AUG 17 1960
L IBRA RA DIRECT-COUPLED MULTIPLIER FOR ACOUSTIC SONAR
by
DAVID ALLEN CAHLANDER
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1960
I1
Signature of Author o.t
Department of Electrical Engineering, May 23,
Certified by
/I
icaN
Accepted by
1960
Thesis Supervisor
1 6
I
Chairman, Department Committee on Graduate(~5tients
J1
F
,
A DIRECT- COUPLED MULTIPLIER FOR ACOUSTIC SONAR
by
DAVID ALLEN CAHLANDER
Submitted to the Department of Electrical Engineering
on May 23,
1960 in partial fulfillment of the require-
ments for the degree of Master of Science.
ABS TRACT
A high-speed multiplier was needed as an essential part of an electronic model of a Bat. Since
no such device was found to be commercially
available I undertook the project of designing one
that would fit the needs.
The resulting device is a four-quadrant multiplier
with a frequency response of DC to 500 KC. The
method of multiplication, reasons for the particular design used, accuracy, and the stability of
the multiplier are discussed.
Thesis Supervisor:
SamuAl J. Mason
Title: Professor of Electrical Engineering
')
_~~_ --_.-----
_____
11_·___1__~~_1_________
... _1
· 1~1 ~_
_
~.~I~
C~~
__
I_
~__~_~ ___~~
Plate 1 Electronic Multiplier
L
-I-
ACKNOWLEDGMENT
The author wishes to acknowledge the encouragement and
motivation that has been provided by Mr. Fredrick A. Webster
and Dr. Donald R. Griffen along the lines of the Acoustic Sonar
model of a Bat; Dr. Donald L. Clark, who originally suggested
this project; Dr. J. J. Gerald McCue, who has been very helpful
in procuring necessary equipment; Dr. Samual J. Mason, who has
been extremely patient and helpful; Mr. Bruce DePalmer, who
suggested several good schemes for the multiplier; Dr. Walter A.
Rosenblith, who suggested the quarted square multiplication technique;
Mr. David G. Kocher, Mr. William H. Levison, and Mr. Albert C.
Schell, who made many helpful suggestions.
Thanks are due to Mrs. Deanne F. Sferrino for the rapid and
efficient typing of this thesis and the staff at Lincoln Laboratory who
produced this thesis.
The writer is especially grateful to Marjorie Mills Cahlander
for her infinite patience, consideration, and encouragement throughout
the preparation of this thesis.
TABLE OF CONTENTS
I.
Introduction
II.
The "Building Blocks" of the Multiplier
1
3
A.
The Input Difference Amplifiers
B.
Output Difference Amplifier
10
C.
The Squaring Operation
11
D.
The Output Cathode Follower
23
E.
Power Supply
28
III.
Interconnections of the "Building Blocks "
30
IV.
Over-all Specifications
36
V.
Concluding Remarks
38
VI. Bibliography
39
VII. Appendix
A.
Analysis of a Difference Amplifier
B.
Electronic Multipliers that are Commercially
Available
C.
Wiring Diagram and Photographs
40
43
46
ILLUSTRATIONS
I.
Drawings
1.
Difference Amplifier
4
2.
Ideal Model for Odd Mode in Difference Amplifier
6
3.
Incremental Ideal Model for the Even mode in a
Difference Amplifier
4.
Hybrid Connection of the 5725's to Obtain the
Squaring Function
5.
12
Circuit for Biasing and Adjusting the Gain for
Squaring Element Simultaneously
17
6.
Cathode Follower Output Stage
25
7.
Ideal Model for the Cathode Follower Output Stage
26
8. Power Supply
9.
10.
Block Diagram for the Multiplier
11.
29
31
Resistive Summing Network Between the Input
Difference Amplifier and the Squaring Element
Input Difference Amplifier Circuit
12. Output Difference Amplifier
II.
8
32
33
35
13.
Ideal Model of a Difference Amplifier
41
14.
Wiring Diagram of Multiplier
46
Graphs
1.
Characteristic Curve for a 5725 Vacuum Tube and a
Parabolic Fit
2,
Characteristic Curve for a 5725 Vacuum Tube
and 5 -er cent Accuracy Limits
13
ILLUSTRATIONS (continued)
3.
Characteristic Curves for 4 Different 5725's
Connected as in Figure 5 with
such that i
4.
o
3 Adjusted
19
= 70 lamp.
Transfer Characteristics for Two 5725's
Connected in the Hybrid Connection of PushPull and Parallel Described in Figure 4.
5.
"Infant Drift" in e 0 for a Typical 5725.
is Adjusted so that i = 70 Vamp.
eo
P
6.
Drift in
III.
22
eo for a Typical, Aged 5725, After Being
off for 24 Hours.
i
21
eo is Adjusted so that
= 70 Ilamp.
23
Plates
1.
Electronic Multiplier
2.
Top View of Multiplier
3.
Under-chassis View of Multiplier
47
-1-
I.
Introduction
The ability of the Bat to navigate by means of acoustic sonar,
even in the presence of relatively intense noise, has been
extensively investigated by zoologist Donald R. Griffin of Harvard.
Dr. Griffin has found that the bat Plecotus Rafinesquii is able
skillfully to avoid wires even when the energy of the bat's echo
from the wire is perhaps as much as 20 db below the noise power
per unit bandwidth of the jamming noise.
This is indeed remarkably
good receiver performance and makes the study of the bat a matter
of great interest to communications engineers.
Many of the parameters in a jamming experiment are difficult
to calculate but should be easy to measure.
An artificial "bat sonar"
is being built for experimental investigation of parameters affecting
anti-jamming capabilities of such systems.
According to information
theory, the best processing of a signal contaminated by addative
gaussian noise is a cross-correlation with the same signal.
Cross-
correlation techniques may also be used to simulate the pulsecompression of an FM signal that is fed through a matched filter.(1)
It is evident that a multiplier is needed for these operation.
The
requirements for this multiplier are: a frequency response of
DC-100 KC and an accuracy of 5 per cent.
An extensive search of
the available literature indicated that it is highly unlikely that
multipliers satisfying these requirements are commercially available.
Many different multiplication methods were considered and
several were tried.
The method that is used in the multiplier described
below utilizes the quarter-square principle.
-2-
That is:
xy =
(x+y)
-
2
(x-y)
(1)
The nearly parabolic plate current vs. grid voltage characteristics
of two 5725 pentode vacuum tubes operating in a hybrid' combination
of push-pull and parallel is used to generate the squaring function.
Difference amplifiers with high common mode rejection are used
for the inverting operations and resistive networks are used for
the summing operations.
-3-
II.
The "Building Blocks" of the Multiplier
A.
The Input Difference Amplifiers
In..a
uiiiltiplier built by Holmes and Dukes
and copied by
Barlow and Brown ( 3 ) , which also uses the quarter-square principle,
the input to the multiplier is AC coupled to a pair of phase splitters
(triodes with similar cathode and plate resistors) and a resistor
summing matrix is used to obtain the sum and difference inputs to
the squaring elements.
This method of obtaining the sum and
difference signals is disadvantagous for a direct-coupled multiplier
since the plate and cathode of the phase splitter are at different
DC potentials.
High frequency AC operation is also impa red since
the impedence level of the plate and cathode outputs of the phase
splitter are different.
This difficulty is overcome by the use of two
difference amplifiers (see Fig. 1) as the iriput stages of the multiplier.
The x signal to the multiplier is fed into one of the inputs of
one difference amplifier and the y signal is feed into one of the
inputs of the other difference amplifier.
difference amplifier is grounded.
The second input of each
The output voltages on the four
plates of the two difference amplifiers are proportional to +x,
+y, and - y respectively (disregarding the DC bias).
-x,
These outputs
are combined using a resistive adding matrix to form (x + y),
-(x + y), (x - y) and -(x - y).
The operation of a difference amplifier can be analyzed with
the use of symmetry arguments.
Considering two different
modes of operation, the common mode and the difference mode or
-4-
Ebb
RL
RL
_
m
12AU7
eg2
'RK
FIG. 1:
DIFFERENCE
AMPLIFIER
-5-
equivalently the even and the odd modes ( 4 ) , the grid and plate
voltages are written:
e
e"
=
e
=g2
+e
go0
(2a)
go
(2b)
2 e1g2
(3a)
ge
e
-e
e
if we define:
e
ge
e
+
9gie
e
-e
egl- e2
go0
(3b)
and
e
e pl
+e
e
e
e
(4a)
po
(4b)
2
(5a)
-e
pe
=
po
if we define:
e
pe
+e
e
= epl
2
e
po
e
2
The gain for the odd mode can be determined by inspection from the
incremental model,
Fig. 2.
The cathode resistor is replaced by
a short circuit because of the anti-symmetry in the odd mode.
-6
epo
2
eg,
2
epo
2
ego
FIG. 2: IDEAL MODEL FOR ODD MODE IN
DIFFERENCE AMPLIFIER.
2
-7-
e
r
p
RL
+R
(6)
L
Rearranging terms, the gain for the odd mode is:
go
p
(7)
L
The gain for the even mode is determined from the incremental
model, Fig. 3.
-
ec = -
(8)
2iRk
and
i Er P +R L
L
+2Rj
=
(9)
ec
and
e =-iR
(10)
L
Combining equations (8), (9), and (10):
e
Pe [rp +RL +2Rk]
ItL
Ep
eP
e
e
Rk]
+ 2
(11)
L
Rearranging terms, the gain for the even mode is:
-- -
e
pe
r
RR L
+R
JLJ
+ Z (t +I) R
K
(12)
The input voltage to the multiplier is:
ein
and
gl
(13a)
-8-
epe
2
epe
2
ege
ege
2
2
FIG 3: INCREMENTAL IDEAL MODEL FOR THE EVEN
MODE IN A DIFFERENCE AMPLIFIER.
I
-9-
eg
(13b)
=0
Therefore, from equations (3a) and (3b):
e
ge
=e
go
=
e.
in
(14)
By substituting equation (14) into equations (7) and (12),
from
equations (4a) and (4b) we find:
e 1
e.
in
4 (rp + RL)
L
E
l
r
p
+
r +R
+ RL
L
+2 •( + I) Rk
(15)
or equivalently:
pRL
e
:-n
e.
In
-4 (r
1
+ RL)
p
(16a)
1 +R
L
S+ r
+R
p
L
and
=+
e.in
4
VRL
(rp + RL)
Z(1j + )Rk
(16b)
1r +,R
p
L
If we define:
1
S(p
+ 1) Rk
r +RL
PL
(17)
and
= RL
G = 4 (r RL
+R )
PL
equations (16a) and (16b) become:
(18)
- 10-
ep.
e.
In
e1
e.
in
= - G (1 +E)
(19a)
= + G (1 - ()
(19b)
When Rk > > (r
+ RL),
E<<
1, and the output signals of the
difference amplifier are very nearly equal and opposite.
When
E <.01 the accuracy of the difference amplifier is limit ed by
the accuracy of the load resistors (RL)(one per. cent deposited carbon
resistors are used for the load resistors,
(
c~.0027.)
A more
complete analysis of the difference amplifier, considering different
.
and r
for the two triodes is given in Appendix 1.
By symmetry considerations,
the output impedance and the
potential of the two outputs of the difference amplifier are to a
first order the same.
All things considered, the difference amplifier
seems to be a very adequate input state for this multiplier.
B.
Output Difference Amplifier
The output difference amplifier is quite similar to the input
difference amplifier.
The only difference is that in the output
difference amplifier the output is taken from only one of the two
plates and inputs are supplied to both grids.
