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LearnAbout
role
PARTl: INTRODUCTION-The
analogcomputerscanplayin engineer'
ing is discussedfor the practicing
comPonents
GomPuter
engineer.
areidentified
Theodore W. Gcdmon and Theodore G. Smirh
University of Maryland, College Park, Md'
Cer-culerloxs, previously done by hand, can now be
completed using coinputers with a 1t?t reduction in calculation time and a marked increase in computation pre'
cision. While such incentives are frequently sufficient to
iustifv the widespread interest in computers, the applica"A*t
äf computen to the solution of engineering problems
tras naa a much more fundamental effect on chemical
and petroleum engineering. Wh-e1 th9-sp:4 *d precision
of m'odertt computers are coupled with their variety, versatility, and large capacity, the reasons soon become
apparent.
The practicing engineer now has at his disposal comorriutioti"t devicJs wüich enables him to handle economiially problems of a complexity ryH"f he. could not have
fta"afäa 10 yean ago. As a result, the class of problems
be expected to handle has been
*t i"tt ttt"
"ttgitt""t-"This extension has resulted in a resreatlv
"*t"ttä.d.
Evalrration of what can be classedengineering calculations'
iü" ,"-"rrul ration is still in progress, but it appears clear
that the engineer of today, and certainly of tomorrow'
;tli b" cons"tantly associatäd with problems once consid,oo comple*, time consuming, or precision oriented
"ä
of solution. Consequently, computers have
i" U. t*i*Ufä
.roi orrly proven to be useful computational tools for the
also provided a means for extending
ift"y have
'engineering
""gi"""'.,
calculations and, in so doing'
in8 t"op" of
engineering itself'
of
scope
the
extended
have
is
Thc increasing complexity of engineering problems
are
today
engineers
that
so
universities
by
recognized
they
b"irri ,o.,tittety trai"ed in the use of comPuters, as
and
rules
slide
of
in tüe past were trained in the use
l
l
4
hand calculating machines. While the degree of comPuter
competence whlch is required of a -placticing engineer
,ouriä *ith his position, some knowledge 9f comPuters
must be consideräd an integral part of his already varied
skills.
For the p'racticing engineer who is- directly associated
and operation of computers, a high
rvith the päg."ttti"!
on the comPuter is. required' A
of
degree
"äp"t"tt""
situation found in many companies is that in
"Ä*o.,
*ni"f, . group of exPerts, highly trained. in the use of
cornputers] is ävailablä for consultation with the practicing engineer. In this case, the practicing engineer can
these experts for assistance in program-ing .?"q
r"iy
of
"i
op"ratiorr. In many cases,these exp€rts will handle all
has
the
engineer
once
solution
a
of
obtaining
ifie details
properly defined his Problem.
If experts are available, the degree of competence
on a
req.rired by an engineer to solve a specific problem
of help
-il;;;
*#"in" coÄputer i"sinversely related to the degree
obtain from the experts' Howwer, frequently the
defined' Assumptions-.of
problems a."ltugtt-ll
"ngi"""ft
be made, the reliability
frequäntly
mu'st
dqqree
.ruirlirrn
tttä, if several comPuters are availConse"iär"-a.üLamined,
uff", *" choice of which to use must be made'
the
with
only
concerned
seldom
is
o.,"rrtlu. the engineer
and'
computer'
a
specific
on
problem
sScific
a
of
t;11#
of
even when computer experts are available, knowledge
practicing
the
aid
significantly
the use of computers can
engineer.
A background of computer knowledge:
o Enables the engineer to communicate more elTecproblem
tively with the experi. It permits him to state his
potential
the
to
anticipate
to the e*peri and
*#
"l"u.ly
may encounter during the detn"
pirf"fft *tti*t
"*p"tt
tails of solution.
o Makes thq engineer aware of the limitations of the
in making the
available computers. This aids the engineer
problem
current
his
to
solve
use
choice of computer to
so as to
solution
of
his'method
gear
to
him
and enables
engineer
The
the imporäce of the limitations'
-irri*i""
comwill also be aware of the versatility of the chosen
relaxing
of
feasibility
to
decide'the
outer and be able
lo*" of his assumptions on an individual Program'
o Perhaps
mo6t imPortant, stimulates the engineer- to
-p."Älems
whicü he could not previously consider'
u,ru"f.
