AD-A237 190

HDL-SR-91 -1
June 1991

User's Manual for DIODE

by Alford L. Ward
Steven L. Kaplan

DTIC
ELECTE I
JUN 25 O

I 0L

U.S. Army Laboratory Command

Harry Diamond Laboratories
Adelphi, MD 20783-1197

Approved frpublic release; distribution unlimited.

91 6 19 180 91-02770
 The findings in this report are not to be construed as an official Department
of the Army position unless so designated by other authorized documents.
Citation of manufacturer's or trade names does not constitute an official
endorsement or approval of the use thereof.
Destroy this report when it is no longer needed. Do not return it to the
originator.
 REPOT DCUMNTATON AGEForm Approved
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Pulf reolngbreP tIsclebnOnoato,=n= repns.Intg the tim torreVteAV ktu~m eer=ilg
=etmaet1aeag=N I hu peW W"~ dataeouroee,

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Da'ns Hli~way, Sutie 1204, Proe. (0704.01) Whngo,020503.
2220*32 andto t Office ofManagement and Budget Paperwork Reduction
1.AGENCY USEONLY (Leave blank) 2.REPORT DATE &.REPORT TYPE AM DATES COVERED

IJune 1991 Interim, from 1Jul 88 to 31 Dec 88

4.TITLE AND SUBTILE 5. FUNDING NU11BERS
User's Manual for DIODEDAP:ILI61014
PE: 62120
6. AUTHOR(S)

Alford L. Ward and Steven L. Kaplan

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) &.PERFORMING ORGANIZATION

REPOIRT NUMBER
Harry Diamond Laboratories HDL-SR-9 1-1
2800 Powder Mill Road
Adelphi, MD 20783-1197
g.spoNSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORtNGIMONITORtNG
AEC EOTNME
U.S. Army Laboratory Command
2800 Powder Mill Road
Adelphi, MD 20783-1145
11.SUPPLEMENTARY NOTES

AMS code: 612120.1400011

HDL PR: I1FE425
12s. DISTRIBUTIONIA VAILABILITY STATEMI NJT 12b. DISTRIBUTION CODE

Approved for public release; distribution unlimited.

13. ABSTRACT (Maximum 200 words)

.- >This user's manual serves two main purposes: (1) to document the present DIODE computer program in
preparation for proposed improvements and (2)to allow others to use the program without personal instruction.
Included are abrief history of the origin of the DIODEcomputerprograni, input datarequirements, description
of the computer code, output, and a discussion of the material parameters. Detailed instructions are included for
the use of a remote IBM terminal on the Harry Diamond Laboratories site. Also included is a list of over 100
papers, reports, and oral presentations based on the DIODE program.

14.SUBJECT TERMS 15. NUMBER OFPAGES

DIODE, computer, user manual, semiconductors, gases, avalanche, mobilities, 7
breakdown, space charge 1.PIECD

17. SECURITY CLASSIFICATION 18.SECURITY CLASSIFICATION 17. SECURITY CLASSIFICATION 20. UMIATION OF ABSTRACT
OF REPORT OFTHIS PAGE OFABSTRACT
Unclassified Unclassified Unclassified UL or SAR
NSN 754001-280-5500 Standard Form 298 (Rev. 2-89)
PrescribedbyANSISlt Z39-18
298-102
 Contents
page
1. Introduction and Background ................................................................ 5
2. Formulation of Computer Program.......................................................... 6
2.1 Basic Equations ............................................................................... 6
2.2 Supplementary Equations.................................................................... 7
3. Input Data........................................................................................ 8
4. Output ......................................................................................... 15
5. Material Parameters........................................................................... 17
6. Using the Interactive Version of DIODE on an IBM Terminal ..................... 19
6.1 Getting Started .............................................................................. 19
6.2 How to Edit/Browse Datasets............................................................... 21
6.3 How to Run the Program................................................................... 22
6.4 Ending the Session.......................................................................... 23
7. The Program.................................................................................... 23
References......................................................................................... 24
Bibliography....................................................................................... 25
Distribution....................................................................................... 71

Appendices
A. Listing of Program DIODE .................................................................. 35
B. Sample Output of Program DIODE........................................................ 61

Figures
1. Circuit model used in DIODE ................................................................ 11
2. Proposed circuit model for DIODE, adding capability for including
series resistance and inductance............................................................ 13

Tables
1. Input format ...................................................................................... 9
2. Parameters in TPPINT LIST (line 21) that cause optional
output data to be printed ...................................................................... 14 -

3. Parameters used for avalanche calculations................................................ 18 3

4. Parameters used for mobility calculations.................................................. 18
5. Other material parameters .................................................................... 19
6. Subroutines of DIODE......................................................................... 23 .

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1. (Introduction and Background
The DIODE computer progra at the Harry Diamond Laboratories
(HDL) is used to compute rent and voltage characteristics in
semiconductors or gases.- t has been used without change for
approximately 12 years; improvements upon the program are over-
due. This report documents the existing program and will serve as a
basis upon which the improvements will be built. A listing of the
computer code is provided in appendix A.
HDL's interest in the basic physical processes of electrical breakdown
in gases arose from the need for sensitive trigger tubes for use in
electrostatic fuzes. The initial theoretical work was based on papers by
Varney et al [1] and by Crowe et al [2]. In the mid-1950's, Professor
William Dow, Chairman of the Electrical Engineering Department at
the University of Michigan and member of the Scientific Advisory
Board of the Diamond Ordnance Fuze Laboratories (the predecessor
of HDL), suggested that the problem could best be solved on the then-
new electronic computers. Ward modified the formulation of Crowe
et al to apply to the rare gases and used the National Bureau of
Standards (NBS)* SEAC (South East Automatic Calculator) computer
to calculate results. In order to extend the calculations to higher
current densities, Ward included the space charge of electrons and
reformulated and encoded the problem for the IBM 704 computer.
Time dependence was included for the first time when B6rsch- Supan
and Oser [3] wrote the basic program that is still in use. Thanks to the
insistence of Irene Stugun, then chief of the NBS programming sec-
tion, the basic continuity equations were used, rather than the partially
integrated equations used previously. Over 100 papers, reports, and
oral presentations (see bibliography) have resulted from the DIODE
program.
In the summer of 1965, a summer student at HDL, Edward R. Berman,
modified the program to include semiconductors [4]. The major
changes were the inclusion of ionization by holes and the use of double
precision because of the higher densities present in semiconductors.
In 1969, under the direction of Burton Udelson, Howard Bloom [5]
modified the external circuit of DIODE in order to study IMPATT
oscillators. Bloom's report includes a full program listing. Since that
time, Ward [6] added thermal effects to the program. Until 1988,
computer cards were used for the input data. Now the program may
be used in the time-sharing mode at a remote terminal.
Before the computer code and its use are discussed, we present the
mathematics that forms the basis of the program. We then discuss in
detail the format of the input data required to run the program; the
*Now the National Institutefor Standardsand Technology (NIST).

5
 input data required consist of physical properties of the material
studied and device descriptions. The information must be supplied by
the user; HDL users can build on datasets previously used, changing
only the information necessary. The output produced by the program
is then described in detail (an example of the output is provided in
app B).
As an aid to users in devising input datasets, we include information
on the material parameters that have been commonly used; this
information includes both data and suggestions on how to choose
constants oased on various earlier efforts.
Finally, we discuss the interactive use of DIODE as it has been
implemented on HDL's IBM 3090 mainframe. These instructions are
particular to HDL users, but non-HDL users may find them useful if
they intend to implement the program elsewhere.

2. Formulation of Computer Program

2.1 Basic Equations

The one-dimensional continuity equations for electrons and holes,
respectively, in a semiconductor are
ean -SI- - / / +R ,(1)
l-

e=- J+
+ al- +,j+- R (2)
wherex and tare the space and time coordinates, eis the electron charge,
n and p are the electron and hole number densities, a and P3 are the
ionization coefficients for electrons and holes, and R is the recombi-
nation rate. The electron and hole current densities,J_ and J+, are given
by
J_ =ne2_E-eDdn-n) (3)
-dx '
J+=pe~u+E-eD+(tf) (4

where p_ and p, are the electron and hole mobilities, D_ and D, are the
diffusion coefficients, and Eis the electric field.
Space-charge effects are determined by the one-dimensional Poisson
equation
dE e(n-p+N) (5)
dx o

6
 where icis the dielectric constant for the semiconductor, eo is the per-
mittivity of free space, and N(x) is the distribution of net fixed charge.
The sign convention in equations (1) through (5) is chosen so that E,
J_, and J are all normally positive quantities.
The one-dimensional thermal diffusivity equation is

4-)JE +kT 0Ei2T ++ fCT

T (2
(6)

where T is the temperature, I is the total current density, p is the

semiconductor density, c is the heat capacity, and kT is the thermal
conductivity.
It may be shown from Lquations (1) through (5) that the total current
density,
(7)
is a constant in space. This merely expresses current continuity in one
dimension. The accuracy of the calculations may be monitored by the
constancy of ] across the width of the diode.
The variation of the intrinsic density with temperature was chosen as

ncr) -2 x 1020( _ 3 exp Lg- (300-1 , (8)

where Eg is the bandgap energy of silicon. The variation of the injected

(thermally generated) current density with temperature was assumed
to be the same as for n(T), since np = n7and the majority carrier den-
sity is fixed by the doping level.

2.2 Supplementaiy Equations

The cathode (x = 0) to anode (x = d) distance is divided into M equal

intervals of width Ax. Initial (t = 0) arrays of M + 1 values must be given
for n, p, and T, and a similar array given for the fixed charges, N = ND
- NA, where ND is the net number density of donors and NA is the net
number density of acceptors.
Two boundary conditions are required for equations (1) and (2). the
electron current density at x = 0 and the hole current density at x = d.
Alternatively, the number densities can be given and the current
densities calculated from equations (3) and (4). At present, the bound-
ary currents are constant in time.
The boundary condition for equation (5) is supplied by the total
voltage across the diode. The initial voltage across the diode must be

7
 given, and the voltage for later times is determined by the external
circuit. Optional external circuits are available, but for this report, the
diode, shunted by a capacitance, C, is in series with a load resistance,
R, and a voltage source, V(t). The voltage source may be constant in
time, have one discrete step, or have an incremental sinusoidal varia-
tion. The last option allows a constant dV/dt value to be closely
approximated.

3. Input Data
The input data for DIODE are formulated on 21 lines, preceded by a
title line, and followed by a variable number of doping distribution
and possibly temperature distribution lines. The final line is an ending
indicator. The data field is justified right and uses standard Fortran
criteria for digital data and integers. The standard line uses five
nmnber fields of 14 spaces each; i.e., numbers end on spaces 14,28,42,
56, and 70. Care must be taken that exponents end at the correct
position, since zeros are read in blank spaces. Some lines have integers
both in space 1 and in space 80.
Table I lists the 21 lines with a short identifying name. In the field
width column, an integer in space I is indicated by "I" and in space 80
by "I*." In the descriptions that follow, consult table I to see the format
and the array of parameters.
Title line. On the title line, the first character must be in space 1; a
maximum of 71 characters is allowed. Typically, such information as
the date of the run and the diode characteristics would be entered in
the title line.
Lines I to 3.On lines 1, 2, and 3, enter the avalanche (that is, ionization
by collision) coefficients for electrons, a (parameters Al to A5); holes,
/(parameters B1 to B5); and gas excitation, 8 (parameters D1 to D5);
respectively. (Since the format is the same for each of these lines, only
line 1 is described.) When a= 0, alpha parameter MODA should be set
at zero, and Al and A5 are irrelevant. MODA should be set to 1 if

a==p Al exp(-Ap) if 1' A5/ (9)

a=pA3exp -A4 P- if p> A5 , (10)

_J
I P
where p, the gas pressure, is set equal to 1 for semiconductors, and E
is the electric field.
If rare gases are being used, MODA should be set to 2; in this case,
dEl/p) 1/ 2 replaces JEI/p in equations (9) and (10).

8
 000

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u q C

0-~i~~Do
U uu u o F.N

t
C4No

en u zu z u

I.' _

L. z

000 00 0

o' o

cuW u
' ~ ~00
u u:2w'

(Ua
zC) N m' o\ N0 NC
M 0 N N
 The program computes A6 internally to ensure continuity in ax at A5;
the result is included in the printed output.
Since
A6 = -A5 [In A1 - In A3 - A2/A5]
one must not set A5 equal to zero. MODA should be set to 3 if
p Al exp -A2 +-p 3e l-A.
a =a~pA~ex('EP!)+pA3exp p-
jEI )

If MODA is set to 4, (p/IEj)1/2 replaces p/JEj.

Lines 2 and 3, for holes and excitations, respectively, pertain to the
same equations as line 1 (for electrons).
Line 4. On line 4, you supply data needed for temperature effects. If the
temperature is constant in space and time, the temperature parameter
MODT should be set to 0; put the temperature value in TEMPK, the
last field in the line. If you calculate that the temperature will change
as a result of power dissipated, set MODT = 1. In CAT, put the
fractional change in Al for a one-degree increase in femperature, and
in CBT put the fractional change in B1. In PWRM, put the power-law
dependence of the low field mobility of electrons upon temperature,
i.e., mo(T) = mo(3O0)TPWRM. Likewise, in PWRP, put the exponent for
the hole mobility. If you are generating an initial distribution of
temperature after the doping concentration lines, MODT must be set
to 2 (see discussion of "Other lines" at the end of this section).
Line 5. On line 5, you enter the material parameters of density, specific
heat, and thermal conductivity. These parameters do not change with
temperature in the DIODE program.
Lines 6 to 10. On lines 6 through 10, you enter data describing the
variation of the electron and hole mobility with field and doping level.
If the hole mobility, lp, should be calculated according to
pp = MUP (1 - C1-E), E5<0C (11)

yp = C2E - 1/ 2 (1 - C9.E) - 3 /2 , E > C3 , (12)

MODVP should be set to 0 (line 6). The computer calculates C9 to
ensure continuity of lip at E = C3, and C9 is printed out.
If you set MODVP to 1,you must enter C9 and C15 in line 10. Then the
computer solves equations (11) and (12) to obtain continuity in pp and
dup/dE over the range from E = C3 to E = C15 for the power law
= Cll + C12.E + C13.E2 + C14.E3 (13)
and prints out Cll to C14.

10
 The last parameter on line 6, C4, is the maximum hole velocity, usually
termed saturation velocity, to be used in the field range for equation
(12). Since a peak velocity at low fields can exceed the saturation
velocity, as is the case for electrons in GaAs, the maximum hole
velocity does not apply for fields less than C3 (eq (11)).
If you set MODVNP to 0, in line 7, the hole mobility is independent of
the doping concentration at that position. If you set MODVNP to 1,the
dependence is given by the other parameters. (This option has not
proven useful for several reasons and will not be detailed here.)
Lines 8 and 9 include the parameters determining the electron mobil-
ity, ,n.The equations are in the same form as equations (11) to (13):
p, = MUM (1 - C5.E) , E 5 C7, (14)
Pn = C6.E- 1/ 2(1 - C1O.E- 3/ 2) , E > C7 , (15)
,up = C16 + C17.E + C18.E2 + C19.E3 , C7 < E <C20 . (16)
Line 10 has been covered above.
Line 11. You would use line 11 primarily for secondary processes in
gases. However, for semiconductors, JMO and JPO are the boandary
values for the electron current density at the negative electrode and
the hole current density at the positive electrode, respectively. These
are used primarily under reverse bias.
Line 12. You enter voltage parameters on line 12. The external circuit
used in DIODE is shown in figure 1 (many of the labels on figure 1
correspond to parameters on line 12; see table 1). USTAT is the initial
source voltage and V is the initial voltage across the diode. At the time
T1, a voltage increment DU is added to USTAT. You may also add a
sinusoidal voltage to USTAT, so that the total applied voltage is
USTAT + VO sin OMEGA t , (17)
where t is the problem time. You may use this option to apply a
constant dV/dt, since for OMEGA t << 1, dV/dt is essentially equal to
VO OMEGA.
R L
Figure 1. Circuit model . IiI
used in DIODE. 2 3
' V0
L,- V C "_ C2 -- C3

USTAT R2 R3

11
 Line 13. You may apply other frequencies, usually harmonics, by using
line 13. PHI1 and PHI2 are the phase lags, with reference to OMEGA,
for OMEGA1 and OMEGA2, respectively. IFOUR may be set at one to
calculate the efficiency of oscillations at the frequency OMEGA and its
second harmonic. If IFOUR = 1, the computer calculates a cosine
instead of sine in equation (17). IFOUR is in column 80 of line 13.
Line 14. Enter geometrical information for the diode on line 14. P is the
gas pressure in torr and should be set equal to 1.0 for semiconductors.
D is the device width in centimeters. S is the device area in square
centimeters. T allows a time other than zero to be the initial time (this
is useful in extending a previous run or changing the phase of a
sinusoidal voltage). DVDT is the initial dV/dt used in the start of the
problem; it determines the displacement current at zero time, but
setting dV/dt = 0 or 1 x 1012, for example, makes little difference in most
problems. JMAX is used to stop calculations when unreasonably high
current densities are calculated, as occurs for instabilities. When JAVE
(see sect. 4) exceeds JMAX, the calculations are halted and the results
of that time step are printed out.
Lines 15 to 17. Lines 15 to 17 are used to describe the external circuit.
The format for each line is identical, and each R, L, and C is shown in
figure 1. VC and DVC are the initial voltage and dV/dt on each ca-
pacitor, used mainly for extensions of previous runs. C in figure 1 is
given by C on line 17. Values of C on lines 15 and 16 are ignored. (The
RLC circuit 2 has proven useful to simulate a resistive load, but not as
a resonant circuit, because transients last too long for the calculation
to be cost effective. We plan to modify the external circuit portion of
the program by removing RLC circuit 3, removing L from the supply
circuit, and adding Li and R1 to the diode leg; see fig. 2.) To remove
C2 and C3 from their respective legs (see fig. 1), set them greater than
1 x 1010 F.
Line 18. Miscellaneous decimal data are entered on line 18. STEPFA, or
F, indicates the fraction of one distance step, Ax, that the electron (or
hole) with the highest velocity, Vmax , travels in one time step, At. This
follows from the calculation of At from
At = F(A x)/Vmax (18)

The factor F may be used to ensure that the calculation time step is less
than the dielectric relaxation time [7]. To ensure symmetric diffusion,
you should set F = 1/2 [8].
DJMAXM is a check on too rapid a current increase in one time step.
If the new current exceeds the old current by a factor greater than
DJMAXM, DT is halved and the computation repeated. The problem
is terminated after three failures. DIELK is the dielectric constant of
the semiconductor. DENSPL is the carrier density of holes, in cm- 3, at

12
 IEXT RS 12
Figure 2. Proposed L
circuit model for
DIODE, adding
capability for includ- CPC2
ing series resistance IINT C
and inductance. _
RI R2

the right boundary, and DENSMI is that of electrons at the left

boundary. In practice, these are used for forward-biased diodes only.
DENS2 is the square of the intrinsic density at 300 K. REC is the
recombination rate for the semiconductor. The lifetime in intrinsic
materiai is given by (2 REC - 1. The minority lifetimes of electrons, t",
and holes, tp, are given by
t, = ni/(REC) p ;tp = ni/(REC) no , (19)
where n i is the intrinsic density and p0 and no are the majority carrier
densities. Finally, EGAP is the energy band gap in the semiconductor
(see eq (8)).
Line 19. Line 19 determines the parameters to be printed out as a
function of distance, x, at selected time intervals. The first two param-
eters are permanently selected to be x and E,the electric field. Users
can choose six additional parameters to be printed out from the list of
20 given in table 2.
Line 20. Line 20 includes 8 integers. M is the number of space intervals,
Ax = dIM. M + 1 values of each distribution parameter must be in-
cluded in the input data (see below, Other lines).
PRTFRQ is the print frequency: When PRTFRQ is set to 1,M + 1 values
are printed out; for PRTFRQ = 2, M/2 + 1 values are printed, and so
on for higher values of PRTFRQ.
N3S is the number of triples to be included on line 21 (see discussion
of line 21).
MODP is the mode of printout: If MODP is set to 1, the printout is
according to time; if MODP = 2, the printout is according to current
(see discussion of line 21).
MODFCH determines whether fixed charges, as for doped semicon-
ductors, are to be included. If MODFCH = 0 or 1,no fixed charges are
included; if MODFCH > 1, you must supply fixed charges to simulate
the device doping.

