Analog Computer Fundamentals: With an Introduction to Matrix Programming Methods
ANALOG COMPUTER FUNDAMENTALS
Engineering
Library
QA
With an Introduction to Matrix Programming Methods
by
Silvio O. Navarro
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x
-
The University of Michigan
Ann Arbor, Michigan
and
Wads worth Publishing Company
Belmont, California
nmvrRsiTY or mmm rnwAPirc
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ANALOG COMPUTER FUNDAMENTALS
With an Introduction to Matrix Programming Methods
by
Silvio O. JNavarro
Associate Professor of Electrical Engineering
and
Director, Computing Center, University of Kentucky
The University of Michigan
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Ann Arbor,
Michigan
This material is distributed by the Project on the Use
of Computers in Engineering Education sponsored by
The Ford Foundation.
It may not be reproduced in
whole or in part without permission of the author.
Additional copies may be obtained from Wadsworth
Publishing Co., Belmont, California
Copyright 1962 by S. O. Navarro
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lv9
t
of Contents
BASIC ANALOG BLOCKS
1.1
Common
1.
The
1.4
1.5
1.6
1.7
\
Table
CxlAi-^>
1.3
i"v
.
1_£ V
2
,
y
1,
ANALOG COMPUTER FUNDAMENTALS
. 1
^2-
^
uses
of the
Components
Multiplication
Analog
Addition
of
Analog Computer
an Analog Computer
by a Constant
Integration
The
Summer-Integrator
Function Generators
Other Analog Blocks
Exercise 1.1
and
Multipliers
Problems
ANALOG SOLUTION
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2. 10
3.
Introduction
Linear Equations
Solution of Differential Equations
of the Analog Circuits
The Non-Uniqueness
Scaling
Differential Equation
Changing the Time Scale
Estimation of Maximum Values
Differential Equations with Forcing Functions
Equations with Variable Coefficients
Simultaneous Differential Equations
Problems
THE ANALOG COMPUTER
Introduction
Potentiometer Panel
3.1
3.2
3.3
The
The
The
The
5
3.6
3.7
3.8
3.9
Patch Board
Output Equipment
MATRIX
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.
4.9
4.10
4. 11
4.12
4.1?
4.14
4.15
4.16
4.17
Panel
The
3. 10
8
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3.
Amplifier Panel
Control Panel
Voltage Supply Panel
Resistance and Capacitance
Function Generator Panel
Function Multiplier Panel
3.4
4.
OP EQUATIONS
a
2.
PROGRAMMING OF ANALOG COMPUTERS
Introduction
The
Algebra of Matrices
Equality of Matrices
Addition of Matrices
of Matrices
Matrix Equations and the Ideograph
The Reflected Ideograph
Standard Form of Ordinary Differential Equations with
Constant Coefficients
Elementary Matrix Transformation and Minimization
of Sign-Changers.
Strategies for the Reduction of Negative Entries
Incorporation of Sign-Changers into the Ideograph
Magnitude Scaling by Matrix Manipulations
Node Elimination
Other Elementary Matrix Transformations
Ordinary Differential Equations with Time -Dependent
Forcing Functions
Equations with Variable Coefficients
Simultaneous Differential Equations
Multiplication
ANSWERS
Problems
TO
PROBLEMS
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Chapter 1.
BASIC
1. 1
Uses of the Analog Computer
Common
Although
ANALOG BLOCKS
analog computers
differential equations,
reason,
may
be used
they are often considered
the analog computer has been called
As we know,
differential
equations
in these systems the important variables
behavior
may
are changing
"transient
behavior"
"differential
of the system.
In other cases
for steady-state
be used
and we
analyzer."
equations.
systems because
the physical
the so called
are not changing with respect
the variables
of the system.
problems as well as
their
so that
In some cases
are asked to find
solution
For this
equation solvers.
are changing with respect to each other,
the "steady-state"
to study
which do not involve
in the study of dynamic
are important
with respect to real time,
to time, and we may want
analog computer may
the
differential
as
mathematically by differential
be expressed
variables
for the solution of problems
for
Although
transient
the
problems,
it is in the latter case in which the computer has been used more frequently.
The
analog computer has been found useful
with constant coefficients such
as
in the solution of ordinary differential
equations
the equation
dt
dt
coefficients a,,...,an are constant, and where P is a forcing function which may be
either zero or some arbitrary function of t. In fact, it is not much harder to generalize the
where the
forcing function P to the form
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»-*<t,3f....,
That
is,
and one
the
*
n.
