impedance i-iatching for alfven wave couplers by a …

IMPEDANCE I-IATCHING FOR ALFVEN WAVE COUPLERS

by

T. M. Rajkumar, B.S.E.E.

A THESIS

IN

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE IN

ELECTRICAL ENGINEERING

Approved

Accepced

I JDean (Sfl the Id raduate School

May 1983

ACKNOWLEDGEMENTS

I would like to thank Dr. M. Hagler and Dr. M. Kristiansen for

their advice and guidance in this work. I would also like to thank

Dr. G. Fredricks for serving on my committee.

The excellent help I received, at every stage of this work,

from my fellow graduate student Mr. Dale Coleman is appreciated.

The financial support offered for this work, by the National Science

Foundation and the Department of Electrical Engineering is also grate

fully acknowledged.

11

CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT iv

LIST OF FIGURES v

LIST OF SYMBOLS vii

CHAPTER

I. INTRODUCTION 1

II. THEORY OF BROADBAND MATCHING 4

A. Broadband Matching Problem 4

B. Youla's Theory 6

C. Real Frequency Technique 8

III. MEASUREI4ENTS OF BIPEDANCE 13

A. Experimental Arrangement 13

B. Procedure 21

C. Results 23

IV. BROADBAND MATCHING NETWORKS 33

A. Lossless Matching Networks 33

B. Lossy Matching Network 34

V. CONCLUSIONS 38

LIST OF REFERENCES 41

111

ABSTRACT

The impedance of an Alfven wave launching antenna has been

measured. An increase in the resistance of the antenna at an eigenmode

has been observed. Various broadband matching networks were tried to

match this antenna to a broadband rf source, over the frequency range

5-15 MHz. It was found that an efficient broadband match is not

feasible in the frequency range of interest.

IV

LIST OF FIGURES

Figure

2.1 Broadband Matching Problem 5

2.2 Detail of Representation of R (oa) 10 q

3.1 Diagram of RF Experimental Arrangement 14

3.2 Block Diagram of Texas Tech Toroidal Plasma Facility . . . . 15

3.3 Detail of Antenna 16

3.4 Detail of RF Impedance Matching Circuitry 17

3.5 Four Channel, 0-360 Degree Phase Detector 19

3.6 Data Acquisition System of Tokamak 20

3.7 Time Variation of Incident and Reflected Power -

Method 1 22

3.8 Variation of I^ with Time - Method 1 22

3.9 Time Variation of Magnitude of B-Dot Probe Signal -

Method 1 22 3-10 Time Variation of Incident and Reflected Power -

Method 2 24

3.11 Variation of I^ with Time - Method 2 24


Method 2 24 3.13 Time Variation of Power, P=VI Cos (6) - Method 1 25 3.14 Time Variation of Power, P=P3-j c ~ ' REF ~ ' ^ ^ ' ^^

3.15 Time Variation of Magnitude of B-Dot Probe Signal -Method 1 25

3.16 Time Variation of Power, P=VI Cos (6) - Method 2 26

3.17 Time Variation of Power, P=PT^C ~ REF ~ ^^^^°^ ^ •^^


Method 2 26

3.19 Time Variation of Antenna Current 27

3.20 Time Variation of Antenna Voltage 27

3.21 Time Variation of Magnitude of B-Dot Probe Signal 27

3.22 Time Variation of Magnitude of Antenna Impedance 28

3.23 Time Variation of Phase of Antenna Impedance 28


3.25 Time Variation of Antenna Resistance, R = (P^„^ - P„^^)/I^ . 30 INC REF

3.26 Time Variation of Antenna Resistance, R = V/I Cos (9) . . . 30


3.28 Time Variation of Antenna Reactance, X = V/I Sin (9) . . . . 31

3.29 Time Variation of Antenna Reactance, X = (Z^ - R^)^/^ . . . 31


3.31 Variation of Antenna Resistance with Frequency 32

3.32 Variation of Antenna Inductance with Frequency 32

4.1 Power Dissipation Capability of "Cantenna'' 36

4.2 RF Characteristics of "Cantenna" 36

4.3 Variation of Antenna Current with Frequency 37 4.4 Variation of Antenna Current with Frequency

(Antenna Inductance Tuned Out at a Frequency of 10 MHz.) 37

5.1 Proposed Narrow Band Matching Network for 30 kW Amplifier 40

VI

LIST OF SYMBOLS

A(s) All Pass Function

C (u)) n Order Chebyshev Polynomial

f Frequency in MHz.