13
Table 2. Parameters
in TPRINT LIST Option Name Meaning
(line 21) that cause No.
optional output data
o1 ALPHA Electron ionization coefficient, a
2 BETA Hole ionization coefficient,/3
Note: A maximum 3 UNIVD Excitation coefficient, 4
of six may be 4 VM Electron velocity
chosen. 5 VP Hole velocity
6 ALPHA * J- aJ-(X)
7 BETA * J+ 0W+(X)
8 NP - NM + DN N+(X) - N_(X) + doping density
9 RECOM RATE Recombination rate
10 JP DIF Hole diffusion current density
11 JN DIF Electron diffusion current density
12 J DISPL Displacement current density
13 J TOTAL Total current density
14 V SUM Voltage distribution
15 DVXDT dV/dt
16 N+(X) Hole number density
17 N-(X) Electron number density
18 J+(X) Hole current density
19 J-(X) Electron current density
20 TEMP Temperature

EXTRAP allows extrapolations to be used instead of the boundary

conditions given in line 11 or 18. If EXTRAP is set to 0,the conditions
in lines 11 and 18 are used; otherwise the extrapolations are used. (The
extrapolation option has rarely been used.)
MS sets a limit on the number of consecutive time steps in which a
negative field is calculated. Calculation is halted when MS is ex-
ceeded, and an error message is given.
Last, IPOINT gives the number of time steps to be used in the
summary printout. See section 4 on output.
Line 21. On line 21 you give instructions for the intervals between the
time steps for which you want data printed out. These instructions,
called the TPRINT LIST, are composed of "triples": two real numbers
(including a decimal point) and one integer compose one triple. (The
number of triples, N3S, was specified in line 20.)
If MODP (also in line 20) is set to 1, the results of the run are printed
out according to time. In this case, each triple is interpreted as follows:
the first number is the time for the initial printout and ends on space
14 of line 21. The second number is the time interval between subse-
quent printouts and ends on space 28. The integer gives the number of
time intervals for that triple and ends on space 31. (The second triple,
if used, gives the same information, but the numbers end in spaces 50,
64, and 67.)

14
 If MODP =2, the results are printed out according to current density.
In this case, the first number in the triple gives the current density for
the first printout, the second number gives the multiplier for subse-
quent printouts, and the integer gives the total number of printouts.
(The spacing of the fields is the same as for MODP = 1.) If MODP = 2,
then JMAX on line 14 gives the maximum problem time in seconds.
Additional lines may be used if more than two triples are desired.
Other lines. Following TPRINT LIST on line 21, you can use a variable
number of lines to simulate the desired initial distributions of holes,
electrons, doping levels, and, if MODT = 2 (line 4), temperature. You
may use M + 1 numbers, 5 to a line, to the various densities, given in
units of cm-3.The first density number must end on space 14, and other
numbers on multiples of 14. Enter the hole distribution first; the
electron distribution follows, starting with a new line. The net doping
concentration follows; a negative sign indicates acceptors or p-type
material. Enter the initial temperatures in kelvins.
Ending line. Following the distributions is an ending line. If you set
ENDING to <10,000, the program executes another run. If ENDING
exceeds 10,000 but is less than 32,767, the program stops.

4. Output
The output of DIODE consists of a printout with three parts: the input
data, the distribution data at selected times, and the temporal results.
Appendix B is a portion of the program output for a sample problem.
The initial page of the input data is headed by the title line, reproduced
from the input. Each of the input lines is then listed in a format similar
to the input and identified by the labels shown in table 1. (There is one
exception in the order of lines: line 20 of table 1 precedes line 19.) The
TPRINT LIST (line 21) prints out the times as determined by the
triples. Following the TPRINT LIST, the values of A6, B6, D6, and C9
to C19 are printed out. (These constants are explained following eq
(10) and also in eq (13).)
The second page of the printout of the input data is headed by the
initial time and voltage. Then the distributions follow in columns
headed by M, N+, N-, DN, and TEMP K. The grid points range from
0 to M, for M + 1 values. N+, N-, DN, and TEMP K are the input
densities of holes, electrons, and fixed charges, and the temperature,
respectively.
The distribution data then follow--one page for each time listed on the
TPRINT LIST. There are four heading lines at the tup of the page. Line
1 includes T (time), V (diode voltage), U (applied voltage), J+AVG

15
 (average hole current density), J-AVG (average electron current
density), J AVG (J+AVG + J-AVG), and GAMMA (used only for gas
excitations).
Line 2 includes DT, the time step; V+MAX, the maximum hole velocity
for that time step; V-MAX, the maximum electron velocity; I EXT, the
maximum current through resistor R; I INT, the maximum current
through the diode; J INT, the average current density (including the
displacement and diffusion current densities); and DV/DT, the rate of
voltage rise across the diode.
Line 3 includes D*ALPHAAVG, the integral of a dx; A*J-AVG, the
average value of aJ_; D*BETAAVG, the integral of b dx; B*J+AVG, the
average value of bJ ; RECOMRATAVG, the average rate of electron-
hole recombination; 123, the sum of currents through circuits 2 and 3;
and DI/DT, the rate of current rise through the diode.
Line 4 includes J+DIF, the hole diffusion current density (calculated
from the Einstein equation and dp/dx and averaged); J-DIF, the clec-
tron diffusion current density; J DIF, the total diffusion current den-
sity; J DISPL, the displacement current (calculated from dE/dT); AVE
TEMP K, the average temperature across the diode; TEM RATE, the
rate of temperature increase, in kelvins per nanosecond; and DENS2T,
the square of the intrinsic density at the average temperature.
The main part of the distribution data at specified times is arranged in
columns, where the number of rows = (M + 1)/PRTFRQ. The first two
columns are fixed and are headed X (distance) and E(X) (field). The
distance is given in centimeters and the field in volts per centimeter.
The other six columns are chosen by the user from the options shown
in table 2.
The page following the distribution data is headed by "capacitor
voltages at last successful printout." It is followed by the voltages and
dV/dT for the capacitors used in circuits 2 and 3. The final page (or
pages) of the printout consists of six columns of data printed accord-
ing to time. The number of times printed is the value of IPOINT (line
20); the times are equally spaced between the initial time and the
maximum time, as selected by the TPRINT LIST. The first colunin is
headed by T and lists the time. The second column is headed by V and
lists the voltages for each time. The third column is headed by I and
lists the diode total current. The fourth is headed by JAVG and is the
sum of the electron and hole current densities. The fifth column is
headed by I1 and lists the current through the external circuit (see fig.
1). The last column is headed by 12 and lists the current through circuit
2. This completes the printout for the program DIODE.

16
5. Material Parameters
The first 10 lines and line 18 of the input data (see table 1) contain the
material parameters for the semiconductor of interest. Table 3 con-
tains the input avalanche parameters for silicon and gallium arsenide
which have been most commonly used. Table 4 contains the param-
eters used for the variation of mobility with field. For silicon, param-
eters are given for high-mobility (ideal) and low-mobility (practical)
material. Other material parameters (lines 4, 5, and 18) are given in
table 5. The temperature parameters are unknown for GaAs, and so
the silicon values are used.
From equation (11), the hole velocity, vp, is given by
vp(E) = MUP. E[1 - (C1)E] , E < C3. (20)
Thus the hole velocity is parabolic with a maximum velocity, Vmax =
MUP/4. C1, at the field Em. x = 1/2(C1). These two equations for C1 are
usually not compatible, and therefore C3 must be chosen less thant EmX
and equation (13) used to fit the experimental curves. Generally, vp
calculated from equation (12) will exceed C4 (Vmax) where Eis slightly
more than C7. The same considerations apply to choosing constants in
equations (14) to (16) to fit the experimental electron velocity versus
field curves.
For fitting the electron velocity versus field curves for GaAs, or other
negative differential velocity materials, the constant C10 in equation
(12) mustbe chosen to be negative. Then equation (12) has a minimum
velocity, Vmin, at the field E(Vmin). Then the fitting equations are
E(vmin) = -(2. C10)2/3 ; Vmin = (3/2)C6[E(vmi,)]1/ 2 . (21)
Again, trial and error will be required to obtain the best fit to experi-
mental data.
The dependence of the low field mobilities of the electrons and holes
upon temperature is given in table 4 as inverse to the 2.5 and 2.7 power,
respectively. However, the saturation velocities vary approximately
inversely with temperature. At present, the saturation velocities (C4
and C8 in table 4) are increased at higher 4emperatures to give rough
agreement to this variation, since the saturation velocity is otherwise
too low. A planned modification to the program will give a single
expression [9] that gives the mobility variation with field, doping, and
temperature.

17
Table 3. Parameters Value for Value for
used for avalanche Variable silicon GaAs
calculations
MODA 1 1
Al 7.03 x 105 5.05 x 106
A2 1.231 x 106 1.95 x 106
A3 7.03 x 105 1.52 x 106
A4 1.231 x 106 1.60 x 106
A5 1.0 x 106 2.86 x 105
A6* 1.231 x 106 1.607 x 106

MODB 1 1
B1 1.582 x 106 2.05 x 107
B2 2.036 x 106 2.49 x 106
B3 6.71 x 105 1.49 x 106
B4 1.693 x 106 1.75 x 106
B5 4.0 x 105 2.82 x 105
B6* 1.693 x 106 1.75 x 106

CAT 2 x 10- 3 2 x 10- 3

CBT 2 x 10- 3 2 x 10- 3
*Computed

Table 4. Parameters Value for Value for Vaufo

used for mobility Variable low-mobility high-mobility Value for
calculations silicon silicon GaAs

MODVP 1 1 0
MUP 4.0 x 102 6.0 x 102 4.0 x 102
C1 2.0 x 10-5 4.0 x 10-5 1.0 x 10-5
C2 1.6 x 104 1.7 x 104 4.47 x 104
C3 2.0 x 104 9.0 x 1037 4.0 x 1047
C4 1.0 x 107 1.0 x 10 1.0 x 10
MODVM 1 1 1
MUM 1.2 x 103 1.5 x 103 8.5 x 103
C5 4.4 x 10-5 4.9 x 10-5 1.0 x 10-4
C6 4.4 x 104 4.4 x 104 1.0 x 104
C7 7.0 x 103 1.0 x 104 5.5 x 103
C8 1.0 x 107 1.0 x 107 1.0 x 107
C9 -1.55 x 107 -1.55 x 107 -4.77 x 106
C10 -1.30 x 106 -1.3 x 106 2.4 x 1047
C15 1.0 x 105 1.2 x 10 5 5.0 x 10
C20 2.0 x 104 2.0 x 104 5.0 x 104
Computed parameters
Cll 2.90x 106 1.78x 106 0
C12 1.11 x 102 2.05 x 102 0
C13 -8.35 x 10-4 -2.16 x 1C- 3 0
C14 1.88 x 10- 9 7.37 x 10- 9 0
C16 1.97 x 106 1,34 x 107 2.63 x 107
C17 6.28 x 102 -1.42 x 103 -1.08 x 103
C18 -1.01 x 10- 2 1.09 x 10-1 2.27 x 10- 2
C19 -1.76 x 10- 7 -2.44 x 106 -1.68 x 10-7
Temperature parameters
PWRP 2.7 2.7 2.1
PWRN 2.5 2.5 1.0

18
TableName Meaning Units Material
material parameters Silicon GaAs
DENSITY Density g/cm 2 2.328 5.32
SPEC HT Specific heat J/gC 0.70 0.35
THRM CND Thermal conductivity W/cmC 1.45 0.46
DIELK Relative dielectric constant - 11.8 13.1 12
DENS2 Intrinsic density squared cm-6 2 x 1020 3.2 x 10
EGAP Energy gap eV 1.12 - 1.43
CAT Proportional change in - 2.0 x 10 2.0 x 10
temperature of a 3
CBT Proportional change in 2.0 x 10 2.0 x 10
temperature of 03
PWRM Mobility dependence on - 2.5 25
power of temperature
for electrons
PWRP Mobility dependence on - 2.7 2.7
power of temperature
for holes

6. Using the Interactive Version of DIODE on an IBM

Terminal
This section describes how to use a dedicated IBM terminal connected
directly to the HDL IBM 3090 mainframe to access the DIODE pro-
gram. Implementations of DIODE on other hosts will differ in detail
from those given here (particularly instructions for logging on, etc).
However, certain aspects of this description may apply to any imple-
mentation of the program.
This section covers procedures for logging on, defining the input data,
and getting the output. It should be noted that this document gives
only a cursory description of the IBM 3090's menu-driven work
environment (ISPF-interactive system productivity facility) and of
its editor, which are necessary to execute the program and specify the
input. A fuller description of each can be found in standard IBM
manuals available through the information management group at
HDL or by contacting Steven Kaplan (ext. 41403).

6.1 Getting Started

These instructions assume a user with an IBM terminal that is directly
wired to the HDL IBM 3090. The first step in using the DIODE code is
to log onto the IBM time sharing option (TSO) and then get into the
menu-driven mode called ISPF. The interactive version of DIODE
currently resides only in the account HK1005. The instructions given
here are easily generalized to include the code's use on other accounts.

19
 The following steps are required to access the main ISPF menu:
(1) Turn on the terminal. Make sure the key (if there is one) is in the
horizontal position. Hit RESET if the keyboard input is not being
accepted. (If there is still a problem, contact user assistance, ext. 42940.)
(2) Type "LOGON" and then press ENTER.
(3) When prompted, type the userid (HK1005) ande then press ENTER.
(4) When prompted, type the password (see Kaplan) and then press
ENTER.
You should now be in TSO. After some system information and
messages appear on the screen, the prompt "READY" should appear
at the left end of the current line.
(5) At the READY prompt, type "SPF".
You should now see the main ISPF menu, which reads as follows:
ISPF PRIMARY MENU
OPTION ===>
0 ISPF PARMS
1 BROWSE
2 EDIT
3 UTILITIES
4 FOREGROUND
5 BATCH
6 COMMAND
7 DIALO(; TEST
8 LM UTILITIES
C CHANGES
S SYSTEM APPLICATIONS
T TUTORIAL
U USER APPLICATIONS
X EXIT
The SPF menu has 14 options. At the arrow prompt on the top line,
enter the number or letter corresponding to the desired option. The
three options that are particularly important for using the DIODE
program are listed below with brief descriptions of their respective
functions.

20
 Option Mode Description
1 Browse Allows user to view a dataset
2 Edit Allows user to view and change a dataset
6 Command Allows user to execute a TSO command
The function (PF) keys at the top of your keyboard perform operations
such as scrolling through datasets and moving between SPF modes.
The PF3 key allows one to exit from the different SPF modes to the
main menu. The uses of the most important function keys are as
follows:
PF1 Help fimction
PF3 Exit current mode
PF7 Scroll up (edit/browse)
PF8 Scroll down (edit/browse)
PF10 Scroll left (edit/browse)
PF11 Scroll right (edit/browse)
The remainder of this section describes using the SPF options to run
the DIODE program, and view or change its input/output.

6.2 How to Edit/Browse Datasets

To create -)r change an input file or examine an output file, you need
to use the edit or the browse (no changes allowed) mode. The follow-
ing describes how to access the edit mode. (To use the browse option,
follow the same procedure, except enter option 1 at the top line of the
SPF main menu.) To edit:
Enter option 2 at the arrow prompt on the top line of the SPF main
menu. You must next specify the file to be edited. A dataset panel
appears as follows:
ISPF LIBRARY
PROJECT ==> (enter userid)
GROUP ==> {enter filename)
TYPE ==> {enter file type)
The datasets that are relevant to DIODE are the following:
INDIOD DATA input dataset
OUTDIOD.DATA output dataset
SKDIOD.FORT : so .Arce code dataset
DIOD.CLIST : command list to run the program

21
 So, for example, to edit the input dataset, the panel should look as
follows:
ISPF LIBRARY
PROJECT ==> HK1005
GROUP ==> INDIOD
TYPE ==> DATA
After the panel is set up to indicate the appropriate dataset, hit
ENTER. The dataset contents should appear on the screen. You can
scroll through the dataset, and change its contents (if in edit mode).
When editing, be sure to save your changes by typing "SAVE" at the
command line prompt at the top of the screen. To exit, hit the PF3 key
(this should also save the dataset, but use the save command to be
sure).
Note: A detailed description of the editor commands, the function
keys, or any other TSO/SPF functions can be obtained by contacting
Steven Kaplan at extension 41403.

6.3 How to Run the Program

In the IBM account HK1005 there is a command list (clist) member

called DIOD(RD) which automatically allocates the necessary datasets,
executes the program, and prints the output. Variations in use of
DIODE, such as incorporating different files for input and output, are
easily accommodated by contacting Steve Kaplan (ext. 41403). To run
DIODE using the existing clist, follow these steps:
(1) Starting from the SPF main menu (see sect. 6.1), get into the TSO
command mode by entering option 6 at the arrow prompt on the top
line.
(2) In TSO an arrow prompt will also appear. At this prompt, enter the
following:
EXEC DIOD(RD)
where EXEC is the command to execute a clist and DIOD(RD) is the
clist to be executed.
(3) When the run is completed, three asterisks (***) will appear. Hit the
PF3 key twice to return to the SPF main menu.

