The Heath Electronic Analog Computer: Its Usage for the Solution of Engineering Problems
THE HEATH ELECTRONIC ANALOG CCl1PUTER
ITS USAGE FOR THE SOLUTION OF
ENGINEERING
PRoBLEMS
A THESIS
Presented in Partial Fulfillment of the Requirements
for the Degree of Master of Science in Electrical Engineering
in the Graduate School of Villa.nova University
By
PAUL J. PIERRE GAYET, B.E.E.
Villanova University in the State of Pennsylvania
1962
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THE HEATH ELECTRONIC ANALOG CCl1PUTER
ITS USAGE FOR THE SOLUTION OF
ENGINEERING PROBLEMS
ABSTRACT
The thesis is a tutorial paper on the basic principles of
analoe computers and analog computation.
A systematic method
for amplitude and time scaling is described.
A number of' sample
problems which were worked on the Heath Electronic Analog Computer
are described and results are shown.
Through the examples,
several interesting aspects of the theory of simulating differential equ~tions on analog computers are discussed.
Brief mention
is made of special techni~ues and circuits which are used with
analog computers.
TABLE OF CONTENTS
Chapter
Page
Preface
iii
I.
The Principles of Analog Computers
1
A.
Types of Computers
1
B.
Analog Computers
2
C.
Components and Operations of
4
Electronic Analog Computers
D.
II.
III.
IV.
Repetitive and Nonrepetitive Operation
8
The Heath Electronic Analog Computer
10
A.
Components
10
B.
Operational Methods
11
C.
Relay Circuits
13
The Principles of Analog Computation
15
A.
Basic Principles of Problem Solution
15
B.
Amplitude and Time Scaling
16
C.
Miscellaneous Considerations
20
Examples of Analog Computer Problems
21
A.
Linear Second Order Differential Equations
21
B.
Linear First Order Simultaneous
35
Differential Equations
C.
A Simple Servomechanism
43
D.
A Frequency Analyzer
50
V.
Special Circuits for Electronic Analog Computers
54
A. Multiplication
54
B.Delay or Dead Time
57
C.
Dead Space
D.
Hysteresis and Backlash
57
58
Conclusion
59
Sources Consulted
61
VI.
PREFACE
This paper is a report of an investigation made into the
principles and uses of electronic analog computers, or as they
are otherwise known, electronic differential analyzers.
More
specifically, an investigation was made of the use of the Heath
Electronic Analog Computer.
It must be kept in mind throughout the reading of this paper
that the investigation was made only into the principles of
electronic analog computers. As such, no attempt was made to
analyze the accuracy obtainable from the Heath computer, or to
dwell excessively on the circuitry of the computer, except in
instances where it was necessary to demonstrate some principle.
The author wishes to extend thanks to all those who have
been of assistance, especially to his wife, Patricia, for her
patient vigil throughout the long months that this work has been
in preparation.
iii
CHAPTER I
THE PRINCIPLES OF ANALOG COMPUTERS
A.
TYPES OF COMPUTERS
In a broad sense, computers may be divided into two classes;
digital and analog.
The former, typified by the desk calculator
and many of the large scale electronic computers, generally
solves problems by numerical methods.
Multiplication is per-
formed by repeated additions, and integration is performed by
summation.
In analog computers, electronic and mechanical components
are used to make physical quantities obey mathematical relations
analogous to those in the original problems.
For example, in the
slide rule, the physical quantity of length simulates a mathematical
variable; in the
ea.th computer, dc vo. tages represent the variables
of a physical system.
In general, digital computers are more accurate and more
expensive than analog computers.
They solve problems in discrete
steps of the input, intermediate, and output quantities, whereas
in the analog computer the variables may be considered continuously
varying quantities.
On the other hand it may be philosophically
argued that the variables in an analog computer are also discrete,
their granularity being determined by such things as the difference
in resistance between two adjacent turns on the winding of a wire
wound potentiometer, and the least count to which meters may be
read.
-1-
2
B.
ANALOG COMPUTERS
Analog computers consist of various types.
One type of machine,
the network analyzer, produces direct simulation of electrical systems
with L, C, and R components.
The high cost of such a machine, however,
has made it rather obsolete.
Another type is the mechanical differential
analyzer, more commonly known as the ball and disc integrator; and
still another is the electronic differential analyzer, or electronic
analog computer.
In addition to the general characteristics of analog
computers, this latter type is characterized by the use of the operational amplifier.