G(oj ) Transducer Power Gain •

I Current

K D.C. Gain n

L Inductance of Antenna

P .-_, Incident Power INC

P Reflected Power

Q Quality Factor

R Resistance of Antenna I

R Generator Resistance S

R (oo) Resistance of Equalizer as Seen from the Load

RF Radio Frequency

s Complex Frequency

Voltage

Reactance of Antenna

Reactance of Equalizer as Seen from the Load

Load Admittance

Load Impedance

p Reflection Coefficient

0) Frequency in Rad/Sec.

V

X

\M

Y(s)

Z^(s)

VI1

w^ Cutoff Frequency

Ripple Factor

Risetime

Vlll

CHAPTER I

INTRODUCTION

In order to reach ignition temperatures in a Tokamak, it is

necessary to provide supplementary heating in addition to the ohmic

heating [1]. Two methods of supplementary heating which are being

widely investigated are neutral beam injection heating and radio fre

quency wave heating. One method of RF heating makes use of high Q,

fast wave resonances, to heat the ions. Once the fast wave propagates

through the plasma, consisting of a single species of ions, it is only

slightly damped, and can interfere constructively with itself and set

up standing waves around the major circumference of the torus. The

formation of the wave resonances is determined by the parameters of

the plasma, the RF excitation frequency, and the physical dimensions

of the toroid [2].

For the RF heating in the plasma to be efficient, it is necessary

to transport as much energy as possible from the source to the coupling

structure. In fast wave RF heating in the Texas Tech Tokamak, the

loading is small (of the order of one ohm). Hence, a low loss matching

network is necessary to transform this load to the source resistance

(usually on the order of fifty ohms).

The RF load is not purely resistive, but has a certain reactance

associated with it. This reactance depends on the frequency and, hence,

the elements of the matching network also depend on frequency. Thus

it becomes necessary to tune the matching network at each frequency of

operation.

The RF heating primarily takes place at the eigenmodes. Hence,

it is desirable for the eigenmodes to last as long as possible. The

frequencies at which the eigenmodes occur depend upon the density.

In a Tokamak the density varies continuously with time; hence, the

frequency of the RF source needs to vary correspondingly to make the

eigenmodes last longer (mode tracking). In order to avoid retuning, as

the frequency is changed, a broadband matching network is desirable.

A broadband coupler also permits the resonant layers to be moved

to convenient locations with different filler gases by changing the

frequency without retuning, and without changing the plasma parameters,

which do change if the resonance is moved by changing the magnetic

field [2]. It thus becomes desirable to be able to match the rf load

to the source in a broadband manner.

The objective of this research was to measure the variation of

impedance of the antenna at an eigenmode. An effort was also made

to match the antenna to a broadband (3-64 MHz, 30 kW for 1 ms) rf

source in a broadband manner.

Chapter 2 discusses the theory of broadband matching networks.

The constraints for matching an arbitrary impedance to the source

are stated. A numerical method for broadband matching, namely the

real frequency technique, is also discussed.

Chapter 3 discusses the experimental setup used. It also dis

cusses the results of the impedance measurements of the antenna.

Chapter 4 shows why broadband matching is not attractive for the

antenna. It discusses a lossy, broadband matching network.

3

Chapter 5 summarizes the results of the research. It also shows

a proposed narrow band matching method.

CHAPTER II

THEORY OF BROADBAtTO MATCHING

A. Broadband Matching Problem

The broadband matching problem can be stated with reference

to Fig. 2.1. The aim is to use an optimum, lossless, two port, network

to match the load impedance Z (s) to a source (represented by a

voltage source and an equivalent Thevenin resistance R ) to achieve g

a preassigned transducer power gain characteristic, G(ui^), over the

frequency band of interest. The transducer power gain characteristic

is defined as the ratio of the average power delivered to the load

to the maximum available average power at the source. The transducer

power gain characteristic, G(a)^), is equal to Is .(joo)!^, where S(s) mj

is the normalized scattering matrix and S ,(s) are the off-diagonal

elements of S(s). Hence the transducer power gain characteristic is

a function of o)^, rather than oo.

Unless the load is a resistor, it is not always possible to

match the load to a resistive generator with a preassigned gain over

the frequency band of interest. This arises due to limitations on

the physical realizability of the equalizer. Hence any matching

problem must include the maximum tolerance on the match as well as

the minimum bandwidth within which the match is to be obtained [3].

