Propagation of the Fast Magnetosonic Wav

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Propagation of the Fast Magnetosonic Wave in a Tokamak Plasma
Citation
Hwang, David Li-Shui Quek
(1979)
Propagation of the Fast Magnetosonic Wave in a Tokamak Plasma.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/f15y-jy67.
Abstract
The propagation of the fast magnetosonic wave in a tokamak
plasma has been investigated at low power, between 10 and 300 watts,
as a prelude to future heating experiments.
The attention of the experiments has been focused on the understanding
of the coupling between a loop antenna and a plasma-filled
cavity. Special emphasis has been given to the measurement of the complex
loading impedance of the plasma. The importance of this measurement
is that once the complex loading impedance of the plasma is known,
a matching network can be designed so that the r.f. generator impedance
can be matched to one of the cavity modes, thus delivering maximum
power to the plasma. For future heating experiments it will be essential
to be able to match the generator impedance to a cavity mode in
order to couple the r.f. energy efficiently to the plasma.
As a consequence of the complex impedance measurements, it was
discovered that the designs of the transmitting antenna and the impedance
matching network are both crucial. The losses in the antenna and
the matching network must be kept below the plasma loading in order to
be able to detect the complex plasma loading impedance. This is even
more important in future heating experiments, because the fundamental
basis for efficient heating before any other consideration is to deliver
more energy into the plasma than is dissipated in the antenna system.
The characteristics of the magnetosonic cavity modes are confirmed
by three different methods. First, the cavity modes are observed
as voltage maxima at the output of a six-turn receiving probe.
Second, they also appear as maxima in the input resistance of the transmitting
antenna. Finally, when the real and imaginary parts of the
measured complex input impedance of the antenna are plotted in the
complex impedance plane, the resulting curves are approximately circles,
indicating a resonance phenomenon.
The observed plasma loading resistances at the various cavity
modes are as high as 3 to 4 times the basic antenna resistance (~ .4 Ω).
The estimated cavity Q’s were between 400 and 700. This means that
efficient energy coupling into the tokamak and low losses in the antenna
system are possible.
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:
Unknown, Unknown
Defense Date:
15 June 1978
Record Number:
CaltechTHESIS:07182014-104710068
Persistent URL:
DOI:
10.7907/f15y-jy67
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
8562
Collection:
CaltechTHESIS
Deposited By:
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Deposited On:
18 Jul 2014 18:23
Last Modified:
26 Nov 2024 22:42
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PROPAGATION OF THE FAST MAGNETOSONIC
WAVE IN A TOKAMAK PLASMA
Thesis by
David Li-Shui Quek Hwang

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

California Institute of Technology
Pasadena, California
1979
(Submitted June 15, 1978)

1978
DAVID LI-SHUI QUEK HWANG

-iiACKNO\~LEDGMENTS

I \'Jould like to express my deep appreciation to my thesis advisor,
Professor Roy W. Gould, for his guidance, encouragement, and many direct
contributions throughout the course of this investigation.

His insight

and knowledge of plasma physics have been extremely enlightening.
I am indebted to Professor William Bridges, Dr. James Long, and
Professor Hardy Martel for the useful discussions and suggestions on both
circuit theory and experimental techniques.

Many of their suggestions

have led to fruitful results.
Special thanks go to Dr. Gary Bedrosian and Professor Paul Bel1an
for their critical reading of the manuscript and useful comments for
improvements.

Special gratitude is extended to Dr. Mario Simonutti for

help and discussions on the initial theoretical part of this work.
I would like to thank Mr. Frank Cosso for his skillful assistance
in the contruction of experimental instruments.

To Mrs. Edith Huang

I would like to extend my appreciation for her help on writing and
debugging many computer programs.

I am very grateful to Mrs. Ruth

Stratton and Mrs. Verona Carpenter for their excellent typing, and
especially to Mrs. Stratton for proofreading the original text.
The generous financial support from the U. S. Departmen t of Energy
in carrying out this work is gratefully acknowledged.
To my wife, Mona, and my sister in-law, Lenita, thank you for the
help in preparing the manuscript.

Finally, I would like to dedicate this

thesis to the members of my family, especially to my parents, for their
constant support, encouragement, love, and understanding wit hout which
this work would not have been possible.

-iii-

ABSTRACT
The propagation of the fast magnetosonic wave in a tokamak
plasma has been investigated at low power, between 10 and 300 watts,
as a prelude to future heating experiments.
The attention of the experiments has been focused on the understanding of the coupling between a loop antenna and a plasma-filled
cavity.

Special emphasis has been given to the measurement of the com-

plex loading impedance of the plasma.

The importance of thi s measure-

ment is that once the complex loading impedance of the plasma is known,
a matching network can be designed so that the r.f. generator impedance
can be matched to one of the cavity modes, thus delivering maximum
power to the plasma.

For future heating experiments it will be essen-

tial to be able to match the generator impedance to a cavity mode in
order to couple the r.f. energy efficiently to the plasma.
As a consequence of the complex impedance measurements, it was
discovered that the designs of the transmitting antenna and the impedance matching network are both crucial.

The losses in the antenna and

the matching network must be kept below the plasma loading i n order to
be able to detect the complex plasma loading impedance.

Thi s is even

more important in future heating experiments, because the fu ndamental
basis for efficient heating before any other consideration i s to deliver
more energy into the plasma than is dissipated in the anten na system.
The characteristics of the magnetosonic cavity modes are confirmed by three different methods.

First, the cavity modes are observed

as voltage maxima at the output of a six-turn receiving probe.

-i v-

Second, they also appear as maxima in the input resistance of the transmitting antenna.

Finally, when the real and imaginary parts of the

measured complex input impedance of the antenna are plotted in the
complex impedance plane, the resulting curves are approximately circles,
indicating a resonance phenomenon.
The observed plasma loading resistances at the various cavity
modes are as high as 3 to 4 times the basic antenna resistance (~ .4 ~).
The estimated cavity Q1 s VJere between 400 and 700.

This means that

efficient energy coupling into the tokamak and low losses in the antenna
system are possible.

-vTABLE OF CONTENTS
I.

II.

INTRODUCTION
1.1 Introduction to Tokamak Fusion and Plasma Heating
1. 2 Summary of Previous \~ork on Magnetosoni c \•lave Heating
in Tokamaks
1.3 General Thesis Outline

COLD PLASMA THEORY AND CIRCUIT MODELING OF THE CAVITY MODES

2.1
2.2

17
25

2.3
2. 4
2.5
2.6
2. 7
2.8
2. 9
III.

IV.

Theory for a Cold Uniform Cylindrical Plasma Cavity
Summary of More Sophisticated Theories of Magnetosonic
Ca vi ty t·1o des
Circuit Model of the Antenna-Cavity Coupling
Transient J'v1eas urements of Steady State Quantities
Impedance Matching
Relations between Circuit Parameters
Antenna Efficiency
Simulation of Cavity Resonances
Q Circles

28
36
38
40
45
47
51

GENERAL EXPERIMENTAL SETUP

3.1 Tokamak Characteristics
3.2 Plasma Diagnostics
3.3 Summary of Plasma Parameters
3.4 Digital Data Acquisition System

EXPERIMENTAL SETUP FOR THE R.F. MEASUREMENTS

4.1
4.2
4.3
4.4

67
71
74
77

Experimental Arrangement for Transmission Measuremen t
Antenna and Matching Network Design
Plasma Loading Resistance Measurements
Phase Measurement

57
64
64

EXPERIMENTAL RESULTS

5.1
5.2

Transmission Measurements
Plasma Loading Impedance in the Absence of the
Cavity t~odes

-vi5.3
5.4
5.5
5.6
5.7
VI.

Plasma Loading Resistance at the Cavity Resonances
Reproducibility of the Plasma Loading Resistance
~1eas urement
Complex Plasma Loading Impedance Measurement
Cavity Q, Antenna Coupling Coefficient, and Antenna
Efficiency
~~atching Impedances at the Cavity Resonances

92
97
98
llO
115

CONCLUSIONS

120

6. l

Summary

120

6.2

Future High Power Heating Experiments

123

Appendix a.

Transmitting Antenna and ~latching Netv.1ork Cons tructi on

126

Appendix b.

Cold Plasma Theory of the ~~agnetosonic Cavity
r1odes

132

Resistivity Loading of the R.F. Wave by Tokamak

142

Appendix c.

\tJa ll

REFERENCES

146

-1-

I.
1.1

INTRODUCTION

Introduction to Tokamak Fusion and Plasma Heating
In order to produce net energy from controlled thermonuclear fusion,

two physical parameters, the product of the plasma density n a ~ rl the
confinement time T , and the ion temperature
satisfy the Lawson criterion.

Ti must simultaneously

The Lawson criterion is a statement of

energy break-even in a thermonuclear reaction, where the energy gained
in the reaction equals the energy lost due to both radiation and particle
losses.

For example, the Lawson criterion for the deuterium and tritium

reaction
D + T ~ 4He (3.5 MeV) + n (14.1 MeV)
is that Ti ~ 10 keV, and nT ~ 10 14 .

(l.l)

Among the many methods under study

to reach the Lawson criterion, on e device that has made a great deal of
progress toward achieving these parameters is the tokamak.
A tokamak is a toroidal magnetic confinement device with a toroidal
magnetic field and an inductively induced toroidal cu r rent (Figure 1.1;
and for details see Section 3.1).
purpose:

The toroidal current serves a two-fold

1) to produce a poloidal field which provides the proper rota-

tional transform for plasma equilibrium; 2) to heat the plasma by ohmic
dissipation due to the plasma resis t ance.
There is a limit to the plasma temper ature that can be rea ched by
heating the tokamak plasma with the toroidal current, becaus e the plasma
resistance decreases with increasing plasma temperature. To dissipate the
same amount of ohmic power, I 2R, in the plasma at a higher tempera.ture,

the plasma current, Ip' must be higher since the plasma resistance R is

-2-

Plasma
Peloidal
Magnetic
Fi~ld

Plasma

Figure 1.1
Schematic of a Tokamak (From Principles of
Plasma P~ics, by N. A. Krall and A. W.
Trivelpiece)

-3-

lower.

The limit on the magnitude of the toroidal plasma current that

can be used for ohmic heating is the condition for plasma equilibrium
which specifies the maximum allowable poloidal magnetic field for a
given toroidal magnetic field (see Section 3.1).
The inverse temperature dependence of the plasma resistance is the
result of the Coulomb interaction of the charged particles in a plasma.
The plasma resistivity, or the Spitzer resistivity, is as follows:
(l. 2)

where A= 12n(E:okBT/e2)3/2 I

;n;' and Te is in keV. Thus, after the

plasma temperature has reached between l and 3 keV,

other plasma heat-

ing methods must be used to supplement ohmic heating and to bring the
ion temperature to the required value.

Currently, the two major pro-

posed methods for auxiliary heating of a tokamak plasma are neutral beam
injection heating and radio frequency wave heating.
The reason for using neutral beam injection to heat the plasma
instead of ion beam injection is that charged particles cannot penetrate
the magnetic field of the tokamak.

The neutral particle injection scheme

is to inject a beam of energetic neutral particles across the magnetic
field.

The neutral particles can deliver energy to the ions by charge

exchange with cold ions in the plasma, thus resulting in energetic ions
and cold neutrals which will escape.
The neutral beam is produced by passing an intense ion beam through
a gas neutralizing cell.

Energetic neutrals can be formed by electron

capture by positive ions, electron stripping by negative ions
ciation of molecular ions [1].

or,disso-

Injection experiments using either

-4hydrogen or deuterium beams with powers up to 700 kW have been performed
in various tokamaks around the world.

A heating efficiency of 57%, for

example, has been reported by the TFR group in France [2].
Although neutral beam heating has enjoyed success in the present
experimental tokamaks, there are doubts about its efficiency in heating
the bulk of the ion distribution in a reactor-size tokamak, which would
be much larger.

Because of the increase in size of the reactor tokamaks,

higher energy neutral particles are needed in order to penetrate to the
center of the tokamak.

At present, difficulties have been encountered

with efficient neutralization of ion beams with energy greater than
120 keV.

Therefore, alternatives to the neutral beam heating must be

studied for the auxiliary heating of a reactor-size tokamak.
The use of radio frequency electromagnetic waves to heat a plasma
was proposed in the early days of plasma physics.

Efficient high power

wave generators in the radio frequency range are at present readily
available, and so the technological basis for using electromagnetic wave
heating is quite sound.

Because of technical know-how in radio wave

generation, the cost of using r.f. heating in a reactor-size tokamak
could be lower than that for neutral beam heating.
The heating of a plasma using r.f. waves has been summarized by
T. H. Stix as follows [3].

First, the r.f. wave must be generated and

delivered to the plasma.

The r.f. energy is then coupled to the plasma

and an ''efficient way to couple the r.f. energy into the plasma is to
match the frequency and parallel wavelength of the driving field to those
of a natural mode in the plasma, thereby exciting a 'coupling resonance'."
The r.f. wave interacts with the plasma through "either linear or nonlinear

-5-

processes". There is some absorption process of the v1ave in t he plasma,
"which competes with eddy current dissipation in the walls". Finally,
there must be "effective thermalization of the energy added to the plasma".
One proposed method for r.f. heating is the use of the magnetosonic wave to heat the ions.

The attractive feature of this method is

that the wave energy should couple directly to the ions, instead of
heating the electrons first, then relying on electron-ion collisions to
transfer energy to the ions.

The propagation of the magnetosonic wave

in a magnetized plasma can be approximately described by the cold plasma
dispersion relation.

The cold plasma dispersion relation indicates that

there are two branches of waves that can propagate when the wave frequency is approximately equal to the ion cyclotron frequency. One branch
is the ion cyclotron wave which is left-circularly polarized (LCP) and
has a resonance, i.e., a large plasma response to the field, at the ion
cyclotron frequency (Appendix b).

The ot her branch is the magnetosonic

wave which is right-circularly polarized (RCP) and does not have a
resonance at the ion cyclotron frequency.
When the magnetosonic wave is propagating in a plasma-filled
metallic container such as a tokamak, the appropriate EM pr oblem can be
thought of as that of wave propagation in a dielectric-filled cavity.
At first sight one would not expect to be able to couple energy to the
ions using the magnetosonic wave, because it has the wrong pol arization.
However, when temperature effects are included in the dispers i on relation, one finds that the magnetosonic wave is no longer purely RCP, but
contains a small left-handed component.

F. Perkins [4] has worked out

the damping decrements of the magnetosonic wave in a finite temperature

-6-

plasma at both the ion cyclotron and twice the ion cyclotron frequencies.
It is found that the damping is not strong, and it is a linear function
of the ion temperature of the plasma.
Wave propagation in a plasma-filled cavity can be described by the
dispersion relation of the wave and the proper boundary conditions.

keep the theory simple and yet retain the essential features of the
physics, some approximations are introduced.
cold, uniform, and magnetized.

The plasma is assumed to be

The tokamak is approximated by a cylindri-

cal cavity with perfectly conducting walls and a periodic boundary condition in the axial direction.

The wave propagation problem is solved in

cylindrical coordinates with the plasma magnetized along the axial direction.

From the cold plasma theory, the dispersion relation, w = w(~). is

obtained, where k is the wave vector (see Section 2.1).

Once the plasma

is placed in the cylindrical cavity, only discrete values of~ which
satisfy the boundary conditions can exist, i.e., the conducting wall
boundary conditions being that the tangential electric field and the normal magnetic field must vanish at the boundary.

By exciting the magneto-

sonic wave at the eigenmode frequencies, w = w(~), standing waves are set
up in the cavity; thus, one has a forced oscillation system which will
enhance the damping of the wave.

With the simple assumptions used here,

there are a few experimental effects that are neglected in the theory,
for example, the toroidal effects, effects due to both density and magnetic field gradients, the poloidal field effects, finite temperature
effects, and the effects of the cavity wall resistance.

Nevertheless,

the simple cold plasma theory describes the propagation of the magnetosonic wave reasonably well (see Section 2.2 for references to other

-7theories that include these effects).
The problem of efficient r.f. heating of a tokamak plasma using
the fast magnetosonic wave can be studied in the following way.

First,

the physics of the cavity modes must be understood experimentally. This
can be done by studying both the standing wave patterns of the eigenmodes in the tokamak and the plasma loading behavior at the transmitting
antenna during the passage through a cavity mode.

Second, efficient ways

to feed r.f. energy into a plasma at the cavity modes need to be examined
carefully.

By knowing the complex plasma loading impedance, t he

antenna and the matching network can be designed so that the generator
impedance can be properly matched to the antenna at a cavity mode.
ensures maximum power input into the plasma.

This

Third, the duration of the

cavity modes used for heating must be long enough during the discharge
to get any significant increase in plasma temperature.

Usually in the

present day tokamak discharges, the duration of the eigenmodes is not
long enough for effective plasma heating.

The reason is that a particu-

lar cavity mode is excited only during the time when both the plasma density and the input frequency satisfy simultaneously the dispersion relation and the boundary conditions.

As soon as the plasma dens ity is

changed sufficiently, this mode no longer propagates in the t okamak, and
so no more wave heating is possible.

one proposed way to tra ck the modes

is by changing the input frequency of the transmitter to compen sate for
any changes in the density which will shorten the duration of the modes.
Finally, high power experiments can be done to study the phys ic s of the
damping mechanism of the wave by the plasma, and the actual temperature

-8-

increase of the plasma due to the r.f. power input.
1.2

Summary of Previous Work on Magnetosonic Wave Heating in Tokamaks
The first series of magnetosonic wave heating experiments in the

United States was done in
Princeton [5,6].

the ST tokamak and the ATC tokamak at

Initially, a low

power experiment was performed on

the ST tokamak to show the existence of the magnetosonic wave and to
study the resistive loading of the transmitting antenna by the modes.
The modes were identified by measuring the standing wave patterns using
a number of probes placed around the tokamak.
mode numbers, m and N, were obtained.

Both peloidal and toroidal

Matching networks were used so

that the r.f. generator impedance could be matched to one of the cavity
resonances [7].

The resistive loading of the transmitting antenna by

various modes was measured, and resistive loading results indicated that
efficient wave generation in the tokamak was possible.
One of the ST tokamak experimental results which was predicted by a
theory worked out by Chance and Perkins [8] was that the m = -1 mode \'/as
split by the effect of the peloidal magnetic field (where the fields vary
as ei(kz+me-wt)), which makes the phase velocity of this mode different
when propagating in the opposite direction along the toroidal axis. In other
words, the dispersion curves for them= -1 modes with positive and negative
N, the toroidal mode number, are different from each other. The splitting of
them= -1 mode appears as "double humps" on the cavity resonance peaks.
High power experiments were done with powe~ level up to 1 MW in the
ST tokamak with a hydrogen plasma, and a typical ion temperature increase
of 100 eV was observed [8].

This corresponded to a heating effictency of

-9-

20%.

High power experiments in a deuterium plasma were also performed

in the ST and ATC tokamaks.

However, the cavity Q of the cavity modes

measured in the deuterium plasma is much lower than the theoretical
predictions.

At present, it is believed that the observed discrepancy

is due to the two ion hybrid resonance effect between the deuteron plasma
and the proton impurities in the tokamaks [6].

In this thesis the exper-

iments are done in a hydrogen plasma; thus, there are no two-ion hybrid
resonance effects.

No further discussion will be made on this effect,

except to refer the interested reader to the latest theoretical and experimental publications on the subject.
Magnetosonic wave heating experiments were also performed on the
TM-1-VCH tokamak and the T0-1 tokamak in the Soviet Union [9,10].

The

TM-1-VCH tokamak is a small device with the major torus radius R = 40 em
and the plasma radius=8 em.

Ion temperature increase of up to 100 eV at

generator power levels of 40 kW was reported.

When a deuterium plasma

is used, the phenomenon of low cavity Q at the eigenmodes was also observed, and the cavity Q was found to be about 10.

Magnetosonic wave

heating experiments in the T0-1 had produced comparable ion temperature
increases.

In the T0-1 experiments, some kind of frequency modulation

had been used to compensate any density variations and thus to remain on
one of the modes for a longer duration [10].
Another experiment at low power level {approximately l kW) was done
on the TFR tokamak in France.

In this experiment careful studies of the

density dependence of the eigenmodes, and tracking of the modes using
frequency modu-lation were performed.

One of the interesting discoveries

-10in the experiment was that the amplitude maxima of the eigenmodes
appeared to be modulated at a frequency around 1 kHz.

This modulation

was due to a periodic density fluctuation in the tokamak of about .5%
atl kHz as observed using soft x-ray diagnostics.

The decrease of the

cavity Q in a deuterium plasma was examined in these experiments, and
some agreements between the data and the two-ion hybrid resonance theory
were found [11].
tempted.

Mode tracking using frequency compensation was at-

The phase information between a local oscillator and a receiv-

ing probe signal was used to frequency modulate the pilot oscillator.
The density in the TFR varied only a few percent for several tens of msec.
The direction of the change in the frequency of the pilot oscillator was
such that it compensated any change in density which would destroy the
cavity resonance effect.