The Heath Electronic Analog Computer is an example of
this t:lpe.
One of the principal uses of an analog computer is to study the
behavior of a real physical system by means of simulation.
It may be
quite impossible or impractical to study the real system itself.
For
example, it may be desirable to study the behavior under various operating
conditions of a new aircraft design prior to construction of a model of
the aircraft.
Simulation on an analog computer of a real physical system may be
. carried out by either of at least two basic philosophies (or a combination of these).
For the first case, suppose that a system exists which
consists of a number of elements, it being possible to mathematically
describe the behavior of each of these elements indiVidually.
Then each
of these elements can be simulated with analog components, and these
comnonents interconnected to simulate the whole system.
4
The response
3
of the system to various input, or driving, conditions may then be
studied; or the simulation model may be adjusted to obtain a desired
response for a particular input function.
This will indicate the
characteristics that the real system must possess to perform as
desired.
The second basic method consists of first describing the operation of the whole system by a set ~f differential equations, and then
solving these equations on the computer.
~lhile this
method usually
performs the simulation with a smaller nllffiber of analog components
than if each element of the real system is simulated individually,
it has the disadvantage that in general, specific components of the
computer are not associated with specific elements of the real
system.
Rather, the adjustment of an analog component changes a
coefficient in one of the differential equations being simulated.
This coefficient may refl€ct the value of several elements of the
system under study.
The utility of analog computers and techniques is not restricted
to the simulation of physical systems.
Special purpose computers,
which may often be conveniently assembled from standard components
of commercially available machines, can themselves serve as control
system elements in some applications
For example, a single commercial
multipurpose computer might be adapted to process signals controlling
several phases of an industrial process.
lKorn and Korn, page 110.
(Source 1)
It has been suggestedl in
4
one case, that analog computers be used for continuous recomputation
of optimum set points as functions of the composition or quality of
raw materials entering a process.
It is conceivable that such
techniques would permit the use of less pure or cheaper raw materials.
C.
COMPONENTS AND OPERATIONS OF
ELECTRONIC ANALOG COMPUTERS
The heart of an electronic analog computer is the operational
amplifier.
The operations most commonly performed on the operational
amplifier are summation and integration.
Specialized circuits
can be used to accomplish a variety of other functions.
The operational amplifier is a very high gain ()O,OOO to 50,000
in the Heath) device.
Figure la shows the symbol for an adder.
The figures inside the adder are the gains associated with the
various inputs.
Figure Ib shows the use of the operational
amplifier and the associated components needed to implement an
adder.
The gain equations are also shown.
an analog computer is numbered.
Each amplifier in
It is customary, when drawing an
analog computer setup, to show the number of each amplifier used,
as in Figure lb.
Figures 2a and 2b show the equivalent notation
for an integrator.
Another component used in the electronic analog computer is
the potentiometer.
Potentiometers are used as adjustable voltage
dividers to multiply a variable by a positive constant less than
one.
The use of potentiometers will be evident with the examples
given later..
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A servomultiplier is a device which produces an output proportional to the product of two input variables.
Several linear poten-
tiometers are grouped together and driven by a servomotor. A
reference voltage is applied to one of the potentiometers and the
output from the arm of this potentiometer is subtracted from the
first input variable.
This tterror" voltage is sent through a high
gain servoamplifier and applied to the servomotor which mechanically
drives the potentiometer in the direction to reduce the lIerrortt to
zero.
This is in effect a servomechanism.
The second input variable
is applied to a second potentiometer, and a voltage proportional to
the product of the two variables is available at the arm of this
potentiometer.
Additional variables may be applied to additional
potentiometers and in each case the output voltage is proportional
to the product of the common first variable and the individual
second variables.
The servo resolver is a special kind of servomultiplier.
In
the servo resolver a precision sine-cosine potentiometer, which
delivers output voltages proportional to the sine or cosine of
the brush angle, is used instead of a multiplying potentiometer
as in the servomultiplier.
Two such potentiometers are mounted
concentrically within each resolver.
The name resolver is derived
from the action of the device as a converter of a vector displacement into its rectangular components
It is to be noted here that the Heath Electronic Analog
Computer contains neither servomultipliers nor servo resolvers.
The comparatively high speeds with which this computer solves
8
problems when in the repetitive mode (section In) would not permit
their use.
Another co~ponent widely used in electronic analog computers
is the vacuum diode.
