Electronic analog multipliers.: NA - Calhoun: The NPS

Electronic analog multipliers.: NA - Calhoun: The NPS
Calhoun: The NPS Institutional Archive
DSpace Repository
Theses and Dissertations
Thesis and Dissertation Collection
1953
Electronic analog multipliers.
Burke, James Adolph.
Monterey, California: U.S. Naval Postgraduate School
http://hdl.handle.net/10945/14139
Downloaded from NPS Archive: Calhoun
Library
U. S. Naval Postgraduate School
Monterey, California
^53
ELECTRONIC ANALOG MULTIPLIERS
J, A,
Burke
ELECTRONIC ANALOG MULTIPLIERS
by
James Adolph Burke
u
Lieutenant, United States Navy
Submitted in partial fulfillment
of the requirements
for the degree of
MASTER OF SCIENCE
IN
ENGINEERING ELECTRONICS
United States Naval Postgraduate School
Monterey, California
1953
This work is accepted as fulfilling
the thesis requirements for the degree of
MASTER OF SCIENCE
IN
ENGINEERING ELECTRONICS
from the
United States Naval Postgraduate School
PREFACE
The purpose of this paper will be to give the reader
a general description of possible methods of multiplication
by electrical analog systems and their possible advantages
and failings,
A report is given on the circuitry and per-
formance of one type of multiplier, which, in the opinion
of the author, best satisfies the requirements for a univer-
sal analog multiplier.
As part of the curriculum in Engi-
neering Electronics at the United States Naval Postgraduate
School, the author spent ten weeks at Gilfillan Brothers,
Inc., Los Angeles, California, working as an engineer in
the computer group.
It was here that the experimental work
contained in this paper was performed.
ijl
TABLE OF CONTENTS
Page
CERTIFICATE OF APPROVAL
i
PREFACE
ii
TABLE OF CONTENTS
iii
LIST OF ILLUSTRATIONS
CHAPTER
I
CHAPTER II
iv
INTRODUCTION
1
POSSIBLE METHODS OF ELECTRICAL ANALOG
MULTIPLICATION
1.
2.
3.
4.
5.
6.
7.
&.
Mathematical principles
Analog multiplier requirements
Electro-mechanical methods
Automatic gain control and modulation
systems
Non linear systems
Cathode ray tube systems
Time division systems
Choice of system for construction
5
7
£
14
1&
2£
31
35
CHAPTER III A TIME DIVISION MULTIPLIER
1.
2.
3.
4.
CHAPTER IV
Precision switch
Circuit design
Output filter
Tests and performance
CONCLUSION
37
40
42
49
51
BIBLIOGRAPHY
52
in
LIST OF ILLUSTRATIONS
Page
Figure
1.
Strain gage bridge multiplier
11
2.
A two dynamometer multiplier
13
3.
Automatic gain control multiplier
15
4.
Step multiplier
17
5.
Photoformer block diagram
21
6.
Quarter square multiplier using a
24
segmented parabolic characteristic
7.
Exponential multiplier
26
£.
Square beam cathode ray tube multiplier
30
9.
Time division multiplier block diagram
32
10.
Precision current switch
3#
11.
Precision potential switch
3&
12.
Time division multiplier with potential switch
41
13.
Time division multiplier schematic
43
14.
Filter band pass requirements
45
15.
Output filter network
45
IV
I
INTRODUCTION
Automatic computers find their use in automatic control
systems and in the solution of scientific and engineering
problems where the mathematical relations may be intricate
or involved or the quantity of data to be handled is too
large to be handled by available skilled personnel.
There
are two separate fundamental approaches to the problem of
automatic computing system instrumentation, usually depending
on the complexity commensurate with the accuracy desired.
Digital computers consist of counters registering and
adding in discreet steps, together with a storage and pro-
graming system in which counting pulses are transmitted between counters in the manner prescribed by the problem to
be solved.
Digital computers perform most mathematical
operations by combinations of additions, for example, multi-
plication is performed by repetitive additions, integration
is by summation and converging series replace non-linear
functions.
Since these indirect computations must usually
be carried to more places than are required in the final
result, the computing elements must have the capacity for
dealing with large numbers and thus contribute to the size
of the installation.
In automatic control systems, shaft
position or electrical voltage information must be trans-
formed to digital form before it is operated upon, then
reconverted to a usable voltage for use in the control
system.
In analog computers the numerical values representing
variables in the equation to be solved are converted to
machine variables upon which the computing operations are
performed.
The machine variables may take the form of
electrical voltages or shaft position, depending upon
whether the system is in the form of a mechanical, electro-
mechanical or strictly electrical system.
In automatic
control systems the inputs are more than likely already in
the form of electrical voltages.
Generally, more accurate
results can be obtained from mechanical or electro-mechanical systems, but since the speed of computation in any
mechanical system is limited by the inertia of its moving
parts, this discussion will be limited to electrical systems.
It is desirable to make a computing machine as simple
as possible.
Accordingly, it is customary to perform more
complicated mathematical operations on the computor voltages
through combinations of a limited number of simple operations
performed by basic computing elements.
The necessary basic
operations are as follows:
1)
multiply a machine variable by a constant
coefficient
2)
take the sum or difference of two machine
variables
3)
generate the product of two machine variables
4)
generate functions of a machine variable
5)
generate the time integral or time derivative
of a machine variable.
High gain direct coupled amplifiers with negative feed back
make it possible to add, substract, multiply by a constant,
integrate and differentiate with high accuracy and speed.
But, the operations required in the generation of complicated
functions and multiplication of variables are more difficult
to perform, especially if both high accuracy and speed are
desired.
Digital computers are inherently capable of much greater
accuracies than analog computers, but these are obtained only
at the cost of added instrumentation complexities; therefore,
where the accuracy of the input data is limited and where
great precision of computer operation is thus not required,
the advisability of using an analog computer is indicated.
As stated above, there are two general classes of multi-
plication used in analog computers.
The first is multipli-
cation, in which one of the variables is constant throughout
a given problem.
This type does not present a very difficult
problem, as there are many devices that can be hand set to
obtain this result.
An example is the high gain D.C. ampli-
fier with controlled feedback.
The second type is multipli-
cation in which both of the quantities may vary in the
solution of a given problem.
this report is concerned.
This is the type with which
This type of multiplier is needed
in the solution of differential equations with variable co-
efficients.
This type of equation is the more difficult to
solve by classical mathematics, so it is important that not
only automatic control systems but practical differential
analyzers should include multipliers of this type.
