather high due to the small antenna area. The

output of the Rogowski monitor led, via a 50 R coaxial cable, to the RF electronics cabinet where it was amplified or attenuated as needed.
During an experiment, the usual standard tokamak diagnostics were
recorded, along with the RF forward and reflected voltages, the antenna RF
current, voltage, and phase, the output amplitude from the Rogowski monitor,
and the signal from a t least one magnetic probe. The desired result from the
experi-ments was the RF current passing through current monitor, normalized
to the total RF current leaving the transmitting probe. Since the antenna
current was measured a t the output of the matching box, approximately 1.2 m
from the antenna element itself, it was necessary to transform the measured
current using the following result from transmission-line theory:

where I, is the desired current at the antenna element, I, is the measured
current a t the input to the transmission line section, 2, is the measured
input impedance, Zo is the characteristic impedance of the section ( 50 R ) , L
is the length of the section, and #? is the wavenumber in the section. Therefore, the antenna input impedance was monitored, as described in Chapter 6,
during these experiments. The actual ratio of the observed current to the
current transformed to the transmitting antenna element turned out in practice to be within -20% of unity.

7.3.2 Experimental R e s u l t s
A typical tokamak shot, showing the raw signals for the plasma current,

the plasma density, the output from the Rogowski monitor, and the output
from the antenna current monitor, is &splayed in Figure 7-29. Note that the
antenna current and the Rogowski monitor signals have similar forms. The
absolute level of the antenna current depends on a number of factors,

PLASMA CURRENT
(KA)

PLASMA DENSITY

1. 0

RF ROGOWSKI
MONITOR CURRENT
(ARB. U N I T S )

RF ANTENNA
CURRENT
(ARB. U N I T S )

0.

5.

10.

15.

T I M E (MSEC)
FIG. 7-29. Raw data from a typical tokamak shot, showing the plasma current,
the line-averaged plasma density, the amplitude of the RF signal from the
plasma current Rogowski monitor, and the amplitude of the RF signal from the
antenna current monitor. For this shot, w/(2n) = 12 MHz and Bo = 4.0 kG.

including the setting of the impedance-matcbng network and the type of RF
exciter used. What is important in this study is the ratio of current passing
through the aperture of the Rogowski monitor to the current leaving the RF
antenna, as this gives information about the spatial distribution of the RF
current density.
Several checks were performed to insure that the observed signal from
the Rogowski monitor was in fact due to RF plasma current. Rotating the
current monitor by go0, so that its major axis was in the poloidal direction,
reduced the output from the monitor during a tokamak shot by more than
40 db. This is consistent with the idea that the major axis of the monitor must
be oriented in the toroidal direction so that plasma may stream through it.
Exciting the antenna with all tokamak fields firing, but with no gas in the
machne (and hence no plasma), ylelded no detectable output from the
Rogowski monitor; hence direct pickup by the monitor was negligible.
Two experiments were performed in order to investigate the spatial variation of the current driven by the RF antenna. For both of the experiments, the
antenna excitation frequency was 12 MHz,the toroidal magnetic field on axis
was 4.0 kG, and the data points were taken following the gas puff when the
plasma density reached a value of

8 x 1012 ~ m - ~ .

In the first experiment, the Rogowski monitor was fixed in location, with
its major axis oriented in the toroidal direction and its leading edge fixed
4.0 crn past the tokamak wall; this corresponded to the entire body of the
monitor being just past the wall. The transmitting antenna element was positioned so that its leadmg face was 2.7 cm past the wall. A series of tokamak
shots were recorded while the transmitting antenna was rotated a full 360"
around the Rogowski monitor; typically 4-6

shots were recorded at each

angular orientation. The records were then analyzed as described above. A
polar plot of the magnitude of the ratio of the current passing through the

Rogowski monitor to the current leaving the RF antenna is shown in Figure
7-30, In this plot, e = 0 corresponds to the orientation with the transmitting

probe directly above the current monitor, and the dashed circle corresponds
to a current ratio of 0.50. The plot shows a dramatic increase in the normalized Rogowski monitor current over a very small range of e, corresponding
approximately to the range over which the toroidal projection of the antenna
on the current monitor intersects the aperture of the monitor. Note that the
peak signal occurs for @-&goo, where the transmitting antenna is aligned
toroidally with the center of the aperture of the current monitor; the magnitude of this peak is -0.43.