This has a variety of uses including limiting,
and the generation of many special functions.
Initial condition power supplies are used for setting initial
conditions on integrators.
Relays in analog computers find application in control circuits.
Such circuits are used to start or hold a problem.
The use of
relays will be made clearer by the examples of this paper.
D.
REPETITIVE AND NONREPETITIVE OPERATION
Although many analog computers are designed to solve a problem
only once when integration is permitted to start, some computers are
designed to solve problems on a repetitive basis, repeating the
solution at a rate of anywhere from 0.1 to 50 solutions per second.
The Heath Electronic Analog Computer can be used in either mode of
operation.
In the repetitive mode it repeats solutions at an adjustable
rate between 0.6 and 6 cps.
The general characteristics of repetitive computers may be
enumerated as follows:
1.
Problems are usually solved in a more compressed time
scale than on nonrepetitive computers.
In general
this means that the operational amplifiers are run
at higher gain.
2.
t is practical to display the solutions on an oscilloscope.
This makes it possible to note immediately the effect on
the solution of varYing the system parameters.
9
3.
The choice of relatively slow speed computing elements is
restricted.
This means that servomultipliers and servo
resolvers are not used.
Hence, these devices for multipli-
cation and t~:i,;gQnome.t,.~c function generation are not found
in the Heath Computer.
Special techniques must be used
if multiplication of two variables is desired.
4.
They are usually only moderately accurate and are
somewhat inexpensive.
This is due in general to the
relatively low open loop gain requirements of the
operational amplifiers, and the less expensive capacitors
which may be used.
Usually mica or ceramic capacitors are
adequate for integration.
Integrating capacitors in
high accuracy (non repetitive type) computers should
be polystryene or of some similar plastic in order to
achieve the high leakage resistance necessary.
CHAPTER II
THE HEATH ELECTRONIC ANALOG CCMPUTER
A.
C0l1PONENTS
The Heath Electronic Analog is equipped with fifteen
operational amplifiers.
Two potentiometers are associated with
each amplifier, and a bridge circuit is provided to set the potentiometers allowing for the effect of amplifier loading on the
potentiometer.
Two additional potentiometers not associated with
the amplifiers each have a vernier dial which allows them to be
set at any desired value.
The resistance of all potentiometers
is 100,000 ohms.
Six independent initial conditions power supplies in the
computer may be set at any voltage from -100 to i 100.
A reference
supply delivers plus or minus 100 volts.
Four relays, each with four transfer contacts are provided.
The relays may be controlled manually from switches on the front
penel or may be driven by a repetitive oscillator at a rate adjustable
from 0.6 to 6 repetitions per second.
In addition, the relay windings
are brought out to jacks on the front panel for other wiring options.
Finally, eight vacuum diodes are included in the Heath Computer.
-10..
11
B.
OPERATI ONAt METH DS
Before using an analog computer it is necesssry that the
o~erational am~liriers be adjusted
input.
to give zero output with zero
The method of accomplishing this is illustrated in Figure 3.
The amplifier is simply switched into the circuitry shown and the
null control on the amplifier is adjusted to give zero output with
a grounded input.
The specific method of accomplishing this by
the various switches on the front panel of the computer is adequately
covered in the Heath instruction manual.
l{hen setting potentiometers for the purpose of delivering a
particular fraction of a voltage to the input of a summing or integrating circuit, it is important to account for the loading ef ect
of the input impedence of the summing circuit.
ortunately, it is
not necessary to hand compute this effect; a built-in bridge circuit
makes it possible to set the potentiometers taking the loading effect
into account.
The principle of accomplishing this is shown in Figure 4.
Rf and ~ are externally a plied resistors.
It is to be emphasized
here that the value of Ri used for setting the potentiometer must be
the same as the value of the corresponding resistor to be used in
the setup for the problem situation.
Suppose that the problem called
for a potentiometer setting at 0.432 feeding an amplifier with a gain
of 10.
Typically, for this case, Rf is 1.0 megohm and ~ is 0.1 megohm.
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13
These values of resistor are plugged into the computer, -10 volts
is ap lied to the potentiometer, and + V is adjusted to 43.2 volts.
The potentiometer is then adjusted until the meter reads zero.
The
details of setting potentiometers are covered in the instruction manual.
C.
RELAY CIRCUITS
Relays are used in analog computers for control purposes such
as setting initial conditions.
