477

v. z NRL Memorand vi1 Rpei tu8

IN*

* /BERTHA A Versatile Transmission Line

t- and Circuit Code

s.. D. D. HINSHELWOOD

__ Plasma Technology Branch

JA YCOR, Inc.Alexandria, VA 22304

~. November 21, 1983

This pwumch was poneored in part by the Defene Nuclea Agency underSubtuk T99QAXLA, work unit 00023 and work uit title "Ion BeanGenergtian- and by the U.S. Department of Energy.

I-T

*~~~.sNo 2 4 m9J

-LL f . ,VLRIEREJAOACR

A.......

4k

SECURITY CLASSIFICATION OF ?NIS PAGE 'When eat Eneed

PAGE READ INSTRUCTIONSREPORT DO~MAE AIM r P AGEBEFORE COMPLETING FORM

1, IREp,RT NUMUIR 12. GOVT ACCESSION NO. 3. RIC:PIENT'S CATA,OG NUMIER

NEL Memorandum Report 5185 h__-Al_______;-_4. TITLE 'and Subtitle) S, I F .1 O T "'I0 COVERED

BERTHA - A VERSATILE TRANSMISSION LINE AND Final report

CIRCUIT CODE Jan. 1, 1982 - Dec. 31, 19826. PERFORMING ORG. REPORT NUMUBER

7. AUTwOR(a) S, CONTRACT Oi GRANT NUMBER(@)

D.D. Hinshelwood*

S. PERFORMING ORGANIZATION NAME ANO ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKNaval Resec L AREA I WO RK UNIT NUMIERSWaval Research LaDoratory DE-AI08-79DP40092;Washington, DC 20375 47-0879-0-3; 47-0875-0-3

11. CONTRIOLLING OFFICE NAME AND ADORESS 12. REPORT oATE

Defense Nuclear Agency Department of Energy November 21, 1983", Washington, DC 20305 Washington, DC 20545 13 NUMBER OF PAGES

:.'. 98

14. MONITORING AGENCY NAME & ADoRESS(II dlferent (treo Controillnd Office) IS. SECURITY CLASS. (of this report)

UNCLASSIFIED130. OECLASSfICATION/ DOWNGRAOING

SCHEDULE

IA. DISTRIBUTION STATEMENT (of tis Report)

Approved for public release; distribution unlimited. "

17. DISTRIBUTION STATEMENT '*I rho abstract entered In Btlc 20. It dtfeent ftome Report) . . :

AIf. SUPPLEMENTARY NOTES / y

*JAYCOR, Inc., Alexandria, VA 22304 - Present address: MIT, Cambridge, MA 02139 ' !.

This research was sponsored in part by the Defense Nuclear Agency under Subtask T99QAXLA,work unit 00023 and work unit title "Ion Beam Generation" and by the U.S. Department of Energy.

19. KEY WORDS (Continue on reerese side It nece.e.ry and Identity by block nlumber)

Transmission lines Magnetically insulated transmission linesCircuit codes Imploding foils

. Pulsed power

20. ABSTRACT (Continue an reere side it nelesary end IdentIlY by block number)

An Improved version of the NRL transmission line code of W.H. Lupton is presented. The cap&-bilites of the original program have been extended to allow magnetically in'sulated transmission lines,plasma opening switches, imploding plasma loads and discrete element electrical networks, forexample, to be modeled. BERTHA can be used to simulate any system that can be represented bya configuration of transmission line elements. The electrical behavior of the system is calculatedby repeatedly summing the reflected and transmitted waves at the ends of each element. Thisprogam is versatile, easy to use and easily implemented on desktop microcomputers.

DO I 0,A, 1473 0IION OF I NOV ,5 IS OBSOLETE$ISN 010 2- 014-6601 SECURITY CLASSIFICATION OF THIS PAGE (WhIni D re Entered)

%'

-*.--.*--**. ---- ----

7.-7 . 71--77

CONTENTS

* INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . 1

BACKGROUND ................................................... 2

DESCRIPTION OF PROGRAM ........................................ 5

EXAMPLES..................................................... 23

LIMITATIONS....................................................54

SUMMARY ....................................................... 56

ACKNOWLEDGMENTS ............................................. 56

REFERENCES....................................................56

APPENDIX A - JUNCTION COEFFICIENTS.............................57

APPENDIX B - SAMPLE PROGRAM...................................60

APPENDIX C - PROGRAM LISTING...................................66

APPENDIX D - PROGRAM SEGMENTS TO CORRECT FOR

CHANGING IMPEDANCE ELEMENTS .................. 91

Accession For -

NTIS CRA&I

A.41

BERTHA - A VERSATILE TRANSMISSION LINE A.ND CIRCMi CODE

INTRODUCTION

* Transmission line codes have many applications ranging from the

simulation of pulsed power devices to the solution of problems involving

electrical networks. BERTHA is an improved version of the elegant code of

W.H. ].upton,l which has been in use at the Naval Research Laboratory and

elsewhere for many years. The capabilities of the original code have been

extended and the general format has been changed to yield greater versatility.

This program is capable of sixmulating any system that can be represented

by a configuration of transmission line elements, including pulsed power

generators, magnetically insulated transmission lines,' discrete element

S electrical networks and their mechanical, thermal and fluid analogs. The

C-. program can handle any numbers (limited only by memory) of line elements (with

arbitrary initial voltages and/or currents and optional shunt and series

resistances), series and parallel tees, self-break and command triggered

switches and reactive components. Arbitrary loads, which may depend on time

as well as various electrical quantities (e.g., a Qiild-Langmuir electron beam

diode with gap closure) may be included, as may variable impedance line

elements. An example of the latter is an imploding plasma load3 which may be

represented by an inductance that increases with time. In addition, external

waveforms may be fed into the configuration.

Of particular note is the fact that this program was written with the now

ubiquitous desktop microcomputer in mind. The program described here was

written in BASIC for the Tektronix 4050 series microcomputers. Of course,

increases in permissible configuration size and speed of execution will result

when the code is placed on a larger machine.

Manuscript approved July 22, 1983.

'47

BACKGROUND

A transmission line supports waves of voltage and current propagating in

two directions and the sum of these two traveling waves determines the

standing wave voltage and current which are physically measured. As the

.4, current and voltage of a single traveling wave are related by the

characteristic impedance of the line, all relevant electrical quantities may

be expressed in terms of the two traveling wave voltages. The instantaneouis

standing wave voltage, V, and current, 1, at a given location are given by

(see Fig. 1):

I-( 1 2

.4 were 1 an V2 are the voltages of the waves traveling out of and into thewhr V ndV

line, respectively, and ZoIs the ling impedance. Likewise, the power flow,

P, out of the line is given by:

2 2 o

A wave incident on an impedance discontinuity will be split into reflected and

transmitted waves according to:

z I-Z0r

2Z 1

4.z 1+Z

2

VI -!

~-V 2

-'SZ

Fig. 1: Definition of the two traveling voltage waves.