22
 6.4. Ending the Session
Starting from the main SPF menu:
(1) Type "x" or "end" at the arrow prompt, then hit ENTER.
The cursor should be at a field marked "PROCESS OPTION" in .the
subsequent panel. Enter one of the options listed in this panel; in
general, either "D"(to delete the dataset containing the session input)
or "K" (to keep the session input-this is usually unnecessary).
(2) You should now see the TSO "READY" prompt. Log off the system by
typing "LO" and then ENTER.

7. The Program
The Fortran 77 program is divided into the main program with 263
source statements and 12 subroutines. The subroutines are listed in
table 6 with the purpose and number of statements for each. Some
statements consist of multiple lines.
The complete listing of the program is given in appendix A. A portion
of the printout for a sample problem is given in appendix B.
If non-HDL readers are interested in implementing DIODE on other
computers, they should contact the authors at HDL for a tape.
Table 6. Subroutines No. of source
of DIODE Title statements Purpose

INPUT 148 Reads and writes input data

INCON 103 Initial calculations
EXTCIR 37 Calculates effect of external circuit
TEST 72 Tests for various conditions
OUTPUT 87 Calculates and prints output data
ABD 8 Calculates ionization coefficients (eq (8), (9))
UNIV 24 Further ionization calculations
MUF 11 Calculates mobilities (eq (10)-(15))
TESTSQ 29 Gas excitation
SIMQ 53 Further gas excitation
EFFCY 98 Calculation of rf efficiency
BLKDT 20 Reserves memory blocks

Statements in rmain program: 263

Statements in subroutines: 690
Total 953

23
References
1. R. N. Varney, H. J. White, L. B. Loeb, and D. Q. Posin, The Role of
Space Chargein theStudy ofthe TownsendIonizationCoefficientsand the
Mechanism of Static Spark Breakdown, Phys. Rev., 48,818 (1935).
2. R. W. Crowe, J.K. Bragg, and V. G. Thomas, Space ChargeFormation
and the Townsend MechanismofSpark Breakdown in Gases, Phys. Rev.,
96, 10 (1954).
3. W. B6rsch-Supan and H. Oser, Numerical Computation of the Tem-
poralDevelopmentofCurrentsin a Gas DischargeTube, J. Res. Nat. Bur.
Stand., 67B, 41-60 (1963).
4. Edward R. Berman and A. L. Ward, FortranIV ProgramforGas Tubes
and Oher Dielectrics,Harry Diamond Laboratories, HDL-TR-1310
(5 November 1965).
5. H. M. Bloom, Computation of Avalanche and Space-ChargeEffects in
Semiconductorsand Gases,Harry Diamond Laboratories, HDL-TM-
69-35 (November 1969).
6. A. L. Ward,An Electro-ThermalModelofSecondBreakdown,presented
orally, IEEE Nuclear and Space Radiation Effects Conference,
University of California, San Diego (27-30 August 1967); IEEE
Trans. Nucl. Sci., NS-23, 1679-1684 (December 1976).
7. A. L. Ward, CalculationsofSecond Breakdown in Silicon Diodes,Harry
Diamond Laboratories, HDL-TR-1978 (August 1982).
8. A. L. Ward, Ionization, Diffusion, and Drift Velocities in a Pulsed
Townsend Discharge,J.Appl. Phys., 36,1291-1294 (April 1965).
9. A. L. Ward, EmpiricalEquationsforDrift Velocities in Silicon, Harry
Diamond Laboratories, HDL-TM-85-9 (October 1985).

24
Bibliography
1. R. N. Varney, H. J.White, L. B.Loeb, and D. Q. Posin, The Role of
Space Charge in the Study of the Townsend IonizationCoefficients and
the Mechanism of Static Spark Breakdown, Phys. Rev., 48,818 (1935).
2. R. W. Crowe, J.K. Bragg, and V. G. Thomas, Space ChargeForma-
tion and the TownsendMechanism of Spark Breakdown in Gases, Phys.
Rev., 96, 10 (1954).
3. W. B6rsch-Supan and H. Oser, NumericalComputationof the Tem-
poral Development of Currents in a Gas Discharge Tube, J. Res.
Nat. Bur. Stand., 67B, 41-60 (1963).
4. H. M. Bloom, Computationof Avalanche and Space-ChargeEffects in
Semiconductors and Gases, Harry Diamond Laboratories, HDL-
TM-69-35 (November 1969).
5. A. L. Ward, Effect of Space Charge Upon Negative Townsend Charac-
teristic, Bull. Am. Phys. Soc. Series 11, 2, 68 (1957).
6. A. L. Ward, Space-Charge Formation and the Negative Townsend
Characteristic,Bull. Am. Phys. Soc. Series 11, 2,82 (1957).
7. A. L. Ward, PulseMeasurementsofSemiconductorDevices, Diamond
Ordnance Fuze Laboratory, R52-57-1 (19 June 1957) (internal
report).
8. A. L. Ward, Basic ConsiderationsRelating to the Breakdown Gap in
Trigger Tubes, presented at the Washington Chapter of the IRE
Professional Group on Electron Devices (21 January 1957).
9. A. L. Ward, Pressure Variation of Metal to Semiconductor Contact
Resistance, AIEE-IRE Semiconductor Devices Research Confer-
ence, Boulder, CO (15-17 July 1957), oral presentation.
10. A. L. Ward, Effect of Space-ChargeFormation Upon ElectricalBreak-
down in Gases, Diamond Ordnance Fuze Laboratory, TR-500
(30 August 1957).
11. A. L. Ward, Effect of Space-Chargein Cold-CathodeGas Discharges,
Diamond Ordnance Fuze Laboratory, TR-592 (28 April 1958).
12. A. L. Ward, Effect of Space-Chargein Cold-CathodeGas Discharges,
Phys. Rev., 112, 1852 (1958).
13. A. L. Ward, Role of Space-Charges in Cold-Cathode Gas Discharges,
Bull. Am. Phys. Soc. Series 11, 4,115 (1959).

25
 14. A. L. Ward, Dependence of Metal-to-Semiconductor Contact Resis-
tance Upon ContactLoading,Diamond Ordnance Fuze Laboratory,
TR-731 (30 July 1959).
15. A. L. Ward and M. J.Reddan, A Unified Theory of Breakdown and the
Glow Discharge, Diamond Ordnance Fuze Laboratory, TR-784
(16 October 1959), reprinted in Diamond Ordnance Fuze Labora-
tory Technical Review 4 (November 1960), 69-85.
16. D. J. De Bitetto, L. H. Fisher, and A. L. Ward, Negative Current-
Voltage Characteristicsin Hydrogen using Plane ParallelElectrodes,
Phys. Rev., 118, 920 (1960).
17. M. J. Reddan and A. L. Ward, Measurements of Negative Static
Characteristicsin Argon, Bull. Am. Phys. Soc. Series 11, 6,126 (1960).
18. A. L. Ward and M. J.Reddan, A Unifled Theory of Breakdown and the
Glow Discharge,Proc. Fourth International Conference on Ioniza-
tion Phenomena in Gases, North Holland Publishing Company,
Amsterdam (1960), IB 169-174.
19. M. J. Reddan and A. L. Ward, Trip Report-FourthInternational
Conference on Ionization Phenomena in Gases, Diamond Ordnance
Fuze Laboratory, R-930-60-1 (15 January 1960) (internal report).
20. A. L. Wa,.d and Eifionydd Jones, ElectricalBreakdown in Hydrogen
at Low Pressure,Phys. Rev., 122,376 (1961).
21. A. L. Ward, Electronic Computationof the Temporal Growth of Cur-
rents in Gases, Including the Effect of Space Charge,Bull. Am. Phys.
Soc. Series II,6, 390 (1961).
22. D. J.Belknap, M. J.Reddan, and A. L. Ward, Oscillationsin theArgon
Subnormal Discharge,Bull. Am. Phys. Soc. Series 11, 6,392 (1961).
23. A. L. Ward, TripReport-FifthInternationalConference on Ionization
Phenomena in Gases, Diamond Ordnance Fuze Laboratory, R930-
61-7 (26 October 1961) (internal report).
24. A. L. Ward, Understanding Electrical Breakdown in Gases, 1961
Annual Report, Conference on Electrical Insulation, National
Academy of Sciences-National Research Council, Publication
973, 91-94.
25. L. G. Schneekloth and A. L. Ward, Impedances andIon TransitTimes
in Glow-DischargeTubes, Bull. Am. Phys. Soc. Series 11, 7,135 (1962).
24. A. L. Ward, Electronic Computation of the Temporal Growth of Cur-
rent in a Gas, Proc. Fifth International Conference on Ionization
Phenomena in Gases, North Holland Publishing Co., Amsterdam
(1962), Vol. II, 1595-1606.
26
27. A. L. Ward and L. G. Schneekloth, ImpedanceandIon TransitTimes
in Glow-Discharge Tubes, Diamond Ordnance Fuze Laboratory,
TR-1020 (30 March 1962).
28. R. D. Reymond, A. L. Ward, and D. J.Belknap, Analysis ofthe Lateral
Current ControlMechanismfor Cold-Cathode Discharges,Diamond
Ordinance Fuze Laboratory, TR-1069 (1 August 1962).
29. A. L. Ward, CalculationsofCathode-FallCharacteristicsJ.Appl. Phys.,
33,2789 (1962).
30. A. L. Ward and H. Oser, Solutions of Problemsin GaseousElectronics
on Electronic Computers; H. Oser, W. B6rsch-Supan, and A. L.
Ward, Numerical Solutions of the Equations of the Electrical Dis-
charges in the Gas Tubes, Third Annual Symposium, Washington
Chapter of the Association for Computing Machinery (20 Sep-
tember 1962).
31. A. L. Ward, Calculations of Relaxation Oscillations in Gas-Tubes
Circuits, Bull. Am. Phys. Soc., 7,459 (1962).
32. L. G. Schneekloth and A. L. Ward, Some Studies of Gaseous Break-
down, 1962 Annual Report, Conference on Electrical Insulation,
NAS-NRC Publication 1080,60.
33. A. L. Ward, Approximate CalculationsofCathode-FallCharacteristics,
IEEE Trans. Electron Devices, ED-10, 255-258 (July 1963).
34. A. L. Ward, Trip Report-Sixth InternationalConference on Ioniza-
tion Phenomena in Gases, Harry Diamond Laboratories, R-930-63-
2 (22 August 1963) (internal report).
35. A. L. Ward and L. G. Schneekloth, Calculationsof Relaxation Os-
cillationsin Gas Tube Circuits,Harry Diamond Laboratories, HDL-
TR-1166 (28 August 1963).
36. M. J.Reddan, A. L. Ward, and D. J.Belknap, Oscillationsin an Argon
Discharge, Harry Diamond Laboratories, HDL-TR-1182 (4 No-
vember 1963).
37. A. L. Ward, Calculations of Electrical Breakdown in Gases, Harry
Diamond Laboratories, HDL-TR-1193 (10 January 1964).
38. A. L. Ward, Effect of Space Chargeupon Transportof ChargeCarriers,
Harry Diamond Laboratories, HDL-TR-1195 (15 January 1964).
39. A. L. Ward, Further Calculations of Current Growth and Voltage
Breakdown, Bull. Am. Phys. Soc., 9, 181 (1964).

27
 40. A. L. Ward, Effect of Space Charge upon the Transport of Charge
Carriers,J. Appl. Phys., 35,469-479 (March 1964).
41. A. L. Ward, Calculationsof Electrical Breakdown in Gases, Physics
Colloquia, Polytechnic Institute of Brooklyn (19 March 1964).
42. A. L. Ward, Gaseous ElectronicsResearch at HarryDiamond Labora-
tories, Physics Club, American University, Washington, D.C.
(10 April 1964).
43. A. L. Ward, CalculationofMidgapBreakdown in Gases, Bull. Am. Phys.
Soc., 9,468 (1964).
44. A. L. Ward, Townsend or StreamerBreakdown, Sixi6me Conference
International sur les Phdnom~nes d'Ionisation dans les Gaz,
P. Hubert, editor, S.E.R.M.A. Paris (1963), V. II, 313.
45. M. Nahemow, N. Wainfan, and A, L. Ward, Formation of the
Cathode-Fall Region of a Pulsed Glow Discharge,Phys. Rev., 137,
56-60 (1965).
46. A. L. Ward, Attempts to CalculateMoving Striations,Bull. Am. Phys.
Soc., 10, 184 (1965).
47. A. L. Ward, Ionization, Diffusion, and Drift Velocities in a Pulsed
Townsend Discharge,J. Appl. Phys., 36,1291-1294 (April 1965).
48. A. L. Ward, Calculations of Electrical Breakdown in Air at Near-
Atmospheric Pressure,Phys. Rev., 138, A1357-1362 (31 March 1965).
49. A. L. Ward, The PurportedTransition From Avalanche to Streamer
Breakdown, Physics, 1, 215-217 (1965).
50. A. L. Ward, CalculationsofMidgap Breakdownin Gases,J.Appl. Phys.,
36,240-243 (August 1965).
51. A. L. Ward, Trip Report-Seventh InternationalConference on Ion-
ization Phenomena in Gases, Harry Diamond Laboratories, R920-
65-5 (29 November 1965) (internal report).
52. Edward R. Berman and A. L. Ward, FortranIV Programfor Gas
Tubes and Other Dielectrics,Harry Diamond Laboratories, HDL-
TR-1310 (5 November 1965).
53. A. L. Ward and J.L. Scales, Computer Programmefor Space-Charge
Effects in SolidDielectrics;J.L. Scales and A. L. Ward, Minority-Carrier
Mobility in the Presenceof Space Charge,Bull. Am. Phys. Soc., 11, 34
(1966).
54. A .L. Ward, Calculations of Relaxation Oscillations in Helium at
Several Atmospheres Pressure,Bull. Am. Phys. Soc., 11, 502 (1966),

28
 presented at the 18th Gaseous Electronics Conference, Minneapo-
lis, MN (20-22 October 1965).
55. A. L. Ward, FurtherCalculationsof MidgapBreakdown in Gases, Bull.
Am. Phys. Soc., 12,226 (1967), presented at the 19th Annual Gas-
eous Electronics Conference, Atlanta, GA (12-14 October 1966).
56. A. L. Ward and J.L. Scales, Computer Analysis of Space Charge In-
stabilitiesin GalliumArsenide,presented orally at the 1967 Informal
Conference on Active Microwave Effects in Bulk Semiconduc-
tors, New York, NY (2-3 February 1967).
57. A. L. Ward, Effect of Ionization, Diffusion and Space Charge upon
MobilityMeasurements,Proc. Seventh International Conference on
Phenomena in Ionized Gases (Becograd, 22-27 August 1965),
edited by B.Perovic and D. Tosic, Gradevinska Knjiga Publishing
House, Becograd (1966), Vol. I, 65-68.
58. A. L. Ward, Calculations of Breakdown in Air Between Spherical
Electrodes, presented at Conference on Electrical Insulation and
Dielectric Phenomena, Pocano Manor, PA (17-20 October 1966),
1966 Annual Report, NAS-NRC Publication 1484, Washington,
D.C. (1967), 30-33.
59. A. L. Ward, Computer Calculationsof Breakdown in Very Long Gaps,
Proc. Eighth International Conference on Phenomena in Ionized
Gases, 1967, Springer Verlag, Vienna, Austria, Contributed Pa-
pers, 193-196, paper 3.2.1.3.
60. A. L. Ward, Calculationsof Voltage Tuning of Gunn OscillatorsDue
to Doping Gradients,oral presentation to the Conference on Solid
State Devices (5-8 September 1967), University of Manchester,
Institute of Science and Technology, paper 6.4.
61. A. L. Ward and B.J.Udelson, Computer CalculationsofAvalanche-
Induced Relaxation Oscillationsin Silicon Diodes, oral presentation,
IEEE 1967 International Electron Devices Meeting, paper 7.7,
Washington, D.C. (18-20 October 1967), abstract printed in IEEE
Trans. Electron Devices, ED-15, No. 6,416 (June 1968).
62. A. L. Ward, CalcoliSul "Breakdown ElettricoNei Gas," La Scuola in
Azione, Estratto Dal Numero 8,180-198 (Agosto 1967).
63. A. L.. Ward, Trip Report-Eighth International Conference on Phe-
nomena in Ionized Gases and Conference on Solid State Devices,
Manchester, Harry Diamond Laboratories, HDL-R920-67-8
(24 November 1967) (internal report).
64. A. L. Ward, Computer Calculationsof Negative Differential Conduc-
tivity Effects in n-Type Germanium, Fourth Annual Informal
29
 Conference on Active Microwave Effects in Bulk Semiconduc-
tors, New York, NY (25-26 January 1968).
65. J. L. Scales and A. L. Ward, Effects of Space Charge on Mobility,
Diffusion and Recombination of Minority Carriers,J.Appl. Phys., 39,
1692 (15 February 1968).
66. J. L. Scales and A. L. Ward, Computer Studies of Gunn Oscillations
in Gallium Arsenide, Harry Diamond Laboratories, HDL-TR-1403
(August 1968), AD 675488.
67. A. L. Ward and B.J.Udelson, Computer Calculationsof Avalanche-
Induced Relaxation Oscillations in Silicon Diodes, IEEE Trans. Elec-
tron Devices, ED-15, 847-857 (November 1968).
68. A. L. Ward, Calculationof Space ChargeEffects in PulsedDischarges,
presented orally at the Second International Conference on Elec-
tron and Ion Beam Science and Technology (17-20 April 1966),
New York, NY, proceedings published 1969, Gordon and Breach,
New York, 75-88.

69. A. L. Ward, A Versatile ComputerProgramfor TemporalSpace Charge

Phenomena, presented orally, Second Biennial Cornell Electrical
Engineering Conference, 1969 Topic: Computerized Electronics,
Cornell University, Ithaca, NY (26-28 August 1969), published
proceedings, IEEE Catalog No. 69C65-Corn., 392-401.
70. A. L. Ward, An Analysis of Measurements of Electron Drift Velocity
in Silicon at High Fields, Harry Diamond Laboratories, HDL-TM-
70-35 (March 1971).
71. A. L. Ward and B.J.Udelson, Computer Studies Toward Improving
PulsedImpatt OscillatorPerformanceby Meansof Double-DriftDiodes
and Second-Harmonic Tuning, presented orally, Third Biennial
Cornell Electrical Engineering Conference, 1971 Topic: High Fre-
quency Generation and Amplification: Devices and Applications,
Cornell University, Ithaca, NY (17-19 August 1971), published
proceedings, IEEE Catalog No. 71C71-Corn., 135-144.
72. B.J.Udelson and A. L. Ward, ComputerComparisonofN+-P-P+ and
P -N-N+JunctionSilicon DiodesforImpattOscillators,Electron. Lett.,
7, 723-724 (2 December 1971).
73. A. L. Ward and B.J.Udelson, TheoreticalComparisonofN+PP + and
P+NN + Silicon ImpattDiodes, oral presentation (18 February 1972),
International Solid-State Circuits Conference, Philadelphia, PA,
published in Digest of Technical Papers, 198-199.
74. A. L. Ward, The Thomson, letter to the Editor, Physics Today, 26,15
(July 1973).
30
75. A. L. Ward, FurtherCalculationsof the Relaxation Mode, presented
orally, Fourth Biennial Cornell Electrical Engineering Confer-
ence, 1973 Topic: Microwave Semiconductor Devices, Circuits
and Applications, Cornell University, Ithaca, NY (14-16 August
1973), published proceedings, Catalog No. 73-Cornell, 391-400.
76. A. L. Ward, An Electro-ThermalModel of Second Breakdown, pre-
sented orally, IEEE Nuclear and Space Radiation Effects Confer-
ence, University of California, San Diego (27-30 August 1967),
IEEE Trans. Nucl. Sci., NS-23, 1679-1684 (December 1976).
77. A. L. Ward, An Electro-ThermalModel of Second Breakdown, Elec-
tron Devices Group, IEEE Washington Section (14 October 1976),
presented at Semiconductor Technology Seminar, National Bu-
reau of Standards (3 November 1976).
78. A. L. Ward, Calculationsof Second Breakdown, Poster Paper, p-24,
Annual Conference on Nuclear and Space Radiation Effects,
College of William and Mary, Williamsburg, VA (12-15 July
1977), IEEE Trans. Nucl. Sci., NS-24, 2357-2360 (December 1977).
79. A. L. Ward, Multipatt and Other Avalanche Oscillations in Silicon
Diodes, oral presentation, Proc. Sixth Biennial Conference on
Active Microwave Semiconductor Devices and Circuits, Cornell
University, Ithaca, NY (16-18 August 1977), 269-278.
80. A. L. Ward, Studies of Second Breakdown, IEEE Trans. Parts Hybrids
Packag., PHP-13, 361-368 (December 1977).
81. A. L. Ward, UnderstandingSecond Breakdown, and A. L. Ward and
Chris Fazi, Avalanche Oscillations and Second Breakdown, both
presented orally, Nuclear EMP Meeting (6-8 June 1978), Univer-
sity of New Mexico, NEM 1978 Record (Abstracts), 37,38.
82. A. L. Ward, Modes of Avalanche Oscillationsin Silicon Diodes, IEEE
Trans. Electron Devices, ED-25, 683-687 (June 1978).
83. A. L. Ward, Ionizing Radiation and Second Breakdown in Silicon
Diodes, Source-Region EMP Technology and Systems Survivabil-
ity Requirements Seminar, Monterey, CA (22-24 August 1978),
Abstract Book, 60-63.
84. A. L. Ward, Harmonic Generation Impatt Diodes, in Numerical
Analysis of Semiconductor Devices, Proc. Conference Dublin,
Ireland (27-29 June 1979), Ed., Browne & Miller, Boole Press,
Dublin (1979), 293-295.
85. A. L. Ward, Doping Profiles and Second Breakdown, 1979 Electrical
Overstress/Electrostatic Discharge Symposium, Denver, CO
(25-27 September 1979), Proc. EOS-1, 109-115.
31
 86. A. L. Ward, ComputerSimulation ofSecond Breakdown in Silicon-on-
Sapphire Diodes, presented Nuclear EMP Meeting (5-7 August
1980), Anaheim, CA, NEM 1980 Record (Abstracts), 96.
87. A. L. Ward, Oscillating Voltage Pulses and Second Breakdown, pre-
sented Electrical Overstress/Electrostatic Discharge Symposium
(9-11 September 1980), San Diego, CA, Proc. EOS-2,130-139.
88. A. L. Ward, CalculationsofSecond Breakdownin SiliconDiodes, Harry
Diamond Laboratories, HDL-TR-1978 (August 1982).
89. A. L. Ward, The Forward-BiasCharacteristicas Predictorand Screen
ofReverse-Bias Second Breakdown, presented Electrical Overstress/
Electrostatic Discharge Symposium (21-23 September 1982),
Orlando, FL, Proc. EOS-4, 71-75.
90. A. L. Ward, Calculationsof Second Breakdown in Silicon Diodes at
Microwave Frequencies, presented Electrical Overstress/Electro-
static Discharge Symposium, Poster Session (27-29 September
1983), Las Vegas, NV, Proc. EOS-5, 102-107.
91. A. L. Ward and J.M. Stellato, Comparison Between Calculatedand
Measured Forward and Reverse Current-Voltage Characteristics,
presented First Annual Electrical Overstress Exposition, San Jose,
CA (24-26 April 1984), Harry Diamond Laboratories, HDL-TR-
2099 (November 1986).
92. A. L. Ward, Calculationof Second Breakdown with Sinusoidal Wave-
forms, presented orally, Fourth NEM Symposium (2-6 July 1984),
Baltimore, MD, NEM 1984 Record (Abstracts), 61.
93. A. L. Ward, Calculations of High-Current Characteristicsof Silicon
Diodes at Microwave Frequencies, Harry Diamond Laboratories,
HDL-TR-2057 (October 1984).
94. A. L. Ward, EMP Damage Physicsfor Electronic Devices and Inte-
grated Circuits,Paper HDL-2, HDL Research Book (1985).
95. A. L. Ward, Modelling of Breakdown Phenomena in Semiconductor
andDielectricMaterials,oral presentation, Electronics Engineering
Department, Lawrence Livermore National Laboratory,
Livermore, CA (28 March 1985).

32
 96. B. Goplen, J. McDonald, A. L. Ward, and J. M. Stellato, A Two
DimensionalCodefor Avalanche Breakdown in Semiconductors, Proc.
Fourth International Conference on the Numerical Analysis of
Semiconductor Devices and Integrated Circuits, Nasecode IV
(19-21 June 1985), Trinity College Ireland, edited by J. Miller,
Boole Press (1985), 299-303.
97. A. L. Ward, EmpiricalEquationsfor Drift Velocities in Silicon, Harry
Diamond Laboratories, HDL-TM-85-9 (October 1985).
98. A. L. Ward, Avalanching in Single-Event-UpsetChargeCollection in
Silicon Diodes, listed by title only, 4th Annual DoD-DoE-NASA
Symposium on Single Event Effects, Los Angeles, CA (8-9 April
1986).
99. A. L. Ward and J. M. Stellato, Comparisonbetween Calculatedand
Measured Forward and Reverse Current-Voltage Characteristicof
Semiconductor Junctions,Nuclear EMP Meeting (19-23 May 1986),
Albuquerque, NM, NEM 1986 Record (Abstracts), 104.
100. A. L. Ward, Avalanchingin Single-Event-UpsetCharge Collection in
Semiconductor Diodes, Nuclear and Space Radiation Effects Con-
ference (21-23 July 1986), Providence, RI, IEEE Trans. Nucl. Sci.,
NS-33, 1552-1559 (December 1986).
101. A. L. Ward, J.B. Deppe, and R. V. Garver, Spike Leakage, Limiting,
and Rectification in Silicon PIN Diodes, High Power Microwave
Technology for Defense Application Conference, Kirkland AFB,
NM (1-5 December 1986).
102. A. L. Ward, Avalaachingin Single-Event-Upset ChargeCollection in
Semiconductor Diodes, Harry Diamond Laboratories, HDL-TR-
2106 (February 1987).
103. A. L. Ward, CalculationsofAvalanche Breakdown in Silicon Dioxide,
Harry Diamond Laboratories, HDL-TR-2112 (May 1987).
104. A. L. Ward, Avalanchingin Single-Event-Upset Charge Collection in
Semiconducting Diodes, Solid State Seminar, Physics Department,
University of Maryland (30 April 1987).

33
Appendix A.-Listing of Program DIODE

35
 Appendix A
C DIODE SPACE CHARGE PROGRAM FOR GASES AND SEMICONDUCTORS WITH DRIFT, A 1
C IONIZATION AND RECOMBINATION AS A FUNCTION OF TEMPERATURE A 2
REAL ICUR,ICURO,IEXT,IINT,IINTO,123,JAVG,JC,JCSAV,JDIF,JDISPL,JDIS A 3
1SP,JINT,JM,JMAX,JMDIF,JMSAV,JMO,JP,JPDIF,JPSAV,JPO,MJDSP,MJM,4JMDF A 4
2,MJP,MJPDF ,MTEM,MUM,MUMO,MUP ,MUPO,NM,NMSAV,NP,NPP,NPSAV,NOM,NOP,L, A 5
3MUF DMUMF ,MUPF A 6
DIMENSION DVXOT(1O1) ,DNPDX(1O1),DNMDX(1O1) ,DIFP(1O1) ,DIFM(1O1), A 7
1GM( 101),GP(101),JDISSP(1O1),DTEM( 101) A 8
INTEGER PRTFRQ,CHECK,ENDING,CHECK2,EXTRAP,CIRCT(10) ,CHECK1 A 9
COMMON /ABC/ MODA,MODB,MODD,MODVP ,MODVM,MODVNP ,MODVNM,MODT,A1 ,A2, A 10
1A3,A4,A5,A6,B1,B2,B3,B4,B5,B6,Dl,D2,D3,D4,D5,D6,ClC2,C3,C4,C5,C6, A 11
2C7,C8,C9,ClO,011 ,Cl2,C13,Cl4,C15,C16,Cl7,C18,C19,C20,MUP,MUM,MUPO, A 12
3MUMO,AVP,AVM,NOP,NOM,VPlM,VMlM,CAT,CBT,PWRPPWRM,TEMP A 13
COMMON /CIRT/ USTAT,U,V,DU,VO,V1,V2,OMEGA,OMEGA1,OMEA2,P111,PH12, A 14
1RD,LD,R(3) ,L(3) ,C(3) ,CP,T1, IEFCY,VNEW A 15
COMMON /DSTR/ NP(201),NM(201),DN(201),TEM(201),E(201),DE(201),VSUM A 16
1(201),VP(201),VM(201),JP(201),JM(201),JPDIF(201),JMDIF(201),JDISPL A 17
2(201),RCMBR(201),ALPHA(201),BETA(201),TPRINT(201) A 18
COMMON /EFCY/ AMP,AMP2,PHASE,PJ,PV,PHASE2,PJ2,PV2,DTP,GNEG,RFP,EFF A 19
1 ,AMPVDC,AMPJDC,DCPWR,AMPV,AMPV2 A 20
COMMON /FUND/ M,MP1 ,N3S,PRTFRQ,NXPRNT(6) ,MODP,MODFCH,EXTRAP,MS, IJ, A 21
1 IPOINT,FM,STEPFA,DX,DT,T,D,S,P,EL,EPSO,DIELK,DNSTY,SPCHT,THCND,DEN A 22
2SPL,DENSMI ,DENS2JDENS2T,REC,EGAP,GAMSEC,GAMI ,GAMMAJ.TAUSEC A 23
COMMON /MJV/ JPO,JMO,JMAX,DJMAXM,JC,JINT,IINT,IINTO,IEXT,123,DIDT, A 24
1DVDT,DVDTO,VC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, A 25
2MJPDF,MJMDF,MJ1SP,MTEM,VPMAX,VM4MAX,CST( 10,3) ,DMTEMDDIDT,DIDTO A 26
COMMON /PLO/ TIME(401),VOLT(401),CURR(401),JAVG(401),CURR1(401), A 27
1CURR2(401),CURR3(401),CLTT(401),TAVR(401) A 28
COMMON /SAV/ NPSAV(1O1),NMSAV(1O1),ESAV(1O1),VXSAV(1O1),VPSAV(1O1) A 29
1,VMSAV(101),JPSAV(1O1),JMSAV(1O1),VSAV,TSAV,JCSAV,GAMSAV,TMPSAV A 30
C A 31
MUPF(Z)=MUF(Z,P,C1 ,C2,C3,C4,C9,MUP) A 31B
MUMF(Z)=MUF(Z,P,C5,C6,C7,C8,C1O,MUM) A 310
2 UNIVA(EA)=UNIV(P,MODA,EA,A1,A2,A3,A4,A5,A6) A 31D
UNIVB(EA)=UNIV(P,MODB,EA,B1,B2,B3,B4,B5,B6) A 31E
UNIVD(EA)=UNIV(P,MODD,EA,D1,D2,D3,D4,D5,D6) A 31F
BOLTZ=8.6E-5 A 32
END ING=10 A 33
10 CHECK1=0 A 33A
CALL INPUT(ENDING,CHECK1) A 338
VNEW=V
IF (CHECK1.EQ.1) GO TO 275 A 33C
C
C SET COUNTERS A 34
C
INSTEP=O A 35
DT=O. A 36
I INT=O
DI DT=O
DDIDT=O
ISET1l A 37
INDEX1l A 38
MINUS=MS A 39

37
Appen,,dix A
MSO=MS A 40
CHECK=O A 41
CHECK2=O A 42
C
C COMPUTE CONSTANTS FROM INPUT A 43
C
CALL INCON(CIRCT,ENDING) A 44
IT=0 A 45
DTP=(TPRINT(IJ)-TPRINT(l))/(FLOAT(IPOINT)-l.) A 46
IF (INSTEP.EQ.O) GO TO 125 A 47
C START OF COMPUTATION LOOP A 48
110 INSTEP=1 A 49
C COMPUTE NEW TIME A 50
DT=STEPFA*DX/VMAX A 51
GO TO 123 A 52
120 DT=DT/2. A 53
123 T=T+DT A 54
C COMPUTE U A 55
U=USTAT A 56
IF (T.LT.T1) GO TO 125 A 57
IF (IEFCY.EQ.O) GO TO 124 A 58
U=U+DU+VO*COS(OMEGA*T)+V1*COS(OMEGA1*T+PH1Ii)+V2*COS(OMEGA2*T+PN 12) A 59
GO TO 125 A 60
124 U=U+DU+VO*SIN(OMEGA*(T-Tl)) A 61
125 CONTINUE A 62
DO 126 1=1,MP1 A 62A
GP( I)=EL*NP( I) A 62B
126 GM( I)=EL*NM( I) A 63
C CALCULATE RECOMBINATION RATE A 64
DENS2T=DENS2*(MTEM/300. )**3*EXP(EGAP/BOLTZ*(l1./300.-i ./MTEM)) A 65
DO 128 1=1,MP1 A 66
NPP=NP( I)*NM( I) A 67
IF (DENS2T.NE.O.) NPP=(NPP-DENS2T)/SQRT(OENS2T) A 68
128 RCMBR( I)=REC*EL*NPP A 69
IF (INSTEP.EQ.O) GO TO 169 A 70
C CALCULATION OF EXTERNAL CIRCUIT - V,JEXT A 70A
CALL EXTCIR (ISET,CIRCT) A 708
C HOLE CHARGE DENSITY A 71
IF (EXTRAP.EQ.1) CO TO 130 A 72
GP(MP1 )=DENSPL*EL A 73
GO TO 140 A 74
130 GP(MP1)=GP(MP1)+(2.*JP(MP1)-3.*JP(M)+JP(M-1))*DT/DX A 75
140 SUM=(UNIVD(E(MP1) )*JM(MP1)+UNIVD(E(1) )*JM(l) )/2. A 76
DO 150 1=1,M1 A 77
GP(I)=GP(I)+DT*(ALPHA(I)*JM(I)+BETA(I)*JP(I)+(JPCI+1)-JPCI))/DX) A 78
150 SUM=SUM+UNIVD(E(I) )*JM( I) A 79
C COMPUTE EFFECT OF PHOTONS HITTING CATHODE A 80
RFAC=O. A 81
IF (TAUSEC.NE.O.) RFAC=EXP(-DT/TAUSEC) A 82
GAMMA=RFAC*GAMMA+GAMSEC*Dx*( 1. RFAC)*SUM A 83
C ELECTRON CHARGE DENSITY A 84
DO 160 1=2,MP1 A 85

38
 Appendix A
160 GM(I)=DT*(ALPHA(I)*JM(I)+BETA(I)*JPI)-(JM()-JM(I1))/DX)+G4(I) A 86
DO 162 I=1,MP1 A 87
GP( I)=GP( I)-DT*RCMBR( I) A 88
162 GM( I)=GM( I)-DT*RCMBRC I) A 89
C TEST FOR NEGATIVE NUMBER DENSITIES A 90
DO 168 1=1,MP1 A 91
NP( I)=GP( 1)/EL A 92
NM( I)=GM( 1)/EL A 93
IF (NP(I).GE.O.) GO TO 165 A 94
GP( I)=O. A 96
NP( I)=O. A 97
165 IF (NM(I).GE.0.) GO TO 168 A 98
GM( I)=O. A 99
N4( I)0. A 100
168 CONTINUE A 102
C ELECTRIC FIELD A 103
169 ESTAR=O.O A 104
E (1)=0. A 105
E(2)=GM(1)+GM(2)-GPC1)-GPC2)-EL*CDN(1)+DN(2)) A 105A
DO 170 1=3,M A 106
170 E()E11+G(-)2*M11+.*MI+M11-PI2-.G(- A 107
1)-2.*GPCI)-GP(1+1))/3.-EL*(DN(I-1)+DN(I)) A 107A
E(MP1)=E(M)+GMCM)+GM(MPl)-GP(M)-GP(MP1)-EL*ON(M)+DN(M1)) A 1078
DO 180 1=2,MP1 A 107C
180 ESTAR=ESTAR+E( I-i)+E( I) A 108
E(1 )VNEW/D-CST(6, 1 )ESTAR/(2.*FM) A 109
E( 1)=E(l)-CST(6,1)*((GM(l)-GP(1)-GM(MP1)+GP(MP1)-EL*(DN(1)-DN(MP1) A 110
1))/(6.*FM)) Al111
C ADJUST E A 112
DO 190 I=2,MP1 A 113
EC I)=CST(6,1)*E( I)+E(1) A 114
190 VSUMCI)=VSUM(1-1)+0.5*CE(I)+E(I-1))*DX A 115
C COMPUTE FUNCTIONS OF ELECTRIC FIELD A 116
IF (INSTEP .EQ.0) GO TO 198 A 117
DO 195 1=1,MP1 A 117A
DE( I)=E( I)-ESAV( I) A 118
JDISPL( I)=DIELK*EPSO*DE( I)/DT A 119
195 DVXDT( I)=(VSUM(lI)-VXSAV( I))/DT A 120
C ELECTRON VELOCITY-VM A 121
198 DO 230 I=1,t4P1 A 122
AOP=E( 1)/P A 123
IF (MODVM.EQ.O) GO TO 200 A 124
IF (AOP.LE.C7.OR.AOP.GE.C20) GO TO 200 A 125
VMO=Cl6+Cl7*AOP+C18*AOP*AOP+Cl9*AOP*AOP*AOP A 126
GO TO 201 A 127
200 VMO=E(I)/P*MUMFCE(I)) A 128
201 IF (MODVNM.EQ.O) GO TO 205 A 129
ADN=ABS(DN( I)) A 130
DMO=C ADN/NOM) **AVM A 131
VM1=MUMO*AOP A 132
IF (VM1.GT.VMlM) VM1=VM1M A 133
VM(I)=(VM1+(VMO-VM1)/C1.+DMO))*(300./TEMCI))**PWRM A 134
GO TO 210 A 135

39
Appendix A
205 VM( I)VMO*(300./TEM(I) )**PWRM A 136
C HOLE VELOCITY-VP A 139
210 IF (MODVP.EQ.O) GO TO 220 A 140
IF (AOP.LE.C3.OR.AOP.GE.C15) GO TO 220 A 141
VPO=C1 1+C12*AOP+C13*AOP*AOP+Cl4*AOP*AOP*AOP A 142
GO TO 221 A 143
220 VPO=MUPF(E(I))*ECI)/P A 144
221 IF (MODVNP.EQ.O) GO TO 225 A 145
DPO=(ADN/NOP)**AVP A 146
VP 1MUPO*AOP A 147
IF (VP1.GT.VP1M) VP1=VP1M A 148
VPCI)=(VPI+(VPO-VP1)/(1.+DPO))*(300./TEM(I))**PWRP A 149
GO TO 230 A 150
225 VP( I)=VPO*(300./TEM(I) )**PWRP A 151
230 CONTINUE A 151A
C COMPUTE MAXIMUM VELOCITIES A 152
VMMAX=VM( 1) A 152A
VPMAX=VP (1) A 152B
DO 235 I=1,MP1 A 152C
VMMAX=AMAX1 (VMMAX,VM( I)) A 1520
235 VPMAX=AMAX1(VPMAX,VP( I)) A 152E
VMAX=AMAX 1(VPMAX ,VMMAX) A 152F
C COMPUTE DIFFUSION CURRENT DENSITIES A 152G
DO 255 1=1,MP1 A 152H
DNMDX( I)=(NM( I+1)-NM( I-1) )/(2.*DX) A 1521
255 DNPDX(I)=(NP(1+1)-NP(I-1))/(2.*DX) A 152J
DNMDX( 1)=(NM(2)-NM(1) )/DX A 152K
DNMD)(MP1)=(NM(MP I)-NM(M))/DX A 152L
DNPDX( 1)=(NP(2)-NP(1) )/DX A 152M
DNPDX(MP1 )=(NP(MP1 )-NPCM) )/DX A 152N
DO 265 I=1,MP1 A 1520
AOP=E( 1)/P A 152P
DIFM( I)0. A 152Q
IF (AOP.NE.0.) DIFM(I)=BOLTZ*TEM(I)*VM(I)/AOP A 152R
JMDIF( I)=-DIFM( I)*DNMDX( I)*EL A 152S
EIFP( I)=O. A 152T
IF (AOP.NE.O.) DIFP(I)=BOLTZ*TENCI)*VPCI)/AOP A 152U
265 JPDIF(I)=DIFP(I)*DNPDX(I)*EL A 152V
C CURRENT DENSITIES-JP,JM A 152W
DO 240 1=1,MP1 A 153
JP( I)=GP( I)*VP( I) A 154
240 JM(I)=GM([)*VM(I) A 155
JP(MP1 )=JP(mp1 )+JPO A 155A
GP(MP1 )=ABS(JP(MP1 )/VP(MP1)) A 155B
NP(MP1 )=GP(MP1)/EL A 155C
DENS IT=DENSM I A 156
IF (EXTRAP.EQ.1) DENSIT=NM(l) A 156A
C COMPUTE INITIAL GAMMA A 157
IF (INSTEP.EQ.1) GO TO 245 A 157A
IF (GAMMA.GE.O.) GO TO 243 A 157B
GAMMA=O. A 157C
DO 242 1=2,M A 1570

40
 Appendix A
24.2 GAMMA=GAMMA+JM( I)*UNIVDCE( I)) A 157E
GAMMA=(GAMMA+CUNIVDCE(l))*Jt4(1)+UNIVD(E(MPl))*JM(MP1))/2.)*GAMSEC* A 157F
lox A 157G
24.3 IF (INSTEP.EQ.O) GO TO 250 A 157H
245 JM( 1)=JMO+GAMI*JP( 1)+GAMMA+DENSI T*EL*VM( 1) A 158
IF (EXTRAP.EQ.O) GO TO 250 A 159
JMPI).tJt(1)+VM(l)*DT/DA*(2.*J14(l)-3.*JM(2)+JM(3)) A 159A
250 GM(1)=ABS(JM(l)/VM(1)) A 159B
NM( 1)=GM( 1)/EL A 159C
C COMPUTE AVERAGE CURRENT DENSITIES A 160
MJP=(JP(1)+JP(MP1) )/2. A 160A
MJM=(JM(1)+JM(MPl) )/2. A 160B
MJPDF=(JPDIF(1)+JPDIF(4PI) )/2. A 160C
MJMDF=(JMDIF(1)+JMDIF(MPl))/2. A 160D
MJDSP=(JDISPL( 1)+JDISPL(MPl))/2. A 160E
DO 4.00 1=2,M A 161F
MJP=MJP+JP( I) A 160G
MJM=MJM+JM( I) A 160H
MJPDF=MJPDF+JPDIF( I) A 1601
MJMDF=MJMDF+JMDIF( I) A 160J
400 MJDSP=MJDSP+JDISPL( I) A 160K
MJP=MJP/FM A 160L
MJM=MJM/FM A 160M
MJPDF=MJPDF/FM A 161
MJMDF=MJMDF/FM A 161A
MJDSP=MJDSP/FM A 161B
JC=MJP+MJM A 1610C
JDIF=MJMDF+MJPDF A 161D
J INT=JC+JD IF+MJDSP A 161E
I INT=ABS(S*JINT) A 161F
IF(CIRCT(l).EQ.5) IEXT=(U-V)/R(1) A 161G
123=ICUR(2)+ICUR(3) A 161H
VNEW=V-RD*I INT-LD*D lOT
IF (DT.EQ.0) DIDT=0. A 1611
DDIDT=O.
IF(DT.NE.0.) DIDT=(IINT-IINTO)/DT A 161J
IF(DT.NE.0.) DDIDT=(DIDT-DIDTO)/DT
IF (MODT.EQ.O.) GO TO 30 A 161K
DO 4.10 I=1,MP1 A 162
JDISSP(I)=JP(I)+JlF.I)+JPDIF(I)+JMDIFCI) A 163
DTEM( I)=JDISSP( I)*E( I)*DT/(DNSTY*SPCHT) A 164
410 TEM( I)=TEM( I)+DTEM( I) A 165
t4TEM=(TEM( 1)+TEM(M.P1) )/2. A 166
DO 420 1=2,M A 167
4.20 MTEM=MTEM+TEM( I) A 168
MTEM=MTEM/FM A 169
DMTEM=0. A 169A
DZT=DT*1 .E9
IF (DZT.NE.0.) DMTEM=(MTEM-TMPSAV)/DZT A 169B
C END OF COMPUTATION LOOP A 169C
C A52-91 INSERTED BETWEEN A169D AND 170 A 169D
C TEST FOR STABILITY AND OTHER CRITERIA A 52
30 CALL TEST(MINUS,INSTEP,CHECK,CHECK2,ISET,MSO) A 53
GO TO (40,120,260), ISET A 54.

41
Appendix A
140 CHECK2O0 A 55
DO 45 I=1,MP1 A 56
ALPHA(I)=UNIVA(E(I))*(l.-CAT*CTEM(I)-3oo.)) A 56A
145 BETA(I)=UNIVB(E(I))*(l.-CBT*(TEM(I)-30O.)) A 568
C TEST FOR PRINTING A 56C
IF (MODP.EQ.2) GO TO 60 A 57
IF (T.LT.(FLOAT(IT)*DTP+TPRINT(l))) GO TO 50 A 58
IT= IT+1 A 59
TIMEC IT)=T A LE0
VOLT( IT)=V A 61
CURR( IT)=IINT A 62
JAVG( IT)=JC A 63
C0Rtl('fT)=ICUR( 1) A 614
CURR2( IT)=ICUR(2) A 65
CURR3( IT)=ICUR(3) A 66
CLTT( IT)=DIDT
TAVR( IT)=MTEM
50 IF (T.LT.TPRINT(INDEX)) GO TO 100 A 67
GO TO 70 A 68
60 IF (JINT.LT.TPRINT(INDEX)) GO TO 100 A 69
C CALCULATE VALUES TO BE PRINTED AND PRINT THEM A 70
C (72 TO 76 REMOVED) A 71
70 CALL OUTPUT(DVXDT) A 77
INDEX= INDEX+l A 78
IF (INDEX.LE.IJ) GO TO 100 A 79
WRITE (6,360) A 80
WRITE (6,350) (VC(I),DVC(I),I=1,3) A 81
IF(IEFCY.NE.1) GO TO 270 A 82
CALL EFFCY A 83
WRITE (6,500) AMP,AMPV,PJ,PV,PHASE,GNEG,RFP,AMPVOC,AMPJDC, A 83A
lDCPWR ,EFF A 83B
WRITE (6,501) AMP2,AMPV2,PJ2,PV2,PHASE2 A 83C
GO TO 270 A 84
100 IF (CHECK.EQ.O) GO TO 110 A 85
CHECK=0 A 86
MI NUS=MINUS-1 A 87
IF (MINUS.GT.O) GO TO 110 A 88
WRITE (6,310) A 89
260 WRITE (6,320) A 90
CALL OUTPUT (DVXDT) A 91
270 WRITE (6,330) A 170
WRITE (6,340) (TIMECIK),VOLT(IK)-LD*CLTT(IK)-RD*CURR(IK),CURR(IK)
1,JAVG(IK),CURR1(IK),CURR2(IK),VOLT(IKY,TAVR(IK),IK=1,IPOINT) A 172
275 READ (1,290) ENDING A 1714
IF (ENDING.GE.10000) GO TO 280 A 175
WRITE (6,300) A 176
GO TO 10 A 177
280 STOP A 178
C A 179
C A 180
290 FORMAT (15) A 181
300 FORMAT (lHi) A 182
310 FORMAT (//51H THERE HAVE BEEN MS CASES OF NEGATIVE VELOCITIES./1 A 183
16H END OF PROBLEM.) A 1814

42
 Appendix A
320 FORMAT (47H PRINTOUT AT TIME OF ERROR FOLLOWS ON NEXT PAGE) A 186
330 FORMAT (lH1,13X,1HT,14X,1HV,14X,1HI,11X,4HJAVG,13X,2HI1,13X,2H12, A 187
18X,7HV(DLEG) ,7X,8HAVG.TEMP) A 188
340 FORMAT (8(lPE15.6)) A 189
350 FORMAT (1P,6E20.8) A 190
360 FORMAT (1Hl,1OX,46HCAPACITOR VOLTAGES AT LAST SUCCESSFUL PRINTOUT A 191
1//17X,3HVC1,16X,4HDVC1 ,17X,3HVC2,16X,4HDVC2,17X,3HVC3,16X,4HDVC3) A 192
C FORMAT (5E14.8) A 193
500 FORMAT (//47H THE AMPLITUDE OF THE FIRST HARMONIC OF JAVG IS, A 194
*lPE20.8/j9H THE AMPLITUDE OF THE FIRST HARMONIC OF V(OLT) IS, A 195
*lPE20.8/54H THE PHASE OF THE FIRST HARMONIC OF JAVG IN DEGREES IS, A 196
*1PE20.8/51H THE PHASE OF THE FIRST HARMONIC OF V IN DEGREES IS, A 197
*1PE20.8/43H THE PHASE BETWEEN V AND JAVG IN DEGREES IS, 1PE2O.8/ A 198
*28H THE NEGATIVE CONDUCTANCE IS,1PE2O.8/16H THE RF POWER IS, A 199
*1PE20.8/28H THE TIME AVERAGE VOLTAGE IS,1PE2O.8/31H THE AVERAGE CU A 200
*RRENT DENSITY IS,1PE2O.8/15H INPUT POWER IS,1PE2O.8/18H THE EFFICI A 201
*ENCY IS,1PE2O.8) A 202
501 FORMAT (//48H THE AMPLITUDE OF THE SECOND HARMONIC OF JAVG IS, A 203
*lPE20.8/50H THE AMPLIlUDE OF THE SECOND HARMONIC OF V(OLT) IS, A 204
*lPE20.8/55H THE PHASE OF THE SECOND HARMONIC OF JAVG IN DEGREES IS A 205
*,1PE20.8/52H THE PHASE OF THE SECOND HARMONIC OF V IN DEGREES IS, A 206
*1PE20.8/43H THE PHASE BETWEEN V AND JAVG IN DEGREES IS,1PE2O.8) A 207
END A 208-
SUBROUTINE INPUT(ENDING,CHECK1) B 1
INTEGER PRTFRQ,EXTRAP,ENDI NG,CHECK1 B 2
DIMENSION TPRNTI(66), TINC(66), NINC(66) B 3
REAL JPDI F,JMDI F,NP,NM,JP,JM,JDISPL,MJPMJM,MJPDF ,MJMDF ,MJDSP B 4
REAL MUMMUP,JMAX,JPOJC,MTEM, L,IINT,JINT,IINTO,ICUR,ICURO,IEXT, B 5
1 123,MUPO,NOP,MUMO,NOM,JMO B 6
COMMON /ABC/ MODA,MODBMODD,MODVP ,MODVM,MODVNP ,MODVNM,MODT,A1 ,A2, B 7
lA3,A4,A5,A6,B1 ,B2,B3,B4,B5,B6,Dl,D2,D3,D4,D5,D6,C1 ,C2,C3.,C4,C5,C6, B 8
2C7,C8,C9,ClO,C11,C12,C13,Cl4,C15,C16,Cl7,C18,C19,C20,MUP,MUM,MUPO, B 9
3MUMOAVP,AVMNOPNOM,VP1M,VM1M,CAT,CBTPWRP,PWRM,TEMP B 10
COMMON /CIRT/ USTAT,U,V,DU,VO,V1,V2,OMEGA,OMEGA1,OMEGA2,PHI1,PHI2, B 11
1RD..LDR(3) ,L(3) ,C(3) ,CP,T1, IEFCY,VDD B 12
COMMON /DSTR/ NP(201),NM(201),DN(2O1),TEM(201),E(201),DE(201),VSUM B 13
1(201),VP(201),VM(201),JP(201),JM(201),JPDIF(201),JMDIF(201),JDISPL B 14
2(201),RCMBR(201),ALPHA(201),BETA(201),TPRINT(201) B 15
COMMON /FUND/ M,MP1,N3S,PRTFRQ,NXPRNT(6),MODP,MODFCH,EXTRAP,MS,IJ, B 16
11POINTFM,STEPFA,DXDT,T,DS,P,ELEPSO,DIELKDNSTYSPCHT,THCND,DEN B 17
2SPL,DENSMI ,DENS2,DENS2T,RECEGAP,GAMSEC,GAMI ,GAMMA,TAUSEC B 18
COMMON /MJV/ JPOJMO,JMAXDJMAXM,JCJINT,IINT,IINTO,IEXT,123,DIDT, B 19
1DVDT,DVDTO,VC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, B 20
2MJPDF,MJMDF,MJDSP,MTEM,VPMAX,VMMAX,CST(10,3) ,DMTEM,DDIDT,DIDTO B 21
READ (1,120) B 21A
WRITE (6,120) B 22
IF(ENDING.LE.3) GO TO 25 B 22A
READ (1,150) MODA,A1,A2,A3,A4,A5 B 23
READ (1,150) MODB3,B1,B32,B33,B4,B5 B 24
READ (1,150) MODD,D1,D2,D3,D4,D5 B 25
READ (1,150) MODT,CAT,CBT,PWRM,PWRP,TEMP B 25A
READ (1,130) DNSTY,SPCHTTHCND B 25B
READ (1,150) MODVP,MUP,C1,C2,C3,C4 B 26
READ (1,150) MODVNPMUPO,AVP,NOP,VP1M B 26A
43
Appendix A
READ (1,150) MODVM,MUM,C5,C6,C7,C8 8 27
READ (1,150) MODVNM,MUMO,AVM,NOM,VM1M B 27A
READ (1,130) C9,C1O,C15,C20 B 28
READ (1,130) GAMSEC,TAUSEC,GAMMA,GAMI,JMO,JPO B 29
READ (1,130) USTAT,DU,VO,OMEGAT1 ,V B 30
READ (1,500) V1,OMEGA1,PHI1,V2,0t4EGA2,PHI2,IEFCY B 30A
READ (1,130) P,D,S,T,DVDT,JMAX B 31
DO 10 ICIR=1,3 B 32
10 READ (1,130) R(ICIR),L(ICIR),C(ICIR),VC(ICIR),DVC(ICIR),CP B 33
READ (1,113) RD,LD
READ (1,140) STEPFA,DJMAXM,DIELK,DENSPLDENSMI,DENS2,REC,EGAP B 34
READ (1,160) (NXPRNT(I),I=1,6) B 35
READ (1,160) M,PRTFRQ,N3S,MODP,MODFCH,EXTRAP,MS,IPOINT B 36
IF (N3S.GT.O) GO TO 20 B 37
WRITE(6,400) B 38A
CHECK1=1 B 38B
N3S=1 B 380
20 READ (1,170) (TPRNTI(i),TINC(I),NINC(I),1=1,N3S) B 39
MP1=M+1 B 40
READ (1,180) (NP(I),I1,oMP1) B 41
READ (1,180) (NM(I),11l,MP1) B 42
GO TO 27 B 43
25 IF(ENDING.EQ.1.OR.ENDING.EQ.3) READ(1,130)USTATDU,VO,OMEGA,T1,V B 43A
IF(ENDING.EQ.2.OR.ENDING.EQ.3) READ(1,130) (R(ICIR),L(ICIR),C(ICIR B 43C
*) ,VC( ICIR) ,DVC( ICIR) ,CP, ICIR=2,3) B 43D
27 WRITE(6,210) B 43E
WRITE (6,230) MODA,Al,A2,A3,A4,A5 B 44
WRITE (6,220) B 45
WRITE (6,230) MODB3,B31,B22,B3,B34,85 B 46
WRITE (6,360) B 47
WRITE (6,230) MODD,D1,D2,D3oD4,D5 B 48
WRITE (6,225) B 48A
WRITE (6,230) MODT,CAT,CBT,PWRM,PWRP,TEMP B 488
WRITE (6,235) B 49A
WRITE (6,190) DNSTY,SPCHT,THCND B 49B
WRITE (6,370) B 49
WRITE (6,230) MODVP,MUP,C1,C2,C3oC4 B 50
WRITE (6,375) B 50A
WRITE (6,230) MODVNP,MUPO,AVP,NOP,VP1M B SOB
WRITE (6,380) B 51
WRITE (6,230) MODVM,MUM,C5,C6oC7,CB B 52
WRITE (6,385) B 52A
WRITE (6,230) MODVNM,MUMO,AV!4,NOMVM1M B 528
WRITE (6,390) B 53
WRITE (6,190) 09,010,015,020 B 54
WRITE (6,240) B 55
WRITE (6,200) GAMSEC,TAUSEC,GAMMA,GA?41,JMO,JPO B 56
WRITE (6,250) B 57
WRITE (6,200) USTATDU,VO,OMEGA,T1,V B 58
WRITE (6,520) B 58A
WRITE (6,510) V1,OMEGA1,PHI1,V2,OMEGA2,PHI2,IEF0Y B 58B
WRITE (6,260) B 59
WRITE (6,200) P,D,S,T,DVDT,JMAX B 60
WRITE (6,270) OP B 61
DO 30 ICIR=1,3 B 62
44
 Appendix A
30 WRITE (6,280) ICIR,R(ICIR),L(ICIR),C(ICIR),VC(ICIR),DVC(ICIR) B 63
WRITE (6,213) RD,LD
WRITE (6,300) B 64
WRITE (6,290) STEPFA,DJMAXM,DIELK,DENSPL,DENSMI,DENS2,REC,EGAP B 65
WRITE (6,310)
WRITE (6,320) (NXPRNT(I),1=1,6)
WRITE (6,330) B 66
WRITE (6,340) M,PRTFRQ,N3S,MODP,MODFCH,EXTRAP,4S,IPOINT B 67
IF(ENDING.LE.3) GO TO 60 B 69A
IF (MODFCH.LE.1) GO TO 40 B 70
READ (1,180) (DN(I),1=1,MP1) B 71
GO TO 52 B 72
40 DO 50 I=1,MP1 B 73
50 DN(I)0O.0 B 74
52 !F (MODT.LE.1) GO TO 55 B 74A
READ (1,180) (TEM(I),1=1,MP1) B 74B
GO TO 60 B 74C
55 MTEM=TEMP B 74D
DO 57 1=1,MP1 B 74E
57 TEM(I)=TEMP B 74F
C CALCULATE AND PRINT TPRINT LIST B 75
60 IF (MODP.EQ.1) GO TO 70 B 76
IF (MODP.EQ.2) GO TO 90 B 77
WRITE(6,410) B 78A
CHECK 1=1 B 78B
RETURN B 78C
C TPRINT LIST FOR PRINTING ACCORDING TO TIME B 79
70 1J0O B 80
DO 80 J=1,N3S B 81
NINK=NINC(J) B 82
DO 80 1=1,NINK B 83
[J='J+1 B 84
X11l-1 B 85
80 TPRINTCIJ)=TPRNTICJ)+XI*TINC(J) B 86
GO TO 110 B 87
C TPRINT LIST FOR PRINTING ACCORDING TO CURRENT B 88
90 TPRINT(1)=0.O B 89
IJ1 B 90
DO 100 J=1,N3S B 91
NINK=NINC(J) B 92
DO 100 1=1,NINK B 93
IJ=IJ+1 B 94
100 TPRINT( IJ)=TPRNTI(J)*TINCCJ)**(1-1) B 95
110 WRITE (6,350) IJ,(TPRINT(l),1=1,IJ) B 96
RETURN B 97
C B 98
113 FORMAT (2E13.7)
120 FORMAT (721IDENTIFICATION CARD HEADING EACH RUN B 99
1 )B 100
130 FORMAT (6E13.7) B 101
140 FORMAT (8E10.3) B 102
150 FORMAT (11,E13.7,4E14.7) B 103
160 FORMAT (2013) B 104

45
Appendix A
170 FORMAT (2(2E14.8,I3,5X)) B 105
180 FORMAT (5E114.8) B 106
190 FORMAT (20X,1P,5E20.7) B 107
200 FORMAT (1P,6E120.6) B 108
210 FORMAT (/16X,4HMODA,18X,2HA1,18X,2HA2,18X,2HA3,18X,2HA4,18X,2HA5) B 109
213 FORMAT (11X,9HDIODE LEG,1P,2E20.6)
220 FORMAT (16X,4HMODB,18X,2HB1,18X,2HB2,18X,2HB3,18X,2HB4,18X,2HB5) B 110
225 FORMAT (16X,4HMODT,17X,3HCAT,17X,3HCBT,16X,4HPWRM,16X,4HPWRP,14X,6 B 110A
lHTEMP K) B 1108
230 FORMAT (120,1P,5E20.7) B ill
235 FORMAT (33X,7HDENSITY,13X,7HSPEC HT,12X,8HTHRM CND) B 111A
2140 FORMAT (ltX,6HGAMSEC,14X,6HTAUSEC,15X,5HGAMMA,16X,4HGAMI,17X,3HJMO B 112
1,17X,3HJPO) B 112A
250 FORMAT (15X,5HUSTAT,18X,2HDU,18X,2HVO,15X,5HOMEGA,18X,2HT1,19X B 113
1,1HV)
260 FORMAT (19X,lHP,19X,lHD,19X,lHS,19X,lHT,16X,4HDVDT,16X,4HJMAX) B 114
270 FORMAT (15X,27HCIRCUIT PARAMETERS WITH CP=,1P,E20.6/16X,4HICIR,19 B 115
1X,lHR,19X,1HL,19X,lHC,18X,2HVC,17X,3HDVC) B 116
280 FORMAT (120,1P,5E20.6) B 117
290 FORMAT (1P,8E15.4) B 118
300 FORMAT (9X,6HSTEPFA,9X,6HDJMAXM,10X,5HDIELK,9X,6HDENSPL,9X,6HDENSM B 119
ii,1OX,5HDENS2,12X,3HREC,11X,4HEGAP) B 120
310 FORMAT CJiX,6HNXPRNT) B 121
320 FORMAT (2013) B 122
330 FORMAT (11X,lHM,6X,6HPRTFRQ,9X,3HN3S,8X,IIHMODP,6X,6HMODFCH,6X, B 123
i6HEXTRAP, 10X,2HMS,6X,6HIPOINT) B 124
340 FORMAT (9112) B 125
350 FORMAT (1214 TPRINT LIST,110,9H- VALUES/(1P,8E15.5)) B 126
360 FORMAT (16X,IHMODD,18X,2HD1,18X,2HD2,18X,2HD3,18X,2HD4,18X,24D5) B 127
370 FORMAT (15X,5HMODVP,17X,3HMUP,18X,2HC1,18X,2HC2,18X,2HC3,18X, B 128
12HC4)
375 FORMAT (1IX,6HMODVNP,16X,4HMUPO,17X,3HAVP,17X,3HNOP,16X,4HVP1M) B 128A
380 FORMAT (15X,5HMODVM,17X,3HMUM,18X,2HC5,18X,2HC6,18X,2HC7,18X, B 129
12HC8)
385 FORMAT (14X,6HMODVNM,16X,LHMUMO,17X,3HAVM,17X,3HNOM,16X,4HVM1M) B 129A
390 FORMAT (38X,2HC9,17X,3HC1O,17X,3HC15,17X,3HC20) B 130
400 FORMAT (75H N3S (NO. OF TPRINT TRIPLES) IS NOT POSITIVE AS REQUIR B 131
lED. END OF PROBLEM.) B 132
410 FORMAT (57H MODP IS NOT EQUAL TO 1 OR 2 AS REQUIRED. END OF PROBL B 133
1EM) B 134
500 FORMAT(6E13.7,12) B 135
510 FORMAT(1P,6E20.6,110) B 136
520 FORMAT(18X,2HV14X,6HOMEGA1,16X,4HPH11,18X,2HV2,14X,6HOMEGA2,16X, B 137
*4HPHI2,5X,5HIFOUR) B 138
END 8 139-
SUBROUTINE INCON(CIRCT,ENDING) C 1
REAL MUM,MUP,NP,NM,JP,JC,JPO,JMO,JM ,IINT,IINTO,IEXT,JINT,JMAX,123 C 2
1, ICUR, ICURO,L,JPDIF,JMDIF,JDISPL,MJP,MJM,MJPDF,MJMDF,MJDSP,MTEM, C 3
2MUMO,MUPO,NOM,NOP C 4
INTEGER CIRCT(1O),ENDING C 5
DIMENSION ONP(1O1),ONM(101) C 6
COMMON /ABC/ MODA,MODB,MODD ,MODVP ,MODVM,MODVNP ,MODVNM,MODT,A1 ,A2, C 7
1A3,AII,A5,A6,B1 ,B2,B3,B4,B5,B6,D1 ,D2,D3,D4,D5,D6,C1,C2,C3,C;,C5,C6, C 8

46
 Appendix A
2C7,C8,C9,ClO,C11 ,Cl2,C13,C14,Cl5,Cl6,Cl7,Cl8,Cl9,C20,MUP,MUM,MUPO, C 9
3MUMO,AVP,AVM,NOP,NOM,VPlM,VMlM,CAT,CBT,PWRP,PWRM,TEMP C 10
COMMON /CIRT/ USTAT,U,V,DU,VO,Vl,V2,OMEGA,OMEGA1,OMEGA2,PHI1,PHI2, C 11
1RD,LD,R(3) ,L(3) ,C(3) ,CP,T1, IEFCY,VDD C 12
COMMON /DSTR/ NP(201),NM(201),DN(201),TEM(201),E(2O1),DE(2O1),VSUM C 13
1(201),VP(201),VM(201),JP(201),JM(201),JPDIF(201) ,JMDIF(2O1) ,JDISPL C 14
2(201),RCMBR(201),ALPHA(201),BETA(201),TPRINT(201) C 15
COMMON /FUND/ M,MP1,N3S,PRTFRQ,NXPRNT(6),MODP,MODFCH,EXTRAPMS,IJ, C 16
1 IPOINT,FM,STEPFA,DX,DT,T,D,S,P,EL,EPSO,DIELK,DNSTY,SPCHT,THCND,DEN C 17
2SPI.,DENSMI ,DENS2,DENS2T,REC,EGAP,GAMSECGAMI ,GAMMA,TAUSEC C 18
COMMON /MJV/ JPO,JMO,JMAX,DJMAXMJC,JINT,IINT,IINTO,IEXT,123,DIDT, C 19
lDVDT,DVDTO,VC(3),VCO(3),DVC(3),DVCOC3),ICUR(3), ICURO(3),MJP,MJM, C 20
2MJPDF,MJMDF,MJDSP,MTEM,VPMAX,VMMAX,CST(10,3),DMTEM,DDIDT,DIDTO C 21
IF(ENDING.GT.3) GO TO 6 C 23
DO 5 I=1,MP1 C 23A
NP( I)=ONPC I) C 23B
5 NM(I)=ONM(I) C 23C
T=OT C 230
DVDT=ODVDT C 23P
VC( 1)=OVC1 C 23Q
DVC( 1)=ODVC1 C 23R
GO TO 55 C 23U
6 EL=1.6022E-19 C 23V
EPSO=8 .854E-14 C 24
FM=M C 25
DX=D/FM C 26
A6=ABD(MODA,A1 ,A2,A3,A5) C 27
B6=ABD(MODB,B1 ,B2,B3,B5) C 28
D6=ABD(MODD,D1 ,D2,D3,D5) C 29
IF (MODVP.EQ.O) C9=SQRT(C3)**3-C3**2/C2*MUP*(l1.-Cl*C3) C 30
IF (MODVM.EQ.O) ClO=SQRT(C7)**3-C7**2/C6*MUM*(l.-C5*C7) C 31
WRITE (6,270) C 32
WRITE (6,280) A6,B6,D6,C9,C1O C 33
C DETERMINE TYPE OF SERIES CIRCUIT C 34
DO 50 ICIR=1,3 C 35
IF (L(ICIR).EQ.O.) GO TO 20 C 36
IF (CC ICIR).GE.1.E20) GO TO 10 C 37
C LC OR LCR CIRCUIT C 38
CIRCT( ICIR)=3 C 39
GO TO 50 C 40
C LR OR L CIRCUIT C 41
10 CIRCT(ICIR)=4 C 42
GO TO 50 C 43
20 IF (C(ICIR).LT.1.E20) GO TO 30 C 44
IF (R(ICIR).NE.O.) GO TO 40 C 45
C OPEN CIRCUIT C 46
CIRCT( ICIR)=l C 47
GO TO 50 C 48
C RC CIRCUIT C 49
30 CIRCT(ICIR)=2 C 50
GO TO 50 C 51
C R CIRCUIT C 52
110 CIRCT(ICIR)=5 C 53
50 CONTINUE C 54~
47
Appendix A
C rOMPUTE CIRCUIT CONSTANTS C 55
55 DO 80 ICIR=1,3 C 56
IF (R(ICIR)*C( ICIR).EQ.O.) CO TO 60 C 57
CST( 1,ICIR)=1 ./(R( ICIR)*CC ICIR)) C 58
60 IF (L(ICIR)*C(ICIR).EQ.O.) GO 10 70 C 59
CST(2, ICIR)=1 ./(L( ICIR)*C( ICIR)) C 60
70 IF (L(ICIR).EQ.O.) GO TO 80 C 61
CST(3, ICIR)=R( ICIR)/L( ICIR) C 62
80 CONTINUE C 63
CST(C4,1) =S*EPSO*D IELK/D C 64
CST(5, 1)=1 ./(CP+CST(4,1)) C 65
CST(6, 1)=DX/(2.*EPSO*DIELK) C 66
CST(7,1)=EL*CST(6,1) C 67
IF(ENDING.LE.3) GO TO 90 C 67A
IF CMODVP.EQ.1) CALL TESTSQ (C1,C2,C3,C9,C15,MUP,C11,C12,C13,Cl4) C 68
IF (MODVM.EQ.1) CALL TESTSQ (C5,C6,C7,C1O,C20,t4UM,C16,C17,C18,Cl9) C 69
WRITE (6,290) Cl1,C12,C13,C14,C16,C17,C18,C19 C 70
C PRINT INITIAL DENSITIES AND TEMPERATURE C 71
90 WRITE (6,250) T,V C 83
DO 110 1=1,MP1 C 84
J=I-1 C 85
110 WRITE (6,260) J,NP(I),NM(I),DN(I),TEM(I) C 86
C COMPUTE U C 105
U=USTAT C 106
IF CT.LT.T1) GO TO 140 C 107
IF (IEFCY.EQ.O) GO TO 130 C 108
U=U+DU+VO*COS(OMEGA*T)+V1*COS(OMEGA1*T+PH11)+V2*COS(OMEGA2*T+PH12) C 109
GO TO 140 C 110
130 U=U+DU+VO*S INC OMEGA*T) C i1l
140 DO 150 1=1,MP1 C 112
150 JDISPL( I)=DIELK*EPSO*DVDT/D C 113
C STORE INITIAL CONDITIONS C 115
DO 220) ICIR=1,3 C 117
DVCO( ICIR)=DVC( ICIR) C 118
VCO( ICIR)=VC( ICIR) C 119
VS=V C 120
IF (ICIR.EQ.1) VS=U-V C 121
ICIRT=CIRCT( ICIR) C 122
GO TO (190,200,200,190,210), ICIRT C 123
190 ICUR(ICIR)=VCCICIR) C 124
GO TO 220 C 125
200 ICURCICIR)=CCICIR)*DVCCICIR) C 126
GO TO 220 C 127
210 ICUR(ICIR)=VS/R(ICIR) C 128
220 ICURO(ICIR)=ICUR(ICIR) C 129
IEXT=ICUR( 1) C 131
DIDT=0 C 133
DDIDT=O
If-3.,=JCUR(2)4+ICUR(3) c 134
IF(ENDING.LE.3) RETURN C 134A
DO 230 1=1,MP1 C 142
ONP( I)NP( ) C 143
230 ONM(I)=NM(l) C 144
OT=T C 156

48
 Appendix A
ODVDT=DVDT C 157
OVC1=VC( 1) C 158
ODVC1=DVC( 1) C 159
RETURN C 162
250 FORMAT (3HlT=,1PE16.6,5H V=,El6.6//4X,1HM,18X,2HN+,18X,2HN-,18X, C 163
12HDN,14X,6HTEMP K) C 164
260 FORMAT C15,1P,4E20.6) C 165
270 FORMAT C/38X,2HA6,18X,2HB6,18X,2HD6,18X,2HC9,17X,3HC1o) C 166
280 FORMAT (20X,1P,5E20.6) C 167
290 FORMAT (37X,3HC11,17X,3HC12,17X,3HC13,17X,3HC14/20X,lP,4E20.7/37X, C 168
13HC16,17X,3HC17,17X,3HC18,17X,3HC19/20X,4E20.7) C 169
END C 170-
SUBROUTINE EXTOIR CISET,CIRCT) D 1
REAL JCIINT.,JINT,IINTO, MJDSPIEXT,LICUR,ICURO,123,JMAX,JMO,JPO D 2
COMMON /CIRT/ USTAT,U,V,DU,VO,V1,V2,OMEGA,OMEGA1,OMEGA2,PHI1,PHI2, D 3
1RD,LD,R(3) ,L(3) ,C(3) ,CP,T1, IEFCY,VDD 0 4
COMMON /FUND/ M,MP1,N3S,PRTFRQ,NXPRNT(6) ,MODPMODFCH,EXTRAP,MS, IJ, D 5
1IPOINTtFMSTEPFADXDT,T,D,S,P,EL,EPSO,DIELK,DNSTYSPCHT,THCNDDEN D 6
2SPL,DENSMI ,DENS2,DENS2T,REC,EGAP,GAMSECGAMI ,GAMMATAUSEC D 7
COMMON /MJV/ JPO,JMO,JMAX,DJMAXM,JC,JINTlIINT,IINTO,IEXT,123,DIDT, D 8
1DVDT,DVDTOVC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, D 9
2MJPDF,MJMDF,MJDSP,MTEM,VPMAX,VMMAX,CST(10,3),DMTEMDDIDT,DIDTO D 10
REAL MJM,MJP ,MJMDF ,MJPDF ,MTEM 0 11
INTEGER CIRCT(1O) D 12
IF (ISET.EQ.2) GO TO 20 D 13
C UPDATE QUANTITIES D 14
IINTO=IINT D 15
DIDTO=DIDT
DO 10 ICIR=1,3 0 16
DVCO( ICIR)=DVC( ICIR) D 17
VCO( ICIR)=VC( ICIR) 0 18
10 ICURO(ICIR)=ICUR(ICIR) D 19
C COMPUTE CURRENT IN SERIES CIRCUITS 0 25
20 DO 90 ICIR = 1,3 D 26
VS=V D 27
IF (ICIR.EQ.1) VS=U-V 0 28
ICIRT=CIRCT( ICIR) D 29
GO TO (40,50,60,70,80), ICIRT D 30
40 ICUR(ICIR)0O D 31
GO TO 90 0 32
50 DVC(ICIR)=CST(1,ICIR)*(VS-VC(ICIR)) D 33
ICUR( ICIR)=C( ICIR)*DVC( ICIR) D 34
GO TO 90 D 35
60 DVC2=CST(2,iCIR)*(VS-VC(ICIR))-CSTC3,ICIR)*DVC(ICIR) D 36
DVC( ICIR)=DV"O( ICIR)+DT*DVC2 D 37
ICUR( ICIR)=C( ICIR)*DVC( ICIR) 0 38
GO TO 90 D 39
70 DI=(1./L(ICIR))*VS-CST(3,ICIR)*ICURO(ICIR) D 40
ICUR( ICIR)=ICUROC ICIR)+DT*DI D 41
GO TO 90 D 42
80 ICUR(ICIR)=V5/R(ICIR) D 43
90 VC(ICIR)=VCO(ICIR)+DT*DVC(ICIR) D 44
C COMPUTE VOLTAGE AND CURRENT OF DEVICE D 45

49
Appendix A
DVDT=( ICUR(1)-ICUR(2)-ICUR(3)-I INT)/CP D 46
V=VliOT*DVDT D 47
IEXT=ICUR( 1) D 48
RETURN D 53
END 0 54-
SUBROUTINE TEST(MINUS,INSTEP,CHECK,CHECK2,ISET,MSO) E 1
REAL NMSAV,NPSAV,JC,MJDSP, JMAX, IINT,JINT,IINTO,123,ICUR,ICURO, E 2
*IEXT,NP,NMJP,JM,JCSAV,JDISPL,JMDIF,JPDIF,MJP,MJM,MJPDF,MJMDF E 3
INTEGER CHECK,CHECK2 E 14
COMMON /DSTR/ NP(201),NM(201),DN(201),TEM(201),E(201),DE(201),VSUM E 5
1(201),VP(201),VM(201),JP(201),JM(201) ,JPDIF(201),JMDIF(201),JDISPL E 6
2(201),RCMBR(201),ALPHA(201),BETA(201),TPRINT(201) E 7
COMMON /FUND/ M,MP1,N3S,PRTFRQNXPRNT(6),MODP,MODFCH,EXTRAP,MS,IJ, E 8
1IPOINTFM,STEPFA,DX,DT,T,D,SP,EL,EPSO,DIELK,DNSTY,SPCHT,THCND,OEN E .9
2SPL,DENSMI ,DENS2,DENS2T,REC,EGAP,GAMSEC,GAMI ,GAMMA,TAUSEC E 10
COMMON /t4JV/ JPO,JM4OJMAX,DJMAXMJC,JINTIINT,IINTO,IEXT,123,DIDT, E 11
1DVDT,DVDTO,VC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, E 12
2MJPDF,MJMDF,MJDSP,MTEMVPMAX,VMMAX,CST(10,3),DMTEM,DDIDTDIDTO E 13
COMMON /SAV/ NPSAV(201)oNMSAV(201),ESAV(201),VXSAV(201),VPSAVC21) E 14
1,VMSAV(201),JPSAV(201),JMSAV(201)oVSAV,TSAV,JCSAV,GAMSAV,TMPSAV E 15
REAL JMSAV,JPSAV,JMO,JPO,MTEM E 17
IF (INSTEP.NE.O) GO TO 60 E 22
C STORE PRESENT VALUE OF VARIABLES E 37
20 DO 50 1=1,MP1 E 38
NPSAV( I)=NP( I) E 39
NMSAV( I)=NM( I) E 40
VPSAV( I)=VP( I) E 41
VMSAV( I)=VM( I) E 42
JPSAV( I)=JP( I) E 43
JMSAV( I)=JM( I) E 44
VXSAV( I)=VSUM( I) E 47
50 ESAV(I)=E(I) E 48
TSAV=T E 49
VSAV=V E 51
GAMSAV=GAMMA E 52
JCSAV=JC E 52A
TMPSAV=MTEM E 52B
ISET=1 E 53
IF (CHECK.EQ.O) MINUS=MSO E 54
GO TO 150 E 55
C CHECK FOR NEGATIVE VELOCITIES E 56
60 Do 70 1=1,MP1 E 57
IF (VM(I).LT.O..OR.VP(I).LT.O.) GO TO 80 E 58
70 CONTINUE E 59
GO TO 90 E 60
80 CONTINUE E 61
CHECK= 1 E 62
C CHECK FOR LARGE CURRENT OR TIME E 63
90 IF (MODP.EQ.2) GO TO 100 E 64
IF (ABS(JC).LE.JMAX) GO TO 110 E 65
C CURRENT DENSITY TOO LARGE E 67
WRITE (6,160) E 67A
GO TO 140 E 68

50
 Appendix A
100 IF (T.LE.JMAX) GO TO 110 E 69
WRITE (6,170) E 70
GO TO 1140 E 71
C CHECK FOR TOO RAPID CURRENT CHt.NGE E 72
110 DJC=ABS( (JC-JCSAV)/JC) E 73
IF (DJC.LE.DJMAXM) GO TO 20 E 73A
WRITE (6,180) E 73B
WRITE (6,190) T,JC,JCSAV,DJC E 73C
MINUS=MSO E 74
CHECK=O E 75
CHECK2=CHECK2+ 1 E 77
IF (CHECK2.LE.2) GO TO 120 E 78
GO TO 140 E 80
C RESET ARRAYS TO SAVE VALUES AND REPEAT INSTEP E 81
120 DO 130 I=1,MP1 E 82
NP( I)=NPSAV( I) E 83
NM( I)NMSAV( I) E 84
VP( I)VPSAV( I) E 85
VM( I)VMSAV( I) E 86
JP( I)JPSAV( I) E 87
JM( I)=JMSAV( I) E 88
VSUM( I)=VXSAV( I) E 91
130 E(I)=ESAV(I) E 92
T=TSAV E 93
\f=VSAV E 95
GAMMA=GAMSAV E 96
JC=JCSAV E 96A
MTEM=TMPSAV E 96B
ISET=2 E 97
GO TO 150 E 98
140 ISET=3 E 99
150 RETURN E 100
160 FORMAT (/29H1 CURRENT DENSITY EXCEEDS JMAX) E 101
170 FORMAT (/18H1 TIME EXCEEDS JMAX) E 102
180 FORMAT (/33H TIME STEP REDUCED,T,JC,JCSAV,DJC) E 103
190 FORMAT (lP,14E20.7) E 103A
END E 1014-
SUBROUTINE OUTPUT(DVXDT) F 1
REAL MJP,MJM,JDIF,MJDSP,IEXT,IINT,JINT,123,L,IINTO,JMO,JPO,JC,JP, F 2
1JM,JMAX, ICUR, ICURO,MJMDF,MJPDF,JPDIF,JMDIF,JDISPL,NP,NM,MTEM F 3
INTEGER PRTFRQ F 14
DOUBLE PRECISION TITLE(140)
DIMENSION XPR(6),DVXDT(101)
COMMON /DSTR/ NP(201),NM(201),DN(201),TEM(201),E(201),DE(201),VSUM F 6
1(201) ,VP(201) ,VM(201) ,JP(201) ,JM(201),JPDIF(201) ,JMDIF(201) ,JDISPL F 7
2(201),RCl.BR(201),ALPHA(201),BETA(201),TPRINT(201) F 8
COMMON /FUND/ M,MP1,N3S,PRTFRQ,NXPRNT(6),MODP,MODFCH,EXTRAP.MS,IJ, F 9
1 1P01'T,FM,STEPFA,DX,DT,T,D,S,P,EL,EPSO,DIELK,DNSTY,SPCHT,THCND,DEN F 10
2SPL,DENSMI ,DENS2,DENS2T,REC,EGAP,GAMSEC,GAMI ,GAMMA,TAUSEC F 11
COMMON /MJV/ JPO,JMO,JMAX,DJMAXM,JC,JINT,IINT,IINTO,IEXT,123,DIDT, F 12
1DVDI,DVDTO,VC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, F 13
2MAJPDF,MJMDF,MJDSP,MTEM,VPMAX,VMMAX,CST(10,3) ,DMTEM,DDIDT,DIDTO F 114
COMMON /CIRT/ USTAT,U,V,DU,VO,V1,V2,OMEGA,OMEGA1,OMEGA2,PHI1,PHI2, F 15

51
Appendix A
1RD,LD,R(3) ,L(3) ,C(3) ,CP,T1, IEFCY,VNEW F 16
DATA TITLE /6H A,4HLPHA,6H ,4HBETA,6H D,4H F 20
1ELTA,6H V,4H-(X),6H V,4H+(X),6H ALPII,4HA*J-,6H BET,4HA* F 21
2J+,6H N,LHET Q,6HRECOM ,4HRATE,6H J+(D,4HIFF),6H J-(0,4HIFF) F 22
3,6H J O,4HISPL,6H J T,4HOTAL6I ,4HV(X),6H DV(X,4H)/DT,6 F 23
4H N,4H+(X),6H N,4H-(X),6H J,4H+(X),6H J,4H-(X),6H F 214
5 TE,4HMP K/ F 24A
UNIVD(EA)=UNIV(P,MODD,EA,D1,D2,D3,04,D5,D6) F 25
C COMPUTE MEAN VALUES F 26
BJPDX=(BETA(1)*JP()+BETAMP1)*JP(4Pl))/2. F 29
AJMDX=(ALPHA(1)*JM(1)+ALPHACMP1)*JM(MPl))/2. F 30
BETDX=(BETA( 1)+EBETA(MPl) )/2. F 31
ALPDX=(ALPHA(1)+ALPHA(MP1) )/2, F 32
RCMBAV=(RCMBR( 1)+RCMBR(MP1) )/2. F 33
DO 10 1=2,M F 34
RCMBAV=RCMBAV+RCMBR( I) F 35
ALPDX=ALPDX+ALPHA( I) F 36
BETDX=,BETDX+BETA( I) F 37
AJMDX=AJMDX+ALPHA( I)*JM( I) F 38
10 BJPDX=BJPOX+BETA( I)*JP( I) F 39
ALPDX=ALPDX*DX F 42
BETDX=BETOX*DX F 43
AJMDX=AJMDX/FM F 44
RCMBAV=RCMBAV/FM F 45
BJPDX=BJPDX/FM F 48
JD IF=MJPDF+MJMDF F 49
I1=2*NXPRNT(l).1 F 53
I2=2*NXPRNT(2)-l F 54
I3=2*NXPKNT(3)-l F 55
I4=2*NXPRNT(4)-l F 56
I5=2*NXPRNT(5)-l F 57
16=2*NXPRNT(6)-l F 57A
WRITE (6,240) T,VNEW,U,MJP,MJM, JC ,V,DTVPMAX,VMMAX,IEXT,IINToJ F 58
1INT,DVDT,ALPDX,AJMDX,8ETDX,BJPDX,RCMBAV,23,DIDT,4JPDF,MJMDF,JDIF, F 59
2MJDSP,MTEM,DMTEM,DENS2T,TITLE(il F 60
3),TITLE(I1+1),TITLE(I2),TITLE(12+1),TITLE(13),TITLE(I3+1),TITLE(I4 F 61
4),TITLE(14+1),TITLE'15),TITLE(I5+1),TITLE(16),TITLE(16+1) F 61A
VSUM( 1)=0. F 62
XPRT=O. F 63
XMPLY=PRTFRQ F 64
DO 230 I=1,MP1,PRTFRQ F 65
DO 220 J=1,6 F 66
NJ=NXPRNT(J) F 67
GO TO (30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,180,19 F 68
10,200,210,215), NJ F 69
30 XPRINT=ALPHA( I) F 70
GO TO 220 F 71
40 XPRINT=BETA(I) F 72
GO TO 220 F 73
50 XPRINT=UNIVD(E(I)) F 74
GO TO 220 F 75
60 XPRINT=VM(I) F 76
GO TO 220 F 77

52
 Appendix A
70 XPRINT=VP(I) F 78
GO TO 220 F 79
80 XPRINT=ALPHA( I)*JM( I) F 80
GO TO 220 F 81
90 XPRINT=BETA(I)*JP(I) F 82
GO TO 220 F 83
100 XPRINT=NP(I)-NM(I)+DN(I) F 84
GO TO 220 F 85
110 XPRINT=RCMBR(I) F 88
GO TO 220 F 89
120 XPRINT=JPDIF(I) F 90
GO TO 220 F 91
130 XPRINT=JMDIF(I) F 92
GO TO 220 F 93
140 XPRINT=JDISPL(I) F 94
GO TO 220 F 95
150 XPRINT=JP(I)+JM(I)+JDISPL(I)+JPDIF(I)+JMDIF(I) F 96
GO TO 220 F 97
160 XPRINT=VSUM( I) F 98
GO TO 220 F 99
170 XPRINT=DVXDT( I) F 100
GO TO 220 F 101
180 XPRINT=NP(I) F 102
GO TO 220 F 103
190 XPRINT=NM(I) F 104
GO TO 220 F 105
200 XPRINT=JP(I) F .106
GO TO 220 F 107
210 xPRINT=JM(I) F 108
GO TO 220 F 108A
215 XPRINT=TEM(I) F 1088
220 XPR(J)=XPRINT F 109
WRITE (6,250) XPRT,E(I),(XPR(J),J=1,6) F 11 '0
230 XPRT=XPRT+XMPLY*DX F i1l
RETURN F 112
C F 113
240 FORMAT (1H1,14X,1HT,12X,4HVNEW,15X,1HU,11X,5HJ+AVG,11X,5HJ-AVG,11 F 114
1X,5HJ AVG, 13X, 1HV/1P7E16.6/14X,2HDT, 11X,5HV+MAX, 11X,5HV-MAX,11X,5 F 115
2H1 EXT,11X,5H1 INT,11X,5HJ INT,11X,5HDV/DT/7E16.6/6X,1OHO*ALPHAAVG F 116
3 ,9X,7HA*J-AVG,7X,9HD*BETAAVG,9X,7HB*J+AVG,5X, 11HRECOMRATAVG, lox, F 117
46H 123,11X,5HDI/DT/1P,7E16.6/11X,5HJ+DIF,11X,5HJ-DIF,11X, F 118
55HJ DIF,9X,7HJ DISPL,6X,1OHAVE TEMP K,BX,8HTEM RATE,10X, F 119
66HDENS2T/lP,7E16.6/9X, 1HX,13X,4HE(X)
7,6(7X,A6,A4))
250 FORMAT (2X,lPE1O.3,1P,7E17.5) F 121
END F 122-
FUNCTION ABD (MOD,Hl,H2,H3,H4L) H 1
C A6 CALLS ABD(MODA,A1,A2,A3,A5) H 2
C 86 CALLS ABD(MODB,B1,B2,B3,B5) H 3
C D6 CALLS ABD(MODD,D1,D2,D3,D5) H 4
ABC=O. H 5
IF (H4.EQ.0. .OR.MOD.EQ.0) GO TO 10 H 6
IF (MOD.EQ.2) H4=SQRT(H4) H 7
ABC=-H4*(ALOG(H1/H3)-H2/H4) H 8

53
Appendix A
10 ABD=ABC H 9
RETURN HI 10
END H 11-
FUNCTION UNIV CXP,MOD,EA,G1,G2,G3,G4,G5,G6) I 1
C UNIVA(EA) = UNIV(XP,MODA,EA,A1,A2,A3,A4,A5,A6) I 2
C UNIVB(EA) = UNIV(XP,MODB,EA,B1,B2,B3,B4,B5,B6) 1 3
C UNIVD(EA) = UNIV(XP,MOOD,EA,D1,D2,D3,D4,D5,D6) I 4
VAR1'(F1 ,F2)=XP*Fl*EXP(-F2*(XP/EB)**XPON) I 5
VAR=O. I 6
IF (MOD.EQ.O) GO TO 70 I 7
EB=ABS(EA) I 8
EAP=EB/XP I 9
IF (EAP.LE.G2/100.) GO TO 70 I 10
XPON1l. I 11
GO TO (10,140,50,60), MOD 1 12
10 IF (EAP.GT.G5) GO TO 30 I 13
20 VAR=VAR1(G1,G2) I 14
GO TO 70 I 15
30 VAR=VAR1(G3,G6) I 16
GO TO 70 I 17
140 XPON=.5 I 18
IF (EAP.GT.G5) GO TO 30 1 19
GO TO 20 I 20
50 VAR=VAR1(G1,G2)+VAR1G3,G4) I 21
GO TO 70 I 22
60 XPON=.5 I 23
GO TO 50 I 24
70 UNIV=VAR I 25
RETURN I 26
END I 27-
REAL FUNCTION MUF(Z,P,G1,G2oG3,GI,G5,MU) J 1
C MUPF(Z) CALLS MUF(Z,P.Cl,C2,C3,C4,C9,MUP) J 2
C MUMF(Z) CALLS MUF(Z,P,C5,C6,C7,C8,Cl0,MUM) J 3
REAL MU J 14
Y=ABS(Z) J 5
IF CY.GT.G3*P.AND.Y.NE.0) GO TO 10 J 6
MUF=MU*(1.-Gl*Y/P) J 7
RETURN J 9
10 Y=SQRTC1./Y) J 10
MUF=G2*Y*(l1.-G5*Y**3) J 11
20 IF (MUF.GT.GII*Y**2) MUF=G14*Y**2 J 12
RETURN J 13
END J 14-
SUBROUTINE TESTSQ (Dl,D2,D3,D14,D5,D6,El,E2,E3,E4) K 1
DIMENSION A(4,l4), B(4) K 2
A( 1,1 )=1. K 3
A 1 ,2)=D3 K £4
AC 1,3)=D3*D3 K 5
AC 1,4)=D3*D3*D3 K 6
A(2, 1)=1. K 7
A(2,2)=D5 K 8
A(2,3)=D5*D5 K 9
A(2,4)=D5*D5*D5 K 10

54
 Appendix A
A (3,1) =0. K 11
A(3,2)1l. K 12
AC 3,3) =2. *D3 K 13
A(3,4)=3.*D3*D3 K 114
A(4,1)=O. K 15
A(4,2)=1. K 16
A(4,3)=2.'*D5 K 17
AC14,4)=3.*D5*D5 K 18
BC1)=D6*D3*(l.-Dl*D3) K 19
B(2)=D2*(SQRT(D5)-D4/D5) K 20
B(3)=D6*(l.-2.*D1*D3) K 21
B(4)=D2*(l1./(2.*SQRT(D5) )+D14/(D5*D5)) K 22
CALL SIMQ (A,B,14,KS) K 23
El=B C ) K 214
E2=BC2) K 25
E3=B(3) K 26
E14=BC14) K 27
RETURN K 28
END K 29-
SUBROUTINE SIMQ(A,B,N,KS) L 48
DIMENSION A~l), BC1) L 49
C L 50
C FORWARD SOLUTION L 51
C L 52
TOL=O.O L 53
KS=0 L 514
JJ=-N L 55
DO 80 J=1,N L 56
JY=J+1 L 57
JJ=JJ+N+l L 58
BIGA=O L 59
IT=JJ-J L 60
DO 20 I=J,N L 61
C L 62
C SE.ARCH FOR MAXIMUM COEFFICIENT IN COLUMN L 63
C L 614
IJ=IT+I L 65
IF CABS(BIGA)-ABS(ACIJ))) 10,20,20 L 66
10 BIGA=A(IJ) L 67
IMAX=I L 68
20 CONTINUE L 69
C L 70
C TEST FOR PIVOT LESS THAN TOLERANCE CSINGULAR MATRIX) L 71
C L 72
IF CABS(BIGA)-TOL) 30,30,40 L 73
30 KS51 L 714
RETURN L 75
C L 76
C INTERCHANGE ROWS IF NECESSARY L 77
C L 78
140 I1=J+N*(J-2) L 79
IT=IMAX-J L 80

55
Appendix A
DO 50 K=J, N L 81
I1=1 1+N L 82
12=1 1+IT L. 83
SAVE=A( Il) L 814
11)=A( 12)
I L 85
A( I2)=SAVE L 86
C L 87
C DIVIDE EQUATION BY LEADING COEFFICIENT L 88
C L 89
SAVE=B( IMAX) L 90
50 A(Ilh=A(I1)/BIGA L 91
B( IMAX)=B(J) L 92
B(J)=SAVE/BIGA L 93
C L 914
C ELIMINATE NEXT VARIABLE L 95
C L 96
IF (J-N) 60,90,60 L 97
60 IQS=N*(J-1) L 98
DO 80 IX=JY,N L 99
IXJ=IQS+IX L 100
IT=J-IX L 101
DO 70 JX=JY,N L1102
IXJX=N*(JX-1 )+IX L 103
JJX=IXJX+IT L 104
70 A(IXJX)=ACIXJX)-(A(IXJ)*ACJJX)) L 105
80 B(IX)=B(IX)-(B(J)*A(lXJ)) L.106
C L 107
C BACK SOLUTION L 108
C L 109
90 NY=N-1 L 110
IT=N*4 L 111
DO 100 J=1,NY L 112
IA=IT-J L 113
IB=N-J L 114
IC=N L 115
DO 100 K=1,J L 116
B( B)=B( IB)-A( IA)*B( IC) L 117
IA=IA-N L 118
100 IC=IC-1 L 119
RETURN L 120
END L 121-
SUBROUTINE EFFCY M 1
REAL L,JAVG,ICUR,ICURO,IEXT,IINT,IINTO,123,JC,JINT,JMAX,JMO,JPO, M 2
1MJM,MJP ,MJMDF ,MJPDF ,MJDSP ,MTEM M 2A
COMMON /CIRT/ USTAT,U,V,DU,VO,V1,V2,OMEGA,OMEGA1,OMEGA2,PHI1,PH12, M 3
1RD,LD,R(3),L(3) ,C(3) ,CP,T1, IEFCY,VDD M 54
COMMON /EFCY/ AMP,AMP2,PHASE,PJ,PV,PHASE2,PJ2,PV2,DTP,GNEG,RFP,EFF M 5
1 ,AMPVDC,AMPJDC,DCPWR,AMPV,AMPV2 M 6
COMMON /FUND/ M,MP1,N3S,PRTFRQ,NXPRNT(6),MODP,MODFCHEXTRAP,MS,IJ, M 7
1 IPOINT,FMSTEPFA,DX,DT,T,D,S,P,EL,EPSO,DIELK,DNST,SPC4T,THCND,DEN H 8
2SPL,DENSMI ,DENS2,DENS2T,REC,EGAP,GAMSEC,GAMI ,GAMMA,TAUSEC M 9
COMMON /MJV/ JPO,JMO,JMAXDJMAXM,JC,JINT,IINT,IINTO,IEXT,123,DIDT, M 10
lDVDT,DVDTO,VC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, M 11
2MJPDF,MJMDF,MJDSP,MTEM,VPMAX,VMMAX,CST(10,3),DMTEM,DDIDT,DIDTO M 12
 Appendix A
COMMON /PLO/ TIME(1401),VOLT(1401),CURR(1401),JAVG14o1),CURR11401), M 13
1CURR2(1bO1),CURR3(401),CLTT(401),TAVR(4O1) M 114
DATA DPR/57 .295780/ M 15
C CALCULATE TIME POINTS M 16
TP=2.*3.14159/OMEGA M 17
TK=TIME( IPOINT)-TP M 18
DO 10 IW=1,IPOINT N 19
I2=IPOINT-I 1+1 M 20
IF(TIME( 12) .GT.TK) GO TO 10 M 21
GO TO 20 N 22
10 CONTINUE M 23
20 CJ=JAVG(12)+(JAVG(12+1)-JAVG(I2))*(TIME(12)-TK)/DTP N 214
VJ=VOLT(I2)+CVOLT(12+1)-VOLTCI2))*(TIME(12)-TK)/DTP N 25
DTP2=TINE( 12+1)-TK M 26
CTO=COS(COMEGA*TK) M 27
CT1=COS(OMEGA*TINE(12+1)) M 28
CT2=COSCOMEGA*TIME( IPOINT)) M 29
CTO2=COS(C2. *OMEGA*TK) N 30
CT12=COSC2.*OMECA*TIMEC 12+1)) N 31
CT22=COS(2.*OMEGA*TIME( IPOINT)) M 32
ST0=51 NCOMEGA*TK) N 33
ST1=SIN(OMEGA*TIME( 12+1)) M 314
ST2=SIN(OMEGA*TINE( IPOINT)) M 35
STO2=S INC2.*OMEGA*TK) M 36
ST12=SINC2.*OMEGA*TIMEC 12+1)) M 37
ST22=SIN(2.*OMEGA*TIME( IPOINT)) M 38
SUMJ1=(JAVG(12+1)*CT1+JAVG(IPOINT)*CT2)/2. N 39
SMJ1=(CJ*CTO+JAVGC 12+1)*CT1)/2. M '40
SUNJ2=(JAVG( 12+1)*ST1+JAVG( IPOINT)*ST2)/2. M 141
SMJ2=(CJ* 'STO+JAVG(12+1 )*ST1 )/2. M 42
SUMJ12=(JAVG( 12+1)*CT12+JAVGC IPOINT)*CT22)/2. N 143
SNJ12=CCJ*CT02+JAVG( 12+1)*CT12)/2. M 144
SUMJ22=(JAVG( 12+1)*ST12+JAVG(IPOINT)*ST22,'2. M 145
SNJ22=(CJ*STO2+JAVGC 12+1 )*ST12)/2. N 46
SUMV1=(VOLT( 12+1)*CT1+VOLT( IPOINT)*CT2)12. M 147
SMV1=(VJ*CTO+VOLT( 12+1)*CT1)/2. N 48
SUMV2=(VOLT( 12+1)*ST1+VOLT( IPOINT)*ST2)/2. M 149
SMV2=(VJ*STO+VOLTC 12+1 )*ST1 )/2. M 50
SUMV12=CVOLTCI2+1)*CT12+VOLT(IPOINT)*CT22)/2. N 51
SMV12=(VJ*CTO2+VOLT( 12+1)*CT12)/2. M 62
SUMV22=(VOLT( 12+1)*ST12+VOLT(IPOINT)*ST22)/2. N 53
SMV22=CVJ*STO2+VOLT( 12+1 )*ST12)/2. M 514
SUMJD=(JAVG( 12+1)+JAVG( IPOINT))/2. N 55
SNJD=(CJ+JAVG( 12+1) )/2. M 56
SUMVD=(VOLT( 12+1)+VOLT( IPOINT))/2. M 57
SMVD= (VJ+VOLT( 12+1) )/2. N 58
122=12+2 M 59
IP1=IPOINT-1 N 60
DO 30 J1-u122,IP1 M 61
CT=COS(OMEGA*TIMECJ1)) N 62
ST=SINCOMEGA*TIME(J1)) N 63
CT2=COS(2.*OMEGA*TIME(J1)) N 614
ST2=SIN(2.*OMEGA*TIME(J1)) M 65
SUMJ1=JAVG( J1)*C1 +SUNJ1 M 66
57
Appendix A
SUMJ2=JAVG( Ji)*ST+SUMJ2 M4 67
SUMJ12=JAVG( Ji)*CT2+SUMJ12 M 68
SUMJ22=JAVG( Ji)*ST2+SU14J22 M4 69
SUMV1=.V0LT(J1 )*CT+SUMV1 M 70
SUMV2=VOLT( Ji)*ST+SUMV2 M 71
SUMV12=VOLT( Ji)*CT2+SUMV12 M 72
SUMV22=VOLT( J )*ST2+SUMV22 M4 73
SUMJD=JAVG( Ji)+SUMJD M4 74
30 SUMVD=VOLT(J1 )+SUMVD 14 75
SUMJ1=2.*( SUMJ1*DTrP+SMJI*DTP2) /TP M4 76
SUMJ2=2.*(CSUMJ2*DTP+SMJ2*DTP2 )/TP M4 77
SUMJ12=2.*(SUMJ12*DTP+S4J12*DTP2)/TP M4 78
SUMJ22=2 .*( SUMJ22*DTP+S14J22*DTP2 )/TP 14 79
SUMV1=2 .*(SUMV1*DTP+SMV1*DTP2)/TP M 80
SUMV2=2.*(CSUMV2*DTP+SMV2*DTP2 )/TP 14 81
SUMV12=2 .*( SUMV12*DTP+SMV12*DTP2)/TP M 82
SUMV22=2.*(CSUMV22*DTP+SMV22*DTP2 )/TP 14 83
AMPJDC=( SUMJD*DTP+SMJD*DTP2 )/TP M 814
AMP VDC=(CSUMVD*DTP+SMVD*DTP2 )/TP M 85
AMPJ=SQRT( SUMJ1**2+SUMJ2**2) M 86
AMPV=SQRT(SUMV1**2+SUMV2**2) M 87
AMP2 =SQRT( SUMJ 12**2+SUMJ22**2) 14 88
AMP V2=SQRT( SUMV12**2+SUMV22**2) M 89
AMP=AMPJ 14 90
PJ=ATAN2(SU14J2,SUMJ1 )*DPR 14 91
PV=ATAN2(SUMV2,SUMV1 )*DPR M4 92
IF(PJ.LT.0.) PJ=360.+PJ M 93
IF(PV.LT.0.) PV=360.+PV M 94
PHASE=PJ-PV M4 95
PJ2=ATAN2(CSUMJ22 ,SUMJ12 ) DPR M 96
PV2=ATAN2CSUMV22,SUMV12)*DPR M4 97
IF(PJ2.LT.0.) PJ2=360.+PJ2 v4'
IF(PV2.LT.0.) PV2=360.+PV2 M 99
PHASE2=PJ2-PV2 M 100
GNEG=AMPJ*ABS(COSC PHASE/DPR) )*S/A14PV M 101
RFP=AMPV**2*GNEG/2. 14102
DCPWR =AMPVDC*AMPJDC*S M4103
EFF=RFP/DCPWR M4104
RETURN M4105
END M4106-
BLOCK DATA N 1
REAL JPDIF,JMDIF,MUP,MUMNOP,NOM,MUPO ,JPO,JMAX,JDISPL,JC,IEXT,hIN N 2
1T,JINT,IINTO,L,JAVG,123,ICURICURO,JMO,MJP,4J1,MJPDF,MJMDF,MJDSP, N 3

58
 Appendix A
2JCSAV,JPSAV,MUMO,NP,NM,JP,JM,JMSAV,NMSAV,NPSAV,MTEM N 4
INTEGER PRTFRQ,EXTRAP N 5
COMMON /ABC/ MODA,MODB,MODD,MODVP ,MODVM,MODVNP,MODVNM,MODT,A1 ,A2, N 6
1A3,A4,A5,A6,Bl,B2,B3,B4,B5,B6,D1 ,D2,D3,D4,D5,D6,C1 ,C2,C3,C4,C5,C6, N 7
2C7,C8,C9,C1O,C1 1,Cl2,Cl3,Cl4,Cl5,Cl6,Cl7,C18,C19,C20,MUP,MUM,MUPO, N 8
3MUMO,AVP,AVM,NCPNOM,VP1M,VMlM,CAT,CBT,PWRP,PWRM,TEMP N 9
COMMON /CIRT/ USTAT,U,V,DU,VO,Vl,V2,OMEGA,OMEGA1,OMEGA2,PHI1,PHI2, N 10
1RD,LD,R(3),L(3) ,C(3),CP,T1, IEFCY,VNEW N 11
COMMON /DSTR/ NP(201),NM(201),DN(2O1),TEM(201),E(201),DE(201),VSUM N 12
1(2O1),VP(2O1),VM(2O1),JP(2O1),JM(2O1),JPDIF(2O1),JMDIF(2O1),JDISPL N 13
2(201),RCMBR(201),ALPHA(201),BETA(201),TPRINT(201) N 14
COMMION /EFCY/ AMP,AMP2,PHASE,PJ,PV,PHASE2,PJ2,PV2,DTP,GNEG,RFP,EFF N 15
1 ,AMPVDC,AMPJDC,DCPWR,AMPV,AMPV2 N 16
COMMON /FUND/ M,MP1,N3S,PRTFRQ,NXPRNT(6),MODP,MODFCH,EXTRAP,MS,IJ, N 17
1 IPOINT,FM,STEPFA,DX,DT,T,D,S,P,EL,EPSO,DIELK,DNSTY,SPCHT,THCND,DEN N 18
2SPL,DENSMI ,DENS2,DENS2T,REC,EGAP,GAMSEC,GAMI ,GAMMA,TAUSEC N 19
COMMON /MJV/ JPO,JMO,JMAX,DJMAXM,JC,JINT,IINT,IINTOIEXT,123,DIDT, N 20
lDVDT,DVDTO,VC(3),VCO(3),DVC(3),DVCO(3),ICUR(3), ICURO(3),MJP,MJM, N 21
2MJPDF,MJMDF,MJDSP,MTEM,VPMAX,VMMAX,CST(10,3) ,DMTEM,DDIDT,DIDTO N 22
COMMON /PLO/ TIME(4O1),VOLT(401),CURR(401),JAVG(401),CURR1(401), N 23
1CURR2(401),CURR3(401),CLTT(401),TAVR(401) ...
:N 24
COMMON /SAV/ NPSAV(201),NMSAV(201),ESAV(201),VXSAV(201),VPSAV(201) N 25
1,VMSAV(201),JPSAV(201),JMSAV(201),VSAV,TSAV,JCSAV,GAMSAV,TMPSAV N 26
DATA MODAMODB,MODD,MODVP,MODVM,MODVNP,MODVNM,MODT,A1,A2,A3,A4,A5, N 31
lA6,B1,B2,B3,B4,B5,B6,D1 ,D2,D3,D4,D5,D6,C1 ,C2,C3,C4,C5,C6,C7,C8,C9, N 32
2Cl0,Cl1,C12,Cl3,C14,Cl5,Cl6,C17,C8,C9,C20,MUP,MUM,MJPO,MUMO,AVP, N 33
3AVM,NOP ,NOM,VP1M,VM1M,CAT,CBT,PWRP,PWRM,TEMP/8*O,53*O.- N 34
DATA USTAT,U,V,DU,VO,V1,V2,OMEGA,OMEGA1,OMEGA2,PH11,PHI2,R,L,C,CP, N 35
iTi, IEFCY/23*O.,O/ N 36
DATA AMPAMP2,PHASE,PJ,PV,PHASE2,PJ2,PV2,DTP,GNEG,RFP,EFF,AMPVDC, N 37
1AMPJDC,DCPWR,AMPV,AMPV2/17*O ./ N 38
DATA M,MP1,N3S,PRTFRQ,NXPRNT,MODP,MOOFCH,EXTRAP,MSIJ,IPOINT,FM,ST N 39
1EPFA,DX,DT,T,D,S,P,EL,EPSO,DIELK,DNSTY,SPCHT,THCND,DENSPL,DENSMI, N 40
2DENS2,DENS2T,REC,EGAP,GAMSEC,GAMI ,GAMMA,TAUSEC/16*O,24*0./ N 41
DATA NP,NM,DN,TEM,E,DE,VSUM,VP,VM,JP,JM,JPDI F,JMDI F,JDISPL,RCMBR, N 42
1ALPHA,BETA,TPR INT11918*0./ N 43
DATA JPO,JMC,JMAX,DJMAXM,JC,JINT,IINT,IINTO,IEXT,123,DIDT,DVDT,DVD N 44
1T0,VC,VCO,DVC,DVCO, ICUR,ICURO,MJP,MJM,MJPDF,MJMDF,MJDSP,MTEM,VPMAX N 45
2,VMMAX,CST,DMTEM/70*O .1 N 46
DATA T IME,VOLT,CURR,JAVG,CURR1 ,CURR2,CURR3/2807*O./ N 47
DATA NPSAV,NMSAV,ESAV,VXSAV,VPSAV,VMSAV,JPSAV,JMSAV,VSAV,TSAV,JCSA N 48
1V,GAMSAV,TMPSAV/813*0./ N 49
END N 50-

59
Appendix B.-Sample Output of Program DIODE

61
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