When an analog computer is used in
the repetitive mode, the initial conditions are reset for each cycle
of operation.
This is accomplished by driving the control relays
from a relay on the repetitive oscillator.
The interconnection of
the components is best described by Figure 5.
evident from the drawing.
Their operation is
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CHAPTER III
THE PRINCIPLES OF ANALOG COMPUTATION
A.
BASIC PRINCIPLES OF PROBL:EM OOLUTION
The usual first step in solving a differential equation on
an analog computer is to write the highest derivative of the depen~
dent variable of the equation in terms of the lower derivatives
and whatever constants or functions of the independent variable may
be included in the equation.
For the present, time is considered
as the only independent variable encountered in the equations;
the extension of the method to other independent variables is
discussed later.
The next step is to assume that the highest order derivative
is available at the output of an operational amplifier.
Each
integrating circuit following this point produces (with sign
changes) successively lower derivatives.
These derivatives,
in turn, are fed back with correct sign and amplitude, to the
input of the first summing amplifier.
Circuits are available for
differentiation, but it is a noisy process and should be avoided
whenever possible
It now remains to connect the external inputs to the problem"
and to set the proper initial conditions at the output of each
integrator.
In general, this means wiring an isolated voltage,
representing the initial value of a variable, in series with a
switch (usually a relay contact) across each integrator.
-15-
The
16
switch opens at the start of the problem.
To avoid noise troubles,
the relay should be at the input end of the amplifier, and ~~e
voltage at the output end.
In this way, any voltages due to leakage
currents in the initial conditions power supplies will be applied
(after the relay contacts have opened) to the output of the amplifier
rather than to the input where their effect would be far greater.
If the initial condition of a variable is zero, the relay contact
is wired directly across the integrating capacitor.
B.
AMPLITUDE AND TD1E SCALING
Although at first glance, it might seem extremely convenient
to set up all analog computer problems so that one volt corresponds
to one unit in the variable being investigated, and one second of
machine time corresponds to one second of problem time, a second
glance will show that such niceties are sometimes neither practical nor realizable.
If a particular variable is expected to
range in value, for instance, from 0 to 200 feet, and one were
to say that one foot equals one volt on a computer Where the
maximum output of an amplifier is 100 volts, there is an obvious
inconsistency.
A real time solution of a differential equation
describing the population growth of a city would take years of
computer time. l
In many instances it is desirable to connect
the output of an analog computer to a pen recorder.
If there
are frequencies greater than two or three cycles per second to
be recorded, the pen will not follow with the desired accuracy_
lIn an article by Smith and Erdley ( ource 6) an economic
system is simulated. One year of real time is made analogous
to 200 microseconds of machine time.
17
The above examples are meant to illustrate the need for both
amplitude and time scaling of problems so that they fit the analog
computer properly and reasonably.
A systematic method of doing
this is desirable.
A number of methods are in use for amplitude and time
scaling.
The method described herein has been worked out by
the author, but no claim is made to originality.
The method introduces the concept of a machine unit.
One
machine amplitude unit is the maximum allowable voltage at the
output of an operational amplifier.
Hence, the output of an
operational amplifier must not be allowed to exceed one machine
unit.
For many analog computers (including the Heath) this is
100 volts.
One machine time unit is one second.
Using this method for scaling, the general procedure is to
transform, by a set of suitable transformation equations, each
of the variables of the original equation (or equations) into
machine variables.
The result is a machine equation which can
be implemented on the computer directly.
The amplitudes and
times of solution will be correct. When the machine solution is
obtained, the inverse of the original transformation equations
are used to obtain the answer in terms of the original problem
variables.
The notation adopted for this method uses lower case letters
for the problem variables and upper case letters for the machine
variables.
Time derivatives of the problem variables are represented
by the operator notation p,p2, etc., in lower case letters.
The
same notation, but with upper case letters is used for the machine
time derivatives of the variables.
18
The procedure for writing the transformation equations and machine
equations is as follows:
1.
Write the equation or equations to be solved with the
proper coefficients.
If a coefficient is to be varied
note the range of the variation.
2.
Estimate the maximum expected value of each of the
variables.
Methods of estimating these values are
available in the literature.
3. Decide tentatively on the relative time of solution to be
used in the problem.
This can be readjusted later i f it
appears that a different relative time will make the simulation easier.
4.