4z,.

. Fg. : Deiniionof te to trvelng oltae wves

. .- . . . . . . . . . .

-.. ... . . . ..... 5 i ~ - . . .. .. .. * . .. .

where ZOis the line impedance, Z1 is the load impedance and r and t are the

reflection and transmission coefficients for the traveling wave voltage.

These may easily be extended to describe more complex junctions and the

reflection and transmission coefficients for the junctions used in this

program are given in Appendix A.

The case of a reactive load would appear more cumbersome to treat since

the reflection from a reactive load depends, in a somewhat complicated manner,

on the time rate of change of the incident pulse. However, the problem can be

simplified by noting that an inductor, for example, is actually a very short

* high impedance transmission line element. The line element impedance, Z, is

related to the total lumped inductance, L, by

L -ZT

where T is the one-way transit time. Likewise, a capacitor is a short, low

impedance line element, with the lumped capacitance, C, given by:

~~C C- /Z

The representation of reactive components by short line elements in effect

replaces the differential equations involved with finite difference equations

and the operations of integration and differentiation are replaced by simple

arithmetic operations.

44

Sq. -. . 4 - -:% ?7 .. . - -1

In fact, entire networks of passive lumped parameter elements may be

represented by transmission line element configurations by using the

equivalences shown in Fig. 2-a. For example, the simple RLC circuit of Fig.

2-b may be modeled by the configuration shown in Fig. 2-c.

-' This code may also be used to model the many analogs to transmission

lines and electrical networks. For instance, multi-layer thin films may be

simulated by strings of transmission line elements, with the indices of

refraction being represented by the line element impedances.4 In general,

then, a large number of physical systems can be modeled by configurations of

transmission line elements and resistors. To predict the behavior of these

systems it is only necessary to account for the waves running around the

configuration.

DESCRIPTION OF PROGRAM

Rather than following particular waves around the circuit, the program

keeps track of the instantaneous values of the two traveling (voltage) waves

* at the two ends of each line element: the wave, V1, that is incident on the

junction from within the element, and the wave, V2, that is leaving the

junction and traveling into the element. V2 is made up of the reflected

*portion of Vi from the same line element and the transmitted portions of Vl's

from the other line elements connected to the junction. For a configuration

comprised of N elements there are thus 4N quantities which imust be

recalculated at each timestep. There are N pairs of equations describing the

transit of waves within line elements: the incident wave at an element end is

the wave that left the opposite end of that same line element at the

5

(a)

HIGH

(b)

CT

(c)

R

Fig. 2: (a) Representation of reactive components by transmission lineelements (b) simple RLC circuit (c) transmission line model.

6L i r

appropriate time in the past, determined by the transit time through the

element. The remaining 2N equations necessary to close the system describe

the reflection and transmission of waves at the junctions.

As an example, consider the circuit shown in Fig. 3. A line element of

impedance Z1, transit time T 1, and initial charge Vo is discharged at t-0 into

an uncharged line of impedance Z and transit time T2 that is terminated by a

resistance R. The eight equations for the two voltage waves are:

Vl(1,t) - V2(2,t-T )

V 1(2,t ) - V 2 (,t-T 1)

V1 (3,t) - V2(4,t-T 2 )

V1 (4,t) - V2 (3,t-t 2)

~v2(11t) -v (1,0t)

Z2 -Z 2Z12 1 00+1

V2(2,t) - V1(2,t) z+z + V1(3,t) 2Z

Z-Z 2Z2

V2(3,t) - Vl(3,t) +- - Vl(2,t) Z 22 1 Z, +Z 2 1 zI+

R-Z

V2 (4,t) - v1 (4,t) R-Z2

4 7

447

I 2 34

.4 Z

R

Fig. 3: The transmission line element configuration used in the example.

The first four equations are in a convenient form for programming; the

last four can be put in a similarly convenient form. The junctions handled by

this particular program join at most three element ends, hence, the general

expression for a wave leaving a junction is:

V2 (l,t) -a 11 V1(lt) + ajl VI(J,t) + 8KI V1 (K,t)

where I, J and K are the element ends adjoining the junction, all is a

reflection coefficient and a and aKI are transmission coefficents. The

values a1 l, 5J1, 8K1 , J and K are sufficient to specify the value of the

reflected wave leaving an end I and they are stored in an array M(2N,5), in

locations M(I,I) to M(I,5), respectively. In the case of a simpler junction,

M(I,4) and M(I,5) are set to 1 and M(I,2) and M(I,3) are set to zero. For the

example of Fig. 3, M would be given by:

M(I, ) M(I,2) M(I,3) M(I,4) M(I,5)I (a) ) (SKI) (J) (K)

1 0 0 1 1

Z2-Z I 2Z12 210 3

Z1+Z2 Z1+Z2

Z C Z- 2Z 21 2 12 0 21

R-Z4 - 0 0 1

9- * **

" * -, . * *- - - - - - - - *4-. .

The equations describing waves leaving junctions are then all in the form:

V2(It) - M(I,1) * VI(I,t) + M(I,2) * VI(M(I,4),t)

+ M(I,3) * VI(I(I,5),t)

1' The memory requirements of this program are reduced in two ways. At each

timestep, Vl(t) is calculated from V2(t- T) and V2(t) is calculated from all

of the Vl's entering the junctions. Thus, it is only necessary to know the

value of Vl for a single timestep, and so VI is dimensioned VI(2N). Also, it

is never necessary to know the value of V2 at a time farther back in the past

than the transit time of the longest element, LO, and so V2 is dimensioned

V2(2N,LO). The entries in V2 are continually being written over in a cyclical

fashion.

It remains necessary to incorporate initial voltages and currents. These

are handled by noting that these initial standing waves are just

superpositions of two traveling waves, determined by:

.V V+I z

V. 0 2

V -I zV 0 0- 2

A as shown in Fig. 4. At the start of execution, the array V2 is back-filled to

10

,., ,-" ," ,' -, . ,.",. , ... .- .-..- ,. -, -' .- .- , , - - .- , .. .-., .. - . ., . .

*VI -----,- V 2

St V. v(t-Zo)-vo- I (t-O)-O

,%

ViV 1(X

.x. V0

2 xV2 (X)

V0

2I. x

Fig. 4-: The traveling wave representation of an initial standing wavevoltage.

N *

V ,--

include these initial traveling waves. For example, in Fig. 3,

if T1=4, r2=6, V =10, the array V2 would be given by:

I V2(I,1) V2(I,2) V2(I,3) V2(I,4)

1 0 0 0 0

2 0 0 0 0

3 5 5 0 0

4 5 5 0 0

5 5 5 0 0

6 5 5 00

At the start, the first row in V2 corresponds to the present time and rows 3-6

correspond to times t--4 to t=-1, respectively. After the first timestep, VI

-. will be given by:

Vi(i) V1(2) V1(3) V1(4)

5 5 0 0

and V2 will be given by:

I V2(I,1) V2(I,2) V2(I,3) V2(I,4)

2ZZ".,1 22

1 5 5*41 5 2 0Z2 +Z1 Z2+Z 1

2 0 0 0 0

3 0 0 0 0

4 5 5 0 0

5 5 5 0 0

6 5 5 0 0

12

4

At this point, the second row corresponds to the present time, the first to

t--I and rows 4-6 to times t--4 to t=-2, respectively.