Typically, a resonance condition which lasted

for .2 msec was extended to a duration of 5 msec [12].

In this experi-

ment, the transmitting antenna was carefully designed for low losses and
good coupling to the plasma.
Recently, magnetosonic wave experiments were done in two of the
smaller tokamaks, the Microtor at UCLA and the Erasmus tokamak in
Brussels, Belgium [13,14].

The results from the Microtor

Showed no evi-

dence of a correlation between the excitation of Alfven (magnetosonic)
resonances and the antenna loading.

Upper bound estimates on these ex-

periments indicate that 70% of the applied power went into the plasma but
less than 5% appeared as resonances

[13].

This result does not agree

with the data presented in this thesis, where it was found that most of
the r.f. power went into the plasma via the cavity resonances.

However,

-11-

not enough is known about the experimental procedures used in the UCLA
experiments to resolve this difference.

The preliminary measurements in

the Erasmus tokamak "does not show a large increase of absorption due to
magnetosonic resonances.

This is in disagreement with the resonance

loading seen on the TFR, T4, and the Caltech tokamak" [14].

In the

Erasmus tokamak experiments the modulation of the resonance peaks by a
periodic density fluctuation which ~1as first reported by the TFR group
was also observed.
The magnetosonic wave experiment was done in the larger tokamak at
UCLA, the Macrotor tokamak (R = 95 em, a = 44 em) [15].

"In this machine

one is able to observe a definite correlation between antenna loading
and magnetosonic resonances for well shielded antenna, and during the
low density portion of the shot."
be between .5 and

ohm.

The resonance loading was reported to

"However, this wave loading is masked by the

parasitic loading during the early part of the shot when the density is
high."

-12-

1.3

General Thesis Outline
This thesis has been devoted to the understanding of how to couple

r.f. energy efficiently into a tokamak pl asma via the magnetosonic
cavity modes.

Special attention was given to the measurement of the

complex plasma loading impedance of the plasma at the cavity modes.
The ratio of the real part of the plasma loading impedance and the
antenna resistance determines the efficiency of the wave generation in
the tokamak at the cavity modes.

The complex plasma loading impedance

contains the information needed to match the r.f. generator impedance
to the antenna during the presence of a cavity mode.

Careful designs

of the antenna and the impedance matching network were found to be
necessary for efficient energy coupling to the tokamak plasma and properly matching the generator impedance to the antenna at a cavity mode.
In Chapter I I, the theory of the magnetosoni c v1ave in an axially
magnetized

cold

uniform plasma filled cylindrical cavity is presented .

Although the cylindrical cold plasma theory is only an approximation
to the experimental conditions in a tokamak plasma, the theory is found
to agree reasonably with the transmission data of the cavity modes.

The

characteristics of the magnetosonic cavity modes are determined from
the cold plasma dispersion relation, w = w(k), and the discrete values
of the wave vector, t, which satisfies the boundary conditions at the
cavity wall.

Each of the discrete cavity modes'is associated with a

particular wave vector which is represented by a set of mode numbers
(t,m,N) corresponding to the three components of the wave vector, ·where
tis the radial modes number, m is the peloidal mode number, and N is

-13-

the axial mode number. An equivalent circuit representation of the
transmitting antenna and the tokamak cavity is used to obtain relations
between various physical parameters of the cavity modes [Section 2.3].
The antenna input impedance has been calculated from the equivalent
circuit with the antenna modelled by a transformer, and each of the
cavity modes represented by a R-L-C resonant circuit.

The various

circuit parameters used in this calculation are either derived from
theory, or measured experimentally [Section 2.8].
In Chapter III, the operating conrlitions and the plasma parameters
of the Caltech tokamak are discussed, and the various diagnostic
tools available on the Caltech tokamak are described.

The time depen-

dence of the plasma density, vlhich is an important parameter governing
the behavior of the cavity modes, is given [Section 3.1].

Typically,

the plasma density increases rapidly dur i ng the first .3 millisecond,
then decays quickly to 20% of its maximum value in the next 2 milliseconds, and stays constant at around 1 x 10 12 particles per cm 3 for
the remainder of the discharge.
Chapter IV is devoted to the experimental apparatus and procedures
of the different r.f. meausrements.

The transmission measurements were

made with a single-turn tungsten transmitting loop antenna an d a small
six-turn receiving loop probe with low coupling coefficient so as not
to load

the cavity modes in the tokamak [Section 4.1].

The i nput

antenna resistance vias determined by measuring the incident and reflected
power into the antenna with a VHF directional coupler, and the r.f.
current in the antenna with a high frequency current probeiSection 4.3].

-14The plasma loading resistance was obtained by determining the additional
resistance present at the transmitting antenna due to the plasma effect.
The complex input i mpedance of the antenna was computed from the data
of the amplitude and the phase difference of the incident and reflected
waves from the VHF directional coupler [Section 4.4]. Considerations that
went into the design of the t wo-turn copper transmitting antenna anct
the impedance matching network using vacuum variable capacitors are
discussed in section 4.2, and the details of the construction of the
copper antenna are given in Appendix a.
Chapter V contains the experimental results of the r.f. measurements and the computed values of the equivalent circuit parameters
from measured data.

The computed equivalent circuit parameters of the

cavity modes include the antenna input impedance under different experimental conditions, the cavity Q of the various cavity modes, and the
antenna coupling coefficient at the various cavity modes.

The data

from the transmission meas urements at the cavity modes appear as volta ge
maxima in the output signal of the receiving probe, and agree reasonably
with the cold plasma theory given in Chapter II when ttle experimental
data are superimposed on the dispersion curves of the cavity modes in
a frequency versus density plot [Section 5.1].

Plasma loading resis-

tance at the cavity modes has been observed to be as high as 3 to 4 times
the basic antenna resistance [Section 5.3].

The complex plasma loading

impedance at the cavity modes fo 11 ows the general

behavior of the

impedance function derived from the equivalent circuit model of the

-15-

cavity modes [Section 5.5].

t~hen

the real and imaginary parts of the

measured plasma loading impedance are plotted on the complex impedance
plane as a cavity mode is passed through, the resultant curve is approximately a circle indicating a resonance effect.
Section 5.6 contains the estimated values of the cavity Q and the
coupling coefficient for the various cavity modes. The cavity Q can be
estimated from the time dependence of the plasma evolution by using the
approximate frequency-density relation for the cavity mode cutoffs
[Section 2.6].

The estimated Q obtained from the density data is the

cavity Q loaded by the impedance of the antenna and the r.f. generator.
The unloaded cavity Q, Q0 , can be related to the loaded Q by a circuit
equation.

Once Q0 is known, the antenna coupling coefficient, K, can

be obtained from the circuit model of the antenna-cavity coupling.
After the parameters of the equivalent circuit have been computed, the
wave generation efficiency, n, of the antenna is estimated.

For

the present antenna design, the efficiency has been found to be as high
as 80%.
In Section 5.5, attempts to match the r.f. generator impedance to
the antenna when one of the cavity modes is resonant are desc 1~ i bed.
Due to the variation of the plasma density \>Jith time, this matching can
only be done for a brief interval.
mode

The ability to match to a cavity

is the ultimate goal of the entire experiment, because i n order

to deliver the maximum amount of power to the plasma, the generator
impedance must be properly matched at a cavity resonance vJhere the

-16-

loading is stronger than when there is no resonance.
Finally, the experimental results and conclusions are summarized
in Chapter VI, and some improvements to the experimental apparatus for
future high power experiments are suggested.

-17II.
2.1

COLD PLASMA THEORY AND CIRCUIT MODELING OF THE CAVITY MODES
Theory for a Cold Uniform Cylindrical Plasma Cavity
The mode structure of the electromagnetic wave in a dielectric filled

cavity with perfect conducting walls can be obtained from Maxwell's equations, equations for the dynamics of the plasma, and the boundary conditions at the cavity wall.

For substitution of the plasma dynamics into

the Maxwell's equations, it is convenient to derive a relation between
the plasma current density and the electric field.

The plasma current

density can be thought of as a displacement current in a dielectric medium,
as shown in equation (b.l) of Appendix b, and the dynamics of the magnetized plasma is represented by a dielectric tensor [16]
E:

jE:

..L

-jE:x

E:
j_

E: II

where the definitions of the components of the dielectric tensor are given
in equation (b.3) of Appendix b.

For the propagation of the magnetosonic

wave in the tokamak, the following assumptions are made to keep the theory
simple, yet contain enough physics to reveal the essential features of. the
cavity modes.

The chamber of the tokamak is approximated by a cylindrical

cavity with perfectly conducting wall and periodic boundary condition in
the axial direction (Figure 2.1).

The plasma is assumed to be uniform,

cold, collisionless, and axially magnetized with a uniform magnetic field,
B.

Several approximations of the dielectric properties of the plasma

can be used to simplify the dispersion relation.

For instance, in the

dielectric tensor of the magnetized plasma, terms of the order (me/m;),

CONDUCTING WALL
()

+-~

2nR

__,

--~~

Figure 2.1
Plasma filled cylindrical cavity with conducting wall. R is the major
radius, and a is the minor radius of the tokamak. Periodic boundary
condition is imposed in the z direction.

0:>

-19-

where me and mi are the electron and ion masses, respectively, are
neglected.

The propagation frequency of the wave is taken to be near

the ion cyclotron frequency, which is much smaller than the electron
cyclotron frequency and the electron plasma frequency.

After including

all the simplifications mentioned above, the resulting dispersion for the
magnetosonic wave is as follows [17]:

st.2 (Jj 2 •

(2.1.1)
where T and k are the radial and axial components of the wave v~ctor, w .
C1

is the angular ion cyclotron frequency, st 1. is w/wC1., VA= 8 /

m.n.
is
1 1

the Alfven velocity in the plasma, and ni is the ion number density. (For
more details, see Appendix b).
All the transverse components of the electric and magnetic fields
can be expressed in terms of the axial electric field, E2 , and axial magnetic field, Hz.

A consequence of the p~opagation frequency being much

smaller than the electron plasma frequency is that Ez is small (see
Appendix b).

For our calculation Ez is assumed to be zero.

The solution

of the axial magnetic field, Hz, as shown in equation (b.31) is
Hz = H J (Tr) ej(wt-me-kz)
o m

(2.1.2)

where Jm is an integer order Bessel function, T and k are the radial and
axial components of the wave vector, respectively. (Note the fields vary as
e-jme --different from the ejme dependence used in some of the references.)
The boundary condition for the cavity is E8 = Hr = 0 at the conducting wall, i.e., at r =a.

As shown in Appendix b, equation (b.32),

the boundary condition can be written as:

-20y2
TaJ•(Ta) +-- mJ (Ta)

where y

(2.1.3)

= k - w ~o El and y 2 = w ~o E x , m = the poloidal mode number,

El and Ex are the components of the dielectric tensor of the magnetized
plasma.
The eigenmodes of the cavity are the simultaneous solutions of
equations (2.1.1) and (2.1.3).

Each of these dispersion solutions is

identified by a set of mode numbers, (1,m,N), where 1 is the radial mode
number, m is the poloidal mode number, and N is the axial mo de number.
The poloidal mode number, m, is the integer order of the Bes sel function
in the solution (2.1 .2).

The axial mode number, N, is related to the

axial component of the wave vector, k, by the periodic boundary condition
in the axial direction.

k = N/R, where N is an integer, and R is the

major radius of the tokamak.

The definition of the radial mode number,

1, can be best described in an example.

Consider the m = 0 modes, the

boundary condition (2.1.3) can be written as

In this case the radial mode number is defined to be the order of the
zeros of J . For instance, the lowest radial mode, 1 = 1, co r responds to
Ta = 3.83, the first zero of J , if J (o) = 0 is not included. (In
waveguide theory it is customary to denote the radial componen t of the
wave vector by T1m corresponding to the (! ,m) mode.

In the ab ove case,

for instance, TlOa = 3.83).
Form f 0 modes the solution is more involved because of the transcendental nature of equation (2.1.3).

Once the values of the independent

variables (the density, the poloidal mode number m, and the axial mode

-21number, N) are imposed, Newton's method for solving a system of equations is used to find the solutions of the input frequency and the
radial component of wave vector, T, which simultaneously satisfy both
equations (2.1.1) and (2.1.3).

For a given set of values for indepen-

dent variables, there are an infinite number of discrete solutions for
the frequency and T.

Therefore, the radial mode number is picked in

the solution by the initial guesses for T and the frequency used in the
Newton method.
Two sets of cavity mode dispersion curves are shown in Figures 2.2
and 2.3.

Figure 2.2 shows the various poloidal and axial modes of the

magnetosonic cavity wave for the lowest radial mode.

Figure 2.3 shows

the various radial and axial modes for the m = 0 poloidal mode.

From

Figure 2.3 one can see that for the parameters in our experiment, i.e.,
density less than 7 x 10 12 particles per cm 3 and w/ w . less than 3, the
Cl

higher radial modes for the m = 0 peloidal mode are not excited.

The

spacings between the various modes with different radial mode number in
the frequency versus density plot are large, so in our experiment only
modes with the lowest radial mode number are excited.

Therefore, only

modes with the lowest radial mode number are used to compare with the
experimental data (Figure 5.2).
Simplifications to the dispersion relation in equation (2.1.1) can
be made under certain conditions for

various

modes as an aid to es-

timating some of the measured physical quantities.

For instance, the

cut-off relation, i.e., k = 0, for the various modes is very useful both
as a guide to the general trends of the dispersion curves in the aensity

CAVITY MODES (£'=I , m, N)

-w

we~ ~

~N=(;
m=1

N=(!

---==-N={~

1 L

m=-1

DENSITY (cm- 3 )

12 X

Id 2

Figure 2.2
Dispersion curves of the magnetosonic cavity modes in a cold uniform cylindrical
plasma filled cavity with conducting walls. l = the radial mode number, m = the
peloidal mode number, and N = the toroidal mode number. Theses are the lowest
radial modes (l ~ 1). For hydrogen plasma, with R=.45 m and a=.l5 m.

CAVITY MODES (l,m=O,N)

3f-

'''

..........._..........._

-w

~ N =(~
1.=2

..........._

Wei

£=I

0~----~------~------~------~------~----~

DENSITY (cm- 3 )
Figure 2.3
Dispersion curves of various radial and toroidal modes for m = 0.
Definitions and parameters same as in Fig. 2.2.

12 X 10 12

-24-

versus frequency plane, and in the estimation of the cavity Q.

The cut-

off relation will be estimated for modes in two frequency ranges. First,
consider w << w . << w
The dispersion relation (2.1.1) with k = 0
c1
ce
can be written as follows:
st.2 w2 .
_1 c 1
(1 -

n~)

-- -

where VA= B0 !I~ 0 m.n.
1 1

I(-)
T2 2
(st.13 w2c 1. )"2 = 0

(1 -

n~)

is the Alfven velocity.

(2.1.4)

For the approximation

that sti << l, the result is
T2 "' st 2.w2 . I v2A

1 C1

If the hydrogen plasma is assumed to be fully ionized, then the electron
number density ne is equal to the ion number density ni.

The relation

between resonant frequency of a given mode and electron density is
(2.1.5)

f "' A)(,mo
f/ne

where A)(,mo
is a constant, and £,m are the corresponding radial and po0
oidal

mode numbers.
Next, consider the region where sti is near one.

(st 3.w 2 . )/[VA(l
- st 2.)] for the lower T modes.
1 C1
can be reduced to

T "' w /VA

Then T /2 <<

Therefore, equation (2.1.1)

-25which is the same as equation (2.1.4).

Since the cut-off relation is

continuous for the frequency range between ~ 1- = 1 and ~ 1- = 3, equation
(2.1.4) should be a fairly good approximation for our purpose.
2.2

Summary of ~1ore Sophisticated Theories of ~1agnetosonic Cavity Modes
The theory presented here is a great simplification of the experi-

mental conditions.

Many physical conditions, such as the toroidal

geometry, density, and magnetic field gradients, finite plas ma temperature, and finite conductivity of the tokamak wall, have all been
neglected.

Therefore, this theory cannot predict all the effects of the

cavity modes, but only can give the general features of the cavity resonances.

There have been several theories developed by different groups,

each including some of the neglected effects.

Perkin s , Chance, and

Kindel have included the finite temperature effects and predicted damping
of the magnetosonic wave by cyclotron damping at both the ion cyclotron
frequency and twice the ion cyclotron frequency, and by electron transit
time damping when the thermal velocity of the electrons is close to the
phase velocity of the wave.

They have also calculated the damping due

to the finite conductivity of the tokamak wall [4].
As mentioned in the introduction, the effects of the pol oidal field
on the cavity modes were first suggested by Chance and Perki ns [8], and
later worked out in more detail by J. Adam and J. Jacquinot [12].

The

poloidal field splits the toroidal mode degeneracy of them= -1 peloidal
modes.

In other words, when the peloidal field is included i n the calcu-

lation, the dispersion curves for the m = -1 modes with positive ~nd
negative toroidal mode number, N, are different from each other.

The

-26-'-

experimental result is the splitting of the cavity modes.

We observed

some modes in our experiments had double peaks; however, no definite
conclusion can be drawn because of two difficulties.

First, there was

not an independent mode identification measurement, other than using
density information to correlate with theory, as to which modes should
appear at a given time in the plasma discharge.

Second, the plasma

density decays bery quickly during the first two milliseconds in the
discharge (see Section 3.1 for detailed explanations), and so the cavity
modes are swept through very fast.

Consequently, it is hard to tell the

difference between a mode splitting and two different modes appearing
very close to each other in time.
The effects of radial density profile on the cavity modes were
studies by Paoloni [18,19].

The first model used in the theory was a

cylindrical cavity \vith a vacuum layer between a uniform plasma and the
conducting wall.

The m

0, ±l modes were studies (where the fields vary

. 8

as eJm ), and the conclusion was that for the magnetosonic wave them= 0
and m = -1 modes each has a definite cut-off frequency; however, for a
sufficiently thick layer of vacuum, them= -1 mode has no cutoff.

In our

experiments the cavity modes disappear when the input frequency is below
7 MHz.

This does not necessarily mean that the m = 1 mode does not propa-

gate below 7 MHz.

Perhaps the transmitting antenna used here does not

couple strongly to this mode at low frequencies.

It is also possible that

the vacuum layer in our tokamak has not reached the thickness requirement
of the theory.

-27-

The second mode 1 used in the theory Has a cyl i ndri ca 1 cavity with
a non-uniform radial density profile [19].

It was found that the radial

variation of the wave fields depended on the assumed radial density profile.

For the low radial and poloidal modes, the fields at the outer

radius of the cylinder are much smaller in the case of the parabolic
profile than in a uniform plasma, where the parabolic and uniform profiles have the same line-average density.

This means that if a loop

antenna is placed at the outer radius of the cylindrical cavity, the
antenna coupling to the cavity modes is weaker for the parabolic density
profile because of the lower field 1i nkage compared to a uniform density
profile.
The effect of the finite conductivity in the tokamak chamber wall
is an important factor in the 1/Jave heating.

As indicated in the surrmary

of r.f. heating by Stix [3], the eddy current dissipation in the tokamak
wall competes with the wave absorption processes in the plasma.

Appendix c, the losses in the stainless steel wall of the Caltech tokamak
have been estimated in terms of the quality factor, Q, of the tokamak
cavity for the ,Q, == l, m= 0, and k == 0 mode.

The quality factor Q is de-

fined as
Q == 2n __e_n_e-:r;-'g""'y-:--s_t_o_r_e_d_,...-

energy lost per cycle

The estimated Q for the particular mode in Appendix c is a l ower limit
for the Q for the various other cavity modes.

Hhen the esti ma ted Q due

to wall loss is compared with the cavity Q measured in the expe riment,
the estimated Q is two to three times the measured Q, indicat ing the absorption processes in the plasma are comparable or higher than the

-28-

dissipation in the wall (Section 5.4).

Therefore, a large part of the

input r.f. energy should be absorbed by the plasma.
2.3

Circuit t·1odel of the Antenna-Cavity Couplino:
For a cavity filled with a linear scalar dielectric, the amplitudes

of the various cavity modes can be described by a set of equations
derived from the t~axwe ll •s equations and the boundary conditions at the
cav·ity vJalls.

This set of equation is the same as those for an R-L-C elec-

tric circuit; hence the cavity can be modelled by an equivalent resonance
circuit [20].

The use of the circuit model of a cavity is only for the

convenience of those who have good intuition about the behavior of electrical circuits.

To justify the use of a simple R-L-C resonance circuit

to represent a cavity filled with a ma9netized plasma would be a very
involved task.

Therefore, v-:e shall summarize the approach used by Slater

[21] to justify the modelling of a linear scalar dielectric filled microwave cavity by a R-L-C circuit, and assume that a similar derivation can
be carried out for a linear tensor dielectric in a cavity.

The validity

of the circuit representation of the tokamak can be tested when the experimental results are compared with the model.
The electromagnetic fields in the cavity can be expressed in terms
of a set of complete orthonormal functions, called the normal modes of
the cavity: {Ee. + F,e_} and {!:!._e.} where V·~_e_ = O, V·H,e. = 0, Vxf__e_ = 0.
The orthonorma 1 conditions are expressed as follows:
E dV
vf I-e.· --m

F dV
vf f,e_. -m

H dV
vf !!_e.. -m

= otm

-tm

-29-

where Vis the cavity volume.

These normal modes are the solutions of

the wave equation,

'iE-1 + k12 I 1 = o
'iH-£ + k£_2 !:!_£_ = 0
and

k£_£. = 'ill/!£_

v21/J£_ + k£_2 1);£_

Associated with each of the eigenmodes is a characteristic angular resonance frequency, w1 , which can be related to the wave number by k12 = q.1w12 .
The fields in the cavity can be expanded in terms of the normal modes with
the following coefficients:

Eo= JE·E odV, Ho = JH·HodV, Fo= JE·F 0 dV

.{..

v- --:{._

.{..

v- -.{_

.{.. v- ...-.{..

I= ~(E£_ I£_+ F£_~ )

(2.3.1)

!:!_= L: H1 H1

(2.3.2)

£_

The solutions of the fields must satify both the Maxwell 1 s equations
and the boundary conditions.
problem:

There are t v10 types of boundarie s in the

conducting surfaces, denoted by S, and insulating surfaces,

denoted by S

Th e boundary conditions are

n X E

-£_

= 0

and

I!.: !:!_1 = 0

(2.3.3)

at a perfectly conducting surface, S, and
and

n·E

-!::,f.

at a perfectly insulating surface S

(2.3.4)
As shown by Slater, if equations

(2.3.l)and(2.3.2) are substituted into the r~axwell S eEJuations, the result1

ing integra-differential equations for the expansion coefficients are as
follows:

-30-

(2.3.5)

These are the differential

equations for simple harmonic motion (terms

on the left-hand side) with dampinqs and external forces(terms on the
right-hand side).

The convenience of these equations is that the bound-

ary conditions at S or s• can be readily substituted into the equations.
To demonstrate the damping terms, consider a cavity filled with a lossy
dielectric represented by a finite conductivity, J = crf. Equation (2.3.5)
becomes

) E

(E:)l(!t2 + CY)ldt

.t
is taken to be ejwt, the following

When the time dependence of
solution for w is obtained
w = ±w.t/ l - (l/2Q)2

+ jw.e_/2Q ,

This equation is analogous

where Q = E:w.e_/a

to a R-L-C circuit if the following equi-

valent circuit parameters are used [ 22]
L.t = )lk .t
c.t = E:/(k1 v)

R.t = CY)lk.e_V/ E:
The 1asses due to the finite conductivity of the cavity wa 11 can be ineluded by substituting the boundary condition

on the conducting surface

S, ~xi= ~(1 + j)lw).l/2a, into the surface integral overS in equation
2.3.6.

The effects of the wall loss in the tokamak are discussed

-31-

Appendix c.
Next let us find the input impedance of cavity using equations(2.3.5)
and the proper boundary condition.

Consider a cavity coupled to an out-

side system by a waveguide or coaxial line.

The input impedance of the

cavity can be obtained from the fields at an insulating surface, s•,
parallel to the cross section of the transmission line near the input
of the cavity.

As shown by Slater, once the boundary conditions of equa-

tion {2.3.4) are imposed, the fields at s• can be expanded in terms of
the transverse components of the normal modes of the wave guide, -n.
Et and
Ht,
i.e.
11

E v .tn. -tn.

where v.e.11 's are the time independent expansion coefficients of the electric field, i

are the coefficients of the magnetic field, and z111 is

the characteristic impedance of the wave guide for the nth mode.

After

some manipulations the surface integral of equation (2.3.5) can be
related to the expansion coefficients

When the above integral is substituted into equation (2.3.5), the following solution of the expansion coefficients of the electric fie l d, E.t,
is obtained

(the transverse electric field at S' )
where

-32-

The quantities ~

and V can be interpreted as the 'current' and the

'voltage' of the nth mode of the wave guide.

Zn.m are the impedance

coefficients of the various modes in the wave guide.
say

If only one mode,

the ;th mode, in the vtave guide is dominating, and loss terms, such

as dielectric and wall losses, are introduced into equations (2~3.5) and
(2.3.6) the resultant cavity input impedance is as follows:

_ !vz;fEw.el
zii -

where

~ j[l - (w~ fw2)]+ l/Ql

1/Q l = 1/Qwall + 1/Q dielectric"

This is J·ust the ea.uation satis-

fied by the input impedance of a R-L-C resonance circuit if the following analogies are made:
w =

and

where Ll , Cl , and Rl are the equivalent circuit parameters of the
lth cavity mode.

vii represents the coupling between the wave guide

and the cavity.

In our experiment, the cavity is coupled to the out-

side system by a loop antenna which is modelled by a transformer 111ith
a certain mutual inductance, ni , to the lth cavity mode; thus, v1e can

make the following analoqy between the coupling coefficient

vi; and ~1~

for high Q cavities, i.e., w ~ w1 , [23]

t1l/ Ll

= vl; I EW l

The equivalent circuit of the antenna-cavity system is shown in
Figure 2.4.

Each of the eigenmodes is denoted by a subscript, for

example, Rp. , Lp. , and Cp. are the equivalent circuit elements of the
th
mode. The subscript 'p' denotes that the cavity is filled with a
magnetized plasma.

Unlike the simple microwave cavity where the circuit

elements can be calculated theoretically, the equivalent circuit elements

-33-

of the tokamak are more difficult to calculate and have not actually
been

computed.

Since the physical quantities measured in the experi-

ments are not the circuit elements themselves, but rather functions of
these circuit elements, such as the Q of the cavity and the resonance
frequency, only the measurable quantities need to be calculated.

particular, one would like to know whether the complex input impedance
of the antenna-tokamak system satisfies the form of the complex input
impedance function derived from the equivalent circuit model.
By using this model, one can get an expression for the input

im~ed-

ance of the antenna, Z , when the various eigenmodes impedances are

reflected into the primary of the transformer.

The contribution to ZL

from the each of the R-L-C circuits is a simulation of the plasma loading.

For the circuit shown in Figure 2.4, ZL can be written as follows:
ZL =Rant+ jwlant + ~ (wtl;)hRp.+j(wLp.- w~ )]

. 1

(2.3.7)

where Rant and Lant are the resistance and inductance of the antenna.
For the convenience of comparison with experimental results, it is desirable to rewrite equation (2.3.7) in terms of the following quantities
which are measured in the experiments.

= w.1 L pi /R pi

Qp.

1/Lp.cp.

Two dimensionless quantities are used for convenience, the co upling
coefficient, K., and the normalized frequency, stp.'

Circuit Model of Toroidal Eigenmodes

I Rant.
M1

= 5on1

llant

Zjn--+-i

GENERATOR

.):::>

ZL __..,

••

..1 ..

.,..
IMPEDENCE
MATCHING
NETWORK

ANTENNA

CAVITY

Figure 2.4
Cirduit model of the antenna-cavity coupling. Each cavity mode
is represented by a R-L-C resonance circuit. Mi is the mutual
inductance between the antenna and the ;th cavity mode.

-352

== ~1./L
pi L an t

P·1

w./w

The real and imaginary parts of ZL can be expressed in terms of these
parameters:

QaK·Qp.Slp·
Ran t [ 1 + l: ---"'2'--'---::2:-'-.;........:_--=-2--=2
SlPi+Qp;(l - Slp;)

(2.3.8)

( 1 - Sl~ i )K Q~ 1.
XL == Xant [ l - l: ----'------=-2
-'--=2J
i rt2 +Q 2 (l - Slp1.)
Pi Pi

(2.3.9)

Near the resonance of the jth cavity mode, equations (2.3.8) and (2.3.9)
can be approximated as
(2.3.8a)
(2.3.9a)
At a particular frequency, only the term with a resonance frequency
closest to the applied frequency vJill dominate the resistive loadinq,
whereas the reactance depends on the couplin~ coefficient and Q of all
the other modes.

Depending on magnitudes of the contribution to the

input reactance from the modes above and below the resonance fr equency,
i.e. Q > 1 or Q < 1, the total reactance from all the cavity ~o des,
XL - Xant, can be greater or less than zero.

If the reactance contri-

buti on from modes Hi th resonant frequency, w.J < w.,
is qreate
:- than
the contribution from the other modes, for instance, the inp ut reactance, XL' ¥Jill show an increase to the basic antenna inductive r~ac
tance from the effects of the cavity modes.

-36-

2.4 Transient Measurements of Steady State Quantities
The impedance measurements made in our experiment are transient
measurements.

The tokamak operates in a pulse mode with the duration

of the plasma current about 12 milliseconds.

Furthermore, as mentioned

in the introduction, the cavity eigenmodes are swept through very
rapidly due to the changing plasma density.
in Section 4.2).

(This point will be detailed

Therefore, the input impedance of the cavity modes is

changing in a very short time.

However, the concept of impedance is

defined for a steady state situation, and so it is appropriate at this
point to examine the conditions under which the impedance concept is
valid.

To get an estimate of how long one must \-iait to achieve steady

state condition in a transient measurement, consider the following
idealized problem.

A R-L-C resonance circuit for one of the eigen modes

is subjected to a step of r.f. voltage input at the resonance frequency ,
of the circuit.

The voltage-current relationship can be written in

the follo vJing integra-differential equation:

c j

L _

jw t

I dt = V0 e

U(t)

(2.4.1)

where
U(t)

t < 0
t > 0

The equation can also be expressed in the following form:
(2.4.2)

-37First, the homogeneous solution to the differential equation is found
using Laplace's transform
s 2 + (R/L)s + 1/LC

and so the solution of the form I e 5 t can be written as

s = -(w /2Q) ± j w J1- (l/2Q) 2

0 '

\'lh ere w02 = 1/LC, and Q = w0 L/ R.

(2.4.3)

In our case the Q is very hi gh and so

the imaginary term is approximately equal to ±w0 .
In the high Q approximation, the general solution to equation
(2.2.5) can be written as follows:
jw t
-w 0 t/2Q
I = (V/R)(l - e
) e

(2.4.4)

From this equation one can see that the time required for the circuit to
reach steady state is 2 to 3 times 2Q/w 0 .

Thus the time, T, to S\-Jeep

through the half power points of the resonance must be longer than 2Q/w .

The longer T is compared to 2Q/w , the more accurately the steady state

impedance can be measured.

The condition for accurate impedance measure-

ment is
T »

2Q/w0

(2.4.5)

Fortunately, the density decay is s l 01'>' enough for this condition to be
satisfied in our experiments.

In Section 5.6, equation (2.4 .5) will be

applied to the experimenta l data and the validity of the impejance measurements will be discussed.

-38-

2.5

Impedance Matching
The impedance matching net1t1ork, consisting of the tltw tuning capac-

itors, c1 and c2 in Figure 2.4, is used to tune out the imaginary part of
the impedance in the antenna circuit, and to transform the real part of
For a particular setting of c and c , only
one value of Lan t and Ran t can be matched to 50 ohms. Therefore, one
must be specific as to the condition under which the antenna is matched.
the impedance to 50 ohms.

The most simple vvay to match the antenna is in vacuum \'/hen no plasma is
present.

However, it is found that once the plasma is formed around the

antenna, the antenna then becomes mismatched.

Even when there are no

cavity resonances present during the discharge, the plasma causes a sufficient change in impedance to the antenna that retunina c and c is
needed. This kind of tuning \vill be denoted as "off-resonance" matching.
A precise definition of the "off-resonance" matching is to match the generator impedance at a specific time in the plasma discharge, when no
cavity mode is resonant. The reason for specifying the time in the discharge is that the plasma condition is changing as a function of time,
and so the impedance contributed from the plasma when no cavity resonance
is present is also changing as a function of time.

From now on the sum of

the "off-resonance" plasma impedance plus the antenna impedance will be
denoted by Zoff = Roff + jXoff"

It is found from the experiments that the

changes in Zoff resulting from the changes in the plasma conditions are
slow enough that "off-resonance" matching for fairly long periods in the
discharge (typically 3 milliseconds) is possible.

In this way, one set-

ting of c and c can ensure that the generator is properly "off-resonance"

-39matched for the first hJO milliseconds in the plasma discharg E: where
most of the cavity modes appear.
There is one more type of matching, namely to match the generator
impedance to the impedance of the antenna plus the added contribution from
the plasma at one of the resonance peaks.

Because the impedance contribu-

tion from each of the eigenmodes is different from the others, only one
mode can be properly matched for a particular setting of c

details of this type of tuning arediscus sed in Section 4.5.

and c .

The

Fo r future

reference, the term "on-resonance" matching is coined to denote this type
of matching.
In the experiment, the directional coupler used has a characteristic impedance of 50 ohms, and it measures the impedance of the antenna
and the plasma loading after being transformed through the matching network.

This measured impedance is the term Z.

shown in Figure 2.4. The

quantity of interest is the impedance looking directly into the antenna,
i.e., ZL (see Figure 2. 4).

The transformation relating these two imped-

ances is readily shown to be
RL

R.1 n XC /D

XL = X

[l -

(2.5.1)
Xc (Xc + Xc + Xin)

(2.5.2)

where D = ( XC+ XC+ X.1n ) 2 + R2.1n and XC
The ideal matching procedure for "on-resonance" matchin g is first
to match the impedance of the generator at the "off-resonance " condition,
\'lhi ch is an easier process than "on-resonance" matching.

From the · measured

-40-

complex reflection coefficient p, ZL can be calculated.
of ZL at the resonance peaks, the XC

and XC

From the values

can be calculated for "on-

resonance" matching, i.e., R., n =50 ohms, X.1 n = 0. This procedure was not followed in this thesis because of the lack of an on-line computer system to
calculate ZL and the nevi c1 and c2 .

The actual "on-resonance" matching

reported in this thesis was done by minimizing the reflected voltage from
the directional coupler at one of the modes through trial and error. r·1ore
discussions on "resonance" matching and data of impedance at "on-resonance"
matching are presented in Section 5.7.
2.6

Relations between Circuit Parameters
The actual physical quantities that are measured in the experiment

are the amplitude and the phase of the incident and reflected voltages
into the matching network from the generator, the antenna current, and
the plasma density.

From these measured quantities, the following cir-

cuit parameters, shown in Figure 2.4, can be calculated:

the input

impedance Zin' the resonance plasma loading resistance r~ w~/Rpi'

the

cavity Q,Qp., the coupling coefficient K, and the antenna efficiency

n·
To obtain the resistance information from the measured incident
and reflected voltage into the antenna and the antenna curren~ requires
some basic equations used in transmission line theory.

The i ncident and

reflected waves into the capacitor matching network are measu r ed by a r.f.
directional coupler, which has a characteristic impedance of 50 ohms.
Since the generator and the directional coupler is also 50 oh ms, the
incident and reflected power into the antenna circuit can be written as

-41-

1nc = v~1nc ;so

(2.6.1)

(2.6.2)
where Pine and Pref are the incident and reflected pm'lers, resp ectively.
If we call the antenna current Ia' then the resistance can be obtained
as
R = (P inc- P ref )/I a

(2.6.3)

To find the complex input impedance, first define the complex reflection coefficient.

The complex reflection coefficient, p, can be

related to the amplitude and the phase of the incident and reflected valtages as
p = (V

ref

inc

) ej ¢

(2.6.4)

where ¢ is the phase between the incident and the reflected voltages
[24].

The complex input impedance can be obtained from the complex re-

flection coefficient by the follo\'ling transformation:
(2.6.5)

where z. = R. +jX. , and Z is the characteristic impedance of the trans1n
1n
1n
mission line, i.e., Z = 50 ohms for our experiment. Using this formula

to solve for R.1 n and X.1 n for our case, the follo\'ling equations are obtained:

x.1n

21 P1cos ¢ + I P I ) J

(2.6.6)

z0 [21 PIs i n

(2.6.7)

Rin = Zo[(l-IPI2)/(l

where IPI is the magnitude of the reflection coefficient.

These two

-42equations are used in Sections 4. 4 and 5.5 to calculate the complex impedance from the experimental data.
The cavity Q can be estimated from the plasma density at the
cavity resonance, the rate of density change as a function of time, and
the 3 dB time vJidth of the resonance peak.

The reason density informa-

tion can be used to get the cavity Q is because of the nature of the
dispersion relation of the magnetosonic wave (Figure 2.2).

As shown in

Figure 2.2, a change in the density can be interpreted as a kind of
frequency sweep in the cavity.

During the plasma discharge, the density

is changing as a function of time (Figure 3.2).

In the experiments where

the input frequency of the antenna is fixed, the cavity modes are swept
through as a series of resonance peaks by the density decay.

Thus, the

cavity Q can be derived as a function of the plasma density. To demonstrate this point, examine the approximate cut-off relation, k = 0,
for the modes.

As shown in Section 2. l, equation (2.1.5) is a good ap-

proximation of the cut-off relation for the frequency range for our
experiment, w . < w < 3w . .
Cl

Restating equation (2.1.5):

The Q of the cavity can be written as

where f

the cavity resonance frequency.

Using the cut-off relation'

the Q can be related to the density as

Such a measurement gives the loaded Q of the cavity QL, rather than the

-43-

unloaded Q, Q , but the two are related as foll01vs.

Consider the rela-

tion between QL and Q of an "off-resonance" matched antenna.

~~hen

the

system is "off-resonance" matched, c1 and c2 are chosen so that Zin
looks like 50 ohms during the plasma discharge when no cavity resonance
is present, i.e., when ZL = Zoff"
defined in Section 2.5.

Zoff is the "off-resonance" impedance

Because the tuning is off resonance, ZL looks

like 50 ohms when transformed through c1 and c . By the same token, the
generator impedance, which is 50 ohms, looks like the complex conjugate of
Zoff (i.e., Roff-jXoff), when transformed back through c1 and c . Finally,
when the generator impedance is transferred through the antenna into the
resonance circuit of the ith eigenmode, an additional resistance of M2w2/Roff
is in the R-L-C circuit.

This additional resistance, as shown in Figure 2.5c,

will add in series with the Rp , thus lowering the Q of the cavity. From the
resonance circuit shown in Figure 2.5c, one can vtrite the loaded cavity
Q as fo 11 ows:
2 2

Lp w/R p (1 + w M /2R0 ffR p

(2.6. 10)

(2.6.ll)
All the terms in this equation are known.

Roff is the antenn a resistance

plus the contribution from the plasma during the "off-resonan ce" condition. w2ti /R p is the 1oadi ng of the an ten na due to the plasma at the
peak of a resonance. Both of these can be determined by expe ri ment .
Now the antenna coupling coefficient can be calculated fo r one of
the cavity modes:

(a)

(b)

zo- 50.Q

Roff

Rant

(c)

Mcf2Roff

Figure 2.5
Circuit model relating the loaded Q and the unloaded Q. (a) is the equivalent circuit of the
antenna, the matching network, and the generator impedance. (b) is the equivalent circuit
looking back at the generator impedance through the matching network. (c) is the circuit in
(b) transformed through the mutual inductance M into one of the cavity resonance circuits.

-45(2.6.12)

2.7

Antenna Efficiency
Another physical quantity of considerable interest is the effici-

ency of the transmitting antenna.

The efficiency n is defined as the

amount of power coupled into the cavity, divided by the total power
delivered to the antenna by the r.f. generator.

can be obtained

straightforwardly by considering the circuit in Figure 2.6.

Here the

plasma impedance has been transformed into the antenna circuit, and the
current i flows in the loaded antenna.

The settings on the matching

capacitors determine the magnitude of the power delivered to the antenna,
with the maximum power transfer when the impedances of both sides are
matched.

At a cavity resonance, the plasma loading impedance is real,
and the value is M2w2/RP. The only dissipative elements in the circuit
2 2
are Rant and t11 w /Rp.

Thus n can be written as

.2,12 W2

1 I'

(2.7.1)

It is more enlightening to write n in terms of K, Qa' and Qp

QaQp

(2.7.2)

This equation is very useful in designing an efficient antenn a system.
After deciding on a particular antenna shape, this equation gives the
directions for improving the efficiency.
problem will be discussed further.

In Section 6.2 the designing

-46-

(a)

X ant

r-7

R +-(wL - )

P H.:

P wCp

(b)

Xant

Figure 2. 6
(a) Impedance of the RLC resonance circuit transforms th r ough
the mutual inductance M into the antenna circuit. i is the
antenna current. Z0 is the qenerator
impedance. (b) At reson2
ance, the transformed impedance is real and equal to t,1 w2 /R p .
The antenna efficiency, n, is
tiw2
R R
+ M2w2
p ant

-472.8 Simulation of Cavity Resonances
It is useful to see what general effects the impedance function
[equation (2.3.1)] will predict before actually discussing the experimental results.

The essence of the discussion in Section 2.3 is that the

form of the impedance function observed in the experiment should be reasonably close to the form of equation (2.3.1 ).

The unknowns are the

various circuit parameters, such as the antenna Q, the Q of the cavity
modes, the antenna coupling coefficient
experimentally.

etc., and they can be measured

The values of the circuit parameters used in the simula-

tion are either estimated from theoretical considerations, or measured in
experiment.
The simulation starts with the equivalent R-L-C circuits for the
resonance cavity.

Since the cutoff relation of the eigenmodes is ap-

proximately f ex: 1/lrl and w0 p ex: 1/vrc-;;-, vvhere w0 p is the resonance angular
frequency of the R-L-C circuit, the change of the capacitor as a function of time is assumed to be proportional to the density.

A typical set

of density evolution data is fitted by a polynomial, n = n(t), and the
time dependence of the normalized frequency ~
tional to this density function, n(t).

, is taken to be proporpi
The proportionality betvJeen the

frequency, ~

, and the density, n(t), for the ith mode is su ch that when
pi
the resonance condition for the ith mode is satisfied at a ce r tain denThe density dependence of the cavity modes is com= l.
pi
puted from the simple cold plasma theory. Each of the cavity resonances

sity value,~

is simulated by one of the R-L-C circuits with its o~tm reson ance frequency.
The resonance effects of all the R-L-C circuits are substituted and
summed in equations (2.3.8) and (2.3.9).

-48-

The coupling coefficient Ki2 and the cavity Q can be estimated
pi
from theory [25]. The values of K ~ and Q used in this simulation are
pi
the same for all modes for the sake of simplicity, even though they are
actually different for the various modes in the experiment.

Equations

(2.3.8) and (2.3.9) are solved on a computer, and the results of the
simulation for ZL are shown in Figure 2.7 for the typical density evolution and "off-resonant" tuning.

The resistance and the reactance. R and X,

shown in the 6th and 7th traces in Figure 2.7 are related to ZL by the
following relations:
Z = R + jX

(2.8.1)
The values of the various parameters used in the computation are
as follows:

Qa = 100, Qp = 400, and K = 8 x 10

-5

Note that the cavity QP

used in this calculation is the estimated unloaded cavity Q0 (see
Section 2.6 for the definition of

loaded and unloaded cavity Q).

The experimental Q which will be compared directly with this calculation
is not the unloaded Q, but rather the cavity Q loaded by the generator
impedance.

Therefore, the estimated loaded QL is computed fo r proper

comparison with the experimental data.

The loaded QL can be re lated

to the unloaded Q0 by equation (2.6.11)

The loaded cavity QL for this calculation is 15U.
To simulate the "off-resonance" tuning effect, the reactance of
the matching capacitor, Xci and Xc , are calculated for Rin = 50 oh ms

-49-

7X10

DENSITY

Or---~--------------------------------------------.6

REF. COFF •

PHASE

-7T

200SJ

50!J
60 Q
X.

-95 Q
1 Q

0 r---------L---~----~~--------~~--~------------

.5 Q

-.5 SJ

1.5

TIME (msec)

Figure 2.7
Computer simulation of various equivalent circuit parameters.
-5
In the computation, Qa = 100, QP = 400, and K = 8 x 10 . .

-50-

and Xin = 0 when the cavity modes are not present, i.e., RL
XL= Xant"

The equations for the capacitive reactances are

= RL (1 + Q2)/[Q
a-

J(l + Q2)RL/R.
1n - 1 J

(2.8.1)

j(l
+ Q2)
R. a RL - 1] Rin

(2.8.2)

where Rin = 50 ohms, RL = .3 ohm= Rant' and Qa = 100.
tion XC

Ran t' and

= 32.5 ohms and XC

= 384 ohms.

For this calcula-

By substituting the r esultant

values of XC

and XC

and the simulated values of RL and XL including the

cavity resonances into the following equations, R.1n and X.1n fat' this model
can be obtai ned:

R.1n

RL/[(RL/XC ) {(XC /RL- Qa)

(2.8.3)

+ 1)]

X.1n = [XL(l-XL/Xcl)- RC/XclJ/[(RL/Xcl)2((xc /RL-Qa)2+l)]-xc2
(2.8.4)
By inverting the conformal transform of equation (2.6.5), the complex
reflection coefficient can be calculated from z. in the following manner:
1n

\'I

here Z0 = 50 ohms .
The time dependence of the density evolution used in the calcula-

tions is shown in the top curve in Figure 2. 7.

Some of the gen eral

features of the computed solutions which will be compared with the experimental data later in Section 5.5 are noted as follows.

First, for a

simple pole resonance, there is a relation between the real and the
imaginary parts of the impedance.

Corresponding to every peak in the

-51real part of the impedance, the imaginary part should go thr ough a stee p
change.

Since the reflection coefficient is related to the i mpedance by

a complex transform, this same behavior should also exist between the
amplitude and the phase of the complex reflection coefficient.

As shovm

in traces 2 and 3, whenever the amplitude of the reflection coefficient
reaches a maximum, the slope of the phase as a function of time also is
a maximum.

Curves 4 and 5 show the similar behavior in the real and

imaginary parts of the impedance.

Second, the direction of the change of

the reactance is a function of the sign of the slope of the density evolution.
2.7.

To clarify this point, consider curves one and five in Figure

The first curve which is the density evolution has a po s itive

slope during the first millisecond \'lhen the density is increasing, and
has a negative slope after the first millisecond when the density decays.
This change in the sign of the slope is reflected in the reactance
curves, traces 5 and 7.

During the density buildup, the reactance goes

negative first, then jumps to a positive value when a resonance is passed
through.

During the density decay, the reactance is positive before pass-

ing through a resonance.
?_:2__Q Ci r:_cl es

Another way to see the simple pole resonance effect of o cavity mode
is by plotting the input resistance of the cavity against the input reactance in the complex impedance plane.

As a cavity

resonance is passed

through, the resultant curve is a circle, known as a Q circl e [20].
Depending on how the resonance is passed through, there is a definite
direction in tracing out the Q circle, i.e., whether it is clockwise or
counterclockwise.

The dependence of the direction of the change of the

-52reactance on the sign of the slope of the density evolution mentioned
in Section 2.8 can be clearly demonstrated by the direction in which the
Q circles are traced out.

The Q circles for the resonances appearing

during the density buildup are formed opposite to the direction of rotation of those occurring at the density decay.

The Q circles of the

experimental data are plotted in Figures 5.7 to 5.9, and this reversal
of direction in which the Q circles are traced out has been observed experimentally (see Section 5.5).

-53-

III.
3. l

GENERAL EXPERIMENTAL SETUP

Tokamak Characteri~tics
A tokamak is a toroidal plasma confinement device ~vhich can be

described as having the shape of a doughnut (Figure l.l).

The vacuum

chamber of the Caltech tokamak is made of stainless steel with the major
radius about 46 em and the minor radius approximately 15 em.

A toroidal

magnetic field is created by a current carrying coil wound on the surface of the torus.

The current in the toroidal field winding is produced

by a capacitor bank containing up to 50 kJ of energy.
A second winding,

the toroidal direction.

known as the ohmic heating winding, is ¥/ound in
The windings are placed on a single surface

above the toroidal field windings.

The purpose of the ohmic heating coil

is to produce a changing magnetic flux linking the plasma, but to have no
field inside the vacuum to disturb the plasma confinement.

By Faraday•s

induction law, the changing magnetic field linking the plasma will induce
a toroidal electric field in the plasma; thus, a toroidal plasma current
will be produced.

This plasma current serves two purposes: First, it will

provide a peloidal magnetic field which, when added to the to ro idal
field, will give a rotational transform to the field as illus tr ated in
Figure 3. 1.

The rotational angle 1(a) at the edge of the plasma
8T _

2nR

(3.1.1)

Bp - ai.(a)
so that

_(3.1.2)

where R = major radius, and a

minor radius.

The safety factor q = 2n/i.

-54-

8 = 0°

_.--- Poloidal cur~nt - - ,

Toroidal ma Qnetic field

"o"'-polo;i' '""•"1__

Poloidal m09netic f~d
from toroidal current
Conductor corryingl toroidal

currH~t

3('\, I

(7.;,.J-

Rotational transform

Figure 3.1
Rotation transform in a tokamak. The pitch
angle 1 = 2nRB /aBT. The safety factor q is
2n/1. 8 = the peloidal angle, and ¢ = the
toroidal angle. (From Principles of Plasma
Physics, by N.A. Krall and A.W. Trivelpiece)

-55must be greater than 2 or 3 for stable operation.

For the Ca ltech

tokamak, q is typically between 5 and 7, depending on the plasma current.
Second, the current will also heat the plasma through dissipation of the
plasma resistance; thus the name, ohmic heating current.

The ohmic heat-

ing winding is energized by a second capacitor bank containing up to 8 kJ
of energy.
Due to the toroi da 1 geometry of the tokamak, the induced toroi da 1
plasma current produces a peloidal field v.Jhich is stronger in the "hole of
the doughnut" than on the outside of the torus.

This results i n a mag-

netic pressure which pushes the plasma out\•Jard.

Therefore, a third set of

coils is used to produce an approximately vertical magnetic field in the
plasma.

This field and the plasma current produce a J x B force which

-v

compensates the outward magnetic pressure.

The vertical field \'linding is

energized by a third capacitor energy supply.

The time dependence of the

vertical field must be designed so as to insure equilibrium throughout the
discharge period, even when the discharge parameters change.

With the

proper vertical field, the plasma current lasts for about 12 milliseconds.
The Caltech tokamak operates in a pulsed mode with a repetition
rate of once a minute, being dictated essentially by the time to charge
the capacitor banks.

As mentioned previously, the energy for th e differ-

ent windings is stored in capacitor banks.

A digital timing u ~ it is used

to control the discharge sequence of the various banks.
toroidal field is created.

First, the

Then, a 16 kHz, one millisecond bu rst, called

the preionization puls~ is applied to the ohmic heating windin g to partially ionize the gas.

This is follo\IJed by discharging the ohmic hea.ting

capacitor bank into the ohmic heating winding, producing a p7asma current

-56up to 15 kA.

Simultaneously, the vertical field is applied to provide

the proper plasma equilibrium.
The vacuum chamber of the Caltech tokamak is cleaned by a process
called "discharge cleaning".

The method employed, first proposed by

Robert Taylor of UCLA, is to bombard the vacuum chamber wa 11 by a rapidly
pulsed (2-3 times a second) low temperature hydrogen plasma [26].

The

object of the process is to reduce the loosely bonded high mass impurities (carbon and oxygen) on the chamber wall so that during the actual
tokamak dischargefewerimpurities \

Such

impurities can be detrimental in a plasma confinement device because
they greatly increase the radiation losses in the plasma.

The rate of

energyloss in the plasma due to Bremsstrahlung radiation is
(3.1.3)

where Te is the electron temperature in keV, ne is the electron density,
Zeff is defined as
(3.1.4)

nk is the density of the kth species ion, Zk is the degree of ionization
of the kth species, and n = ne/Zeff [27].
One can see that to minimize the Bremsstrahlung radiation power
loss in a plasma, the Zeff must be minimized.
using discharge cleaning.

This is the reason for

The Zeff of the Caltech tokamak plasma is

believed to be quite low as the result of lm'l power discharge cleaning.
A side effect of the discharge cleaning is that the plasma density
drops very quickly after the initial plasma density buildup.

The exact

cause of this behavior in the plasma density is not completely understood

-57and is currently under investigation. As shm·m in Figure 3.2, the
electron density peaks at 7 x 10 12 particles per cm 3 in the first .3
millisecond, then drops to 1 x 10 12 particles per cm 3 in the next two
milliseconds.

This behavior in the plasma density has important con-

sequences in the wave excitation experiments.

From the dispersion

curves in Figures 2.2 and 2.3, one can see that for an input frequency
between one and three times the ion cyclotron frequency, no cavity mode
can propagate in the Caltech tokamak beyond the first two milliseconds
in the plasma discharge when the plasma density falls below 1.5 x 10 12
particles per em -3

This means that all the impedance measurements of

the cavity resonances must be made within the first t\vo milliseconds in
the plasma discharge.
3.2

Plasma Diagnostics

Plasma Current and Toroidal Field Measurements
The plasma current is measured with a Rogowski coil placed on the

vacuum chamber surface.

The Rogowski coil is made by winding a coil on

a long plastic tube, which then encircles the plasma.

By Faraday's in-

duction law, the voltage measured from the coil is

v -_ Trp 2 ( [N ~
dt )

(3.2.1)

where p is the radius of the tube, L is the length of the tube, N is the
number of turns of the v.Jire, Ip

is the plasma current.

To get the

plasma current, the signal is electronically integrated.
The toroidal magnetic field can be accurately calculated from the
toroidal \vinding current which is measured with a Rogowsk-i coil. · The

8X10

PLASMA

(MICROWAVE

DENSITY

INTERFEROMETER )

co
N8

TIME ( msec)

Figure 3.2
Electron density evolution as a function of time (4 mm
microwave interferometer).

-59toroidal field variation as a function of the major radius, R, is an
inverse relation, i.e., BT a: l/R.

From the dimension of the Caltech

tokamak, R = 45 em and a (the minor radius) = 15 em, the toroidal magnetic field varies by a factor of two from the inner vo~all to the outer
VIa 11 .

One-Turn Voltage
The voltage induced by the ohmic heating coil to drive the plasma

current is another important quantity.

To measure this voltcge, a

single turn wire is placed around the outside of the vacuum chamber in
the direction of the plasma current.

It encircles the hole in the

"doughnut", thus enclosing all the flux produced by the ohmic heating
air core transformer.

The voltage from this one-turn loop is just the

EMF produced by the changing ohmic heating flux.
One of the purposes of the so-called one-turn voltage is to
infer the average electron temperature of the plasma through a measurement of the plasma resistance.

The plasma temperature is related to

the resistivity of the plasma as follows:

(3.2.2)
l2rr (t: k T;e 2 ) 3i 2
0 8
where n is the resistivity of the plasma, fl. is
, and
112

ne
Zeff is the effective charge of the plasma due to high mass imp urities
in the plasma [28].

The Zeff defined in equation (3.1.4) fo r a hydrogen

plasma is greater than one.

Although we do not have a direct measure-

ment of the Zeff' the Zeff in the Caltech tokamak is believe d to be
quite lov1 because of the discharge cleaning.

-60-

Plasma "Magnetic" Pas it ion Measurement
One would also like to know the position of the plasma column with

respect to the vacuum chamber wall in order to keep the plasma well centered.

This position is measured by placing two coils, the in-out coil

and the up-down coil, on the torus.

The in-out coil is a cosine coil, so

named because it is a Rogowski coil with the number of windings per unit
length following a cosine function of the poloidal angle 8 (see Figure 3.1).
The up-down coil is a sine coil.

The cosine coil is wound on a plastic tube

such that there are more turns near 8

0 and 180°; moreover, the direction

of the winding is changed at 8 = 90 and 270°.

Therefore, the signal from

the left half of the windings is of opposite sign to the right half.

the plasma moves toward the one side of the chamber, the signal picked up
by the coil on that side will increase.

Thus the total output voltage is a

function of position [29]
dl

V = f(r,8) ~

(3.2.3)

By electronically integrating the signal with respect to time, the
output is a position signal.

The sine coil works the same way except it

is rotated 90" in the poloidal direction from the cosine coil.
Because of the toroidal geometry the magnetic flux produced by the
plasma current is greater at 8 = 180° than 8 = 0°, i.e., it is stronger on
the inside of the torus than on the outside; therefore, the winding density
is no longer symmetric with 8 for the proper calibration of the output
voltage.

The cosine coil has less windings on the inside of the torus,

i.e., e = 180~ than the outside, e = 0~

-61d.

Line Average Electron Density t~1eas uremen t
The line average electron density in the Caltech tokamak is meas-

sured by a microwave interferometer (Figure 3.3).

The phase shift

between the reference signal and the signal through the plasma contains
the density information.

The plasma density is a function of the posi-

tion, and so the average phase difference between the two legs of the
i nterfero!Teter is

L'lw = Lk -

zI n

(3.3.4)

where L is the width of the plasma, np is the index of refraction of the
plasma, and k is the free space wave number.

For an ordinary '1·/ ave, i.e.,

the electric field of the wave is parallel to the d.c. magnetic field,
the index of refraction can be written as follows:

l -

t}(x)
p2

so that
Lk - ~

~2(x)

I" l -

{3.2.5)

Since w2 ;w2 a: m./m , the contribution is mostly from the electrons~
pe pi
When the applied wave frequency is much greater than the elec t ron plasma

frequency, i.e., w!w

by the fo ll mvi ng:

(x) »

l, the above equation can be ap J roximated

~ wpe
2c

(3.2.6)

\"'here

wpe

wpe(x) dx

is the average electron plasma freq uency.

phase shift can be seen as a series of interference fringes at the

This

-62-

WAVE
GUIDE

• •• • •• • •• • •• • •• • •
•• • •• •• • •••• •• • •
• • • • • • • •
• • • • PLASMA • • • • •
• •• • •• • •• • •• • •• • •• • •• • •

ATTENUATOR

KLYSTRON
~----------~ATTENUATOR

Figure 3.3
Interferometry arrangement for microwave measurement of t he
plasma density. ( 4 mm m·i crow ave interferometer)

-63detector output.

One fringe corresponds to a phase shift of 6~

2n. The

corresponding average electron density ne = L1 JL ne ( x) dx, is

(3.2. 7)
when

w2 f w2p2 (x) >>

is imposed.

Thus the electron density is a linear

function of the phase shift or the number of output fringes when the
micro1t1ave frequency satisfies the above condition,

(r};w~e(x)) » 1. The

frequency of the mi crmvave interferometer used on the Cal tech tokamak is
60 GHz. and the maximum average electron density is about 7 x 10 12 particles
per cm 3 , which corresponds to an electron plasma frequency of 24 GHz.
If it is assumed that the density profile is a parabolic function of distance, the relation between the peak density and the average density is
npeak = (3/2)navg·

So the peak density corresponding to our case is approximately l x 10 13 particles/cm 3 , 1t1hich gives an approximate electron

plasma frequency of 36 GHz.

Therefore, the assumption of (w2/ wpe(x))
>> l

is a good one even for the peak density.
The fringe counting for the microwave system on the Caltech tokamak
has an uncertainty factor of ±l/4 fringe.
comes from the noise
tector.

The source of the uncertainty

superimposed on the interference signal from the de-

The origin of the noise is not completely understood.

may be due to actual fluctuation in the plasma density.

Some of it

By carefully

matching the fringes for the initial density buildup with the decay fringes,
the time dependence of the plasma density can be determined fairly well.
e.

Langmuir Probe Measurement
The conditions at the edge of the plasma are mild enough that

Langmuir probes can be used to measure the local electron density and

-64temperature.

Data have been taken for the first 5 em into t he plasma

by R. Kubena [30] without any major probe damage.

Th e results when

extrapo 1a ted agree fairly we 11 \'lith the density me as uremen ts from the
microwave interferometer mentioned in Section 3.2c, and the electron
temperature data from the plasma resistance measurement depicted in
Section 3.2b.
3. 3

Summary of Plasma Paramete rs
From the diagnostics just described, the Caltech to kamak plas ma

has the follovJing characteristics:
Toroidal field:

3 to 6 kG (4 kG on center) at R = 30 em
and R = 60 em, respectively

Plasma current:

15 kA (peak)
12 msec (duration)
12

cm- 3 (decays during

Line average electron density:

7x 10

Average electron
temperature:

50 to 100 eV (assuming Zeff = 1 .5)

to 1.5 x 10

the first two msec)

where R is the major radius of the torus.
3.4

Digital Data Acquisition System
All experimental data from the Caltech tokamak experi ments, such

as the signals from various diagnostics, the crystal detecte 1:! r.f.
signals, etc., are recorded on a multi channel digital transi ent recorder
which converts the various analog signals into digital data t hat are
stored in its semiconductor memories.

Each of the 16 channels of the

-65-

transient recorder has a 1024 word memory with 8 bits amplit ude resolution per word.

Four of the channels have a one-microsecond per word

clock rate, so the maxi mum frequency response with four-word reso 1 uti on
is about 200 kHz.

The rest of the channels have a clock rate of 5

microseconds per word, so the frequency response \'Jith four-word resol uti on is about 40 kHz.
The di gi ta 1 output signa 1s from the transient recorder memories
can then be used in several ways.

Analog signals can be reco nstructed

with 0-A converters for continuous display on scope monitors after
each plasma shot.

The transient recorder can also drive an analog

pen plotter, so that hard copies of the signal can be produced.

calculations need to be done with the data, the digital data can be
written on magnetic tape for later processing at the Caltech central
computer facility (IBM 370, model 158).

-66-

IV.
4.1

EXPERIMENTAL SETUP FOR THE R.F. MEASUREMENTS

Experimental Arrangement for Transmission Measurement
The first step in the study of the magnetosonic cavity modes vias

to observe them \vi th a receiving probe located 180° toroi dally from a
transmitting antenna (Figure 4. 1).
A simple single-turn transmitting loop antenna made of tungsten
vias first used (Figure 4.2).
dimension of 3.75" x 1".
three factors.

The race track shape antenna had the

The design of the antenna Has governed by

First, it must fit into a 4"x l"x 6" port.

Second, to

get good coupling with the plasma, the loop area should be maximized.
Finally, the antenna should be kept a~vay from the center region of the
plasma where most of the damage to the antenna will occur.
the shape long and narrovJ.

This made

R.F. signals are carried to the tungsten

antenna by parallel copper wires enclosed in a glass-to-stainless steel
transition tube.

The stainless steel tube provides the mechanical feed-

through from the outside into the vacuum chamber.

The glass is to give

electrical insulation for the antenna from the tokamak.

The measured

resistance of the entire antenna structure is about 2 ohms at 10 MHz.
The antenna can be moved radially in and out of the plasma
through a vacuum 0-ring seal.

All transmission measurements are done

with the antenna located no more than 1.25 inches into the vacuum chamber in order to prevent any plasma damage to the antenna.

This is the

lm"l density region in the tokamak, according to Langmuir probe data,
(n < 5x lOll particles/cm2 ).

TRANSMITTING
ANTENNA

300WATT
AMP.

0"\
'-1

AMP. & PHASE
DETECTOR
R. F.

OSC.

CURRENT
PROBE

REF. SIG. FOR
PHASE DETECTOR

Figure 4.1
Experimental arrangement of the transmission measurement.

16 GAUGE
TUNGSTEN

STAINLESS
TUBING

STAINLESS STEEL
TO GLASS

TUNGSTEN
COPPER
GLASS
TRASITION /TUBING

TRANSITI~N

~,_

3.751N.

GLASS TUNGSTEN
VACUUM SEAL

10 IN.

5 IN.----

Figure 4.2
Single- turn tungsten antenna used initially as the transmitting antenna in the
transmission measurement.

0'1

-69-

A matching network consists of a variable series capacitor used
to tune out the antenna inductance, and a R.F. transformer to match the
antenna impedance to 50 ohms.

An ENI 300-watt wide-band amplifier

driven by a Hewlett-Packard 8601A sweeper oscillator is used to excite
the \'lave.

The input r.f. frequency to the transmitting antenna is fixed

for each plasma discharge.

This 1t1ay only one variab l e, the plasma den-

sity, is changing during the experiment.

To study the frequency depend-

ence of the cavity modes, the input frequency is changed between plasma
shots.
To detect the cavity resonances, a small six-turn loop probe is
placed in the tokamak.

The receiving probe is kept small so that it

couples weakly to the cavity.

This way the probe does not influence the

cavity VJhile it is measurin g the r.f. signal.

As sho•.lfn in Figure 4.1,

the receiving probe is located 180° toroidally from the transmitting
antenna.

The output of the probe is passed through a tunable bandpass

filter, with a band\'lidth of 300kHz; thus any broadband noise from the
plasma can be reduced.

The r.f. signal is then split into tvJO branches.

One branch goes into a square law crystal detector which has c:n output
operational amplifier with a slew rate of 4V in 2 ~sec , for a ~plitude
detection.

The other line is fed into a phase detector which can

respond to a 2n phase shift in 4 ~sec so the phase between the transmitted and the received signals can be examined.

As mentioned in

Section 3.4, the output of the phase and amplitude detectors i s digitized and recorded in the multichannel transient recorder.

The

experimental data of the transmission measurements are presented in
Section 5.1.

-704.2

Antenna and Matching Network Design
The initial measurement of the plasma loading resistance was made

with the single-turn tungsten antenna, which has a resistance of 2 ohms
at 10 MHz .

The impedance matching circuit consists of a series of air

varia ble capacitors used to tune out the antenna inductance, and a
broad-band ferrite core r. f. transformer made to match the antenna resistance to the amplifier impedance.

With this setup, only a minute

amount of plasma loading at the cavity resonances v~as detecte d. However,
the large increase in the transmitted signa l measured by the six-turn
probe at the cavity resonances led us to think that there must be better
pm'ier coupling betvteen the antenna and the tokamak at the cavity resonances than when there were no cavity modes.

This effect should show up

as antenna loading by the plasma at the cavity resonances.

It was be-

lieved that the sum of the resistance from the antenna, the matching
network, and the r.f. transformer \'~as so high that the plasma loading
was overshadowed.
To understand the effect of the antenna resistance on the plasma
loading resistance measurement, consider equation(2.3.8a) at one of the
cavity mode frequencies:

where it is assumed that the various cavity modes are separate d far
enough in their eigenfrequencies that only one mode dominate s in the
resistivity loading.

From this expression, one can see that in order

to measure the plasma loading effect, QaKiQp.

Let us estimate

the magnitude of this factor for the

tungsten antenna.

For the

-71-

tungsten antenna, Qa is around 10, K 2 is estimated to be 5 x 10 -6 , and
Qp is assumed to be 500, so their product is 2. 5 x 10 -2 , which is much
smaller than one.

Furthermore, consider the efficiency of the antenna

in equation (2. 7.2)

for one of the modes.

In order to have efficient wave generation in the
tokamak, the sarre inequality, i.e., K 2QaQp > l, must be satisfied in

order to generate more energy in the tokamak than is dissipated by the
antenna.

Therefore, the antenna and the matching network v1as redesigned
to improve the factor K2Qa.
The coupling coefficient K2 can be increased by increasing the
antenna size.

However, as mentioned in Section 4.1, the loop area of

the antenna is determined by the port size on the tokamak, and the maximum distance the antenna can protrude into the plasma without suffering
damage to the antenna.

Therefore, the coupling coefficient of the an-

tenna cannot be increased very much.
antenna Q, Qa:

There are two ways to increase the

either increase the inductance or decrease the resistance.

The antenna inductance is increased by going from a single-turn loop to
a two-turn loop.

The maximum number of turns on the loop antenna is de-

termined by the size of the conductor used and the width of the port,
which is one inch on the tokamak.

The antenna resistance is decreased

by using material v·lith better conductivity, and by increasing the size of
the conductor.

The conductor used in the antenna is changed from 16

gauge tungsten wire to l/8-inch diameter copper tubing. To preven~ plasma

-72-

damage to the copper antenna and to insulate the antenna electrically
from the plasma, the copper antenna is enclosed in pyrex glass.

The

measured Q of the bare copper antenna is about 130 at 10 MHz, and the
inductance of the antenna is about .46 microhenry.
Ho~tJever,

once the copper antenna is placed in a glass-to-stainless

steel transition tube which provides the mechanical feedthrough from
the outside to the vacuum chamber, the antenna Q drops by a factor of
two.

The additional losses come from the eddy current losses in the

stainless steel tube which has a 50 times higher resistivity than copper.
To reduce the eddy current losses, a copper lining of .025 inch thick
is placed on the inner wall of the stainless steel tube, thus reducing
the eddy current losses.

vJith the copper lining, the Q of the antenna

is about 100 at 10 MHz, and the inductance of the antenna is .46 microhenry.
It is just as important to reduce the losses in the impedance
matching netvwrk.
netvJOrk.

There ~'Jere b'lo problems \'lith the original matching

First, the equivalent series resistance of the air variable

capacitor and the added resistance from the transformer are quite high.
Second, the winding ratio on the transformer is fixed, thus the impedance
of the generator can be matched only at one frequency, since the antenna
resistance is a function of frequency.

Therefore, the improved matching

network must have two essential features.

It must have low resistance

and it must be able to match the antenna and the generator for the entire
range of frequencies of interest.

It was finally decided to use vacuum

variable capacitors which have low series resistance and multiturn

-73adjustment capability that assures precise tuning.
is shown in Figure 4. 3.

The ne!.N matching network

This particular circuit was chosen because of

its simplicity and the minimum number of circuit elements needed.
Details on the dimensions of the antenna and the values of the capacitors
in the matching network are covered in Appendix a.

4.3

Plasma Loading Resistance Measurements
As shown in Section 2.7, the plasma loading resistance at one of

the cavity resistances, RL = r,·1 i;Rp' is a crucial quantity i n determining the efficiency of the antenna in delivering the r.f. pov1er into the
tokamak.

The efficiency,

n, depends on the plasma loading resistance and

the antenna resistance, Rant' in the fo1lowing v1ay [equation (2. 7.1)]

n =
Therefore, in order to have good efficiency in wave generation in the
tokamak, it is essential for the resonance plasma loading resistance to
2 2
be greater than the antenna resistance M w /R

> R

t•

And so the plasma

loading resistance must be measured in the experiment and compared with
the antenna resistance.
One way to obtain the p"lasma loading resistance is to r.e asure the
incident power, the reflected power into the antenna, and the antenna
current.

As indicated in equations (2.6.1) and (2.6.2), the i ncide nt

and reflected pm·Jer into the antenna can be derived from the i ncident
and reflected voltages measured with a VHF directional coupl e r placed
between the generator and the antenna matching network.

P.1nc

v?1 nc ;so

P ref = Vref/ 50

-74The antenna current is measured \vith a high frequency Tektron ix current
probe.

Once the antenna current is knovm, the plasma loadinq resistance

can be calculated as follows:
R= (P.

1 nc

-P

ref

)/I 2 -R

(4. 3.1)

ant

where I is the antenna current, and Rant is the antenna resistance.
The experimental setup for the plasma loading resistance measurements is shown in Figure 4.3.

As mentioned previously, the incident and

reflected voltages are measured by a VHF directional coupler with a characteristic impedance of 50 ohms.

The directional coupler is placed

between the r.f. amplifier and the antenna impedance matching network,
and so any change in the antenna resistance due to the plasma would shmup as a change in the reflected voltage.

The output of the directional

coupler is fed into a r.f. crystal detector and a phase detector .

The

crystal detector measures the amplitude modulation on the r.f. signal
coming from the directional coupler.

The output of the crystal detector

is fed into the multichannel transient recorder to be digitized and recorded.

The phase measurement of the incident and reflected voltages is

for obtaining the complex plasma loading impedance, and the de t ails of
this measurement are covered in the next section (Section 4.4) .
When the low resistance copper antenna is used, the r.f . current
in the antenna can get as high as 30 amperes.

The r.f. curre nt probe used

is only linear up to 2 amperes, so a 15 to 1 current divider i s placed in
parallel vJith the antenna.

The current divider is simply a piece of small

diameter \'lire with resistivity 15 times higher than the l/8 inch copper
used in the antenna.

Since the current probe is mounted on th e divider,

CRYSTAL
DETECTOR

w I°
TO
I BW

E~~AT!Sj

DIRECTIONAL~o WATTS
COUPLER
ANP.

TRANSIENT
RECORDER

ft511

TAPE
RECORDER

-....J

HIP 8601A
OSC.

I I

CRYSTAL
DETECTOR

Figure 4.3
Experimental arrangement for plasma loading resistance measurement. The input r.f.
power into the antenna is measured by the VHF directional coupler. The antenna
current is measured by a high frequency current probe.

-76which is in parallel with the antenna, any added resistive losses du e
to the current probe has little effect on the antenna resistanc~. The
results of the plasma loading resistance measurements are presented in
Section 5.3.
4.4

Phase Measurement
To obtain the complex loading impedance of the plasma at a cavity

resonance, the phase difference beb1een the incident and the reflected
voltage into the antenna must be measured.

As shown in Section 2.6,

the ratio of the amplitudes of the incident and reflected voltages into
the antenna gives the magnitude of the reflection coefficient, and the
phase difference between the incident and the reflected voltage into the
antenna gives the phase of the reflection coefficient.

= (V

ref

where '+'~ = ~'~'ref - ~'~'inc·

/V.

1 nc

) ej¢ =

IPI ej ¢

(4.4.1)

The complex input impedance can be obtained from

the complex reflection coefficient by a complex transform. The resistance,
Rin' and the reactance, \n• measured at the antenna matching network
(see Figure 2.4) are related to the complex reflection coefficient by
equations (2.6.6) and (2.6. 7).
As shown in Figure 4.4, the signals from the directional coupler
which measures Vre f and V.1 nc are split with one branch goin g to the
crystal detectors, and the other going to a phase detector.
detector is built to measure phase in a pulsed system.

The phase

The phase detec-

tor is capable of following a 2n phase shift in 4 microseconds.

As shown

in the block diagram of the detector (Figure 4.5), the input r.f. signals

CRYSTAL
DETECTOR

OIRECTIO~AL

COUPLER

EN I
300 WATTS
AWP.

PHASE
DETECTOR

HANSIEMT
RECORDER

TAPE

ucoaoER

TO
IBN
370

-(158)

.......

HIP &SOIA
OSC.

CRYSTAL
DETECTOR

Figure 4.4
Experimental arrangement for complex plasma loading impedance. The phase between the
incident and reflected waves is measured by a r.f. phase detector (see Figure 4.5).
The amplitudes of the incident and reflected voltages are measured by crystal detectors.

R.F.
MIXER
(SN7514)

ZERO CROSSING
COMPARATOR

I MHz
BAND PASS
FILTER
+SV ~

REFERENCE
SIGNAL
INPUT (f)

D.C.
THRESHOLD
ADJ.

_r
-=

LOCAL
OSCILLATOR
DETECTOR
SIGNAL
INPUT (f)

. R.F.
MIXER
(SN7514)

LOC

• f + 1MHz )

D. C.

+ 5V

74LSOO

74LS04

THRESHOLD
DETECTOR
CLEAR
74 LS74
EDGE TRIG . .,...._...,
FLIP-FLOP
CLOCK

1-j

THRESHOLD
THRESHOLD I
ADJ. =
DETECTOR
I MHz
BANDPASS r-; I ZERO CROSSING
FILTER I I COMPARATOR

200 KHz
LO\V PASS
FILTER

PHA

SE
I OUTPUT

"'-.1
(X)

74LSOO

Figure 4.5
R.F. phase detector (5 -50 MHz). Zero crossing comparators are used to shape the 1 MHz sinusoidal
signal into a square wave. The threshold detector has no output if the input 1 MHz signal is below
the voltage set by the threshold adjustment, thus disabling the phase detector. 200 KHz low pass filter
is a 5-pole Butterworth filter with 10-90 %risetime in 4 ~sec.

-79are mixed down to 1 i"lHz \'ii th a 1oca 1 osci 11 a tor, so that the output
voltage of the detector is not frequ ency dependent.

A s ens itive zero

crossing comparator is used to ensure that the phase output is not
amplitude dependent.

The input frequency range of the detector is be-

tvJeen 5 and 50

VJhich covers the frequency range of interest, 7 to

20 MHz.

i~Hz,

The output voltage of the detector is a linear function of the

phase, and the detector is capable of measuring phase shifts up to 2n.
When the complex reflection coefficient is calculated from the measured
amplitude and phase of the incident and reflected voltages, t he complex
plasma 1oadi ng impedance can be obtai ned from the con forma 1 tY'ans forms
in equations (2.6.6) and (2.6. 7).
The quantity of interest in th e experiment, as indica ted in
Section 2.6, is the plasma loading impedance, ZL (see Figure 2.4).

can be obtained from z.

by substituting the measured values of c and
of the impedance matching net\!lork into equations (2.5.1) and (2.5.2).
1n

The plasma loading impedance, Z, can be derived from ZL by subtracting
out the antenna impedance, Zant:
Z = ZL - Zan t

( 4. 4. 2)

The experimental results of these measurements are presented in Se ction
5. 5.

-80V.
5.1

EXPERIMENTAL RESULTS

Transmission Measurements
The toroidal eigenmodes were first observed in transmission.

described in Section 4.1, the transmitted signals were detected by a
six-turn loop probe located 180° around the toroidal axis from the
transmitter (Figure 4.1).

The input frequency into the transmitting

antenna was held constant.

The cavity modes were swept through by the

change in density as a function of time.

The modes appear as a series

of peaks on the r.f. output of the receiving probe.

The received r.f.

signals were passed through band-pass filter with 300 kHz bandwidth
and then fed into a crystal detector for amplitude detection.

The

output of the crystal detector is just the amplitude modulation on
the r.f. signal, i.e. a series of peaks.
A few of the typical transmission measurements for various input
frequencies are shown in Figure 5.1.

The top curve in Figure 5.1 is a

trace of the electron density evolution as a function of time for a
typical plasma discharge.

The density evolution for different plasma

shots is not completely reproducible, so the purpose of this trace is
only to give the general features of a plasma discharge.
The density values at which the cavity resonances are swept through
are found to be a function of the applied frequency.

At the lower ap-

plied frequencies, the transmission peaks cluster near the high density
region, whereas they become more spread out and appear in the low density
region at the higher applied frequencies.

The reason for this behavior

can be understood by studying the dispersion relations of the magnetosonic
wave.

From the dispersion curves in Figure 2.2, one can see that in

-81-

PLASMA DENS lTV

10X1Q

R. F. TRANSMISSION

8MHZ

10MHZ

12 MHZ

13 MHZ

14 MHZ

16 MHZ

...

PLASMA START

.a
TIME ( msec )

Figure 5. l
Transmission measurements versus a typical density evol .ution
as a function of time.

-82-

order to excite a particular mode at a given frequency, a certain plasma
density is required.

To excite the same mode at a lower frequency means

the plasma density must be higher.
tal result.

This is just the observed experimen-

When the input frequency is low, the resonance peaks gather

around the high density region, and as the input frequency increases, the
peaks move into the low density region.
This inverse relation between frequency and plasma density can be
simply summarized by the cut-off relation of the modes, i.e., k = 0.
The approximate cut-off relation is expressed in equation (2.1 .5),

where 1 is the radial mode number, m is the peloidal

mode number, N = 0

is the axial mode number, and ne ·is the electron density.

The equation

shows that for higher density, the cut-off frequency is lower; by the
same token, the low frequency modes propagate only near the density maximum.

One of the observations in the experiment is that no cavity mode

was observed at frequencies below 7 MHz.
To compare with the theory for a cold uniform cylindrical plasmafilled cavity model, the cut-off curves for various peloidal modes are
superimposed on the experimental data in a density versus frequency plot
(Figure 5.2).

The data points in the figure are obtained by the follow-

ing procedure.

The transmission peaks and the plasma density are re-

corded as in Figure 5.1 for a series of plasma discharges, typically
between 4 and 6 shots, with the same input r.f. frequency.

The time at

which a cavity resonance appears during the discharge is recorded ~

The

THEORETICAL CUTOFF FREQUENCIES
(UNIFORM PLASMA CYLINDER)

m .. 3

m= 2
m:-3
.,

Wei

m .. -2
m= o

+++ ...

1-1

ms-1

EXP. DATA
EXP. ERROR

12X10

DENSITY

Figure 5.2
Theoretical magnetosonic cavity mode cutoffs (i.e. toroidal
mode number N=O) in a cold plasma versus experimental data.
These curves are for the lowest radial mode (l =l).

-84-

values of the density at which the cavity resonances appear can be obtained from the measured line average density values at these recorded
moments in time.

Once both the frequency and the density for the modes

are known, a point can be plotted on the density-frequency gra~h (Figure
5.2).

The input frequency has been normalized to the ion cyclotron at

the center of the tokamak, i.e., 6 MHz.
is, peaks that appear
shots, are used.

Only the consistent peaks--that

the same general density region for all the

To get the frequency dependence of the modes, the input

frequency is changed between series of fixed frequency shots.
The agreement between the experimental data and the theory as shown
in Figure 5.2 is fairly good, though not perfect.

There are data points

below the general region of the cut-off curves.

The reason for the small

number of discrepancies between theory and experiment is the
simplicity of the theory used.

The toroidal effects, radial density pro-

file, poloidal magnetic field effects, and many others have not been
properly accounted for in the theory.

There is also a small amount of

uncertainty in the experimental data as indicated in Figure 5.2.

This

comes from the experimental errors in the electron density measurement
where the density uncertainty is about ±% fringe at the output of the
microwave interferometer (see Section 3.2d).
Since the plasma-filled cavity can be modeled by the equivalent
R-L-C circuit which has a simple pole at resonance, there must be some
relation between the amplitude and phase of the transmitted signal at
the cavity resonances.

Passing through a resonance, the phase should

undergo rapid change whenever the amplitude shows a peak.

This effect

-85-

can be detected by measuring the phase difference between the received
wave and the input oscillator signal at the transmitter (see Figure 4. 1).
The result of a typical phase measurem2nt and amplitude signal versus
time is shown in Figure 5.3.
Several properties of the amplitude and phase detectors, and the
experimental conditions can aid in the understanding of some of the
features of the data shown in Figure 5.3.

The amplitude signal is

inverted because the crystal detector used in the experiment i s inverting.
As shown in Figure 4.5, the phase detector has a threshold detector
where if the input signal is below a preset

d.c. value, the output of

the phase detector sits at the highest output level (corresponding to
the zero value shown in Figure 5.3).

This is the reason that the phase

signal always returns to zero when the amplitude drops below a certain
level.

The phase measurements were done with approximately 20 watts

r.f. power into the transmitting antenna, so the received amplitude
signals were rather low.

As the signal level approaches the d.c. threshold

level of the phase detector, there is a transition region of 10 mV around
the threshold voltage \'/here the phase detector output is an asci 11 ating
signal.

This is the result of the TTL transition region for :he nand

gate ( 74LSOO) when it switches between zero and one states.

This can

explain some of the noise-like oscillation when the phase de tec tor is
turning on and off.

Furthermore, the phase detector has a "dead" region

-86-

of 20 degrees when the phase goes beyond 360 degrees and returns at
zero degree.

Finally, there were cases of the data where the peaks

in the amplitude do not occur exactly at the same time as the steepest
rate of change of phase.

There is no good explanation for such cases.

One proposed way to identify the various poloidal modes, which
are separated fairly far from each other, is to reduce the step size
of the change in input frequency between shots.

This waY a resonance

peak seen at a particular density for a given input frequency can be
identified with a peak at a slightly different density, when the
input frequency is changed by a small amount.

In other v.JOrds, to

decrease the size of the frequency step taken between plasma shots
so that the data points in Figure 5.2 would be more closely spaced
along the frequency axis.

In priciple, as the frequency steps are

reduced to small values, the peaks that belong to the same mode can
be picked out.

(In our experiment some correlations between peaks

can be made).

By overlaying the theoretical dispersion curves on the

data one can guess that a particular set of data corresponds to a
certain mode.

However, this method is not used here because t he

various theories are not adequate for such a detailed compari so n.
For example, the dispersion curves depend on the assumed radi al density
profile used in the theory.
The theory used in this thesis is for a uniform plasma den sity.
However, if a vacuum region is introduced between the plasma and the
cavity wall, the locations of the dispersion curves would shi f t . . If the

RECEIVED SIGNAL

AMPLITUDE

PHASE

1t'
2'1t'

TIME ( msec)

Figure 5.3
Amplitude and the phase of the received signal in the transmission measurement.

-88vacuum is large enough, as was shown by Paoloni [18], them= 1 mode has
no low frequency cut-off (see Section 2.2).
The unambiguous method for mode identification is to use probes to
measure the spatial dependence of the fields in the tokamak.

This mea-

surement was not made in our experiment because of lack of time, so there
is no definite mode identification.
5.2

Plasma Loading Impedance in the Absence of the Cavity t1odes
In this section the experimental results of the "off resonance"

antenna impedance for different input frequencies are presented.

The

"off resonance" antenna impedance, Zoff = Roff + jXoff' is defined in
Section 2.5 as the sum of the antenna impedance and the plasma loading
impedance during the absence of any cavity resonances.

Roff and Xoff

are determined by substituting the capacitance values, c1 and c2 , of
the impedance matching circuit used to match the generator impedance
"off resonantly" into equations (2.5.1) and (2.5.2).

(The condition

for impedance matching in these equations is when Rin = generator
impedance= 500.)

To obtain the "off resonance" loading impedance due

to the plasma alone, 6Z = 6R + j 6X, the antenna impedance must be subtracted from zoff'
(5.2.1)
(5.2.2)

An interesting experimental finding is that the "off resonance" plasma
loading reactance, 6X, is greater than zero, i.e.
tance is increased by the plasma effect.

the antenna induc-

A possible exp.lanation for the

-89increase in the antenna input inductance with the onset of th e plas ma is
given in Section 2.3.

As indicated in equation (2.3.9a), if the reac-

tance contribution from the cavity modes with resonant frequencies higher
than the applied frequency is greater than the contribution for the other
modes, the basic antenna inductive reactance will shm-1 an increase from
the effects of the cavity modes.

The values of ~ R and ~ X for

different input frequencies are given in Tables 5.1 and 5.2.
The experimental procedure for measuring the "off resonance"
impedance, Zoff' was as follows.

First the antenna was matched to the

generator impedance in the absence of the plasma.

The matchi ng process

was to adjust the capacitors c and c so that a minimum in the re1
flected voltage from directional coupler was observed (see Figure 4.3).
Once the plasma was formed around the antenna, the generator impedance
was no longer matched, and the reflected voltage from the directional
At this point c and c2 were readjusted to mini mize
the reflected voltage in the presence of the plasma. The values of c
and c2 were recorded and then substituted into equations (2.5.1) and
coupler increased .

(2.5.2) so that the "off resonance" impedance could be obtained.
The effects of "off resonance" plasma loading impedance for various
input frequencies are summarized in the 7th and 8th columns of Tables
5.1 and 5.2.

The data in these tables were taken under simil ar condi-

tions but on different days, and so they serve as a comparis o~

for

each other.

Corre-

The first column indicates the input frequencie s.

sponding to each frequency, the input impedance of the anten na was
measured with and without a plasma, as indicated in column 2.

First the

antenna impedance was obtained in the vacuum chamber by measu r ing the
values of the tuning capacitors c1 and c2 (see columns 3 and 4).

The

-90-

antenna resistance, Ran t' and inductance, Lan t' can be deriv ed by substituting c1 and c2 into equations (2.5.1) and (2.5.2).

As shown in

the tables, when the plasma is present the capacitors must be retuned,
and the "off resonance" impedance Zoff = Roff + jXoff is computed from
the retuned values of c and c .

c (pf)

c2 ( pf)

LYJI)

R(Q)

~R(Q)

~~( Q)

Vacuum
Plasraa

531
502

45.7
47

.44
.462

.31
.36

.05

1.38

.5 - .9

Vacuum
Plasma

360
347

36
38

.445
. 46

.4
. 485

.085

1.13

.6 - l .

Vacuum
Plasma

270

28.7

.456

255

30.7

.433
.453

.57

.114

l . 76

.8- 1.5

207

.43
.46

.514
.668

.154

.9 - 1.2

.42

.6
. 82

.22

4.41

Freq.
(Mliz)

Condition

"10

Zan t is shown on the top line of

Vacuum
Plasma

190

23.5
25

Vacuum
P1asna

165.6
148.8

20.5
22

.459

cur

(Q)

.9 - 1.2

Table 5.1 Summary of the plasma loading impedance for 'off res onant'
matching condition. R and L are the input resistance and indu ctance of
the antenna measured under various conditions. The top line of each double
rows is the data taken in vacuum and the second line correspon ds to data
taken in the plasma. 6R = Roff-Rant and 6XL = Xoff-Xant are th e 'off resonant' plasma loading impedance. Rcur is th2 range of the ~eak l oading
resistance obtained by using equation 4.3. l. The antenna is l. l inches into
th e tokamak ch amber.

-91-

Freq.
(MHz)

Cond.i tion

c1 (pf)

c2 (pf)

L( )JH)

R ( ~)

fiR(~)

fi x(~)

Vacuum

525
490

44. 5
45.2

.445
.474

.3
.350

.05

1.823

. 4 - .8

364
352

35.1
37.5

.441
.453

.38
.456

.076

. 86

. 8 - 1.2

264
244

27.8
28 . 2

.444
.476

. 448
. 53

.082

2.77

. 6 - l.

200
192

23 . 9
26

. 443
.455

. 563
.7

.137

1.2

.6 - l. 3

161
150

20
24 . 9

.432
.448

. 604
.4

1.81

.7- 1.2

Plasma

Vacuum
Plasma

Table 5.2

cur

(r2)

Summary of another data set taken under similar conditions

as the data presented in Table 5.1.

This data set was taken on a diff-

erent day than those given in Table 5.1.

-92columns 3 and 4 for each set of data with a given frequency, and Zoff is
on the bottom line.

The contributions from the plasma alone to the "off

resonance" impedance are shown in columns 7 and 8, where 6R = Roff- Rant
and 6X = Xoff- Xant.
Two points must be emphasized about the condition under which these
data were taken.

First, as shown in Figure a.l of Appendix a, the anten-

na impedance is a function of its distance into the tokamak vacuum
chamber.

This is because when the antenna is out of the vac uum chamber

it sits in a 6 x 4 x 1 11 stainless steel port.

This port can influence the

antenna impedance by lowering its inductance and increasing its r esistive losses through eedy current losses in the port wall.

Also, the

plasma loading depends on how far the antenna is into the chamber, since
it is a function of the coupling coefficient of th e antenna .

When the

antenna is completely out of the tokamak chamber and into the port, for
example, the plasma loading is zero.

The data presented in Tables 5.1

and 5.2 were taken with the antenna approximately 1.1 inches into the
vacuum chamber.
5.3

Plasma Loading Resistance at the Cavity Resonances
The experimental results of the plasma loading resista nce , R, at

the various cavity modes are presented in this section .

The plasma re-

sistance is obtained from the power-current measurements dis cu ssed in
Section 4.3.

The equation used to compute the plasma loadi ng resistance

is reiterated here for the convenience of the readers (equat ion (4.3.1)):

inc

Pref )/I a - Rant

-93-

where P.

and P fare the incident and reflected power, re s pectively,
re
Ia is the antenna current, and Rant is the basic antenna resistance.
1nc

The presentation of the experimental data for the plasma loading
resistance at the cavity resonances is divided into two parts.

First,

one set of experimental data taken at a given input frequency is presented as an example of the measured data and the computed results of
the loading resistance.

The general features of the data for other fre-

quencies are described.

Second, the magnitudes of the plasma loading

resistance for the various cavity modes at different input frequencies
are summarized in Tables 5.1 and 5.2.

For each input frequency there

are many cavity modes excited, each with a different loading resistance.
Therefore, only the range of peak loading resistance for various cavity
resonances at each input frequency is given in these tables.

This

range of the "resonant" loading· resistance is denoted by Rcur , where
the subscript 'cur' is to identify the power-current method used to
determine the resistance, and to differentiate this result from the
range of "resonant" loading resistance Rres obtained from the measured
complex reflection coefficient.

The computed values of Rcur and Rres

from experimental data are compared in Table 5.3.
Figure 5.4 shows a typical plasma discharge, where th e input
r.f. frequency is ll MHz and the antenna is "off resonantly" t uned with
the tuning capacitors c1= 424 pf and c2= 44 pf (see Figure 2.4 ).

The

"off resonance" tuning condition is indicated in the reflec ted voltage
data, the 4th trace, where the reflected voltage is a minimum between
the cavity resonances.

Under the "off resonance" tuning condition, when

MHZ

TUNE FOR OFF RESONANCE
1.35
1/)

RESISTANCE

o.o

1.0
...,.

RECEIVED

ANT. CURRENT

REFLECTED VOLTAGE

TIME

1.5

(MSEC)

Figure 5.4
Calculated plasma loading resistance from the input power
into the antenna and the antenna current data. Antenna is
matched to son in the absence of the cavity resonances.

-95-

a cavity resonance appears, the generator impedance is no longer ma tched,
and so the cavity resonances show up as increases in the reflected voltage from the directional coupler and decreases in the antenna current.
These are the observed behaviors of the measured reflected voltage and
antenna current as shown in traces 3 and 4.
The relation between the reflected voltage and the antenna current,
i.e., an increase in the reflected voltage corresponds to a decrease
in antenna current, can be understood as follows.

Since the incident

voltage is relatively constant throughout the plasma discharge, the
amplitude of the reflection coefficients,

!PI = (Vref/Vinc) should be

proportional to the amplitude of the reflected voltage.

The antenna

current can be expressed in terms of the reflection coefficient as:

V/Z
(5.3.1)

where equation (2.6.5)is used to related Z and p, and Z0 is the characteristic impedance.

For a mismatch condition, the reflected voltage

increases so the magnitude of the reflection coefficient will increase
accordingly.

The term 1/(1 + p) of equation (5.3. 1) can be appr9xi-

mated by (1-p) if the mismatch is small.

Thus equation (5.3.1) can be written

in the following approximate form:
I "' v
50 ( 1 - 2p)

where Z0 is taken to be 50 ohms.

From this relation one can see that if

-96-

there is an increase in the reflected voltage, there must be a corresponding decrease in the antenna current. As a confirmation of this
relation, the antenna current in trace 3 of Figure 5.4 has a minimum
whenever the reflected voltage shows a maximum.
The general features of the time dependence of the peaks of the
resonant loading resistance correlate well with the time dependence
of the peaks in the transmitted signal.

~~henever

a peak in the trans-

mitted signal occurs, a corresponding peak in the loading resistance
appears at the same time.

Moreover, the density dependence of the

resistive loading peaks is the same as the transmission peaks .

~Jhen

the input frequency is low, most of the resistive peaks occur near the
density maximum, whereas at the higher frequencies, the peaks become
more spread out.
An observation in the experiment is that the modes with the largest
transmission amplitude are not necessarily the ones that show the largest
input loading resistance.

This is because the input loading resistance

measures the power delivered into the cavity, whereas the receiving probe
only detects one component of the field.

Depending on the ca vity mode

that is excited, strong input loading does not necessarily co rrespond to a
strong field component measured by the probe.
In Tables 5.1 and 5.2, the ranges of the peaks of the resonant
loading resistance, Rcur , for the different modes at various in put frequencies are summarized.

The resistance values are calculated using

equation 4.3.1 in the same fashion as the data shown in Figure 5.4..

Only

-97-

loading peaks that are substantially above the noise level are kept.
In the same tables, the 11 0ff resonance 11 plasma loading resistances,
i.e., 6R = Roff- Rant' for the same plasma shots are presented as a
comparison to the peak 11 resonant 11 plasma loading resistance, Rcur·
5.4

Reproducibility of the Plasma Loading Resistance Measurement
Even vJith the same input frequency, the magnitude of the resistive

loading at the various cavity modes has been observed to be different
for different plasma shots.

For two consecutive plasma discharges, the

resistive loading may be strong at certain cavity modes on one shot,
yet appears weaker for the same modes on the next shot.

One possible

explanation for this behavior in the plasma loading is that the radial
density profile of the plasma is not completely reproducible for different plasma discharges.

As shown by Paoloni in a recent paper, the

coupling coefficient of the transmitting antenna depends on the radial
density profile [18].

For the low radial and poloidal modes, which

are believed to be the observed modes here, the radial dependence of
the r.f. magnetic field for a uniform density profile is quite different
than that of a parabolic density profile.

The strength of the magnetic

field components at the outer radius of a cylindrical cavity is weaker
for a parabolic density profile than a uniform density profile.

(The

uniform density profile used here has the same line-average density as
the parabolic profile.)

Since the antenna is located at the outer edge

of the tokamak, the coupling coefficient of the antenna should be higher
for a uniform density profile than a parabolic profile.
tails of the theory see the paper by PaoToni).

(For more de-

The position of the plasma

-98-

column in the tokamak in some sense can be thought of as a r adial density
profile. The coupling coefficient will depend on whether during the
first tvw msec of the discharge the plasma column is near the outer wall
of the tokamak where the transmitting antenna is located, or it is
formed initially near the inner wall. Since there is no radial profile
measurement in our experiment, this is only a possible explanation of
the fluctuation in the magnitude of the resonance loading resistance. It
must be pointed out here that there were some judgmental factors in the
data taking.

Only shots with strong cavity mode loading

those with weak loading were discarded.
mode loading

were kept and

Therefore, the ran9e of the cavity

presented in Tables 5.1 and 5.2 are examples of the strong

loading cases.
5.5

Complex Plasma Loading Impedance Measurement
As indicated in the introduction, the main emphasis of this thesis

is on the measurement of the complex plasma loading impedance of the
cavity modes.

The real part of the impedance, as mentioned previously,

is important in the determination of the efficiency of wave generation
in the tokamak.

The complex loading impedance is important i n determin-

ing how to match the generator impedance to one of the cavity resonances.
Only by matching the generator impedance to a cavity resonanc e can the
maximum power be delivered to the cavity when the plasma loa di ng is
highest.
The experimental data in this section are presented in t he same way
as in Section 5.2.

Experimental results for two input frequ encies,

11 MHz and 16 MHz,are shown in Figures 5.5 and 5.6, and the general behavior of the complex input impedance for the various input frequencie s
are discussed.

Then the complex impedance data for the various input

11 MHZ

\D
\D

REFLECTED VOLTAGE

PHASE

2rrt
0.

1.5

TIME (MSEC)

Figure 5.5
Measured phase different between the incident and the
reflected waves using a VHF directional coupler.

-100-

16 MHZ

.64

REFLECTION COEF.

0 r------------------------------------------------1.2 fJ

-.8fJ

1.5

TIME (msec)

Figure 5.6
Computed complex plasma loadinq impedance from the complex .
reflection coefficient. The three identified peaks correspond
to the three Q circles shown in Fiqures 5.7 to 5.9.

-101frequencies are summarized in Table 5.3.
The physical quantities actually measured in the complex plasma
loading impedance experiment are the incident voltage, the reflected
voltage, the phase difference between the incident and reflected waves,
the antenna current, the transmitted signal, and the plasma density. The
incident voltage, as mentioned earlier, is fairly constant during the
plasma discharge.

The antenna current is measured so that the resistive

loading can be calculated using the power-current equation as a check for
the real part of the complex loading impedance.

Figure 5.5 shows a set

of experimental data for an input frequency of 11 MHz.

When the third

and fourth traces of Figure 5.5 are compared with the second and third
traces in Figure 2.7, one can see that there is general agreement between
theory and experiment as to how the reflected voltage and the phase between the incident and reflected \'laves pass through a cavity resonance.
Here it is assumed that the incident voltage is constant enough so that
the reflection coefficient follows the trends of the reflected voltage.
As expected, corresponding to every peak in the reflected voltage, there
is a steep change in the measured phase.
Figure 5.6 contains a typical set of circuit parameter results computed from the measured data for an input frequency of 16 MHz.

This set

of data was taken with the antenna "off resonantly" matched.

The first

trace in the figure is the amplitude of the reflection coefficient, and
it is calculated from the incident and reflected voltage data using equa(4.4.1).

The phase data in trace 2 are the direct output of the phase

detector.

From the complex reflection coefficient, the complex input

impedance, Z.1n , can be calculated from equations (2.6.6) and (2.6.7).

-102The results of this calculation are shown in traces 3 and 4.

When

these data are compared with the calculated values shown in the fourth
and fifth traces in Figure 2.7, the general features of the data seem
to agree well with the computed results from the circuit model.

For

the experimental data in Figure 5.6, the plasma density reaches a maximum at

time = 1 msec.

At this time the slope of the density evolution

reverses in sign, so as stated before, both the phase and the reactance
should reverse in direction.

From the second trace of Figure 5.6 one

can see that the direction of the change in phase is reversed at around
time = 1 msec.

Furthermore, the direction of change in the reactance

shown in the fourth trace of Figure 5.6 is very similar to the fifth
trace in Figure 2.7.

The measured reactance goes negative before pass-

ing through a resonance during the density buildup and goes positive
before passing through a resonance during the density decay.

This is

the same behavior as the computed results using a similar density evolution.
The next step in the calculation is to compute the plasma loading
impedance

Z = R + jX by transforming Zin across the matching network

using equations (2.5.1) and (2.5.2) and subtracting the imped ~ nce of the
antenna, Zant·

Before making this calculation, the capacitance s of the

elements in the matching network must be measured.

For this set of data

c1 = 190 pf and c2 = 34 pf.

The results of the computation are shown in

traces 5 and 6 of Figure 5.6.

The maximum resistive loading is about

1.8 ohms here

in comparison with 1.2 ohms, the value of the maximum

plasma resistive loading obtained from the power-current method. · From

-103Appendix a, the antenna resistance at 16 MHz is about .56 ohms, therefore the plasma loading is between two and three times the antenna
resistance.

This corresponds to a wave generation efficiency n between

70% to 80%.

The ranges of the complex plasma loading impedance of the various
input frequencies are summarized in Table 5.3 in a manner similar to that
of Tables 5.1 and 5.2.

The format of Table 5.3 is the same as Tables 5.1

and 5.2, except the plasma contribution to the "off resonant" loading
impedance, ~R and ~X, is not shown.

They can be obtained by taking the

difference of the values in the two lines under columns 5 and 6.
input frequencies of the experiments are given in column l.
under which the data were taken is shown in column 2.

The

The condition

The capacitances

of c and c2 needed to match the generator impedance to the antenna and
the antenna plus "off resonant" plasma are shown in two lines , columns
3 and 4, respectively.

Column 7 shows the range of the peaks of the load-

ing resistance Rcur for the particular data set.

It is obtained from the

input power and the antenna current in the same fashion as the data presented in the 9th column of Tables 5.1 and 5.2 (see equation (4.3.1) for
this calculation).

This range of loading resistance Rcur is presented

here as a comparison to the real part of the complex loading impedance
calculated from the complex reflection coefficient measured in the experiment.

The complex plasma loading impedance

Z . is calculated from

equation (2.8.1), where the antenna impedance has been subtracted.
real and imaginary parts of the impedance are given as

R = RL - Rant

The

Freq .

c1 (pf)

c2 (pf)

L( )lH)

R(IG)

Vacuum

532

44 . 5

. 44

. 293

Plasma

509

46 . 1

. 457

. 34

Vacuum

364

34 . 8

. 442

. 375

Plasma

352

. 453

. 445

Vacuum

262

28 . 4

. 446

. 472

Plasma

256

. 45

. 61

Vacuum

199

. 446

. 53

Plasma

189

. 459

. 77

Vacuum

161

20 . 8

. 43

. 65

Plasma

146

. 461

. 98

Conditi on

(MHz)

Rcur( IG)

Rres(IG)

Xr es(IG )

. 5 to l.

. 6 to 1.1

-.52 t o . 6

.4 t o . 7

. 75 to 1. 1

-. 6 to . 8

.7 t o 1 . 3

. 9 t o l. 46

-.7 t o l.
___,

..j:>

l. t o l. 5

1. 2 to 1.8

-. 8 t o 1. 2

. 82 to l. 7

. 95 t o 2 .

-. 9 to. 8

Table 5. 3 The ranges of the complex plasma 18ading impedance , R
and X
, at the
various cavity modes are given in columns 8 and 9 . The resistiv§eloadingr~~ the same
cavity modes measured with the power- current method is shown in column 7. c1 and c2
ar e t he val ues of the tuning capacitors . R and L are the antenna resistance and
induct an ce with or without plasma . The data were taken under ' off resonant ' tuning
condition and the antenna is at 1.1 inches into the tokamak chamber .

-105where RL and XL are given in equations (2.5.1) and (2.5.2).

The range

of the complex loading impedance maxima at the various cavity resonances
are given in columns 8 and 9.

The 8th column shows the range of the

plasma loading resistance maxima, Rres' for the various cavity modes.
These ranges seem to agree generally with the data, Rcur , measured using
the antenna current.

The discrepancies between the data obtained from

these two methods as shown in columns 7 and 8 can be attributed to errors
in the calibrations of the instruments used in the measurements.

Column

9 shows the range of the reactance, Xres' for the largest resonance peak
measured in the particular plasma discharge.

As shown in the sixth

trace in Figure 5.6, this reactance changes sign rapidly as a cavity
resonance is passed through.
As indicated previously, the cavity resonance effect can be seen
more readily when the complex cavity input impedance is plotted in the
complex impedance plane.

When the real and imaginary parts of the imped-

ance are plotted against each other as they pass through a resonance,
the resultant curve is a circle, known as the Q circle.

Figures 5.7,

5.8 and 5.9 are the experimental Q circles of the three major peaks which
occur at time= .5 msec, time= .9 msec, and time= 1.1 msec in Figure
5.6.

The time between the points in the Q circle plots is 2 micro-

seconds.

They are approximately circles, although there are some dis-

tortions.

The distortions in the Q circles can be divided into two

classes, depending on how fast the density is changing as a function of
time.

The first class is when the density is changing slowly enough with

time that the condition for meaningful impedance measurement, the .
inequality (2.4.5), is satisfied:

-106-

16 MHZ

(PEAK 1)

X in
(Ohms)

-25

-50L-------------------------L---~------------------~

100

150

Rin (Ohms)

Figure 5.7
Q circle for the first peak shown in Figure 5.6. The circle
is traced out counterclockwise, and the time between consecutive
points is 2 ~sec.

-107-

16 MHZ
(PEAK 2)

100

X in
(Ohms)

-50

100

~ ~ ~

150

200

Rin (Ohms)

Figure 5.8
Q circle for the second peak shown in Figure 5.6. The circle
is traced out counterclockwise, and the time between consecutive
points is 2 ~sec.

-108-

100

16 MHZ
(PEAK 3)

t!)

50
t!)t!)

t!)

t!)
t!)t!)

X in
(Ohms)

t!)

(!)
(!)

I!!(!)

I!)

t!)

(!)

l!le

I!)

t!)
t!)

I!)

t!)

I!)

-50

t!)

I!)
., 'I!)

t!)

100

150

200

Rin (Ohms)

Figure 5.9
Q circle for the third ~eak shown in Fioure 5.6. The circle
is traced out clockwise, and the time between consecutive
points is 2~sec.

-109-

T >> 2Q / w
L o
where T is the time between the half-power points of the resonance
w0 is the resonance angular frequency, and Q is the loaded Q

peaks,

of the cavity.

(This inequality is examined in detail for various

cases in Section 5.6, where the Q of the cavity is calculated.)

this case the cavity modes are swept through much slower than the response time of the output operational amplifier of the crystal detectors,
4V oer 2 usee, and the response time of the phase detector, 2n per 4usec.
The Q circles sho~n in Figures 5.7, 5.8, 5.9, are observed under this kind of
condition where the distortions from a circle are not large.

The distortions

of the Q circles shown in Figures 5.7 through 5.9 can be partly attributed
to errors in the calibration of the square law crystal detector, the
linear phase detector, and partly to density fluctuations.

The effects

of the density fluctuations are most obviously observed in Figure 5.9
where small oscillating points are superimposed on the main circular
curve.
The second class of distortion in the Q circles is when the density changes so fast that the condition T ~ 2Q/w0 is approached.

this case the limits of the response times of the phase and crystal
detectors are also approached.

The resultant Q circles are greatly

distorted, usually becoming a very flat ellipse.

Data of these kind

are discarded, since the impedance information from them is not meaningful.
As mentioned before, the direction in which the Q circles are
traced out as the cavity resonance is passed through is a function of

-110-

the sign of the slope of the density evolution.

The first two peaks shown

in Figures 5.7 and 5.8 are traced out counterclockwise as the cavity
resonance is passed through.

When the time is around one millisecond in

Figure 5.6, the density reaches a maximum, so the slope of the density
evolution is reversed in sign after this point.

Therefore, the resonance

shown in Figure 5.9 is traced out in the clock1t1ise direction, opposite to
the previous two peaks.

This behavior is what the circuit model has

demonstrated.
5.6

Cavity Q, Antenna Coupling Coefficient, and Antenna Efficiency

Cavity Q
By using the approximate density-frequency relation for the cavity

mode cutoffs, the Q of the plasma-filled cavity can be simply estimated.
The method used to estimate the cavity Q is described in Section 2.6, and
the Q is related to the density by the following equation:

As mentioned in Section 2.4, the Q obtained from the density measurements
is the loaded Q of the cavity, and the unloaded Q of the cavity can be
related to the loaded Q by equation (2.6. ll) at a cavity resonance by

All the quantities in this equation have been measured experimentally or can be calculated from experimental data.
density measurements.

QL is derived from the

Roff is the antenna resistance plus the res.istive

contribution from the plasma 1t1hen the cavity J~esonances are not present.

-111-

It can be calculated from the values of the capacitors that are used in
the matching network to tune the antenna "off resonantly".

Roff is de-

rived from the following equation:
X )2 + R2. J
2 [(
o ff = R.1n XC I XC + C
1n

are the capacitive reactance used
to tune the antenna "off resonantly".
The quantity w2M2/R is merely the

where Rin is 50 ohms, and XC

and XC

loading resistance measured at the resonance of one of the cavity modes.
From equation (2.2.2a) this factor can be related to the difference between
RL and Rant at the resonance frequency of one of the modes, i.e., at rlp= 1:

(RL - Rant )max
The 3 dB drop-off points in the reflected voltages can be obtained from the
experimental data.

As an example, the QL and Q0 for the three major reson-

ance peaks are given in the second and third column or Table 5.4, respectively.
Cavity
Resonance
Peaks

Loaded
Cavity Q

Unloaded
Cavity Q

[QL]

Antenna
Efficiency

[Qo]

Coupling
Coefficient
[K2]

240

470

5.6 X 10- 5

70 %

240

560

170

400

6.5 X 10-5
8.3 X 10 -5

[n]

77 %

75 %

Table 5.4 The estimated loaded Q, unloaded Q, antenna coupliDg
coefficient, and the antenna efficiency for the three resonance
peaks shown in Figure 5.6. The antenna Qa used in the computation
is 90.

-112It must be emphasized that this way to obtain the Q of the cavity
is only an estimate. Furthermore, there is an experimental error in the
measured density, as stated in Section 3.ld.

That is, there is an uncer-

tainty factor of ±1/4 fringe in the fringe counting method of the density
measurement with the micrmvave interferometer.

The estimated Q for the

cavity resonances with different input frequencies does not differ
greatly.

For the frequency range used in the experiment, between 10 and

16 t·1Hz, the range of the 1oaded Q is between 120 and 250, and the range of
the unloaded Q is between 400 and 700.
The contribution to the measured cavity Q can be divided into two
parts: the damping of the wave by the plasma, and the energy losses due to
the finite conductivity of the tokamak wall which is made of stainless
steel.

As noted in the introduction, there have been several theories on

the damping mechanism of the magnetosonic wave by the plasma [4,8,25].
Th e theories are quite involved, so it is left to the interested reader to
look up the references.

It is important here to estimate the losses in

the tokamak wall to see whether the wall los s is a dominating factor.

The

calculated cavity Q for a cold plasma-filled cylindrical cavity with
stainless steel \t.Jall is presented in Appendix c.

The estimati on is for

them= 0 poloidal mode, the low radial and the axial modes.
lower limit on the estimated cavity Q.

This is a

The calculated cavity Q 'tlith wall

loss is about 1300, which is two to three times higher than t he measured
cavity.

Therefore, although the wall loss is not small, it is not the

dominating term in the measured cavity Q, and so a large part of the wave
energy should be absorbed by the plasma.

-113Since the Q in the experiment has been obtained, it is appropriate at this point to go back to the inequality (2.4.5) and see \1/hether
the impedance measurements in our transient system are valid.

Restating

(2.4.5),
T »

2Q/w 0

where T is time between the half voltage points in the resonance peaks,
and w0 is the resonance frequency.

Every resonance datum taken in the

experiments has been substituted into this inequality to check for the
validity of the impedance measurement.

Those data that do not satisfy

the inequality because the resonances are swept through too quickly by
the density are discarded.
in Figure 5.6.

As an example, let us check the three peaks

Since the experiment was performed with the input antenna

coupling strongly to the tokamak, the Q used in the calculation is the
loaded QL of the cavity.
For the first peak, T = 24 ~sec, QL = 240, and the angular frequency
w0 = 2n x 16 x 10 6 rad/sec. Therefore, 2QL/w 0 = 5 ~sec, which is smaller
than T, and so the impedance measurement of this resonance is valid.
For the second peak, T = 30 ~sec, QL = 240, and the angular frequency is the same as above.

Therefore, 2QL/w 0 = 5 ~sec, rthich is again

smaller than T.
For the third peak, T = 50 ~sec, QL = 170, and w0 is the same as
above, therefore 2QL/ w0 = 4 ~sec, vo~hi ch is smaller than T.
As one can see, all three of the peaks in this data set satisfy the
inequality, and so the impedance measurement is valid.

~~hen

this test is

given to other data at various input frequencies, there are cases where
the density changes so fast that this inequality is no longer satisfied.

-114-

These data can usually be picked out during the experiment and discarded
right av1ay.

In future experiments this constraint will not occur, because

a gas puffing system is presently being installed on the tokamak to keep
the density higher and more constant as a function of time (see Section 6.2
for details).
b.

Antenna Coupling Coefficient
With the complex impedance and the Q measurements, the coupling coef-

ficient of the antenna can be calculated using equation (2.6. 12)
2 =

where again this is for an "off-resonantly" tuned system.

Since the Q of

the cavity is an estimate, the coupling coefficient should be called an
estimate as well.

As mentioned in Appendix a, the coupling coefficient is

a function of the distance that the antenna protruded into the vacuum
chamber.

For the different cavity modes at various input frequencies, the

coupling coefficient of the antenna when it is l.l inches into the tokamak
chamber has a range between 3 x 10- 5 and l x 10- 4 . As an example, consider
again the data in Figure 5.6 which were taken with the antenna at l.l
inches into the tokamak. The coupling coefficient, K2 , for th e three peaks
shown in Figure 5.6 are given in the third column of Table 5.4 .
c.

Antenna Efficiency
Once both the cavity Q and the antenna coupling coefficient are es-

timated, th e wave generation efficiency, n , of the present tv-m-turn
copper loop antenna can be estimated from equation (2.7.2),

-115-

The antenna efficiency for the three resonance peaks in Figure 5.6 are
given in column 5 of Table 5.4.

The efficiency for the present antenna

system has been observed as high as 80%.

Possible ways to increase the

antenna efficiency are elaborated in Section 6.2.
5. 7

~·latching

Impedances at the Cavity Resonances

For future high power experiments, it is essential to be able to
match the generator impedance at one of the cavity resonances where the
resistive loading of the cavity is high.

This process is much more dif-

ficult than matching "off resonantly", because during the pass age through
of a resonance both the real and imaginary parts of the impedance are
changing very fast.

Very precise tuning is required to trans form the re-

sistance to 50 ohms and tune out the reactance at one of the cavity
resonances.

In our experiment the difficulty is compounded by the fast

density decay as a function of time.

This makes tuning "on resonance"

harder because sometimes it is difficult to tell whether the tuning is
exactly on resonance or just slightly mistuned, because the resonance
peaks are so sharp.

Before presenting the data in our experiment, some

improvements to the measurement system so that "on resonant" t uning will
be easier are discussed.
First, if the change in the density is slo\tJer, the tuni n;J process
would be easier.

Recently, a gas puffing system has been ins ta lled on our

tokamak to puff neutral gas, which can diffuse across the con f inement
magnetic field, into the tokamak plasma.

The neutral gas is i onized in

the plasma, thus increasing the plasma density.

By puffing t he gas at

the appropriate time in the plasma discharge, the fast decay in the density after the initial buildup as shown in Figure 3.3 can be compensate d,

-116and the density evolution can be kept constant to about
stantial portion of the discharge.

10% for a sub-

This way the cavity modes are swept

through much slov1er by the density evolution, and so the resonant peaks
appear broader.

Second, the better procedure for "on resonant" tuning, as

mentioned in Section 2.5, is first to match the impedance of the generator
at the "off resonant" condition, which is an easier process than "on
resonance" tuning.

If an on-line computer system is available, the plasma

loading impedance ZL at a cavity resonance can be calculated from the meai¢

sured complex reflection coefficient, p = V f/V.
re
we e ' where ¢ is the
phase difference between the incident and the reflected waves. By using
equations (2.5. 1) and (2.5.2), the input impedance at the antenna, ZL, can
be calculated from the input impedance at the impedance matching circuit,
Zin' and the values of the capacitors,
tune the antenna.

c1 and c2 , used to "off resonantly"

Once the values of ZL at the various cavity resonances

are known, the capacitances
one of the cavity modes.

c1 and c2 can be recalculated for matching to

Recently, a minicomputer was acquired for on-line

operation with our tokamak.

vJith the aid of the computer, "on resonant"

matching will be easier for future experiments.
The actual "on resonant" tuning reported in this thesis VJas done by
minimizing the reflected voltage from the directional coupler at one of
the cavity modes through trial and error.

The experimental data of the

reflected voltage and r.f. current for the "on resonant" matching experiment appear just opposite to the data from the "off resonant" tuning experi ment.

Under the "off res on ant" matching condition, the reflected

voltage is minimized and the antenna current is maximized between the
cavity modes; v1hereas for the "on resonant" matching condition, the

-117reflected voltage is minimized and the antenna current is maximized at
one of the cavity modes.

For the "on resonant" matching condition, the

reflected voltage is high and the antenna current is lov1 betv1een the
cavity modes, because the antenna is mismatched to the generator without
the cavity modes.

Since the matching conditions for the various cavity

modes are different, only one mode can be exactly matched for a given
setting of c1 and c .
Two of the "on resonantly" tuned cases are given in Figures 5.10
and 5.11.

As indicated before, only one mode is properly matched for

each case.

For the data in Figure 5.10, the tuning capacitances needed

to tune "on resonantly" are c1 = 210 pf and c

38 pf, and for the

traces in Figure 5.11, c1 = 336 pf and c2 = 49.5 pf.

15 MHZ
MATCHED

c;)

AT RESONANCE

P1- Pr

1.2

R=~

RESISTANCE

:E
J:

oo.o~

.. r~

~-~v~

--'

ANT. CURRENT

1.5

TIME

(msec)

Figure 5. 10
Antenna is matched to 50 ohms at one of the cavity resonances.

12 MHZ
MATCHED AT RESONANCE

1.0

en
::;:

0.0

.....
.....

\.0

1.5
TIME (msec)

Figure 5. ll
Antenna is matched to one of the cavity resonances.

-120VI.

6.1

CONCLUSIONS

Summary
This thesis has presented the results of some low power experiments

in the propagation of the fast magnetosoni c cavity modes in a research
tokamak.

A great deal of attention has been given to the study of the

complex input impedance of the antenna, the antenna design, and the design of the impedance matching net'IJOrk.

These measurements are of great

importance to future high power experiments where efficient coupling of
pal'ler to the plasma is essential.

Through the high power heating experi-

ments the feasibility of using magnetosonic v1aves as a method to heat the
plasma to fusion ignition can be evaluated.
The toroidal cavity modes could be readily observed in transmission
measurements, where they appeared as a series of maxima in the transmission amplitude.

The measured eigenmode dispersion relation seemed to

agree qualitatively with the results from the simple theory for a cold
cylindrical uniform plasma cavity. Although mode numbers were not determined,

the manner in which the phase between the transmitting and the

receiving signal changed indicated that when passing through a cavity
resonance the received amplitude peaks were due to cavity rescnances.
After carefully designing a low-loss transmitting antenr a and a
low-loss matching network, the cavity resonances were seen in t he input
impedance of the cavity.

When the antenna was "off resonantl y" matched,

the cavity modes appeared as maxima in the reflected voltage detected by
the input directional coupler, and as minima in the antenna current.
dividing the input power by the antenna current squared, the loading

-121-

resistance of the plasma was calculated.

The loading resistance at the

various resonances was observed to be as high as three to four times the
basic antenna resistance.
The phase difference betvJeen the incident and the reflected input
voltages has been investigated.

The phase information, along ~tJith the

amplitude of the incident and reflected waves, gives the complex reflection coefficient.

The complex input impedance was derived from the complex

reflection coefficient.

The real part of the complex impedance determined

this way agrees well with the results from the loading resistance from
pm'ler-current measurements.

The complex impedance follm11ed the predicted

characteristics of a circuit model often used in microv1ave cavity theory.
The model, along with a set of reasonable assumptions, gave the general
features of the measured impedance function.

The measurement of the com-

plex plasma loading impedance is crucial to the understanding of hovJ to
match the generator impedance to one of the cavity modes.
In order to deliver the maximum amount of power at resonance, it is
necessary to match the antenna impedance plus the plasma loading impedance to the generator impedance at one of the resonance peaks.

This way

the maximum amount of power can be fed into the tokamak when the plasma
loading is high.

Although this was a difficult experimental task, due to

the fast changing nature of the impedance near resonance because of rapid
decay of the plasma density, we were able to match impedances at a few of
the resonances.

In future experiments the "on resonant" matching of the

generator could be aided by improvements recently acquired.

First, an

on-line computer has been acquired so that once the complex plasma . loading
impedance of the cavity modes is measured under the "off resonant" matched

-122-

condition, the required values of the circuit elements in the matching
net~-Jork

can be readily computed.

By resetting the matching circuit ele-

ments to the new values, the generator impedance can be matched to the
antenna impedance at one of the cavity modes.

Second, the density of

the plasma can be held more constant by gas puffing, so that resonances
would be swept through much more slowly.
From the approximate cut-off relation of the cavity modes in cold
plasma theory, the loaded QL of the cavity could he estimated.

The un-

1oaded Q of the cavity caul d be derived from the QL by a circuit trans0

The measured unloaded Q0 of the various cavity modes ranges
from 400 to 700. Finally, the antenna coupling coefficient K 2 \-Jas ob-

formation.

tained from the plasma loading impedance and the estimated cavity Q. The
range coupling coefficients for the various cavity modes are between
3 x 10- 5 and l x 10- 4 for the 2-turn antenna.
The general conclusions for these experiments are that the possibility for efficient pmver coupling into the plasma-filled cavity looks very
encouraging because of the reasonable plasma loading resistance found at
the cavity resonances.

The matching of the generator impedance to the

antenna impedance at resonance does not seem to be a serious problem. The
loading resistance at the cavity modes has been observed to be as high as
three to four times the antenna resistance, and the generator impedance
has been matched to a fevJ of the cavity modes, if only briefly
the changing density.

due to

This means that with the present antenna design,

as mu ch as 80 % of the power can be delivered into the tokamak via the
cavity resonances, and only 20 ~1o of the input pmver vJill be lost in the
antenna.

-123-

6.2

Future High Power Heating Experiments
From the results of the low power experiments, one can see t hat a

few improvements of the experimental setup must be made before an effici ent high pm'ler heating experiment can be performed.
First, in order to improve the efficiency of the wave generation in
the tokamak, the antenna design can be improved in the followin g ways.
Looking back at the equation of the antenna efficiency, n

the
Since the cavity Qp is not a controllable quantity, to increase n
product K2Qa must be maximized. The coupling coefficient, K , can be increased by increasing the length of the antenna.

The ultimate size is

limited by the size of the tokamak chamber and the locations of the ports.
The antenna Q, Qa = wlant/Rant

can be -increased in two 'days, i.e., de -

crease the antenna resistance or increase the antenna inductance.

decrease th e antenna resistance, a bigger conductor for the antenna should
be used.

The limit on th e size of the conductor is the size of the ports

on the tokamak.

The inductance of the antenna can be increased by in-

creasing the number of turns on the loop antenna.

Since the i nductance

increases approximately as the number of turns squared and the antenna
resistance increases linearly as the number of turns, the anten na Q
should increase linearly with the number of turns.
For the heating experiment it is essential to be able t o couple
to one of the cavity modes for a substantial amount of time.

The present

plasma condition in the Caltech tokamak makes the heatin g experiment difficult becaus e of the fast density decay causing th e cavity modes to be

-124swept through very quickly.

To improve the situation, the plasma density

must be kept as constant as possible.

This can be done by gas puffing,

where during the discharge a small amount of neutral gas is introduced
into the tokamak.

The gas is ionized, thus increasing the plas ma density.

By programming hov1 the gas is puffed into the system, the plasma density
can be tailored to specification.
Moreover, from the approximate cut-off relation of the ma gnetosonic wave, f£mn ~ 1/~, one can see that the input frequency can be
swept to compensate for the change in the density.

There are several

ways to track the modes by frequency modulation.

One 1t1ay is to use a

phase locked loop, i.e., by using the phase information from the transmission measurement as the control signal for a voltage controll ed
oscillator.

As the density moves away from the required value for a

cavity resonance, the phase of the transmitted signal would shift. This
shift in phase can be used to change the input frequency so as to return
to the resonance condition.

Another method is to use positive feedback,

i.e., to use the transmitted signal picked up by a receiving probe as
the input for a broad-band amplifier driving the antenna, thus making
the cavity resonance the frequency determining element of the oscillation system.

If the gain of the amplifier is higher than the loss

through the cavity, positive oscillation is excited.

This oscillation

\vill adjust its own frequency in order to stay on the cavity resonance.
At first sight the high Q nature of the input antenna might appear
as a limitation for mode tracking because of the narrow bandwidth of the
antenna with its tuning nehvork.

However, at the cavity resonances the

plasma loading resistance increases substantially, and so the loadin g

-125-

resistance on the antenna will decrease the input Q.

In the low power

experiments, plasma resistive loading was observed as high as four ti mes
the antenna resistance.

Thus for an antenna with Q = 100 , th e loaded Q

at the cavity resonance is only 20.

To compensate a change in the plasma

density of 10%, a 5% change in the input frequency is required, or an
antenna Q (Q ~ 2n /6n ) of 20 is needed at the cavity resonance.

-126-

Appendix a
TRANSMITTING ANTENNA AND HATCHING NETWORK CONSTRUCTION
The transmitting antenna is a two turn loop made of l/8 inch
copper tubing.

The copper tubing is enclosed in a 1/32 inch thick

layer of pyrex insulator. The functions of the glass are to protect
the copper from plasma damage and to
plasma electrically.

The approximate loop area of the antenna

is 3.5 inches by 1 inch.

Because of the glass coating, the antenna

never intrudes more than 1.25
(Figure

insulate the antenna from the

inches into the tokamak vacuum chamber.

a.l)

The glass coating surrounding the copper is joined to a l/2-inch
OD Cajon (G304-8-GM-3) stainless steel-to-glass transition tube.

The

transition tube provides the mechanical feed-through for the
antenna to go from vacuum to the outside.

The glass portion is needed

to give the antenna electrical insulation from the stainless steel
wall of the tokamak.

The transition tube goes through a vacuum o-ring

and attaches to a 5/8" OD copper tubing.

Finally, an Amphonel twin axial

connector is screwed on the copper tubing to make the connection to the
matching network.
A copper inner lining of 0.25 inch thick is pressed inside the 3
inches of Cajon stainless tubing to minimize the eddy current losses
due to image currents produced inside the feed-through.

The length

of the entire antenna is kept to a minimum so that the antenna re~is
tance can be reduced below the plasma loading resistance.

The entire an-

.tenna measures 12 inches. The extra length is duetoa mechanical carriage

GLASS
COATED
COPPER

1{£s.S. TUBING

wnH

COPPER
INNER UNING

1/SCOPPER

H\GH POWER
TW\N AXIAL
coNNECTOR

-'
-..I

S.S. TO
GLASS.
TRANS\T\ON

31N.-I-31N.-

5 IN.-----t

Figure a.l
oesign of the two-turn copper antenna.

-128-

made for the antenna to move it smoothly in and out of the plasma.
The ultimate limitation on the antenna size and the feed-through
length is determined by the tokamak port size which is 4 x 6 x linch.
The antenna resistance and inductance are a function of the distance that the antenna protrudes into the vacuum chamber of the tokamak.
The reason for this dependence on the distance into the vacuum chamber
is because of the 6 x 4 x l inch stainless steel port where the antenna
sits when it is completely outside the tokamak vacuum chamber.

The

effect of the stainless steel port is to lower the antenna inductance
and increase the antenna resistance through eddy current losses in the
port wall.

Therefore, as the antenna moves out of the port and into

the vacuum chamber, the antenna inductance should show an increase
with distance, and the antenna resistance should show a decrease of
the distance.

Data for the antenna impedance as function of the dis-

tance into the tokamak chamber are shown in Figure

a.2.

In Figure

a.2, r = 0 corresponds to the case where the antenna sits just outside the tokamak chamber and completely inside the port; thus, r is
the distance that the front surface of the antenna is inside the
tokamak.
The antenna inductance measured in the experiment is appro ximately independent of the input frequency, and the antenna resistance
increases with an increase in the input frequency.

The frequency

dependence of the antenna resistance is shown in Figure

a.3.

The

data weretaken with the antenna at 1.5 inches into the tokamak ch.amber

-129(a)

.5
.4

.3
Lont(fLH)
.2
EXPERIMENTAL
RANGE

.25

.5
.75
1.0
r (inches)

1.25

1.5

(b)

18 MHz

16 MHz
14 MHz

.4
Rant

12 MHz

cnl
.3

10 MHz
8 MHz
EXPERIMENTAL
RANGE

•I

.25

.5
.75
1.0
r (inches)

1.25

1.5

Figure a.2
The antenna inductance and resistance as a fun ction of
radial position into the vacuum chamber. r == 0 is the
position that the antenna is just outside the chamber.

-130-

112

DEPENDENCE

----I
-- I

___ !-- r---r-----

--------I

.2 I

I -DATA
QL-------L-------~------~------~------~

Frequency f (MHz)
Figure a.3
Antenna resistance as a function of frequency. Data taken with th e
1:
antenna at 1.5" into the tokamak. A Rant a: f 2 curve is sup eri mpos ed
on the data.

-131-

thus minimizing the effects of the stainless steel port. Superimposed
112
on the experimental data is an Rant a:f
dependence fit, which is the
expected frequency dependence from skin effect calculations.
The matching network is a two capacitor arrangement shown in Figure
2.4.

Because of the lo~tJ resistance and accurate tuning capability

required, fifteen-turn Jennings vacuum variable capacitors are used.
For tuning the antenna in the frequency range between 6 and 20 MHz,
the capacitor in parallel with the antenna, c1 , ranges between 30 to
2000 picofarads, and the capacitor in series with the generator
impedance ranges between 15 to 300 picofarads.

The values of c1 and

c2 for various tuning conditions are shown in Tables 5.1. 5.2, and 5.3.

-132-

Appendix b
COLD PLASMA THEORY OF THE MAGNETOSONIC CAVITY MODES
(based on unpublished memorandum by R. ~J. Gould, 1960)
Consider a uniform cold collisionless plasma, axially magnetized
in a cylindrical geometry (Figure b.l).

The axial magnetic field

makes the plasma anistropic; thus, the dielectric property of the
plasma must be expressed as a tensor quantity.
Define a general displacement, D, with ejwt time dependence for
the plasma [16].
jwQ

j wE+
L.J =jws ·E
-n
= -

(b.l)

where ~ =

rnq nz n-n
v is the current density of the nth species of particles,

zn is the ionic
. charge, q n is the sign of the charge, and s is the
dielectric tensor.

Subtituting into equation b. l the momentum equation

where~ has ejwt time dependence,
(b.2)

into the current desnity -n
J , the dielectric tensor becomes the following

.1.

j sx

= - j E: :X

E.i

E:
E:

where

sII

(b.3)

/CONDUCTING WALL

+f<

27TR

--'

....J

------~Yl

Figure b. 1
Plasma filled cylindrical cavity with conducti ng 1vall. Periodic boundary
conditi on is imposed in the z direction to simulate the closing of the
torus. R = the tokamak major radius, and a = the tokamak minor radius.

-134-

€0

E:..L = 2

(R + L)

R= 1 -

LT (w + ~ w )

n w

, L

n en

In the tensor, two frequencies, wpn and wen' have been defined as
follows

wen =
w =
Pn
where wen and wpn are the cyclotron and plasma frequencies respectively.
With the dielectric tensor, Maxwell's equations are cast in the
following form

VxH=-(cE)

(b.4)

a {,,~""o-H)
V x E = at

(b.5)

When ej{wt - me - kz) dependence of the fields is assumed in the
cylindrical geometry, the following set of equations are obtained:

-135-

(b.6)

(b. 7)

(b.8)

(b.9)

(b.lO)
(b.ll)

After some manipulation, the above equations can be reduced to two
second order differential equations involving only the - longitudinal
components of the fields, Hz and Ez.

Following the notations used

by R. W. Gould [17].
(b.l2)

(b.l3)
where Ez = ¢1 + ¢ 2 , and Hz = a 1¢1 + a 2¢2 .

Here a 1 and a 2 are constants.

Also d = - w~ 0 y 1 /g, c = ky 2/g, g = y 12 - y 22

and y 1= k2 - w2~ 0E~,

Y2 = w 2 ~ 0 Ex. If a radial component of the wave vector, T, is defined as
r 2 = w~ /(d-jac), then equations (b.l2) and (b.l3) are just the s·essel's

equation, and the solution of¢ , where n can be 1 or 2, is as follows

-136-

where ¢0 is a constant.

The dispersion relation can be expressed as

follows

(b.l4)
As indicated in this equation, for every value of T2 there exist four
possible solutions of k.
Since only frequencies near the ion cyclotron frequency are of
interest the dispersion relation can be simplified by the following
approximations. For w rv O(uJ Cl. ) , B0 = 4 kG(fc 1. = 6 MHz), density =
5 x 10

cm- 3 , Z = 1 for hydrogen, then
wpe rv 1.5 x 1011 rad/sec
wpi rv 3 X 1o

rad/sec

w.rv 4 X 10 7

rad/sec

wee rv 7 X 10
Cl

where wpe and

wpi are the electron and ion plasma frequencies, wee and

wei are the electron and ion cyclotron frequencies.
wpe >> wCl.. Let n.1 = w/w Cl..
can be simplified as follows

Therefore, wee>> wei'

The components of the dielectric tensor

-137-

w2 .
pl

= £

w w w.
pe ce c1

w .[ 2
Cl
w w2 (1-~)

wee

..ii.pl ,}.Cl .

2 2
w (w .-w )
Cl

(wee - w

(b.l5)

(b.l6)

For Ell very large compared to the other terms in the dispersi on relation
(equation b.l4), one can make the followin g approximation
-+

By substituting this approximation into equation b.l4, the si mplifi ed

-138-

dispersion relation is as follows:
(b. 17)

Substituting the values of y

and y 2 the solution of k in terms of T

is
k2 = w2E.Lf.lo - ~ +./ (T2 /2)2 + (w2f.loE)2

(b.l 8)

-I

(b. 19)

I_
(T2 I 2 ) 2 + ( w2ll E ) 2
w EJYo
2 .
0 X

A consequence of Ell being 1arge compared to the various other
quantities in the differential equation, is that E is small (see equaz
tion (b. 11)). In our approximation we will take E2 = 0.
As mentioned before, the magnetosonic wave is right circularly
polarized; thus, it has no resonance at the ion cyclotron frequency.
Since there are two branches of the dispersion relation (b.l7), one can
check the polarization of the waves propagating along the longitudinal
d.c. magnetic field by letting T + 0.

It is easier to find the polari-

zation of the wave in rectangular coordinates.

As T + 0, the cylindri-

cal solution should reduce to the same solution.
The polarization of an electromagnetic wave can be expresse d
as follows:
"E
J X Ey -

±l

where +l corresponds to a right circularly polarized vvave, and -1 corresponds to a left circularly polarized wave.

The polarization of an

oblique wave propagating in a cold magnetized plasma is

-139"E
J X Ey -

n -s

(b.20)

where the d.c. magnetic field is in the z direction,
index of refraction, S = E~/E 0 , 0 = -Ex/E 0 .

n = kc/w is the

For propaQation along the

d.c. magnetic field, there are two solutions:
n2 = R

(b.21)

(b.22)

When these solutions are substituted into equation (b.20) and the definition of the dielectric tensor (b.3) is used, the following polarizations are found for the two branches.
.E
J X - R-S
Ey- -0-

·E

X ~-

L-S = -1
-0-

n2 = R is a right circularly polarized wave and n2 = Lis a left circularly polarized wave.
For frequencies near the ion cyclotron frequency, the fo ll ovJi ng
simplification to the dispersion relation can be made.

For w ru w . the
C1

frequency is small compared to the electron cyclotron frequency and to
the electron plasma frequency.

Therefore, R can be approximat ed as

follows:
R ru

where VA= Bo;/~ 0 n.m.
1 1
netic field.

is the Alfven velocity, and B0 is the d.c. mag-

Thus for the right circularly polarized wave

"' v2 ( 1 + r~. )

(b .2 3 )

-140and so there is no resonance at the ion cyclotron frequency, i.e.,
rti = l.

For the 1eft circularly polarized ~<~ave
c2

L "' 2(1- rt .)
VA

and

(b.24)

k2 ~~ _1__

- v2 ( 1 - rt. )

so for this 'vJave there is a resonance at the ion cyclotron frequency.
As T ~ 0 in equations (b.l8) and (b.l9), we have
k2 = W2lJ [ E_l + E ] = ~L
k2 = W2ll [ E - E ] = ~R
0 ...!..

(b. 18a)
(b. 19a)

And so equation (b. 19) is the magnetosonic branch that is of interes t.
Since the tokamak has a conducting wall, consider the solution of
the magnetosonic wave in a cylindrical cavity.

To simulate the closing

of the tokamak on itself, periodic boundary condition in the axial
direction is imposed (i.e., k = N/R where R =major radius of the
tokamak, see Fig. b. 1).

At r =a, a perfectly conducting wa l l is as-

sumed; thus the tangential electric field Et and the normal magnetic
field Hn must vanish.

Since the approximation of E 11 ~ oo imp li es that

E = 0 all the remaining fields can be written in terms of H .

.k
T2

aH
ar

=-~ [-z + -

!!! H ]

1 r

(b.25)
(b.26)

-141-

(b.27)
(b.28)
Note that

E8 = - k J.l o Hr

( b • 30 )

The solution of (b. 12) is the integer Bessel's function.

H = H J (Tr)ej(wt-me-kz)

o m

Thus, H is

(b.3l)

The boundary condition is E8 = Hr = 0 at r = a, or from (b. 25) we have

Ta J' (Ta) + m

m J (Ta) = 0

(b.32)

[For more details on the fast magnetosonic cavity modes see references

31 to 35].

-142-

Appendix c
RESISTIVITY LOADING OF THE R.F. WAVE BY TOKAMAK WALL
The r.f. energy generated in the tokamak by the transmitting
antenna can be assumed to be either dissipated in the plasma, or lost
in the tokamak wall which has a finite conductivity.

It is important

to estimate the resistive loading of the wave due to the finite conductivity of the tokamak \vall, and to compare the calculated value with the
measured resistivity loading in the low power experiment.

If the esti-

mated loss in the tokamak \vall can account for most of the resistive
loading effects measured in the low power experiments, then the validity
of the high power experiment becomes questionable, because the r.f. wave
will tend to heat the tokamak wall more than the plasma.
The most convenient method to study the effect of the tokamak wall
loading is to compare the estimated Q of the cavity, due to ~1all losses
alone, with the measured Q of the cavity.

If the estimated Q due to the

wall, denoted by Q , is comparable to the measured Q, then the wall loadw

ing is the dominating dissipation factor in the tokamak.
One approach to estimate Qw is to calculate the damping decrement,
y, of the cavity modes due to the finite resistivity of the cavity wall.
The damping decrement is defined as the attenuation per unit time of the
electromagnetic ~"ave in the cavity. If the time dependence of the ith
jw. t
ei genmode is assumed to be e 1 , vJh ere wi = w0i + jw2 , then the damping
decrement yi is just w .
decrement as

The cavity Qw can be related to the damping

wo./2y.

( c.l)

-143v1 h ere w i 1. s t he 1. th e1. genmo de f requency of th e cav1. t y.

The damping
decrement due to the cavity wall can be calculated for the various modes

by using a finite conducting wall boundary condition at the cavity wall.
The new boundary condition is

f.= n x !:i_(l+j) J2a

where a is the conductivity of the wall and n is the outer normal to the
wall [36].

If the tokamak is again approximated by a cylinder with

periodic boundary in the axial direction, then the boundary can be written in terms of E8 and Hz'
E8/H = ( 1 + j) jw1,1 /2a

(c. 2)

The real part of the term on the right is the wall resistance, and the
imaginary part is the additional reactance from the 111all.

From Appendix

b, the solution of Hz for a plasma-filled cylindrical cavity is as follows:
Hz= H J (Tr) ej(wt-m8-kz)
o m

(c.3)

E is related to Hz in the follm•Jing manner (b.28)
Y2 m
E = (j w1,1 o/T )[ aHz (Tr)/ar + --y
r Hz (Tr)]

(c.4)

where Tis the radial wave number, k is the axial wave number, and m is
the azimuthal mode number.
r = a.

For the perfectly conducting wall E = 0 at
Now the boundary condition at r = a is
. (c. 5)

-144-

where Ta is replaced by ~. and H'z = aH z ;a~.

Rearranging the above

equation, obtain a dimensionless equation,
HI

y1 ~

l [ ~ + _1_ !!!] = (1 - j)

r=r=_

(c.6)

·J2awf.l 0

lj2 1

For a highly conducting wall, the term on the right side, a

awf.l

<< 1.

Thus, the wall resistance contributes a small imaginary term to ~ and
the wall reactance adds a small real term to ~~

If ~ is written as

= ~ 0 + jo, where o << ~ 0 • then the wall resistance adds a damping term

to the radial wave number T, and the wall reactance will shift the resonance frequency w.1 by a small amount.

To find the damping decrement,

solve the complex T = ~/a and substitute into the dispersion relation
(b.ll):
2 2
s-2. w .

2 2
s-2. w .
(-)
+ 1 C1
V~(l - s-2~)

For the purpose of this section where only an estimated damping decrement
is needed, the approximate dispersion relation can be used.

For the

lm•Jest few axial mode numbers N, \of the torus, the dispersion relation can be approximated in the following
way:
W·

w + jy. = VA

JT 2 + k2

where VA is the Alfven velocity.

(c. 7)

The simplest mode to estimate is the

k = 0, m = 0, and the lowest T mode.

From equation (b.32), one can see

that the lowest radial mode form= 0 corresponds to the first zero of
the first integer order Bessel's function, J 1 .
reduced to the following:

Equation (c.7) can· be

-145-

( c .8)

where T0 = ~ 0 /a.

The following parameters for the Caltech tokamak are

subs t ituted into equations (c.6) and (c.8):
conductivity of stainless steel = l X 10 6 mho m-l

a = minor radius = • 15 m

major radius = .45 m

f = input frequency = 12 MHz

0 = first zero of J 1 = 3.83

The following value of Qw is obtained
Qw = 1300
which is two to three times the various measured cavity Q in the experiment.
For the higher radial and axial modes, both T0 and k will be larger,
which means that Qw should be higher.

Therefore, this estimated Q for

m = 0, k = 0 , and lowest radial mode is a lower limit for the higher
modes.

Although the wall loading is not negligible, it does not

account for all the measured loading in the tokamak; thus, r.f. energy
should be dissipated in the plasma.

-146REFERENCES

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