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DESfiGN 0? AN ANALOG
COMPUTER

Thesis for the Degree ef M. S.
MiCHESAN STATE COLLEGE
Nyia Eugane Sash
1:945

 

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thesis entitled

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Design of an Analog Computer

presented by ;_ (.

Nyle Eugene Bush

3 has been accepted towards fulfillment.
f of the requirements for

M.S. degree in EOE.

 

 

 

Major fl) r

WW3 P , ,
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Date May 223 1948

 

 

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DESIGN or AN ANALOG 00333333

by
Kyle Eugenepgueh

A maesxe
Submitted to the Graduate School of Michigan
State College of Agriculture and Applied
Science in Partial fulfilment of the
requirements for the degree of

MASTER or scramcn

Department of Electrical Engineering
1948

THESlS

 

ACKl-IOIEVGEQEITT

The author wishes to express his appreciation
to Dr. Strelzoff and other members of the Electrical
Engineering Staff who assisted in the development of

this TheSiSe

201340

TABLE OF COUTEHTS

Analogs, Integrators and Differentiators
Synthesis of Integrator and Differentiators
into Dynamic Systems

Design of an Operator Amplifier

Solution of a Differential Equation
Experimental Results

Bibliography

10
15
28
35

I. Analogs, Integrators and Differentiators

The ordinary differential equation may be interpreted
as a mathematical model representing a physical system
with lumped circuit parameters. This mathematical
system.acts as a common denominator expressing the
action of several wholly different types of natural
phenomena by one equation. Such systems are said to
be analogous,or analogs of each other.1

A common illustration of analogous systems is that

of,a torsion.pendu1um.subjected to a driving force,
and a series RLC circuit subjected to a voltage.

Jé +Ré-+K6=7%fl%wf
z " +4“; +13 :5. If» “’7‘
on this illustration it can be seen that

electric charge and moment of inertia, resistance
and friction, torsional stiffness and inverse capacitance,
are analogous. Other analogies between electrical and ,~
mechanical systems may be made in.like manner and a
table of these appears in the literature.2’3

The foregoing material suggests the possibility
of exploring phenomena on either an enlarged or reduced
scale by means of analogies from a different field. It is
the object of this paper to explore such possibilities

and to design an analog computer.

 

1. Murray F. Gardner and John L. Barnes, Transients in Li%a'
systems. John Wiley and Sons, Inc., New”YorE, 19453'p.

2. Louis A. Pipes Applied Mathematics for ineers and
physicists. Mcéraw Hill Taco c‘o'.","T9"'"4e.” p. i‘e“"""5 ""'"'

33 Gardner and Barnes, op. cit.’p. 6

If we add to our repertoire of analogs a unit called,
for lack of a better name, an Operator unit, and later
called Operational amplifier,4 we have an analog for the
mathematical operator 9 = dAt. This operator also performs
the Operation p'1 = dt.

Such an Operator unit can be formed electro-mechandtalw'
using a watt hour meter which performs the operation [P dt.5
‘where‘P is power in watts. If voltage, V, is constant then
the Operator becomes vy/i dt. Performance of the Operation
IE‘Ly dx is accomplished by associating x with time, T,
and causing current i to vary with time in the same way
I varies with y. One disadvantage of this system is the
slowness of Operation caused by inertia of the
mechanical elements. Another disadvantage is lack of
flexibility. These difficulties are overcome if electronic
methods are utilized.

A differentiating circuit, which forms the basis
for an electronic differentiator is shown in fig. 3.6

Assume that the output circuit has a small impedance
compared with R, (z.<<R)’and that V(T) is well regulated

or in other words supplied from.a low impedance source.

3?;<:<: /?M+'-L-¥

Jouc

 

4. J. R. Ragazzini, R. H. Randall, F.A. Russell, Analysis
9;,Problems in_gynamics p1 Electronic Circuits, Proc.
I.R.E., vol. 35, p. 444, May, 1947.

5. Varney, R. N., "An all Electric Integrator for Solving
Differential Equations", Rev. Sci. Inst., vol. 13, 1942.
p. 115.