Electronic analog computers call for multipliers with
various requirements, depending upon the special applications.
It would be desirable to have a universal multiplier which
satisfies the most rigid of all the varied requirements.
It will be the aim of this paper to discuss examples
of possible methods of electrical analog multiplication,
while keeping in mind the requirements of a universal analog
multiplier.
After determining the method which best satis-
fies these requirements, one circuit which was built and
tested by the author will be presented in detail.
4
II
POSSIBLE METHODS OF ELECTRICAL
ANALOG MULTIPLICATION
1.
Mathematical principles used in analog multiplication.
There are three important mathematical identities that
have found use in analog multiplication.
These identities
permit the operations of addition and subtraction to be used
instead of the more complicated operation of direct multiplication.
This is a very useful substitution since the
operations of addition and subtraction are easily performed
in analog computers compared to direct multiplication.
1)
Logarithms
Logab m Loga f Logb
This method is inherently restricted to positive numbers and
requires the use of non-linear logarithmic elements.
2)
Quarter square
ab - 1/4 [(a*b) 2 - (a-b) 2 ]
This method requires the use of a non-linear square law element
which will accept both positive and negative values.
3)
Integration by parts
uv -y^udv f^/vda
This method is of use only in mechanical computers, since all
integration in an electronic system must be performed with
respect to time.
The above expression requires integration
5,
with respect to two other machine variables.
Integration
with respect to a machine variable in an analog computer
can be accomplished by using the identity
yxdy
s
J(x dy/dt) dt
The procedure involves differentiation, multiplication, and
then a final integration with respect to time.
As can be
seen, multiplication of two variables is required to perform
the integration.
In addition to the mathematical identities mentioned
above, several geometrical theorems are applicable for analog
multiplication.
1)
The area of a rectangle
Area - a x b
Where a and b are the lengths of two sides of a
rectangle.
In an electronic system one of the
variables must be converted to time; this auto-
matically restricts the multiplier to two quadrant
operation.
An averaging technique is then used to
obtain the desired answer.
2)
Altitude of a Right Triangle
Altitude 5 (slope) x (base length)
This method is very similar to the rectangular area
method in that one machine variable must be converted to a time for base length.
Instead of
averaging to obtain the result, an accurate peak
detector is required.
2.
Analog multiplier requirements.
The discussion of existing methods of electrical analog
multiplication which follows will be based on:
1)
Complexity of circuitry involved.
As is true in
all electronic circuitry, increased complexity
results in increased initial cost and usually greater
maintenance problems
2)
Rapidity of solution.— In many differential analyzer
applications the speed of solution is of minor importance, but in automatic control systems, and
repetitive computers, solutions are often desired
in a few milliseconds.
3)
Accuracy of result.— The accuracy of multipliers is
important since the overall accuracy of a computor
system is generally limited to the accuracy of the
associated multipliers.
It follows then that the
desired accuracy of multiplication is limited only
by the accuracy of the input data.
In physical
systems, this is often limited to about one percent.
1+)
Polarity of acceptable input signals.— Often only
one or possible two quadrant operation of a multi-
plier is required for a given problem.
-7
But, when
both plus and minus variables are expected for the
inputs and the algebraic product is desired, a four
quadrant multiplier is necessary.
Switching methods
for providing the required input polarities and
placing the proper sign on the resulting product are
possible, but become more complicated than existing
four quadrant multipliers.
Four quadrant operation
can also be obtained by combining two simpler multi-
pliers capable of two quadrant operation (1), but
as in the above, the resulting combination of cir-
cuitry becomes excessively complicated.
5)
Dynamic range of input variables and output product.—
The dynamic range of the multiplier is important since
signals must over-ride any undesired noise in the
system and often the theoretical absolute accuracy
obtainable in a system is a function of the dynamic
range
3.
Electro-mechanical methods of analog multiplication.
In electro-mechanical systems high accuracy commensurate
with slower speed can be obtained by using Ohm's law in a
variable conductance network.
Since Ohm's law is a natural
product, it seems quite logical that it should be used in an
analog multiplier.
The basic equation is:
E = IR
E - voltage across the circuit
where
I - current in the circuit
R - resistance of the circuit
A common application of this principle is the potentiometer
(2, 3, 4 and 5)
A linear potentiometer is used for this pur-
•
One of the variables is the voltage impressed across the
pose.
potentiometer, and the other variable is the position of the
sliding contact.
^in f I $ max.
The voltage E
Where
is given by the equation E
max is the maximum possible angle of
rotation.
This method requires that one of the variables be a me-
chanical position.
If both of the variables are voltages, a
servo system can be used to convert one of them to a mechanThe single potentiometer requires that one
ical position.
of the variables always be a positive quantity.
This can be
overcome by the use of a center tapped potentiometer with a
push pull output.
The balanced bridge offers another method of obtaining
a product (4)
The bridge is kept balanced by a servo system
•
which adjusts a rheostat in one of the legs of the bridge.
The other three legs contain rheostats that are set by the
variables of the problem.
Ri
R1R2
R..
*
R2
~
_
R
3
9
or
Ri
s
The condition for balance is
where R^, R2 and Ro are proportional
R3
to the input variables and
R>
is proportional to the desired
9
-
n
'
,s
I
1
.
:
Xo
.
"
i
9J
-
I
I
i
"
5
i
product.
It is apparent that this method can be used for
either multiplication or division, but is limited to one
quadrant operation,
A variation of the balanced bridge utilizing low inertia
strain gages has been developed to meet the requirements of
.1 percent
accuracy and an effective time constant of about
1 mil second (6).
a-c voltages
;
The variables to be multiplied appear as
one of the voltages , X sin w_t , controls
through an amplifier, the mechanical movement of a loudspeaker voice coil, whose movement produces fluctuating
strains in a strain gage bridge (Fig. 1).
excited by a constant amplitude voltage, V
This bridge is
sin w c t.
The
bridge output is fed back to the amplifier input in a negative sense.
The mechanical displacement of the strain gage
bridge is thus proportional to the variable voltage (X).
A second strain gage bridge is also coupled to the same
loudspeaker voice coil so that its strain is also proportional
to X.
However, the second bridge is excited by the other
variable voltage, Y sin w c t.
Hence, the output of this
second bridge is proportional to the product XY.
But since
the input variables are modulated a-c, the strain gage multiplier is a one quadrant device and requires considerable precision construction in order to obtain accurate results.
The dynamometer can be used with two electrical inputs
to obtain a product as a mechanical rotation (7).
