impedance i-iatching for alfven wave couplers by a …
TRANSCRIPT
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
3.12 Time Variation of Magnitude of B-Dot Probe Signal -
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 ~ ^^^^°^ ^ •^^
3.18 Time Variation of Magnitude of B-Dot Probe Signal -
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
3.24 Time Variation of Magnitude of B-Dot Probe Signal 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
3.27 Time Variation of Magnitude of B-Dot Probe Signal 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
3.30 Time Variation of Magnitude of B-Dot Probe Signal 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
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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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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
• • • • • •
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20
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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-.
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
Fig. 3.11 Variation of I^ With Time - Method 2.
V ^
^ .000
24
B 2
D 0 1
001-
0 10Gfc-
0 500 1000
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
Fig. 3.17 Time Variation of Power, ^^^^^^ ' ^REF " ^^^^°^ ^'
B 2 L
0 1 T
001-
00 =-
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
Fig. 3.20 Time Variation of Antenna Voltage.
V
:22Q
B £03:-
0 10(t-— I—
' P P
0 ^wNJ
500 .C00
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
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
TIME IN MICRO SECONDS
Fig. 3.29 Time Variation of Antenna Reactance, X = (Z^ - R^)^/^
=h • . •! • ' U
500 1000
TIME IN MICRO SECONDS
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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