Thus, even though the transmitting antenna was

some 5 cm away in the toroidal dzrection from the current monitor (at
e=9O0), greater than 40% of the current leaving the antenna passed through

the aperture of the monitor. The plot is nearly symmetrical about @=On,
whch shows that the current flows equally in both directions and is highly
localized along the toroidal direction. Thus, nearly all the current leaving the
antenna is accounted for.
The second experiment consisted of a radial current scan. The transmitting antenna was fixed in position, 2.7 cm past the wall and a t an angle of

e= 90" and the current monitor was inserted, starting a t a position withdrawn
from the plasma. Again, some 4 - 6

tokamak shots were recorded a t each

position of the current monitor, and the data records were analyzed as before.
The results are shown in Figure 7-31; here ( is the distance from the outer
face of the transmitting antenna copper plug (i.e., the face a t larger minor
radius) to the inner face of the current monitor. Thus, ( = O corresponds to
the position where the toroidal projection of the antenna on the current monitor just begins to intersect the aperture of the monitor. The results show that
the normalized current increases substantially a t just t h s point, consistent
with the idea that the current does indeed flow largely in the toroidal direction.

TRANSMITTING
PROBE
(ROTATES)-p I

-TOKAMAK
WALL

CURRENT
PROBE
(FIXED)

FIG. 7-30. Polar plot of the magnitude of the ratio of the RF current passing
through the plasma current monitor aperture (IRocowsxr)to the RF current at the
transmitting probe (IANTENNA),as a function of the angle of rotation of the
transmitting probe about the current monitor. The dashed circle corresponds to a
ratio of 0.50. The geometry of the experiment is shown beneath the plot; the
Rogowski current monitor is fixed in position while the transmitting probe rotates
around it. The position 8 = O 0 corresponds to the transmitting probe being vertically above the Rogowsln monitor.

TOKAMAK
WALL

TRANSMITTING
PROBE
(FIXED)

' (TRANSLATES)
FIG. 7-31. Plot of the magnitude of the normalized RF current passing through the

aperture of the plasma current monitor, as a function of the coorhnate # . The
geometry of the experiment is shown beneath the plot; the transmitting probe is
fixed a t 8=0° while the current monitor translates radially into the plasma, # is
the distance between the outer face of the transmitting probe (the face a t larger
minor radius) and the inner face of the current probe, as shown in the diagram.
For # < 0 , the toroidal projection of the transmitting probe intersects the aperture
of the current monitor.

Note that at the maximum insertion of the current monitor, when the toroidal
projection of the antenna element is completely within the aperture of the
current monitor, greater than 40% of the current leaving the antenna passes
through the monitor.
These experiments offer convincing evidence that the current being driven
by the electric field antennas flows largely along the toroidal direction and
maintains t h s structure some distance away from the antenna. Although a
theoretical explanation for this phenomenon is not yet a t hand, the results do
suggest a mechanism for the good coupling to toroidal egenmodes observed
with these antennas. The highly localized current density flowing from the RF
antenna results in a poloidal magnetic field compoent, and it is likely that this
component is responsible for the coupling. Note from Figures 2-10 (theory)
and 7-1 1 (experiment) that, although traditional loop antennas couple to the
toroidal ( z ) component of the eigenmode magnetic field because it is the largest component in the outer plasma, the poloidal ( 8 ) component is also
significant at the plasma edge and coupling to the wave fields through it should
be possible.

CHAPTER VIII
Summary and Conclusions

This thesis has discussed a &verse array of topics pertaining the coupling,
propagation, and damping of ICRF waves in a tokamak plasma. As in many
research endeavors, the original questions gave rise to many more. This
chapter briefly reviews some of the more important conclusions of this work.
Theoretical models were pursued largely to explain experimental results.
The model discussed in Chapter 2, however, of the effect of r a l a l density perturbations was of interest in its own right. It was found that even a very small
coherent radial density perturbation could greatly enhance the left-hand circularly polarized component of the wave electric field; i.e., the component
responsible for second-harmonic cyclotron damping. The model used a uniform magnetic field, however, and in the tokamak the magnetic field gradient
causes cyclotron damping to occur near a resonant surface consisting of a
cylindrical shell. Thus, more work needs to be done to assess the potential
effect of density perturbations on wave absorption in an actual tokamak.
The experimental work was lvided into two major areas: antenna
impedance measurements, and wave propagation investigations. The two were
not entirely unrelated; for instance, the antenna impedance due to eigenrnode
generation could be explained qualitatively and quantitatively in terms of wave
measurements of the cavity Qs.
An extensive study of the impedance characteristics of five different RF
antennas was performed by directly measuring the R F antenna voltage and

