Copyright by Hugh C. Kirbie
Transcript of Copyright by Hugh C. Kirbie
Copyright by Hugh C. Kirbie
1978
^
DESÎ3N AND CONSTRUCTION OF ~H£
TEXAS TECH TCfC^MAK
by
HUGH C. KIRBIE, B.S. 1n E.E
A THESIS
IN
ELECTRICAL E/lGriElRING
Sdbr,^tt d -0 the Gradijate râcj^ of Texas Tech Univer:-, ty in
Part ia l Ful f i i lr ient cf the r.equiren^encc for
cne Degree of
MASTER O^ SCIE:ÍCE IM
ELECTRICAL ENGIflEERI'iG
Acproved
>'av, i^.c
•97^ ACKNOWLEDGMENTS
I wish io thank Professors M. 0. Hagler, M. Kristiansen and
B. Duran for serving on my committee, and I also wish to recognize
the valuable contributions of Or. Rodney Cross during his visit froTi
the University of Sydney, Australia. A great deal of credit for the
design and completion of the tckamak facility goes to my gcod friend
and fellow student Steve Knox. Steve's excellent engineering •'nsicht
and spirit of teamv/ork insured the success of the device. I also wisí"
to recognize the outstanding craftsmanship of the undergraduate labor
force during the construction of the facility. The undergraauate
team included Ken Mikkelson (now a graduate student), Steve BeckerTch,
Ed Myers and Pete Davis. Special acknowledgement also goes to Gary
Froehlich for the unselfish contribution cf his phctographic talents,
I would also like to thank the Naticnal Science Foundation and
Texas Tech University for their continuing financial support.
I wish to offer a final tribute to Wiliiam Pete Davis who, at th€
age of 21, met an untimely death on Monday, February 20, 197S. 'ue ha\
lost a close friend, but his memory lives on in his work.
n
TABLE OF CONTENTS
ACKNOWLEDGMENTS ^^
ABSTRACT ^T
LIST OF TABLES .--
LIST OF FIGURES / i i i
I. INTRODUCTION ^
Philosophy 1
History and Purpose •;
Construction and Cost 2
Performance 3
Cymator 3
II. BASIC TOKAMAK THEORY 4
Introduction ã
Magnetic Confinement 4
Equilibrium . , 7
MHD Stability and Piasma Beta 10
Stable Operat"ing Regimes 15
III. THE DESIGN OF SMALL T KAMAKS ^9
Discjssion 19
Toroidal Field (TF) ]g
Vacuum Chamber 21
rlasma Parameters 22
Ohmic Hesting (OH) 23
Vertical Field (VF) 36
Radial Field [R^)
i i i
1 /1
IV. CONSTRUCTION 46
Introduction A6
Vacuum Chamber 46
Toroidal Field 54
Ohmic Heating 55
Vertical Field 64
Radiai Field 58
Preionization 74
Discharge Cleaning Oscillator 74
Systems and Control 80
A Pictorial Review 83
V. DIAGNOSTICS 88
Introduction 88
Major Field Currents 39
Locp Voitage 91
Plasma Position 96
Plasma Current and Toroidal Field 103
Microwave Interferometry 106
Spectroscopy 107
VI. MACHINE PERFORMANCE 111
întroduction 111
Major Currents and Fields lii
An Early Performance Simulation 114
Refined Performance 11J
Summary 119
i V
APPENDIX i^^
A. COMPUTER PROGRAMS 122
B. CONSTRUCTION SUPPLEMENT 136
C. COST ANALYSIS 173
D. DERIVATIONS 177
LIST OF REFERENCES 196
ABSTRACT
A small, circular cross section research tokamak (R = 45.7 cm,
a = 16.2 cm) has been designed and constructed at Texas Tech Universitj
The basic facility includes a 130 kJ toroidal fiela (B ) bank, 18 kJ
ohmic heating bank, and a 2 kJ vertical field bank. A 40 kW, 35 kHz
discharge cleaning oscillator produces a low energy plasma ('3 1 - 2 pps
to prepare the chamber walls for tokamak discharges. After several
hours of discharge cleaning, the tokamak typically produces discharges
of 10 - 12 ms, with I = 20 kA, B^ = 0.95 T, and V|_ = 3.5 V at
dl/dt = 0. Design and construction information for the basic device
is presented together with plasma positioning coil development, loop
voltage (V.) interpretation, and current probe design. Presently
efforts are directed toward the completion of a 4 mm microv/ave density
diagnostic system and measurement of Doppler broadened impurity 1ines
for determining the ion temperature.
VI
LIST OF TABLES
Table Page
5-1 A Tabulation of the Normalized B and B 99 Component Fields ^ ^
6-1 Operating Parameters ]2i
C-1 Grant Expenditures "174
C-2 Encumbered Equipment 175
VI 1
LIST OF FIGURES
Figure Page
2-1 A toroidal chamber indicating toroidal, poloidal, 5 vertical and radial coordinate directions.
2-2 The cross ssction of plasma current showing the 6 current direction and poloidal flux.
2-3 Graph of the rotational transform with toroidal 8 coordinates.
2-4 Cross section. of the plasma current with OH 9 solenoid and return flux.
2-5 Plasma current cross section with possible 11 locations of vertical magnetic field coils.
2-6 Plasma current cross section with possible 12 locations of radial magnetic field coils.
2-7 Graph of the stable operating regime in the 17 density-current parameter space.
3-1 A plot of the "integral of H" vs e for the ohnic 25 heating windings.
3-2 A plot of the "integral of H" with partitior.s and 26 the OH transformer fcrm with the associated wire locations.
3-3 Inductive energy storage scheme with CH primary 28 current, plasma current and basic circuit diagram.
3-4 The primary and plasma current waveforms fcr d 29 capacitive energy storage scheme.
3-5 Schematic diagram of the OH capacitor banks, OH 31 transformer, plasma inductance and mutual inductance.
3-6 Basic vertical magnetic field coil arranger^^ent 40 fcr the Texas Tech tokamak.
3-7 A p1ot of the"integral o-*" H" vs 8 for the vertica; di field windings.
VI n
3-8 The partitioned plot of the "integral of H"and the 42 OH transformer form with the VF winding locations. Note the addition of six extra turns (*) to approximate the computer-predicted wire placement.
4-1 Diagram of each type of port with dimensions and 47 access area.
4-2 Location of ports on the vacuum chamber (viewed 48 from above).
4-3 View of a female vacuum flange (90° location of 50 Fig. 4-2) with lathe cuts.
4-4 Three perspectives of the steel support stand 51 for the vacuum chamber.
4-5 Schematic diagram of the vacuum station and 53 toroidal vacuum chamber.
4-6 A block diagram of the toroidal magnetic field 56 system.
4-7 Computer simulation of the current in the toroidal 57 magnetic field coils.
4-8 Oscilloscope photograph of B.. 53
4-9 One OH transformer form with the ohmic heating 59 coil locations only.
4-10 Cross section of the toroidal magnetic fieid 61 windings with OH forms.
4-11 Three-step construction of crescent-shaped air 62 bags.
4-12 Block diagram of the ohmic heating system. 63
4-13 Computer simulation of the current in the OH 55 transformer (neglecting plasma coupling).
4-14 Oscilloscooe photograph of the current in the 66 CH transformer without plasma.
4-15 One OH transformer form with the vertical magnetic 67 field coil locations only.
4-16 Block diagram of the vertical field system. 59
IX
4-17 Computer simulation of the current in the vertical 70 magnetic field coil.
4-18 Oscilloscope photograph of the vertical field 71 current.
4-19 Location of the two hoops that produce the radial 72 magnetic field.
4-20 Block diagram of the radial magnetic field system. 73
4-21 Computer simulation of the radial magnetic field 75 current for three values of shunt inductances.
4-22 Cross section of the torus with preionization 76 solenoid inserted in the center.
4-23 Block diugram of the preionization system. 77
4-24 Circuit diagram of the preionization capacitors 78 and ignitron.
4-25 Oscilloscope photograph of the single loop voltage 79 response to the preionization flux.
4-26 Block diagram of the discharge cleaning oscillator. 81
4-27 Oscilloscope photograph of: 82
(Top) Single plasma current pulse from DCO.
(Bottom) Toroidal magnetic field pulse for discharge cleaning.
4-28 Block diagram of the entire tokamak system. 84
4-29 Block diagram of the remote control system. 85
4-30 Photographs of Texas Tech tokamak: 86
(A) Top view of torus with preionization solenoid.
(B) Top view of torus without preionization solenoid. Note the radial field hoop and poloidal distribution of the OH transformer windings.
(C) View of the torus and vacuum station. 87
5-1 Equivalent circuit fora Rogowski currenr transformer. 90
5-2 (A) Location of loop voltage probes with respect 92 to the plasma and vacuum chamber.
(B) Model of the loop voltage probes using two concentric hoops and a filament current.
5-3 Straight wire plasma model of radius b and length l. 95
5-4 (A) Sine and cosine coil pair. 98
(B) Plot of K vs X and K vs y.
5-5 (A) Diagram of sine coil compensation network. 101
(B) Diagram of cosine coil compensation network. 102
5-6 Equivalent circuit of Rogowski belt. 104
5-7 Diagram of plasma current probe with passive 105
integration.
5-8 Basic microwave interferometer. 108
5-9 Block diagram of the "zebra-stripe" electronics. 109 6-1 Oscilloscope trace of the toroidal field current 112
vs time.
6-2 Oscilloscope trace of the toroidal field vs time. 112
6-3 Oscilloscope trace of the ohmic heating primary 113 current without plasma loading.
6-4 Oscilloscope trace of the vertical field current 113 vs time.
6-5 (Top) Cosine position coil response. 115
(Bottom) Plasma current without vertical or radial fields applied.
6-6 (Top) Sine position coil response. 116
(Bottom) Plasma current without radial field applied.
6-7 Loop voltage response for a properly adjusted 117 plasma current.
6-8 (A) Calibration; "no-plasmâ" trace. 118
xi
(B) (Top) Cosine coil response. 118 (Center) Sine coil response. (Bottom) Plasma current properly adjusted.
6-9 (Top) D„ radiation from the plasma. 120
(Bottom) Plasma current properly adjusted.
A-1 Diagram of the plasma, vacuum chamber, and ohmic 123 heating flux showing the circular field line passing through all the wires.
A-2 List of the OH winding distribution program. 125
A-3 Basic toroidal field circuit and its SCEPTRE 129 circuit model.
A-4 Modification of the SCEPTRE circuit model of Fig. 130 A-3 to include the radial field netv/ork. Note that the switch, Si, is modeled the same for both Figs. A-3 and A-4.
A-5 List of the SCEPTRE program for the toroidal 131 field circuit model of Fig. A-4.
A-6 Basic ohmic heating circuit and its SCEPTRE 132 circuit model.
A-7 List of the SCEPTRE program for the ohmic heating 133 circuit model of Fig. A-6.
A-8 Basic vertical field circuit and its SCEPTRE 134 circuit model.
A-9 List of the SCEPTRE program for the vertical 135
field circuit model of Fig. A-8.
B-1 Top view of the stainless steel vacuum chamber. 137
B-2 Cross section A-A of the vacuum chamber. 138
B-3 Cross section B-B of the vacuum chamber. 139
B-4 Cross section E-E of the vacuum chamber. ;40 B-5 Cross section I-I of the vacuum chamber. l-il
B-6 Exploded edge view of the male and female main 142 flange halves (see Fig. 4-2) with the 3/16" annular bakelite insulation.
xii
B-7 Side and edge view of the stainless steel material 143 used in manufacturing the rectangular channels for the 8.0" dia. Varian ConFlat ports.
B-8 Side and edge view of the stainless steel material 144 used in manufacturing the rectangular channels for the 6.0" dia. Varian ConFlat ports.
B-9 Circuit diagram and single capacitor section of i45 the toroidal field capacitor bank.
B-10 12 kV DC power supply for the toroidal field 146 capacitor bank.
B-11 One of six HV diode modules for the TF 147 power supply.
B-12 Remote actuator and control connections for the 148 TF power supply.
B-13 Toroidal field switching system. 149
B-14 PVC transmission line connecting the TF switching 150 system to the toroidal magnetic field windings.
B-15 Ohmic heating circuit with fast bank and single 152 slow bank section.
B-16 Ohmic heating fast bank power supply. 153
B-17 Ohmic heating slow bank power supply. 154
B-18 AC circuits (with remote control connections) 155 for both fast and slow bank OH power supplies.
B-19 Ohmic heating remote control actuator and 156
control connections.
B-20 Ohmic heating switching system. 157
B-21 Ohmic heating ignitron trigger circuit. 158
B-22 Vertical field circuit diagram with fast and 159
slow capacitor banks.
B-23 Vertical field fast and slow bank power supplies. 160
B-24 Vertical field power supply remote control 161 subchassis.
xm
B-25 Vertical field switching system. 162
3-26 Vertical field ignitron trigger circuit. 163
B-27 15 kV DC power supply for the preionization 164 capacitor bank.
B-28 Ignitron trigger circuit for the preionization 165 system.
B-29 40 kW, 35 kHz discharge cleaning circuit. 166
B-30 Screen and control grid bias supplies for the 167 discharge cleaning oscillator.
B-31 AC control circuits for the discharge cleaning 168 oscillator. The fans indicated cool the 4CX15000A tube and support components.
B-32 Circuit diagram of the solid-state screen pulser. 169
B-33 Circuit diagram of the solid-state toroidal 171 field pulser with fail-safe spark gaps.
B-34 Solid-state SCR trigger circuit for the toroidal 172 field pulser.
D-1 Loop with current I and the coordinate system. 178
D-2 Coaxial loops separated by a distance z. 180
D-3 Plasma modeled as a circular wire of minor radius 184 b and uniform current density.
D-4 Diagram of the gas pressure p acting on the ends 189 of a plasma section producing a radial force F ..
D-5 Diagram of a plasma section with internal and 191 external fields that produce toroidal tension forces dF^.
D-6 Diagram of the plasma cross section with the 192 coordinate system.
XIV
CHAPTER I
INTRODUCTION
Philosophy
The national research effort, in the field of thermonuclear fusion,
has recently focused on the tokamak as one of the promising devices for
reactor investigation. A sudden interest in small research tokamaks
has, therefore, evolved.
Those interested in establishing a tokamak facility will encounter
a surprisingiy smal 1 collection of information describing the design and
construction of these devices. This thesis attempts to bridge this
information gap by providing, in a single document, a simple design
outline for small tokamaks. It illustrates also the general design
criteria by describing the construction of the tokamak at Texas Tech
University.
History and Purpose
The basic design of the tokamak at Texas Tech is not original. but
simply a modification of several preceding c-ircular cross section
devices with air-core, ohmic heating transformers. The first tokamak
of its kind in the United States was Versator I, which was con-
structed at the Francis Bitter National Magnet Laboratory under the
direction of R. J. Taylor. Similarly,the torus at the California 2
Institute of Technology, recently completed under the direction of
Professor R. M. Gould, closely resembles the Versator I design. The
Texas Tech torus reuresents a third-generation, circular cross section
device following the design legacy of the Versator and Caltech
machines.
The Texas Tech tokamak was designed to produce a high temperature,
low density, long duration plasma for the investigation of wave propa-
gation and plasma heating with waves in the ion cyclotron range of
frequencies (ICRF). A simple loop coupler will be used to launch the
fast or compressional Alfvén wave in a deuterium plasma. Wave
damping, at the second harmonic of the ion cyclotron frequency
(o) = 2 w .)» will be studied with emphasis on defining the damping
mechanism. To obtain enhanced coupling to the fast wave, a "tracking"
of the toroidal eigenmodes, by varying the excitation frequency, will
be investigated.
Construction and Cost
The tokamak was constructed in a 13 month period with a labo>
force of two graduate students and three pre-baccalaureate, part-time
employees. The machine was financed under a grant from the National
Science Foundation (NSF grant ENG-76-05897) with support from Texas
Tech University. The operational tokamak facility was completed for
less than $60,000 in combined grant expenditures for labor, services,
and capitâl equipment. However, a significant amount of equipment
from past experiments was also incorporated in the tokamak
system. Including this previousîy acquired equipment, the basic
tokamak facility required approximately $90,000 in construction costs
and capital investments. Appendix C contains a tabulated assessment
of the expenditures encountered during the construction period.
Performance
The Texas Tech tokamak has a 1.0 T (10 kG) toroidal field with an
18 kJ ohmic heating capacitor bank and a 2 kJ vertical field capacitor
bank. The all-stainless steel vacuum chamber maintains a base vacuum
-8 of 2.0 X 10 Torr with a three-pump vacuum station described in
Ch. IV. The tokamak produces a 20 kA (peak), 10 ms duration plasma.