10
The rotation
—
<?
Y j//?
—tMnmni
cut
r
•
/W\AAr
-\A/VW-
T
}
*
wvw-
-VWVv
Cfv-qes
srra/n
EBgggg| -,vo/ce
macf/iet
I
L
ty
MsY
\ co/7
d
\
\
Strain
W\A„V
gages
VV\M
N
£
1
-sA/VW"
a/7?£>//Y/er
cS//7
^
c
/•
Srra/n Ga^e Br/ doe Ma/tip/ier
F/qure
11
1
Ys/nuj<r
of the movable coil is given by the equation:
9 K Im Is
s angle of coil rotation
where
K s proportionality constant
Im S current in movable coil
Is 2 current in stationary coil
This method is good for four quadrant operation and the
angular output can be converted to an electrical voltage
by a servo system,
A more refined, higher speed version of the dynamo-
meter method utilizing two dynamometer movements rigidly
connected on a common shaft is shown in (Fig. 2).
(1)
The torque from one movement is proportional to (i]^) an<^
from the other movement i3ii
When the two torques are
•
equal and opposite, the rotational acceleration will be
zero and the shaft will assume a position such that K^i^i2
s
-K2'i')i.L
¥
»
In other words, the sum of the shaft torques
will be zero when i^ 2 «K
i i
l 2
__.
proportional to X and ±2 to
1
and
Therefore, if i^ is made
ii
is a constant reference
current, io will represent the four quadrant product XY.
product ±i ±2 causes a torque in the shaft resulting in a
rotation of the mirror.
As the mirror rotates from its
center position, the voltages derived from the photocell
outputs are compared, amplified and used to generate a
12
The
|
<5j
13
current, i^, proportional to the difference voltage.
The
shaft will oscillate about its new position and settle down
i li 2
with io s -K
The speed of response is limited by the
-
h
inertia of the moving parts and the inductance of the coils.
Accuracy is effected by the linearity of the conversion from
unknown voltage to current, the coulomb friction in the movement and the linearity between the coil flux and current.
4.
Automatic gain control and modulation systems.
These systems may give accuracies of about 0.1 percent
of the range of the output; that is, with a dynamic range
of 100 volts, 0.1 percent accuracy can be obtained, and have
a response time as low as ten micro-seconds.
A typical ex-
ample of an automatic gain control multiplier is shown in
(Fig. 3)
(#)•
A standard reference signal of 500 Kc is put
through a variable gain amplifier.
The 500 Kc component of
the amplifier output signal is compared with one of the input signals,
V]_,
and the difference is fed back to control
the amplifier gain.
The result is that when the loop is in
equilibrium, the gain of the amplifier is proprotional to V^.
A second signal,
V"2,
modulates a 200 Kc carrier, which is fed
to the input grid of the amplifier.
The output voltage at
200 Kc is then proportional to the product V1V2.
The chief
advantage of this method of multiplication is that it does
not depend (to the first approximation) on the tube character-
istics and does not require unique components.
the system is only one quadrant.
14
The output of
K
^
V
3
"
\
~$$
^»
10
«
<o
V
o
o
.CS
v.
15
Another example of a modulation system (9) utilizes
a balanced modulator.
An accuracy of better than one per-
cent has been obtained; it uses a double modulation and
subsequent detection scheme.
A balanced modulator is used
to produce side bands through modulation of a carrier by
one of the variables.
form E^ cos (wet).
The resulting voltage is of the
This voltage is used as the carrier
for a second modulation which gives a side band output of
E^E2 cos (wet).
The carrier voltage is suppressed in both
cases through the use of a balanced modulator.
is then accomplished by a varistor bridge.
Detection
This system is
also limited to one quadrant operation.
A variation of the automatic gain control amplifier
is the so-called step multiplier (10), which used relays to
vary the input conductance by steps in a negative feed back
amplifier.
Although the system is not entirely electronic,
the speed of response approaches that of an electronic system
by the use of fast acting switching relays.
The system makes
the value of the input conductance proportional to one vari-
able and the other variable is applied across the network.
This is illustrated in (Fig. 4).
reversible binary counter.
The relays are set by a
The counter is made to count pulses
from an oscillator whenever the voltage fed back from the conductance network is different from the input variable voltage.
The system was developed to obtain greater accuracy than is
16
osc.
and
pu/se
former
po/ar/fy count> re vers/b\e
S ens/t/ vc GLtfd
>~ 6/ nary
yare
COUnfQr
SUbrracr
,,
r
r
re /cl y
operated
/r
con due fane
net work
'\\\
\\
re /ay
ope roared
C On o/u ctandc
K
ne>tworK
vww
XI
St ep
/
/
A/fu /r/p//'er
a ttr-
e
^r
17
possible with potentiometers.
High speed, 100 micro-second,
relays are used with an oscillator frequency of 1000 cps.
There are 1024 steps in the conductance network, so that the
output can go from minimum to maximum in about one second.
The system accuracy is very good, but the one-second response
must be considered slow.
§.
Non-Linear systems.
The use of the quarter square identity and logarithms
for analog multiplication was mentioned earlier.
There are
several applicable methods for obtaining the square law non
linearity required.
Among these methods are the use of spe-
cific tube characteristics, or non-linear materials.
The
transfer characteristics between grid voltage and plate current
in a vacuum tube provides a somewhat approximate method for
obtaining the square of a voltage.
The instantaneous plate
current of a triode with negative grid voltage expressed in
a power series in terms of the grid exciting voltage e is:
ip =
a-^e
^ a3 e
f &2 e
•••
For certain tubes, the plate current versus grid voltage
characteristic is parabolic in form over a limited range of
negative grid voltage values.
Thus the voltage across the
output resistence will be proportional to the square of the
grid voltage.
Suitable tubes must have a substantially con-
stant Gm over the required range where
Gm - dip/de
Id
.
-1
This results from the fact that the coefficient a 2
*s
given
by
a 2 - 1/2 d 2 e/di 2
p
A more precise parabolic transfer characteristic may be
obtained by using two triodes in a balanced circuit designed
to cancel odd power terms in the series expansion.
Circuits have been developed (11) in which the loga-
rithmic relation between a low level applied voltage and the
resistance of a rectifier, such as a selenium cell rectifier,
is used to generate an output voltage proportional to the
square of the input voltage.