current and the phase between the two. The very small phase changes
observed during the plasma shot necessitated the design of an extremely sensitive phase detector with wide dynamic range.
The impedance observations separated into two regimes - those antennas
whose conducting elements were insulated from contact with the plasma, and
those which were not. The loading resistances of the insulated antennas (the
ceramic-covered and Faraday-shielded loop antennas) displayed peaks coincident with the generation of eigenmodes; this was expected and was modeled
reasonably well by a simple theory. Those antennas which were uninsulated
(the bare plate, bare loop, and "T" antennas) did not show t h s eigenrnoderelated impedance form. Rather, their loading resistance was dominated by a
much larger continuous background resistance which depended primarily on
the plasma density at the antenna location; t h s was, in fact, the "anomalous"
or parasitic ICRF antenna loading. An initially surprising result was that the
bare loop antenna exhibited a loa&ng resistance whch increased with density,
whle the bare plate and "T" antennas &splayed loading resistances with the
opposite density dependence. A simple model was developed whch explained
this behavior. The model does not include wave propagation and treats the
bare plate and "T" antennas as RF-driven Langmuir probes. The plasma sheath
presents a non-linear impedance whch can be modeled with the classical
probe I - V

curve, Using experimentally obtained probe curves, the RF

impedance of the driven antenna was calculated. Since the current drawn by a
probe is directly proportional to the plasma density a t the probe location, it
follows that the shunt resistance seen by the probe or antenna depends
inversely on plasma density; t h s result was confirmed experimentally. A more
severe test of the model involved the dependence of the loading resistance of
the bare plate or "T" antennas on the excitation level; good agreement with
the model was obtained here as well. Finally, the density dependence of the

bare loop antenna impedance was explained in a consistent fashion by considering a model with the plasma shunting the loop a t its midpoint.
Thus, the so-called anomalous density-dependent antenna loading was
adequately explained in terms of a very simple model based on particle collection through the plasma sheath. The above results suggest that particle collection by the antenna is an important effect which should be considered in the
design of ICRF systems. These effects have thus far not been included in any of
the sophisticated coupling codes used to study the design of high power ICRF
wave launchers [Bhatnagar et al., 1982; Elet and Chiu, 1982; Ram and Bers,
19821.
The wave experiments described in Chapter 7 began with stu&es of the
eigenmode &spersion characteristics.

The simple cold-plasma, periodic-

cylinder theory was seen to qualitatively explain many of the results. For
instance, the observation, based on phase measurements of the wave magnetic
fields, that the first 4-5 eigenmodes to appear after the initiation of a gas

puff were all m = 1 poloidal modes of low and successively increasing toroidal
mode number is exactly what the theory predicts. The toroidal mode number
could not be identified absolutely, but the first observed mode was likely an

N = 1 o r N = 3 mode. Similar results have been previously seen in other
laboratories [Takahashi, 1979; Coleman, 19831.

A new and surprising feature of the study, however, was the discovery that
the very simple electric-field antennas (the bare plate and "T" antennas) could
excite the ICRF eigenmodes with comparable efficiency to the traditional loop
couplers, in terms of wave amplitude normalized to antenna current. The
eigenmodes generated with these antennas were shown to have approximately
the same form and dispersion characteristics as those generated with the magnetic antennas; it is likely that they were in fact the same modes. This result
is believed to represent the first report of ICRF fast wave excitation with an

electric-field structure; other laboratories employ loop couplers exclusively.
Although the electric field antennas exhibited good coupling to the eigenmodes, it is not suggested that they are candidates for wave launchers for
heating experiments. The coupling efficiency was defined in terms of the
antenna current; since the electric field antennas also exhibited a large background loading resistance, much more input power was required to excite a
mode to a given amplitude than was needed with the loop antennas. Nevertheless, these coupling structures may prove useful for exciting ICRF waves for
diagnostic purposes; their extreme compactness is an advantage in a tokamak
environment where port space is a t a premium.
Wave darnping measurements showed the anomalously high &ssipation
seen in other laboratories, i.e., the eigemode Qs predicted from theory were
much hgher than those observed in the experiments. The damping was investigated both from the point of view of the cavity resonances (by measuring the
cavity Qs) and from the point of view of the waves being attenuated as they
travel axially around the torus (by measuring the wave damping length). The
two damping measurements are related by the group velocity, which was calculated from the two independent observations. One distinct feature of the
damping measurements was that the cavity Qs increased as the toroidal mode
number increased; this this is in contrast to the dependence predicted by
theory, but the effect has also been observed in other tokamaks [Bhatnagar et
al., 197BaI. Also, the wave damping was observed to increase as the fundamental ion cyclotron resonant layer entered the plasma, an unexpected result
since fundamental damping in a one-component plasma is usually neglected.
It is clear from the wave measurements that, although the wave propagation is described reasonably well by the theoretical models, the simple damping
mechanisms &scussed in Chapter 2 do not adequately explain the observed
damping. The disposition of the energy lost in the antenna-plasma system is

