Handbook of Analog Computation (Including Application of Digital Control Logic)
HANDBOOK
O F
SYSTRON &==g/D > DONNER
co 6 © G:F A YS ON
The publication of this professionally prepared handbook could only be
realized at a considerable investment in time and money. To defray
some of the publication costs, a nominal charge of $5.00 per copy is
made. Of course, this handbook is supplied free of charge to all Systron-
Donner Computer users and to persons who participate in a Systron-
Donner Analog Computer Seminar.
To order your copy, please address your request to:
Analog Computer Operations Group
SYSTRON-DONNER CORPORATION
888 Galindo Street
Concord, California 94520
©1967 SYSTRON-DONNER CORPORATION
HANDBOOK OF
ALOG COMPUTATION
(INCLUDING APPLICATION OF DIGITAL CONTROL LOGIC)
Prepared by
Maxwell C. Gilliland, Ph.D.
COMPUTER RESEARCH, INC.
and
Analog Computer Staff
SYSTRON-DONNER CORP.
JUNE 1967
SYSTRON DONNER
cORPORATION
Table of Contents
Chapter Title
1 The Motivation for Analog Computers
2 Basic Analog Computing Elements
3 Elementary Analog Programming
4 Block Programming for Physical Systems
5 Scaling
6 Computer Operation
7 Logical Algebra
8 Basic Operation of Digital Logic Elements
9 Circuits for Simple Logical Functions
10 Circuits for Simple Linear and Non-Linear Functions
1 Simulation of Constant Coefficient Transfer Functions
12 Control System Simulation
13 Fundamentals of Vector Analysis
14 Partial Differential Equations — Part 1
15 Basic Iterative Programming
16 Sampled-Data System Simulation
17 Partial Differential Equations — Part II
18 Correlation Analysis
19 On-Line Data Analysis Programs
20 System Optimization
2] Medical Applications
22 A Practical Approach to Adaptive Control
APPENDIX
|. Glossary of Abbreviations
Hl. Uniform Graphics for Simulation
CHAPTER 1
THE MOTIVATION FOR ANALOG COMPUTERS
Analog Computation, based on the modern electronic
analog computer, is of fairly recent date. The first
commercially available general purpose electronic
analog computers appeared on the market in the 1940's.
These early machines were an outgrowth of an emerging
electronics technology anda critical needfor automatic
computing machines that could solve. complex dynamic
problems. Slide Rule and manual equation solving could
no longer be relied upon as a practical approach to
seeking engineering solutions. Therefore, analog com-
puters became important tools in the design of air-
craft, jet engines, atomic reactors, oil refineries,
chemical plants, etc.
Many types of analog computers have evolvedover the
years. The family has included the mechanical differ-
ential analyzer, electromechanical differential ana-
lyzer, and most recently, the iterative differential
analyzer.
The analog computer has always had several advantages
compared to a digitalcomputer. These are primarily
speed, more simulation capability per dollar, an ability
to integrate, and an excellent man-machine interface.
The mainfeature of the analog computer is that it can
integrate time-varying voltages. There is no easy
way to differentiate. Consequently a mathematical
model of a physical system which is expressed in terms
of differential equations cannot be solved with the
machine directly. It is necessary to reformulate the
mathematical model in terms of integral equations,
either implicitly or explicitly.
The analog computer can integrate only with respect
to time. Thus, a mathematical model which contains
partial integrals (corresponding to partial derivatives)
with respect to several variables must be approximated
by a set of ordinary integral equations with respect to
time. Of course, computer time need not correspond
to time inthe physical world, although it usually does.
Before 1959 the analog computer was a synchronous
machine; all its integrators operated in unison. In
1959 the DYSTAC! was introduced. The name was an
acronym for ‘dynamic storage analog computer’. This
machine was the forerunner of the iterative differential
1
DYSTAC is a registered trademark of CSI.
Two Large SD 80 computers are used in this engineering laboratory to stimulate the behavior of
drone helicopters with different load configurations and vartous automatic flight control sys-
tems under a wide range of operating conditions.
Stmulatton saved time and money, reduced the
exposure of personnel and materiel to posstble damage when actual flight tests were made, and
it also eliminated the necessity of waiting for special environmental (weather) conditions.
(Photo courtesy, Gyrodyne Company of America, Inc.)
1-1
Analog computers have long served in the
ftelds of chemistry and process control as
a conventent, low cost means to observe,
analyze, control, and predict the effect
of varying parameters in a dynamic problem.
Some of the problems may pertain to enzyme
reaction, chemical kinetics, continuous
distillatton, heat transfer or transport
delay -- just to mention a few baste appli-
eations of analog computers. The university
student shown here is performing a research
problem involving the effect of potential
barriers on kinetic energy levels.
(Photo courtesy, University of California)
analyzer which appeared in1960. The iterative differ-
ential analyzer is an asynchronous computer; the
integrators need not be controlled in unison. They
can operate independently either in groups or singly.