Considering now amplitude scaling of each of the dependent
variables, determine a scale factor such that:
1
I maximum expected value of variable, i I
For example, if the maximum absolute value of py is not
expected to exceed 4 feet/second, then:
a py = 0.25 machine units/feet/sec.
It is important to point out here that the value of a py. is
not necessarily related to that of a y ,
2y ' etc.
8p
5. For the time variable write p ·ettf,. or T = ex t t. In
this equation OCt represents the factor by which 8
problem is slowed down or the time of solution increased.
By this definition, a value ofqt less than 1 indicates
that problem solution time is decreased.
19
6. Using the scale factors of steps 4 and 5, the transformation equations are written using the relationship:
problem variable = (olt )k
ai
machine variable,
where k is the order of the derivative of the
dependent variable.
For example, if a py • 0.25
machine units/foot/sec. and c{t = 10, then
py =
py = 40 Py
If the prob1em is solved in real time, then the transformation
relationship reduces to:
problem variable =
machine variable
It should be clearly understood that these transformations
are performed on the variables; not their coefficients.
7. Replace each of the variables in the problem equation by their
equivalent in machine language.
The machine equation has now
been formulated.
8. Write the relationship between the various derj~atives of each
variable if this relationship is other than one to one.
After the above is carried out, the problem is ready to be put on
the computer.
These rules will be clarified in the next section by the
examples that were actually worked on the computer.
20
c.
MISCEI.LANEOUS CONSIDERATIONS
As a final word on scaling, two miscellaneous items require C1ar"
ification.
Suppose that the independent variable of a differential equation
is not time, but is, for example, x.
The most straightforward way
of handling this situation is to pretend that the variable is t.
standard rules apply, 8nd at the completion of the solution it is
merely necessary to substitute x for t.
The remaining situation is the case where the independent
variable occurs explicitly in the differential equation.
For
example:
py .. -y -+- t - sin 2t
In this case it is necessary to define a new machine variable
r
defined by the transformation equation
r = att
The scale factor at must not be confused with the time scale
factor C(t.
r is generated as a function of T which is the
independent variable in the machine.
consider
It is therefore proper to
r as a dependent machine variable ..
The
CHAPTER IV
EXAMPLES OF ANALOG Cm·1PUTER PROBLEMS
This chapter illustrates the use of the analog computer by
solving some typical problems on the computer.
A.
LINEAR SECOND ORDER DIFFERENTIAL EQUATIONS
The first example consists of the solution of several some~
what similar linear second order differential equations.
They
are inserted here to illustrate in particular the proper use of
amplitude and time scale factors.
The equations to be solved are:
p2 y ... 2y • 200
(in real time)
p2 y of- y/5 .. 20
(in real time)
p2y -.> 2y = 200
(in less than real time)
AI. Consider first the equation
p2y + 2y .. 200
(1)
This may be written
t
py" ~ (200 - 2y)dt
(2)
o
It is estimated that the maximum values of y and py do not exceed
200.
.
2
One need not be concerned about the IDBXlffiUrn value of p y,
since if equation (2) is solved, p2 y never appears.
following scale factors may be formulated.
1
ay •
2'50
a py =
1
200
-21-
Hence, the
22
A logical choice appears to be to run the problem in real time.
Hence:
t = T or p ... p
The transformation equations are consequently:
y ... 200 Y
py = 200 PY
Equation (2) may now be written
J
(200-400Y) dT
J
(1-2Y) dT
T
200 Py ...
o
T
or
py =
o
Figure 6~ shows a block diagram of the logic to solve equation
0).
Figure 6b shows the implementation and the amplifiers used
The quantities Y and -PY were displayed on the
to run this problem.
scope.
The results obtained (with -PY inverted) are shown in Figure
6c.
Consider for the moment equation (3).
This may be solved by
classical methods.
py ...
f
T
(1-2Y)dT
0)
o
p2.y ~ 2Y
II:
(4)
I
m2 ~ 2 .. 0
m=:.i-...[2
Y = A cos
V2 T .. B Sin -{2 T
...
Yp
Let Yp ... C
Substituting into (4)
2C ~ 1
Y • A COB
C "" 1/2
V2
T
...
B sin
V2 T
...
1/2
-y
ion
A
A
y
00'/
-"
6
•
25
at T ... 0 Y .. 0
py
•-..r;
J• .. -1/2
sinV2
T
+
2
at T .. 0,
Hence:
B =0
PY .. 0
y... 1/2 - 1/2 cos
V;
and PY...
sin
2
BV2 cosV'2 T
V2 T
V2' T
(5)
(6)
An inspection of Figure 6c shows that the curves fit the above solutions.