The versatility of this program stems from the fact that the

configuration is alternately considered to be a collection of separate

" elements and then a collection of separate junctions. This approach is also

used to facilitate data entry. First the properties of each element are

entered in turn, then the junction properties are entered. The junctions that

. can be used in this program are shown in Fig. 5. The properties of the

junctions are stored in an array J(N+l,4) (a configuration of N elements will

• .have in general N+l junctions). J(I,l) is the junction number and J(1,2) is

* the first element end entered. J(I,3) is the second element end (0 for a type

I junction) and J(I,4) is either the third element end or the value of the

junction resistance (or 0 for a type 2 junction). For the example in Fig. 3,

J is given by:

I J(ll) J(l,2) (J(I,3) J(1,4)

1 1 1 0 106 (or any large number)

* 2 2 2 3 0

3 1 4 0 R

413

4%

* 13

.,,

- -~ . -. -

-. 4

*~4~

I..j4

X~ll 2'.4

-.4

-. 4

.4

-. 4 (4-Ad'~--( 4

LJ

I 2

.~p.

6

Fig. 5: The junctions used in this program.

4,

.4,

14

/4

A-

-. -4..-... - - -. . -. -. --*~f ~ 2 - * ~ -= S.- A * . *

For the purpose of illustration, a simplified version of the program is

*shown in Appendix B. As can be seen, the following variables are used:

N Number of elements

Ti Total time

WO Number of plots desired

L(N) Transit times of elements

Z(N) Impedances of elements

VO(N) Initial voltages of elements

10(N) Initial currents of elements

WI(WO) End numbers for which plots of voltage are desired.

The program is easy to use. The first step is to number the junctions,

elements and element ends. The only restriction on numbering is that element

I must have its ends numbered 21-1 and 21. After entering the numbers of

elements, timesteps and plots desired, the element properties are entered

sequentially, followed by the junction properties and finally the element ends

where plots are desired. The mnemonic given for the type 6 junction (line 490)

* means that the first element end to be entered is the one that has its hot

* connected to an opposing hot and its ground connected to an opposing ground,

etc. The element ends for the type 6 junction shown in Fig. 5 would be

entered in the order 1,2,3.

* This method of data entry allows straightforward input of arbitrarily

complicated configurations. It can be seen that the longest section of the

program is the transformation of J to M, which only needs to be done once.

* 15

The actual program execution is contained in the short section at the end. In

that section TO is a pointer which indicates the row in V2 corresponding to

the present time; it cycles from 1 to LO continually.

The version of BERTHA currently in use has several features in addition

to the sample program described above that permit configurations to be

modified during program execution. Junction types 7-9, shown in Fig. 6, are

switches. The closing [opening] switches are just resistances whose values

change from infinite [zero] to zero [infinity] at a preset time or voltage.

These are simple to incorporate as switching just corresponds to a

multiplication of the reflection coefficient by -1. It should be noted that

the inductive and/or capacitive characteristics of a real switch may be

modeled by adding suitable short line elements as shown in Fig. 7. To take

into account more detailed switch characteristics such as resistive risetimes,

however, variable resistances (described next) must be used.

The values of the resistances in Junction types 1, 3 and 4 may be varied

during program execution by external subroutines. The time, voltage across

and current through each resistance are supplied to the subroutines at each

timestep. The line element impedances may be varied in exactly the same

fashion.

External waveforms may be fed into the configuration from any Type 1

Junction. In this case, the injected waveform is added to reflections from

the Junction resistance. By convention, the program assumes that the supplied

waveform is the open circuit voltage, V 0(t), that would have been measured at

the end of a line of impedance equal to the Junction resistance R. The

16

.................................

177

voltage that is actually fed in from the junction is that which would have

been transmitted oaln elemento impedance ZO i.e,

zV (t) 00 +

0

as shown in Fig. 8.

Output plots are available for the standing wave voltage, current,

impedance, power flow and energy flow at any element end. It is also possible

to save for plotting any values calculated in the external subroutines, such

* as the radius (at each timestep) of an imploding foil.

The full program listing is shown in Appendix C.

Variable impedance elements are indicated to the program by typing a"V

after the element (initial) impedance. Variable resistances are indicated in

* the same manner. If an external waveform is to be fed into a Type 1 junction,

an "I" should be typed after the junction resistance. (Both features may be

combined at a Type 1 junction by typing "VI" or "IV".) (This particular

version has a limit of nine input pulses, variable resistances and variable

impedance elements, each.) It is assumed that the subroutines for variable

* resistances and impedances and the values of the external waveforms are stored

on disk prior to running the program; it is the file names that are entered.

The starting time and plotstep (line 430) refer to the output plots - valuebi

are only saved and plotted at every plotstep. The Type 10 junction (line

1210) is a minor convenience which allows a string of Type 2 junctions to be

entered at once by entering the end numbers at the beginning and end of the

string. When a configuration includes loops (Fig. 9) the number of junctions

will be less than N+l. In this case, zeros should be entered as the junction

types for the missing junctions.

18

(0)

Z=R o V V t)

v= V (t)

zo0

R

V2 VI R+Zo+ V R+Zo

Fig. 8: The waveform supplied is assumed to be an open circuit voltage (a);

the injected waveform is this open circuit voltage multiplied byZo/(R+Zo).

19

L% ". '. ''" ."" . "" , "" . ". .'.",. ', ,',". ." " " .'".. ..'. . . . . . . . . ..,. . ..". ..-'. .-. '.'.' .' . , ,"-' - "- - ' -

---..-... S.0..%......~*..-......-S.. . S S S* a-..-.

.4

~SS

'p.

4.

4-.

4

-4.

Fig. 9: An example of a configuration which includes a loop.

4..

A

.5.5,,

4..

5,*44

.4

4*

a.

AN

*~6

S

4.

20

-a,

**~*f***5** . . * . . . .~ . .5

-- . 7 .- . e-~7.

For each output plot, the end number should be entered, followed by "V",

"I", "Z", "P" or "E", depending on whether the plot is to be of (standing

wave) voltage, current, impedance, power flow or energy flow, respectively.

Zero should be entered as the end number of plots of values calculated in the

*external subroutines, followed by a single symbol which will serve to identify

the plot. For each of these values to be plotted, a program line is needed in

the subroutine, and the values are then saved for plotting in the order in

which they were calculated. This is all illustrated in the fourth example'I

given in the next section.