thus obscured. One possible explanation recently considered by Thompson
[1982] involves mode conversion of the fast wave near the second-harmonic

resonance layer. This damping mechanism does not, however, appear to
explain the large increase in wave damping observed when the fundamental
cyclotron resonance layer enters the tokamak. More work is certainly needed
in this area in order to identify the wave damping processes and to clarify the
eventual destination of the RF power.
In a different approach to the investigation of ICRF waves in a tokamak, a
wave-packet technique was used to study the propagation of waves in the torus
on very short time scales, unencumbered by the eigenmode resonances. Exciting the antennas with short bursts of RF current ( 4 - 5 cycles) launched
identifiable wave-packets which could be observed to travel many times around
the torus; the time delay between passes gave a direct indication of the
toroidal group velocity. These experiments represent the first observations of
ICRF wave-packets in a tokamak.
The group velocity was investigated as a function of a variety of plasma
parameters. The results were compared with theoretical models and, in general, reasonable agreement was noted. The observed increase in group velocity
as the density decreased could be explained with a model which included a
vacuum layer a t the plasma edge to simulate the very low-density layer that
probably exists a t the plasma-wall interface.
Finally, the RF Rogowski current monitor experiments demonstrated for
the first time direct measurement of RF particle current in a plasma. The
study showed that the current driven by the electric field antennas in the
outer tokamak plasma is spatially localized along the toroidal field lines. Thus,
it is proposed that the electric field antennas couple to the toroidal eigenmodes through the poloidal ( e) component of the magnetic field generated by
the current flowing along the localized toroidal path. I t would be of interest to

pursue these studies further to see, for example, how far away from the
antenna the localized current path extends and how it eventually returns to
the wall.

APPENDIX A
Coupling Efficiency t o ICRF Toroidal Egenmodes
and Transnrission between Two Identical Antennas~

Measurements by Hwang [I9791 indicate that with careful design of the
antenna and matching network so as to minimize losses in both, a t least 75%
to 65% of the available input power can be deposited in the antenna-plasma
system a t an eigenmode resonance. The disposition of this power, however, is
not clear: is it carried globally throughout the plasma by the fast wave fields,
or is the power perhaps being deposited locally near the antenna or in the
antenna-plasma sheath? Knowledge of the wave fields everywhere within the
plasma volume would permit calculation of the energy carried by the wave;
this measurement, however, is not practical experimentally. Magnetic probes
are routinely used to investigate the magnetic fields associated with the waves,
but they are restricted to the outer few centimeters of the plasma and to a few
specific locations around the tokamak.

If the energy being deposited in the plasma a t an eigenmode resonance is,
in fact, being transported globally by the fields, then the question arises as to
whether that energy can again be extracted with high efficiency by another
antenna. The transport of power from one antenna to another across the
tokamak would provide a convincing demonstration.

t This paper was presented a t the 21st Annual Meeting of the Division of Plasma Physics,
Boston, Mass. [Greene and Gould, 19791. The equivalent circuit model was originally
derived by R. W. Gould.

The expected transmission efficiency between the two antennas will not
simply be the square of the wave-launching efficiency since energy is also dissipated in the plasma. In the following derivation, the addition of a tightlycoupled receiving antenna into the plasma is found to reduce somewhat the
coupling efficiency.
Consider the simple circuit model of the system' shown in Figure A-la.
Here Ro is the internal resistance of the generator (50 R), & is the intrinsic
resistance of the antenna, and L, is the antenna inductance. C, and C2 are
the variable capacitors which provide the necessary impedance transformation. Each cavity eigenmode is represented by a resonant circuit with parameters selected for the proper resonant frequency and quality factor Q ; the coupling between the ith mode and each antenna is represented by mutual inductances M i . The transmitting and receiving antennas are assumed to be identical and to be located 180 degrees apart azimuthally along the torus.
When one high Q toroidal eigenmode is resonant, the non-resonant modes
have Little effect and will be ignored. In t h s case, the circuit can be redrawn
using Thevenin's theorem as shown in Figure A-lb. The form of the quantities