If the inverse of the transformations equations are now s'l1bstituted into the sol~tions (5) and (6), the solutions are obtained in
terms of the original variables
y .. 100 - 100
py • 100
cos
V
t
V2' sin -{2 t
It is easily verified that these are the exact same solutions obtained
if equation (1) is solved directly.
!,g.
Consider nOl'T the second equation:
p2y ... y/5 • 20
or py.
It
(20 -
y/5) dt
o
Using the same scale factors as in the previous problem, the machine
equation is:
py ._l~_
10
f
o
T
(1-2Y) dT
26
Figure 7a shows the logic to solve this equation, and Figure 7b shows
the implementation. Figure 7c shows the solutions for I and PI
recorded on the oscilloscope.
y •
It appears that the expressions
-1- - -1- co;V ~ t
12
----r V-ro sin-V io t
and
1 ""\
PY I :
fit the curves of Figure 7c.
Transforming, the solutions to the
original equation are
A3
-V l~
-V io -V l~
y I:
100 - 100
py.
100
cos
sin
t
t
Consider again the equation of the first example:
(1)
This equation will now be solved in a ma.nner such that one second
of problem time is equal to 10 seconds of computer time, or c(t • 10
Hence, the solution will take ten times as long.
to be used are as follows:
1
26"0
• ...L
200
a Py
2 •
p • 10 .P
1
200
or
T = lOt
The scale factors
y
-'(
qu ti n
(-
i'y
I
j
1 _ '2Y
r
/
----, R
~
y
"1
I
•
•
29
The transformation equations are:
y
= 200 Y
py • 2,000 PY
p2y • 20,000 p 2y
Substituting into equation (1),
py.
!
1
100
T
(1-2Y) dT
It would have also been possible to write
PY R
J
t
(200_2Y)dt
2,000 PY •
PY co ...!...
100
J
T
JT
d'T'
(200-4ooy) 10
(1..2Y) dT
Since we are really solving the same equation as in Al, the value of
py ought to be the same, but the scale factors make the value of PY
1/10 of that obtained when the problem was run in real time
Accuracy
considerations make it undesirable to run the output of operational
amplifiers at such low voltages.
Hence it is better to solve the
equation
Figures 8a and 8b show the logic and results obtained for this equation.
The implementation is identical to that of Figure 7b with potentiometer
12a set at 0 1.
Y•
The solution fits the equations
~- ~
cos ( "
~o )
y
-y
•
or c
j agram
o
..
for SolI
I1
o
J
. 11 of
(1 I-
c
o
32
-IOPY =
v;
- -2
or Py.
sin
In terms of the pr.oblem variables the solution is
y ... 100 - 100 cos 1f:2t
py • 100
sin...y:;- t
It· is satisfying to note that the same solution was obtained l'1hen
the problem was solved in real time.
It .is interesting to note that although Py does not have the
same maximum value that it did when the problem Nas solved in
real time, 10 PY in this solution has the same maximum value as
PY in the real time solution, and in both cases Y has the same
maximum value.
This seems to suggest that if a problem is
properly amplitude scaled when the time scale is changed, if PI
is replaced bY<tPY" p2y bYC(t2p2y, etc." at the output of the
integrators.
This also suggests an alternative method of time scaling.
The problem can be set up as planned for a real time solution, and
the gain of all integrators then reduced by the factor qt-
The
result shown in Figure 9 is the logic of Figure 8a with the parameter
values of Figure 6a.
In substituting into the equations for the
solutions to this problem, however" the transformation equations
y = 200 Y
py .. 200 PY
T ... 10 t
would be used.
(rather than 2000 PY)
'!"i
9.
r
~i
6~
'lith .Intelc, 'ator
R
uc d t
'rh ir Val n •
i
r
1.D.
Lo'~ic
fc r
lu I;,i on
10
~o oJ (1 _ 2!)
tion
1
34
There is one othEr interesting aspect of this example which
might be inserted here for academic interests.
The logic of the
last mentioned method of solution may be examined without reference
to a specific problem or scaling.
If the output of the first
integrator is called - PY, and the output of the second called
Y/10, as in Figure 10" it appears that the machine is solving the
equation
PY • ..1...