Before execution, any necessary subroutines and waveforms are pulled from

the disk. At every timestep during execution, the switches are examined and

flipped if necessary, input pulses are added in, values are stored for plots

if needed and the variable resistances and impedances are changed. Then the

J P- M segment of the program (which is written here as a subroutine) is called

for every junction for which the reflection and transmission coefficients will

have changed - i.e., those junctions that contain variable resistances or that

border variable impedance elements. The bookkeeping necessary to keep track

of these junctions is handled by the arrays P1 (a list of junctions where

pulses are to be injected), Q1 (variable impedance elements), RI (variable

resistances) and Jl (a list of junctions whose coefficients must be

updated). The intermediate arrays J9 and Q9 are used to form Jl. The switch

.parameters are stored in the array S(SO,3), where SO is the number of switches

in the configuration. S(I,l) is the end number of the switch, S(1,2) is 1

for a command triggered and 0 for a self breaking switch and S(1,3) is either

the switching time or the breakdown voltage. At each timestep, either the

time or voltage (depending on the type of switch) is compared with S(1,3). If

the switch is to be flipped, the corresponding reflection coefficient is

21

,!!I ,/ .. u . , - - -- i -. r % 1 ."-' -'- ' ' '/' ' ,° :}-." "

- . ' ? ' : : . '" ,' -" '-" -' -- .' - " ' ''-

multiplied by -1 and then S(I,3) is set to a very large number to prevent the

switch from being flipped again.

The variables Ni, N2, R2, V6, and 16 are furnished to each variable

resistance sub-outine. NI and N2 are the two end numbers in the junction

containing the resistance (NI is the first junction number that was entered

and N2 has no meaning for a type 1 junction.) R2 is the current value of the

junction resitance and V6 and 16 are the voltage across and current through

the resistance, respectively. The new value of the resistance calculated in

the subroutine should replace the old value in R2. The variables Ni, N2 and

Q2 are furnished to each variable impedance subroutine. NI and N2 refer to

the end numbers of the variable impedance line element and Q2 is the current

., value of the element impedance; again, the calculated new value should replace

this in Q2.

When the impedance of a line element is changed, the values of the

traveling wave voltages on that element must in general be changed to take

into accout the IdL/dt and VdC/dt terms. Since the change in element

*. impedance is assumed instantaneous at each timestep, no charge or magnetic

flux can flow into the element during the change. The charge, Q, and

flux, , are given by:

Q - CV = V+

%'V

- LI = r(V1-V 2

'2

".

22

-. , , . ." .". ,* .', --.-.,-. . . , ..... -.,.,... . . .. .. . . - -.. ,.: ,..,/ ,- .- . .

The requirement of charge and flux conservation determines the new values of

S1 and V12:

(Zv 1/2 [V1 ~-- 2 Z

1/2 N + I V1 1

2 2 '9z

where primes denote the new values. V, and V2 must be corrected for each

timestep of the element length. A general subroutine for this is given in

Appendix 4. This correction is greatly simplified when the variable impedance

element is used to model a changing lumped parameter element. In this case:

V -V

1 1

.

V2 '12

for an inductor, and

V =V z /Z

- 2 - 2 z/Z

for a capacitor.

EXAMPLES

Consider the simple RLC circuit shown in Fig. 10a, with the corresponding

line element configuration shown in Fig. lOb. The data entry and output are

shown in Fig. 10c. The exact analytical solution is also shown on the output

plot, and the agreement is seen to be excellant. The accuracy of

approximating reactive components by short line elements is related to the

*- parameter T/T, where r is the line element transit time and T refers to the

23

time constant of the circuit. A larger T/T Will give a better approximation

* at the expense of added execution time, since in general the line elements

corresponding to reactive elements will be one timestep long. For the

* configuration of Fig. l0b, T/T - 10 .The output for a similar configuration

* with 5 ns long elements and correspondingly different impedances, for

which T/r - 2, is shown in Fig. 10d for comparison.

A second example -an RC circuit with a changing capacitance - is shown

in Fig. 11. Note that the traveling wave voltages on the variable impedance

element are changed at each timestep in the subroutine. Again, the exact

solution:

V - (1 + .t-

* is shown on the output plot and the agreement is excellant.

As a third example, consider the hypothetical magnetically insulated

transmission line experiment shown in Fig. 12a. A charged pulse line is

connected to a short section of unzharged line at t=o by the closing of an

* output switch. This short line is connected to a short magnetically insulated

section which is terminated by an inductive load. The parameters of the

various segments are indicated in the figure and the experiment could be

modeled by the configuration in Fig. 12b. The resistor in junction 5

- represents losses in the magnetically insulated section. Assume that the

- following simple model is to be used for the magnetically insulated section:

24

the loss (i.e. shunt current through the resistor) is zero if the current

through the line, Io, exceeds the self limit ng current, 5 which happens to be

02

I SL = 56,600 y ln [y + ( 2-1) 1/2

y = 1 + V[MT]/.510

for this case. If I < ISL' then the loss current will adjust itself so that

the current into the line, IIN , is equal to ISL, i.e.,

I shunt o ISL

ISL o 0 I SL

The variable resistance subroutine for this model is shown in Fig. 12c. The

data entry and output for this example are shown in Fig. 12d.

An imploding foil driven by a capacitor bank 3 is given as the last

" example. The circuit inductance may be expressed as:

L - L Cn(R/r)

where r is the foil radius and R is chosen to include both the foil and

external circuit inductances. The equations describing the implosion are

then:2 12'"d r .1h I

dt 2 m r

dL .2h drdt r dt

25

where h is the foil height in cm, m is the foil mass in gm, r is in cm, L in

nH, I in MA and t in is. For the example here, r(t=0)=7, h=2, =.02 and the

foil is driven by a 200 pF capacitor bank which is charged to 70 KV (these are

typical of the parameters in Ref. 3). The circuit, configuration, foil

subroutine, input and output are shown in Fig. 13.

Cumulative errors may result when the element impedances are changed

during the course of program execution. For example, in the present case,

assume that the calculated foil radius is too small (and the inductance too

large) at a given timestep. The larger inductance will tend to decrease the

calculated foil current over the correct value on the next timestep, while the

smaller radius will result in a greater foil acceleration for a given

current. These effects will compete but for a sufficiently small radius the

latter will dominate and the calculated solution will diverge from the correct

result. As a test for this, the conservation of energy and magnetic flux are

checked in this example. The foil (kinetic and magnetic) energy is calculated

in the subroutine and compared to the flow of electrical energy into the foil

that is calculated in the main program. The instantaneous value of the flux,

LI, is compared with the integrated flow of flux into the foil, fVdt, both of

- which are calculated in the subroutine. The variable W3 refers to the row in

the plot array, W, for each particular calculated quantity. Note that during

* input the subroutine plots are entered after the standard plots.

K Runs for two different timesteps are shown, corresponding to a foil

collapse in 30 and 300 timesteps. Note that the energy and magnetic flux are

more closely conserved for the smaller timestep but there is little change in

the calculated values of the foil radius and current as the timestep is

decreased.