For low voltages, it has been
found that when R is the rectifier resistance, e the applied
voltage and K and q constants of the rectifier
R « K E -q e
The current through the rectifier is therefore
i •
£
m
R
1
E qe
K
The exponential may be expressed in series form as follows,
neglecting quadratic and higher order terms on the assumption
of small voltages:
E<*
Hence
i a
e - 1* qe4
l/K (e
4-
qe 2 )
By subtracting the linear e/K term, a voltage proportional
o
to e
will result.
Appropriate vacuum tubes Can also be used to generate
a logarithmic function directly.
19
Variable mu tubes such as
!
-
-
-
I
:
1
*
,
-
the 6SK7 produce plate current proportional to the logarithm
of the grid voltage.
It has also been found (12) that
by-
operating a 6SK7 in an inverted circuit, that is, with input
voltage applied to the plate and current withdrawn at the grid,
the output current is proportional to the antilog of the vol-
tage at the plate,
A technique which uses biased diodes to switch approp-
riate conductances into a parallel circuit makes a non linear
transfer characteristic by developing straight line approximations to a given curve.
Circuits with accuracies as good
as 0,4 percent and 12 micro-second response time have been
built, using this method (13)»
The number of straight line
sections used determines the closeness of the approximation
to a desired curve.
For example, a square law curve can be
approximated to within two percent by three line sections
and to within one percent by five line sections.
Another type of non linear function generator, called
the photoformer (14, 15), is essentially an electronic servo
system which makes a cathode ray beam follow the edge of an
appropriate mask (cut to represent the desired function),
simplified block diagram is shown in (Fig, 5).
A
The hori-
zontal deflection represents the input variable and the vertical deflection, the output variable.
Here the input variable
is received from a signal source and applied to the hori-
zontal deflection plates so that the spot is positioned on
20
'
t
_
.
.
-
-
.(
--...."
fane
+/0/7
/77CZS A
/77u/f/'p//er-
fu6e
ouf/>a/
r
fu/icr/o/?
B* SCA)
PAoto form e r*
S/Vr?p////ed
3/ocA
D i ao-ram
/v our e
5
21
the edge of the mask.
The precision is limited by the sharp-
ness of the beam focus, as the spot cannot accurately follow
details which are smaller than the spot itself.
By using a
lens system to improve the focusing and reduce parallax,
accuracies better than one percent have been obtained.
This
accuracy may be limited by the signal to noise ratio in the
feed back path.
Cathode ray tubes especially designed for producing a
single transfer function have been developed, which operate
essentially like the photo former, but the external mask and
photocell are replaced by a target and collector electrode
inside the cathode ray tube envelope.
One such a device,
called the mono former (16), employs a standard cathode ray
tube base.
The target is an aluminum disc of one inch
diameter, on which the function to be reproduced is printed
with a coating of carbon ink.
The operation of the tube
depends on the fact that aluminum and carbon have different
secondary emission ratios.
An electrode is provided to col-
lect the secondary electrons emitted from the target plate
as a result of impingement by the electron stream.
If the
beam, in sweeping across the target, tends to ride too far
into the uncoated area or into the coated area, it causes
a variation in secondary electron emission.
The correspond-
ing variation in target current produces a variation in vol-
tage drop across a load.
This error signal is fed back
22
-
through a network. to the vertical deflection plates of the
The result if to keep the electron beam directed
tube.
against the boundary between the coated and uncoated areas
of the target.
The accuracy of the unit is considered to be
one percent without amplification in the feed back loop.
A
response time of 400 micro-seconds to a step input is developed.
With sufficient amplification in the feed back
loop, the response time reduces to one micro-second.
The above methods of generating non-linear functions
can be used in multipliers based on the logarithmic or
quarter square identities.
A block diagram of a quarter
square multiplier is shown in (Fig. 6).
An error of less
than one percent of maximum operating range has been obtained, with a solution time of about 50 micro-seconds (13
)•
In this circuit, only one squarer is used on a time sharing
basis.
When two function generators are used, they must be
very nearly identical.
The error due to a small discrepancy
in the squarer characteristics will be considerable if one
of the inputs is large and the other small, because here the
difference of the square of two large quantities is involved.
The parabolic function generator used is of the biased diode
network type.
In the system, the sum term is generated by
adding the X and Y input variables.
But since the input to
the squarer must be positive, a method is provided to insure that the difference term is never less than zero.
23
The
-
>
:
-
!
,
'
'
•
.
5
3
fe
^
O
%t
k
<-0
O
24
X and Y inputs are compared in an amplitude discriminator
and fed to an electronic switch which passes only the smaller
of the two inputs.
The smaller of the variables is then
chopped into equal on and off pulses, multiplied by two and
subtracted from the sum term.
Thus, the input to the para-
bolic function generator alternates between the sum and
difference terms, but is always positive.
The difference
between the amplitudes of the adjacent squared sum and difference pulses is determined in a difference detector and is
equal to the output, 4 X Y.
The circuit can handle only
positive input variables and has a dynamic range of
volts.
to 25
The circuit is rather complex, with five d c ampli-
fiers, twenty-six diodes and fifteen other tubes of various
types, exclusive of the squarer and pulse generator.
Exponentials can also be used for analog multiplication.
A circuit based on the following mathematical expressions
gives promising results:
if Vi : -A e I**
T
and Vo s -A
e
Z$l
T
then
V^
2 e
s A
-Ux
* ty)
T
The circuit shown in (Fig. 7) and described below will give
the response - A e ~( tx *
^
t
thus differing from its de-
sired answer by a scale factor which can easily be taken into
account.
25
'
-
I
!
'
C
.
-
.
.
K
•2
A.
it
PL
kl
26
Figure 7 shows the schematic diagram and the idealized
wave forms present at various strategic points in the circuit.
At point A, a square wave with a one millisecond period is
fed to the grid of V\a, which is a normally fully conducting
tube.
volts.
Point B normally resides in the neighborhood of -400
When, however, tube
V-^
is driven beyond cut-off by
the negative half cycle of the square wave input, point B
will proceed to rise exponentially towards ground potential.
The duration of its rise will depend on the period of the neg-
ative half cycle and the rate at which it will rise will depend on the RC time constant in the plate circuit of tube V^a.
The input variable voltage, ex, will set the cathode potential
of tube V2 by virtue of the cathode follower action.
potential at the grid of V2, point
B,
When the
reaches the grid con-
duction region, it will be prevented from further rising by
the low effective grid to cathode resistance.
will be as shown in (Fig. 7B).
The wave form
This signal will be ampli-
fied and inverted by tube V2 and applied to the control grid
of V4A.
A similar action will take place here,
it will be initiated at a later time.
except that
The resulting signal
will be amplified and inverted by tube V3 and applied to the
control grid of tube V5a.