10
J
2Y
(1 - 10 ) dT
2Y
1
100 = 10
or
The solution for this equation is:
y = 5 - 5 cos ... h,. or
lO
V-50
sin
-Py·
These values for -PY and
cos
Y
T
-{SO
T
V50
io are the outputs" respectively of the
first and second integrators.
They are identical to the values for
the output of the first and second integrators of Figure 8a" although
they were known by different names in that case.
The fact that they
are the same is not surprising, for examination shows the logic of
Figure 8a to be identical with that of Figure 10.
35
B.
LINEAR FIRST ORDER SHIDLTANmUS DITFERENTllL EQUATIONS
Consider the pair of simultaneous differential equations:
px ... 4x .... 4 py'" lOy = 6
x ... py'" 3y = 0
If, at t = 0, x = 0, and y = 0, the solutions are:
x = _12e- t + 3e- 2t + 9
px = 12e-t _ 6e- 2t
y • 6e- t - 3e- 2t - 3
py = -6e- t ... 6e- 2t
It is desired to show how these solutions may be obtained on the
analog computer.
First it is estimated that the maximum absolute values of the
variables involved do not exceed the following:
lxl
10
lpxl
10
lyl
lpyl
<:.
4
2
The transformation equations are:
x
= lOX
px .. lOPX
y ""' 4y
py • 2PY
t
and since y =
J
=T
pydt;
Y•
~
J
1
FYdT
lThis apparent contradiction is inherent in the method of time
scalingo used. No inconsistencies result if all steps are followed
correctly.
36
The machine equations are now
10 PX + 40x + 8 PY + 40 Y
III:
6
10 I ,;. 2 Py... 12 Y = 0
If the first of these is solved for PY, PI becomes one of the
components of PY.
There appears to be no way to obtain PI from
the second equation without differentiating.
Consequently, the
first equation is solved for PX, and the second for PYa
PX
III:
0.6 - 4 I - 0.8 py -
py :: - 6Y -
4y
5X
In the present problem it is not required to record PX.
Hence the
machine equations may be rITitten
x
C
ofT (0.6 -
4 X - 0.8 PI - 4 Y) dT
FY=-6Y-5X
The logic diagram and the implementation used in the solution are
shown in Figures 11a and lIb.
Before discussing the solution to the problem there are two
points worth mentioning. First, it is noticed that the number of
operational amplifiers in any simple loop is always odd
a negative gain around the loop.
This produces
A positive value (greater than one)
of loop gain would cause oscillation.
The other item concerns initial conditions.
The initial values
of X, Y, and PY are zero, hence it is not necessary to insert initial
conditions.
Normally the initial value of a variable is inserted
by connecting an initial conditions power supply in series with the
break relay contact across the integrating capacitor.
One may ask,
-0.
x
x
-y
y
_-----_ _-----_
........
.....
...........
i
IT
J' a
'~---,.,
\
,.. Y -
A
--1
. .. f»)r
x
~,_Il
> ()
h rl
y
A
0.\,,1 )
•
F
38
however, how initial conditions are set in a variable which appears
at the output of an adder where no integrating capacitor is available.
The answer is that such an initial condition automatically
sets itself.
This was confirmed in the present example by using
a slightly more elaborate setup (not shown) than shown in Figure
lla.
PX was generated and its value measured before the start
of the problem.
Its measured value was 60 volts or 0.6 machine units
which is in agreement with.the calculated value.
Figures lla and llb show the use of an integrator to (driv~
the scope.
The scope was set up so that 0 volts of horizontal
drive positioned the beam on the left side of the face, and 100
volts placed it on the right.
Applying -25 volts to an inte-
grator which has a gain of 1 results in a linear increasing out~
put which rises to 100 volts in 4 seconds. Hence, the horizontal
trace on the scope represents 4 seconds.
The results are shown in Figure llc.
A quick inspection shows
that Y and PY are in correct relation in that PY appears to be of
the form of the derivative of Y.
Further analysis shows that
these drawings very closely fit the analytical solutions of the
machine equations for both the transient and steady state conditions.
The solutions to the original equations are easily obtained by multiplying the values of X by 10, those of Y by 4, and those of PY by 2.
Before going ahead, an observation is in order.
The equations
under discussion were also implemented in a slightly modified form
of Figure llb.
The 0.167 meg resistor was replaced by a potentio-
meter set at 0.6 (including loading effects) feeding into a 0.1 meg
resistor to give the amplifier a gain of 10.