26

-t (a )

* IOnH.. '-0001

10 nF 191

a-. (b)(b) 3 4

0 0

Fig. 10: (a) an RLC circuit (b) the transmission line model.

27

4.. . . . . . . . . . . ..o

0.0

CD N0

-E-4

4) L~ o(A 0- -CD 00 u L

0 0-4- - tj. -u

L 3 CC - C *-

N 0 (Uw 0 C

-w 0)1 0L M~ L J'-0 CJ- L*-0L0

a# 40 0*0 0 0 0. '

E - 0 301-

COL.~ LLI CL L L > 0

-. ~ '- '-L L JJ

44-9- c-.'. m1 03 01-n c~ 0n20 2

.00 0 E0 e0'.-. C -CC-o v

M- 0 > CC~ C 0 u9- - - .0" M mc 0 -0

z- .C.Z C. Zc r z

LLL. LL LLL.LLL L bo

WWW WW WWWWWW w

28

z

z (0 1

cr *1-e.

M++ ~ (MJMM

- .L (J + +

cr - 0 (

z cr (o ZI~ cU 4c:Z <

ow + ILoo) 0

z WP"

z p n L) - . 6

U) -1 zJai N nV D4 m - l

Z :3 :) )

CI C C -j Im

z- m- N -

>- a.: r 4

AC2 <S -W*-J~

U) C + + V - 0 00IC -j (0--J~ - L - -- L-

I-- w5 L- CLLa- I e )c E 0 z a,-->>(0 - m WM

0 IC 03 0 .0 /* I-~ 0IL 0- 0..L -'

Ci w L. '00LLL L 03t-Jm - '-5w-',-waw CL-\ >~ > j Az -CL. Zl~ GGffnll z fa~ C-

1 30 aCE-cIE>Es 0 V)-~0

Wi 0 no - O D Z z _j-im L 0 z1- z z zz w a- - Qo.

29

-7. C

00

0 (n 0

'.4

4

4-I

* 0

444

300

zwm

wI

0

4

0

rr

311-

lI

C=5+t 1 1l

IT

(b):".5 ns

Fig. 11: (a) A circuit with a changing capacitance (b) transmission linemodel.

3760 REM ------- SUBROUTINE CTESTI3761 03=0.5/(5+T*T2)3762 FOR 11=1 TO LO3763 V2(NI.I1)=V2(NI. J1103/023764 V2(N2,I11=V2(N2,11)*O3/023765 NEXT 113766 02=033767 REM ----- END OF SUBROUTINE

* Fig. 1ic: The subroutine used to model the capacitance.

32

C . - - *" . .%. . . - •

-7 6A A 77777=

w >P 0

> 0~ CL 4

A Lr4 -C- 04 DOin w -:

41 C1 L - 000 0 L aNc -10 c - 44

61 0G0 01 w C:LMCE .- C - C

a) 0 -u 00

0 - --- -)04CL-0 0 L 0IL6 CU 44I.4

ca.L L - OLOL > ua @640. 0 L %- %..82.0C 0 -0 -0 V

0 L ** 000.0 0 4)

-. 0 -C -L -L L '

0 3 CO.0 c "

-C -. 0 20 2- :)0615a- C *-c Ccn - 0 >6 do 0uE s.- ".aE CVCV c V2 0 0 3 c 3c cC-. C -,61--t6 W w

00

L LL L L L LL L L*we 6150 wwww 616 v1 6

33

z C9

o ~~ cW 0)D w ~

A to "~WW M-

Z~ 00

oL z l to-PUz <I

II 0ea 1 -4

>W N()( 00- 44

0004 w 0

zZ 0 1.

z9c w (n

cD wo,Zw N- 4j-o~

M 0 z C ISEla N w <Z

0 - w L

zto 0 N a_ 0U) -

amo w

E- 0- w

z w 0 ZN 0

z E 0.-14 44 W49- 000(D--CFEO OO M (M (n Z

I-- Wi L0 L4 C- G U 0

6-4 1-- V ED-. W VC W WW-CON' --w CZ W W ~ n rO. uCL ) CaZ, <gS>s E- w- .- 0- Ws

w a' L..-OOD LL- _j _

to 4qu 0 - - Z W4 CL0

* I )0-0.EOOOOO~ 34

Z-LAJw co 0

w-o-4

o~r-0

-4

0 1LWU

04)

w 0

,6Jx

--

0-

C r-

35.

, . ., . , . -, -: :-- T- T T -. .- . . . ,. . . .. :' - . - .* .--.- . ' . . , - . .- *.. .* . :. '.- . . . ' -.- .... . . - .

2n 5nHn H 50nH

360kW

18ns - n-3nss Ins ------ <Ins-

(b)

(D 2 3 4 5 6 7 a 9 10

18n .5n 3n In I

vo ' 360kV

Fig. 12: (a) Short magnetically insulated transmission line experiment (b)transmission line model.

4660 REM ------- SUBROUTINE MILOSS4661 R2=1000004662 17=-(VI(N2)-V2(N2,TO)I/Z24663 CI=I+(V6 MAX 0)/5100004664 G2=(C1t2-1)1t0.54665 I8=510000/9*G1VLOC(C1+G1*G2)-4666 IF 17>18 OR I8=0 THEN 46684667 R2=V6/(18-17)4668 REM-----END OF SUBROUTINE

Fig. 12c: The subroutine used to model the MITL loss current.

*36

'p f

*........... - * - *. . - *

- C,

0 (%sP- CR0 -0 "

x. EDL-'LC

CD -L n )- 0

V) - CU

0 - -- -- 0 O 0C0) L ) 9 0

N. - ~ LO CLL

@-% 0) -'JO .- -CDi- 3~ - c D (a 0 "

LO CCC- C v- W C

(A E EE C- C - 1

o "0- V 4D - ' - -NLI) -n LL -- --)

L) W L- 00 LL0

0000 - r"ri- C C -s- -0 C-.4 -~ 0 c00 0 ww 0

weG w.-u-wo-cIDc- c

G w- U3iWuG 0I~ C -O cu w 0UC0 3 0 m- - -- 0 c1 D 03 i D --

C .. 0. 0 0W -LL L 0 * . FC. LL OOO *-OL 4FIr rI0 :.a3- 00000>> IL. .- C'-L.

0 3JcCG'JO>oIC CC

0.- 01W w. 0 IG

o C C C C C Cf flflcw ~ ~ ~ E -04.0 w 22 jwU JWuiwu iu iu

LGD ~wr~' *--0-~-J37

z _ 1 11do0o - I I I a:- Mm) IAI I I c 9 z LC

at I~ I IZ Wf rM M 2 MI I IUr

z OG G VCD ED~ Luj

~ -'LWWWL + + + <

-2 0 (JU) -0n-0 - 0

z I EE E Ei1o Z I :)3: )I ( 00

u < I VVVV I < -

G MM I C C CC IJw G+++++ - I WLLWLJI

>WWJ WWJ~~ PoC'

m pnU C-0

z wZt w 0W@ c.