Also, at the beginning of the en-
tire sequence, the square wave input is applied to the control
grid of tubes Vlb and V4b.
The negative half cycle of the
square wave will initiate an exponential rise at points D
27
.
and E, while the signal to the control grid of tube V5a from
tube V3 will end it.
The signal to tube V4b is to discharge
the storage capacitor between reading times.
Results obtain-
ed from this circuit to the present time have not been con-
clusive, but it is felt that further refinements in the de-
sign could make this a good and simple one quadrant multiplier,
6.
Cathode ray tube systems.
The crossed field electron beam multiplier (17) uses
an electro-static deflection cathode ray tube in conjunction
with a feed back amplifier and a photo multiplier tube.
The
electron gun of the cathode ray tube generates a sharply
focused beam of electrons.
The force on a stream of electrons
moving with average velocity,
v,
at right angles to a mag-
netic field H is counteracted by an electro-static field E
proportional to the product vH.
If the adjustment of E were
automatic and instantaneous, then its value would be a continuous measure of this product.
In practice, v is propor-
tional to a voltage V x applied to the horizontal deflection
plates of the cathode ray tube; H is proportional to a cur-
rent
I
through a coil wound around the vertical deflecting
plates of the cathode ray tube; and E is automatically adjusted by means of a mask, phototube and amplifier (as in
the photoformer discussed above).
Then E will be propor-
tional to the product IV X . *This system gives four quadrant
23
;
-
.
.
'
I
..
—
I
J
operation, an absolute indication of zero (freedom from zero
drift) and is independent of normal changes in electrical
characteristics.
As the important parameters are geometric,
the prospects of high stability are good.
The inductance of
the magnetic field coil is the main factor which limits the
speed of response of this device.
The solution time is about
500 micro seconds with an accuracy of two percent.
Another principle of multiplying by cathode ray tubes
makes use of a square beam of electrons which is deflected
horizontally by the X input voltage and vertically by the Y
input (16).
The deflection causes the beam to fall eccentric-
The current
ally on four square collector plates (Fig. S).
from each plate passes through a load resistance.
If the
beam current density at impingement is uniform over the area
2L x 2L, the current through each load resistance is propor-
tional to the area of impingement on the corresponding col-
lector plate, which in turn is a function of both the X and
the Y voltages.
These areas are:
(L-x)
(Liy) = L 2 4 (y-x) L-xy
1
(L-y)
(L-x) = L 2 - (y+x) L*xy
2
(L+x)
(Wy)
z L
(L*x)
(L-y)
= L
2
2
Uxy
3
- (y-x) L-xy
4
+
(ytx)
If equations two and three are added, one and four subtracted,
the net current is found to be 4 x y, or the same as the result of the quarter square multiplier.
29
Although good accuracy
•
•
-
±
^
ZZ7
r-4*
&mp
T
>Afw
(
W\W
^\/vw
A
Pk
i— *2£
£
/c
~-i
i
1
r
—
t
i
x—
3
A
a
^>o uare LJecLm
L/afhode f\OLy / uh>
Multiplier
r/o
ur e
O
30
was not obtained with the first experimental tubes, the speed
of response is not limited by an inductive circuit, as in the
crossed field multiplier.
Time division multiplier systems.
7.
Pulsed attenuator multipliers using a combination of
pulse width and pulse amplitude modulation have been used
successfully (3).
They use the area of a rectangle technique
mentioned earlier, where the average value of a pulsed voltage
is equal to the product of the pulse amplitude, and the pulse
width averaged over a full cycle.
The pulse amplitude is
made proportional to one input variable and the ratio of the
on and off time of the pulses is proportional to the other
input variable.
This type of multiplier is limited to one
quadrant operation and the switching tubes must be carefully
matched in order to obtain the high accuracy of 0.2 percent,
which has been obtained.
An extension of the pulsed attenuator technique called
time division (19, 20) makes four quadrant operation possible
and eliminates the necessity for carefully matched tubes.
The basic principle of operation is that the algebraic product
of two variables is formed by averaging several cycles of a
quasi rectangular wave form.
The duration and amplitude of
the wave form are functions of the input variables, as shown
in (Fig. 9).
Ti
s
The amplitude of the portion T^ is
4-Y,
where
-£— seconds, and the amplitude of the portion To is -Y,
Z-X
31
I
;
*
•
i
...
i
sun? n?/hy
e/ec/"^ on/'c
7 */- -^
>
*
SW/'tc/)
tz,-z
//?reyraror
*"
/
j
l'
A
j
>
*
/
2.
mu/tJ1/
M
r
o urp u t
e/ec Tronic
V
/
v
/bra tor
amp
V.
S* tch
^
and
+Y,-Y
XX
^-
fitter
z
/</ea//ze&
/n tepra tor /njou t
Wa ve forms
XtZ
tz
1
<
•
i
~z
77
/nfeyrator output
M.
V.
Ml/
out /
out
F///er
7~/t7?e
2
+r
jr?pu.f
/?/'y/s/o/?
-Y
A//u//l//o//e/- /S/oc/r /J/'ayra/7?
f~/y
are
9
>2
where T 2 *
-JL. seconds.*
The average value of the complete
Z4-X
cycle is
(Tl - T2J
or
Ti * T 2
XY
,
The basic wave form is pro-
Z
duced by the switching process shown in (Fig. 9) and described
below.
The pulse timing is dependent on input variables X and Z.
It is controlled by a closed loop feedback system consisting
of switch 1, the integrator and bistable multivibrator as
shown in (Fig. 9).
The multivibrator changes from one of its
stable states to the other whenever the output of the integrator
reaches e± or
eously.
e2
and actuates switch 1 and switch 2 simultan-
For simplicity of explanation, the period when switch
1 is open will be called T,
and when it is closed T ? .
The
output of switch 1 during T^ is -Z, and during Tg is +Z.
There-
fore, the input current of the summing integrator during T^ is
x - z.
Switch 1 remains open until the output of the integra-
tor reaches en, at which time the multivibrator changes states,
closing switch 1 and changing the integrator input current to
x +
e-,
z.
The output of the integrator then changes from &2 t0
and continues to repeat the switching cycle.
The transition
time T^ is computed from the integrator response to a step
function, as follows:
Let (ei-e 2
)
=
voltage excursion required at input
of bistable multivibrator to change
its state
30
•
•
noJ
<
1
t
'
:
C -
capacity of integrating condenser
in farads
R : input resistor for variables in ohms
x s X
— = input current to integrator due
R-£
to variable X
z
-
±L
- input
current to integrator due
Rz
to Z.