The combined gain was
40
then the required 6. With this setup, the voltage X reached a
steady state level of about 0.63 machine units, rather than 90.
A small correction in the potentiometer setting to about 0.56
gave the correct results.
This result was obtained about four
times, running the experiment on different dates with different
amplifiers.
It was concluded, first of all, that the equation solu...
tion was quite sensitive to the voltage at this point. Secondly
it appears that one must be extremely cautious when loading
potentiometers with values of resistance as low as 0.1 meg.
It
is better to avoid possible difficulties by using the arrangement shown in Figure }JLb.
The same set of equations were also solved at 10 times the
speed of the first solutions, or with LA. t • 0.1
factors are:
x = lOX
px •
PI
1&
(fol
(lCUC)
y = 4Y
py • 0.2 PY ..
t
also
= lOT
y.
~
or
-J
IfJ (2Y)
p = pllO
PYdT
Substituting in the original equations:
PX of- 40 X -to 0.8 PY + 40 Y .. 6
10 X ... 0.2 PY ... 12 Y • 0
For this, the scale
41
Solving the first equation for X and the second for ~ :
x • 10
py
10
of
(6 6 - 4 X - 0.08 py - 4 Y) dT
.... ,X-6Y
The logic diagram for this setup is shown in Figure 12.
The figure is identical to Figure lla except for the fact that the
integrator gains have been increased by a factor of 10, and the
output of one of the amplifiers is now PY, rather than PY.
10
the logic is implemented by simply replacing the 1.0 JlF
Hence,
capacitors of Figure lla by 0.1 tJF capacitors.
This applies
also for the amplifier driving the oscilloscope.
The results ob-
tained were identical to those shown in Figure llc.
x
-O.l,
-y
Fi
12
1,0 ic for So'
ion o~ r achi
x
6y
y
43
C.
A SI
:I.E SERVOMECHANISM
Perhaps the most extensive and fruitful applications of
dc analog computers have been in the field of automatic control
engineering.
c analog computing elements lend themselves
naturally to the representation of feedback loops analogous to
those used in control systems.
Figure 13al shows a simple servomechanism designed to
position a load so as to follow the motion of a control dial.
potentiometer type pickoff device measures the output error
t"" xi ... X o
and produces a dc error voltage
el "" al€" al(xi - x o)
which modulates a 60 cps ac carrier.
The modulated ac is
amplified and controls the torque of a two-phase servomotor
so as to reduce the error.
The equation of motion of the system
is
2
(IL n Im) p2xo + n2rpxo • nala2a3(xi ... xo)
The identification of the constants and typical values are as
follows:
1 dapted from Korn and Korn, page 92.
1 a
45
\
Symbol for
Constant
Meaning of Constant
Typical Value
of Constant
n,-
moment of inertia of load
0.05 slug ft 2
moment of inertia of motor armature
2xlO-5 slug ft 2
n=
gear ratio between motor and load
100
r •
motor damping coefficient
10-4 (ft-lb-sec)
feedback coefficient
20 volts/radian
amplification factor of servo
amplifier-modulator
25
motor stall torque constant
2 x 10-5 ft-lb/volt
It is desired to study the effect on the servo performance of
varying the motor damping coefficient r between 0 and 2 x 10-4 ftlb-sec.
Physically this might be cb ne by changing the resistance of
the motor armature winding.
The response is desired when the
input receives a step function of one radian.
Two cases are
considered, namely starting the problem after an equilibrium has
been established, and starting the problem during an unstable
period when the output is already one radian behind the input.
Using the above values for the constants, but leaving r as
a variable, the equation of motion becomes
O.25p2xo + l04rpx o • Xi ... Xo
The limits of xo, Xi' and€ are not expected to exceed plus or
minus 2 radians in this problem, and the limits of pXo not to
exceed an absolute value of 8 radians per second.
factors are therefore:
Suitable scale
46
.lL_
-.xc)
a
1
•
PXo
"2
a::
t • T
1
"S
8nd the transformation equations are
Xi • 2Xi
with
~. 4
Jk
odT
Thus the machine equation is
Figures 13b and 13c show the logic and implementation of
the machine equation.
The input ~Tas connected to an initial
conditions supply of -50 volts.
The response (X o ) was obtained
for zero initial conditions and for an initial displacement of
minus one radian by using an initial conditions voltage of 50
volts across amplifier 13.
The results are shown in Figure 13d
for