From the physical realizability condition of the equalizer Fano

[4] has developed a set of constraints in integral form, with proper

weighting functions depending on the load impedance. However, since

+

i--CM

B

o u

00 C

AyVV

cr + <3>

a

(/) U) o

ossI

^

(U N

• M B ^

a cr LU

T3 C C8

«J O U

PQ

CM

4^-

+

00 •H PC4

6

they are in integral form it is cumbersome to use them.

B. Youla's Theory

Based on the principle of complex normalization, Youla [5]

developed a theory which gave the constraints for the physical

realizability of the equalizer as a set of algebraic constraints.

For a given load admittance y(s), g(s) = y(y(s) + y(-s)), is said

to be the even part of y(s) [6]. A closed, right hand side, zero of

multiplicity K of the function w(s) = g(s)/y(s), is said to be a

zero of transmission of order K of y(s). For a prescribed transducer

gain characteristic of G(u) ) the lossless equalizer load side reflec

tion coefficient p(s) is related to 0(0) ) = l-lp(ja))|^ and can be

written as

Y22(s) - y(-s)

'^'^ = ^^'^ Y^,(s) + y(s)

where A(s) is the regular all-pass function, defined by the poles of

y(-s) for Re(s)>0, and Y22(s) is the input admittance at the output

port when the input port is terminated in the generator resistance.

The zeros of transmission can be divided into four mutually

exclusive classes.

1. class 1 contains all those with Re(s) > 0

2. class 2 contains all those on the real frequency axis which are also zeros of y(s).

3. class 3 contains all those on the real frequency axis

for which 0 < |y(ja)Q)| < "

4. class 4 contains all those for which |y(jajQ)| = «,

where SQ = JOJQ

7

The restrictions are formulated in terms of the coefficients of

the Laurent series expansions of the following quantities, about a

zero of transmission SQ = a + ju) of order K of y(s).

00

m P(s) = Z p^ (s-s„) m=0

A(s) = E A^ (S-SQ)°^

m=0

F(s) = 2 g(s) A(s)

= \ ^m (-^0>" m=0

Then the basic constraints on p(s) are stated as follows: For

each zero of transmission s of order K of y(s), depending on the class

of the zero of transmission, one of the following four sets of con

ditions on the coefficients must be satisfied.

(1) Class 1: A = p for x = 0, 1, 2, , K-1.

(2) Class 2: A = p for x = 0, 1, 2, , K-1, and

(VPk>^\+i ' - "• (3) Class 3: A = p for x = 0, 1, 2, , K-2, and

(A^_^) - Pi^^i)/\ 0» where K > 2,

(4) Class 4: A = p for x = 0, 1, 2, , K-1, and ^ ' X X

F /(A^-Q ) > a_,, the residue of y(s) at the

pole JCOQ.

8

C. Real Frequency Technique

Youla's theory becomes very difficult to apply if the load con

sists of more than one or two reactances and resistors. Also it is

necessary that the analytic form of the transfer function (load plus

equalizer) be known to realize the equalizer.

In the real frequency technique [7,8] it is not necessary that

the analytic form of the transfer function be known. It makes use

of only the real frequency impedance data.

Z (ja)) = R (ja)) + jX^(a)),

determined, say, experimentally in the frequency band of interest

Let the Thevenin impedance of the equalizer, as seen from

the load, be

Z (jo)) = R (o)) + X^M q - q q

The transducer gain G(a) ) = 1-1 p

4 R, (to) R M 1 q

|z (ja)) + Z (ja))| 2

where

is the complex, normalized reflection coefficient at the load-

equalizer interface. The aim is to find the R (o)), X (03) that maximize q q

the minimum transducer gain G^ to optimize G(cj2) over the frequency

band, where G(aj ) is assumed to be approximately flat over the pass-

band.

The unknown real part R (00) is represented as a number of

straight line segments in the frequency band, i.e. semi-infinite

slopes with frequency break points at 0 < ooi < u)2 < . . • • < w^.

The quantity Z (w) is also assumed to be a minimum reactance function

whose real part R (o)) = 0, for 00 > oo .

The u) break points are chosen by examining the load data or

can be divided evenly over the frequency band of interest. The

equalizer resistance can then be specified as a linear combination

of the individual total resistive excursions of each of the straight

line segments. Figure 2.2 shows the details of this representation.