In 1962 the analog computer was augmented with digital
logic. This innovation first appeared in a machine
called the HYDAC! The HYDAC had a very large
quantity of synchronous digital logic. Since that time
the inclusion of asmaller complementof asynchronous
digital logic has become accepted practice.
Digital logic can be used for the implementation of
logical decisions, These are based on results obtained
from the analog portion of the computer during the
solutionof the problem. Digital logic also can be used
for mode control of the analog computer. All of the
Systron-Donner 10/20 and 40/80 series analog com-
puters can be operated in the iterative mode and can
be augmented with digital logic.
Analog computers have found widespread acceptance
in virtually every area of scientific investigation. This
growing interest in analog computers has created a
need for complete software, specially designed for the
beginner and less experienced user. Also, the recent
addition of digital logic control has greatly improved
the problem-solving capability of analog computers.
How this new feature can be used inanalog computation
is thoroughly illustrated in this publication.
It is the purpose of this handbook to provide students
as well as experienced computer users with compre-
hensive and up-to-date analog computer software.
Chapters 2 to5 develop the basic fundamentals of com-
puter operation and illustrate the solution of elementary
problems. Chapter 6 provides a detailed description
of the operating controls and computer logic of the
SD 10/20 and 40/80 series computers. This informa-
tion serves as useful reference material to problem-
solutions illustrated inthe more advanced discussions
1
HYDAC is a registered trademark of EAI.
1-2
An SD 80 computer mounted inside a Boeing
experimental jet transport, selected for
the NASA sponsored Supersonic Transport
(SST) Program. In this actual in-flight
application, the SD 80 is inserted between
the pilot's controls and the aerodynamic
control surfaces of the jet plane. This
permits the total control system to assume
the dynamics of any of a wide vartety of
SST types.
(Photo courtesy, the Boeing Company)
A student in mechanical engineering ts shown
how to stmulate a mass-spring-damper system
on an SD 3300 analog computer. Stating
equations ts unnecessary. Ustng block
programming techniques, illustrated in
Chapter 4, the student need only understand
the basie relationship of physteal variables
and constants. Following the program block
schematic for a given system, the student can
eastly program the problem on the computer,
observe results on the osetlloscope, and make
further parameter adjustments to seek an
optimum solution.
which relate iterative programming techniques to the
Systron-Donner computers. Chapters 7 to 22 develop
more sophisticated programming techniques and appli-
cations on a progressive basis.
On the Apollo Program, Douglas Aireraft
coupled an SD 40 computer to a large centrt-
fuge. The computer calculated and integrated
errors in human performance during Apollo
lifting body reentry simulatton studtes. The
Systron-Donner computer is seen next to the
programmer, in upper left portion of picture.
(Photo courtesy, Douglas Aircraft Co.)
Analog computers are now standard computing equipment for classroom teaching and research work
tn colleges and universities. Students in the departments of Electrical Engineering, Mechan-
teal Engineering, Chemistry and Biosciences receive instruction in the use of desk top analog
computers as baste electronic model builders of dynamic problems. The ease and swiftness of
presenting a solution on a readout (oscilloscope, XY recorder), and the ability to vary problem
parameters and observe immediately their corresponding effects, have made the analog computer
an important teaching aid.
(Photo courtesy, University of Santa Clara)
1-3/4
CHAPTER 2
BASIC ANALOG COMPUTING ELEMENTS
This chapter shows how electronic equipment and
circuits are used to implement mathematical ‘rela-
tionships in an analog computer.
OHM'S LAW
Ohm's law describes the relationship between the
current through, and the voltage across a passive
impedance. A passive impedance is a collectionof
passive elements such as resistors and capacitors
connected together inanarbitrary way. Such an impe-
dance is generally considered (in analog computing)
to be a two-terminal network which can be denoted by
where Z is the dynamic impedance of the element.
Ohm's law states
E = ZI
for
where I is the time-varying current (in the direction
indicated) through the passive element generated by
the impressed time-varying voltage, E (with the
polarity indicated).
In order to simplify what follows, transform notation
will be used where
s f(t) = 4 f(t)
and
a; f f(t)dt
:
For a resistor Z = Rand Ohm's law is
E = RI
R 1,
—was
+ E -
For a capacitor
dE _
Cc dt = I
so that Ohm's law is
1
E = Gs I
or E = ZI
where Z= ao
~ Cs
I
—- ,, ©
1
+ E -
SERIES and PARALLEL IMPEDANCE
Impedances are additive in series:
Z7*=Z,+Zo
This canbe proved in a simple way. The total voltage
from point a to point b is Ey = Ey, + Eg and by
Ohm's law
E, = Z,1 + Z
T 1 gi = (2, + Z
9) I = Zyl.
When impedances are connected in parallel the total
impedance can be found as the reciprocal of the sum
of the reciprocals:
2-1
I
—__
Z)
Ig
2 Zr Z; Zo 23
Tz
=~
Z3
This again can be proved by simple application of
Ohm's law:
Pet +h +h
1 2 3
ail at, ty
Zz, EB | 2, 2, * 2,
The total impedance of two parallel impedances has
the simple formula:
ne ne te
T 1 1 Z,+Z5
Z1 +2
DIODE
A diode is a non-linear resistor. Its resistance or
impedance depends on the direction of the current flow-
ing through it. It is denoted by
+E-
where
E = ZI
Z = R,; E>0
= Ro, E <0
and generally
Ro >> Ry
2-2
Typically, Ry ranges between 1 ohm and 100 ohms;
and R2 between 100, 000 ohms (100K®) and 1, 000, 000
ohms (1M). Thus the diode is an approximation toa
switch for which
R, = 0
Ryo = #
OPERATIONAL AMPLIFIER
An operational amplifier is of the type that is called a
d-c amplifier; it amplifies not only time-varying volt-
ages, but also d-c or constant voltages. It is charac-
terized by its excellent stability and extremely high
low-frequency gain (amplification). It is denoted by
where
These voltages are measured with respect to ground
(zero reference). The gain, A, is usually frequency
dependent and will decrease with increasing frequency.
As will be seen presently, this becomes a limiting
factor in the use of a computer at high speeds. An
equivalent circuit for the amplifier is
where, generally, R; is greater than 108 ohms and
Ro less than 1074 ohms (closed-loop) at zero fre-
quency. However, the amplifier is current limited,
That is, it will only perform satisfactorily if the out-
put current, I,, is less than some value. The cur-
rent limit for the S-D! amplifier is +25 milliamperes
,at +100 volt output. The amplifier is also voltage
limited; it will not function satisfactorily if the output
voltage, €9, is greater, in absolute value, thansome
upper limit. The limit for theS-D solid-state 100 volt
amplifier is 105 volts. Since the output of these amp-
lifiers contains unwanted noise whose magnitude typi-
cally can be 1072 volts, their effective useful rangeis
about three and one-half decades (5x10~2 to 102 volts).
GENERATION OF TRANSFER FUNCTIONS
In what follows, it will be assumed that the input
impedance of the operational amplifier is infinite, the
output impedance is zero, and the gain is infinite.
1 Abbreviation for Systron-Donner.
These assumptions introduce negligible error at zero
frequency and are a good approximation at mid-fre-
quencies, At high frequency, the assumptions cannot
be made (particularly for gain).
Consider an amplifier with input and feedback (from
output to input) impedances and applied input voltages
as shown.
y
ei Zz; ae Zt
{"
I Tr J
e z 2 | Te ste (SN
2 2 ej wea w— @o
es) 233 Se
From Ohm's law
; _ -e.) . (ey -e.) . (eg -e,)
T Z, Z5 Zs
5 “en
IL =
Z
f f
pr th
Now if the input impedance, R;, of the amplifier is
assumed to be infinite
ej
oR =o
i
for finite e . Thus
Ip = 1
and
(e, -e.) (e, -e.) (e, -e.) e. -e
1 2 3 f
Further, if the gain, A, of the amplifier isassumed
to be infinite, then
e
= 98 -
&; = Ta 0
This is a reasonable assumption at low frequencies
since as noted above €p, at most, will be in the neigh-
borhood of 100 volts in absolute value. Thus, equation
(1) becomes
oO
“2
Z Z
e
1
BD
iv)
Hh
Ze Z Zz
e = - —_—_—
(s) Z
_
no
eo
The simplest case is
ej Z; eo
e = - te
oO Z. i
i
which can be expressed
SOL as woe e ene e eee (3)
e; Z. .
By a suitable choice of impedances many desired trans-
fer functions can be generated. (Henceforth, the units
of megohm and microfarad will be used for resistance
and capacitance respectively. )
SUMMER (ADDER)
Consider the configuration shown below, which is
called a summer.
Ry Re
e—§ Or
Re
eo (> ey
R3
es—W
If, in equation (2), the substitutions
1
Zo = Ry
23 = Ry
Zs = Ry
are made, then
° -- pie -phe-ghe,
9 1 Re 3
2-3
A set of typical values (in megohms) for these resistors
’ jn a summer in a computer is
R, = Ry = R, = 1.0
Rg = 0.1
so that
e, = “ey 7g -10e, .
In a computer, access is usually provided to the input
(summing) junction so that additional input resistors
or feedback components can be added to the summer
circuit externally. Thus, the summer circuit is
| 10 1.0
e| WwW
| 1.0
eo WN = eo
10 Ol J
No
where the gains (multiplying factors) are indicated at
each input, and the summing junction terminal by J.
The program symbol for the summer is
=|>—
where the input gains are omittedif they are unity. If
itis desired to indicate the junction or summing junc-
tion (high-gain input), the symbol becomes
=>
INTEGRATOR
The high-gain input is labeled only if pertinent. Con-
sider the configuration shown below, which is called
an integrator
2-4
Substitution in equation (3) of
4
Z = RZ = G%
leads to the transfer function
So . 1
e. RCs
1
or
_ 41 _ 41
©) =" RGs % re f e(tat
If R=1, C=1, then
t
e, = ff ete
oO
so that the output voltage, e_, of the amplifier will be
the integral with respect totime of the time-varying
input voltage, e;
If a switch is included as shown
Ke)
ej Ww—e eo
and if
1) there is an initial charge stored on the capacitor
which results in the voltage, & (0)
2) the switch closes at t=0,
then
t
e, = 6, (0) - f e, (t)at .