the speed, precision, variety' versatility'
of
;; b.t.ä aware
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Although it is not shown in Table 1, one of the most
importänt components of the computer is the constant
DC power supply. Depending upon the design of
voltage
the
approximate
only
can
which
values
terms of discrete
the computer, the power supply is usually either 100 or
exact solution. Furthermore, the computationscan be car10 volts. This DC voltage is called the reference or maried out at practically any speed desired. This permits
voltage and its stability will determine to a large
in
chine
parameters
of
range
the rapid exämination of a wide
degree the äccuracy of the solution that one obtains'
u rpu"ä of time that can be significantly shorter than the
The first element in Table 1 is an attenuator comfixed time of calculation on the digital. Analogs do not,
called a pot-an abbreviation for potentiometer'
monly
for
however, have the capacity for memory, the capacity
of päts are commonly found in an analog
types
Two
precision
of
degree
nor
the
a large number of operations,
One is referred to as a grounded pot and the
.o-pni"..
of the digital.
pot' The grounded pot enablesthe
ungrounded
other an
Programing an analog does not require knowledge
outPut voltage which is some fracan
to
öbtuin
operator
of a hlghly specialized language as does the digital. The
of the input voltage' The un1
and
0
between
ti,on
details äf programing are generally quite similar to classigrounded pot performs the function shown in Table 1
cal methods of solution with which the engineer is familänd is generally used in the construction of special funciar. Analog computers find their greatest application in
tion circuits. Attenuators are usually adjusted manually
the solution of differential equations, such as occur in the
to
set the value of a constant K. This value is determined
dynamic analysis of processing systems,in which the
by measuring the output voltage from an attenuator with
arnount of algebraic and logical operations required is
the voltage measuring device built into the computer'
rather limited.
The next component listed in Table 1 is the high gain
A hybrid computer is the combination of a digital and
amplifier. The function of the high gain amplifier is -to
computer in which the solution of a problem
un unälo.g
muitiply an input voltage by a large constant, generally
-between
the two. By containing both an analog
is shared
of thä order o1 10s. The high gain amplifier is seldom
and a digital, the solution of a problem on the hybrid
used by itself, but usually forms an integral part of -other
analog components. The high gain amplifier is
computer can incorporate the advantages of both' The
.o-*on
capacmemory
Iarge
provides
a
generally designld so that it provides very stable operation
digital computer portion
ity" and peimits rapid logical and algebraic operations
t th" -ugttitude of the output voltage does not exceed
the comprfter reference voltage. If the output voltage of
*trite tfte analog permits continuousintegration' The
and
the
other
to
from one portion
transfer of infoÄition
the high gain amplifier exceeds the machine voltage the
beyond
hybrid
the
of
versatility
the
operatlo.t"of the iomputer will be non-linear' For this
back again extends
,öu.o.,, the output of a high gain amplifier during oPerathat of the analog or the digital alone. Hybrid comPuters
tion should not exceed that of the computer reference
are finding application in the solution of complex probis
exsolution
voltage. Most computers have a built in alarm system
lems wherä thä time required for a digital
or
capacity
the
possess
not
do
whicü warns the operator when the output of an amplicessiveand where analogs
for
a
solution.
required
fier exceedsmachine voltage. This restriction on the outprecision
the
.o*ponent in which a high
put voltage holds for
"n"ry
in
analog
term
The
gain ampiifier aPPears.
Analog Computer Components.
analogiomputer is really a misnomer. That is, the electriThe next component of interest in Table 1 is the suman
is
not
problem
particular
to
solve
a
used
ciicuitry
cal
ming ,circuit. e'i its name implies, the- summing circuit
electrical analog in the conventional sensethat electrical
adds two or more voltages and gives the resulting su-m'
current is analogous to fluid flow, voltage is analogous to
The summing circuit cönsists of a high gain amplifier
pressure, electrical capacitance is analogous to mass, etc'
with a feedbaik resistorand three or more input resistors
The electrical analog iomputer is actually a device which
The input resistors and the feedback resistor, in a sumhas been designed to perform certain mathematical operaming circuit, are usually fixed not variable- resistors' The
tions, such as addition, multiplication, integration, etc' on
inpu't voltage to a summing circuit is multiplied by. the
consecomputer
an
analog
of
use
The
of the feedback resistor
specified voltages.
neeati
-the re tulio of the resistance
For most comPuters
resistor.