O -j-mvmm 000 -EEEEEE W -L L 0_- CL 0~ w 0 OGG & z 03 3 3 D I lL L 0

.S . .* . . <CZZZzzz Z :)j(a- NWWWWW m 4 LIGL) U I

(a . wwwwww 0-Z K) N- N 0- N r_4C CD G m

ct I w

0 r- M5 Ca--) 00c

(.2c OGG& W L~~-OJW 00o0 + I + + I CL 0 - - - C0Z

wa 05 - T) ( _.jwWWWWW> )--0.0a.0.- W_j t E 0l 'D 0 &MM I-WE ECW0- - wa 00 w CCaMMM w _- _---Z -4

rc 4. E0 r_ . . .

U E5 00--' -tw0W -1

w L - 0 L.-O0L- -- _N\JNI0..J WA 0 - C 00ait - -CJ

Z AC 0) 00.L 0 V)

4- *-.L I- Q.j '-0 -@-(O.'.WW W-10-W E\1~Il E I- 00m V

P-- Li. Lf0 5 5 5..ff. L CL IS 0L.Li 0 0 -J LL Z m-4--

I- OLLI- zZ Z La.Inn n. u D*QCD

38

VO . IV' - tO7

Zol- z 0 -w *., W LwiLW

Cc I--

0

-4

z0

0

(n-

JJ

4-i

439

3nH

(a)

20OuF L (t=0) =4nH.07 MV

(b)

L =.051.005 L =.051 *005

Z = 2.5x10- 4 225x 10- 5 Z=.L-,l 1.4

* V= .07

Fig. 13: (a) Imploding foil circuit (b) transmission line model.

%' 40

.4.I

N c

-. (9 --4

+ -N N 0L

M -- - 0

w (Z 4+

0 rLJ ZF- ' N* - N Z~-O--0- N.Z ir

W ~ - L.) r- 0t~ JN

_j - W 4c W U x *," .- )- 0N -- -z

4c-u).-.u N *. NN + -N WU m0rc ~ D Z 0' Z M V-.C Z -j (

_fl M-'if) N * N < LaJ* >Zu- -L.

m - N .ix N 0_ -=- +n 0- 0

Z ZZZZWu -SX N -J -*0z l*O)- -CL 1 f f 1 1 1 w 0 x- - - Z

>- xx4 - ->Nm- L. IL I- --L I..~ II I I NL)JU -*NNN.J-.I IIZXI liI II. I I I (m i >-( + I+I II W -- - ~ - x - -

I l I I I - AN -X)C iN fI +' it- I

I I11 I I (m1 r I -m I-N-N I m I4 1 N'I

U.JwLw-jW-LL-N-N-NW-W-W--W-w e

41

.4J

ILL

- a. xL -C-I 0.

0..-W~aJL0. -4

c.0 1S

0 > - 00000..~. 0 0

6- W. - 000000LCI CL 'A

N -L Ar4 .. 0 L LL L LLW'C 003fl -WN - 000000

'0 0to -C V K) L L LL L L

0 L 0 000000 .

C -N M C C C-w4

L0 C (E -C.- o wiw'w*Co @1W 00 0 Q.0O..O.L2 C C 'C C ...

-V 0 WW- 0 - u - o~ C-, - L (L CL *-. C-q 4

a - C -C s

0. 0" LJL--) - -n - ---4------

cCL. L L- OL0 -O0L > >> > --& 414. 00OL 4- .-N'.E 0 4... 0 10 'o -0 VV -0. D 00 0 > wC c SwcC C C CC C.c

@16 W L. >%W>l L0O -C.0 0 0JO J'.C 4L 0-L 4L L LL L L L

0 c o .ncc m .nnnmnm n.0C E 0 0FE06E EE E C-E . *

W)E - GG .C -C4- C c c cCC cn0- 0 >> w 0 0 0

3 0 0 'c C c Cc c c c* %C W w al a)WwwWW (

O N N 11035

a. 0o. r- z. z jc~r C Z.C r rm

L LL L LL L L LL LL L L L LLL ;u

I- " q- 0- V - -0- -" -0- 4 - "- 0- 4- "- 4-

WWLL LJWW WWWWWW JWWWLLILJ

42

-z w 1 0U) .- I- 0)

toI ~. Iw to z

z Gtsl - I

0" .N - Q - - 19 + + - CM ( + 4-

w >- soEl .

IC9

0 wW Pt)

z r: G

0- 4

-u Z I 4.cr. ii

* ~~ C) enC ca~czz z

r~~ w cc

2 I N - N -C

Z w to 44-

0 -j- NC W 0 0

C* 0. M* - -LLM L

I I, LU 0)2

w0 W w.- Nw0-QC 0 W-WV 0

Z I~ 01w CL(

r~~c M-- Eo

-0-.EE O O MM w.......

0 z M-00) M C3

w 0 0Z-, -ZZ Z w 5, 0r- ( 940

4c. 43t O

z V

WSWo

w Low

W -0 II 0

W44

41

sw

0

4.4

0:

'-44

0

0

C.,

444

C+N% J

04U,4

WZ'4) 4~4

4..4

4.'

4.z

4JJ

0.

0:

45U

ZN INIWNW NL W

10 0Wi- o c 4

w4 0- S x(

-S4 1.1

0

.4., -44

P-4

"4 S.z44

-. 4 . -46

w U) w N

IZI.- H Z 1-II 0IWO-X '~

L. 0- It

t.-4

cn ..

S 4-4:

m =

a S-

C: 0 0-

00-S I-

Si l

H US I-.

47U

cm

OL .. " -4 U. 0"

N - 0 L O C -- C

9- - L. - -

to ID ) m1 L-NIL LWLL

cm co 9- w-9 -99

0 L 0 00 0 0 00. - N--- M (C c

N- 0...000 0 o0 0w w uw

alw C -D - 00 00 -

E E C1 - C C- WNJ-Jto ~ se- t-f

01 -- C"P -- - N-- N) L.4LC- -0 -C C- t-L 01 041 W 4 LWWMiL WJW

9-4 CL LU --0L0 0 L >>

No 411 00 00 L .O.o..4.a .