The plasma diagnostics are not complete, but the expected te rperature
(T + T.) and electron density (N ) are 200 - 300 eV and
13 -3 0.5 - 1.5 X 10 cm , respectively.
Cymator
It has become a tradition to christen various plasma research
devices with some "name" deemed appropriate by its creators (e,g,
Rector, Versator, Microtor, Macrotor, e t c ) . As a mark of tradition,
the tokamak at Texas Tech has been christened "Cymator," from the 4
Greek adjective vyaxo, meaning wave-like. As the name implies, the
tokamak facility is devoted to investigations of plasma-wave
interactions.
CHAPTER II
BASIC TOKAMAK THEORY
Introduction
In the search for a magnetic confinement scheme, the toroidal
geometry (Fig. 2-1) has some advantages over linear devices.^ External
magnetic field coils produce toroidal field lines (B. or B J parallel t 'b
to the flow of current. As a result, there are no "enrís" through which
particles may escape the plasma volume. In tokamak devices the plasma
current itself produces a magnetic field (B or B ) in the pololdal p (f)
direction (Fig. 2-2). The combination of these two magnetic field
components results in helical field lines which spiral arounú the
torus. Since the magnitude of B increases as r increases to the
plasma edge, the pitch of the helical field lines increases as a
function of r. The change of pitch vs r causes "shear" of the toroidal
fieid. The helical twist and shear of the magnetic field lines make
the tokamak stable against most MHD (magnetohydrodynamic) instabilities,
Tokamaks (Russian acronym for "toroidal chamber magnetic") were
developed by the Soviet Union in the early 1960's. Tokamaks are
generdlly long confinement time, low density, high temperature plasma
devices. The combination of long plasma confinement and high temper-
ature make the tokamak an attractive candidate as a contrclled
thermcnuclear fusion reactor.
Magnetic Confinement
The maanetic ccnfinement scheme in tokamaks is m.ore coniolicated
TOROIDAL IRECTION
CHAM8£R
POLÛIOAL Q DIRECTTCN
>VERnCAL å CIRECTION
RACIAL DIRECnCN
MINOR AXiS
MAJCR AXIS
ScCnCN A-A
Fig. 2-1. A tcroidal chamber indicating toroidal, ooloidai, vertical and radia: ccorcinate directions.
CJ
c
tr. Oco
2<
U
'X.
í-
t
than just any arbitrary combination of toroidal (B.) and induced w
poloidal (B ) field components. For instance, the Kruskal-Shafranov
6 7 9 limit ' ' requires that the helical field lines must not complete a
spiral (in the 9 direction) before traveling the circumference of the
torus (in the f direction). Stated differently, the rotational
transform, i (iota; Fig. 2-3), must remain less than 2^. The
rotational transform, R B
1 = 27; - ^ , 0<r<a (1)
may be rewr i t ten in terms of the safety fac tor , q ( r ) , where
27r r B^
q(r) = r = R r • (2) P
The Kruskal-Shafranov l i m i t requires I<27T or q( r )> l for s t a b i l i t y of
^ t the plasma. This c r i t e r i on implies a l i m i t on the ra t io of D~ ^S a
funct ion of minor radius.
Since ports must protrude through the toroidal •^'ield c o i l s , s l i gh t
var iat ions in the toroidal f i e l d strength occur. These toroidal f i e l d
" r i pp les , " of only a few percent of B., cause a non-symmetric trapping Q
of particles and their subsequent loss to the chamber walls.
The poloidal field is produced by the plasma current which is
induced by the ohmic heating flux. To avoid disturbing the plasma
equilibrium, the chmic heating return flux should circumvent the plasma
volume as shown in Fig. 2-4.
Equilibrium
The sheared toroidal field alone is not enough to contain the
plasma ring. Since the poloidal field component is stronger on the
8
Í I 2 3 p O = g= § UJH-UJF
xo xo i - y i - u j
Q: Q:
í55í2û co
C
TJ
O
rr C
r-, 1
X Z3
O û.
9 o 2 LU - I
o
3
GJ
wn
~ O
ui
C
c
10
inside of the plasma ring than the outside, a net J x S force is
directed outward along the major radius. To halt the radial expansion
of the plasma, a small vertical magnetic field (S" ) is required to
produce a J x B^ force directed radially inward. The vertical magnetic
field can be produced by external field coils, Fig. 2-5, or generated
for short periods of time by image currents induced in a copper 9
stabilizing shell.
Imperfections in the toroidal field geometry not only trap
particles but also produce a small radial field component
(10 - 20 X 10 T) which can cause a vertical plasma drift. The
average toroidal field error, plus small radial field contributions
from the vertical and ohmic heating fields, must be corrected to
maintain plasma equilibrium. The coil arrangement of Fig. 2-6 can
generate radial magnetic fields adequate to control any vertical plasma
column displacements. As a final equilibrium requirement, the olasma
kinetic pressure may not exceed a small fraction of the imposed
poloidal magnetic pressure.
MHD Stability and Plasma Beta
The equilibrium of a toroidal plasma can be established by various
vertical and radial fields and, according to ideal MHD theory, the
plasma itself ( a fluid of infinite conductivity) will be stable if
q(r)>l for all 0<r<a. There exists, however, plasma mcdes where the
safety factor, q(r)>l, is not a sufficient condition for plasma
stability.
11
Û _ J UJ LJL.
<
o
uj o > o 0
U
U
OJ c cn fO
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^ UJ o cr o
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IT3
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13
The MHD instabilities have space dependencies of the form
f(r) exp [i(no + nt^)],
where
m = 0, 1, 2, 3, ...
n = 0, 1, 2, 3, ...
are the mode numbers. The control of the common helical modes (n = 1;
m = 1, 2, 3, 4) depends on the location of the "singular surface" at
the radial location r , where q(r ) = m/n. If the singular surface
falls into a region of "perfectly conducting" plasma the m > 1 modes o
are suppressed, and only the m = 1 mode can cause stability problems.
If the Kruskal-Shrafranov condition (q(r)>l) is violated between the
singular surface and the minor axis, 0<r<r , a local m = 1 disturbance
will occur. Even if the condition q(r) > 1 is satisfied, unstable
kink modes (m > 1) may still exist if the singular surface is in the
vacuum outside the plasma or within a region of poor plasma 7 q 1 .-
conductivity. These finite-resistivity, kink ("tearing") modes' ' ' can be prevented if the safety factor at the outer plasma edge has
5 the value q > 2.5. This condition limits the maanitude of B which, ^ ~ - p '
consequentiy, restricts the magnitude of plasma current.
Impurities affect the plasma stability by altering the plasma
conductivity. Impurity injection from the walls decreases the current
channel radius due to a decline of the plasma conductivity near the
plasma column periphery. The narrowing of the current channel is
believed linked to the m = 3 helical instability.
The plasma 6 (beta) is the ratio of plasma kinetic pressure to the
14
magnetic pressure. In the case of the poloidal magnetic component, the 5
poloidal beta, 3 , is defined as
P B P
where T and T. are the electron and ion temperatures, k is Boltzmann's
constant, and n is the particle density. For equilibrium,the poloidal 5
field should support at least a fraction a/R of the kinetic pressure.
This restricts 6 to be less than the aspect ratio (6 ± R/a) of the
tokamak. To calculate the maximum total 6 of the tokamak, both
toroidal and poloidal magnetic field components must be considered.
The ratio of
3 B 2 B 2 1
- = ( ) = . ? o = 2 (' ^n B ' B / ^ B ^ 1 + B. P t p ,U
^B '
with
P
q(a) = - ô^ > 2.5, (5) R Bp
and the restriction of
3 1 R/a (6)
yield a limit on the total 6,
R/a ^ a /7^ 3 < - Y I g R ' ^ ^
1 + 6.25(^) 5
that can be obtained in tokamaks with circular cross sections.
Non-circular plasma cross sections permit higher values of s
15
than Eq. (7) defines. Certain Doublet and D-shape cross sections also
permit a high plasma current density and an increase in the poloidal 9
flux so as to extend the plasma confinement time.
Stable Qperating Regimes
The preceding equilibrium and stability comments identify the
restrictions on both the safety factor, q(a) > 2.5, and the total
beta, 3 < In* of the tokamak. These results lead to the more tangible
restrictions of maximum plasma current and critical particle density.
A stable operating regime is therefore defined over a current-density
parameter space.
Consider the result of Eq. (2) at r = a
a B. q(a) = ô R^> 2.5, (5) R B
P
and the value of B ,
p Z7Ta
where I is the plasma current. Combining these equations, the
safety factor,
27T B . a^
q(a)=7-^\-, (9) Po
specifies a maximum plasma current.
?" B ?
- q(a) u S
16
that can be permitted if MHD stability is to be maintained. Furthermore,
the total beta of a tokamak,
nk(T + T.)2 y a 3 = %^ '- < - (11)
B^ 6R
where
2 2 2 B = B^' + Bp' , (12)
limits the density to a critical value (n ),
a B^ "^ " c = 6 R k ( T ^ ^ T . ) 2 p , ' (13)
for stable tokamak operation.
Two types of abnormal tokamak discharges border the region of
stable operation shown in Fig. 2-7. The first abnormality is the
disruptive instability ' ' which is characterized by an explosive
expansion of the plasma column. The disruptive instability occurs
when the q value at the plasma edge has dropped below the threshold
for m = 2 instability formation. The disruptive instability limits
both high and low current regimes. An increase in plasma current (I),
for a fixed plasma radius and B., causes a decrease in q due to the
increase of B . The same effect, a decrease in q, occurs at the low P
current boundary due to the shrinking discharge radius at low currents 2
Since q « — , the decreasing plasma radius (r) can outweigh the de-
creasing current magnitude and lead to the development of m = 2, 1
instabilities. The disruptive instability appears again as n is
approachea, since high density also tends to shrink the discharge
17
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CJ rO
;5; O
c
I
cn c O) -o CJ
o
rî o
CVI
O
ro 1 6 u m
c
CT) OJ
^ ' ^ • r— 4 - >
^ ^ ' 1 '
o" <L
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</)
o
I
18
radius. The second abnormality, electron runaway discharge, ' '
appears only at low densities. The electron runaway discharge is
characterized by a zhin high velocity electron sheath, which carries
the bulk of the plasma current and produces intense x-ray pulses.
Since small tokamaks are relatively low temperature, collision
dominated devices, the particle transport can be described by the
Pfirsch-Schliiter theory, which is a modification of the classical
15 diffusion rate. Hence, the interesting "neoclassical diffusion"
regimes, due to particles trapped (banana orbits) by the variation of
the toroidal field magnitude, are not applicable to small research
tokamaks.
CHAPTER III
THE ESIGN OF SMALL TOKAMAKS
Discussion; Ambitious Economics
The establishment of a tokamak facility, of any size, is an
economically ambitious undertaking for most universities. The vacuum
chamber, vacuum station, toroidal field, and ohmic heating systems are
undoubtedly the most expensive investments in basic equipment (see
Appendix C). In addition, a set of "second-generation" expenditures,
for the Texas Tech tokamak, will include a computerized data acquisition
system and a solid-state vertical field feedback network. Together,
these new expenses will exceed the original S57,000 equipment investment
for the entire facility. (Obviously, tokamaks are not econcmically
stable.)
For small research tokamaks, the plasma current magnitude, chamber
size, toroidal field, etc. are usually limited by the astronomical
increase in cost for a relatively small increase in tokamak performance.
In this chapter, the general design criteria for small tokamaks is
presented with an emphasis on economic a1ternatives. To accompany the
general design presentation, certain critical design steps will be
illustrated by specific calculations for the Texas Tech device.
Toroidal Field (TF)
Since the maximum plasma current is proportional to the toroidal
field strength, a large investment in the toroidal field system would
be wasted unless followed by a proportional investment in the ohmic
19
20
heating system. Typically, the magnitude of the toroidal field is
1.0 - 2.0 T for a tokamak with a major radius less than or equal to
0.5 m and an aspect ratio greater than or equal to 2.5. The production
of such a field requires a moderately expensive 100 - 300 kJ capacitor
bank. Furthermore, the field strength should be constant and near
maximum during the tokamak discharge. A reasonably constant TF current,
from a capacitive energy storage system, can be accomplished by crow-
barring the TF coil at the current maximum. This technique requires
the TF coil to have a large L^p/R ratio (> 50 ms) and the crowbar
ignitron(s) to handle the large currents during the crowbar interval.
The TF Coil
Which one of two prevalent toroidal field coil styles is chosen
depends on the size and shape of the vacuum chamber. The small
rectangular cross section devices typically use a continuous coil
comprised of copper "picture frames" bolted together. ' This
particular low resistance, low inductance coil system operates at a
high toroidal field current level and consequently demands a higher
amp-sec rating of the crowbar ignitron(s).
The small, circular cross section machines, like Cymator,
generally have a continuous TF coil comprised of several layers of
large cable(l/0 AWG or larger) wrapped directly on the vacuum chamber.
This coil style is inexpensive, simple to construct, and operates at a
relatively low ( 3 - 6 kA) current level.
Since ports must protrude through the toroidal field coi1, an
effort should be made to minimize the toroidal field ripple. The
21
compensation for toroidal field ripple requires an increase in
toroidal field winding (or current) density near the ports.^ The
continuous cable type of TF coil has an obvious advantage for TF
ripple compensation, since adding a few extra turns near the ports
is a simple task.
The TF Capacitor Bank
The toroidal field capacitor bank is the largest energy storage
system of the tokamak facility and deserves special engineering
considerations. An efficient utilization of space and funds requires
the TF capacitor bank to operate at a high voltage level (10 - 20 kV|^.w)
Such a voltage requirement demands special attention to personal
safety and a large kVA, high voltage supply. To charge the bank
quickly and efficiently, the TF power supply should also be a constant
current source. As an example, the 130 kJ, 10 kV toroidal field bank,
for the Texas Tech tokamak, is charged from a 24 kVA, 2.0 A constant
current power supply described in Ch. IV and Appendix B.
Vacuum Chamber
Both rectangular and circular cross section chambers have their
respective advantages. The circular cross section style is easier
to construct, requires fewer vacuum welds, and can employ conventional
Varian ConFlat' flanges for ports. Stainless steel 90 pipe sections
comprise the bulk of the vacuum vessel (described in Chapter IV),
which can be purchased, ports cut, and assembled in a short period of
time.
22
The rectangular chamber style, although í ore difficult to fabricate,
provides the best access ports. If the ratio of the cross sectional
height to the width is larger thanone (usually 4/3), the variation of
B. over the plasma volume is smaller than for the circular style.
The toroidal vacuum chamber, regardless of style, must be separated
by one or more insulating voltage gaps. These insulators allow the
induction of current within the plasma rather than in the vessel wall.
For the Texas Tech tokamak, two annular bakelite insulators separate
the circular cross section, vacuum chamber halves (see Ch. IV).
Plasma Parameters
The chamber size, together with the toroidal field strength,
determine the remaining machine parameters. For instance, the chamber
size determines the aspect ratio (R/a), and for a specific toroidal
field and plasma current the safety factor, q(a), can be evaluated at
the chamber wall. The specific calculations of the maximum plasma
current and plasma inductance require a model of the plasma current
19 profile. The most convenient current model for simple design
calculations is a uniform current density for a plasma of minor radius
b, which is somewhat smaller than the chamber radius. For this
approximation, the maximum permissible current in the tokamak can be
calculated from Eq. (10) of Ch. II. The calculation,
277 B^ b^
- q(b) UQ R
gives a conservative upper current limit for a plasma of radius b.
23
For the Texas Tech tokamak,
b = 0.1 m
h =1.0T
R = 0.457 m
q(b)= 3.0
which require I to be less than 36.5 kA. For fixed major and minor
plasma radii, the toroidal plasma inductance can be calculated from
the formula,
L = yo R(ln ^ - 7/4) , (2)
for a uniform current density over the plasma cross section.
(Equation (2) is derived in Appendix D.) Equation (2) yields
L = 1.06 yH for this tokamak.
Ohmic Heating (OH)
The ohmic heating transformer, described in Ch. II, should induce
a plasma current and not perturb the plasma equilibrium. The small
Soviet tokamak devices channel the OH flux safely away from the plasma
volume via a laminated steel core. Many small U.S. tokamaks (pioneered
by Versator I) use an air-core OH transformer with special field
shaping coils that exclude the OH flux from the plasma.