Thus,

n

k=l

= r^ + [aj^[r]

where r = R (0) and r, is the unknown resistance excursions of the 0 q k

k straight line segment between break points 03 and f^-^^-^- Since

R ((D) = 0, for 0) > ojj , it can be seen from Fig. 2.2 that

k=l

10

«n=<4> Frequency

Fig . 2.2 D e t a i l s of Representa t ion of ^ M

Hence, with r given, there are n-1 unknowns, i.e. the

11

r ^ , k = 1 , 2 , n - 1 ,

T h e r e f o r e

a, = 1 CO, < 03

k

0 3 - 0 3 k - 1

\ - \-l \-l < ^ < \

= 0 03 < 03 k - 1

Since the r 's are real and provided R (03) 0, the impedance Z (03)

is passively realizable.

The minimum reactance function X (03), as a result of the above q

representations for R (03), can also be represented as a linear

combination of the same unknown resistive excursions r, . Thus, k

n X^(03) = Z b (o3)r

q k=l ^ '

The b, (03) are independent of r and, for any frequency, 03 are given

as

, X 1 r k T ly + tol J (03) = r— j In -r^ dy. , /03, - 03, i W <' y - 03

The line segments describing R (03) are then approximated by a

''•* WfF

12

rational function R (a)) = R (a3) by the method of least squares. The

equalizer impedance Zq(ju)) can thus be realized as a Darlington

reactance, 2-port, with a resistive termination of the source impe

dance [7],

CHAPTER III

MEASUREMENTS OF IMPEDANCE

A. Experimental Arrangement

In order to be able to design a matching network, it is necessary

to know the impedance of the antenna as seen by the source. The

impedance of the antenna varies with time, depending on the plasma

parameters and changes when an eigenmode occurs. Hence, the

antenna impedance was measured to investigate the change in impedance

at an eigenmode.

The experimental setup used in the measurement is shown in

Fig. 3.1. The experiments were carried out on the Texas Tech Tokamak,

which is a small (R(major radius) = 46 cm, a (minor radius) = 16 cm),

research Tokamak with a circular cross section. A block diagram of

the machine is shown in Fig. 3.2. The machine was built primarily for

investigating fast Alfven wave propagation in the plasma. The machine

design, construction, and performance are documented elsewhere [2,9].

A broadband (1-200 Ifflz), linear amplifier capable of delivering

up to 500 watts is used as the rf power source. This source is

connected to the Alfven wave launching antenna, through a matching

network (4.5-17.5 MHz) consisting of two, low loss variable capacitors,

The details of the launching antenna used are shown in Fig. 3.3, and

the matching network in Fig. 3.4.

The antenna current is monitored by a 30 MHz bandwidth current

transformer. The antenna voltage is monitored using a conventional

13

14

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e 03 60

c CO u u < CO

c 0) e

•H

03

a X w 0:5

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— (

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o - — (J 5 cz :D

O

15

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18

voltage probe (Tektronix 6015) with a 1000 to 1 attenuation ratio and

a 75 MHz bandwidth. A directional coupler is used to monitor the

incident and reflected power. A magnetic probe [10] is used to

monitor the B fields of the Alfven waves for the occurence of the

eigenmodes.

The output of the magnetic probe is amplified and fed through

a video detector (x = 5 ys) to a digitizer. The outputs of the

directional couplers (the incident and reflected powers) are also

passed through a video detector (x = 5 ps) and fed to the digitizer.

The measured current and voltage are passed through power splitters

and divided into two equal parts. One part of the signals is fed

through a video detector (x = 5 ys) to the digitizer to get the

magnitude information. The other part of the signals is used for

retrieving the phase information, and is fed to the inputs of a

phase detector, with the current signal being fed to the local

oscillator port and the voltage signal to the rf port. The phase

detector (shown in Fig. 3.5) has a frequency range of 2-32 MHz and

is capable of giving a 0-360 degree phase resolution with an accuracy

of ±5 degrees. The outputs of the phase detector are fed to the

digitizer.

The digitizer used was a LeCroy 2264 (8 channel) digitizer, and

all data were digitized at the sampling rate of once each 2.5 ys. The

digitized signals were then stored on a floppy disk in a PDF 11/34

computer using a data acquisition system [11]. A schematic of the

data acquisition system is shown in Fig. 3.6.

19

• • • • • •

il a 2.

CO

i_U ^ O -

CVJ f O ^ o • • * e *

^

c *= O ® t c C 3 O

< o 2

o 4-1

a 03 4-1 03

O

03 cn CO

0) 03

00 03 Q

O

m I

o

03 C

c CO

O

5-1 3 O fa in

CO

00 • H

fa

20

CO o -J <

< =

cn H -3 0 .

z

J UI z z < r o (0

_J UJ z z < X CJ <M rt

- I UJ Z Z <

UJ

< U UJ

o < <

_J o (T h-z o u

UJ o I - 2 > Ui 03 S

^ cr

Ui

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< X o

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NE

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ON

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fa

21

B. Procedure

The measurements were initially done at a frequency of 10 MHz and

at a Deuterium gas pressure of 8 x IQ-S torr. The measurements were

done under two conditions.

1. The matching network was tuned so that the signal from the

magnetic probe is maximum during the eigenmode. Under such a

condition, although it seems to be matched to the eigenmode,

it is observed that the instantaneous reflected power from the

antenna increased at the eigenmode, while the instantaneous

incident power remained constant, leading to an actual decrease

in the power absorbed at the eigenmode. The average incident

power in this case was on the order of 500 Watts and the average

reflected power was on the order of 200 Watts. The average

power absorbed by the plasma was on the order of 300 Watts.

2. The matching network was tuned again so that a decrease in

instantaneous reflected power occured at the eigenmode, while the

instantaneous incident power remained constant. Thus the power

absorbed at the eigenmode increased. The average incident power

in this case was on the order of 500 Watts, while the average

reflected power was on the order of 400 Watts. Thus the average

power absorbed by the plasma was on the order of 100 Watts.

Hence the average power absorbed by the plasma under this condi

tion was less than the power absorbed under the matching condition

of method (1). This is due to the fact that a larger portion of

the power was reflected.

The incident and reflected powers are shown in Figures 3.7 and

22

500 1000

TIME IN MICRO SECONDS Fig. 3.7 Time Variation of Incident and Reflected Power - Method 1,

TIME IN MICRO SECONDS

Fig. 3.8 Variation of I^ With Time - Method 1.

B Z

D 0 T

200 z-

1 0 ^

0 Aj\il

500 1000

TIME IN MICRO SECONDS Fig. 3.9 Time Variation of Magnitude of B-Dot Probe Signal - Method 1

23

and 3.10 for method 1 and method 2, respectively. The variation of

I with time and the variation of the signal from the magnetic probe

(placed at 180 degress toroidally from the antenna) with time are

shown in Figures 3.8 and 3.9, respectively, for method 1 and in

Figures 3.11 and 3.12, respectively, for method 2.

The power absorbed by the plasma is also calculated for both

the methods as P = VI Cos(9) and P = P^^_ - P„^^. The variation of INC REF

power absorbed, as calculated by P = VI Cos(9) and P = P,„^ - P T- .,

INC REF

for method 1 is shown in Figures 3.13 and 3.14, respectively, and in

Figures 3.16 and 3.17, respectively, for method 2. It is found from

the figures that though the power absorbed for the two types of

calculation differ in detail, the general shapes match fairly well

for both the methods. Figures 3.15 and 3.18 show the variation of

the magnetic probe signal with time for method 1 and method 2,

respectively.

C. Results

Figures 3.19 and 3.20 show the variation of current and voltage

of the antenna with time. Figure 3.21 shows the variation of the

magnetic probe signal with time. Figures 3.22 and 3.23 show the

variation of magnitude and phase of the antenna impedance. Figure

3.24 shows the variation of the magnetic probe signal with time.

The resistance of the antenna was calculated by two methods:

(a) R = (Pjjjc - ^REF^/I'

(b) R = (V/I) cos (e)

Incident Power

Reflected Power

500 ' •

1000 i-.


Fig. 3.10 Time Variation of Incident and Reflected Power - Method 2.

I

300.

C0C

2 E

10£

\ A / V ^ ^ ^ V • -VwvArAT^ V

h}\

0 500


Fig. 3.11 Variation of I^ With Time - Method 2.

V ^

^ .000

24

B 2

D 0 1

001-

0 10Gfc-

0 500 1000


Fig. 3.12 Time Variation of Magnitude of B-Dot Probe Signal - Method 2,

500 1000

TIME IN MICRO SECONDS Fig. 3.13 Time Variation of Power, P=VI Cos (9) - Method 1,

P 30a—

0 I E ZQS^ R t

W 1 0 A T T S 0

^.ww-*^^^

rM f \

1 I I

u -AT \ ^ , / ^ ^ / W vV

' ' •00 1000

TIME IN MICRO SECONDS Fig. 3.14 Time Variation of Power, P=P3- (n - Pj gp " Method 1,

25

B Z0Ctl-Z

0 13EE-

0"" ^ ^ L

500 1000

TIME IN MICRO SECONDS Fig. 3.15 Time Variation of Magnitude of B-Dot Probe Signal - Method 1

26

'\^^,/ffMh¥^'t^^

500 •

1000


Fig. 3.16 Time Variation of Power, P=VI Cos (9) - Method 2

P Z0a-0 I w i5eF E R

100t"

T T S

\ ^A•' ^vv^v'\A.; v^vv //v^^^^ eF ' ' 0 500 1GO0


Fig. 3.17 Time Variation of Power, ^^^^^^ ' ^REF " ^^^^°^ ^'

B 2 L

0 1 T

001-

00 =-


Fig. 3.18 Time Variation of Magnitude of B-Dot Probe Signal - Method 2.

27

•V;VV--V^-"^""'M

500 1000

TIME IN MICRO SECONDS Fig. 3.19 Time Variation of Antenna Current.

0 40C1-

T 30ei-

G 20C

lOGfl V ^ ^

0

/*A^AA.V ^ ^ . ^ ^ • ^ ^ " ^ "

•>

500


Fig. 3.20 Time Variation of Antenna Voltage.

V

:22Q

B £03:-

0 10(t-— I—

' P P

0 ^wNJ

500 .C00


Fie. 3.21 Time Variation of Magnitude of B-Dot Probe Signal,

3Qr-M

0 G H N M I Z$r S T

U D E 20

'.S

r Hft^^Wy--'^''^^^ .,;y^_/»/

0 ' ' I L .

500 ' ' • '

1000


Fig. 3.22 Time Variation of Magnitude of Antenna Impedance.

SGU

0 E == G H R A 95 E S E E S

sq 0

' - " V w - ^ .

. _ ! 500

' ' ' •

1000 I I

TIME IN MICRO SECONDS Fig. 3.23 Time Variation of Phase of Antenna Impedance.

3 2G2£-7 C

D E 0 IOGE-T P

0 I \

" \

5-00 « /

1000

Tir€ IN MICRO SECONDS Fig. 3.24 Time Variation of Magnitude of B-Dot Probe Signal.

29

These are shown in Figures 3.25, 3.26, respectively. Figure 3.27 shows

the variation of the magnetic probe signal with time.

The reason the values of case (a) and case (b) seem to differ by

a factor of 2, is probably due to the errors in phase angle measurement.

The phase angle as measured by the detector is about 2 degrees lower

than the phase angle obtained if calculated from 9 = cos"^ (P-..-_-P„„„)/V I, INL Khb

(The accuracy of the phase detector is to within 5 degrees of the actual

phase angle). Hence the resistance as calculated from the power

measurements seems more accurate. Figure 3.25 shows a noticeable

increase in antenna resistance during the occurence of an eigenmode.

However the increase shown, of approximately 300 m^, was the largest

change observed in any of the cases investigated.

The reactance was calculated from X = (V/I) sin (9) and

X = (Z^ - R^)°*^. Figures 3.28, 3.29 show the variation of reactance

with time, as calculated by the two methods. Figure 3.30 shows the

variation of the magnetic probe signal with time.

The measurements were repeated at the following frequencies:

6.5, 8, 12, 14, 15.25 MHz. Figure 3.31 shows the variation with fre

quency of the resistance calculated by method (a). The inductance

is calculated as L = (Z^ - R2)0-5/27rf and Fig. 3.32 shows the varia

tion of the inductance with frequency. The inductance seems fairly

constant with frequency. The resistance seems to increase with

frequency. This may be due to the fact that the radiation resistance

of the antenna is proportional to the frequency [12].

0 H M S

R E S I S T A N C E

30

V

wA/\Wi y^HirJt^

500 '

1000


Fig. 3.25 Time Variation of Antenna Resistance. R (P - P )/l2 ^ INC REF^^ •

E U S 1.5H_

0 I ^ H S • M T S P

N C r I I

500 •

1000

TIME IN MICRO SECONDS Fig. 3.26 Time Variation of Antenna Resistance, R = V/I Cos (9)

B Z 200—

D 0 1 2 ^ T E

0 ." J| iV^/^-v^

500 1000


Fig. 3.27 Time Variation of Magnitude of B-Dot Probe Signal.

30-R E

0 P H C M T 25j— S A

N C E 2q.

0

31

^^^^^vyh.n^^v^

.J L I

500 1000


Fig. 3.28 Time Variation of Antenna Reactance, X = V/I Sin (9).

3a-R L E

0 ul H : M T 23-S H

N C E 21

0

,,.! .^^f^yMj^^y^^f'^r^ ^ ^ A

I I I i 500 1000


Fig. 3.29 Time Variation of Antenna Reactance, X = (Z^ - R^)^/^

=h • . •! • ' U

500 1000


Fig. 3.30 Time Variation of Magnitude of B-Dot Probe Signal.

32

R E S I S T PI

N .51 C E

0 H M S 0|

5 10 15

FREQUENCY MHz.

Fig. 3.31 Variation of Antenna Resistance with Frequency,

I . N D U . C T A ., N C E .1|

HH J

•e-

5 10

FREQLENCY M H z . Fig. 3.32 Variation of Antenna Inductance with Frequency,

< c

CHAPTER IV

BROADBAND MATCHING NETWORKS

A. Lossless Matching Networks

Two types of lossless matching networks which are commonly used

are the Butterworth and Chebyshev matching networks. The Chebyshev

matching network has a transducer gain of the form.

k G(u)2) = 0 < k„ < 1

(l + £2c^2(^/^^)j n

where G(a) ) is the n order Chebyshev transducer power gain charac-

teristic, C M is the n order Chebyshev polynomial of the first

kind, K is the maximum passband gain, oj is the cut-off frequency, n ^

and real e(ripple factor)<1. The ripple factor specifies the minimum

passband gain as YT72 ' ' ^ specified as decibels below K^. For

a series R-L load, Chen [13] has shown that the maximum passband gain

K is given as

K = 1 - e^ sinh^fn sinh n

sinh a -2R sin Y^

LtOc

where

Y^ = 7T/2n, a = - smh -

33

34

In the frequency range of 5-15 ^ z , the impedance of the antenna

is approximated as a resistance of 0.5 ohms with an inductance of

0.41 uH in series. It is desired to match this impedance to a 30 kW

rf amplifier, with an output impedance of 50 ohms, in a broadband

manner through a lossless netowrk. It was tried to match the impedance

with a fifth order, low pass, Chebyshev network with a 1 db ripple in

the passband and a cutoff frequency co = 10^ rad/sec. Hence,

n = 5

10 log (1 + e2) = 1 (jb giving e = 0.50885

R = 0.5 ohms

L = 0.41 pH.

The maximum passband gain K that can be obtained is 0.0736. This

implies that the Chebyshev type of matching network can transfer a

maximum of 7.36% of the power and the rest would be reflected.

The efficiency that can be obtained with a Butterworth type

of matching network is even lower. The efficiency that can be

obtained by the Real Frequency Technique is also on the order of

7%. Hence a lossless broadband type of matching network is not

attractive for this antenna.

B. Lossy Matching Network

Since a lossless matching network is not feasible, a lossy

broadband matching was tried. A 50 ohm resistor was connected in

series with the antenna (resistance of 0.5 ohms, inductance of

0.41 yH, in series), so that the 50 ohm load would dominate and a

broadband match would be achieved.

35

The resistor used was a Heathkit model HN-31, "Cantenna", dummy

RF load, capable of handling one kilowatt of power. Figure 4.1 shows

the power dissipation capability as a function of time. The oil-

cooled, temperature stable, resistive element provides a very low

voltage standing wave ratio up to 400 MHz. Figure 4.2 shows the

radio frequency characteristics of the resistor, measured using a

rf vector impedance bridge.

This load was connected to the 30 kW amplifier and the antenna

current was measured as a function of frequency. The variation of

antenna current with frequency is shown in Fig. 4.3. It is found

that the antenna current has a peak of 43 Amps at 6 MHz and drops to

about 10 Amps at 14 MHz. Thus the match is not found to be broad

band. This is due to the presence of the inductance. The reactance,

being proportional to frequency, changes the impedance of the antenna

at each frequency- This leads to an increase in the reflection

coefficient at higher frequencies, hence reducing the antenna

current at higher frequencies.

The inductance of the antenna was then tuned out at a frequency

of 10 MHz with a variable capacitor in parallel, and a 50 ohm resistor

was connected in series to this parallel circuit. The antenna current

was measured again, in the frequency range of 5-15 MHz. Figure 4.4

gives the antenna current as a function of frequency for this case.

The antenna current reached a peak of 50 Amps at 6 MHz and dropped

to about 5 Amps at about 14 MHz. Thus, this setup was also found

not to be broadband. Hence, a lossy type of broadband matching net

work is also not attractive for the antenna.

36

a o .J o ^ ^'Z 3 < 2 * — Z « "" Ui 5 O a.

1000 900 800 700 630 500

400

3CQ

200

100

V \ \

\ J - . . .

\ 1

\ M •• \ " ^

f^

I 1

\ \{ 3 2 0 y

' ' I j t

1 I i 1 1

f"

0 4

1

0 s

' I M E

a .

0 6

IN

1

1

0

M I N U T E S

1 1 1 1

1 1

1 1 1 1 1 1 1

1 12

Fig. 4.1 Power Dissipation Capability of "Cantenna" (From Heathkit Data Sheets)

120

100

80

2. 60

R/Rdc -

1

Z/Rtfc /

- M.

~ • —

10 16 Frequency MHz.

23 30

Fig. 4.2 RF Characteristics of "Cantenna".

37

FREQLENCY MHZ.

Fig. 4.3 Variation of Antenna Current with Frequency.

Fig.

24

FREQLENCY M H Z . 4.4 Variation of Antenna Current with Frequency

(Antenna Inductance Tuned Out at a Frequency of 10 MHz.)

CHAPTER V

CONCLUSIONS

The impedance of the antenna as a function of time was measured

for various frequencies. It was found that it did not undergo any

significant change when an eigenmode occured.

Broadband matching networks, both lossless and lossy were tried

and found not to be attractive. This is because of the low value

of the series resistance (0.5 ohms) and the presence of the antenna

inductance (0.41 yH).

Only narrow band matching of the antenna appears possible, in

the frequency range of 5-15 MHz. The present matching network

consists of two low loss, tunable, vacuum capacitors, as shown in

Fig. 3.4. It is necessary to modify this narrow band matching

network to be able to match it to the 30 kW amplifier.

One limitation of the present matching network is the reactive

voltage developed at the high voltage end of the antenna, causing

the breakdown of the air gap (between the conductor and glass

insulation) of this antenna, at the feed-through to the Tokamak,

when more than 500 watts is fed in.

It would be necessary to insulate the air gap with epoxy or

some other suitable material to avoid the breakdown. A low loss

tuning capacitor may be connected in parallel, at both the ends of

the antenna, instead of grounding one end as at present. This

enables the ends of the antenna to float electrically, and choose

38

39

a ground in the middle, reducing the reactive voltage developed at

both the ends of the antenna. The proposed setup is shown in Fig.

5.1.

40

•H

<N O

Vj

to o

^ h ^tHi

cr\:^

CO

00 o cc

a. I o

O

u o :2

00

c •H O 4J CO

s C nj

PQ

o u u CO

2:

OJ CO o a o u

PL,

LO

&0 •H

LIST OF REFERENCES

1. T. H. Stix, Nuclear Fusion 15 , 737 (1975).

2. S. 0. Knox, "Phase Measurements of Fast Wave Toroidal Eigenmodes," Ph.D. Thesis, Dept. of Electrical Engineering, Texas Tech Univ. (1979).

3. W. K. Chen, Theory and Design of Broadband thatching Networks (Pergamon Press, New York, 1976).

4. R. M. Fano, Journal of Franklin Institute 249, 57 (1950).

5. D. C. Youla, IEEE Trans. Circuit Theory IJ , 30 (1964).

6. W. K. Chen, Electronics Letters J^, 337 (1976).

7. H. J. Carlin, IEEE Trans, on Circuits and Systems 23, 170 (1977). —

8. H. J. Carlin, et al, IEEE Trans, on Circuits and Systems 28 , 401 (1981).

9. H. C. Kirbie, "Design and Construction of the Texas Tech Tokamak," M.S. Thesis, Dept. of Electrical Engineering, Texas Tech Univ. (1978).

10. P. D. Coleman, "Probe Measurements of the Magnetic Field Structure of Toroidal Eigenmodes," M.S. Thesis, Dept. of Electrical Engineering, Texas Tech Univ. (1980).

11. S. R. Beckerich, "A Computer Based Data Acquisition System for the Texas Tech Tokamak," M.S. Thesis, Dept. of Electrical Engineering, Texas Tech Univ. (1980).

12. T. H. Stix, Third S3miposium on Plasma Heating in Toroidal Devices, Varenna, Italy, p. 156, (1976).

13. W. K. Chen, Electronics Letters 12 , 412 (1976).

41

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impedance i-iatching for alfven wave couplers by a …

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