Then:
e
x
- e 2 = -l/C
J^
'(x-z)
dt
This equation assumes a high gain amplifier and an integrating
capacitor with a high leakage resistance.
Then, assuming that
X and Z are constant during the period:
7"--
C(e,-ez)
and similarily:
T~ = P
Ce, -
e
J
The variable Y is switched through the use of switch 2, which
is actuated by the same switching pulses as switch 1.
The
rectangular wave form of (Fig. 9) appears at the output of
the final amplifier if filtering is not performed.
age of this wave form, E
rto
,
The aver-
may be computed as follows:
+Y7T-YT,
77-71
34
•
.
:
fc a
-
Z-X
z.
+
x~~
Thus the average value of the output voltage is proportional
to the product of the two variables, X and Y.
Since d-c
amplifiers with the required dynamic range and stability are
available, the accuracy, response time, and dynamic range of
the multiplier are dependent upon the type of electronic
switch used.
Circuits of this type with accuracies within
0.1 percent have been built.
8.
Choice of system for construction.
It is evident from the discussion of electrical analog
multipliers that the requirements of a universal multiplying
device are difficult to fulfill.
In order to satisfy all of
the requirements, the multiplier must be a four quadrant de-
vice which combines a short response time with high accuracy
over a large dynamic range.
The requirement of high speed
limits the selection to an all electronic device.
The re-
quirement for a four quadrant device limits the selection
even further, unless added complex circuitry is acceptable
to convert a one or two quadrant multiplier to four.
The
only electronic analog multipliers which are fundamentally
four quadrant devices are the crossed field cathode ray tube
35
•
-
c
I
•
multiplier, the square beam multiplier, the quarter square
multiplier with a photoformer squarer and the time division
multiplier.
To the authors knowledge, the square beam
multiplier is not an accurate device in its present stage
of development and other cathode ray types tend to be bulky
and complex.
Through a process of elimination, the time division
multiplier is the only type which approaches the requirements set down for a universal multiplier.
A circuit of
this type was constructed and tested by the author and will
be discussed in more detail in the following chapter.
36
Ill
A TIME DIVISION MULTIPLIER
1.
Precision switch.
The basic principle of operation of the time division
type of multiplier was stated in the previous chapter.
It
and proper pulse timing is
is obvious that high accuracy
dependent upon an electronic switch with excellent high speed,
precision characteristics.
The switch and its associated cir-
cuitry must have the following characteristics:
when the
switch is in one condition, the current or voltage out must
be a linear function of the voltage input; when in the alter-
nate condition, the current or voltage out must be the negative of that in the first condition; in addition, the char-
acteristics of the switch must be independent from normal
variations in the tubes employed, must have a large dynamic
range, a high input impedance and a low output impedance.
Two switches suitable for the application will be discussed.
The first is a so-called current switch, rather than a po-
tential switch.
(Fig. 10) is a diagram of the electronic current switch
(20).
Switching signal voltage levels are applied to the
grids of the triodes V2 and Vo.
If the control grid of V
is positive with respect to the control grid of V~ by a
sufficient amount, the plate current of the pentode,
will flow through tube V2 and the output voltage, E Q
-
37
V-^,
,
will
j
-
.
rWW-i
V,
Vs
arm
I—
/y-ec/s/o/?
Curre/7/
Sw/Vc/?
/O
Fi'yure
4
-v%Vv\r-
n
v,1
C71
[A?
^_j_
-^AAAs-
vAa/v-
-nAA^
<2 <V*.£^
*i
£«
Precis/ on rofe/?r/a/ Sw'/~c/?
Figure
//
33
a
.
I
c
have a negative polarity.
If -the control grid of V\ is
positive with respect to the control grid of V by a suffic2
ient amount, the plate current of V^ will flow through
rather than through V 2
,
V-,
and E Q will have the same magnitude
as before, but it will have a positive rather than a negative
polarity.
can thus be made to flow
The plate current of V,
in either of two external circuits.
Since the switch is uni-
directional and the range of operation is limited, it is
necessary to add a fixed voltage to the variable voltage at
the input.
The unwanted component derived from the fixed
voltage is eliminated from the output of the multiplier by
a bridging system.
(Fig. 11) is a diagram of an electronic potential switch
with suitable characteristics (19)
•
It consists of a d-c
amplifier with two alternately switched feed back impedances.
The switch tubes in series with the two feed back impedances
are connected so that when one is conducting, the other is
cut off.
V 2 and R 2 .
The output voltage is taken from the junction of
When
V-^
is on and
V*
2
is off, the output voltage
is zero, since the junction of R^ , R 2 and
R<>
is maintained
at ground potential by the high gain negative feed back ampli-
fier.
When Vx is off and V2 is on, the output voltage is
equal to - R 2/Ri Ein »
When a linear function of the voltage
input equal to l/2 R 2 / R^ Ein is added to the switch output,
a rectangular wave form symetrical about the zero axis is
obtained.
39
•
•
I
I
I
-
:
:
.
Although both switches would perform satisfactorily in
the circuit, the potential switch was selected since it is
generally less elaborate than the current switch.
It re-
quires one less d-c amplifier and no battery bias supply,
and does not produce an unwanted component in the output.
The results of tests indicated that the desired high
speed, accurate switching could be accomplished with this
rather simple circuitry, if the effects of distributed capacitance were kept to a minimum.
This indicates that the
feed back resistances must be kept small and the amplifier
gain high.
A compromise had to be made in determing the
size of the resistance, however, since too small a feed
back impedance reduces the dynamic range of the amplifier.
2.
Circuit design.
A block diagram of the multiplier, incorporating the
potential switch, is shown in (Fig. 12).
ratio resistances, b,
c,
The size of the
and k are established as a result
of several considerations.
Zmin/4b must be greater than
Xmax/c, since the output of the integrator must reverse
direction.
Larger resistances will reduce the loading ef-
fect on the previous stage, but since the integrator is based
on switching wave forms with short rise times, the effects
of distributed capacitance will be minimized by using small
resistances.
The lengths of the times
T-.
and T^ are also
affected by the magnitude of the ratio resistances.
40
-
'.