Oo
It is common practice to call this initial voltage the
‘initial condition’ for the integrator. The derivation
of this terminology is obvious from mathematics.
Itis necessary to find a practical way to establish the
initial condition or initial voltage for the integrator.
To accomplish this, the circuit below is used.
R R
~€1c
eo
Here
1 1 _ RCs +1
7 = Cs + R R
f
R
2 RCs +1
Z = R
and substituting in equation (3)
ae ne _ 1
eo = CResad (Ic) = Res +i “Ic
or
de,
RC a + ey = eh
The solution of this differential equation is
e. = Aexp (=t) +e
fe) RC IC’
where A is a constant depending on the initial voltage
stored on the capacitor before e1q Was applied. Then
Lim
to £5 = ater
For practical cases, it is only necessary that t>10RC,
since e~10<, 0001, whichis compatible with the accu-
racy of the circuit. The result in simple terms is:
to guarantee the establishment of the initial condition
it is necessary to wait 1ORC seconds after the voltage,
“e1@ has been applied to the circuit.
The two previous circuits, combined with appropriate
switches as shown below, constitute a practical inte-
grator
-eIc AW ; Nm
1.0
ej} ——VWn—_o Oey
The operation is as follows:
With Sg closed and Sj open, eo = ejc after
1.0 second (or 0.1 second if C = 0.1, etc.).
With Sg open, eg still equals eyo, since the
current through the capacitor is zero. (This
must be true since both Sy andSq are open and the
input impedance of the amplifier is assumed to be
infinite. )
With 8) closed at, say, t=0, then
t
e,(b) = €1q 7 f e, (t)dt.
fe)
Ifatt = T, Sy isagain opened, eg will stop chan-
ging and remain at the last value before 8S,
opened, namely
T
e,(T) = eq - f e, (that.
ie)
If S2 is closed again while Sy is open, the out-
put of the integrator will return to
& = fc .
Each integrator in a computer has these switches as
part ofitscircuit. They are open or closed depending
on what the programmer wants the integrator todo.
The state of these switches is called theintegrator
mode. If all the integrators are controlled in unison,
the switch states are determined by the main computer
mode.
The modes have simple names with obvious interpre-
tations. During the R, or reset (IC) mode, initial
voltages are impressed on the integrator capacitors.
During the compute or C mode, the integrators inte-
grate input voltages. During the hold or H mode, the
integrator outputs remain constant at the last value
achieved before entering the hold mode. Thus, the
mode permits the programmer to stop the computation
at any time, enabling him to evaluate what has happened
thus far in the calculation. The table below shows the
states of Sy andSg in the various modes. Thenumeral 1
indicates the switch is closed (logical 1) while 0
indicates the switch is open (logical 0).
MODE s 1 So
R(IC) 0 1
H 0 0
Cc 1 0
Finally, the general integrator circuit is shown below
together with a definition of its transferfunction for
various modes.
2-5
MODE TRANSFER FUNCTION
R ey = 10
Cc eo (t) = &10
t
- S[ex® + ea (t) + 1040] dt
Oo
H e,(T) = €10
T
- ff + eg(t) + 104(0) dt
oO
Note that the gain (multiplying factor) of the computing
inputs is noted next to the terminals and is determined
by the value of the input resistor. The SJ terminal is
provided to allow the other external input resistors to
be added. TheIJ terminal provides the ability to gen-
erate an initial condition which is the sum of several
voltages. The J terminal permits the connectionof
additional feedback elements around the amplifier (to
be discussed in chapter 10). Also, these terminals
provide an external connection for the use of solid-
state switches in place of the mechanical switches
(relays) S; and Sg. In this case Sy and So are con-
strained to be open regardless of integrator mode and
thenecessary switching is done by the external solid-
state switches.
The program symbol for the integrator is
The input gains are usually omitted if they are
unity.
POTENTIOMETERS
It is necessary to have a device for entering constant
parameters in the computer program. This is accomp-
lished with a potentiometer. The circuit for this ele-
ment is shown below.
eo
If the output voltage, ep, of the potentiometer (pot) is
applied to an input of another element of the computer,
then the input impedance, RL, of the other element is
connected from the pot output to ground.
2-6
Then
I, = L + IT
so that
*i7 fo _ So | 20
Ro Ri Ry
and
ae = e _1 + a + 1
Ry (a) R, R, Ry
Since
R, + R= R
where R, = total potentiometer resistance,
R, = R- Ry
and
e = 1 e
an ek ee
Ri RL OR,
RR,
e = > 2
fe} R. (Ry + R,) Ry
e = ae,.
19) 1
Now, obviously, it would be time-consuming to deter-
mine Ry, with the knowledge of Rt, Ry,for eachnew a.
Consequently, in practice, the pot is set with the load
connected, by reading the output voltage, e , with a
meter, foraknown input voltage, which is ustially 100
volts.
Thus the meter reading is equal to 100 eand the pot is
changed until the desired value for e@ is obtained. The
symbol for the potentiometer is
HV.
nw, 1,
where His the notation for the high end, or input. The
His usually omitted since it is obvious from the com-
puter program which side is the input to the pot. The
element discussed above is called a two-terminal pot
since it has two available terminals (or connections)
on the computer program board (patchboard). Some-
times it is desirable to connect the bottom (low) end
of the pot to some computing- element instead of ground.
In this case the low end is made available at the patch-
board. The circuit is
and the program symbol is
)
or
Hf H
A
where notation for the arm, A, is omitted when it is
obvious from the computer program.
ARBITRARY FUNCTION GENERATOR
Many mathematical problems to be solved with a com-
puter require the generation of an arbitrary function.
This is accomplished with a device called an arbitrary
function generator (or frequently, diode function gen-
erator, since the internal circuitry uses diodes). This
device allows the programmer to approximate the de-
sired function with straight linesegments. Anexample
is shown below
f’(e, )
ow” fle)
Here the function f(e,)is approximated by f’(e,) with
three line segments. In general, each function gen-
erator, depending on howitis used, will provide either
10 or 11 line-segments with maximum slope changes
of 20r2. 5:1. Several of these devices can be used to-
gether if more segments are required. The location
of the slope discontinuity is called the breakpoint and
is adjustable. Detailed instructions for the setup of
this element appear in Chapter 6.
The principle of operation of the function generator
depends on the use of a non-linear input impedance
for an operational amplifier. That is, the impedance
generates an input current proportional to the function
whichin turn constrains the output voltage to have this
functional relationship to the input voltage. A simpli-
fied circuit is shown below.
eer €
Ke)
| 7 _|t#
\ 7
1, =-f (e,)
€, = f’(e,)
The program symbol for an arbitrary function gener-
ator is
FIXED FUNCTION GENERATOR
Fixed function generators are used to generate often-
used functions such as sine, cosine, log. etc. These
are similar to arbitrary function generators in opera-
tion, but do not permit the programmer to change the
parameters within the device. Generally, a fixed
function generator is more accurate and has better
frequency and noise specifications than an arbitrary
function generator. The program symbol for a sine
generator, for example, is
Sin x(t)
x(t) ———| Sin
MULTIPLIER
The multiplier used with the S-D computer is calleda
quarter-square multiplier. The name derives from the
equation
XY = 4 [x + y)- ax? |
The actual equation to be implemented with hardware
in order to provide multiplication is
€1 &2 = 700 (lex * eo” - | ey 7 e, "| (4)
The multiplier module contains two fixed function gen-
erators. Ablock diagram for one of the generators is
2-7
oO
i Lit
[ezon omx—n |
-@5
(Note that both polarities are required as inputs for both
ey and eo: ) The transfer function is
le, + e, |”
I = K—700 ’
where K is aconstant that determines the feedback
resistor of the output amplifier (to be shown below).
The block diagram for the other is
}wzon omx—7 |
|
e|
ep @—
-e,
ee
The transfer function is
The multiplier module block diagram is
F
e,e——¢ }
x
E
— D
F R
Ny
1
-@o@ «
}Z207 oOmx—7
2-8
The transfer function is
e,e
Ts 460 2 + el” - Je - e,|”| - Kg
A simplified block diagram for the module is
—
e, @ M R
U
-e,e—lL
T
——_
e,e—— | 1
0
-e,e——_|0
2 |__f
If the module is connected to an output amplifier
U
2 T
e
QO
e,e——M
0
_,.e——|0
2 _
then
-3- dll +e,|7 -le - e,|" | K °1%9
R 400 |{I"1 2 1 2 100
and if K -i
~ R
7&1 &2
e =
° 100
The program symbol for the multiplier module (with-
out the output amplifier) is
e|
eo “ep
With the designations
M: The module is used as a multiplier
I: The output is a current
X, Y: Bipolar inputs
If the bipolar inputs are not naturally available from the Note that the ''minus sign" associated with the ''M" indi-
program (e.g. other variables) then summers can be cates a negative current is generated for a positive
used to generatethem. This can be done in two ways: product.
Case I: The program symbol for the multiplier module together
with an output amplifier is
e| M
-e; x 1}. ex x &o
ex
Y ey
e2 | - 2x &y
ex ey fF -— OD
tT =———
100
100
CIRCUIT As before, summers can be used to generate the bi-
polar inputs if they are not otherwise available.
Case I:
M
(+) -
e x I \ Se eo
ey —4 t y J
ey eo | ey, ey
e = ——s
° 100
PROGRAM SYMBOL ey—
CIRCUIT
Note that the "plus sign" associated with the ''M" indic-
ates apositive current is generated for a positive pro-
duct, exey. The sign is generally omitted since this
is the normal condition.
Case II:
e) M
xX l-_e—.
£2
ey ™ “eo PROGRAM SYMBOL
—@x ey
~ 100 (Note that here the sign inversion is due to the output
amplifier. )
CIRCUIT
Case II:
M
e x 1’. (+)
x 1-4 — So
Y ey~—¢ el Y
ey -~ eo ee ey
° 100
PROGRAM SYMBOL CIRCUIT
2-9
PROGRAM SYMBOL
A multiplier module canbe used to make a divider with
the same output amplifier. (This connection can be
made conveniently at the patchboard. )
-ey
Cc
Y
R €o
e@,0—W- IT x
I -e
M °
Here
ee e,
T= Krag oR
so that for R = os
K
_ _ Soty
& ~ ~ {00
or
100e,
e = -
fe) e
y
The programmer's symbol is
ex | + eo
| lOO e,
ey ~ ey
From the circuit it can be seen that the condition ey>0
must hold in order to avoid instability. This is easy
to show. The multiplier module can be considered to
be a non-linear feedback resister, R,. The sign of
R¢ is the same as the polarity of e x’ Suppose in the
circuit
2-10
R¢<0. Then, Ig will be in the direction shown, for
9 > 0, and I¢ will produce ane; <0. Since the gain
of the amplifier is negative and large, the circuit will
be unstable. An additional restriction, |e,| <|ey|, is
necessary to ensure that the output amplifier not ex-
ceed the voltage limit.
The divider can be used to generate square root in a
simple way. Ife, = ey
— I x
Y
ey, Co
CIRCUIT
then
e
eg = -100 *
fe)
or
e 7 = -100e
fo) x
so that
= 10 e .
ey /-e,, » @<0
The diode insures that the system will not saturate in
the wrong direction if e x B0es positive inadvertently.
This circuit has the program symbol
v eo 1Ofex
FUNCTION SWITCH
A function switch is a manual switch which can be oper-
ated from the controlconsole. It is used to change the
computer program during execution. The switch has
three positions: up, down, andcenter (off). Its circuit
is shown below.
a
tL.
This is the program symbol as well. The terminals
of the switch are located at the patchboard.
FUNCTION RELAY
Afunction relay is usedto make program changes auto-
matically during execution. It is energized (logical 1)
for an input equal to or greater than +28 volts and is
de-energized (logical 0) for an input equal to or less
than 0 volts. The circuit is shown below
Ux
n —~
X__9"o-__Ux
U U,=X if U=0
U,=X if Ue
FRIN x if U=1
This is also the program symbol where N is the relay
number.
ELECTRONIC SWITCH
An electronic switch is used in much the same way as
afunction relay; the only restriction being that it must
be in series with the junction of an amplifier. Its speed
of operation is much greater (10-5 sec) than a function
relay (10-3 sec). It requires the same input voltages
(logic levels) as the function relay (i.e. logical 1:
28<e,_<100, logical 0: -100<e, <0).
The program symbol is
qa Ug
where N is the switch number.
COMPARATOR
Frequently, itis required to determine the sign of the
sum of two variables.
A circuit which accomplishes this is
D2
eS 068
If (ey + eg) >0, then eg is limited to approximately
Ovolts by Dj because of its low resistance when it con-
ducts (logical 1). If (ey + eg) < 0, then eg is limited
to approximately +28 volts by D, (logical 0). The
latter is true since
1) Ip mustbe in the direction shown (e, + e€,>0)
and es > ey
2) €9 = 28 volts since ey must be approximately
0 volts due to the high gain of the amplifier.
The program symbol for the comparator is
Dn
=! if (at+b)<O
=O if (atb)>0
If both logical comparator outputs are required, then
€, is complemented with abiased analog inverter. The
program symbol is
U
Qa
U
and the analog circuit is
|} U
30
ce
-100
2-11/12
CHAPTER 3
ELEMENTARY ANALOG PROGRAMMING
All dependent variables in an analog computer are vol-
tages. Consequently, it is necessary to equate these
voltages to physical variables. Since the elements of
the computer are voltage limited, scaling will be re-
quired. The art of scaling is discussed in Chapter 5.
In this chapter scaling is ignored since the necessary
concepts can be developed without reference to scaling.
As pointed out inChapter 1, itis necessary to remem-
ber that an analog computer cannot differentiate easily.
It is a machine whose main feature is its ability to
integrate. Consequently, problems which are defined
by differential equations must be reformulated in terms
of integral equations either explicitly or implicitly.
Consider the problem
y = y(x)
dy _
ak * By = 0
y(0) = A.
This can be rewritten as the integral equation
For this problem computer time will represent x, and
y(t) + Bf yat = O,t=x
y(0) = A
An integrator, as shown in Chapter 2, integrates from
t = 0 onward. It also requires an initial condition:
namely the value of the integral at t = 0, which in
this case is A. Thus, the problem is formulated for
the computer as
t
y(t) = ~ f By(t)at, (1)
0)
y(0) = A
The right half of equation (1) can be generated by
A/I0O0
- 100
t
+y(t)> ()—- -f By (dt
Le]
where the constant B and the initial condition A are
inserted by means of pots. Remember that the inte-
grator inherently has asignchange. Equation (1) states
that y(t) is equal to the negative integral. All that is
needed, to complete the computer program for this
problem, is to connect the output of the integrator to
the input of the pot set to B. Hence, the computer
program is
A/I00
-100
y (t), t=x
The independent variable, y, can be recorded from the
output of the integrator.
In this example the differential equation was reformu-
lated explicitly in terms of an integral equation. It is
frequently possible to do this in an implicit way, as
the next example will show.
Consider the problem
y + Ay + By = f(t), y = y (t) (2)
y(0) = C
y(0) = D
This is equivalent to
y = f(t) - Ay - By
y(0) = Cc
y(0) = D
First, y can be generated from y by
3-1
(Note that again the pots are used to generate initial where equation (2) is satisfied by applying the correct
conditions.) Using this circuit, the complete program inputs to integrator 1.
can be generated by
>
~Ay
D/100
+100
Cc/100
Next, the use of non-linear elements is demonstrated.
-100 Consider
y + Ay + ty = cost, 1<A <10,
y(0) = D.
The computer program (except for scaling) is
+100
Ae)
cos ¢
+100 ~100
Cc D
106 Tere)
10) my
[m0
wr
sint \
-cos ¢
-ty A + Ol
< >< -100
3-2
As another example, consider
y¥y + f(y = 0
y@) =A
y(0) = B
The computer program is
+100 -100
AL 8
100 100
7
Py
-y f(t)
f(t)
i .Ol
The next example shows how the DCU's (Chapter 8) can
be used to determine the solution of a differential equa-
tion for a specified value of the independent variable.
Suppose it is required to find y (.17) for
y+y =0
y(0) = 100.
From above, the analog program which generates y is
-100
In order to determine y(.17) the computer is allowed
to compute for 0.17 seconds and then put into hold.
The latter can be accomplished by applying +28 volts
to the "problem hold" terminal at the patchboard after
0.17 seconds. This will halt the entire computer and
in particular the computation carried out by the above
program. The digital program which will do this is
shown below:
f fi
<RH
\ ioms _!OOMS
211 SY ous Ce ou 4
CLOCK
100 MSJO [7 | ~ R
2|2 T PROBLEM
IMS s HOLD
INT cu 5 TERMINAL
Ov
Wo —<FRT’
The operation is as follows: The clock pulses are not
applied to DCU 1 until the machine is in compute be-
cause the output of the RH' bus is a logical 0 in I C
and this is an input to gate 1. The FF is initially in
the R state whenthe machine is in compute having been
reset by the FRT' Logic output. Divider 1 reduces the
clock frequency of 1000 cycles/see to 100 cycles/sec.
The output of the 0. count of counter 4 is 10 cycles/sec
and is connected to the input of counter 5. The 7 count
output of counter 4 and the 1 count output of counter 5
are connected to gate 2. Thus, whenever t = 0.17 sec
the output of gate 2 will change from 0to1, which will
change the FF tothe Sstate. At this time the computer
will go to the hold mode due to the S output of the FF
being connected to the problem hold terminal at the
patchboard. Also, the clock andcounters will be reset
because of the logical i applied to the OV terminal.
As soon as the machine goes to the reset mode the
FRT' logic output will cause the FF to return to the
R state. Thus, the logic circuit is ready to be used
again as the computer is manually put first in the reset
mode and then the compute mode.
ALGEBRAIC EQUATIONS
The first examples were intended to illustrate the basic
approach to programming. The following illustrates
a more serious application of the analog computer.
Suppose it is desired to solve a set of simultaneous
algebraic equations which are expressed in matrix form
by
AX =C (3)
where the n x n matrix A and the column matrix C
are known. For thepurpose of illustration it will be
assumed that A is of rank 2, although the derivation
which follows is perfectly general. Consider
X+AX=C (3a)
3-3
When the system has reached steady-state (assuming
there is one)
x = 0
AX = C.
Thus the solution to equation (3) is obtained from the
steady-state solution of equation (3a). The latter can
be written
Ky + yyXy + AyoXo = C
1
Xp + ApyXy + AgoXy = Cy (4)
and the analog program is
xX, Xo
ony One are 920
Cc; Co
100 100
-100 -100
Generally, integrator capacitors are chosen equal to
0.001 ufd to decrease the solution time. However,
there is no guarantee that the system of differential
X|
equations, (4), is stable. A different approach will
provide an unconditionally stable system of differential
equations. If
AX = C
it follows that
A'AX = A'C (5)
where A' is the transpose of A. The solution of equa-
tion (5) is the same as that for equation (3). Again at
steady-state the solution of
X + A'AX = A'C (6)
will be the solution of equation (5) and therefore equa-
tion (3). The stability of equation (6) is determined by
X + A'AX = 0. (7)
If equation (7) is multiplied by the row matrix X, then
the result is
XX + XA'AX = 0.
It is well known that XA'AX is positive definite (i.e.
non-negative for all values of X). Unless X = 0,
XX < 0
and stability is guaranteed. The computer program for
equation (6) is
-100
100
3-4
In this program it is easy to change individual coeffi-
cients because each pot is associated with only one co-
efficient. The program can be simplified so that it
requires no more equipment than the previous pro-
gram if the following coefficient changes are made.
OLD NEW
2,2, 2
a bl 414 04
419 444242 + 494299
94 414742 * 491499
2 2
A990 agg 7 Arp
1 4Cy + AgyCy
Cc Cc
2 Aigtg * agave
This simplified program saves equipment but is in-
convenient to use when the solution is required for
several different values of the coefficients.
ARBITRARY CLOSED FUNCTIONS
Many analytic or closedfunctions can be generated by
representing them in terms of their generating dif-
ferential equations. Thefollowing examples illustrate
the technique.
f(t) = 1/t, t>a:
This function has the generating equation
df 2
a7
1
fa) = =
The program which solves the generating equation is
(l00a)7!
-100
—w f(t)
f(t) = A(t + a)"*t >0:
ain,
dt tt+a
£(0) = Aa’
Generating Equation
Aa”
ete)
-l00
10 4(1)
-100
a
100
n/lo Ol
< -100
Program
f(t) = In (t+ a, t >0:
df 1
dt tt+a
£(0) = Ine
Generating Equation
Ol
Ol
_—()}—__—
Program
3=5
f(t) = A cosh(t), t>0:
f-=f
#(0) = A
Generating Equation
-100
A cosh (t)
Program
Mean value:
The mean value, ¢ (t), of a function, f (t), over the
interval [Tr T, + t] is defined by
T+
ot) = 2 [7 tide, tr0. (8)
qT
The generating equation for ¢ can be obtained by dif-
ferentiation:
P= Zit, +8 - 9} (9)
The program is
-f (1!) ——
01
-—(_)}-—— - 100
,
Suppose the integrators 1, 2are in the initial condition
mode until t = Ty, at whichtime their mode is changed
to compute. Then, theoretically, the program will
generate the solution of equation (8). However, the
program willnot work because the right side of equa-
tion (9) is indeterminate for t = 0. Thus the divider,
in the program will have an unstable output. Practi-
cal limitations require the Y-input to be at least 3
volts for useful operation. Consequently, it is neces-
sary to choose a different initial condition:
Tyte
o(e) =+ f tas,
T,
3-6
This program is
S
—e
-f (t')
So (t)
Ol
s, i 2 -100
As before, when t' = T,, the mode of integrators 1,
2 is changed from initial cOnditionto compute. Initially
S1 is closed and S2 is open. The output of integrator
1 will not be ¢ (t) untilt = «. Whent = «, the cor-
rect initial condition, namely ¢(«), will have been es-
tablished for integrator 1. Atthis time the comparator
will be activated, which will open S1 and close S2.
Also the output of integrator 2 will be «. Thus, the
generating equation is implemented by the program
starting when t =«’. The output of integrator 1 will
be ¢ (t), t>e.
COORDINATE TRANSFORMATION
Suppose two coordinate systems have the same origin
and are displaced by an angular rotation, @:
An X-coordinate point, say x, will have the coordinates
x’ X COS @
y’ = -x sin 6
in the x', y'-coordinate system. Similarly, for a
y-coordinate, say y,
x’ = y sin @
y’ y cos 6.
Thus, a vector, R, in the X, Y-system with components
X, y, Will have the components
iT
x! x cos @ + y sin @
-X Sin 6 + y COS 6
y’
in the X’, Y’-system. In matrix notation
R’ = TR
where R, R’ are the representations of the vector in
the unprimed and primed coordinate systems respec-
tively, and where
T = | C089 sin @
-sin 6 cos 6}.
In the three-dimensional coordinate system
positive rotations 6, ¢,\ respectively about the X, Y,
Z axes correspond to the transformations
x Tr 0 0 |
R, = |0 cos 6 sing@
Q-sin @ cos 4
Tcos g 0 -sin ¢|
Ry =/|0 1 #0
® | sin 0 cos ¢|
z [ cos A sin A 0 |
R, = -sin A cos A 0
0 0 1 |
Any arbitrary three- dimensional rotation can be repre-
sented asa product of these three matrices. The super-
scripts refer to the axis about which the coordinate
system is rotated and the subscripts to the name of
Zz the angle of rotation.
ar
The transformation of a vector, V, by R* is
Vv’ = RV
vo = VV;
La x x
(—Y
+ Vv’ = V_cos @ + V_ sine
y y Z
vio = -V_sind + V_cosé@.
8 Z y Z
x
The program is
Vz Vy Vy
\ \ v/
@ -WCOS@ x
H-VySIN@ ;
SIN EyZSIN “Vz Vy
—~
cos
near a
/
Viz
Note that 6 is limited by the range of the sine and cosine function generators.
3-7
The program symbol used frequently is
Wy x x! Wy.
R*
Vv, ——Y y/_——v,/
8
The transformations RY, R” have similar programs.
MATRICES
A matrix, A, is a collection of elements:
A =f{ayhil uh,
(in what follows it is assumed that n = m). Thus, if
n