V' , Ri , and C' as functions of the circuit parameters need not be displayed
explicitly for the present &scussion. The loop equations for t h s circuit can be
solved and simplify considerably if we assume that both antenna circuits and
the

toroidal

eigenrnode

are

simultaneously

resonant,

i.e.,

that

Lac'=Lp Cp = I.
Then the circuit model can be further simplified as shown in
u2

In the absence of a receiving antenna, the input impedance is just

&+T

we@

. The latter term represents the resistance reflected back into the

1. For justification of this equivalent circuit approach, see Collin [1966].

I MATCHING I
TRANSMITTER, NETWORKI

FIG. A-1. a) Equivalent circuit model for the system. Each toroidal eigenmode is represented by an R-L-C resonant circuit. b). Equivalent circuit
when one toroidal eigenmode is resonant. c). Equivalent circuit when both
antenna circuits and the toroidal eigenmode are simultaneously resonant.

antenna due to the toroidal eigenmode while R, is the ohmic resistance of the
antenna. To maximize power transfer from the generator to the eigenmode,
C1 and Cz are adjusted so that R; =&

w 2 ~ 2
, and the ratio of power
+-

delivered to the eigenrnode to power available from the generator is then

w2
where y = -

With the ad&tion of the second antenna, it can be shown from Figure A-lc
that the ratio of the power delivered to the receiving antenna load to that
available from generator is

Rb

where x = 1+ -. For a given mode and antenna design, y is fixed, but x can

R,

be adjusted (by tlsning the variable capacitors CI and C z ) so as to maximize

the above ratio. This maximum occurs when ,y = 1 + (1+ 2 ~ ) note
~ ; that this
condition is n o t the same as that which leads to optimum power transfer to the
eigenmode in the absence of the receiving antenna. The optimum transmission efficiency is found to be

A comparison between the optimum efficiency of power transfer between
the two antennas, ?tnzns, and the efficiency of power coupling to the eigenmode, qCmp
, as functions of the parameter y is shown in Figure A-2. Note
that qtnrns is substantially smaller than r]c,up

for all values of y . I t is of

interest, under these circumstances, to give an accounting of the disposition of

F'IG. A-2. Plots of the optimum efficiency of power transfer
between the two antennas, qtrans,and the efficiency of power coupling to a single eigenmode in the absence of a second antenna,
qctrup
, as functions of the parameter y .

the available power for, say, y = 4 , corresponding to the case where, in the
absence of the receiving antenna, 80% of the available power would be
transferred to the elgenmode. Of the available power, 33%is dissipated in the
transmitting antenna, 33% is dissipated in the plasma, 8%is dissipated in the
receiving antenna, and 25% of the power is delivered to the receiving antenna
load.
The experiment performed on the Caltech tokarnak is shown schematically in Figure A-3. The glass-covered two-turn loop antennas which were used
were described previously by Hwang [1979]. Two identical antennas were positioned in two tokamak ports separated by 180" toroidally. Two identical
L-network impedance-matching boxes were constructed using hgh-& vacuum
capacitors. One of the antennas was connected, through a matching network,
to a 300 W broadband amplifier which was driven at a frequency of 18 MHz with
a signal generator; the other antenna was connected, through the other
matching network, to a 50 R load. Directional couplers monitored the power
being delivered to the antenna and to the load. Crystal detectors were used to
monitor the RF signal amplitudes; each detector was calibrated over a wide
range of input voltages and these calibrations were used during the data
analysis to properly unfold the sgnals.
The experiments were carried out in the first few milliseconds of the
tokamak discharge during the initial plasma density rise and fall; no gas
puffing was used. The impedance-matchng networks were tuned, from shot to
shot, to maximize the fraction of input power which was delivered to the load
for a particular eigenmode. A shot showing the highest power coupling which
was achieved is shown in Figure A-4. The maximum value of the power coupling
efficiency for one eigenmode is about 23%which agrees well with the prediction
of Figure A-2 for y~ 4 .

TOKAMAK

DIRECTIONAL
COUPLER

DUAL
DIRECTIONAL
COUPLER

fl

50 Q
LOAD

AMPLIFIER

RF
DETECTORS '

TIMING
ELECTRONICS

S$kt!

FIG. A-3, Schematic hagram of the power transfer experiment.

200.-

tr

" 100.-

I-

0. 0.

1.

2.