•
6/stabte
mu/t/'vt bra tor
U
I
output
electronic
switch
Y
\,o
xr
amp
%
and
2
f/'/fer
A
£+%6
%-%&
integrator input
/nteyrator output
mu
J t/'v /bra/or
m u/t/ vibrator
out
l
out Z
%«
//7put
f/'/ter
T/'rn
e
-#*
D/y/s/on
S/oc/c
Ma/ ftp/*ter
Dtag ram
r/gure /Z
41
A schematic diagram of the multiplier circuit is shown
in (Fig. 13), where the triangles represent high gain d-c
amplifiers.
The amplifiers have a differential input, a
stage with regenerative feed back and a cathode follower outThe decision to use amplifiers without automatic bal-
put.
ancing was prompted by the desire to make the circuitry as
simple as possible.
The dynamic range of the multiplier is
limited by the amplifiers used and, in this case, is from
minus fifty to plus fifty volts.
Certain scale reduction is necessary in an electronic
multiplying circuit, in order to reduce the maximum swing
of the output to the dynamic range limit of the final amplifier.
The scale factor of the final amplifier here is made
equal to kc/b, so that the multiplier will give the output
XY/Z.
It is evident from the relationship for the output
voltage that the circuit can also be used for division by
the variable Z.
However, when Z varies, the frequency of
the switching wave form varies widely and increases the diffi-
culty of filtering the output.
When the circuit is to be used
only for multiplication, the variable Z should be set at a constant reference value, thereby establishing the desired scale
factor for the final amplifier.
3.
Output filtering.
If the input variables to the multiplier are varying
d-c voltage, or very low a-c,- (that is, if the required
&
•'
Si
'
•
.
.
43'
frequency response is very low), a simple high pass feed
back network around the final amplifier is all that is
If the output of the multiplier is to be fed
necessary.
to an integrator, then no output filtering is necessary at
But if a high frequency response combined with good
all.
absolute accuracy are desired, the filtering problem beThe following calculations show the ap-
comes difficult.
proximate requirements put on the filter and the carrier
wave forms.
Expressing a square wave in its fourier
expansion:
(0
Y=
£f (s's?
x +j- s/n
3X
+"• +n stn s?a)
n odd
and realizing that the rise time is dependent upon the high
frequency components of the wave form, consider the pass
band requirements for one percent accuracy combined with a
3
The function must be expanded
millisecond response time.
until the accuracy is within one percent; or, in other words,
the harmonics of the fundamental must be unattenuated until
the accuracy of the resulting wave form is within one percent.
The so-called fundamental frequency is determined from
(Fig. 14) and the following:
arc
(2)
when
st'n
/ "* <s
t' 3 /77i'//t seconds
f -d5.4(
cyc/es
per second
'
_
'
•
;
i
:
•
.
2 m/'/sec.
1
\
\
\
\
/
\
1
1
\
1
V
1
/
1
1
/
\
/
\
/
\
\
/
\
N
~" --
/ //fer hanc/pass
/^
/-/jure
Output
/^eyu/renienrs
f/'/fer-
/^
/~/'qure
45
network
Since consecutive harmonics add and subtract at the point
of interest, and the series is convergent, the desired har-
monic of the fundamental frequency can be obtained from
equation (1) as follows:
%r
%v
a Y =
(4)
(5)
.
r
0/
2.
J
n
-
63
Therefore, the filter must pass the 63rd harmonic of 33.4
cycles, or 5»25 kc.
The second step in determining the necessary filter
characteristics is to find the attenuation required at the
In order to obtain a
carrier or switching repetition rate.
maximum product term of 50 volts in the form of XY/Z, the unfiltered carrier level must be 250 volts.
One percent ac-
curacy requires that only 0.5 volts of the carrier remain
after filtering.
Therefore, approximately 54 decibels atten-
uation at the fundamental of the carrier frequency will be
required.
A low pass filter with an attenuation of 60 db/decade wnl 1
be examined for the desired 3 millisecond response time with
one percent error.
Since 54 decibels attenuation is required,
and assuming the attenuation to be zero to 5»25 kc, as deter-
mined above, the smallest carrier frequency allowable can be
determined from the following:
(r\
{OJ
(7)
f
/carr/pr* ~
fcorner
fearer
j:
T^^atten uat&</
/ axccn nation
(
~\
r £<}«"<*/ J
'
s
\
M (&*<#>)(-£&&»)
=
£Z5
-
4 7.25
AC
46
N
ft/re n aCfe/i u*t'or?J
I
:
l
:
I
I
;
This means an average switching time of only 10.6 microseconds; 47^25 kc is almost 6.5 times the highest frequency
with which one percent accuracy can be obtained in the present circuit.
Using a similar approach, it would be possible to attain
an £ millisecond response time with two percent accuracy with
the existing 70 microsecond switching wave forms.
This re-
quires a 60 db/decade low pass filter with a flat response up
to 900 cycles.
A filter with these approximate character-
istics can be designed as follows:
A low pass filter with a 60 db/decade attenuation can be
obtained from the following network transfer function:
IS)
Cfs)
where
n f"
(JTS
=
T
=
-
^
&nft
being the breakpoint for the composite attenuation versus
frequency curve, or in the present case, 900 cycles per second,
7= /.7T
*
/O' 4
In order to make selection of components easier, let
thus making
(9)
Gcs)
=
CJJL'Q'fs *-J±
(Zx/o'+S
+
0( 4xJO~ G S* +/o-*s
47
+
/)
/*"»
2.*/0~
.
'
:
i
The above transfer function can be obtained from the network
shown in (Fig. 15)
In the circuit with the high gain d-c amplifier, the vol-
tage at point 1 is effectively zero and 1^ - -Io»
§ r£
-
(10)
00
Z,
=^K-OfsC,s
02)
z£
=Rz
(,sj
pr , -jr.
"
x
&
°
i
O
*
TRTcTs^rrTRi
,
{R<C,s
(RsG
s t
+
Q
i)[(RzR«
C,
Ct )S* + CQR +RiRJs
4
* /J
fti3
Now choose values for the components of
Z-,
and Z«, to make
the transfer function equivalent to the desired transfer
function, keeping in mind that a gain of 10 is desired in
the output stage.
If R^ is made 1 megohm, which is sufficient-
ly large to reduce the effective load on the output amplifier, allowing a dynamic range of +50 to -50 volts, then R,
+ Re must equal 0.1 megohm to satisfy the gain of 10.
It is
also desirable to have as many components as possible with
standard values and to make
R»
reasonably large, so as to
keep the input impedance to the amplifier high.