24 The Air-Core Transformers
Two distinct styles of air-core transformers have been used in
small U.S. tokamaks. The first type is a simple solenoid with a set
of extra windings that detour the return flux from the vacuum
20 chamber. This solenoid transformer style is preferably employed on
the rectangular cross section machines. The second type of OH
transformer is a set of windings poloidally distributed around the
vacuum chamber on a constant minor radius (see Fig. A-1 of Appendix A).
These conductors are positioned to approximate a computer-generated,
toroidal current sheet at the winding radius. The fictitious current
sheet varies in current density, as a function of 6, to generate OH
field lines that avoid the plasma volume.
Appendix A contains a brief description and listing of the
computer program used for the OH transformer design for Cymator. The
computer program in Appendix A calculates the tangential value of H^^
along a circular field line at the winding radius for each degree of
e e [180 , 360°] (see Appendix A for further explanation). The
collection of these tangential values is proportional to the cumulative
sheet current density over the e interval. The numerical integration
of the sheet current density (integral of H) over the e interval is
shown in Fig. 3-1. If the integral of H is partitioned into equal
segments, as shown in Fig. 3-2, then each segment represents a wire
location at a particular e angle. It should be noted that a
poloidally distributed transformer, like that on Cymator, is easily
fabricated but characteristically yields a much lower plasma coupling
25
co
c "O c
cn c (T3
u •f—
E
o
<u
o *<-
CD
>
<P ^ Q 0) co coi cvi ^ <o 8 8
<0 « ^ ^ S 2 o o ÍV ^. <D CO íO cvi - o
cn
OJ x: 4->
O
-M O
I cn
cn
H iO "TVd93iNI
26
x:
2
o
E i -O
4-to c
o
XJ c
</) c o
!T3 Q .
J = 4->
e lO fNJT lO
o O 00
2 ( -o z UJUJ Q-UJ 1 o a:<
Ui_) ^^
8§i
H dO 1V«931NI
M-O
p —
na i -CT o; +-> c
•^ r
(U ^ 4->
M-O
-M o
r—-Q .
«a;
CVJ
co
. to c o
• t —
•M 03 (J O
r—
<u s-
• 1 —
2 "O (U
-«-> <T3
•r-U O cr, co (T3
0 1
27
efficiency than the solenoid transformer type.
The OH Energy Storage
Regardless of the transformer styles, solenoid or e-distributed,
there are two basic methods of energizing the primary windings. The
rate-of-change in OH primary current provides the loop voltage that
drives the unidirectional plasma current. The plasma current can be
dlQu '^^H
initiated by a rapidly increasing ( + -gT-) or decreasing (-—^)slope of
the OH primary current. An inductive energy storage scheme slowly
transfers the energy from an ohmic heating capacitor bank to the CH
transformer. The switching circuit of Fig. 3-3 causes a sudden change
in OH primary current (by introducing a small resistance into the
circuit) when s- opens, forming the plasma current, and continuing to
sustain the plasma until the primary current decays to zero. The
circuit of Fig. 3-3 produces long plasma currents {<_ 100 ms) but
requires a wery expensive, high-current opening switch (s^).
A capacitive energy storage scheme initiates and sustains the
plasma current only during the positive slope of the OH primary current.
The v/aveforms, of Fig. 3-4, demonstrate that the plasma current in-
creases only until zhe OH primary current nears the peak. The
remaining OH primary current pulse serves no useful purpose.
The Texas Tech tokamak employs capacitive energy storage since
this method is relatively inexpensive and easily implemented. The
primary current waveform (of Fig. 3-4) can be produced from a double-
stage capacitor bank system described in Ch. IV. The "fast" bank
stage initiates the rapid bulk ionization of the preionized filling gas.
\
\s. CROWBAR
INDUCTIVE ENERGY
STORAGE
28
OHMIC HEATING TRANSFORMER
PRIMARY CURRENT
CLOSE S2
OPEN S.
PREIONIZE TiME
PLASMA CURRENT
TIME Fiq. 3-3 Inductive energy storage scneme witn OH primary
current, plasma current and basic circuit diagrarr
29
\ \
FAST BANK
\
CAPACITIVE ENERGY
STORAGE
•'*' SLOW "- BANK
CROWBAR OHMiC HEATiNG TRANSFCRMER
CLOSE S3
PRIMARY CURRENT
PREIONIZE
PLASMA CURRENT
TiME ,-ig. 3-4. The primary and plasma current wavefcrms for a
capacitive energy storage schene.
30
The "slow" bank stage increases the magnitude and duration of the plasma
current until the slow bank reaches a current maximum. Under perfect
equilibrium conditions, the peak plasma current then decays in time
with a waveform similar to exp [- R t/L] , where R is the average P P
plasma resistance and L is the plasma inductance. Since the temporal
location of the OH primary current peak ideally determines the plasma
current duration, the capacitance of the slow bank and the inductance
of the air-core OH transformer should be as large as physically or
economically possible.
The OH Bank Design
Presumably the chamber size and shape, toroidal field magnitude,
and OH transformer style have been selected. Since Eq.(l) has
established the maximum plasma current, the OH banks should be
designed to induce this calculated current. Figure3-5, and tne
following equations, describe the relationship between the OH prim.ary
and secondary plasma currents:
di, dip
di, di« ^ -. R^i^ . L ^ ^ . (4)
and
= -^ (5)
where
M dt
M = mutuai inductance,
L-, = OH transformer inductance,
31
co <
OJ Q:
j2
o.
GJ
c to
c
l / l
c
o
(-! •
CL O ra c (_• f O
-4.J
~ u O 3
QJ C
o z:
ro C •»- O • o
o u • f - c -1 - ; trs rz -t-> E O Oy r
-C "O u c
I
m
O l
32
Lp = inductance of the plasma,
R- = resistance of the OH primary circuit,
R^ = resistance of the plasma,
i-j = OH primary current,
i^ = plasma current,
and q = charge on the OH capacitor(s). The following analysis also
assumes that some calculated or measured value of L-. is available and
that the calculation of L^ (Eq.(2)) has been performed. If the OH
transformer is properly designed (BQM = 0 in the plasma), all of the OH
flux lines link the plasma. The mutual inductance is therefore (see
Appendix D)
M = / , (6)
where N is the number of primary turns in the OH transformer.
Neglecting the plasma's resistance (R^ = 0) Eq.(3) shows that the
maximum plasma-to-primary current ratio will be
i M L
l MAX ^2 ' 2
Equation (7) can be very useful if modified slightly to include the
effect of a finite R^. Equation (8) is an empirical relationship,
0.7 M i. i , = •. ^ , (8)
that estimates the plasma current magnitude in the Texas Tech tokamak
for most discharge conditions. VJith this value set equal to the plasma
33
current maximum of E q . ( l ) ,
27r B b^ 0.7 Mi^
a maximum OH primary current can then be estimated from Eq. (9) which
becomes
27r L^ B^ b^
'l " 0.7 q(b) yo M ^R"'- ( '
The fast bank should be designed to attain the primary current
maximum of Eq.(lO). The value of the OH fast bank capacitor should
be chosen to limit the rise time of i-.. Experimentally, i-. should not
rise faster than 10.0 to 13.0 MA/s for a major chamber radius < 0.5 m,
a toroidal field < 1.0 T, and a mutual inductance (M) of 10 to 20 uH.
If i-j rises too fast, an excessive toroidal electric field is produced
which can drive a normal discharge current past the Kruskal-Shafranov
21 limit or lead to the abnormal electron runaway discharge. The voltage
of the fast bank should be chosen to attain the desired i-j ^^, and
can be estimated from the initial fast bank energy, when the circuit
resistance is ignored. Therefore,
iáC^V^^ = h L ^ i / (11)
gives the current value
at the first quarter - period. In Eq.(12) C^ is the fast bank
capacitance and V^ is the initial charging voltage. The result of
Eq.(lO) can be substituted into Eq.(12) to yield a conservative voitage
34
requirement for the OH fast bank.
The OH slow bank should, at least, maintain (and possibly increase)
the magnitude of plasma current established by the fast bank. The
value of the slow bank capacitance is limited, in practice, mainly by
economic considerations. For the slow bank to keep i^ constant, Eq.(4)
requires
di
dt" ' 2'2 uniLia! ly;
22
M;n:i = R^i^ + 0 == 5 V , (13) (initially)
where the initial 5 V value of R^i^ is typical for small tokamaks.
Equations (3) and (13) determine the voltage of the slow bank,
di
^s^^l^^hdt V. = R^ i ^ + L , - ^ + 0 (14)
= R^1VL^(-^) (15)
= Riii + L (|) , (16)
necessary to overcome the OH primary current voltage drop (R-.i-i)
and maintain a constant plasma current. Computer solutions of Eqs.(3)
and (4) can refine the OH bank design if a plasma resistance model can
be found. The empirical computer model of the plasma resistance is
R« = 4 . 0 fi , ( 1 7 ) '^ 8Ô~rT^FxTÔ3"
which accurately predicts the plasma current formation in the Texas
Tech tokamak. The variable t, of Eq.(17), is the progression of time
(seconds) which begins with the initiation of the plasma current.
35
The Cymator Calculations
Listed below are the OH circuit parameters for the Texas Tech
tokamak:
L = 500 yH
N = 48 (number of OH windings)
L^ = 1.06 yH
M = 10.4 yH
R, = o.no n
Cf = 120 pF, 10 kV^^^
C3 = 93 mF, 600 V^^^
i^ = 36.5 kA (see Eq.(l)).
From these specified parameters, the following parameters can be
calculated:
{S) = 9.81 (7) 1 max
0.7 Mi ip = r — - = 0.0/ "I1 1 = 6.87 i, A (8)
i = 5.31 kA (10)
V^ = 10.8 kV (10) and (12)
(fast bank charging voltage)
= 584.1 + 240.4
= 824.5 V (slow bank charging voltage).
36
Due to economic constraints during construction, the 600 V (V ) OH ^ max s
slow bank prevents the ohmic heating system from attaining the design
values.
Vertical Field (VF)
The vertical field, described in Ch.II, produces a net J x S force
directed radially inward, that establishes the radial plasma equilibrium,
Image currents, induced in a copper stabilizing shell, produce the
desired force on a transient basis in some tokamaks. However, quasi-
steady equilibrium of the plasma column in small tokamaks usually re-
quires an externally applied vertical field. To quench the radial and
vertical n = 0 axisymmetric MHD modes, the externally applied vertical
field must have a positive radius of curvature, R , (Fig. 2-5) greater
than 2/3 of the major radius (R > 0.67 R).'''^
The vertical field magnitude, at the minor axis, can be calculated
from the formula (derived in Appendix D)
^ = 4 ^ f l " ( f ) ' ^ ' ^ -2/2] . (18)
where
and
I = plasma current,
R = major radius,
b = radius of the plasma column,
3- = poloidal beta, 6
L. = internal inductance/unit length of the ^ toroidal plasma.
37
Dynamic vs Programmed Vertical Fields
A dynamic feedback system can be incorporated in the VF network to
insure proper radial plasma position during the temporal development of
the plasma current. Yet, plasma position and stability need not be the
only task of a dynamic VF. The ATC device (Adiabatic Toroidal
Compressor) uses the vertical field to heat the plasma, by adiabatic
compression, while preserving a nearly constant aspect ratio of the 9
plasma. The Doublet II A device employs a set of VF coils to shape
the plasma cross section favorably, which yields an increase in plasma s, g
current density, and confinement time.
Most small university tokamak programs cannot afford these expensive
VF feedback networks and must rely on a simple, programmed VF waveform.
The VF current should be a scaled reproduction of the plasma current
waveform to preserve a constant B /I ratio for equilibrium (Eq.(18)).
During the construction of the tokamak, the plasma current waveform
cannot be known apriori . Therefore, the VF current is designed ic be a
scaled version of the OH current as a crude approximation of the
plasma current.
The Vertical Field Coil
The VF coil, unlike the OH transformer, does not have a strict set
of winding placement criteria. The VF coil must simply provide a
sufficient B , of the proper curvature for equilibrium, while minimizing
the mutual inductance between the VF and OH coils. The coil configu-
ration of Fig. 2-5 is a possible candidate, and can be applied to a
specific machine size with the help of a computer to map the field
38
magnitude and curvature.
The VF Capacitor Banks
The calculation of the VF fast and slow capacitor stages must be
based on the assumptions that a style of VF coil has been chosen and
the measured or calculated value of the VF inductance is known.
Furthermore, the ratio of the vertical field at the minor axis to the
B vertical field current (j^) must be known for the selected VF coil
V
style.
The VF fast and slow bank capacitances must be chosen to match the
respective fast and slow bank quarter-periods of the OH current wave-
form. The voltages of the fast and slow banks must be chosen to provide
sufficient B (Eq.(18)) for the predicted plasma current of Eq.(8). The
basic VF design procedure is outlined below:
(a) Calculate or measure the VF inductance, curvature, and
B /I ratio. V V
(b) Calculate the plasma current of Eq.(8).
(c) From Eq.(18), calculate the necessary B to maintain plasma
equilibrium.
(d) The B /I ratio will yield the requirad VF coil current,
Iwp max.
(e) Choose the fast bank capacitance to match the OH current
quarter-period.
(f) From the inductance and capacitance of the VF fast bank,
the required fast bank voltage can be estimated from
Eq.(lO).
39
(g) Steps (c) through (f) can be repeated for the plasma
current peak due to the OH slow bank, provided the
resistance of the VF winding is also considered.
The Cymator VF Coil Design
The vertical field coil, for the Texas Tech tokamak, is simply a
set of conductors, on a constant minor radius, that crudely models a
copper shell (Fig. 3-6). The initial coil scheme had an insufficient
radius of field curvature which was corrected with the help of the
winding distribution program used to design the OH coil.
The OH program was instructed to generate a field line with a
radius of curvature R = 0.75 R at the VF winding radius. The compu-
tational results are shown in Fig. 3-7, which is a plot of the integral
of H vs 9 for 9 £ [270°, 450°]. The "integral of H" represents a
numerical integration of the tangential Hyp values, from a properly
curved VF field line, (at the VF winding radius) over the 9 interval.
The actual VF and OH winding distributions are determined in the same
manner. The equipartition of the plot in Fig. 3-7 gives the desirea
wire placement, as a function of 9 (Fig. 3-8), which would generate the
same curved field line. Since the wire placement along the 3 interval
(270°, 450°) was already equally spaced, six extra windings (also
Fig. 3-8) were added to approximate the proper current density vs e.
These additional windings greatly improved the vertical plasma stability,
The present VF winding configuration, including the six extra
-5 turns, produces approximately 1.7 x 10 T/A of vertical field at the
minor axis. In addition, the measured value of the VF inductance is
40
<z
u GJ
00
X OJ
CJ
u •í-i
o
>
UD I
o
41
ÍT,
± o • «D
^
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O
'T
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V
O • æ
O • «o
lO
340
O (Vj
ro
§ +
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NGL
<
S
eld
>+-
r— (T3
rti
OJ 2»
or
the
M-
o
«/) >
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^» íC S-
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o + (O
CVJ
O
0<M o o
H do "ivasaiNi
OJ
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4 -O
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o Q.
<c
I
crs
42
OJ -t->
; % <fi
î : o w UJ
(r (T Ui <r-
o
(O
ir o Ui
o
o CM
H iO nVM931NI O
CVi
o Ui
Uir^
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Zo: o *
3 E S-o
<+-s> OJ E s-o
4 -to c «3 s_
-M
C o OJ
x: +-> -o £Z 03
3
^ M-
f —
(T3 S-
X o s Q . Q . rt3
O -(->
-
to c s. 3
-M
S--i-> X O)
X •r—
(/) t+-o sr o
•^ • ( -> • 1 —
T3 o>-o (U
+J £=
•^ 3
(U J = •+J
•+-o
-t->
o (— Q .
"O 0) c o <
•r— -M •r--M s - • rtî Q . -
<L)
(O
OJ . C - f J
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• -M C o; e <u u «3
r—
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CU S-
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cn -»-) c t—
XD c Í2
ZJ û . ^ O u OJ
CX) I
ro
O )
43
350 yH ((3 400 Hz).
VF System Calculations
The calculation of B for equilibrium must follow from several
general assumptions regarding small tokamaks;
(a) A uniform current density in a plasma of radius b.
(b) B^ » Bp
(c) 200 eV < (T + T .) < 400 eV
19 -3
(d) n = 1 X 10 m (particle density),
which are true for a modest size, ohmically heated device. Under
these general assumptions, the magnitude of B can be calculated from
Eq.(18),
where
B V 4Tr R
2TT L,
D 6 u„ - 3/2
L. = STT
(19)
is the internal plasma inductance/unit length, for a uniform current
density, and
^e = nk (T + T.) 2TT b'
e 1
10-^ l' (20)
23 is the poloidal beta^"* from Eqs.(3) and (8) of Ch. II.
Given the realistic parameter set for Cymator,
I = 20 kA peak
B. = 1.0 T » B^ t p
44
Tg+T. = 300 eV = 3.48 x lO^ ^K
n = 1 X 10^^ m"" ,
Eqs. (20),(19), and (18) yield the values
Bg = 0.756,
L . = 50 nH/m,
and
B^ = 1.36 X 10"^ T
for equilibrium. The 1.7 x 10 " T/A ratio of the B coil requires
lyp = 800 A. In order to produce similar I^p and IQ,^ waveforms, the
VF coil was connected to a 500 pF fast bank and a 110 mF slow bank.
The charging voltages of the fast bank (800 V) and the slow bank (160 V)
are required to produce the 800 A VF current and waveform necessary for
plasma equilibrium. It is significant to note that the finite vertical
field penetration time through the stainless stee vacuum chamber
requires the initiation of the VF current 300 - 400 liS before the start
of the OH current.
Radial Field (RF)
The vertical plasma column motion, due to erroneous fields, can be
controlled by an externally applied radial field. Since most of the
vertical drift is caused by slight toroidal field imperfections, the
external radial field should follow the same temporal development as the
toroidal field. A simple inductive current divider, which samples the
toroidal field current, can be used to drive the radial field coil. A
two (or four) hoop system, described in Ch. II (Fig. 2-6), can provide
45
the necessary radial field for plasma equilibrium.
The Cymator Radial Field Design
The two hoop radial field system, for the Texas Tech tokamak, is
described in Ch. IV (Fig. 4-19). The two turn RF coil produces
'v 4.0 X 10" T/A of radial field at the minor axis. The magnitude of
the radial field is programmed by the selection of a toroidal field
shunt inductor, which typically diverts 1.0 kA of toroidal field current
through the radial field coil.
CHAPTER IV
CONSTRUCTION
Introduction
A circular cross section tokamak (R = 0.457 m, a = 0.162 m) was
constructed at Texas Tech University under the general design procedures
outlined in Ch. III. The basic device includes a 600 turn TF coil and a
poloidally distributed, air core, OH transformer. The 130 kJ toroidal
field bank, 18 kJ ohmic heating bank, and 2 kJ vertical field bank
comprise the three major energy storage systems. The torus-support
devices include a 35 kHz, 40 kW discharge cleaning oscillator, a solid-
state toroidal field pulse generator, and a simple remote control net-
work. The entire facility was operational after a 13 month construction
period with federal and university grant expenditures of $57,000.
Vacuum Chamber
The basic toroidal vacuum chamber consists of four type 304 L,
stainless steel 90° elbows. Each elbow has a major radius (R) of
45.72 cm (18.0 in.), a minor radius (a) of 16.19 cm (6.38 in.), a
0.4 cm (0.156 in.) wall thickness, and an aspect ratio (R/a) of
approximately 3. Prior to assembly, the 90° elbows were individually
milled to accommodate either circular or rectangular ports.
The various types of ports attached to the toroidal chamber are
shown in Fig. 4-1. Each port was manufactured entirely of 304 stainless
steel with interior welds at the chamber walls and port flanges
(Details of all port assemblies are given in Appendix B.). Figure 4-2
46
47
íO
lO
2: o
CVi E u
Vi </) Ul O o <
tn c o to c o
CNi
E u r-V m
<
AR
E
cn (0 Ui u 0 <
< i—
CNi CVi
</> H-(S
£ u. 0 (T
^ Z O Z
^ >-0
§ ^ —
0 <n Ui ? o
S-0 CL
' . 4 —
^ OJ
' > í
-M
. C 0 03 0
M—
0
' « ^ ^ •3^ (T3
• ^ <-
. '—
<^
. i ^
r
rz
</> i / ) O) 0 s—^
TJ
• 0 ^ (T3
cn
48
PORT C
PORT E
POflT a
PORT A
NOTE. !"i í2" STAíNLESS STEEL ROD TO SUPPORT TT3WJS FTROM THE BOTTOM (S ftOGS T o m u
Fig. 4-2. Location of ports on the vacuum chamber (viewed from above).
49
illustrates the location of the various ports, viewed from above. The
vertical port types are symmetric with respect to the major radius. The
90° elbows are joined with full penetration welds to produce two 180°
sections. These sections are joined by two 50.8 cm (20.0 in.) dia
304 stainless steel vacuum flange assemblies with 0-ring seals.
These two flange assemblies are each comprised of three major
components. The male and female stainless steel halves are separated
by a 0.48 cm (3/16 in.) thick annular bakelite insulator to prevent
induced toroidal currents in the chamber walls. Each three-layer flange
is joined by twenty-four 1.27 cm (0.5 in.) dia ,threaded G-10 fiber-
glass rods. The rods compress the 0-rings within the flange assemblies,
by means of aluminum nuts and washers, to form a high vacuum seal. The
female flange of Fig, 4-3 depicts the 24 holes and 0-ring lathe work
that protects the 0-ring from the plasma and seals the machine (see
Appendix B).
The entire chamber is supported upon six 2.54 cm (1.0 in.) dia
threaded, stainless steel rods. These rods are seated, with nuts and
washers, on the six plastic insulators inserted in the stand of Fig. 4-4,
These insulators prevent induced currents in the stand and chamber.
Measured from the major radius, the stand supports the chamber 130.18 cm
(51.25 in.) above the floor. Note from Fig. 4-4 that half of the
supporting stand rolls away for machine disassembly.
The vacuum chamber walls are cooled with chilled water flowing
through 0.32 cm (1/8 in.) QD copper tubing. Fifteen meters (50 ft )
of continuous copper tubing per 180° is arranged in parallel rows and
50
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52
secured with silver solder at 15.24 cm (6.0 in.) intervals. The
soldering was performed with the torus under vacuum to prevent oxidation
of the clean inner walls.
Between the rows of copper cooling coils, 5.08 cm (2.0 in.) wide
strips of 0.32 cm (1/8 in.) thick neoprene insulation were cemented to
the vacuum chamber. These strips provide a soft cushion for the two
layers of toroidal field coils. Over the neoprene and cooling coils,
the entire chamber is poloidally wrapped with two layers of 5.08 cm
(2.0 in.) wide fiberglass reinforced mylar tape (3.0 kV/ layer, 155° C)
for insulation. The neoprene strips and insulating tape, permit the
toroidal field coils to be wound directly on the chamber.
Vacuum Station
The vacuum station is basically the three pump system shown in
Fig. 4-5. The mechanical roughing pump and the 2400 l/s oil diffusion
-8 pump maintain a base vacuum of approximately 2.0 x 10 Torr. During
idle periods, the vacuum is retained by the 140 £/s, stand-by ion
pump. The stainless steel bellows, between the vacuum chamber and pump
station, allow small movements of these structures without endangering
the vacuum quality. At present, O^, Ar, and He are metered from a
central gas manifold and delivered to the chamber via 0.64 cm (1/4 in.)
OD stainless steel tubing. The filling gas is gated at the torus by
the valve shown in Fig. 4-5. The chamber pressure is measured by a B-A
(Baynard-Alpert) style ionization gauge and an auto-range selecting
millitorr gauge. The residual gas spectrum is monitored, during dis-
charge cleaiing, by a monopole residual gas analyzer and strip chart
58
0.2 T/cm
Toroidal Field
2 ms/cm
Fig. 4-8. OsciUoscope photograph of B^
53
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j = u
>
<T3
-o o S-
c •o
c c o
u > o
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54 recorder.
Toroidal Field (TF)
The toroidal field coils are actually four separate 150 turn coils
of 1/0 AWG cable wound directly on the insulated chamber. Each 180°
section is wound with two of the four coils connected in series to pro-
duce a double-layer, 300 turn coil. Likewise, these two 30O-turn coil
sections are connected in series to complete the 600-turn TF coil system.
The inductance of an N turn, uniformly wound toroid, of major radius
R and minor radius a, is given by
L^P = u^ N^ [R- (R2 - a^)^/2]. (1)
For the particular dimensions of R = 45.72 cm and a = 16.40 cm, Eq.(l)
yields L.|.r = 13.76 mH. The stainless steel vessel contributes an
insignificant field penetration time constant of 400 us, and causes
a slightly lower inductance (L. p = 13.2 mH; f <_ 50 Hz) than p'^eviously
25 calculated.
Approximately 615 m (2018 ft.) of stranded, 1/0 AWG size, 600V THnN
cable was used in the fabrication of the TF coil. For this length of
cable, the series connected coil system has a total resistance of 194 m:..
The toroidal field coil is energized by a 130 kJ, 0 - 10 kV
capacitor bank. The four bank sections, 0.65 mF each, are coaxially
connected (RG-17 A) to a central ignitron switching system. A single
20 kV, 100 kA ignitron (Type A) switches the entlre 2.6 mF bank to the
TF coil through a custom m.ade PVC-insulated transmission line. At the
current maximum, a 25 kV, 300 k.A ignitron (Type D} crowbars the 'f
system. Details of the switching system, capacitor arrangements. aro
55
PVC transmission line are given in Appendix B.
The toroidal field bank is charged by a three phase, 24 kVA,
0 - 1 0 kV, constant-current (2.0 A max) supply. The solid-state supply
features a motor driven Variac, programmed upper and lower voltage set
points, and remote charge controls. A circuit diagram is shown in
Appendix B.
Figure 4-6 is a block diagram of the toroidal field systen. The
2fi results of a simple SCEPTRE computer simulation of the Tf system are
shown in Fig. 4-7. These results compare favorably with the measured
values of TF current and B. in Fig. 4-8.
Ohmic Heating (OH)
The ohmic heating transformer is a poloidally distributed, air core
solenoid designed to maintain BQU = 0 within the plasma volume. To
accomplish this, the computer program described in Ch. III (and
Appendix A) dictates an optimum distribution of the toroidal sheet
current density as a function of poloidal angie. An N-integer equi-
partition of this toroidal sheet current yields the angular placement cf
N turns of wire.
Figure 4-9 is an example of the eight OH transformer coil forrr.s,
illustrating the OH winding distribution. The form is an annular
insulator of 1.27 cm (0.5 in.) thick XX paper reinforced plastic with
forty-eight 0.95 cm (3/8 in.) dia holes. Each hole supports a single
turn of^4AWG, 600 V THHN cable. Approximately 87.8 m (288 ft ) of
cable was used in ccnstructing the 500 yH, 104 mf.: OH transformer.
In order to seoarate the five outer turns of the OH tr^nsfcrTier
UJ
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57
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58
^^^^ww:
0.2 T/cm
Toroidal Field
2 ms/cm
Fig. 4-8. Oscilloscope photograph of B^
59
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60
(and free the inner chamber for disassembly),a high current, low re-
sistance connector was devised. This connector is simply a 1/4 in. ID
27 SWAGELOK brass union which joins two pieces of #4 bare cable as if
these pieces were copper tubing of the same OD (see Appendix B).
The eight annular OH transformer forms must be supported directly
on the toroidal field windings. Note that the cross section of the TF
windings, Fig. 4-10, is not circular but "egg-shaped" and slightly
inconsistent over the toroid's circumference. To support the trans-
former forms, the crescent air bags of Fig. 4-11 were constructed to
cushion and secure these eight forms, regardless of variations in the
TF winding shape.
The primary current of the ohmic heating transformer is supplied
by a double-stage, 18 kJ capacitor bank. The "fast bank" stage is a
120 yF, 0 - 5 kV capacitor section that produces 2.5 kA in 400 ys. At
the current maximum of the fast bank, the 93.0 mF, 0 - 600 V electro-
lytic "slow bank" ?tage is passively (diode) switched into the circuit.
The entire bank is diode crowbarred at the slow bank current maximum,
providing the circuit is underdamped. Details of the ignitron
switching and passive diode crowbar stack are presented in Appendix B.
The solid-state "fast" ånd "slcw" bank power supplies, unlike the
TF supply, are resistively current limited with a programmed voltage
maximum for each stage. Both OH bank stages can be remotely charged
and dumped from the screen room.
The ohmic heating system, previously described, is outlined in
Fig. 4-12. A SCEPTRE computer simulation of the OH system (without
61
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63
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64
plasma coupling) is shown in Fig. 4-13. The measured "no-plasma "
current of Fig. 4-14 compares well with the computer-predicted results.
Vertical Field (VF)
The vertical field halts radial expansion of the plasma column
during tokamak operation. The vertical field is created by two sets of
windings, parallel to the OH windings, carrying equal but oppositely
directed currents. The six additional outer windings of Fig. 4-15 add
the necessary VF curvature described in Ch. III. The entire 38 turn,
350 yH, 90 mfi VF coil was wound with #4 AWG, 600 V, THHN cable, and
supported by the OH transformer forms. The twenty outer windings of the
VF coil åre spliced with SWAGELOK fittings to access the mobile half of
the vacuum chamber.
Since the vertical field is required to contain the radial expansion
of the plasma column, the current producing the VF should be a scaled
reproduction of the plasma (ohmic heating) current. To obtain this
waveshape, a dual-stage, 2.0 kJ vertical field capacitor bank was
constructed. The fast bank stage is a 500 yF, 0 - 1 kV capacitor section
that produces 600 A in 650 ys. Just as in the ohmic heating bank, the
110 mF, 0 - 200 V, electrolytic slow bank stage is diode switched into
the circuit at the current maximum of the fast bank. The entire bank is
passively crowbarred Sfnce the circuit is underdamped.For details of the
capacitor sections (fast and slow), the ignitron switching, and diode
configuration, refer to Appendix B.
The solid-state VF power supplies are resistively limited with a
programmed charging voltage for both the fast and slow capacitor stages.
65
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66
700 A/cm
Fig. 4-14.
1 ms/cm
Oscilloscope photograph of the current in the OH transformer without plasma.
67
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o u "O
u
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68
As in ohmic heating, the VF supply and capacitors may be remotely
charged and dumped.
A block diagram of the vertical field system is shown in Fig. 4-16.
Figure 4-17 is the SCEPTRE computer simulation of the VF system. The
computer-predicted VF current profile closely resembles the measured
profile of Fig. 4-18.
Radial Field (RF)
The radial field is an error correction field compensating for the
imperfections in the toroidal field geometry. The ports, which protrude
through the TF coil, cause a perturbation (ripple) in the toroidal
field. This results in a local leakage of TF flux near the ports. The
total leakage field may be idealized as a superposition of small vertical
and radial component fields on a perfect toroidal background field. The
vertical components of this field are overshadowed by the applied VF;
yet the radial component, if left unchecked, can cause a vertical drift
of the plasma column.
The two hoops of #4 cable (Fig. 4-19) produce enough radial field
(5 - 10 X 10" T) to cancel the plasma's vertical drift. The current
in the RF hoops, for proper compensation, should be a scaled reproduction
of the TF current waveform. Therefore, the 4.0 yH, 2 mfi radial field
system samples current from the toroidal field coil via an inductive
current divider. A selection of several shunt inductors (7, 13, and
25 yH) program the Rf (radial field) current necessary to stabilize the
plasma.
The basic outline of the radial field system is given in Fig. 4-20.
69
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71
145 A/cm
Fig. 4-18.
1 ms/cm
Oscilloscope photograph of the vertical field current.
72
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OJ
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74
A SCEPTRE program of the toroidal field system was modified to include
the radial field. Figure 4-21 illustrates three computer-predicted
radial field currents for the three shunt inductor values.
Preionization (PI)
Early investigations of tokamaks in the Soviet Union demonstrated
that preionization of the fill gas prior to the initiation of the OH
current. eliminated electron runaway discharges.^^ Many small U.S.
22 tokamaks employ an L-C ringing current in the OH windings to commence
early ionization. In the case of Cymator a separate preionization
coil was constructed. Flux parallel to the major axis, without regard
to Bpj = 0 in the plasma volume,is the only requirement of the PI system,
To avoid sharing the OH transformer, a separate 515 yH, 102 mfi
preionization solenoid was constructed. The 60 turn solenoid of #8 AWG,
600 V, THHN cable, on a 25.4 cm (10.0 in.) dia, PVC coil form, is
supported by the eight OH winding forms shown in Fig. 4-22. A 0.2 yF
capacitor bank is charged from a 0 - 15 kV power supply (Appendix B) and
ignitron-switched to the PI coil. The resultant 16 kHz osciliations
last for 1.8 to 3 ms.
The block and circuit diagrams of the preionization system are
shown in Figs. 4-23 and 4-24 respectively. The response of a single-
turn voltage loop to the PI flux is depicted in Fig. 4-25.
Discharge Cleaning Oscillator (DCO)
Preserving a low impurity level in small tokamaks is a constant
problem. The efficient elimination of low - Z impurities, through low
75
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79
50 V/cm
Fig. 4-25.
1 ms/cm
Oscilloscope photograph of the single loop voltage response to the pre-ionization flux.
80
energy discharges (< 10 eV), has been previously demonstrated by Oren
29 and Taylor. The plasma necessary for the conversion of oxygen and
carbon to HpO and CH, at the vacuum chamber wall may be produced by a
low frequency oscillator.
The 40 kW, 35 kHz oscillator is a Colpitts design employing a
4CX 15000 A tube and utilizing the preionization coil as the tank
inductor. During operation a small negative bias supply (-400 V screen;
0 to -700 V control grid) determines the off-state voltages of the
screen and control grids. To initiate the oscillation, the screen grid
is gated with a square, 0 - 1 kV, 5 - 1 5 ms, periodic screen pulse. The
plate current is supplied from a 66.7 yF, 15 kV capacitor bank, which is
continuously charged from the TF power supply. A small toroidal field
-2 pulser, described in Appendix B, provides a 70 ms, 6 x 10 T (peak)
toroidal field during the discharge cleaning pulse. With the help of
this small toroidal field, the DCO induces an 800 A p-p pulse of current
in the filling gas. Appendix B contains a detailed circuit description
of the oscillator chassis, screen grid pulser, and bias supply.
The basic block diagram of the DCO is presented in Fig. 4-25.
During normal oscillator operation, a constant flow of 0«, with a
-4 pressure of 2.8 to 8.4 x 10 Torr, is provided. The bursts of plasma
current recur at 1 - 2 pps for 7 - 1 0 hours before tokamak operation.
Figure 4-27 is an example of a single plasma current pulse and the
associated toroidal field pulse.
Systems and Control
The union of the systems previously described is schematically
81
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UJ LLÍ
rr o co
S -
o -M 13
U (A o
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S -
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U
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82
200 A/cm
0.02 T/cm Bottom
2 ms/cm
Fig. 4-27. Oscilloscope photograph of:
(Top) Single plasma current pulse from DCO.
(Bottom) Toroidal magnetic field pulse for discharge cleaning.
83
illustrated in Fig. 4-28. Large systems, such as this one, should
utilize a remote control network to make daily operation a humanly
comfortable task.
In particular, the toroidal field, ohmic heating, and vertical
field banks are all remotely charged. A transistor-activated, D.C.
relay module, within each bank, simultaneously receives a +5 V charge
command from the screen room. A mechanical 30 s timer and 5 V power
supply presently provide the necessary charge command signal. However,
the bank control modules can directly interface with a more complex
digital command device. In the case of the vertical and ohmic heating
banks, an additional +5 V signal simultaneously controls the dump
circuitryof each bank. The basic block diagram of the control system
is given in Fig. 4-29, and a more detailed description of these control
modules can be found in Appendix B.
A Pictorial Review
To enhance the description of various components, the photographs
of Fig. 4-30, (A)through (C), present several views of the completed
torus. Figure 4-30 (A) is a view of the torus from above showing
the deck of the preionization solenoid resting on the eight OH
transformer forms. Figure 4-30 (B) is the same view without the
preionization solenoid, which shows the poloidal variation of the OH
winding density and one of the radial field hoops. The final photograph,
Fig. 4-30(C), is a side view of the torus with the vacuum station, sup-
port stand, and wall-mounted gas manifold.
84
a o
i • v> N
2 (L
fsl (D
o
c
OSo
û < d o o
< 0 2 ^ 0 " S<íw2 (9
o SÍ2>
o
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85
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86
Fig. 4-30(A). Top view of the torus with preionization solenoid.
Fig. 4-30(B). Top view of the torus without preionization solenoid. Note the radial field hoop and poloidal distribution of the OH transformer windings.
87
Fig. 4-30(C). View of the torus and vacuum station.
CHAPTER V
DIAGNOSTICS
Introduction
The early development of a basic diagnostic system, during the
machine's construction, can minimize the time spent in plasma
optimization once the facility is complete. Common voltage and current
measurement techniques can be employed to describe the "external"
performance of the tokamak. These gross exterior measurements include
the toroidal, ohmic, and vertical field bank currents, loop voltage,
plasma position, B., and the magnitude of the plasma current. Devices
to measure these quantities can be developed during the design and
construction of the tokamak and installed as the various systems near
completion. Prior to the initial tokamak operation, diagnostic devices
should be assembled or obtained to measure the "internal" machine
behavior. These devices should include a microwave interferometer to
determine the electron density (n ) and a spectrometer to monitor
impurity line radiation.
Doctor Rodney C. Cross, a visiting professor from the University of
Sydney in Australia, designed and constructed all the external diagnostic
equipment presently in use on the Texas Tech tokamak. An NSF technical
report (NSF ENG-7303941-2) was compiled by Dr. Cross describing these
diagnostic devices. His report, and supplementary references, provide
the basic foundation of the information presented in this chapter.
88
89
Major Field Currents
The currents from the ohmic, toroidal, and vertical field capacitor
banks to their respective coils are all monitored with current
30 transformers. The basic current transformer circuit is shown in
Fig. 5-1. The magnetic field from the current, I, induces a voltage,
VQ, across R^, which is proportional to I provided (r + R ) « uL where
L is the inductance of the transformer and w is the radian frequency of
I . From Fig. 5 - 1 ,
and
i = f (1)
* = W (2) 27r R
is the to ta l f l ux l i nk ing a toroidal N-turn transformer of major radius R,
and a cross sectional area of A. The inductance of a toro id with a
31 large aspect ra t i o can be approximated by
UoJI^A (3^ L 2 7T R • ^^'
From Eqs. (1) - (3 ) ,
i = (4)
and the output voltage
is a scaled reproduction of the current to be measured.
Since the frequency of the current to be measured is small
90
S-3J
O < + -co C
s.
c Ol s . S-
u
o C7)
o
<rs
S-
o
«• «• I I
u
c OJ
ÍT3
> 3 o-
I
C7)
91
(145 - 168 rad/s)> a transformer with large L and small r is
desirable. Such a current transformer was constructed from a large
Variac autotransformer with a laminated steel core. As an example,
the ohmic heating bank's current transformer was constructed from a
213-turn Variac with a cross sectional area of 5 cm x 3 cm. The wire
resistance r = 0.85 n and the external resistance R = 0.152 í2 yield 0 -^
an Y~ í atio of 1.40 kA/V. This current transformer has a lower -3 dB 0
frequency of -v 0.1 Hz and will not saturate provided the product of
average current and pulse duration remains below 90 amp-sec.
Loop Voltage
The voltage response of a single loop in the plane of the major
radius is an external measurement that aids in the calculation of useful
plasma parameters. The loop voltage response can indicate the average
plasma resistivity (if the magnitude of the plasma current is known) and
identify the descriptive instability by the characteristic negative
voltage spike.
For the Texas Tech tokamak, the location of both loop voltage probes
are shown in Fig. 5-2 (A). The plasma is assumed to have a radius (b)
of 10 cm and a uniform current distribution of magnitude I and to be
centered at the major radius R. Consider the response V of the loop
at r-j,
where S is the vector sum of ff^^ and ff^ fields. For convenience, the 5, On y
5 L,> 3nd t. fields are assumed to intersect normal to the major radius Un 6
92
tc UJ
< X
Q. o o
r—
u-o
^ -M • r -
2 CO
«3 E to <T3
p —
. s> (L)
Q . ^
d) oj x: C - Û
o •^* 4-> <T3 U O
!
^--•» < ""—^
• CM
i r j
o s.
-M
O a.-M (U O î <t>
-M
o >
-M U OJ Q . ( / }
o; S-
£ 03
U
S 3 3 U !T3 >
• o c (T3
4-> Q . c O O ) CO O) O C Q . i . — • • - O S-
(/1 O 3 3 x: u o
-M to U -M OJ -r- C
^ Æ S_ <D O O -M £
S- C (T3 Q . OJ r—
U • ! -(U C M-CT) O «3 U (T3
d) • o o
^ r - O -O CQ O 2 C
> -M (T3
C\J I
CD Oî
93
plane, which permits computation with scalar field values. The emf in
the plasma is given by,
-" = Í ío ^e 2-'"dr = Rp , (7)
where R is the average plasma resistance. Combining Eqs. (6) and (7),
the calculation of the plasma emf may be written as,
^'"' = h-ít /r, e ^'''' (8)
since Bgjj = 0 between loops r.. and r^. The integral of Eq.(8) can be
approximated in three parts.
Imagine two concentric circles located at position r-. and the
plasma edge (r^) as shown in Fig. 5-2(B). The flux linking the circle
at r-j from a filamentary current I at r = R is given by,
(J)-, (r < 25.7 cm) = M^ I (9)
32 where M-, is the mutual inductance of two concentric hoops and, in this
case, has the value
M-, = 0.748 i R. (10)
(computed in Appendix D as an example)
Likewise, the flux linking the circle at radius r^ = R - b is given by,
(1,2 (r < 35.7 cm) = M^I (11)
and
M^ = 1.511 y R (12)
(also calculated in Appendix D)
94
for these dimensions. Hence the flux between r-, and r« = R - b is
simply,
(j>2 - (p-i = <\>2 (25.7 cm < r < 35.7 cm) (13)
o r
<}>3 = 0 . 7 6 3 po RI = 4 . 3 8 x 10"'' I . (14 )
The flux within the plasma of radius b (10 cm) must now be determined.
Neglecting toroidal effects, the plasma can be modeled as a straight
wire of radius b and length l (Fig. 5-3). For some radius r' within the
wire (0 < r' < b), B. is given
B, = '-^-^ (15)
and the poloidal f lux is necessari ly,
*4 - !l%^ dr' (16)
-. ji> ^.ZA^ r'dr' ( 17 )
^o I l (18) 47T
If a slightly smaller radius than R (40.7 cm) is chosen to fit the
linear plasma model to a curved geometry, the average £ becomes
£ = 2Tr (0.407) cm, (i9)
and the flux within the toroidal plasma is simply
^. (35.7 cm < r < 45.7 cm) = 2.56 x10 " I. ^ (20)
95
- • - )
CT> C <v
-o c «T3
co
" O
S-
O
OJ X3 o
«2 «/) (T3
s-
(T3 S-
-4-»
(>0
I
O í
96
The approximation of the integral of Eq.(8) is therefore
/^ BQ 27rrdr - <^^ + ^^ = 6 . 9 4 x 10"'' l . (21)
With this information and Eqs.(7) and (8), the response of the voltage
loop at r-,,
V^ = R I + 6.94 X 10"^ , (22)
can be expressed as a function of the plasma resistance. By a similar
argument, the response of the loop at r^ is given by the relationship
2 -Vk í e 2-^ (23)
= Rpl + 5.4 X 10"^ . (24)
Note that if one seeks the plasma resistance when -nr = 0, the
evaluation of the integral in Eq.(8) is not necessary. However Eqs.(22)
and (24) are necessary to calculate the plasma resistivity (temperature)
as a function of time.
Plasma Position
Since the location of the plasma is controlled by external fields
(B and B ), a diagnostic method to determine the plasma location is
necessary. The early measurement techniques employed a set of Mirnov
coils that indicate relative plasma displacement vertically or
radially as a function of time. A cosine Rogowski coil, with toroidal
34 corrections, and a saddle coil are described by Hugill and Gibson as
another position measurement scheme. Similar poloidal field
97
measurements on Alcator utilize an entire system of pick-up coils
with cos me sin me winding density variations. With this coil system,
plasma position and a Fourier analysis of the poloidal magnetic field
(outside the plasma) is instantaneously available. A position diag-
nostic, however, need not be a magnetic measurement of poloidal field '3'] OC
variations. For example, the Microtor device * detects the intense
luminosity of the plasma edge as an indication of position. For the
tokamak at Texas Tech, a simple sine and cosine coil pair provides the
vertical and radial position information. This coil pair is shown in
Fig. 5-4.
The poloidal flux generated by a toroidal plasma current, of major
radius R, is greater near the major axis (r < R) than at the outer plasma
boundary (r > R). To compensate for this toroidal effect, the cosine
coil requires more turns near e = 0° than at e = 180°. A calculation
of the relative field magnitudes of a filamentary current of radius
R = 45.7 cm, evaluated at the points R - 20.6 cm; e = 180° and
R + 20.6 cm; e = 0°, will yield the necessary correction factor.
Table 5-1 is a normalized calculation of the B and B component fields
relative to B. (at the point P) for a current carrying filament of
radius R. For the cosine coil (z = 0, R = 45.7 cm) the B^ field is
3.93 times larger at e = 180° than at e = 0°. The resulting correction
factor requires the cosine coil to have - 3.9 times as many turns/cm
at e = 0° as at e = 180°. Several pairs of points can be calculated,
from Table 5-1, to yield relative winding densities for all
e e [0°, 360°]. Similarly, the sine coil, which has a sinusoidally
cn
98
UJ
o z
E o
(O o' <VJ
<o ^™
_ i
CO
II
2 o _ (Ô o 0 .
^ u < UJ
u. o o: (u H-UJ
< O
<o Q .
,— • r "
o u (U c
•r— CO
o t_)
XJ c <T3
OJ
c • r " (>0
- » - ^ «C ^—'
• ' d -
1 uo
• a> u_
co >
^
-o c ÍT3
X
(/1 >
:^
t+-o
• M
O rw> ( 2 .
..— ca
TABLE 5-1
A TABULATION OF THE NORMALIZED B^ AND B^ COMPONENT FIELDS
99
z - 0
r/R
0.000
.0557
.1270
.1716
.2922
.3820
.5195
.5345
.7522
.3181
.8961
.9387
.9802
1.000
1.020
1.Q55
1.115
1.222
1.329
1.576
1.925
2.618
3
T
2.973
2.776
2.652
2.378
2.188
1.920
1.711
1.503
1.386
1.240
1.153
1.060
1.000
.9401
.3465
.7699
.5458
.5547
.1102
.2895
.1691
At P,
wnere
r » R
Z/R
2.000
1.3094
1.0000
0.6667
0.4588
0.2358
0.2010
0.1098
0.0633
3.0200
0.0000
(P ' z]
0.3560
0.-1559
0.-1324
0.4654
0.4131
0.3290
0.2680
0.13Q1
0.1214
0.G499
0.0000
0.243
0.4410
0.5717
0.7367
0.3440
Û.92Í6
0.9570
0.9847
0.9918
0.9989
. 0000
B,-.aãg
ir-^B^
39 = "- i i
4t 2 = 0
At j = 0
At r = ^
p^R-r
a - ^ - r
•»= l
(r<^)
' r > w 1
ZTP
100
distributed winding density, must correct for the same toroidal effect
by using fewer turns between e = 90° and e = 270°.
Each 300 turn coil is wound on a 130 cm length of plastic tubing
with a minor radius of 0.69 cm and a major radius of 20.6 cm. To insure
that each coil responds primarily to changes of plasma position rather
than changes of plasma current magnitude, the winding direction of each
coil appropriately reverses direction (Fig. 5-4). Both coils were
calibrated by spatial variations of a 1.0 A, 50 kHz current, to
simulate a moving plasma. The output voltage from each coil will be
VQUT = a î ' ' 2 ^ '
where K is a coefficient determined during calibration. A plot of
K vs X and K vs y for each coil is also shown in Fig. 5-4. With a 50 ?.
termination the -3 dB frequency for both coils is roughly 300 kHz.
These coils not only respond to the change of plasma position, but
to the various externally applied fields (B^, B^, B^ leakage,and BQ^)
as well. Therefore, the coils must be compensated to yield a zerc
output voltage when the plasma is centered and stationary. Each external
field current is sampled, scaled, and electronically summed with the
sine and cosine coil voltages such that the output is zero with no
plasma. This compensated signal is then actively integrated. A block
diagram in Fig. 5-5 schematically illustrates the electronic compensation
and integration network.
10-
u r^V>
í. o
•M <U
c c o
13
c d) Q. E o u
<u c
•r-
o
<T3 S _
uo I
O )
102
I
(T O
tn
< <n z Ui CL o o
* £ •*< < h 3 d Z U. Ul _
£ 1
( )
g 1 z Ui
(
|AN
O
FIL
TE
R
)
H* I 3 Z UJ
o •M
C o
<o (/) c d)
Q .
5 u o u <u c </> o u
^ -o E <T3 %-Oî fT3
CQ
l O I
t f )
Oî
s
cn
o o a:
103 Plasma Current and Toroidal Field
To monitor the magnitude of the plasma current, a passively
integrated Rogowski belt was constructed. Figure 5-6, from Huddlestone
30 and Leonard, is the equivalent circuit for a Rogowski pick-up belt
with an RC integrator. The circuit equation for Fig. 5-6
reduces to
^ = iR„ - 1 ; î i dt (27) dt 0 C ^o
when R >> OÛL. If the time of desired integration, t, satisfies
t << R C, Eq.(27) can be reduced to
^-r% (28) 0
so that the output voltage, V , across C is given by
0
Since the Rogowski bel t is basical ly a N-turn toroidal transformer, the
to ta l f l ux l i nk ing the toro id from the current I (F ig. 5-7) is simply
* ( t ) = ^ (30)
where A is the winding cross sectional area and R is the major radius of
the Rogowski belt. Equations (29) and (30) define the voltage response
from the belt as
^ . o NAI (^.. ^O " 27r R R C* ^ '
104
f "f o > 11
(1) .Q
C zn o ce: o
oz
"O t"o
3 u i-u
c <v
03
> zz
U3 I
cn
105
c O
•r— •M fC í. (31 OJ
4-> C
OJ
>
to <T3 Q.
OJ
o
dJ
u 2 E co 03
« 4 -
O
(T3
s . O í 03
I LO
O î
106
The particular Rogowski belt constructed is a 685 turn torus, wound
on a 133 cm length of flexible tubing, with a major radius R = 21.17 cm, 2
a cross sectional area A = 1.5 cm , and an R C integration constant of
200 ms. For these parameters, the probe has a 2.06 mV/kA sensitivity.
The measurement of the toroidal field, B^, is accomplished by a
simple 12 turn coil wound outside the toroidal field coils and
passively integrated. Under the assumptions R >> wL and t << R C,
defined for Eqs. (27) and (28), the integrated output voltage is given
30 by the relation
NAB. o = F ^ ' (32)
0
where N is the number of turns and A is the cross sectional area
encompassed by the N turns. 2
The 12 turn toroidal field probe has an area A = 908 cm with an R C integration time constant of 1.089 s. - . Equation (32), for these
parameters, yields a 1.0 V/T probe sensitivity.
Microwave Interferometry 37 Standard microwave interferometric techniques can be used to
determine the average electron density (n ) in a tokamak plasma. If the
plasma is free of violent instabilities, the electron density within the
discharge wiU change slowly with time. Since high temporal resolution
is not a requirement, sophisticated density interpretation schemes, such
37 38 39
as the "zebra-stripe" and digital phase comparator, ' are in
common use.
The basic interferometer for the Texas Tech tokamak is shown in
107 37
Fig. 5-8. The zebra-stripe technique, outlined by Heald and Wharton,
is employed to decipher n as a function of time. A block diagram of
the zebra-stripe electronics is given in Fig. 5-9. Once the present
zebra-stripe measurements have been accomplished, a slight modification
of the electronics will be made to convert the same interferometer to
a digital phase comparator scheme. With a digital phase comparator, a
direct indication of n vs time is given without the need to count the e ^
zebra-stripe fringes.
Spectroscopy
The plasmas in small research tokamaks are generally of low
13 3 density {'^ 10 cm ) and high temperature (200 - 400 eV; T + T.)
with moderately long confinement times (10 - 50 ms). Due to the high
plasma temperature, special ultraviolet (UV) windows and UV spectroscopic
techniques must be employed.
Heavy impurity atoms, such as oxygen, carbon and chromium, are
injected into the hot plasma from the plasma wall interaction. This
impurity influx not only influences electron density, temperature, and
discharge stability, but provides an early temporal estimate of the
plasma temperature evolution. Investigations by Oren and Taylor on the
40 Microtor/Macrotor tokamaks illustrate the ionization sequence of
oxygen (01 thru OVI) as the machine's temperature increases with time. 0
If a spectrometer of sufficient resolution {<_ 0.5 A) is available, 30 41 42
Doppler-broadening ' of these same impurities may be used to
determine the ion temperature. Spectroscopic diagnostic facilities for
the Texas Tech tokamak are presently under construction to examine the
108
MJ<o UJCO
^iP O-11}
LJ
<
^ :
U J O
U J S U.Í2
2
f 1 58 S^ «n
UJU
Ji
11
*<
UJ
a. Q: • -co I
< c: CD UJ N l
Q:
a-gj
5 oi
V -r
ÔJ 4-)
c d) > 03
O r
U
u tn
CO I
o i
0
cr UJ - j - j
U J Z O. <D
<D <n
109
z o
o o o
c;
>
tn
.
U
c o s-
+J u <D
<V
<U Q .
to I
<T3 S -
- Q O) M
Z
IC UJ
^ a.
OJ
o
03 S-
íT3
U
o CQ
(
O î
UV impurity radiation. At present. however, only the visible D.
{4859.6 h radiation of the filling gas has been measured. A 2^54 c.
dia sapphire window transfers the D^ plas.a light to the 0.5 m
Oarrell-Ash spectrometer via a 1.0 . flexible light-pipe. Observation
of the visible deuteriun, radiation indicates full ionization of the
fi'lling gas by the plasma current in 'v- 1.5 ms.
CHAPTER VI
MACHINE PERFORMANCE
Introduction
The performance data for a stable, well designed tokamak is readily
43 44 45 46 available, ' ' * but examples of the early, non-optimal operation
of a basically "normal" machine are more difficult to acquire. In the
early stages of construction, the incomplete tokamak will undoubtedly
be tested without vertical or radial field compensation. The short
duration plasma, produced by this premature operation, has a predictable
current magnitude, duration, and wave shape. As an example, a
simulation of the early Cymator tests, without vertical or radial field,
has been included in the following data.
Necessarily, the tokamak performs better provided the compensation
fields (B , B ) are properly adjusted. A data summary and analysis of
major field currents, toroidal field, plasma current, loop voltage, and
D^ radiation for a stable plasma confinement are provided in this p
chapter.
Major Currents and Fields
The oscilloscope traces of Figs. 6-1 through 6-4 are each an
example of toroidal field current, toroidal field (B.), ohmic heating
primary current, and vertical field current. Each current is typical in
magnitude of that which occurs during stable operation but was recorded
independently of the other respective fields. Therefore, the traces in
Figs. 6-1 through 6-4 do not show the interaction of the plasma or other
111
112
825 A/cm
0.2 T/cm
Toroidal Field Current
2 ms/cm
Fig. 6-1. Oscilloscope trace of the toroidal field current vs time.
Toroidal Field
ms/cm
Fig. 6-2. Oscilloscope trace of the toroidal field vs time.
113
700 A/cm
Ohmic Heating Current
1 ms/cm
Fig. 6-3. Oscilloscope trace of the ohmic heating primary current without plasma loading.
145 A/cm
Vertical Field Current
1 ms/cm
Fig. 6-4. Oscilloscope trace of the vertical field current vs time.
114
confining fields.
An Early Performance Simulation
The vertical and radial fields of Cymator were disconnected to
simulate the early tests during construction. The short {<_ 1.5 ms)
intense plasma pulse (20 - 30 kA peak) produced without these fields
is shown in Fig. 6-5. The short plasma life is primarily due to the
unrestrained radial expansion of the plasma ring. The cosine coil
response (also Fig. 6-5) indicates the rapid excursion of the plasma
radially outward (+in, -out).Figure 6-6 is an example with vertical but
no radial field. Since the radial field controls the vertical plasma
position, the sine coil response indicates a slow drift of the plasma
column downward (+up, -down). A "no-plasma" base-line trace was
provided for the sine/cosine coil pair because, without vertical or
radial field currents, these coils are not properly compensated (see
Ch. V).
Refined Performance
With all the confining fields properly adjusted, the tokamak
consistently produces the plasma current and loop voltage (V, ) shown in
Fig. 6-7. Figure 6-8(A) is a set of "no-plasma" sine, cosine, and plasma
current reference traces. Figure 6-8(B) shows the plasma current together
with typical sine and cosine coil responses. The sine and cosine coils
are invaluable in estimating the proper magnitudes of the vertical and
radial field currents. However, these measurements are intended to be
qualitative, since the most sensitive indicator of proper plasma
115
Arb. Units
10 kA/cm
Top
Bottom
1 ms/cm
Fig. 6-5. (Top) Cosine position coil response.
(Bottom) Plasma current without vertical or radial fields applied.
116
Arb. Units
10 kA/cm Bottom
1 ms/cm
Fig. 6-6. (Top) Sine position coil response.
(Bottom) Plasma current without radial field applied.
117
5 V/cm
V,=0
10 kA/cm
Loop Voltage
Plasma Current
1 ms/cm
Fig. 6-7. Loop voltage response for a properly adjusted plasma current.
118
Cosine
Sine
Plasma Current
1 ms/cm
Fig. 6-8(A). Calibration; "no-plasma" trace.
Arb. Units
Arb. Units
10 kA/cm
Top
Center
Bottom
Fig. 6-8(B).
1 ms/cm
(Top) Cosine coil response.
(Center) Sine coil response.
(Bottom) Plasma current properly adjusted.
119
position is the maximization of plasma current and duration. Figure 6-9
is a typical trace of plasma current and D^ radiation wh.ich indicates p
the rapid ionization of the fill gas.
Summary
All the data presented were recorded by the diagnostic instruments
discussed in Ch. IV. Before recording the data, the machine was
discharge cleaned by a 10 ms, 800 A p-p plasma current pulse for
approximately 5 hours at 1.5 pps. The host gas, for discharge cleaning,
was a deuterium flow at a pressure of 3.0 x 10 Torr. After discharge
cleaning, the information was recorded with the tokamak pressurized
-4 with a light deuterium flow at a pressure of 2.8 x 10 Torr with a
60 s ' waiting period between shots. Table 6-1 is a brief summary of
general machine parameters that are representative of the usual
operatinq conditions.
120
Arb. Units
10 kA/cm Bottom
1 ms/cm
Fig. 6-9. (Top) D radiation from the plasma
(Bottom) Plasma current properly adjusted.
TABLE 6-1
OPERATING PARAMETERS
121
Toroidal Field
Toroidal Field Bank
1.0 T (10 kG)
2.6 mF, 8 kV, 84 kJ
Ohmic Heating Bank Fast: 120 yF, 4.5 kV, 1.22 kJ
Slow: 93 mF, 450 V, 9.42 kJ
Vertical Field Bank Fast: 500 yF, 800 V, 160 J
Slow: 110 mF, 150 V, 1.2 kJ
Radial Field Shunt Inductor 20 yH
Pressure 2.8 X 10"^ Torr D
Plasma Current 18 - 20 kA peak
Pulse Length 10 - 12 ms
APPENDIX A
COMPUTER PROGRAMS
Introduction
A sumjiiary of computer programs employed during the design and
construction of the Texas Tech tokamak is presented. The programs
were executed on the IBM-370 computer facilities at Texas Tech
University. The winding distribution programs, for both ohmic heating
and vertical field coils, are in FORTRAN IV language. The SCEPTRE
circuit analysis programs are available as a scientific subroutine
package (SSP) on the IBM-370 system.
Winding Distribution Programs
The ohmic heating winding distribution program was designed by
M. Hedemann and R. W. Gould at the California Institute of Technology
for use on the Caltech torus. Doctor Gould graciously provided us with
an unpublished draft of the program design which briefly outlines the
concept, features, and purpose of the program. This draft served as a
guideline for the design of the OH transformer on the Texas Tech torus.
The ohmic heating transformer should induce a plasma current and
avoid disturbing the plasma equilibrium by preserving BQ^ = 0 in the
plasma volume. If a winding configuration positioned on a constant
minor radius could be designed so that a single circular field line
passes through all the wires then, according to Ampere's Law, there will
be no B^u within the plasma volume (Fig. A-1). The computer program Un
generates such a circular field line, at the ohmic heating winding
122
123
UJ (D
<
o OJ
X
o
C
o 00
Z5
o>
GJ J =
U .— ^
J T
o •o c rtí
^ f
a J D
1)1
.•T3 J Z
u / • »
13 •^ U TS >
r i
t — </) rc
r ^ O
OJ ^ - * - >
^ -o
U)
rC ^ Cn rc
• co (U U
• t —
S
QJ J C ••->
1 —
r— 'O
J Z .*/ • ^ o s_ r—
^
O í
c •r—
a1 (.0 <T3 ;11_
QJ C
•.— '—
_> r— d)
• — >4-
^ C3
r— :3
u ' j —
Q U
I
124
radius, from a few ring currents near the equitorial plane of the torus.
Once the circular field line was generated, the magnitude of the
tangential HQM field was calculated for each point around the circle.
Conceptually, the same circular field could be produced by a toroidal
sheet current, at the ohmic heating winding radius, with the current
magnitude (as a function of e) proportional to the value of the
tangential HQJ, field. The cumulative sheet current (proportional to
the integral of the HQM values) can be partitioned into N equal
segments, and each segment represents a wire location at a particular
position of 9. Figure A-2 is a listing of the ohmic heating winding
distribution program, with subroutines and input parameters.
After completing the construction of our tokamak, a vertical plasma
positioning problem was encountered. The vertical field did not have a
sufficient curvature, causing the plasma column to drift up or down,
randomly, late in the discharge pulse. The desired VF curvature of
0.75 R required the addition of six extra windings in series with the
existing vertical field coil. The number of wires (6) and their
location was determined by a modification of the ohmic heating winding
distribution program. Basically, the program generated a field line,
of the proper curvature, at the proposed radius of the extra VF
windings. The resulting integration of the tangential H values vs 9
dictated the extra winding locations.
SCEPTRE Programs
Prior to their construction, all of the capacitor banks were
simulated on the computer to determine the expected wavefcrms and
125
MAIN
IMPLICIT REAL*8(A-H,C-Z) REAL*4 TMAX,TO,XR,XL,YMAX,YMIN,TOP,BOT CALL PL0TS(0,0,9) REAL*4 RN(101),TN(101),HIN(101),DD(3), HN(lOl),RMR0N(10l) REAL*4 HSUM(lOl) COMMON A,B,C,D,S,DSQ DATA RAD/ 1.7453292519943296D-2/ DATA PI/ 3 1415926535897932/ TW0PI=2.0*PI
50 READ(5,1,END=100)A,B,C,D,E,R0,T0,S 1 F0RMAT(8F10.0)
READ(5,2)N,M 2 F0RMAT(215)
IF(N .GT. 100) GO TO 100 WRITE(6,7)A,B,C,D,E,R0,T0,S,N,M
7 FORMATCl A = ' , F 8 . 4 , ' B = ' , F 8 . 4 , ' C = ' , F 8 . 4 , ' D ' , F 8 . 4 , , ' E = ' , F 8 . 4 , ' RO = ' , ^ 8 . 4 , ' TO = ' , F 8 . 4 , ' S = ' , F 8 . 4 , ' N = ' , 1 2 , , ' M = ' ,12 /1H0,7X, 'T ' ,12X,
'R ' , 13X , 'U ' , 12X , 'V ' , 10X , 'R -R0 ' ,10X , 'F ' , ,10X, 'F PRIME' ,8X, 'H ' ,10X, '1 /H' / ) DSQ=D*D T=TO DT-180./M U=RO*DSIN(T*RAD) V=RO*DCOS(T*RAD)+C 1=0 FO=F(RO,TO) FP=(F(RO+E,TO)-FO)/E H=FP/(TWOPI*V) HI=1.0/H HN(1)=H TN(1)=T0 RN(1)=R0 HIN(1)=HI HSUM(1)=0 RMRO=0 RMR0N(1)=0 WRITE(6,6)T0,R0,U,V,RMR0,F0,FP,H,HI
6 F0RMAT(2X,9F 13.6) DO 5 1 = 1,N T=TO+DT*I R=RN(I)
10 CONTINUE FN=F(R,T) FP=(F(R+E,T)-FN)/E DR=(FO-FN)/FP IF(DABS(DR) .LT. l.OD-4) GO TO 20
Fig. A-•2. List wind" prog)
of ing "am.
the dis
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126
^"^•'•^'^ FÍQ A - 2 í c o n t ) GO TO 10 ^ * icont . ;
20 TN(I+1)=T RN(I+1)=R U=R *DSIN(T*RAD) V=R *DCOS(T*RAD)+C H=-FP/(TWOPI*V) HI=1.0/H HN(I+1)=H HSUM(I+1)=HSUM(I)+.5*(HN(I)+HN(I+1) H I N ( I = 1 ) = H I RMRO=R-RO RMR0N(I+1)=RMR0 WRITE(6,6)T ,R ,U,V,RMRO,FN,FP,H,HI
5 CONTINUE N1=N+1 XL=TO XR=T0+180.0 XMARK=HSUM(N+1) GO TO 50
100 WRITE(6,30) 30 F0RriAT(lHl,5X,'T', 10X,'HSUM'/)
WRITE(6,70)(TN(I),HSUM(I),I=1,37) 70 F0RMAT(2X,F10.4, F10.6)
CALL PLOT(0.0,0.5,-3) CALL SCALE(TN(1),12.0,36,1) CALL SCALE(HSUM(1),12.0,36,1) CALL AXIS(0.0,0.0,16HANGLE IN DEGREES,-16,12.0,0.0, ,IN(37), IN(38)) CALL AXIS(0.0,0.0,13HINTEGRAL OF H,13,12.0,90.0, ,HSUM(37),HSUM(38)) CALL LINE(IN,HSUM,36,1,2,1) CALL SYMBOL( 7.0,3.0,0.21,10HWINDING DISTRIBUTI0N,0.0,20) CALL SYMBOL(1.0,20.0,0.14,7HCYMATOR,0.0,7) CALL PL0T(15.0,0.0,999) STOP END
SUBROUTINE
FUNCTION CEI (K,A,B) IMPLICIT REAL*8(A-H,0-Z) THE GENERALIZED COMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND REAL*8 K DATA PI4,EPS/.7853981633974483,1.00-9/ EPSI=1.0D0-EPS U-1.0 V=DSQRT(1.0D0-K*K)
127
l\:^ Fig. A-2. (cont.) 10 UV=U*V
AV=AI*V AI=AI+BI/U U=U+V V=2.0*DSQRT(UV) BI=2.0*(AV+BI) IF(U*EPS1 .GT. V) GO TO 10 AI=AI+BI/U CEI=PI4*AI/U RETURN END
SUBROUTINE
FUNCTION PHI(U,V,A) IMPLICIT REAL*8(A-H,0-Z) REAL*8 K RT=DSQRT((A+V)**2+U*U) K=2.0D0*DSQRT(A*V)/RT PHI=2.0D0*A*V/RT*CEI(K,-1,0D),1-0D0) RETURN END
SUBROUTINE
FUNCTION F(R,T) IMPLICIT REAL*8(A-H,0-Z) DATA RAD/ 1.7453292519943296D-2/ COMMON A,B,C,D,S,DSG TRAD=T*RAD U=R*DSIN(TRAD) V=R*DCOS(TRAD)+C F=PHI(U,V,A)+S*(PHI(U+D,V,B)+PHI(U-D,V,B)-2.0*PHI(U,V,B))/DSQ RETURN END
DATA
A = 0.63 B = 0.75 C = 1.00
D = 0.01 E = 0.001 RO = 0.611 TO = 180.0
S = 0.180 N = 36 M = 36
128
current magnitudes. Figure A-3 is the basic toroidal field circuit and
its SCEPTRE model. Figure A-4 is basically the same model with the
radial magnetic field coil and shunt inductor added. Figure A-5 is a
listing of the SCEPTRE program for the toroidal field system and radial
field network. Figure A-6 is the basic ohmic heating circuit and its
SCEPTRE model, while Fig. A-7 is a listing of the SCEPTRE program for
the ohmic heating system. Figure A-8 is the basic vertical field
circuit and its SCEPTRE model with the program listing shown in Fig. A-9
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TOROIDAL FIELD SYSTEM
SCEPTRE SYSTEM INPUT - 07/71 VERSION - S/360
CIRCUIT DESCRIPTION ELEMENTS Rl, 3-4 = 0.190 Cl, 2-1 = 2.6 Ll, 4-5 = 13.6 R2, 5-6 = 0.002 L2, 6-1 = 0.007 R3, 5-7 = 0.003 L3, 7-1 = 0.004 R4, 3-1 = Tl(TIME) R5, 2-3 = 0.01 OUTPUTS ILl, IL3, IL2, PLOT INITIAL CONDITIONS VCl = 7500.0 FUNCITONS Tl = 0,10,0E06, 9.45,10.0E06, 9.50, 0.001, 25.0, 0.001 RUN CONTROLS STOP TIME =20.0 INTEGRATION ROUTINE=IMPLICIT RERUN DESCRIPTION (4)
ELEMENTS L2 = 0.013,0.010,0.025,0.032 END
Fig. A-5. List of the SCEPTRE program for the toroidal field circuit model of Fig. A-4.
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OHMÎC HEATING SYSTEM
SCEPTRE SYSTEM INPUT — 07/71 VERSION - S/360
CIRCUIT DESCRIPTION ELEMENTS Cl, 2-1 = 0.120 C2, 4-1 = 03.0 Rl, 2-3 = 0.01 R2, 3-5 = 0.1 R3, 4-3 = Tl(TIME) Ll, 5-1 = 0.5 OUTPUTS ILl, VCl, PLOT INITIAL CONDITIONS VCl = 5000.0 VC2 = 500.0 FUNCTIONS Tl = 0,10.0E06,0.355,10.0E06,0.360,0.01, 25, 0.01 RUN CONTROLS STOP TIME = 10.0 INTEGRATION ROUTINE=IMPLICIT END
Fig. A-7. List of the SCEPTRE program for the ohmic heating circuit model of Fig. A-6.
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VERTICAL FIELD SYSTEM
SCEPTRE SYSTEM INPUT - 07/71 VERSION — S/360
CIRCUIT DESCRIPTION ELEMENTS Cl, 2-1 = 0.5 C2, 4-1 = 110.0 Rl, 2-3 = 0.01 R2, 3-5 = 0.150 R3, 4-3 = Tl(TIME) Ll, 5-1 = 0.350 OUTPUTS ILl, VCl, PLCT INITIAL CONDITIONS VCl = 850 VC2 = 160 FUNCTIONS Tl = 0,10.0E06, 0.570, 10.0E06, 0.578, 0.01, 25.0, 0.01 RUN CONTROLS STOP TIME =10.0 INTEGRATION ROUTINE=IMPLICIT END
Fig. A-9. List of the SCEPTRE program for the vertical field circuit model of Fig. A-8.
APPENDIX B
CONSTRUCTION SUPPLEMENT
Introduction
Appendix B supplements the construction information of Ch. IV
by providing mechanical and electronic details of all major systems.
The collection of diagrams and circuits follows a presentation format
parallel to that in Ch. IV.
Vacuum Chamber
The illustrations of the vacuum chamber, beginning with an overall
chamber diagram (Fig. B-1) followed by sectional views of the various
port configurations, are shown in Figs. B-2 through B-6. Figures B-7
and B-8 are diagrams of the material used to construct a rectangular
channel connecting the Varian ConFlat ports (6.0 in. and 8.0 in. dia.)
to the vacuum chamber body. The two stainless steel "u" channels are
joined with interior, fuU-penetration welds.
Toroidal Field
Figures B-9 through B-12 describe the toroidal field capacitor
configuration and power supply. Figure B-13 is the toroidal field
switching system. Note that if the toroidal field crowbar should fail
to fire, the TF capacitors ring with the TF coil and charge with reverse
polarity. The 15 kV fail-safe diode and 800 íi resistor discharge the
TF capacitor bank to prevent inadvertent connection of the TF supply to
the improperly pre-charged capacitor bank. Figure B-14 is the TF
transmission line.
136
137
PORTE
POfiTC PORT B
P0RT6
PORT A
NOTE- I x 12 STAJNLESS STEEL ROO TO SUPPORT TDRUS FROM THE BOTTO*! (6 Rcos "rerrMj
Fig. B-1. Top view of the stainless steel vacuum chamber.
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Ohmic Heating
The ohmic heating capacitor configurations, power supplies, and
remote control network are shown in Figs. B-15 through B-19. Figures
B-20 and B-21 describe the ohmic heating switching system. The switch
tube of Fig. B-21 is a gas filled "Krytron" switching device from the
E6 & G Corporation.
Vertical Field and Preionizer
The vertical field system is described in Figs. B-22 through B-26.
The switching circuit of Fig. B-26 is a subchassis of the vertical field
power supply and obtains the AC, for the HV bridge rectifier circuit,
from the* secondary of the VF fast bank transformer.
Figures B-27 and B-28 are the power supply and trigger circuit for
the preionization system. The trigger module of Fig. B-28 differs from
the OH and VF trigger circuits because the trigger current, through the
pulse transformer, is switched with an SCR.
Discharqe Cleaning Oscillator and TF Pulser
Figures B-29 through B-31 describe the discharge cleaning
oscillator and bias supplies. The solid-state screen grid pulser, of
Fig. B-32, produces a 1 kV (max) square pulse in a three-step process.
The SCR pair, triggered as "pulse charge," transfers a current pulse
from the 160 yF electrolytic bank which charges the 60 yF capacitor.
Secondly, the SCR pair, triggered as "SCR start," switches the 60 pF
capacitor to the 50 kfo load. The third SCR pair commutates the start
SCRs by delivering a high voltage pulse to the 50 kíî load.
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O S-
o u cu
<u
Q. Q. 3 to
<U
C Q.
<U
-T^ U
S-
I ca
162
o Q:
UJ o o u. >
o
•fA
uf o
l i .
5 -VVv-
in
- V A -UJ
o > <5 CNJ h - W
- H
-vw-c: JC
o
-vw
i
i ro Z
OJ - M CO >. «/)
o> c
u -M •^
2
"O
<T3
u
S-
<u
cvj I
ca
ii'
+ . . Í
i I
I I
I •
zx > C D 2
163
i Í C t -
I <DO - J O O
< O) lO I
(D Q:
3 U S-
•^ u s-(U o> cn
o S-
-M •r— C cr,
"O
<V
<+-
<t3 U
S-O)
<sO Cvj
Cû
<zn
S
164
s-o
•4->
u
u c o +J
M
c o
Q.
O .c
s-O
Q. Q-
to
s-
o c.
C_)
t n
CM I
CQ
> O
165
8 (O rL
> o o c\j u. 4.
(U +-J to >> to
d
p .
(T lU S (T o u. to z < í UJ (0 -J
2
«
«> z
? >•
DAR
z o o bj
^ co z (T 3 )->-< z tr (L
<T3 N
•r—
c o •^ (U S-Q .
<V sz +-> S-o
<4-
-M •^ 3 U
-.-• n -
u S-OJ CD o>
• I —
S--M
C o s-
-i-> •T—
c 1—1
. CO CM
CQ
« o>
•r— U_
> o (M
cn f<n i
o t
4
166
3 U
s-
u O)
•r—
c <T3 (U U
(U C7> S-
2 "u to
"O
tsi zc J>c
in co
o
C\J I
o>
167
O) c • p -
c <T3 Qi U
<V O) i-<T3
. C u to
<v .sz •M
i-o
< < -
to
<v
Q . Q . 3 to
to
<T3
. Q
• c
S. O)
o i-
+-I c o u • o S-c o fO +->
<aj •>-s. u u to
ÍJT) o
o ro
I C3
C?>
168
V <
LUZJ
ê^ SS\ ZOD
2 o J
5^
.j° ir\
COQ Z i n ^ C <v \n
^ s j
> U J XGC
/ ^ M 1 1 viy ^^^ > 2 X O
•
h-S.
FIL
AM
R
ELA
Y
^ / \ ( ) v:]
- ^
•v^ / \ < ) [></.
^
>
í ^ ^ j
/•
1
Q < > >
O <0 « M
1 ^ >
o: UJ
LJU.
<i -J^S r ^ H
f V
03
^^ •
'N
fO
T / • ^ f l M víy
ZO
-I
)
• o (U +.) <T3 U
•^
? 1 ^ <c
t l _
<u .c t—
s-o
• M <T3
•T—
U to c C7> tO C -M
1 ^ *
C OJ <T3 C (T O
u E o
c; u S- + J <>3 S-
-c o U Q .
•<— 3 "O to
CJ • c x: c +-> o s- c
uit
s
to
OOA
tub
U o S- uO
- r - —^ U X
'— «^
^ <u —> .c c +-O U r—
o c^ o <: u
._, C^
1
• o '
1 J
lO
169
s .
3
o. c CJ <u u (/»
(U •M <o
+ J </) I
-o o to
<v
E <B S-
cr> <T3
3 U
C\J <m
I CQ
o>
170
The toroidal field pulser and SCR trigger module are shown in
Figs. B-33 and B-34. The trigger diode and Triac network of Fig. B-33
modify the capacitor charging current waveform to permit the SCR to
recover between shots. The 15 kV diode (Fig. B-33) permits the TF
pulser to remain connected to the toroidal field transmission line
during normal operation of the TF capacitor bank.
171
Í2 m
<T3 < + -
S -
<u (/, 3 SZL
• o
<u
<T3 • o
O s-o
• » - 1
a» •»-> <T3
-(-> 00 I
-o
o
(U
<+-o
II)
IT:
s_ CT> <o
• r -• o
+J • r -3 U S-
• r -C_)
ro co CQ
to CL <T3 C7>
.: : S-<o Q . 00
<U <i-<T3 </>
C7>
•f T < lO CVJ
> .3C
UJ
>C3
< > «
— ÍNJ
t I
172
" U U U L / -
j'yy^^
s. o u. cr
co CVJ
cs j)c
4. lO
o
—Hr-
+'
CJ to
3 C L
T3
O
"O • r -O S-O
<v
cî .ac lO
CJ C>sJ C\J 00
CVJ
e 21°-.
\ I -vw +•
u '-. u S -
<u
cs GO O
<_;> </o
OJ
to
C
oo
O
03 * -
I co
O )
> l O CVJ
APPENDIX C
COST ANALYSIS
Introduction
The tokamak at Texas Tech was constructed under the daily direction
of two graduate students employing an undergraduate labor force of
three students each working about 15 hours/week. The tokamak was
operational after a 13 month construction interval, beginning with the
appropriation of funds and ending with the first "tokamak" plasma, at an
estimated cost of $56,700 (equipment, services, and salaries).
Grant Expenditures
Table C-1 is a list of general expenditures on NSF Grant ENG-76-
05897 (June 1, 1976 - May 31, 1977) and local research support funds
from Texas Tech University.
Encumbered Equipment
Table C-1 is an accurate account of labor and equipment expenditjres
on both NSF and university grants. However, a significant amount of
capital equipment, previously acquired by our laboratories, has been
devoted to the Texas Tech tokamak facility. Table C-2 is a brief
appraisal of the encumbered equipment.
173
TABLE C-1
GRANT EXPENDITURES
174
Labor and Equipment
Salaries
(2) Graduate students
(3) Undergraduate students
subtotal
Vacuum Chamber
Raw Materials
Components
Mfg. Services
Chamber Insulation
Residual Gas Analyzer
Cost
$10,748
6,300
17,048
3,436
2,473
925
352
4,885
subtotal 12,071
Diagnostics
Klystron
Microwave Components
R.F. Amplifier (500 W)
Display Monitor
Camera/Film
Spectroscopic (UV) window
Probes
subt ta l
3,300
4,036
7,700
1,175
1,129
214
722
18,276
175
Capacitor banks and Field Coils
Electrolytic capacitors
Raw Materials
Switch Devices
Wire
Hardware/Services
subtotal
Electronic Components
Misc. (Tools, books, hard-ware, shipping, services)
Total
1,967
481
1,172
2,240
764
6,624
1,832
790
56,641
TABLE C-2
176
ENCUMBERED EQUIPMENT
Toroidal Field System
(40 ea) Capacitors ('v $400 ea)
TF power supply components
subtotal
Vacuum Station
Mechanical Pump
Diffusion Pump
lon Pump and Control Unit
Pneumatic Valves, and ionization gau unit
LN^ trap, ge control
subtotal
Total
16,000
2,000
18,000
525
1,400
2,650
4,000
8,575
26,575
Comment: This total does not include scopes, tools and general hardware.
APPENDIX D
DERIVATIONS
Introduction
The summary of derivations presented in Appendix D indicates the
origin of the formulae necessary for both machine and diagnostic coil
designs. The derivations from the selected references are reproduced
in mks units (courtesy of Professor Cross) to be consistent with the
previous calculations in Chs. II through V.
32 Vector Magnetic Potential and Field of a Circular Loop (see Fig. D-1)
t has only the single component A .
% = (f)'/I''t^ ^ ' " " ^ V V de (1)
* TTk r ^o 2 ( _ k ^ . Zgj-a
where
2 ^ r , (2) Given the complete elliptic integrals of the first and seconcí kind,
K = / ' T — ( ) ° (1 - k^ sin^e)^
and ^/2 0 ? u
E = / (1 - k^ sin'^e) ^ dø , (4) 0
Eq.(l) becomes
ol ,a^h r/. k^ \ = ^ ( F ' ^ f ( ^ - ^ ) K . E ] . (5)
177
178
<v •*->
to >> to
(U • M <D C
•o S-o o u <u
-o c
c <u s_
u
Q . o o
I Q
179
Since S = 7 X Â, the vector magnetic potential yields three components
of S in cylindrical coordinates (r, <(), z ) .
B = —i-r 8z (6)
%'-0 (7)
h- r 3r('"' 4,' ' (8)
where
and
B„ = v'ol
' 2.r [(,.,)2 , ,2^^. -K +
2x 2^ 2 a +r +z " x2 ^ 2 (a-r) - z
(9)
z 2T[
1 [(a-Hr)2 + z^f^
K + a^-r^-z^ f N2 ^ 2 (a-r) + z . (10)
32 25 Mutual Inductance of Coaxial Loops ' (see Fig. D-2)
The current in loop 1 produces a flux through loop 2 of (p^-i = f oi
*21 " ^s ^l * ^^2 = /s ^ ^ ^ * ^^2 (11)
= ^ X-. • dîpj by Stokes' Theorem,
loop 2
(12)
or
*21 = \l 2^^ = "21^- (13)
180
— co
Nl
<V u c ItJ
•<->
to
•o
<T3
•o <v
+•> <c s -<T3 Q . <U to
to
o o
X <T3 O
I
o
cn
If Eq.(l) is evaluated for r = a of loop 1, A,-, becomes
181
%1 = ^ ( f ) ' [ 0 - ^ ) K - E] (14)
vjhich gives
"21 = ^ - ( f ) ' ' Cn - ^ ) K - E] (15)
^ (Ra)'' [(1 - | - ) K - E] (16)
and
4 Ra
[(R+a)^ + z^] (17)
I f the loops are concentric (z = 0) , Eq. (17) gives
4 Ra
(R + a)' (18)
As an example, consider the calculations of the mutual inductance
for the two concentric hoops and the filament plasma current of Ch. V,
Fig. 5-2(B). For the hoop at r-. = 0.257 m and filament plasma current
of radius R = 0.457 m, Eqs. (16) and (17) yield the mutual inductance
« i=V^(«^ i ) ' (1 - ) K - E (19)
For r, = 0.257 m and z = 0,
(R+r^)' = 0.9215, (20)
182
and from Abramowitz and Stegun, the evaluation of the elliptic integrals
becomes,
K = 2.690
E = 1.088
which yield
M^ = 0.4298 yH = 0.748 y^ R. (23)
For the loop at r^ = 0.357 m, Eqs.(16) and (17) yield
M^ = n p - ( ' ^ r )" ^ [(1 . ^ ) K - E]. (24)
For r^ = 0.357 m and z = 0,
P 4 R r« V. = ^ = 0.9849 (25)
(R+r^)"^
and
which yield
K = 3.619 (26)
E = 1.022, (27)
M^ = 0.8676 yH = 1.511 y^ R- (28)
31 Mutual Inductance of OH Transformer and Plasma (see Fig. 3-5)
The inductance of the OH primary produces the flux
h 'i *i = - i r ^ (29)
where N 1s the number of OH transformer turns. The mutual inductance (M)
between the transformer primary and the plasma is simply
183
M = j ^ (30)
where (»-.0 is the flux linking the plasma from the OH primary coils.
The OH primary coil design specified that all of the primary flux (4>-j)
link the plasma since 8«^ = 0 inside the plasma volume.
Therefore,
which yields
h '*1 *12 ~ *1 N — ' ' '
M = -hrr- = TT ' (32) i-j N N
32
Self Inductance of the Plasma (see Fig. D-3)
The inductance of the plasma can be calculated if the plasma is
modeled to be a wire of minor radius b, and major radius R. The total
inductance is therefore the sum of (A) the internal inductance of the
wire and (B) the mutual inductance between the wire and a loop along 32
the wire's inner edge.
(A) A wire of radius b, permeability \;, and carrying current I has
an internal magnetic field
B = ^ . (33)
The energy stored magnetically within the wire of length l is simply
"int = k !, ^\ ''
184
to
c "O
c <v S -S -
S -
o
-o c
.Q
v>
• o <T3 S .
o c
QJ S -
s^ <o
Z3
u
u <T3
tr) 03
\-^ OJ
cu "O o E <T3
E 0 0 <T3
t Q
o>
185
"int = TÍT /O '"' 2. rdr (34) 07T D
2 = - M — ^ = a L. ^ I^ (35)
16Tr ^ int '
re L. . is the internal inductance of the wire. From Eq.(35)
and, since
^ = 27T R, (37)
whe
Eq. (32) yields
^int = f (3«)
for a uniform current density.
(B) The mutual inductance of the wire and a loop of radius R-b
follows from Eqs. (16) and (17),
M = l ^ [R(R - b)]'' [(1 - ) K - E] (39)
for a = R - b , z = 0 ,
and ^ 2[R(R-b)]^^ (40)
2R - b • ^ ' The external inductance calculation,
.2 I k' ext
= M = Uo(2R - b) [(1 - ) K - E], (41)
186
reduces to
1-ext = 2 Uo R ['s l " r • ^ (42)
= p, R [In l^ - 2] (43)
8R for b << R, k = 1.0, E = 1.0 and k " - • The total inductance then
becomes
L = L. ^ + L ^ (44) int ext ^ '
= y^ R [ I n f ^ - 2] + H i . (45)
Note that if y -> VIQ for the plasma, Eq.(45) reduces to
L = lio R [In ^ - 7/4]. (46)
Derivation of the Vertical Field Requirements '
The centering force for radial equilibrium per unit arc length is
proportional to the product of the current I and the curved vertical
field B (-j— = B,,I). This force must balance three terms due to v^ds V
(A) poloidal field, (B) kinetic pressure and (C) toroidal field.
Poloidal Field Forces
The force due to the poloidal field is derived from the radial
change of stored energy in the plasma inductance, where
W = h L I^ (47)
is the stored energy in the poloidal magnetic field from the plasma
current I.
187
Therefore,
F - ^ ra • dR (48)
is the radial force.
"^a R = dW = hl^éi + LI dl (49)
If the loop is perfectly conducting, the expanding loop attempts to keep
the poloidal flux constant.
Therefore,
L dl + I dL = 0
or
LI dl = -l2 dL
and Eq. (49) becomes
(50)
ra or
-F_ dR = H I^dL - I^dL
f'ra dR = h I^ dL
F = k T2 dL
(51)
(52)
ra ' •' dR • (53)
Since the mean length of the plasma is 2TT- R, the radial force (dF ) ra
per u m t arc length (ds) of the plasma.
dF I^ dL ra *-ds 4TT R dR ' (54)
can be evaluated with the help of Eq. (45).
188
dL d_ dR " dR ,,R[ln 1 ^ - 2 . 0 ] + ^
8R y. [R( ,)+ ^nf- -^] (56)
= Po [In 8R
] (57)
Kinetic Pressure Forces
2 A force, F = TTb p, acts on the ends of the plasma section (Fig.D-4)
due to the gas pressure p = nk(T + T.). Since the arc length ds = Rdcj!, c I
the radial force per unit arc length can be written as
^ ^ b (^^^P) ^ (2) TTb^p ds Rd<í) R (58)
Toroidal Field Forces
The curvature of the toroidal field yields a force which can be
found hy integrating the magnetic stress tensor over the plasma surface
The pertinent terms are
(a) parallel pressure
P =
2 2
2 ;i^(R+b cos e) (59)
and
(b) the perpendicular tension
T =
2 2
y^^R+b cos e)' (60)
189
u O)
<5
î < T :
IT3
l/> • o
c cu <v —• -~ c o CT>
.~ • — -^ u <c c
OJ S-^ to to
u L.
• ^ ;._
U_
o; u S-
o cm-</> r
fO
o>-f ^
— • o
d 'C
a
<T3
^ cn TS
-^ Q
•
1
U Z3
•o o S-
^
190
where (Fig. D-5)
B. =
B_ =
B. =
field inside the plasma,
external toroidal field,
toroidal field at the minor axis without plasma.
and B_ = ^o^ (61)
From Fig. D-6,
r = (R + p cos ø) ,
and Eq.(61) becomes
B 2 2
2 li,
(62)
2 Mo (R+p cos e) (53)
The net force at the surface of the plasma is the difference of these
two magnetic pressures integrated over the plasma cross section
dF^ = R^ .2TT ,b ( o "^' '
• ^
0 ' o (R + p cos e) 2 pdpde (64)
_R /g 2 _ g 2x rZi: rb pdpde
HD (R + p ccs e)' (65)
2TT R ^ ( B / - B . ^ )
2 y.
1
[1-L
( ) ] "" - 1
The radial force per unit arc length ds is (Fig. D-6),
(66)
dF = 2dF ^ rc c 2 (67)
191
r<3 <T3 C
E s-to 0) <T5 - ^
— C Q . 1 -
<9 . C -«-)
< - ^ - r —
O S
E C <T3 O ^ 'r—
o> -<-* <T3 U
..- cj Q to
t n 1
Q
Li
-o r—
<v •r—
' • * -
r—
<o c '._ <v
•!->
X
<v N M T
c <
IT:
^ • • —
w. k. o •*->
0) u o w /
o S-r->
.<—> <T3 r—
-«J
. u
Lk .
XJ
to
u S-o
t , -
c - r—
to c OJ
•«-J
t3
<
AS
M
_ j CL
LJ O cn z
o - J UJ u. I I
UJ
u.
IDA
L
O Q: O
- j < z i r UJ H X UJ I I
03 SQ
192
<v
"O S-o o u <D
2
u <v (/> </> i/>
(/)
Q.
<v
<T3 1 -
*~ o
0> -f-' <o
•r~
Q
1
to >> i/>
o>
and 193
ds = Rd;í>
^^rc " !;?^í^^ component of the toroidal f ie ld forces/unit length
2dF d< dF c _ c
Rd(í) R
a l t e r s the appearance of E q . ( 6 6 ) , where
dF 7TR rc _ _
ds ij^
2 . 2, ( o - B / )
L [1 - (^)^]^^ - 1
For the approximation of | « 1, Eq.(70) becomes
(68)
(69)
(70)
dF rc
ds
1 TTR b ^ .
(o) ÍBJ ]^r R Bi^)
2 uc R
(71)
(72)
The Net B Equilibrium
The sum of the poloidal, kinetic pressure, and toroidal f ie ld
forces (per unit length) must be met by the vertical f ie ld force.
dF
_ l' 4 7T R
^ ^ r h dF
ds ds
2 9 - ^^ + ^b p _ TTb"
dR R 2 d_ R
(73)
( B , ' - B . 2 )
(74)
Note that
B _ ^ I e 2TTb
194
( 7 5 )
a t the plasma edge , y i e l d s
1 = ! : ^ (76)
and
P 4TT^ b^ B ^ r = e
Hô T ( 7 7 )
which can be combined with Eq.(74) to yield
B = — B ' 2R ^
1 dL y^ p (B/-B.'^) 2 n 2 0 "1
L yo dR B, 2 B e
The pressure balance equation.
J (78)
^ j_ B. P + 1
^^ 2 Po 2 Uú'
can be rewritten in terms of the poloidal beta
3a = 2 Mrj p (B^^ - B.^)
= ô — = 1 + — 2 TJ. B e B
e
(79)
(80)
which yields
B = V
^ô I
2R 27Tb
2 . 2
1 dL 1
Mo dR 2
JB^--B/) ^ (B / -B / )^
2B e 2B,
( 8 1 )
195
from Eqs.(75), (78), and (80).
The simplified expression of Eq.(81)
• v 4TTR ^— 4k-+ 1/2 + 3. -1 Po dR e
can be combined with Eq.(57) to give
or
R - ^-2-1 • v 4TTR
R = } ^ ^y 4TTR
In f ^ - 3/4 -H 3Q - 1/2
I n f . 3 3 - 5 / 4
(82)
(83)
(84)
49 50 for a plasma of uniform current density. Some references ' prefer
to express Eq.(84) in terms of the internal inductance per uni t length
L. (L. = ]p ) so that B can be calculated for various current p ro f i l es .
or
^v 47TR
^v " 4 ^
8R 2Tr L
u. 1 - 1) -1/2 (85)
8R 2TT L . - - 3/2
V
(86)
LIST OF REFERENCES
1. D. S. Stone, et.al., "Preliminary Results on the Versator Tokamak," Tech Report, RLE-117, MIT, Res. Lab. Elect., January, 1976, Cambridge, Mass.
2. R. W. Gould, Private communication, 1977.
3. H. Takahashi, "ICRF Heating in Tokamaks," Tech. Report, PPPL-1374, Princeton Plasma Phys. Lab., Oct. 1977, Princeton, N.J.
4. P. Christiansen, Prof. of Classical and Romance Languages, TTU, Private communication, 1978.
5. M. 0. Hagler and M. Kristiansen, An Introduction to Controlled Thermonuclear Fusion (Lexington Books, Lexinaton, Mass., 1977), pp. 59-109.
6. L. A. Artsimovich, Nucl. Fusion }2_, 215 (1972).
7. S. 0. Dean, et.al., "Status and Objectives of Tokamak Systems for Research," USERDA Report, WASfí-1295, 1974 (available from the Superintendent of Documents, U.S. Government Printing Office).
8. A. 0. Anderson and H. P. Furth, Nucl. Fusion 12, 207 (1972).
9. H. P. Furth, Nucl. Fusion 15, 487 (1975).
10. T. H. Stix, Phys. Rev. Letters 36, 521 (1976).
11. V. A. Vershkov and S. V. Mirnov, Nucl. Fusion ]±, 383 (1974).
12. L. A. Artsimovich, et.al., Plasma Phys. 7 (J. Nucl. Energy C), 305 (1965).
13. V. S. Vlasenkov, et.al., Nucl. Fusion l^, 509 (1973).
14. I. H. Hutchinson ana J. D. Strachan, Nucl. Fusion 14, 649 (1974).
15. F. F. Chen, Introduction to Plasma Physics (Plenum Press, New York 1974), p. 288.
16. R. J. Taylor, Private communication, 1976.
17. R. C. Cross, Private communication, 1977.
18. ConFlat is a registered trademark of Varian Associates, Vacuum Division, Palo Alto, Calif.
196
197
19. H. P. Furth, et.al., Phys. Fluids 16, 1054 (1973).
20. L. E. Zakharov, Sov. Phys. Tech. Phys. 20, 660 (1976),
21. B. Richards and D. S. Stone, "Parameter Space Available to fníf r y Tech.Report, RLE-118, MIT, Res. Lab. Elec, July, 1976, Cambndge, Mass.
^^' S;-r'^^n?^^^^' et.al., "Small Tokamaks," Tech Report, PRR-77/21, MIT, Plasma Res. Lab., August, 1977, Cambridge, Mass^
23. H deKluiver and H. W. Piekaar, "Heating and Containment of a Plasma in a Small Tokamak Device," Rijnhuizen Report, Instituut Voor Plasmafysica, Nederlands, Dec, 1974.
24. V. G. Welsby, Theory and Desiqn of Inductance Coils (John Wiley and Sons, London, 1960), p. 45. ~
25. R. C. Cross, "Construction of Diagnostic Equipment for the Texas Tech Tokamak," Tech Report, NSF-EN6-7303941-2, TTU, Plasma Phys. Lab., April, 1977, Lubbock, Texas.
26. J. C. Bowers and S. R. Sedore, SCEPTRE; A Computer Program for Circuit and Systems Analysis (Prentice-Hall, Inc, New Jersey, 1971j, pp. 34-85.
27. SWAGELOK is a registered trademark of Crawford Fitting Co., Solon, Ohio.
28. V. S. Mukhovatov, Nucl. Fusion U, 509 (1973).
29. L. Oren and R. J. Taylor, "Trapping and Removal of Oxygen in Tokamaks," Tech. Report, PPG-294, UCLA, Center for Plasma Phys., March, 1977, Los Angeles, Calif.
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