q.rentty involvis the specification and completion oj the
input
resistanceof the
to
1
or 10' Since a sum0.1,
Äathematical operations required to solve a particular
is
usually
ratio
this
of
the value
set of equations and is entirely independent of the fact
mer contains a high gain amplifier, the output from a
that the equations may describe a flow system, a heat
summing circuit rnust not exceed the reference voltage'
exchangeproblem, a mass transfer problem, or a reactor
One äf t-he most important components of an analog
system.
computer is,the integrator. The circuit fo.r an integrator
Since most of the analog comPuters being used today
is essentiallya high gäin amplifier equipped with a capacitype
this
to
restricted
is
discussion
this
are transistorized,
tor in the ieedbäck-line and two or more input resistors
of analog computer' This is not a large restriction because
The output from an integrator is -the -negative sum of
most of ihe older vacuum tube computers operate in the
the initiäl condition and the integral with respect to comsamefashion with the exception of a few mechanical computer operating time of fixed ratios of the input voltages
ponents.
äs indicäted in Table 1. Since the output of the integrator
is time varying, there is a possibility that this value may
A list of the components found in a modern electronic
be greater ihaä machine ,röltage at. sometime during the
analog computer is given in Table 1. The first column
'IiUte 1 contains a symbolic representation of each
pro6l"rn. Therefore, special precautions must be taken to
in
insure that this does not happen' These precautions will
component, the second column contains an electrical repbe discussedlater in the series.
,eserrtutio.tof each component and the third column shows
The componentsdiscussedto this point are commonly
the mathematical operalion that each component performs'
...
TEARNABOUTANATOG COMPUTERS
6
cailed linear components and the principle of superposi:ion holds. The next group of componentsto be discussed
a:'e commonly called non-linear components.
Nonlinear mathematical operations, including multi: r l i c a t i o n ,d i v i s i o n , e x p o n e n t i a t i o ne, t c . , a r e a c h i e v e d
through the use of specially designed components. Generallv a particular component can be used to obtain both
the designedoperation and its inverse depending on the
:rarticular patching used. Thus, what is termed a multipiier can be used to achieve either multiplication or division. The details of patching and the electronic circuitry
r.ariesfrom one type of analog to another. The usually
acceptable programing s1'rnbolsare in Table 1.
A discussionof the internal construction of the nonlinear components will be covered in a later part. The
purpose here is to present some of the more common
ronlinear components and to examine their lirnitations
and the precautionswhich must be exercisedin their use.
\\rhen it is desired to generate a function such as
) : x2, where * is the input and y is the output, a nonlinear component is used to achieve this operation. Transistorizednonlinear components can be uied to produce
an output which is a seriesof straight-line approximations
as indicated by the dotted lines in Fie. 1.
Generally, for economic reasons,sufficient line segments
are not used to closely approximate the desired function
over_the entire input range. The best responseis usually
obtained when the input is near its maximum maenitude
(that is, the machine reference voltage) .
A high gain amplifier is an integral part of nonlinear
components. Because of this, the output from a nonllnear component can not exceed the reference voltase.
Numerical factors are included in nonlinear co*po.r"rrt,
so that when the input is near its maximum maänitude
the output is also.
Becausecertain nonlinear operations such as squaring
may_ be more easily obtained than other opeiations,
mathematical relationships such as Equation ( I may
)
form an integral part of a nonlinear component. perhaps
the most common of these is the relationihip used in the
quarter-squaremultiplier
^. ^t -, _
(x*y)'-
(x-y\z
4-
FROM HYDROCARBON
PROCESSIN6
Qr
Q2
R2
Fig.,2-. Ä.n analog.computer circuit is developed to compute
the height of liquid in this tank.
(t)
which is used to obtain a product because squaring is
fairly easily achieved.
Due to the internal construction of nonlinear com_
ponents, they will draw a varying amount of current depending on the value of the input. It is, therefore, very
important that the output from a pot never be used as
an input to a nonlinear device sinie a varying current
produces a varying voltage distribution from'the- pot. In_
puts to nonlinear components should always be outputs
from high gain amplifiers contained in other analog äm_
ponenß.
The size of an analog computer is usually gaged by
the number of high gain amplifiers that it contÄsl Mosi
commercial computers normally run from ten to several
hundred high gain amplifiers, most of these high gain
-r*ä11",
amplifiers are normally summers. A somewhat
number are integrators, which may also be used as sum_
mers. The remainder of the high gain amplifiers are
usually integral parts of specialized,,orrli.,ea,components.