E-. 0 0 10 **) 14 m -0 1.4 .-0 1

* 0C - CL 0 0 000 0 0 0a s GoW0 > 0 >.LfL1-----------

LL -Z -L -L L -OL L>L>LL>L>

SW E 4C do W do CC GoWW0'I C~ .0 c-m0.0 .000000 n.0

0n E- 0- L- aL LL 0

L0 0 Df~lf n.f.Ol 0-3 333: ):

41£F.- MCD -C--CC cCCC c.0-0 > >4 0w

E - - a..i.E CVC C 10VV c 0c1 0 0mM3 0 0 3CC~ c3 C cC C cc

41-O 4C 4)44444 4n1w141414w41 0

r-~ 0 -r .c rrzrz rrrrzr -

L LL L LL L L LLL L LL L LL L

48

z i

0- i 0) 4

G~ IC

z G C2~ 0 (9 I o 0 W

4 4)

UJ - M W& & W -+ +- ACwo "W w --

Ix >- wo ( l nC

0 z11 I r.D

Li c

w w - -

Z tiD r 2

P a -C Zj w>-z N 4w .

o - -i - tv o M& -E E a >CL L M W -0-o

IA - < I4 + ZZZ Z :3

to cr Go VV00 0 L.r o 4 OI ~c c w - 4(.0 I >7 'W LIJ ..C..=Z Z N - NY3r 0 -4

(0 0

O w

0 E0. Pwcon I-~0 'C Mw Q. O 0)-C

r ~ C 2O.C - l CL-1 LU E CL :--W 0 U)L( t0 1- 0 Eca < 0- - w - CL wL0w0 ( -0

-- q c - EO0.C C . -'m-nm 04c1 Lu Go 0C U' L - C.-Iz~) 0

z) L -l 0 0 (1)

Ix. '-a -C - CL -L >-'000000 z

4. 40 '-i .- O C- C t -- - -.-Z-Z-Z -

0-~ CL ZOS Elffff a 0 0 M )---

4C 0 -L .4LL L Z iz z

U. Z 0a CZ 1 0-73)Z.Z3 .0 .0 .0 r u_-, & - Q-.0--

49

Wr%om.Jz 1%02

I- XI-

0

410

0 4

0-Li

4

4

500

Z-x

Dz-

%-4

0 -4

0 -4

0

'If

41 0

Ia.

0

"4

(0

51

Dom

I% COD

WN W X

LJ.-4-4

L3

44

0 0,

Ln

0

0

A LJ.4

52j

NW X W

0

*0 -

a 0

4- 04

4- 0

C0

530

LIMITATIONS

An idealized transmission line element is one dimensional, supporting

only TEM waves. Errors may result when this code is used to model two

dimensional structures, such as that shown in Fig. 14a, as the solutions to

these problems will in general involve higher order TE and/or TM modes. Often

these two dimensional effects may be simulated by a clever choice of the

transmission line element configuration. For example, in Ref 6 it is shown

that the structure in Fig 14a can be approximated by two shunt capacitors as

shown in Figs. 14b and 14c.

The finite difference nature of this program can cause errors when the

values of resistors and element impedances are fu-ztions of the electrical

* quantities. These errors may be in the form of a calculated solution that

diverges from the correct solution, or in the form of large numerical

instabilities. The imploding foil in the previous section is an example of

the former, while the latter problem can occur, for example, in magnetically

insulated transmission line simulations. A situation can arise where the

shunt current across a line that is not insulated at a given timestep will

* .. load down the line voltage on the next timestep to the point where the line

becomes insulated, thus shorting off the shunt current, which will raise the

line voltage back up on the succeeding timestep, and so on. This problem has

9... 7been remedied in practice by iterating within the shunt resistance subroutine

to find a self-consistent shunt resistance. In general, when resistances or

element impedances are varied, it is a good idea to check that the calculated

solution does not change greatly when the run is repeated with a smaller

times tep.

54

• (a)

I _I zo

(b)I I

(C)

Fig. 14: A two dimensional structure (a) is modeled with shunt capacitors(b); transmission line representation (c).

55

-9w1

9..

:!! (c5

SUMMARY

This program is quite powerful, permitting the simulation of a wide

variey of systems subject to the limitations described above. It is easy to

use and readily implemented on a small microcomputer.

ACKNOWLEDGMENTS

The author is happy to acknowledge very helpful discussions with W.H.

Lupton, J.R. Boller, G. Cooperstein and A.E. Robson (NRL), J.D. Shipman, Jr.

- (Sachs/Freeman), J.M. Neri, R.A. Meger and Shyke A. Goldstein (Jaycor) and G.

Bekefi (MIT).

REFERENCES

1. J.K. Burton, J.J. Condon, M.D. Jevnager, W.H. Lupton and T.J. O'Connell,

IEEE Cat. 73CH0843-3NPS, p. 613 (1973).

2. K.D. Bergeron, J. Appl. Phys. 48, 3065 (1977).

3. See for example W.L. Baker, M.C. Clark, J.H. Degnan, G.F. Kiuttu,

C.R. tClenahan and R.E. Reinovsky, J. Appl. Phys. 49, 4E94 (1978).

"V 4. G. Bekefi, personal communication.5. J.M. Creedon, J. Appl. Phys. 48, 1070 (1977).

6. J.R. Whinnery and H.W. Jamieson, Proc. I.R.E. Feb. 1944, p. 98.

7. J.M. Neri, personal communication.S.

56

°.o* * -

,... 1 . _,I .. P ,, . i F * . .... . -... .. . ,- ...... -, ..- • -. - . -. --

, .

APPENDIX A - JUNCTION SCATTERING COEFFICIENTS

Again, the incident and reflected waves at an element end are related to

'a the standing wave voltage and current according to:

2 VI = V + IZ (A-ia)

" R a V - VI (A-ib)

For the parallel tee (see Fig. 5):

.

V M V V (A-2a)

I + 1 +1 = 0 (A-2b)1 2 3

Writing Eq. A-la for each element end, adding the resulting equations and

usingEq .A-2a and Eq. A-2b gives:-.

2V 2V2 2V31

*li/ZI + I/z 2 + 1/z 3

'J Combining this with Eq. A-lb then gives:

VIR - VI(2K -1) + V2 1 (2K 2 ) + V3 1 (2K3 )4.

V2R = V1I(2K1 ) + V2 1 (2K2-1) + V (2K 3 )2R 1 1 1 231 3

V3R "V 1 I(2KI ) + V2 1(2K2) + V3 1 (2K 3 -1)

57

• %% %% ' ,''.-L -2 '. ..m , m.- " - ---- ."'. ". " -" -"% """'" . . . . "L ' . ' ".". °- " "-"-"'"

• "where

K/z 1

1 "/Z I + I/Z 2 + 1/Z3

K- 2K 2 i/Z 1 + I/Z2 + i/Z3

K-3 I/Z + I/z + I/Z1 2 Z3

For the series tee (see Fig. 5) (noting that the Junction is not symmetric):

-I, 1 2 3 - I (A-3a)

V1 - 2 - 3 0 (A-3b)

Again, writing Eq. A-la for each element end, adding the resulting equations

and using Eq. A-3a and Eq. A-3b gives:

-2VlI + 2V21 + 2V312 3-z zI + z 2 + Z3

And once more using Eq. A-lb gives:

VIR VII (1-2KI ) + V21(2KI ) + V3 1 (2K I

t 2R V 1I(2K2) + v21(1-2K2) - v3 1 (2K 2

V3 R - V1 (2K 3 ) - V2 1 (2K 3 ) + V3 1 (1-2K3 )

58

-U,. . . . .-lw- - .-- .'.- -.. - ..--.- ,,.. - . - - ., . ; , ., •" . - . - •

*.1

where

z

K2 T1 + Z 3K- 12 TZ + Z2 3

+ Z2 + Z3

iz

The other junction types are all special cases of these two.