4S
Utilizing
'
.
|
-
t
-
•
the above restrictions,, the following identities can be used
to find the desired component values:
(14)
\
(15)
R5 C
(16)
(17)
R
C
2
± z
X 10 -5
- 2 X 10
*
R^ C x C 2 - 4 X 10 8
2
R5 1 Ri = 5 X 10 ^
The network then takes the following form:
R 1 - 50 K
ohms
C
1000
x 8
uuf
M
ohms
C
2
- 2000
uuf
R3 - 250 K
ohms
C3 3 4000
uuf
- 20 K
4 ~
R c - 50 K
ohms
R2 =
1
R.
4.
ohms
Tests and performance.
Both the accuracy and response time of the multiplying
circuit were tested.
Tests of the accuracy of the multi-
plying circuit were made by applying different values of con.
stant voltages, ranging from -50 to f50 volts, to both the
X and Y inputs with a constant -50 volts Z input.
For pre-
cision testing, a zero reading micro ammeter was used, with
a calibrated helipot across a known voltage source.
Peak
errors were found to be about one percent of the maximum
output voltage and less than two percent absolute error at
any output voltage.
This accuracy was obtained when the
multiplier was operated with a five kilocycle switching frequency and a simple high pass feed back network around the
49
final amplifier.
The unstabilized d-c amplifiers were bal-
anced about every ten minutes to attain this accuracy.
In
order to improve the response time to a step input to five
milliseconds, the repetition rate was increased to 7.2 Kc,
with a corresponding decrease in accuracy to two percent of
the maximum output voltage.
The speed of response of the multiplying circuit was
tested by applying a constant voltage to one input and a
square wave to the other and viewing the output on an oscillo*
scope.
A response time of eight milliseconds with two per-
cent accuracy was attained.
It is obvious that the response
time can be decreased by increasing the switching frequency
and adjusting the filter characteristics correspondingly,
but this reduces the accuracy of the multiplier.
Experi-
mental results showed five percent accuracy with three millisecond response time.
In order to improve the accuracy at
a higher switching frequency, the transient time of the
switching and carrier wave forms must be shortened.
Sug-
gestions for these improvements, without increasing the circuit complexity, are to select tubes for the square wave
amplifiers with smaller input capacitance and higher transconductance tubes for the bistable multivibrator.
50
IV
CONCLUSION
The experimental results show that the time division
multiplier circuit which was tested approaches the requirements set down for a universal analog multiplier.
It
four quadrant device with a relatively simple circuit.
i s
a
One
percent accuracy can be obtained, but only at the expense
of lowering the response time.
Greater accuracy and faster
speeds could be expected from the circuit after certain re-
finements.
Automatically stabilized amplifiers must be used
to eliminate the necessity of frequent balancing.
There is still a need for further development of simple,
accurate, all electronic multipliers of low cost for use in
computing operations on fast time scales.
51
BIBLIOGRAPHY
1.
Korn and Korn, "Electronic Analog Computers"
McGraw-Hill 1952
2.
D. A. Bell
3.
"Compact Analog Computer"
S. Frost,
Vol. 21, pages 116-122 July 1943
4.
M.I.T. Staff "Electronic Instruments" Rad Lab
Series No. 21 McGraw Hill pages 43-60 1943
5.
D. J. Mynall,
"Electrical Analog Computing"
Electronic Engineering Vol. 19 June, July, August,
September 1947
6.
Albert C. Hall, "A Generalized Analog Computer
For Flight Simulation" A.I.E.E. Vol. 69 1950
7.
V. Bush, F. D. Gage, H. R. Stewart,
3.
Chance, B., Hughes, V., MacNichol, E. E.,
Sayre, D., Williams, F. C, "Waveforms"
McGraw-Hill 1949
9.
McCann, G. D., Wiles, C. H., Locanthi, B. N.,
"Electronic Techniques Applied to Analog Methods
of Computation" Proc. I.R.E. 37 August 1949
"Reactive Circuits as Computers and
Analogs" Electronic Engineering Vol. 22
page 232, 235 June 1950
Electronics
"A Continuous
Integraph" Journal of Franklin Inst. Vol. 203
Pages 63-34 Jan. 1927
10.
Goldberg, Edwin A., "Step Multiplier in Guided
Missile Computer" Electronics 24 August 1951
11.
Draper, H. P.,
"A Square Law Circuit"
Scientific Instruments 24 1947
12.
Snowden, F. C, Page, H. T., "Electronic Circuit
"Which Extracts Antilogs Directly"
Review of
Scientific Instruments 21 February 1950
13.
Chance, B., et al "Quarter Square Multiplier
Using a Segmented Parabolic Characteristic" Review
of Scientific Instruments 22 September 1951
52
Journal of
14.
MacKay, D. M., "A High Speed Electronic
Function Generator" Nature 159 March 22, 1947
15.
Mynall, D. J., "Electronic Function Generator"
Nature 159 May 1947
16.
Munster, A. C.,
Engineering 44
17.
Macnee, A. B., "An Electronic Differential
Analyzer" Proc. I.R.E. 37 November 1949
IB.
Somerville, A., "A Beam Type Tube That Multiplies" Proc. of Nat. Elec. Conf. Vol 6 1950
19.
Morrill, CD., and Baum, R. V., "Stabilized
Time Division Multiplier" Electronics Dec. 1952
20.
Goldberg, E. A., "A High Accuracy Time Division
Multiplier" Cyclone Symposium #2.
"The Monoformer"
October 1950
53
Radio Electronic
2
2
JUL
NOV
JAN
DE
B
INOERY
2
5
173
1
1
1
1
U 6
457
Thesis
B683
06
2075
urice
Electronic analog multipliers
B
INOERY
NOV
JAN
DE 457
I
Thesis
3883
i
Burke
««-.«.«
2
1
5
2
1
5
55
173
1
1*606
^D 0J53
Electronic analog multipliers
Library
U. S. Naval
Postgraduate School
Monterey, California
thesB883
Electronic analog multipliers.
3 2768 002 07946 9
DUDLEY KNOX LIBRARY
fwjmmm
ffimMmfri
^^B 93
^^H
"' ;
»
aJ88IS8
''.
'}')''
:
JBEmKHk
isireiiiiii
JmM
MwUa»'ra
••>!-
!,;
•
''/'''V.'i
$|pwl|
ISpl
<''
5$HH&]
jM$|
HJW
£83
H
bSbw
'::',{
ayuffi
SKSy
K
-..
'tf
:
'
;
-
::
'--
Kiss 3£$3
P^MS;fe;''
'
:
~-
'
••-.,--
--'•
'•
:
''*
.
'
•:.'.. ..;.,'.... •
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement