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.equirenêncc 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^ênt 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

> æ 4 —

^ UJ o cr o

<tj

>

o

c

UJ

) -O < 0 UJ-J

25 Q O

•J-.

c c

c

OJ

u

IT3

© 1

o _J Ul

12

^

u o Q: o u>

CQ OJ

c

rt3

C

^

(/> O

CJ

I

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

d, u

<X) +-> OJ

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

< .

</)

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





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—

 <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« ^ -. Rî^ . 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 = Rî^ + 0 == 5 V , (13) (initially)

where the initial 5 V value of Rî^ 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

^

460

O

'T

O • CJ

V

O • æ

O • «o

lO

340

O (Vj

ro

§ +

co u u a: o Ui o z "• UJ

NGL

<

S

eld

>+-

r— (T3

rti

OJ 2»

or

the

M-

o

«/) >

— . -^

O

^» íC S-

CNi

o + (O

CVJ

O

0<M o o

H do "ivasaiNi

OJ

-M

4 -O

•»->

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^

O . UJ

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

(U -t->

o •zi

• n c o 1 —

+-> (T3 (J O

• -M C o; e <u u «3

r—

Q .

CU S-

' r ~

3 XJ d '

-M (J r--

• o 0) s_ Q . 1 s_ a;

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

ê = 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

-M

U

OJ JC

03

C>0 I

O

(T3 U O

o O <T>

CJ o c

• t ;

2 ã)

o

5;

I

s . CJ

E

51

2 -j

z

2 aí

1:

Ui

Ui S < û

I cv

ut

,T—r

l > — -

U

>

s. o

'+-

-o <ô </>

o Q. I—. Z3 l/l

aj O)

-t-> co

OJ

4 ^

CL

o

=

t r

9 UJ

>

&

</1

> -t-> u OJ c •y» s. o

O) (3J

s -

I

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

s -OJ

j = u

>

<T3

-o o S-

c •o

c c o

u > o

^3

s _ fT3

U • I — • « - >

5 u

I

O î

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'êviously

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

IJ

56

Q

í co

u

OJ

c T:

JiC

8 Z < m

I < o u.

»0

p o

•«->

(D

lO

o

(O

- —

{VJ

—

" V T " - 3 JC

32.5

UJ co ^

•• H

<T3 S- • 05 E ro QJ

"O oo

U

o -r:

I

Oioco

- ! •

o o Q

O

t

57

(/>

o 'Ô C4

o u - o

<v

o cá u

Q (O

<v c Oi

o

o cû

O d

Q ^

O (d

•o

o S -o

-«->

OJ

o

o cvi

Q o fC o

lO •0 CVi

Q CNi

lO m (3 o

o lN32JtJn3

c (D

O

I

CT>

58

^^^^ww:

0.2 T/cm

Toroidal Field

2 ms/cm

Fig. 4-8. Oscilloscope photograph of B^

59

c o

c o

<T3 U o

o u

03

U

cu sz

sz •r—

- . r* — « í o ' * « ^ Q Q ^ t^ ai cv * i ^r'irîO*> o io A J£ <£ 2 íg t t í^ s5a03G

o to

s -

OJ c o

I

CT

' UJ

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

s a

H »-5 oui

O

3Z O

co CT c

• f "

-o c 2:

•o

'0 •p—

u •^ •M OJ c 01 Í3

O s-o

c o

u

o S -

I

cn

62

co

<T3

ro

-o OJ

.c 00 I

+-> c 0 u to OJ S-

u

bJ

Uj

^l -—3

M

Vi I O <

zts

<o Ui (T O.

o X < > Ui Ol

t -oc Ui 2„

Oi

z 2

<

c O

u 13 S-

- l - J to c o u Q. <U -M ;/) I

aj j LU TT

3: S ô-í~} 5^ û-

P ^

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

o ai

..o

o (C

..o lO

o

O)

u o;

OJ c

s-<u E s-o to

c

o OJ

(U s-s. o u

<u

lM3iåUnO

c o

iTJ O^

to r7 o s- u a;

• M (T3 3 E C (/) E 03 O r—

o a.

cn I

«5-

Oî

66

700 A/cm

Fig. 4-14.

1 ms/cm

Oscilloscope photograph of the current in the OH transformer without plasma.

67

X X o

Ui

c O (/» c o

u O

o u "O

u

c (T3

03 u

>

a;

\ ^ / /

/ 7

/ 1 - '

S3< 55 U J O

2 S ãS uût ( T X < t -(n _io 2 0 ( 0 X 3 : < <

^ 0

4 -

S. OJ »

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M-(/) c ítJ S-

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: : ; 0

0) c 0

lCi r—

1 «a-

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

2

< r

Hi'

ûj

to >^ «/) -o

o •z Z g 5 t "^ 2 5 C) > — W

^ 1

i^ 2 < OD ^

VF

SL

(1.7

i

i-- 3

FAST

(2

00

u. >

>

-.111

IGH

.T

AGE

IkV

-g i

03 U ^ 1

s-(L) >

f th

e

' o

f S-O i

"O

JL

Blo

ck

VO 1—t

1

u. >

Q: >J

Û- ÍO

•r— U-

70

o u

OJ

c fT3

<TS

u

s.

>

(U

XN3HdnC

CJ s. s-o u

OJ

o c c

•f—

to

s . cu

Q .

O

o

I

u.

71

145 A/cm

Fig. 4-18.

1 ms/cm

Oscilloscope photograph of the vertical field current.

72

•o (U

d)

c ra

• o

S-

OJ

OJ

u •o o s_ c

^3

{/1 Q . O O

o 2 o;

c o

<r3 u o

(T>

I

O î

u.

73

cn _» o > 00

-o OJ

u

o

o >. .J _l ui a.

> C % 2 5 I a cc

cr

S UJ

ii |i o

n3

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d)

S-

•r—

T3

U O

co

o CVJ

I •53-

C1

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

X -5 a. J^ *n O N 2 esí

C9

(/) o Z3

<X3

>

o s.

S-

o

iNBMdno

CJ

s . 13

u • c

 O

C\J I

'=3-

76

o 5 z Ui

§ 5

( 0 < . .

lUJ

in<

(U •M C cu u

(U

-o (U

-M s . OJ LO c

o c OJ o to

c o

-M «3 N

C

o •r" O) s. c

(/)

c

d)

u dJ lO

co (/» o

CJ

CVi

I

C î

77

CD I

O

cr O-llitzHl

< M 2 9 -J 14J o

<D cn

Í3 I

t ií _ < S "=

s -

b CL 0 .

> 2 ^ co i2 £r 6 ^

2

cu -M

co

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CU sz -M

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-c

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( ^ CVI

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o

78

o

o 5 ^ i£ O UJ cr Q.

c cn

• o c 13 t/) S-

o

u o. u

-1-J <c Nl

(1» S-Q .

<3J

E

C3

03

3 U

(_)

C^vJ I

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

I

o û _J o co

UJ LLÍ

rr o co

S -

o -M 13

U (A o

c «3 QJ

<U

S -

~ (/)

(U

<V3

U

o

I ^3-

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

< 8l3> « W O J 5 2

H ítoJOi —

( C ^ UuJ

(tsí a:

U i 3 0 (CcoO

o

ii

'

oc IT

RO

N

RO

WB

A

Xo<J © 2

(E OT < «n t 2

-J

o ( j

i - j

1 (C IC

Ê l - i < - l O

2

E iU

-M to >> co

IB

O

ÛJ

s-

c c

(U

O

iT3

U

o CQ

CO C\J

I

CD

85

2

> to

o s.

-M c o u 0)

s.

ÍT3 S -

o>

u o

CQ

(D (Ni

I

(Tî

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

UoJIÂ (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 ê 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) = MÎ (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 )

ô 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

 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 ô

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 (^.. Ô " 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

îP 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.


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


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 Bû 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

OH tribL jtion

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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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.

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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 ô 2 ( _ k ^ . Zgj-a

where

2 ^ r , (2) Given the complete elliptic integrals of the first and seconcí kind,

K = / ' T — ( ) ° (1 - k^ sinê)^

and ^/2 0 ? u

E = / (1 - k^ sin'ê) ^ 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

înt = 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.

"â 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_ = ô^ (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)

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