6- Otto H. Schnidt and W- B Tollis, "Electronic Differentiation"
Rev. Sci. Inst. vol. 13, 1942. p. 117

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F / 6. 3 - Ac 77 vs DIFFERENT/A TING N£7WORK

Then (I) 1; = C ‘flé/c/f
where Va 3: C(V’E/
then .Z'c :: C o/{V—‘E/

. a/f
f :: RC a/(V‘f
/f

If V775
(2) EU) : RC é/VfflA/f:/?C/b V/f)

where /b : p//a/f

If V775, this means that
41:77
M. R . .R
lR + (/wc. :: ’00!

(30 £44, :: ..<DL/’/?<:
which gives the frequency limit for accurate differentiation.

The frequency limit may be increased by decreasing the
RC product at the expense of decreased voltage output.
Another means of increasing both frequency response
and output level is by use of electronic amplifiers.
It is necessary that, for accuracy in differentiation at
low frequencies, the frequency range of such an amplifier
extend down to zero frequency. This can be accomplished
by means of a direct coupled amplifier. Problems peculiar
to the d. c. amplifier will be considered later.
If the box marked A in fig. 3 represents a d. 0. amp-
lifier and no d. c. path frmm grid to ground exists
62=Aev

Continuing the analysis on a node basis:7

 

7. H. W. Bode Network analysis and Feedback Am lifier
Design, D.’Van Nostrand Co., New YorE, 1945. p. l3.

(4) E W) + melee +WH€z = f
6f: «fa/A
l= nob/e,

WW mobile/A meme-m.)
/€;a Iva‘fcl; 3,:" £2£k21, I
M e //+(P/[I+ l/A[I+K'(Pl/V;(p)}]
(9’) 52-: [If (fl/I?(1zl]€a f; A>>I

Y represents admittances of the Laplacian transform

 

type.
This is a very general form and the admittances
could consist of resistors, capacitors, and indictanees

or networks formed from these elements. In the case of

a differentiator: I’, (f! :7 Cf
[fin/v) : 6; = l/R,
Thus 92 : Rf C p 9'

This equation is similar in form to eq. (1). However
the R in eq. (1) is replaced by the input resistance of
the amplifier,8 more R; z c" /('

Then fickle.- + flaws; = (

6" -'-"- et/A
”(P/5“ 1" Knob/I4 6’; z: I.
Rf 3' I Rf

 

f’WP/L—HAJ I +A
If A is large then
R(' 3 Rf /A

The frequency cut Off then becomes

2“ 2: .o/A /:77R,».c

 

8. D. J. Mynall, "Electrical Analogue Computing", Electronic
Engineering l_9_ p. 178, 1947

h:

.\

 

The cut off frequency is thus seen to be directly
proportional to the amplifier gain while the Output
voltage is directly preportional to the feedback
resistance Rf. For a given gain, A, the output voltage
is increased at the expense of decreased cut off frequency.
There is also a lower limit on Rt which determines f0 maximum
imposed by conditions for stability in feed back
anplifiers.9 This will be treated in the study of «1.0.
amplifiers. The basis for the electronic differentiator
analogous to the Operator p is thus formed..Any number
of such Operators may be connected in cascade to form.the
operation pn.

Equation 5 shows that the electronic Operator may
also be used as an integrator.

Let 50:) r: (/9

and If (,a/ = //R

then 8.. = { //RC/b]€,
The basic integrating circuit is shown in.gig. 4

Here the output voltage is taken across c.
I: Ke/R scl‘et/R
631.: .2747 za-el/RC/a
C” 7782..

6.. a: Mac/ad)“

For sufficient accuracy of integration is is
assumedé: >> etmeans that e, 2.. loo eghis, together with
the R0 constant fixes a lower frequency belOW'Wthh the

accuracy of integration fails.

 

9, Bode, op. cit., chap 8

 

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¢
-

F/é. 6 Jfl/‘ffl/Nc; AMPL/F/f/i’

This lower limit is determined frmm
(R + (/wc 2" /oo ('/wc

64/2 :3 577h/CA?C:

Again, this performance may be hmproved by use of
a direct coupled amplifier.

Using the same analysis made for the electronic diff-
erentiator ff [/0/ .2: W5}

Lu l{.(I'I-A)
The capacitor in t 6 basic circuit is this:made longer

by a factor of A.and the frequency cut off becomes for

the electronic integrator, a/c :- 77/RCf/4

Although it is theoretically impossible to extend
W0 to absolute zero, this ideal condition is approached
with a large amplifier gain. A high amplifier gain also
affords a reasonable output level. Thus the electronic
,integrator which performs the Operationzp'l, is formed.
Its frequency gain characteristic decreases 20 loglo(2)
or 6 DB per octave.

Synthesis of electronic differentiators and integrators
requires a summing operation.which can be performed by
use of a d.c. amplifier connected as in fig. 6.10

The coefficients 096’, I are scale change

coefficients determined by the raticjfgfl/g etc.
’3‘ ’3-

 

10. D. J.‘Mynall, Op. cit., p. 180, June 1947

The use of admittance notation implies that the use
Of scale changing components is not limited to
resistors. If capacitors are used for Y2, Y5, etc. , the
Operation of differentiation is performed at the same
time a scale change is accomplished. Another advantage
of using an amplifier in this:manner is that it acts 7
as an isolator. The output circuit of the feedback
amplifier does not reach back on the input circuit.

An economy of amplifiers is effected by simultaneous
performance of several functions by a single amplifiers,
in synthesizing a dynamic systan.

Sign changing of the dependent variable function
may be needed frequently. This, of course, means there
should be a 180° phase shift between the input and output
of the electronic Operator and is accomplished byusing

an odd number of stages in the electronic operator.

10

II. Synthesis of Integrators and Differentiators

Into Dynamic Systems.

The methods of synthesis considered here are the fin lined1

technique and the node method for electronic differentiatoral
The node method is synonomous with the method known as

"current Junetion"12technique but is preferred because of

brevity. The node method is based on Kirchoff's current

law and therefore corresponds tc>'current Junction" techniqueiz
Ineline electronic differentiator synthesisl4 amounts to

a summation of functions of the independent variable.

The independent variable is the input voltage to a system

of electronic operators each of which compute a function of

the input and then are summed in a summing amplifier. lhe

output voltage is the desired dependent variable. Such

a system As demonstrated by the equation

3 (T) is the independent variable voltage input and
x.(T) the dependent variable output. The constant coefficients
can_be varied to give a desired.x(ll for a given y(l)

 

ll. Bagazzini, Randall, Russell, op. cit., p. 446
12. Ibide. P0 447

13. Gardner and Barnes, op. cit., p. 58

14. Ragazzini, Randall, Russel, op. cit., p. 447

.LJ.

Fig. 7 represents one possible combination of
amplifiers which will solve for 1(T).

The node method depends on summing a number of
currents at a node in the system. ‘The simplest system

of this type corresponds to a single electronic differentiator
solving the equation/ 81. = K/a 6,

This analysis may be extended so that

”We. = WM e ’ + I? 0-): e” ’2 Wu ‘3
Taking the example for "in line” technique and

applying the node method, let
erlbeIé’,
By applying the proper scale changes at the

summing amplifier A, and applying a preper sign to the
input function the equation

lq/bfigf/1t£%/>11V' f-Cafiar'1~17f"-)(:=C7 ,«rjcv/Vflf

kahkteb N\<\ V - >\\ I x. 6 \K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15

 

th§0€§Qb WQO>\ I Q GNK

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

£76. 9 - aAr/c o. c. AMPA/F/ER

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FIG. /0 -— fowl/AA E/Vf (mam 7' FOR
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15

III. Design of an Operator Amplifier

An amplifier which is to serve as an Operator must
meet the following requirements:

(1) Uniform frequency response over operator range

(2) Frequency response down to O c.p.s.

(5) Stability

(4) Freedom from drift

(5) High gain

(6) Low output inpedance

The requirement of zero frequency response compels the
use of a d.c. amplifierf’A.potentiometer type of coupling
was selected. Fig. 9 shows the basic circuiti4ihe equivalen:

circuit for quiescent conditions is shown in fig. 10.

(:79 :3 lC/qub (9%: =:‘//6:Lz
C" = I/R. 6: : HR:

The node equations are:

(7) /2;*e +1971F6Ez)£i.-é;1153 :=’€;115}

(;Z, -‘;¥LAE9 ‘*‘(2;a.1FC§31L£i3 ::"€33 £52
am I)

(7;) 2.1: GB; 15/156,46J]

.1

 

fafiouwfazj

(za) 6,, :: 61, 3;?—
" .9

 

13. E. L. Ginzton, "D. G. Amplifier Techniques", Electronics,
M3:., 1944, p. 98

 

l4. Artzt, Mhurice, "Survey of D. C. Ampfi, Electronics,
Ange 1945, Fe 112

 

16-

RI:

 

 

 

F/6.//-— AMPL/F/ff? EQU/VAZE/W’ ClRCU/T

R2
Pl 9 62

 

5’ R3

 

F/é - IZ. COUPL/IVG CIRCUIT

 

3+

9|

 

a. con/PENIA T" R

 

 

 

 

 

a}

F/G. l3

1?

Solving (2a) for G3 and substituting the value of
G2 from eq. (13).

(3) 5,_ 51/19:.- -E.. )_ 613.5, E:(/+6,jél]

Solving eq. (1)! for E6,

(;7l(£;-4£%L) :: ’féafi»1L¢;1n)15c "C;1453

 

a, = _(4..c6.)£. 4.5 : §,£.+6z(5-£,)
“£0 E,‘£°

Substituting the value of 62 from eq. (2a).

 

(4) 6/ :67 é'£2+éfi/—éLA—]
5,—5

Although the preceding equations were more conveniently
derived be means of a node representation with equations in
terms of conductances, a more useful form is in terms of

resistances. Inversion of eqs. la to 5 will give this.

Thus,

(2;) /?é_-‘/?I“—4éELZAEéL-—-3 /? ,AE§;:J§:_
E’-£;(/*fihflfikJ IES‘N£;(/VLAmflfi})

7 R =R ["5“-
( ‘) .3 I,” OflhfIRZ/OSbu, :1

lb

(g) R: 7" R 'fi 5-5 ‘1
3 £3 “52+R3/Rffo

 

(7) R2 1': R3 5‘5?-
11-2-5
Equations 6 through 9 are the design equations where

either R1 and all supply voltages are knwon or R3 and all
supply voltages are known. E0 is the potential at the
chosen operating point and 10 the quiescent plate current at;
this point andfizée , RF is not the same as the
dynamic plate resistance.

The equation for gain may be obtained in much the same
:nanner as for R.—C. coupled amplifiers.

Fig. 11 shows the equivalent circuit. includes the
gr id coupling network as well as shunt capacitance.

E0 = output voltage

Eo -‘-’ If;

£1: = IRR

I=_flfj ~4IR5
r, + Rx +171.

JEN/4R3 /(I;. +RK+Z..)] =xq59/KII;+RK+ i‘.)

A" -"-‘-' 5/137 :: "f ”it

This represents the expression for the complex gain.
At low and middle frequencies 21 is simply R]. , which
c3C>Iisists of the coupling networks as shown in fig. 11. The
acillivalent circuit for R1 is shuwn in fig. 12.

From fig. 12

R. _-_- LRL (R..+R,_Z_

Substituting for R1.

19
fl Rlfifia +Rll’

’Vb '+'I?KF(kWW‘/) iL/RDIVQHSfJQ32::
Rp+RL +R3

:: ¢__ /¢”’Aaleb ~—*i_
(R: +Rt+R3){r/n +Rk(x4+/)]+R. lam.)
Output voltage is derived across R3, therefore the gain

becomes

A=K Re
. /az'*1?3

 

Z __ MRI R3 __
V (R, +R.+R3)[I;, +Rxm+ll} +Ra(Rz+Rz)

Ifgt+g3,R' I R‘may be neglected and

,4 :: M41 # .12....
1;. +RKIAI+U+R0 55+“?
This result confirms that given in the literature,15

 

although the exact equation is the one which was used in
computing the gain of the experimental amplifier.

Steady state drift is one disadvantage of direct
coupled amplifiers. The main causes of this drigt are

variations in cathode temperature, and variations in plate
voltage supply. Any variation of this type is amplified by
each succeeding stage so that it is quite apparent a

well regulated plate power supply is needed.

 

15. Artzt, op. cit., p. 112

The drift due to cathode temperature variation or
filament voltage changes may be considerably reduced

by use of a Iiller compensation circuit in at least the firs:

16

stage. This circuit works on the principle that two

similar triodes may be so connected that changes in one

conpensate change in the other. The circuit is shown in
fiSe 13e

The balancing action occurs when the voltage drop across
R1, due to drift current causes a change in grid bias on
t
he compensator tube, T2 . The compensating current’-41'R']m’
causes a voltage dr0p of ‘(4 Ifijlptacross R2 . If the
disturbing voltage, 41-3., is balanced by the drop across

R :_due to change in current,

AIR,- (AIR,j.../R.. == 0
and R1: l/ju,

Voltage stabilization with the type of direct coupling
euployed is extremely important. Any variations in voltage,
or drift from the steady stage, is amplified from the
input of the second stage to the output of the final stage.
Batteries would provide the most stable voltage supply..A
voltage regulator was choyen as the most practical as

batteries would be costly to maintain.

 

16. Ginzton, op. Cite, Fe 98

21

In general, Voltage stabilizer circuits may be
classified as amplification bridge circuigs, mutual
conductance bridge circuits, degenerative amplifier
circuits, or combinations of these.17 For power supply
stabilization here, a degenerative amplifier with an
additional pentode amplifier stage was chosen. The
fundamental degenerative amplifier circuit is shown in

£13. 14.18

The equation for effectiveness of regulation can be
in terms of the differential plate voltage,aJZiladd the
differential plate current 4/4 .19

d5. -.-. ail-.015 +25%. cap—alt“. 4.0/12
27£ii 512% /44 jinn

AE‘ z 45¢. "" A5.
6705; 1="H/(4515} .IVV98713 /<E: .flfi,//Qq;,p/?,_

"/(13153 :: 1315i/jén. - (ZI£?"15£;L)/(:;1

43,6; :: fAQHET’I/kaci 'fl'13a15L_z//V'i}h
/+//K/4 r

.Jldljvflflfarfheib AQJEB :: 1315;,/¢?L

 

17. I. V. Hunt, R. W. Hickman, "Electronic Voltage Stabilizers",
Rev. Sci. Inst. 19, 1939, p. 9.

18. W. L» Emory, Ultra High Erequency Radio Engineering,
' the Mac Millan 00., 1944, p. 15.

19. Ibid.’ p. 16

 

22

AEo/AE? 1' Kym R1. +/
Kz/ij Re.
/9( ‘/?;Z»y/€z J>}’/'

13~£;’/423£5’ :: /;/gt>¢r

Let 8 33—2.. [or large values of I, J“: K”

The circuit used for the two regulators is shown
in fig. 1’.

The gain for the éfflamplifier is A =fmg‘ :4/3
Only a third of the output voltage variation appears at
the grid of the 6.5277 so that K: 4/3/3 = [38

J‘ : 138x44. .2580

Negative feedback may be used.for many purposes in
feedback amplifier design. One of the most important uses is
in decreasing the sensitivity of the amplifier gain to
changes in the tube characteristics. Feedback is used here,
however, to improve the frequency response and output of an
R0 differentiator or integrator. This type of feedback is
known as shunt or voltage feedback. The other most common type
is series of current feedbackfin

It is necessary that there be no feedback voltage
component in phase with the input of the anplifier or

oscillations will occur.

 

20. Bode, op. cit., Chap. 3

q f“ .71
L {'3

|l
£2.

AE.
R; L

(/1 ran
IVE Rféa
-— géiNtA’Ar

F l6. M 0

Grid of flexf ffajc

 

24

The middle frequency phase shift in a vacuum tube amplifier
is 180° and would be the same at low frequencies for a
ddrect coupled amplifier. However, at high frequencies
interelectrode and wiring capacitance causes a falling off
of the gain characteristic with an accompanying phase shift
which approaches -90°. The same effect prevails in an RC
coupled amplifier at high frequencies. The quuist criterion?1
for stability may be applied to determine whether the
amplifier will escillate. However, the result which may

by obtained from a quuist diagram for three identical
stages is given in the literature .22 es A648

where A is the amplifier gain without feedback and 8 is the
feedback factor. It is possible to use larger values of
feedback provided the rate at which gain falls off at the
high frequencies is not too rapid.23 Local feedback.may be
applied to the individual stages to accomplish.this purpose.
Interstage design also is of help and in this connection,
two terminal networks are generally most useful since a

four terminal network.produces excessive phase shift under

24

most conditions °£ “3" One type of two terminal interstage

 

21. BOdC, op. Oi‘e, Pe 151e

22. Terman, F. 3., Radio Eggineer's Handbook, MbGrew Hill
Book.CO., 1943, p. 398

23. BOdC, Op. Ci‘e, i‘ 291

24. Ibid., p. 403

25

which might be used to shape the gain characteristic at
high frequencies is shown in fig. 15.
Fig. 1! is the circuit diagram of the complete amplifier.

 

26

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m [\ llll'll
NHHW
/
o w.
o o
W W U.§\.M.W
M
m < < Iml
chm: ohm... \
{MW \ \«NKKAH.

es s cow.»

2?

IV. Solution of a Differential Equation

A simple type of differential equation to which the
technique of operator amplifiers might be applied would be

of the form d dx + fix :: F77")
a’f

.As explained at the beginning a differential equation
may represent the mathematical model of several widely
different physical systems. The electrical analog of the
chosen equation is that of an.inductance and resistance in
series with an applied driving force or voltage, 1? (1-),
as shown in fig. 18.

A d/Va/r +29; =F(r/
The mechanical analog may be of two forms, translational and
rotatinnal. The translational model is of the form?5
M a’V/a/r {wt/V = F/f/
where N: mass
,4/’: coefficient of friction or mechanical
rectilineal resistance
The rotational model is of the form
J's/6%” +1449 = F/f’w
where I 2 moment of inertia

/60b is the coefficient of friction or mechanical

Venture/resistance. F(f)ris the applied torque function.

 

25. H. F. Olson, Elgamical Analogies, D. Van Nostrand 00.. luck
1945, Fe 1080

 

 

29

Taking the mechanical translational model for the
problem let it be required to find the velocity when a certain
force function is applied to the mass M. A Whole series of
solutions may be obtained by variation of the parameters
H and,“ . Although this is a simple case which can be easily
solved by mathematical methods the principle may be applied
to more complicated systems.

The general form of solution using operational

°81°nlu826 is MV’H‘) +M ”fl = F/fl
.1 [ MK’M +fl l’ffI] :— ,[ NH
M; H!) -— f1 [70/ +M Hr} =F(.r)

assuming y(o) :- a
MJ‘V +/4V = Fr!)

-, Fl!) _
V: f [7‘11444

The simplest form forF(f)is obtained whenFIfl=l from which

 

FIJI: ’fi'by Laplacian transformation. The inverse Laplace

transform then becomes f'
V: WH-F’W")

A somewhat more complicated function results if a sinusoidal

function is substituted forF/i‘l.
Then /[F(H} : («J/1‘19“)"

—l w
V : __ __
J [(I‘+w‘-}(M.r+.al }

26. Gardner and Barnes, op. cit., p. 127

 

30

Solution of this expression yields

.. -nw .
V '- [Mk ”"7" conur +__.."’ 1»? an“
,a fsmut/‘1“ w
If the force function is Foosut the result can be shown to

 

be
-1!
V: F00 _ Mf/nwf-A/w/(Olwf"’c 4'17]
M‘+w"M"
In general, however, the force function may have any
initial phase angle and the value of the transient portion

of the solution will depend on the initial phase angle?7

The complete solution is V: F[f/h/wrfa-dl-I/‘nlg-f/F’A/"fl .
Via '- 4- («NW ‘-

 

wheree = initial phase angle
¢ : fan"{ 60%}
F13. 19 shows the arrangement of amplifiers for solution of
this problem.
The scale factors may be chosen by relating the values
of feedback.and input admittances to the system of units
in.which the problem is to be solved.

For example:

444:: J7awmg/Creuc.
Then
M7MI. = K Inlet
M finest/lee. = M M40!
1 (T) volts :- 1! dynes

The coefficients K, m, and n, have dimensions necesesyy

 

27, H. H. Skilling, Transient Electric Currents, Macraw 3111
Book 00., 1937, p. 189

bl

 

FH) R

Fla /6’

 

 

 

 

 

 

 

 

 

FIG. /9

 

32

to make them consistent with the quantities they represent.
for Ag, 6¢'/5+ 3 ll“

1‘3I'./42V, JP.:' 64% I 463;: ka' /?€7;: /

The output conductance for A3, 6-: ll, will have an actual
value in mhos of M times the input conductance of A1.

For the sake of this problem is has been assumed
that the initial conditions are such that the dependent
variable and its derivatives are equal to zero at time
T g 0. Initial conditions, for values greater than zero,
could be applied by application of a constant voltage
at the input of the desired amplifier. The complete
sedition of the problem then.expends from time T‘: O
‘to T :.~D

Several means may be used to indicate the desired
solutions. If only the steady state solution is of interest,
a high sensitivity vacuum tube voltmeter will suffice.
.A cathode ray oscilloscope may be used to study the wave
shape of the solution. unless some means, such as taking
(a photograph,zais available, however, the transient portion
of the solution will not appear. If the frequency is not
too high (usually a few thousand cycles)29, an.oscillograph

may be used to record the solution.

 

so. Ibid., p. 342
29. Ibid., p. 342

,‘Jt

 

33

V. Experimental Results

A curve of E3 vs. ES is shown in fig. 20. This was
obtained by varying the grid bias on the second stage
of the amplifier and reading the corresponding output on
a vacuum tube voltmeter. This curve covers the operating
range of the amplifier and is linear. The gain at low
frequencies is about 37. This provides an amplification of
over 1000 over the last two stages, which is the amount
by which variations in the power supply are amplified.
These voltage variations amounted to as much as 10 volts
at the output of the amplifier. This emphasizes the
severity of requirements for voltage stabilization.

Much difficulty with 60 cycle hum was encountered.
Grounding the filament circuit of the amplifier and using
a battery supply for the filament reduced the hum consid+
erably. The rest of the 60 cycle input seemed to be picked up
on the leads to the amplifier, the laboratory being
located in strong fields from nearby e. c. machinery.
With the saplifier gain reduced to about 1000, the hum
output was about 3 volts which means there was a hum input df
approximately .003 volts. With much higher gain this is
enough to overload the last amplifier stage. Although

input voltage of about .04 volts were used the hum voltage

was still noticeable when viewed on the oscilloscope.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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35

The gain of the last amplifier stage was reduced to
nearly unit? by use of voltage feedback over the last
stage. The sign changing property of the amplifier was
thus retained while at the same time the output impedance
was greatly reduced.

Attempts at differentiation were preceded by determining
the effects of 01 and Br shewn in fig. 2E.

With the set-up shown in fig. 21 and a frequency
input of 1000 G. p. s.‘A minimum value of .5 ufd. was
Obtained for 01 without oscillation of the amplifier
occuning. Fig. 22 shows the variation of output voltage
and gain with change of Hi. The curve is relatively linear a
only for fairly low values of Rf. This result is predictable

from the equation

e; = , 1
{r {f ”/0986 [I +M4[/+/7'/r//IMI]

A larger linear range could be Obtained with larger values

 

of A and smaller values of Y1(s).
Differentiation accuracy of the amplifier may be
determined by taking the gain characteristic as a
differentiator. It is more easily determined whether
the amplifier is functioning as a differentiator by use
of a square wave input. Fig. 25 shows the perfectly differeniated
square wave as it appears on the oscilloscope.
With a square wave input of 1000 c. p. s. and Br out of the

circuit, the wave shape appeared as in fig. 24.

 

 

 

 

F/é. 2/

 

 

F/é. 23

 

 

 

Fla. 2 4

FIG. 25'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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7 p . I 7 . . - 7 . 7 7 . .r . . 7 7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

with Rf connected the wave appeared as in fig. 25.
As if Was decreased the wave became narrower finally appearing
as in fig. 26.
The oscillosc0pe horizontal amplifier gain was greatly
increased to obtain the spread of the pulses. The small
amplitude pulses and distortion of the main pulses are due
to the 60 cycle component.

with a sinusoidal input voltage or .04 volts an output
or 2.75 volts was obtained at 1000 c. P. s. The inut
Caycuiu7h78 was 1 ufd. and if has 74000 ohms. Ihe output

voltage should be eg

‘3 74.900 x/o"x.‘277x loco 77/,“ ,6”,
A. difference of 0", volts is noted which
is not negligible. This indicates there may have been a
lower frequency voltage superimposed on the input voltage.
If the input voltage had been composed of two parts, namely,
.03? volts at 1000 c. p. 8; input and .005 volts at 60
c. p. ss’fhe cupput components would be 1.72 volts and
.8 volts respectively.
Integration of a square wave produces a triangular
wave. Attempts to do this failed because of oscillations
in the amplifier when the feedback admittance was a
capacitor.
One feature of the differentiator is that the output
is directly coupled to the input thru the feedback resistor.
This practically eliminates drift due to variations in

supply voltage.

'1

a

 

 

 

I

Fl‘ 26

3?

The output of the electronic integrator is net directly
coupled to the input for direct currents so that the
voltage supply stabilization must be much better than
for the electronic differentiator. For this reason,

it is advantageous to use diiferentiators in the syn-
thesis of dynamic systems.

Combination of the operator amplifiers into the
dynamic system on page 28 was unsuccessful, partly
because of the high a. 6. hum level, and partly because
of instability. Thorough shielding of all leads to the
amplifiers, and of the inbut circuit, should eliminate
hum. Oscillations, or instability, may be elininated
by redesigning the interstage nettorks, and feedback

circuits.

BIBLIOGRAPHY

Books

Bode, Hendrik W., Network Anal sis and Feedback Amplifier
design, D. Van Nostran o., New“York, lglfi.

Emery W.L., Ultra-Hi hpFre uenc Radio Engineering, the
NaoMillan 50., New Yorfi I944.

 

Gardner, Murray I; and Barnes, John 1., Transients in Linear
systems, John Wiley and Sons, New Ybrk, 1942,'731. I.

Olson, Harry 3., Dynamical Analogies, D. Yan.NOstrand Co.,
New York, 1943,

Pipes, Louis A. A plied mathematics for En ineers and
Physicists: Writ, e+F—Iw or, 91's:

Skilling, Hugh Hildreth, Transient Electric Currents, MbGrssk
Hill Book Co., New York, 1937

Terman, E. E., Radio Engineers Handbook, McGrawhHill book
00., New York, 1943

Magazine Articles

Artzt, Maurice, 'D.C. Amplifier Techniques", Electronics,
Vol. 18, p. aaz,(Nagus;,1945).

Ginzton, E. L., "Survey of D. C. Amplifiers," Electronics,
Vol. 17, p. 98, (March, 1944).

Hunt, F. V. and Hickman, R. N., "Electronic voltage
Stabilizers", Review 3; Scientific Instruments, Vol. 10,
P. 9, (F‘b. 1959).

Mynall, D. J., "Electrical Analogue Computing" Electronic
Engineering, Vol. 19, p. 173, (June, 1947).

Ragazxini, J. 3., Randall, R.H., Russell, E. A., "Analysisibt
Problems in Dynamics by Electronic Circuits", Institute
of Radio Engineers Proceedings, Vol. 36, p. 444, (May, 1947);

 

Schmidt, Otto I}, and Tolles, W. E., "Electronic Differention"
Reviewag£_Scientific Instruments, Vol. 13, p. 117, 1942.

Varney, R. N., "An all Electric Integrator for Solving
Differential Equations", Review 2: Scientific
Instruments, Vol. 13, p. 115, 1942.

Aug 29 '50

 

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