TIME (MSEC)

En

w x
LLw

Z>
C T Z

I-w

1.
TIME (MSEC)

FIG. A-4. Records of a tokamak shot showing the bghest poiver
RF pulse ends a t
transfer acheived. Tokamak fires a t t ~ 0 . msec;
t 1.9 msec. a) Power incident on transmitting antenna. b) Poiver
extracted from receiving antenna. c) Ratio of extracted to
incident power.

In summary, a simple equivalent circuit model has been developed to
predict the power transmission efficiency between two identical antennas,
given the coupling efficiency from a single antenna to an eigenmode. The
experimental results agree reasonably with this estimate and demonstrate
convincingly the global transport of power by the eigenmode.

APPENDIX B
Transient RF Heating of a Conducting Cylinder

When an RF current flows along a conducting wire or rod, joule heating
occurs due to the finite resistivity of the metal. At low frequency, the current
is uniformly distributed throughout the cross-section of the cylinder, and the
temperature profile is also uniform. At sufficiently h g h frequencies, the
current distribution is peaked a t the surface of the rod because of the skin
effect, and the temperature of the surface rises more rapidly than that of the
interior. It may therefore be important to include the skin effect in the thermal design of high power RF antennas and systems.
This appendix presents a calculation of the transient RF heating of a uniform conducting cylinder of infinite length, assuming that ohmic losses are the
only heat source and that there is no conductive or radative cooling. The
geometry of the problem is shown in Figure B-1. The heat equation with the
appropriate source term is Laplace-transformed, and a Green's function for
the equation is found. The solution of the problem is then obtained by inverse
Laplace-transformation and integration of the Green's function.

B-2 RF Current Profile
The current density ( j ) in an isotropic conductor is given by Ohm's law:
j = aE, where a is the conductivity and E is the electric field. Assuming that

all fields vary as e - i o f , the Maxwell equation for the magnetic induction B
becomes

RF
CURRENT

FTG. B-1. Geometry of the problem. RF current
flows axially through a cylinder of ra&us a ,conductivity u,diffusivity k , and heat capacity Cp,

where p and E are the permeability and dielectric constant of the conductor,
respectively, and c is the velocity of light. For all reasonable conductors at RF
frequencies, the inequality a>>wc: is satisfied, so the displacement current
may be neglected. Using Faraday's law then ylelds an equation for j :
Pj = -i(r,uwj

(B.2)

The solution in cylindrical geometry which is finite as T -, 0 is

whch can also be written in terms of

hero and b e i o functions [Ramo and

Whinnery, 19441.

The quantity 6 =: -is called the skin depth and is a measure of how

././.w

far the RF fields penetrate into the conductor. For the frequencies and
materials considered here, 6 <

Integrating equation B.4 over the cross-section of the cylinder ylelds the relation between the constant A and the total current I. f l o w in the cylinder:

where y =

and terms of order (Wy)%d

higher have been neglected. The

joule heating energy input to the conductor, per unit volume and unit time, is
then given by

- 423 where 7 is the resistivity of the conductor.

13-3The Heat Equation
The heat equation is given by

where u ( r , t )is the temperature, k is the thermal dlffusivity and

is the heat source term. Here C;, is the heat capacity per unit volume a t constant pressure, and g ( t ) is a function of time. The RF current d l be assumed
to be turned on at time t = 0 for a period t o ,so that

if
if
if

t

Ost sto

t >to

and the temperature a t time t = 0 will be assumed to be zero. The boundary
condition at r = a is the vanishing of outward heat flux:

(B. 10)
and the other necessary boundary condition is that the solution be finite as
r40.

B.4 Laplace Transformation
The Laplace transform of u ( r,t ) is defined by

where

is restricted to the half-plane R e ( s ) > O . For convenience, we

introduce a new function f ( T )=

such that q ( 7 . t ) = f ( ~ ) g ( t )Taking
the

CP

Laplace transform of the heat equation then ylelds

( B .1 2 )

where g^ ( s ) =

1 - e W 6 '0

, and the boundary condition at T = a

becomes

(B.13)

Defining new variables A2 = - and F ( r ) = -- f ( T ) , equation B.12 may be cast
into the form of an inhomogeneous modified Bessel equation:

( B .14)

B.5 Green's Function
A solution f (r ,T ' ;s ) to the equation

( B .15)
where 6 is the Dirac delta function, is a Green's function for the inhomogeneous equation. Let the solution of equation B.15 in the region O < r called f l , and in the region T ' it follows that

and

) =

D( A T )

for 0

(B.16b)

where B and D are constants, and I and K are the modified Bessel functions.