(1.1.2)
right hand side of the equation may be a function of the independent variable t
or more of the derivatives.
The computer may
coefficients
for solving linear differential
also
be used
J|
... + f-(t) 4\- P(t),
such as
Mt)
where
■
dt
+
equations
with variable
(1.1.3)
dt
the functions
f^(t) are functions of time rather than constants.
of the analog computer is in the solution of non-1inear
One of the most useful applications
equations, which in general do not lend themselves
linear differential
ivatives
equation
is
one
in which
to analytic
solution.
the dependent variable
appear raised to a power other than unity or as the argument
of non- linear ordinary differential
equations
are:
As we
recal1,
a
non
or one or more of its der
of
a
function.
Examples
differential equations, linear and non-1inear,
Certain types of partial
solved with the electronic
a)
Parabolic
V2
equations, such
as
^ft
*2
=
kl
+
Hyperbolic equations, such
which governs
+
most
important are:
"diffusion equation"
*3'
d2
ox 5~
>
physical systems involving
many
can be solved more
d 20
+
the wave equation
as
Elliptic equations such as Poisson's
of
The Components
the propagation of waves.
equation
-~ .
2
;s
y
o
conveniently with less expensive
treated with electronic
1.2
the
The
is important in the study of heat transfer.
which
b)
computer.
analog
can also be
passive analog networks
and
are seldom
computers.
analog
an Analog
Computer
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Suppose that the equation
is
to be solved with an electronic
this equation,
a)
we
multiply
conclude
a
that
variable
w '
a2
analog
components
computer.
are needed
or a function times
^y
of
the derivatives
c)
generate
arbitrary functions
such as
d)
multiply
two
functions
f-^(t)
e)
as
a
such as
g,
the addition
to:
constant,
as
in
variable,
generate
perform
a
observing the operations required in
,
etc-,
'
b)
and
By
and
subtraction
required by the left-hand
f^(t),
of variables
side of the equation.
-2-
and
functions
In an electronic
analog computer these basic operations are performed
electronic "black boxes" which accept voltages at their input terminals,
voltages continuously,
and produce
voltages
on
operate
voltages at their output terminals that are
by-
on these
some
function
of the input voltages.
The
involved in the use of this computer is the analogy which the user
analogy
up between the physical variables
such as displacements,
various voltages present in the computer.
simple,
Individual
1.3
will be explained in Chapter
and
computer components
Multiplication
One
which perform
of setting
this is done,
angles, pressures
up
and
the
this analogy is very
however,
the required fundamental
must
we
study
the
operations.
by a Constant
may
be performed
If the constant is less than 1, the operation
or attenuator
(sometimes called
"pot")
a
as shown
on a
may
voltage is multiplying
be performed
with
a
it by
potentiometer
in Pig. 1.3.1
circuit diagram in Pig. 1.3.l£) shows that if a voltage E1(t) is applied at the input
terminal,
then any
moving the
slider
calibrated
dial
fraction K of this voltage may
or down.
up
which
Pig. 1.3.3£).
This
The
is connected
It is more convenient
any reference to
fraction K is
to the
to think
is a one-1ine
to the machine reference ground)
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procedure
Before
2.
of the simplest operations which
a constant.
The
The
velocities,
set
must
be
always less
slider.
A
than 1, and
diagram of the
of the potentiometer
may
dial is
be
terminal
by
indicated by a
shown
in Pig.
third terminal which is connected
which conveys the idea of the operation performed without
Pig. 1.3.1
a)
-3-
1.3.3(b).
in terms of its block diagram shown in
diagram (we do not show the
circuit details.
-
transmitted to the output
The
potentiometers used in most analog computers
of ten turns.
turn counter in the dial displays
The
to the tenth digits
turn number corresponds
of K.
parts which represent the hundredth digits of K,
representing the thousandth digits of K.
accuracy
of one part in a thousand.
ponding
to the
The
Pig.
input
and output
voltages
it is always implied.
but
and each
is a helix
This
in a small window.
dial is divided into ten
of these is divided into ten
the constant K
be
may
the block diagram
set with
parts
an
of the "pot" corres
general functions of time.
E2(t) are in
and
the symbol
and d
The
same
In the
indicating time dependence (t) is dropped
thing
will be done in some of the block
in these notes.
potentiometer is
constant K can never
be
a
passive device
greater than
1.
An
amplification is the £onstant_multi2lier
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face of the
The
1.3.^ shows
E-^(t)
for simplicity,
block
element which
dial setting shown in Pig. 1.3. l(b).
of Pigs. 1.3.1c
The
resistance
the turn number
In this manner,
condensed diagrams
diagrams
a
have
is implemented
and can
"operational
amplifier,"
an
to the
right of the diagram indicates
and
is, the
block of Pig.
1.3.2.
Figure 1.3.2a
shows
how
this
from an .amplifier, usually
called
of the block,
that
active device which provides both attenuation and
Pig.
input voltage times
only provide attenuation,
a
two
and
1.3.2
resistances
and
that the output voltage
constant equal to the ratio
it may be adjusted to values
gain is K, the one-1ine
diagrams
of Pig.
1.3.2b
circuit oriented.
notation of Pig.
1.3.2c
The
RQ
4
RQ/R^.
The
is equal
This ratio
greater than
and c
F^.
as
well
operational
to the negative of the
may
as
equation
be
called
less than
are more convenient
1.
the gain
If this
to use and are less
will be used in these notes.
It should be noticed that the circle and the triangle of Pig.
should not
The
be confused
with the symbol of
figure below
shows
the difference
Pig. 1.3.3
shows
four examples of the
a constant voltage of 5 machine
sign,
and the
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use
1.3-3d
In Fig.
is multiplied
and reversed
In Pig. 1.3.3c
This last block occurs
so
who does not have
a
often,
by a
gain of 3,
1.3- 3a
in
In Pig. 1.3.3b the input voltage is
the time-varying function
is used to change
the analog block
Pig.
voltage E.
amplifier in series with a potentiometer.
of the "constant multiplier."
of -15 machine units.
gain is less than unity.
multiplied by -a, and in Fig.
and
between these symbols.
units (5 volts)
to produce a constant output
negative
operational
an
1.3.2c form a symbol
P(t) is
the sign of the
1.3.3
that it has
been
the special
given
name
of
slgn^.
£h anger.
The
reader
with words
confused
computer does
such as
diagrams which are
connect
the resistances,
important
"amplifier," "gain," "resistance,"
not require such a background since
logical
phase
circuits
strong background in electronic
circuit independent.
capacitances,
and
we
is
of the problem-solving process
5
The
not be
use of a modern
analog
can set up our problems
in the form of
practice,
can learn to
After
amplifiers
etc.
should
some
anyone
to implement the diagrams.
the construction
The
inter
most
of the logic diagram.
The
actual wiring of the computer
One
thing
leve1,
gram
and
R^
limits
this is that all
and
which
may
be used
and
installat±ons
and many
even when
we
are programming at the
restrictions
certain
have
For example, there are limits
in connection with a particular amplifier design.
and maximum gains
of a block and on
their operating
on
resistances
size of the
on the
logic d±a-
also
There are
the minimum and maximum value
the
of*
voltages.
output
gain and voltage are .01 6 R * 10
ranges of resistance,
Common
however,
analog components
satisfied.
on the minimum
input
in mind,
be kept
must
ranges which must be
and
is the job of a technician,
provide this service to the user.
may
RQ
components
-100 * V* +100
volts.
consult the computer
reader
should
values
of resistance
The
(megohms),
manual
1/50 * Gain -& 50,
for the actual ranges
in the machine at his disposal.
Notice that the
or 10^ ohms).
This
often see the
ohmic
Rq
= R,
the
megohm
a convenient
in units of megohms
unit for analog circuits.
value given in "megs," so that
in Fig. 1.3.2,
the multiplying
ratio of two resistances
proper
potentiometer may
a
sign-changer
may
show
(a
million
ohms
reader will
The
of
the values
be used
RQ
and
R^.
constant K has to
Whenever
in order to adjust
two
be
adjusted by selecting
such resistances
are not
the
available,
the gain to the proper value.
1.3. 1
Example
A
value
of K = 2.56
is needed,
Other gains such as
available.
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makes
are given
= 1 meg.
As seen
a
common
divide
K by. one
result.
The
of the available
gain K'
may
but two resistances
which provide
this ratio
.01, .1, 1, 2, 5, 10, etc., are available.
gains
then be set
greater than K
such that a gain K'
in a potentiometer.
in Fig. 1.3-5
Fig.
1.3.5
6
Two
ways
The
are not
procedure
less than
of doing this are
1
is to
will
shown
1.4
Analog Addition
addition of two machine voltages E1(t) and E2(t) is accomplished by the adder bl£ck
or simply the adder or s^uraner^ The diagram of a two-input adder is given in Pig. 1.4.1a. The
The
one-line
diagram which
an n-input
adder
will be used in these notes is given in 1.4.1b,
and
the block diagram of
is shown in 1.4.1c.
E.lt)
a)
E. * - (&,£, + G,EJ
-
Et
G;
(G.E.- •
-
Ro
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b)
Fig.
1.4.1
Notice that the adder multiplies each input variable
products
and
changes the sign
of the sum.
Pig.
1.4.2
analog variables.
-[S+E(t)]
a)
Pig.
1.4.2
7
by
its corresponding gain,
shows
two
adds
these
examples of the addition
of
Subtraction of analog variables
a
sign-changer
may
be done
with
the sign of the subtrahend
by changing
in Pig. 1.4.3.
as shown
- (&,A -GZB)
1. 5
Pig.
Integration
Before we discuss
to review
block which
an analog
rb
function f(x)
The
lirolts-of. integration.
aixsb, and if f(x) is the derivative
fb
,b
'a
=
is called
the integrand and the
if f(x) is continuous in the interval
Now,
of another
f(x)dx = F(x)
\
function called P(x),
F(b)-F(a).
(1.5.2)
'a
functions
Corresponding
it is worthwhile
First, we remember that the symbol
(1.5.1)
is called the defini.te_integral^ of f(x).
a,b are called
for integration,
be used
f(x)dx
\
/a
numbers
may
of integration.
basic concepts
some
1.4.3
f(x)
F(x)
and
be found
may
in tables of indefinite_integrals_
of the form
F(x)
=
jf(x)dx
Notice that the indefinite integral symbol does not
if f(x)
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For example,
of
derivative
sin(t).
we
b
ja
In analog
cos(t)
j
F(o) term may
recall
we
any
from memory,
limits of integration.
or find in a table,
is the
that cos(t)
then write
may
.b
cos(t)dt = sin(t)
sin(b) -sin(a)
=
(1.5-3)
' a
computer work the most
t
The
=
have
integrals
common
f(t)dt = F(t)
it
=
are time dependent,
F(t) - F(o).
such as
(1.5.4)
|^
be
called the initial £ondition of the function represented by
the
indefinite integral.
Now
we
shall describe
the analog
block which
may
Integration of a machine voltage with respect
or integrator shown in Fig.
resistance,
and
discharged)
when
a
1.5.1a.
capacitance.
The
to time
for the evaluation of integrals.
is done with the integration block
integrator consists of an operational amplifier,
If the voltage across
the input voltage E1
be Used
is applied,
8
the capacitor
the output
voltage
a
is zero (the capacitor is
will be equal
to a constant
times
the
definite integral of the input voltage with respect to time.
to the -1/RC ratio where
Pig. 1.5.1b shows
R
is in ohms
and C
is in farads,
begins,
process
an
is called
constant is equal
the gain of the integrator.
In order to specify that the capacitor
the block diagram of the integrator^
is discharged when the integration
and
The
oval with
a 0
in it is shown
below the
triangle.
a)
For example :
suppose that
Pig.
in Fig. 1.5.1 E. (t)
. -tsin(t)dt
■„(*>
=
=
If the capacitor is not discharged
For example,
voltage
-
=
k = 1.
Sin(t) . and
*-
(-cos(t))
as
a
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=
cos(t)
-1
process, the voltage
constant on the output of the integrator.
if the voltage across the capacitor is 10 volts as shown in Pig. 1.5.2, the output
will be equal
cos(t)-1 + 10.
to
't>v
e4 . cosu)
Fig.
output
obtain
j
-cos(t) + cos(O)]
SIN It)
On the
|
We
at the beginning of the integration
will be superimposed
across the capacitor
-
b)
1.5.1
-i +to
1.5.2
other hand,
if the voltage across the capacitor is reversed, as shown in Fig. 1.5.^ the
will be equal
to
cos(t)-1 - 10.
Fig.
1.5.3
The
action of the capacitor voltage la
symbolically
shown
/
(kV
Pig.
1.5.4
In this figure the capacitor voltage is indicated inside
the capacitor
At the beginning
by means of a switch.
is closed and the capacitor is charged
switch
voltage is held constant
the voltage
condition
in the oval is called
voltage is applied to
initial condition
with
and equal
input
an
As
This voltage is appl±ed to
the oval.
of the integration
to the voltage
constant term added to the definite integral.
output
in Fig. 1.5.4
in the oval.
the initial £ondition
the capacitor.
of the initial condition voltage
the sign
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Fig.
so
that
a
which produce
use the diagram
A
a
sign reversal
should
when we
negative voltage -V
1.5-5
take
value
as
this fact into
of Fig. 1.5-5-
fact that will help us later
In this case the
1.5-5, which is computer oriented,
will write in the oval the actual initial condition
of computers
This
1.5-5
Rather than using the convention of Fig.
we
±nitlal
the
is similar to the other amplifier inputs except that a switch in series
will appear as a positive constant +V at the output^ as shown in Fig.
notes
a
reason
amplifier input called the initial condition terminal.
it may be used to connect or disconnect this input from the amplifier.
amplifier reverses
For this
In some computers
voltage.
as
This appears
is closed the integrator
long as the switch
to the voltage across
(t = o) , the
process
start using the integrator
in these
in Fig. 1.5.4.
account
Users
or may prefer to
is that if the initial
condition voltage is equal to the factor-F(o) of equation (1.5.4), then the integrator output
is simply the indefinite integral of
the input voltage.
Fig.
1.5.6
10
This is
shown
in Fig. 1.5.6
Example
1.5.2
sin t
a)
sin t dt
(-coat)
+
cos
cost
-
efcdt
=
When
as
using the integrator
the output
wise,
integral.
1.5.2
showed
two cases
cos
+
0
£sin t dt
-e
=
-e*
the
to the value
of P(o).
Other
evaluating the definite
be found by
indefinite integral resulted
because the
initial condition voltages were set properly.
If, on the other hand, the initial condition
voltage in Example
volts,
is set equal
1.5. 2a
EQ(t)
=
-
(
to -10
is given
by
cost -
11
an
integrator
will be when one
unit (1 volt) of different polarities is applied to
the input.
Notice that only in the
The examples of Pig.
machine
then the output
sin t dt -10 = cost - 1 - 10
=
top example
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0
that the indefinite integral will appear
voltage must
in which
0
e° -e°
only if the initial condition voltage is adjusted
Example
cos
0
-jefcdt
it must be remembered
the expression representing the output
cos
o
cost
+
+
may
we
use
the
1.5.7
show what
the output
indefinite integral for the evaluation of the output.
+0
-,-©-
'-©-
of
Fig.
1.5.8
shows
involving
an example
varying input voltage.
a time
Volt*
-50
cos t
50
sin t
E.
(real time in seconds
degrees
are analogous
angle in
in this example)
and
Fig.
In most
capacitance.
citances.
so that a
Fig.
computers
This
The
most
ratio is adjusted by varying
the l/RC
is done
because
it is easier to measure
capacitor used has
common
ratio of unity
1.5.8
may
be obtained
by
a
using
resistances
capacitance of
a 1 megohm
1
rather than the
accurately than capa
microfarad (1x10"^ farads)
(1x10^
ohms)
resistor
as seen
1.5.9.
«t
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the resistance
(U/dfcoU,)(U/0-'f-arads)
-©Fig.
Other
ratios may
potentiometers
Example
be obtained
as was done
by
in Example
1.5.9
increasing
or decreasing the value of R or by using
1.3.1.
1.5.1
Give the block diagram of an
that integer gains
of 1, 2,
and
5
integrator with a gain of l.58.
are
available,
solutions.
12
the diagrams
Solution:
of Fig.
If we assume
1.5. 10 are equivalent
in
Fig.
1. 6
The Summer-Integrator
The
voltages
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1.5.10
weighted
may
be
sum
obtained
of the integrals with respect to time of several time-varying input
by
in Pig. 1.6.1.
the summer-integrator or adder-integrator
Ec
- - \ (GE. -
Pig.
1.6.1
13
-G„E„)dt
*-
K
of the use of thls block are given in Pig. 1.6.2.
Examples
Pig.
Function Generators
1.7
and
Multipliers
In paragraph 1.2 we discussed
one
or more variables
which are changing
pliers
functions
for these
are used
two
Fig.
Function Generators:
of time.
1.7- la shows
v2,'*',vn
v^,
arbitrary functions of
generate
Devices
called
of two or more variables
function generators
and
function multi
operations.
block diagram of a function generator.
varying voltages
for devices which
the need
devices which produce the instantaneous product
and
as
1.6.2
an<*
the general
In its most general form this
pr°duces
an output
device
voltage which is
accepts
n
time-
prescribed function
a
of
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the input voltages.
In some cases
function generator may consist of
a
function generator is
In genera1,
however,
components.
In commercial
by
a
setting dials, inserting
The
most
common
single variable
of the function,
internally
by
as
so
types
shown
with the
may
be
electronic
set or changed
same shape
diodes
as
by
various
that the generator needs
integrating a constant voltage
Fig.
no
input voltage, since time
a)
b)
Fig.
1.7.1
14
a
1.7.1c results when time
as was done
resistances.
means,
the graph of the function,
of function generators are those which produce
in Fig. 1.7.1b;and c.
and
piece of equipment containing
a complicated
models the function
templets
a few
may
many
such as
etc.
function of a
1s
the argument
be
supplied
in Fig. 1.5.7.
o
Common
wave,
square-wave,
Fig.
these
generators of the type
examples of function
1.7.2
examples
and
shows
triangular wave oscillators
two examples
shown
available
in Pig. 1.7.1c
are the
in any electronics
of the generation of functions.
laboratory.
reader
The
sine-
should
study
carefully.
V0
( liEAL TIME IN SECONDS
IN
\H
***t> AUGL.E
THIS EXAMPV-E.")
ARE. ftNMJDGOU^
I(kNS
^1
l(l+t)
sec
Pig.
Function Multipliers:
Each
The
general block diagram of
is in general a continuous
input variable
voltage equal to
develops
an output
voltages.
The
will show
the constant inside
Pig.
multiplying
1.7.3b
1.7.2
shows
a
a
voltage changing with time.
constant K may
multiplier
The
either positive or negative.
be
in Pig. 1.7.3a.
shown
of the input
constant times the instantaneous product
In these
notes
we
the block.
a common
type of function
multiplier.
general multiplier which allows the multiplication of
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function multiplier is
two
This
variables
is a special
only,
case
and has a
of the
scale factor
of -.01.
2
y.
1
KV,2
-.01
a)
Pig.
Another
in which
common
a
type of
variable
multiplier
Z may
be
usually of the "servo" type,
1.7.3
might be represented
multiplied
by each
15
the block diagram
of figure 1.7.3c,
of n other variables y,,...,yn.
then provides n outputs
of Z with each of the n inputs.
by
which are proportional
This multiplier,
to the product
of the use of the two-input multiplier is given in Fig. 1.7.4.
An example
voltages
=
v-y
to -.01 (10
lOsin©
sin 9) (10 9)
are multiplied
= 10©
and
V,^ IOSf/V<»)
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vt=
the angle
is
angle
©
is
equal
d-c voltmeter
equal
ing
have
popular methods
on the
on the
The
»oe
equal
— - e sin id)
-.ot
Pig.
1.7.4
Pig.
1.7.5
lOsin© the analogy between radians and
its analog is equal
between ground and
most
been
v1
to 1.5708
terminal would
this termina1,
volts is implied.
volts and
be 10
When
the sine of the
volts.
If an ordinary
it would read 10 volts at the time
are provided
common
for obtaining function multiplication.
developed
by
values
particular multiplier.
frequency
to produce an output
volts.
Many methods
multiplier.
=
x,
w
to 1, so that the voltage at the
when © is 1.5708
most
v-^
to tt/2 radians
is connected
input
sin 0.
= -©
Notice that in the function
instantaneously
Two
the
servo-multiplier
and the square-1aw
for the constant K in Fig. 1.7.1
The
of the input voltages.
accuracy
+ 1
the
electronic
and +
.01,
depend
with which multiplication is performed depends
Static accuracies of
16
are
Perhaps
. Vf> are
common
and
accuracies
of .01J6 or better
ates
as
may
the frequency
servo-multiplier.
in more
increases,
and
detailed
A
expensive
reader
might notice
function generator which
several input voltages
1.5.7
explanation of the different
A
dynamic accuracy
deterior
multiplier
in the
than
for obtaining function
methods
that the function generator is the most general
accepts
a
and develops
the product
how the
The
it is better in the electronic
since all of the blocks discussed are special
it develops
equipment.
is not within the scope of these notes.
multiplication
The
be obtained
functions
voltage
cases
their sum,
t+5,
For example,
multiplier is similar
a
etc.
integrator is
an
its time integra1,
and generates
of the input variables,
t,
of it.
a matter
As
t-5 may
be generated
with
be used
for division
by
analog block,
an adder accepts
to an adder except
of fact,
a
we
showed
that
in Fig.
integrator.
an
Multiplier used for Division
A
in Fig.
function multiplier
1.7.6a.
may
Notice that the amplifier
to the multiplier.
Fig.
serves
as
connecting it to
an adder whose
1.7.6b gives the block diagram of
amplifier as
an
output
voltage
shown
is fed back
divider.
a
R
E.tt)
e»tw
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a)
E,
b)
1.8
Fig. 1.7.6
Other Analog Blocks
There are many analog
which have
been
presented, however,
which are tackled by analog
The
output
computer components
we
have
for solving
a
not mentioned here.
large percentage
The
blocks
of the problems
techniques.
devices, that
answers are explained
are enough
which
is,
in Chapter
meters,
3, where
recorders,
etc. which are
a sample computer
17
needed
is described.
to obtain the
final
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Exercise 1.1
DRILL EXERCISE
ON
ANALOG BLOCKS:
18
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19
PROBLEMS
1.1
Generate the function
a single analog block.
1.2
Notice that
Use
*
i
*
-j-(©+/5)
"
•
'A
the functions
0 , /* , and-<* by, using,
+ < from
from
the functions
/*
0
••
and-<*
by using
this idea to generate function multiplication from two identical function generators
the squaring function plus any additional analog blocks needed to form
which generate
the terms
1.3
Draw
to be squared.
the block diagram of an analog
assuming
that the function
which accepts the voltage
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1.4
Draw
v^
T
circuit for generating
ln(v1)
may
and produces
the block diagram of an analog
and generate
1.5
vQ
=
.
be implemented
the voltage
the function -t-ln(B),
with a function generator
vQ.
circuit which will accept 5 variables x^, ...
Generalize
the block diagram to accept n variables.
has operational amplifiers which are pre-wired
as shown
amplifier
has a selector switch S which converts it into an adder
in the drawing. The
or 1.1. The
if the switch is in position A, a summing integrator if in positions
following Jacks are available for connections: the seven input Jacks, four common
output Jacks, a Junction Jack, and an initial condition Jack. Inputs which are not needed
Thus, this amplifier may be used as a universal analog blockmay be left disconnected.
Constant 1 megohm resistors are available and may be connected to the Junction Jack
when more inputs are needed.
A
certain analog
computer
Il
20
1.5 (cont'd)
Show how the
following blocks may
a.
b.
c.
d.
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e.
be implemented:
sign changer
constant multiplier (How many different constants can you
get without the use of attenuators?).
three input adder with gains 1,2,5 (Hint: resistances may
be paralleled or attenuators may be used) .
an integrator with unity gain.
a summing integrator with gains 1,.5,2
21
Chapter
ANALOG SOLUTION
2
OP EQUATIONS
Introduction
2. 1
In Chapter
discussed the basic operations which
1 we
components of analog computer
in the computer
the variables
and
In the discussion
components rather
Now
in certain
less of the type of analog computer
simplest equation that
The
example,
set up between the voltages
be
this,
will be useful regard
the diagrams
is solved.
the problem
may
solved in
be
is
computer
an analog
linear equation.
a
which
is recognized as
generated
the equation of a straight
of the
by means
(2.2. 1)
analog
block in Pig.
Pig.
The
equation may
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mathematical
shown
variables
in Pig. 2.2.1
and
be
line with intercept
solved by assuming that
voltages
two
variable y.
to the mathematical
study
in the way in which
the problem variable
to the problem
yield
negative
the
diagrams
and
the machine variables
b
analogous
to the
are adjusted as shown
they are analogous
should
X and B are
the gains
Other computer connections which
reader
may be
If these voltages are fed to the two inputs of an adder as
in the figure,
The
and
slope m,
x and b.
cording to the given linear equation, is equal to -Y.
variables^
and
2.2.1
times
voltage analogous
b
2.2.1.
will be equal to the negative of the sum of the inputs
2. 2. 2d
For
the equation
y = mx + b,
The
by the
Linear Equations
2. 2
a
voltages
to use the block diagram of the analog
By doing
on which
on
of problems.
types
shall prefer
we
circuit diagram.
than the
be performed
shall show how an analogy may
we
which follows
may
is
made
analogous
voltages are used for X
the
The
the output
their repectlve gains,
output
and Y are
distributed.
in order to eliminate
is then
2.2.2.
differ from Pig. 2.2.1
For example, in 2.2.2a
a
voltage,
and
the sign changer.
in
This
easily done, since analog computers have a voltage supply which provides both polarities
22
ac
the machine
are given in Pig.
solution of (2.2.1)
to the gain B rather than to
voltage.
adder
and y.
prove to himself that they only
and B
which
of the sign changer
voltages X, B,
variables. x, b,
are defined and
of the
of
is
Preference for one of the diagrams in Pig.
2.2.2 depends
the magnitude of a voltage or the magnitude of a gain.
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being
on whether
This depends
it is easier to change
on the computer which
is
used.
Pig. 2.2.2
The
problem
problem variables
for which
equation which
we
x, y,
the equation
remember
m,
and b are
is a mathematical
from elementary
given in units which
model.
physics
y = vt + yQ
depend
For example,
one
on the
physical
form of the
is ,
(2.2.2)
23
linear
total distance
which gives the
v and
traveled during the time t by an object with
y
initial distance yQ from a reference point.
an
hour and t
in hours,
voltage
is analogous
T
between the physical
the analogy
to miles
voltage Y is analogous
to miles y
to miles/hr
Pig.
may
Obviously,
ask:
What
problem variable
that
and
one-to-one correspondence
designed
to operate
at a maximum voltage
variables
in Pig. 2.2.3
of t,
miles of y
v
in miles
per
will be:
YQ
v
2.2.3
are equivalent
to hrs.
and
will
always be within
the range
and yQ?
and each
for which
the
between t and T might damage an analog
block which
is
of + 100 volts, since this would necessitate feeding
volts to one of the inputs of amplifier 1 in Pig. 2.2.3.
in more detail
Let us investigate
machine variables
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and problem
in miles,
If for a particular problem t is equal to 2000 hrs, it is
designed.
obvious
2000
voltages T, Y,
so that the machine voltages
analog blocks were
a
are
set up proportionality constants between each machine variable
must
we
If y and y
yQ
non-dimensional gain v is analogous
we
constant velocity
of time t
to hrs,
voltage YQ is analogous
Now
a
Example
may
be found.
how the
relationships between problem variables
will do this with a simple
We
and
example.
2.2.1
Given:
v =
2miles/ hour
yo=
100
for
miles
Find :
y
0 * t « 200
Solution:
Equation (2.2.2)
hrs .
becomes
y = 2t + 100
maximum value
The
Y
max
(2.2.3)
of y is
= 2
t max
= 2 x
+ 100
200 + 100 = 500
miles
(2.2.4)
If this equation is to be solved by analog techniques, the machine variables Y, T, and YQ must
If a one-to-one
never exceed the maximum operation voltages or gains of the analog blocks.
is set up between machine and problem variables,
correspondence
is used,
the output
of amplifier
1
(T)
of amplifier
would require
2
would be required to
a maximum
voltage of
24
and
if the diagram of Fig. 2.2.3
deliver 500 volts and one of the inputs
200
volts.
Assuming that our computer
does not handle voltages above
volts,
+ 100
than a one-to-one
relationship between
reaches a maximum
value
of
500,
problem units of y (miles),
t (hrs.)
each
must use a
we
problem and machine
set of appropriate scale .fact£rs. rather
unit of Y (volts)
machine
For example, since y
variables.
may
be made
unit of T (volts) equivalent
each machine
to
2
equivalent to
5
problem units
of
This is written
t,
2T =
5Y = y,
and corresponds
(2.2.5)
to the process
When these relationships
of changing variables in mathematics.
are substituted
5Y = 2»(2T)
into (2.2.3)
we
obtain
+ 100
and
0^2T<200,
which
(2.2.6)
after simplification, yield the machine
'/
4
Y = £
5
equati£ns
• T + 20
o£T<100,
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and
the
(2.2.7)
circuit given in Pig. 2.2.4.
Pig. 2.2.4
We must
of volts
must
keep
be changed
example, when T
and
in mind that in order to interpret the analog results,
= 50
into