.5,

is

4U'

., -

2. "" """"" "" " "" ""'" i+""" """" " "' '"'"+""-" > "'""+""""

APPENDIXK B -SAMPLE PROGRAM

The following is a simplified version of BERTHA which illustrates most of

the essential features of this program. Both this and the full program in

* App. C are written in BASIC for the Tektronix 4050 series microcomputers,

although translation into other languages should be very straightforward.

460

* &

000L 0

0.

0L

C L.

** C .5 0a* a - C - 40 C

N.. C- -: 0r. ai a- - t U - G

O. 00 D4- A acc 1.- 0 Go ( C c 3

U 0.1 -A to - '-0 -- 0-rz 0 CL 4j a0 >0 .C- do40 - -'- L. L c'-' L 0)

w L .- @1 fa -4 = C -w4 c~ (D1 C 0.' CL -01 0-4ZI 0 -. 0 4)0 0 C a4 0 L

CL. c C.zC CL IVG C. 0 410a > 0 0 N. - S00 C

(A- A o -D" c . to 0 do C0 do

w4 0 Z1 Co C 0V 0. EE 0Ac u Z a 4 0 0) - C C -o C

x Sx W- 0 ). -C C0M- c o0 00m.10 c.9.0 4-, U '- CIO C.o CL C '

W. -C L - 0'- 0 C WC - Cl -0 C- 3C

wU 1- ) -Z LIC CC-. L G VC)M -ow AZ n. cu N (9GI '41t M N ~4

0 a (OCR- 1 - 0) -do- NW.CU 01'- tna1.N- > -'C ~ C Z L. Z - Z - Oc-

-4N.J I C - w _.j- L W ~ co1-,-) Q1 -" 1-'" 01-*4 I +0- CL CL CL 0. - 0. - 0 -

N> I L > Z i Z LN% I ZL - -X XL >.L.- XL'- X...L'A4 1 0) %"1 41- 1 41--SQNC fi-SN)c I-Sdo i-SN

I '- N.1 0'-- 1 C3 o.- ,..--. I o- , 1- g ,*I CI C - -- I P- C' ~ C C (m I C'-0 I C'-. C'-** I U .-- W- I LU-- I -- I L--i W-- iw-

* * I Ii II _j 4 I I I ' I I -

RN. I z m P.-. z m-.I.- I z mI-- i -I Z =I-- I Z M I-i

wwwwwwmZW-wmmZWW ,,oxZoWcxZow~xZ0W~ZowcrZMW m mL o ML L- M L-LU%- M L- XL O O

61

4,2.

- ~ ~~~~~ V~ I*~ * . 4. .

JL

4. L

-. -Ld~~ 00s G

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APPENDIX C - PROGRAM LISTING

The full BERTHA program, as it currently exists, is listed here. (The

various routines are accessed by pressing the special function keys on the

4050 series machines). The start and execution routines together comprise an

expanded version of the sample program in App. B. The remaining routines

pertain to I/0 and disk storage and, as they will be highly dependent on the

requirements of the user, are listed here without comment (additional routinr:s

for waveform storage are omitted for clarity).

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>. E: -. EEa2E N I-- rN 1 0-

1 0 .2 Zz - NZZZZZ a N~ MI IZ . 00 : z zC2==2 2W w 0.. 0~ 0 1 c -

I L -)0 - - -- N CL 0 NNj C% r- N%I I-n *Z z: a =)-:: z::. X( >- XD N-I x x

4I taiC 0 MOM -0000000 KE) -In) -T N NI _.j Co _j LaJ V TIWI L -C (m m- 0 0I CD.. -i N X x V0xX CO- -- NI >10 o:mmm X - -N-<~ x xI INCz NNONNNNNNN z ICo W: N) .NN- r- N W

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1 00w- -~<CC C N' 01j C - N) r- N NDII 63 -I- 2 x = - - X T )I X .@ --L--N -- N: z q- .- Z -f- D CI 1-0-'< u:3:3:33:3W 0 Z- W -o L 0 -I mZZ o *I.LZz zz~ r z M" 4--z-Z -z zI W C O "0404P0 "0900 ZZ Z ZZ Z . "Z 0- .- * - - MZ -0Z 6

WZZ4JZZZZZZZZZZZZZZZ Z--ZZWz CAZ-ZzI-Zt--z

w mm L ua.eLcL o-a.cLa- cL Ea. K o.. a- -Kz Zx w- "& - L2a .G.-G

----------------------- N N N N NN NN N N N C J

87

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3N 0- z. I nmX ts cw to c a -

- N Ifl (x (m x x x. z z x

C4- N)- oI) X i 0 z -r. N. W. =

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< x X) cwz - - v 0 0 0. - a. 0 ..(a - N :)< +++N N N N Xz C

X- L-1u-xI N q- x - - 0- x -

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w x x (A 0- r --) -~ x - N L MDN - (As- --- N -0 X0( X (*

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Wi A-M Z NVWNU.- - -. V i..Mw M M M 0 MW aIC Li. -U U u - CL W

I ~ ~ ~ ~ ~ m m' m- - z -L

I OC NU*-~*- : 3- Q~'90

I UW D -L.) ~ 0- LJ '7

4W;

APPENDIX D -PROGRAM SEGMENTS

The following program segments will correctly adjust the traveling wave

voltages as the element impedance is changed. The first is a general program

for any element, the second is a simplified version for an element of single

timestep length and the third is a simplification for a purely capacitive

element (Again, no correction is needed for a purely inductive element).

* 91

1~-1

zCLW

W

U

z

NZ N-0

0 U - -

m 4c - - -

IxI

cr. cr. -N l 0

I--. ea - ZZN -

<(=G) - N-Z NN-

-(< - - NO - O>->COLL. -i, -j- Ztl - ti

I ~ 1I 00 -* > z- Z

WWW-VOVMNNL NNW 9

- - ------ NNN NN

zro It b-. - 92

z z

W& w

W w

4 I.- Ld-

Oz ow

(- -- -L

w I N

-N -0

" W C-

5.z U)* U

r-W =Pr I-OLU uO >~

Z~flD

0 W + +-W 0WAc CL IV ooin

UM- X: u cr.W, q r0:LL >- CrLLir

OZ<c I--0 0z -- 0u a r.N - r -u 0N Ix~-

"Na I-- - a.ZZMW -N L U -m z

4.W- N"O < w 0- -- 0

.OeLCC. - -I liZ 00 -1-1 zII>UDxO --- Li >UX0 W-LI I <-NZ Mmi' I-II ir< II- - I

I II N -N l I---'I I IOO>ZZ I 1 00 Zzi- I

4% ~ -lOW'fl(.!-~ -N M)4 moc r-a2

93

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