The solutions

f l and p2 in the two regions must be joined a t T =T'.

Integrating the differential equation B.15 from T

to T

=TI-E

=TI+&,

and taking

the limit E + 0, shows that the slope of the solution is discontinuous a t T = T ' :

(B.17)
A trial solution of equation B. 15 valid throughout the range 0 < T < a is

~ ( T , T ' ; s=) ~ f ~ ( r < ) f ' ~ ( r,, )

(B.1Ba)

where G is a constant, and T, and T, are defined by

(B.1Bb)
and

(B. 1Bc)
Substituting equation B. 18a into equation B. 17 then yelds
(B.19)

where

W[^rl,^rz]= rl dT, - FFZ

WIKo(C),Io(()]= -

[Abramowitz

is

the

Wronshan

and

Stegun,

19721, it

Noting

that

follows

that

DB. The Green's function solution can then be written as
W[T,(T'),&(T')]= -T'

where

and

B.6 Inversion
The Green's function solution is now inverse Laplace-transformed, using
the complex inversion integral [Mathews and Walker, 19701,

where t > O and q is real and chosen so that all the poles of the integrand lie
r the complex s-plane.
to the left of the c ~ n t o u in
Since l o f z ) is an entire function of z , and K o ( z ) is regular in the z -plane
with a branch cut along the negative real axis, an appropriate contour for the
evaluation of y , ( t ) = L-'fy^,(s)j is shown in Figure B-2a. The integrals along
contours B and F can be shown to vanish as R+rn, while the integral around
contour D vanishes as f + 0 . Furthermore, the integrals along paths C and D
cancel. Since

it follows that y l ( t )= 0.
The inversion of G 2 ( s ) is more complicated because of the poles from the
modified Bessel function I I in the denominator of the integrand. These poles
are simple, and occur a t s =s,

r-ewhere h, is the nth zero of the
a2

Bessel function J1. Since h, must be real [Abramowitz and Stegun, 19721, all
the poles lie on the negative real axis in the complex s-plane.

FIG. B-2. a) Contour for the evaluation of L-' [y^l(s)j. b) Contour
for the evaluation of L-l ly^z(s)j.The poles occur a t s = -g k /a2.

The contour for evaluation of L-ltQ2(s)j is shown in Figure B-2b. Again,
the contribution from the paths B and F vanish as R
from the integral around path D, in the limit

-j

=. The contribution

< + 0 , is

The integrals along paths C and D contribute

where 4 is the residue of the integrand a t the pole s,. Evaluation of the residue ylelds

and the inverse transform is then

The inverse transform of equation B.%Uais then given by the convolution
theorem:

Using equations B.26 and B.9, the above reduces, after some algebra, to

where t , is the lesser of

, and t , is the greater of

The desired solution of the original heat equation, U ( T , t), may now be
obtained by integrating the Green's function:

where

and we have defined a new function,

B.7 Convergence
The convergence of the surnxnation in equation B.29a can be investigated

by examining the behavior of &, the nth term in series, as n -+m. We note
t h a t for n large, An -(n + &)T, and for z large (where z is a dumrny vari4

able),

Then a upper bound on the magnitude of S, is

where K is a constant independent of n
Using a integral representation for Jo,the function W,(y) can be written
as
00

2 ysin (h,cosht) - h,cosht cos()\, cosht)
Wn(?) = F.jdt .
4 y +
coshZt

(B.32)

Expressing the sine and cosine terms in complex exponential form, the
integral can be evaluated asymptotically for large n using the method of stationary phase [Copson, 19651. The result is

--

The magnitude of the nUL
term in the series then decreases as n ', for n -, m,
and hence the series does converge.

B-8 Discussion
The term involving the summation in equation B.29a can be neglected for
large times ( >>to),so that

It is easy to show that the energy per unit volume absorbed by the rod at any
time t is just