REPRINTED
x
Fig. l-The nonlinear relation shown by the solid line can be
approximated by a series of linear relations.
htNrr/Ref.
* Ref.
Fig. 3-Steps are combinedto give the finishedcircuit.
The complexity of the problem which may be solved is
very closely related to the number of high gain amplifiers
available on the computer. A problem is usually solved
on an analog computer by connecting the various com_
flow out of the tank is linearly related to the height of
fluid in the tank through the flow line resistanceRr'
...
LEARNABOUTANALOGCOMPUTERS
(4)
Qr: h/R,
Then our relation becomesupon substitution
arrangement
* Ref.
dh/dt:
Fig. 4-Numerical
values are added to the circuit.
Fig. S-The analog circuit
is simplified.
ponentson a patch panel with external leads' This process
is called patching.
As an example of how an analog computer circuit
may be set up we will consider the case of the change
in üeight of liquid in a tank having a cross-sectionatea A'
A mass balance on this systemyields:
Accumulation- InPut - OutPut
A (dh/dt) :
et-
Qz
(2)
(3)
of
We will assume that Qt is constant and that the rate
About the qulhors
Tsnonoen W. ClolraN is assistant pro'i'n
d'eiiito,
'partment the clzemical engineering
(JniaersitE of Maryland'
of
'CoUese
Park, Md.' where he teaches
oraduate and undergraduate courses
in oro""r, control, separation processes,
mathematical modeling, and computer
applications. Dr. Cadman also conducts
rii"arch in process contt^ol, computer
applications, and process modeling' Befiie receiuing h-ry !h. D-. .desr.ee in
(J
clwmical engineering f r om Carnegie-MeUon- ni'aersi'tg' he
with St' JosephLead Co'' Allied
i"A""iiiri-t""r'* Trortrtiont
and Deuelopment.Co'
öi"*liät Corp.,'and Gutf Resert"t"ch
of AAAS',ACS, ISA, AICLE,NSPE' Si'sma
i;-;;;-*"*bLr'
Xi and Tau Beta Pi.
Tuoooone G. Snrrtn i's associ,ate prof'ilepartment
essor in the chemi cal engineering
of [JniuersitA of Maryland',
Coltege Park, Md' He conducts reundergraduate
anil instructs
,noril,
anil grad'uate courses on polEmet' phgsics, ieactor desi'gn, and mass t'r'a'nsfer'
Dr. SrwitlL receiaed B.S. and M'S' deorees in chemical engineering from
D"'Sc'
tlniuersitg and a D.'Sc'
iohns Hopkins Uniueriitg
iolms
ehemi,cal enginee'r'ing from
in ch.emical
degree ii
Wäshington (Jniaersitg. He has tt;orked
du-P^2nj de Nemou't's
department of E'-!'
in tlw-plastics
'and
and
'i's a' member of ACS, AICILE, A A A S
A Co.
Sigma XI.
(q,/A) -
and re-
(h/AR,)
(5)
The analog diagram may be constructed from the differential equation by applying the following.steps:
Step 1. Write the'diffÄrentiat equation with the highest
o.d"räd derivative by itself on the left side of the equation'
Step 2. Begin a diagram by showing the highest ordered
derivative at the output of a summing amplifier'
Step 3. Perform as many integrations as a-renecessaryto
for.n ätt of the variables on the right side of the equation'
Step 4. Use the generated variables to form the original
hishest ordered derivative.
"Step
S. Insert any forcing functions.
Thä sequential construction of the program is shown
in Fig. 3.
Thä pot settings can be determined when the values
of parÄeters in the problem are known' Assume thal
hini.rot :
,4 :
41 :
4 ft'
10 sq. ft.
10 cu. ft./min'
Pot1 : qt /10A :O.l
P o t2 : l / A R " : O . l
P o t3 : h r n i r r ; / L 0 : 0 . 4
Rr:1min.,/sq.ft.
The final diagram with pot settings is given in Fig' 4'
Although the arialog diagram aPpearssatisfactory, it may
in<ieed-beunsatisfactory for one or both of two rearsons'
. lffr" output from'a high gain amplifi-er may exceed
the referenc" ,roltage and hence yield a f-aulty solution'
o The output föm a high gain amplifier may be so
in
small that inierent limitatiäns-of the analog resultsmonitoring
in
u fu.ttty solution or difficulty is encountered
-the value.
a knowledge of the nature of the numerical solufto
output, hi fro amplifier 2-has an initial value
the
tion,
a final'value of L0 v', and asymptotically uPof +
".,
orou.h"r'th" final value in an exponential fashion' Hence
fOO v. or a 10 v. machine, amplifier 2 will not be
i".
dhf dt
"
overloaded. Similarly amplifier 1 with the output
of 0'6
value
rvill not be overloadeäb".ä,,t" it has an initial
v'
100
a
Ilowever,.for
and decreasesmontonically to 0'
and
small
quite
ate
dhldt
änd'
machine, the values of. h
the cirfor a 10 v. machine dhldt is quite-small' Since
probwill
solution
the
.o*pott"tttt,
nonlinear
no
cuit uses
difficult'
il satisfactory although monitoring 1n?y !9
-"i1;;;iJ
;üiu
be noied that;if the value of the highest.derivative does not need to be available for monitoring'
diagram
step 2 aborremay be eliminated' In this case,the
example
chosen
t" nig. 5 is obtained for the
;h;;'
with a net reJuction in the number of analog components
required for a solution.
be prein the next article of this series,techniques will
obtained
be
to
,"nr"a which permit an analog solution
direct solurrfr"" ,t aforementioned reasons prevent a
variables
"
computer
satisfactory
which
by
ii.". fn" process
so that the outputs from amplifiers are mainvoltage' is
"r"-.nit"ti
i.l.r"a near, but do not "*ce"d the reference
termed amPlitude scaling.
:e1'"*
,-n'TH:,i"ii}:ru-,''*$i1".ä,;19,"'?äJ*lL
:li;tilf,
LearnAbout Analog Computers
are maintained near, but do not exceed, the
SCALING-The amplifiers
PART2: AMPLITUDE
reference voltage. This process is termed amplitude scalirg.
outputs from high gain amplifiers
Amplilude Scoling. This is accomplished by expressing
in an analog computershould be
equations in a normalized form, wherein the normalized
maintainednear the machine's
variables are to be the outputs from high gain amplifiers
referencevoltage,but not exceedit
Theodore W. Godmon and lheodore G. Smitl
University of Maryland, College Park, Md.
Fon serrsrecroRy opERATroN,the output from a high
gain amplifier contained in any analog component must
not exceed the machine reference voltage. On the other
hand, the outputs should be maintained near their maximum permissible values for the most satisfactory oPeration of most nonlinear components and for ease and precision of monitoring. Thus a method is needed to scale
an analog problem so that the outputs from high gain
with magnitudes which approach, but do not exceed, the
reference voltage. The analog solution is consequently the
solution of the normalized form.
The processof amplitude scaling is perhaps most easily
-f I
achieved if the reference voltage is considered to be
-+
-r
volts.
100
10 volts or
unit, even though it may be
Scaling accomplished using this generalization is termed
unity scaling and has several advantages over scaling in
which the value of the relerence voltage is implicitly used.
First, unity scaling permits an analog comPuter Program to be obtained. which can be used on any analog
computer regardlessof the value of the reference voltage.
Second, the normalized variables which are outputs from
**"*"1**ft1
*%"*,Y;
i+r
1e$.{ryj
T A N K2
T A N KI
-j:-,;tfl
ärli
iä'
ffi
Fig. 6-Unity
e lr:
i ..:..!
scaling is applied to an engineering problem before it is solved on the analog computer.
. . .
LEARNABOUTANA,IOGCOMPUTERS
variable by its estimated maximum magnitude. Note theie
on the list made in Step 3.
Output3 ond Potentiometer
lABl,E 2-Scollng
Amplifier
Component
Output
or
Value
Estimated
Maximum
Value
Scaled
Value
hL
_h
ht*
h x
ha/hJ
-hJhax
h:z
-h2
h"*
Amp I
Amp 2
Amp 3
Amp 4
Pot I
Pot 2
Pot 3
Pot 4
Pot 5
Pot 6
Pot 7
Pot B
Seftlngs
h2/h2*
h *
-h2/h2*
fr'(o)
h'(0) /h'*
q1/Ar
qr/Arh:
l/AaR1
| /aaR1
h,"(0)
h'(0) /h,*
q2/A"
r f r
_ t _ I _ l
A2L\'
q2/42h2*
1 l
1 [ r , r l
4 L r y- & J
Rz)
(#) (#)
(#) (#-)
I
-7n
I
ZF,
high gain amplifiers are simply the value of the original
variable divided by the maximum magnitude of the original variable. Third, and perhaps most significant, the
process of scaling non-linear components is greatly simplified.
For example, if r and y are the inputs to a multiplier,
the output expressed in volts for a l0-volt machine is
xylß. For a 100-volt machine, the output is ry/100 expressedin volts. But in terms of unity scaling it is ry
regardless of the machine reference voltage. The simplification encountered for other nonlinear components is
similar.
Unity scaling can be achieved by mathematical manipulation of the original equations before any programing is attempted. However, scaling is frequently more
easily achieved and more clearly visualized if it is done
after some preliminary programing. In particular, the
following seven steps have been found to be a most convenient method of achieving unity scaling.
down the differential
SteP 2. Sketch a_preliminarv analoq diasr
eglecting the prob-lffiTl-F&1lng, which connects all of the required analog components together. Clearly mark the
output from each amplifier and the value of each pot as
it appears in the problem equations.
Step 6. Insert additional pots in the amplifier input
lines and/or modify the values of the pots which were
needed to complete Step 2 so as to compensate for the
change in scale from one amplifier to the next in series.
Again modify the list completed in Step 3.
Step 7. Sketch the completed analog diagram using the
normalized variables and the new pot settings. Note that
this step is easily accomplished by referring to the preliminary sketch and to the list rvhich has been completed.
If the programer closely follows these rules and systematically lists each variable and the pot settings during the
scaling process, scaling can be readily achieved, errors
easily detected, and any further scaling rapidly completed.
In addition, a completely general analog computer pre
gram can be obtained by syrnbolically retaining the maximum magnitude of the problem variables rather than
computing their numerical values. The following example illustrates the use of this seven-step procedure for
achieving unity scaling.
An Exomple. Consider a hydraulic transient problem involving two tanks as illustrated in Fig. 6.
Assume that the cross-sectional area of tank 1 is .,4t
and that of tank 2 is Ar. Further assume that the volumetric flow of fluid through the valves is lineaily related
to the difference in liquid level across the valves. Given
initial values f.or h' and hr, it is desired to determine the
variation in these levels with time. The equations which
describe this system are given as follows:
For tank 1
For tank 2
A a @ h L / d t ): e t - e s
(6)
A z U h 2 / d t ): e z * 4 e - 4 *
(7)
For resistance I
Qs= (hr-
(B)
h,)/R1
For resistance 2
q4:
h2/R2
(e)
h,
-_r 7 r h\ "
ArR,
(t0)
Or
d h r -: 4e-,
dt
_
And
dh,
4z
dt
A2
---:--:1
J-
'
h ,-
h2
h2
A"R,
ArR,
ArR,
(lt)
Steb 3. Make a list of the vari4bles that are outouts
from amplifiers" These are the variables which must be
'äfiFlifiGiEilAd.
Also list the values of the pots. These
may be changed as a result of the amplituC.e scaling
process.
Step 4. Estimate.the maximum magnitude of each of
the varrables wnrch are outputs lrom amDllhers.
les for
the outputs from the amplifiers by dividing the problem
REPRINTED
FROM HYDROCARBON
PROCESSING
il
hr(o)/hr*
- REF.
- REF.
Fig. 7-This analog circuit representsthe variables in the
two-tankexampleproblem.
Fig. 8--Amplitude scaling determinesthe proper setting for
the amplifiersand potentiometers.
with initial conditions of /z' (0) and hz (0) respectively.
Assuming that gr and g. are positive constants,a preliminary analog diagram for solving Equations ( 10) and
( 11) for h, and ä2 versus time can be drarvn. Prior to
specification of numerical values, the diagram in Fig. 7
and the list of variables which are outputs from amplifiers
and the pot settings in Table 2 are obtained.
Letting a superscript* indicate the maximum value of
the output from the amplifiers as indicated in Table 2,
the protess of unity scaling can be symbolically carried
to completion. The results obtained are given in the final
colunn of Table 2. The output fron each amplifier becomes the value of the problem variable divided by its
maximum value and some of the values of the Pots are
changed to compensate in the change of scale from one
amplifier to the next. The latter changes perhapl become
moie evident if the final analog diagram given in Fig. B
is considered.
In readjusting pot values follou'ing the scaling of outputs from amplifiers, four casesarise:
Case 1. The pot value must be divided by a maximum
expected value. Examples are pots 1, 2' 4 and 5 wherein
the maximum value is included so that the inputs to the
amplifier contain the same scale factor as do the outputs'
Case 2. The pot value remains unchanged. Examples
are pots 3 and 6 which remain unchanged because the
input to the pot contains the same scale factor as is desired in the outPut.
Case 3. The pot value is changed by a ratio of expected
maximum values. An example is pot 7 which is changed
by hr* f h1* becausethe input to the pot contains the factor lf h2* whereas the output must contain the factor
I f hrx. Pot B is another similar example.
Case 4. Additional pots must be incorporated into the
- hz been
circuit. This would arise, for example, had fr'
-hrf
hrx and h2f hrx would
desired.In this caseinputs of
t2
- h1/h1*
have beenavailableand an outPut of.(h' - hr) I &' - hr)*,
where (ä, - hr)* * hrx or /r2*, would have been the
scaled output from the summer. Additional
Pots, not
would
diagram,
prelirninary
in
drarving
a
found necessary
have been needed in the input lines to compensate for
the change in scale factors.
Choosing the numerical values given in Table 3, a nunrerical solution can be obtained on the analog computer
once /i,* and,h2x are estinated. Thcse may be estimated
by calculating the final valucs of /r, and /2, at equilibrium,
noting that these values are larger than the initial conditions, and conscrvatively choosing maximums which are
about nvice the steady state values to account for the
possibility of oscillation before the final values are ar'
tained. The numerical values of the variables and the pot
settings for this numerical example are given in Table 4'
It should be noted that the analog diagram for the
foregoing example does not include physical constraints
which may be imposed on the actual situation. Consequently, for certain Pararneter values, the analog solu-
TABLE 3-Porometers
Used for the Exompte
Physical Parameter
PrirtRnre
Numerical Value
R2
2 sq. ft.
4 sq. ft.
B cu. ft.,/min.
6 cu. ft.,/min.
2 min.,/sq. ft.
I min.,/sq.ft.
h1(o)
lJ
hr(o)
h, at equilibrium
ftr* - 60 ft'
10ft.
30 ft.
ft, at equilibrium
h^x - 30f.t.
14 f.t.
ar
A2
Qr
Qz
R1
.**q
lt.
.
q €
IABIE lt-Seltings
Three precautions should be exercised when using this
rule.
Used for the Exomple Problern
Numerical Value
Component
h1/60
-h1/60
h2/30
-h2/30
0.2500
0.6667
0.2500
0.3333
0.5000
0.3750
0 . 12 5 0
0.2500
Amp 1
Amp 2
Amp 3
Amp 4
Pot I
Pot 2 (through gain of 0'1)
Pot 3
Pot 4
Pot 5 (through gain of 0.1)
Pot 6
Pot 7
Pot B
3. The rule assumesthat the problem has a stable solution. If the programer is doubtful of his problem's
stability, the stability is easily examined by using the
Routh criterion.
The maximum magnitudes for this example were estimated from a knowledge of the behavior of the physical
situation. In other cases,one must resort to other techniques for estimating maximum and minimum values.
One method is to obtain these values by a trial computer run. Another technique is that discussed by Jacksonl which is called the equal-coefficientrule. For differential equations of second order or higher, this rule may
frequently be used to estimate the maximum magnitude
of the dependent variable and of its derivatives. Given
an n-th order diflerential equation of the form
o*
dy
dN-1y
o,#+aor-f(t)
ä*at-t#+...+
(12)
where l(r) is the forcing function, the maximum magnitude of y,
dv
7l'"
' '
dNv
dtv
are estimated as follows:
Replace f (t) by its maximum magnitude, denoted by
A. The dependent variable and its derivatives are multiplied and divided by their maximum magnitudes to yield
r -o*,r f
d*Y1
| | z,Fl,_..+ ."rrt_lffil:a (13)
-|
"'L-*)
l.#l
dzhtt[
dts
'
r
*'
LlrR,
I
ArR,
t
*'
I
A2Rz)
dh,
dt
*'
,
AaRLA2R\
t;^n,+;o,[+.
+f"
o.:
(16)
Using the equal coefficient rule and evaluating using the
numerical values in Table 3:
[är]
* : 2 (qrRz*
l+1.
9
z
+
'
_AaR\A2
I
I
LA]R,
-' t
f a " n , 1 * : - - - q! -" + , 1 t =
l-:l
'
L dt, _J
A1R1A2
q r R z* { r R r ) , : 6 0
(17)
Q t _ + Q r
A1A2RL' A:A|R2 _q
/rq\
I
r-l
ArR, ', ArR,J
AlA2Rr
+' jA: L-A12 R. 8
75
|
\
^ v /
(19)
The magnitudes decrease as the order of the derivative
increases, a check of the initial conditions indicate that
they are less than the estimates, and the solution is
known to be stable. Hence the indicated values may be
used as estimates for the maximum magnitudes.
In many problems the computer operating time, in
addition to the output voltage from amplifiers, must be
scaled. If the physical problem takes hours or days for
completion, the analog solution must be speededup both
for convenience and precision. Likewise if the physical
problem is completed very rapidly, the analog solution
must be slowed down. In the next article of this series,
procedures which enable time scaling to be achieved will
be presented.
LITDRATURE CITED
*
1*
f a n y l d , - l,-ldznp' r=
,
REPRINTED
As an example of the equal-coefficient rule consider
the hydraulic transient problem and the determination
of. hf . Combining Equations tro)and trr), eliminating the
dependence of är on hr, the following equation is obtained.
'{ä".[lri,*ä;
Having numerical valuesfor A, ao, at, . . . , a7y,the
maximum magnitudes are estimated by assuming
" rl r , ) :
1. The estimated maximum magnitudes must either
continually decrease or increase as the derivatives are
considered in order. If this is not the case, the magnitudes should be estimated in another manner.
2. The rule assumeszero initial conditions. For a problem with non-zero initial conditions, the estimated
magnitudes should be compared with the initial conditions and the larger of the two in each case used
for scaling purposes.
tion will predict negative values of. ht andfot hz. In a
similar vein, values of. ht andf or h, may be obtained
which would represent an overflowing of the tanks. The
manner in which such constraints can be built into an
analog program will be discussedin a later article.
dNy
(15)
a oI y ) * : 2 A .
and
. . .
TEARNABOUT ANALOG COMPUTERS
ay1*
: ' , 1f ; - A ( r 4 )
): "'
)
FROM HYDROCARBON
PROCESSING
f;. tAi]"r
conputation, McGraw-Hill Book co.
Inc., New
Indedne Term: Analoes-9.Circuits-l0, Computations-4,Computers-9,Descriptiois-$ Electricity-l-0,Engineering-4,Progr:-ing-10, Simulation-4.
LearnAbout AnalogComPuters
PART3: TIME SCALING-Whenthe
actualtime tor a processchangeis
very short or very long,time scalingis
used to make it better suited to
analog computerspeed
Theodore W. Codmqn and Theodore G. Smith
University of Maryland, College Park, Md.
Trrenn ARE A LARGENUMBERof physical problems in
which the variable of interest (the dependent variable)
changes either very rapidly or very slowly. Reaction kinetics problems in which the concentrations of some of the
of seconds are often
components change in fractions
found in the hydrocarbon processing industry. The combustion of a hydrocarbon is an example of a problem in
which it may be very difficult to determine the reaction
rate or follow concentration with time by using a computer running at the same speed as the problem (real
lime) . For such a problem, voltage changes occurring in
the computer may be so rapid that some of the comPutational elements may not accurately follow the changes
because of the limited precision of analog components
for high frequency voltage fluctuations.
Problems which require a great deal of time to e><hibit the phenomena of interest . are also important to
practicing engineers. The study of the dynamics of large
icale process equipment falls in this category. Becauseof
the excessive amount of computer time which may be
l4
involved and the effect of noise and amplifier drift on
a solution's validity, the analog simulation of such processesin real time may not be satisfactory. The ability
to either speed up or slow down the solution of analog
simulation is one of the important characteristics of analog computation. The process of scaling the independent variable is called time scaling.
Recording The Solution. Quite often it is desirable to
record the results of an analog computation. The problem frequencies must then be adjusted by time scaling
so that the recording device can follow the results' There
are in general three types of recording devices commonly
used to record analog results; cathode-ray oscilloscopes,
galvanometer-type recorders, and servo-motor driven re