Hence u,(t,)

corresponds to the temperature that would be obtained if the

same energy that was deposited in the cylinder by the RF current a t time t

had been deposited unif orrnly over the cross-section.
The function u(r,t ) has been evaluated numerically for a variety of
parameters. A n example which shows some prominent features of the solution
is shown in Figure B-3. The peak in the temperature profile always occurs a t
the surface of the rod. As expected, the surface temperature rises more
quickly than the interior temperature. After the RF pulse ends at time to,the
temperature profile begins to flatten out; the temperature in the outer part of
the rod decreases while that of the inner part continues to rise. The profile d l
eventually become uniform with a temperature u = u,(to).
To investigate the solution over a wide range of parameters, it is desirable
to recast equation B.29 in terms of dimensionless variables. The scale lengths
appearing in the problem are a , the cylinder radius, and 6, the skin depth.
The ratio of the two, y , has already been defined. A characteristic time for
a2
heat flow is tc = - We introduce new &mensionless variables:
k '

t> rc = -t, ,
tc
tc

7, = -

TO

= -,
tc

m.The quantity O

= - p = -,7 and O[p. T ) =
UW(t<)
tc

represents the ratio between the physical temperature and the temperature
that would result if the same amount of energy were deposited uniformly; it is
a measure of the importance of the s h n effect on the temperature profile.
Mahng the above substitutions, equation B.29 becomes

Since we are mostly interested in peak temperatures, we shall limit our further
discussion to the case T S r OSO
, that T , = T.
A numerical evaluation of O is plotted as a function of p, for various

times T , in figures B-4a (for y = 10) and B-4b (for y = 100). The temperature
distribution is sharply peaked at the surface (p = 1) for times r << 1, and it

F'IG. B-3. Typical solution for u ( r , t ) .
The RF pulse begins a t time t = 0 and
ends a t t = t o . Here y = 1 0 0 , a ~ 0 .em,
t o= 0.7 msec, w / 2 7 r = 10 MHz, ~ = 5 . B lo4
mho/cm, k =1.14 cmz/sec and Cp =3.46
joule/("C-cm3). Vertical axis units are
arbitrary.

FIG. B-4. Plot of Q ( ~ , T a) s a function of p , for various T .
a) Evaluation for y = 10. b) Evaluation for y = 100.

gradually flattens out to a nearly uniform profile for T 2 1. At a given time T,
the distribution is more sharply peaked for larger y since the corresponding
skin depth is smaller and the energy is being deposited in a narrower region a t
the surf ace.

A quantity of some importance for the design of RF systems is the peak
surface temperature, O(1, 7). It is important to limit the peak temperature,
both to prevent excessive impurity evolution from adsorbed gases on the
antenna surface and to prevent possible vaporization of the surface.
A plot of O(1, T) as a function of T is shown in Figure B-5 for a wide range

of y. The skin effect causes greatly enhanced surface temperatures for T<< 1,
i.e., when the energy is deposited on a time scale short compared to the
characteristic heat diffusion time. O(1, T) is again seen to increase as y
increases.

As an example, consider a copper rod, 0.63 cm in radius, carrylng an RF
current at a frequency of 10 Mhz. Then y

300., and since k

1.14 cm2/sec

for copper, it follows that t, =0.35 see. For a 1.0 msec long RF pulse,
~ 0 = 2 . 9 lo9,

and, from the numerical evaluation, O(1, TO) = 10.5. Then

u ( a , to)= 10.5u,(to)

= 1.0 x 10-'1$ ("C), where u, has been evaluated using

7 = 1.72 x l o * R-cm

and 5 = 3.46 joule/ (" c cm3). So, for example, an RF

current of 50 kA (rms) would yleld a peak surface temperature rise of about
900 " C.

For parameters characteristic of the experiments described in t h s thesis
( a = 2 em, w / 27~.= 12 Mhz, t o= 5 msec, I
= 350 A
u,(to)

(rms)), the results

are

=0.51 " C and O(1, T ~ =2.65,
hence u ( a , to)= 1.35"C. Therefore heating

of the antenna structure from ohmic currents will be negligible.

FIG. B-5. Plot of normalized surf ace temperature @(I, T )
as a function of 7, for various y.

B-8 Conclusion
A solution has been presented for the transient temperature distribution

of a conducting cylinder heated by an RF current and has been solved numerically for a wide range of parameters. Significantly enhanced peak surface temperatures are found for short times ( T < <1). The effect is very small for
parameters appropriate to the experiments performed on the Caltech
tokamak but may be important for systems operating in other regimes with
very h g h currents or hgher frequencies.

CES
Abramowitz, M. and Stegun, I. (eds.), Handbook of Mathematical F'unctions
(National Bureau of Standards, Washington, D.C., 1972), passim.
Adam, J., et al., "Wave Generation and Heating in the ST-Tokamak at the Fundamental and Harmonic Ion Cyclotron Frequencies," in Hasma Physics and
Controlled NUclear FIUSion Research 1974 (Proc. 5'h Int. Conf., Tokyo,
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