Using the same
analysis that was used for the input difference amplifiers, the output
from equation (4b):
signal is,
e
e
=
-e
pe
po
from equations (7) and (12)
(20)
-
- i RL
o
e
e += RL + Z( + 1)Rk
2
i RL
r + RL
+
ge
11-
go
substituting equations (3a) and (3b) into (21)
-
LR
p
L
p
r +R
LL
e
e
ee(
(22)
Rearranging terms and substituting from equations (17) and (18):
e
= G [eg
(1 -
) - eg
(1 ()]
+
(23)
Since the absolute value of the gain is of little interest here,
and the output is taken from only one of the plates, 5 per cent
resistors are used in the plate circuit (RL).
If
<< 1, this
circuit actually does take the difference in the voltage on the two
grids and amplifies this difference by G.
C.
The Squaring Operation
The squaring operation is performed with two 5725 tubes
connected in a hybrid combination of push-pull and parallel (see
Fig. 4).
When the control and suppresser grids of a single tube
are tied together, the plate current-grid voltage characteristic
curve is parabolic over a wide range (see Graph.l).
Replotting
graph 1 on log-log graph paper, Graph 2, we see that the 5725 tube
characteristics fall well within 5 per cent of the desired second
power characteristics.
-12-
-y)2
(X+y)
-(x+y)
5725
(6AS6)
5725
(6AS6)
FIG. 4: HYBRID CONNECTION OF THE 5725's TO
OBTAIN THE SQUARING FUNCTION.
- 13 -
10.0
9.0
8.0
7.0
0
E
z
cr
tar
5.0
w
0-
4.0
3.0
2.0
1.0
-7.0
- 6.0
- 5.0
- 4.0
- 3.0
eg (volts)
J
GRAPH 1
- 2.0
-1.0
0
- 14 -
10
9
8
7
6
5
4
ci
d
E
3
0
_a
2
w
a:
I
_cr.a
ILU
o•
U
C,
z
I4
1.0
.9
.8
.7
.6
.5
0J
.4
.3
.3
.4
.5 .6 .7 .8.9 1.0
3
SIGNAL VOLTAGE (eg-e 0 )(volts)
GRAPH 2
4
5
6 78910
- 15 -
The main disadvantage of using the characteristics of a
vacuum tube to accomplish the squaring operation is the necessity
of biasing each tube individually to the point near cutoff around which
Even after
the plate current-grid voltage curve is parabolic.
this bias point is reached, it would seem, the gain must be
Selecting tubes for this operation is unreasonable since
adjusted.
the tube characteristics should fall within several per cent of
Even after the bias and gain of the tube have been
of each other.
set by some means, what is to assure us that the tube won't drift
with time ?
Fortunately things are not as bad as they seem.
The plate current-
grid voltage characteristics for a 5:725 can be approximated by a
Taylor series:
i
p
i
o
+ i
1
(e
g
- e ) +i
o
)
(e
g
- e )
+
(24)
.
By changing variables:
(e g -e oo ) = pe
(25)
The Taylor series becomes:
ip =io + i
(e
+ i 2 (e
s))
+ ".
(26)
It follows that there are two conditions that must be satisfied, for
satisfactory operation of the multiplier.
Condition 1.. e
Condition 2
o
i2
must be chosen such that i
1
= 0.
2 must have the same value for all the
squaring tubes in the multiplier.
- 16 -
Experimentally, it was found that with 5 tubes that were
= 70 p.amp when Condition 1 is satisfied and
carefully tested; i
i2 e
2
is a constant for all of the tubes tested when Condition 2
= y
is satisfied.
These experimental facts suggest a method of biasing the tube
-and setting the gain at the same time (see Fig. 5).
Let:
R
2
R1 + R
2
(27)
than:
e =j (es - a)
where
(2'8)
is the signal Vltage, and -a is a fixed bias.
es
Approxi-
mating the Taylor series, equation (24), with the first three terms
and subject to Condition 1:
ip = i
+ i (e
- e
2
(29)
Substituting equation (28) into (29):
ii =i +i2
But P is chosen
i
p
=i
(e - a) - e ] 2
)when
(30)
es = 0:
.(31)
o
Therefore
i2 [A (0 - a)-
e
2
=0
(32)
which means:
e0 = - a
(33)
-17
-57
(es
FIG. 5: CIRCUIT FOR BIASING AND ADJUSTING THE
GAIN FOR SQUARING ELEMENT SIMULTANEOUSLY
-
- 18 -
or
e
0
(34)
substituting into equation (30)
(35)
ip i0 +i22 [---a (es -a)- e0
or
e
i
p
=i
o
+i
(
2
e
-2
2
(36)
s
but
i 2 eo
(37)
=
is the same for all tubes, so finally
p
(38)
+4( ) e
i =
o
sa
In words, this means that if the circuit in Fig. 5 is used to
couple the signal voltage to the grids of the squaring element,
when 0 is adjusted such that i
= 70 jIamp when es = 0, both Condition 1
and Condition 2 are satisfied !
This result is not unreasonable.
that the expansion for i
Essentially all that is said is
may be approximated with the first three
terms of equation (24), and that when the grid voltage is equal to
zero, the plate current for any 5725 is approximately the same.
Graph 3 illustrates the accuracy of equation (38) for various 5725
tubes.
Two of the 5725 tubes a.re connected in :the hybrid combination of
push-pulland 'paraliel shown in Fig. :4.
The input to each pair of
- 19 -
10
9
8
7
6
5
4
0
E
3
z
w 2
--crZ
LI.
(-
.1
.9
z .8
w .7
crc:: .6
H
0
I0Z
=,,
w
.5
.4
.3
.2
.3
.4
.5
.6 .7 .8 .9 .1
3
4
5
6 7 891
SIGNAL VOLTAGE (arbitrary units)
GRAPH 3 CHARACTERISTIC CURVES FOR 4
5725'S CONNECTED AS IN FIGURE 5 WITH
SUCH THAT io = 70
-- ---~I-~'
p. amp.
DIFFERENT
(8 ADJUSTED
- 20 -
grids is the connection described in Fig. 5, with 0 adjusted so
that i
When the input signal voltages to the two sides
= 70 Lamp.
of this circuit are equal and opposite,
a two-sided parabolic plate
current-signal voltage characteristic is obtained (Graph 4).
A drawback that tubes in general have for use as non-linear
elements, is their inherent drift of characteristics with time.
In
the case of the difference amplifier, the effects of the temporal
instability in the triodes characteristics tend to be cancelled due
to the symmetry of the circuit.
With the squaring device, some of the
drift effects cancel due to the symmetry of the circuit, but even so,
the effect of drift in the squaring amplifier is much more pronounced
than it is in the difference amplifier.
to have very nice drift characteristics.
Fortunately, the 5725's seem
When turned on for the
first time, their characteristics drift for a period of a day or so.
(See Graph 5).
After this initial "Infant drift", the characteristics
of the 5725 seem to be stabilized.
In other words, if the tube is turned
off and then turned back on, at some later time, after a short warm-up
period, the characteristics assume the value that they had at the
time the tube was turned off.
These characteristics are then quite
stable (see Graph 6).
This property of the 5725 suggests that when first adjusting the
multiplier, one should age the 5725's at least 40 hours before
attempting to allign the unit.
tubes would "age"
Experimentally it was found that the
just as well with heater voltage applied as they
would with both the heater and plate voltages on.
It is therefore
-
-10
-8
-6
-4
-2
2
0
4
6
8
10
e s (arbitrary units)
GRAPH 4 TRANSFER CHARACTERISTICS FOR
TWO 5725 TUBES CONNECTED IN THE HYBRID
CONNECTION OF PUSH-PULL AND PARALLEL
DESCRIBED IN FIGURE
4.
'- -- "--I"
21 -
~I-~~
·-
·---.
-------~-~-i;-~-c------·-.L·,I.~~_
._ii·--~-Y-l.~l~ -·IIIYI·L~I
-~~ -~I~IIX1I-II~X-I.i^i*.l-.iiL
I~
--
7.0
6.8 -
6.6 F6.4
6.2
6.0 -0
5.8
Ebb = 120 volts
5.6 H
eg 2 = 120 volts
5.4 H-
eg1 = eg2
ip = 70
5.2 FH
5.0
.001
I
I
.05
.4
I
.002
GRAPH 5.
)2
"INFANT
DRIFT"
Imp.
I
I
.2
.5
TIME (hours)
= eo
oa mp
I
I
I
I
I
4
2
5
10
20
IN e o FOR A TYPICAL
5725. e o
IS ADJUSTED
50
100
c
(.U
6.8
6.6
6.4
6.2
v
1111
6.0
.-
T
m
1L---T
-4
5.8
Ebb = 420 volts
5.6
eg2 = 120 volts
5.4
eg
= eg 3 = e o
5.2
ip
= 70p.amp
5.0
.004
.002
I
.005
I
.01
I
.02
I
.05
I
.1
I
.2
ISI
.5
II
1
II
2
I
5
I
10
I
20
50
TIME (hours)
GRAPH 6. DRIFT IN e o
24 HOURS.
FOR A TYPICAL,
AGED 5725, AFTER BEING OFF
IS ADJUSTED SO THAT ip = 70/•amp.
FOR
100
- 2-4 -
age the 5725- tube s.a This
p ossible:'to. build a uni-t:specifiplly'rtoq
A: tube socket, and a. filament transformer.
unit would consis:t.-of ncthing riore :t:han.
D.
The Output Cathode Follower
For low impedance output, a cathode follower Fig. 6 was used.
This is the only section of the multiplier where symmetry is not used
The gain of the cathode follower along with the
to give stability.
dependence of the output on the DC supply voltage can be computed
from the ideal model, Fig. 7.
e
e
-0 e C +ir
=
Eb
e
-E
p
By inspection:
=Ebb
(39)
k
(40)
+ iR
or
i =
e
Rk b
(41)
=e
(42)
+e
substituting equation (41) and (42) into (39)
r
e
- i (e
- eo) + (e
- Eb) R
(43)
= Ebb
k
rearranging terms:
r
e
r
- )R =Ebb +Eb
(1 + L +
k
P
+
es
(44)
k
and finally:
r
I (Ebb +_
bb
e
I +
mema
L,
_
_
-------
-
R
Eb) + e
k
r
b
(1 + P )]
SR
k
s
(45)
-25-
Ea.
6C4
+
+
tb
FIG. 6: CATHODE FOLLOWER OUTPUT STAGE
-
--
-Ig
- 26 -
Ebb
+
g
RK
Eb
IODEL FOR THE CATHODE FOLLOWER
FIG. 7: IDEAL IV
OUTPUT STAGE.
- 27 -
is divided into two terms; the signal term
The output voltage
eo,
and the bias term.
In this circuit, since symmetry is not used,
both the gain for the signal term and the value of the bias term
muse be investigated for instability.
The incremental gain of the
cathode follower is:
Ae
0
Aes
e
1
(46)
1 +- ( +
V
Rk
The change in the gain due to a change in
ALe
a(
r
(1 +-
o )
s
= +
k
V is:
)
(
rP
rJ
1+ 1
2
(47)
Z
Rk
R
or approximately:
Ae
Ae
s
=
1
(48)
In this multiplier the tube used in the cathode follower is a
6GC4, which has a
change in
V of 20.
Equation (48) indicates that a 20 per cent
V causes less than a 1 per cent change in the gain of the
cathode follower.
The change in the bias term may be calculated by differentiating
equation (45).
This is approximately:
E
o
__
·
r
1
V
bb
I
VRk
b
-28are 150 volts in magnitude and are regulated within
Ebb and E b
1 volt.
is approximately 7K,
r
a 5 per cent change in
.
V = 20, and R k = 150 K.
will change
e
0
Therefore,
only by 0.3 volts, a 1-volt
change in Ebb will change eo by 50 mv, and a 1-volt change in Eb
will change
e
by & 5 mv.
This cathode follower circuit appears
to be an adequate output circuit for the multiplier.
E.
Power Supply
Power is provided by a regulated power supply of standard
design.
(See Fig. 8).
The plus and minus 150 volt supplies are
controlled with shunt gas diode regulator tubes.
The +270 volt
supply uses a series regulator tube controlled by a high gain DC
amplifier, with the +150 volt supply as the reference standard for
this supply.
iSAl1c01
+•-7oY
I
A..mpr
P cge;~
q~ciarT
ji
-
C47-148
Fig. 8
_
.~l~i--·-
---
Power Supply
/
-
III.
30-
Inteconnections of the "Building Blocks"
Now that all the building blocks have been assembled and
analyzed, we are ready to interconnect them to realize the complete
multiplier.
The block diagram for the multiplier is shown in Fig. 9.
The input difference amplifier must be connected to the squaring
elements through some adding network such that the bias for the
squaring element can be correctly adjusted by the.:method stated
earlier (see Fig. 5).
This is done by direct coupling the plates
of the two difference amplifiers to the grid of the proper squaring
element through resistors with a variable resistor connected
between the grid and cathode of the squaring element (see Fig. 10).
Ifthe bias voltages on the plates of all the difference amplifiers are
the same, the gain and bias point of the 5725 squaring element are
both adjusted correctly when the variable resistor is adjusted so
that the plate current through the 5725 is 70 lamp.
It'.
is unreasonable to expect the characteristics of the triodes
used in the input difference amplifier to be exactly alike.
Adjustable
resistors are added to the input circuit so that the plate voltages
of the tubes in the input circuit may be adjusted to be identical
(see Fig. 11).
The "x bal. " control is adjusted so that both
plates are at the same potential, and then the "x bias" control is
adjusted so that the potential on both plates is 12 volts less than
the 150-volt supply voltage with the input shorted.
Both adjustments
are then repeated for the y input amplifier.
In order to obtain high frequency response, 10 Vtif capacitors
are added in parallel to the 200 K summing resistors, and an
i
-(x+y)
2
-x
xy
y
-I
RESISTIVE
SUMMING
NETWORK
-
(X - y
-y
12AU7
FIG. 9:
I-----------------------~-------~~-··---
6C4
BLOCK DIAGRAM FOR THE MULTIPLIER
- ~;----~ ;-;-r-;--;~ -;----~--i---~
-·-~·----
-- -- ~--.~~---.. 111~-.-_.
+ 270V
+ i50V
40K
1%
200 K
1%
12AU7
X INPUT DIFFERENCE
AMPLIFIER
+ 150V
-150V
5725
5725
1OK
1%
200K
1%
SQUARING ELEMENT
12AU7
+150V
Y INPUT DIFFERENCE
AMPLIFIER
-150 V
FIG. 10 RESISTIVE SUMMING NETWORK BETWEEN THE
DIFFERENCE AMPLIFIER AND THE SQUARING ELEMENT
_I, 1
,3·-- r.-l1·;-_ir;--·-··--··-_-l·--·r_.--·-------·~.i'-~-.-.'*"-~-i~-~~~.~l~l-I
INPUT
-~.^.--pll" "i-~-E'"~~li~·· ·~i-~-I---,~,~ ..~I~I~
- 33 -
+150 V
+ 150V
1OK
1%
+150 V
1OK
00OK
1%
-X
IN
100K
I2AU7
1K
100K
47K
)V
00 K
lOOK
-150V
FIG. 11
INPUT DIFFERENCE AMPLIFIER
CIRCUIT
-
"~
I
- 34-
adjustable 20 ý+f capacitor is connected betwern.ithe grid and cathode
of the squaring element.
The adjustable capacitor is adjusted, after
the bias point for the squaring elements, has been adjusted, so
that the RC time constant for the 10 V±f capacitor in parallel with the
200 K resistor is equal to the RC time constant for the parallel
combination of the input capacitance of the squaring element, the
adjustable capacitor, and the variable resistor.
The output difference amplifier uses a similar resistive divider
network for its input. (See Fig. 12.)
The capacitors are adjusted
to cancel the effects of stray capacitance in the circuit.
In this
circuit the balance of potential on the two plates is not essential
since the output signal is taken from only one of the two plates.
Since the plate of the second half of the output difference amplifier
is connected directly to the output cathode follower, the value of the
plate voltage is critical.
The plate voltage is adjusted by means of the
variable resistor in the cathode circuit.
This resistor is mounted
on the panel of the multiplier for accessibility.
TO PLATES OF SQUARING AMPLIFIER
7Hpýf
500 K
1%
1OK
1OK
-
500 K
7/L
1%
f
Ioo0
6C4
12AU7
150K
1%
OUTPUT
DIFFERENCE
150 K
1%
OUTPUT
CATHODE
FOLLOWER
AMPLIFIER
47K
-150 V
-150V
OUTPUT 0
BAL.
0OOK
-150 V
FIG. 12 OUTPUT
-bcls~i~tp~lisrari~in;rrt~-
DIFFERENCE
AMPLIFIER
QnI
- 36 -
IV.
Over -all Specifications
Up to this point, accuracy and frequency response have been
referred to without mentioning what they mean or how they were
measured.
Determining a good working definition of accuracy and
frequency response for a multiplier is a problem.
With an amplifier,
the terms frequency response and accuracy, or linearity, if you
wish, both have well defined meanings.
With a multiplier, this is
not the case. The output signal is in general at a different frequency
than the input signal, and linearity has little meaning for a non-linear
device.
A simple definition for frequency response, is the frequency
response of the multiplier with one of the inputs connected to a DC
source and the other input connected to a sine wave signal generator.
Using this definition both the frequency response and the rise time
of the multiplier can be measured.
They are:
Frequency Response
DC-5 00 KC
Pulse Rise Time
0.6 Vsec
x input - Signal Generator
y input - DC Source
However, the main use of this multiplier is for multiplying two
signals that have about the same frequency.
For this use, the
frequency response that is of interest is the response of the multiplier
with both inputs connected to a sine wave oscillator.
characteristic is:
x input DC-200 KC lv RMS:
yinput DC-200 KC lvrRMS
xy output
:,L.5
+
DB
DC-400 KC . 8v RMS ..
This response
- 37 -
With an amplifier one measures dynamic range.
With a
multiplier dynamic range measurements have to be made for the
inputs and output separately.
The dynamic range for this multiplier
is:
inputs
3 v RMS maximum or + 4.5 v peak
output
2 v RMS maximum or + 3 v peak
To avoid clipping, both the input and the output voltage must be
within these limits at all times.
A useful measurement of the noise level of the multiplier can
be obtained by shorting one of the inputs and applying the maximum
signal to the other input.
A common way of expressing this figure
is to compare it to the maximum allowable output signal.
multiplier, this "noise level" is -35 DB.
For this
- 38 -
V.
Concluding Remarks
This multiplier is actually better than I had hoped it would
be.
The Acoustic Sonar requires a multiplier with a frequency
response of only 100 KC. This multiplier has an almost flat frequency
response out to 500 KC. Several models of this multiplier were built
and they already have been used successfully in an Acoustic Doppler
Radar and an Ultrasonic communication link.
The operation of the
multiplier seems to be very stable and entirely satisfactory for our
us e.
For improvements the only thing that I might suggest, is the
use of two 6AR8 vacuum tubes as the multiplication element.
I built
a rough circuit using one of these tubes, and was bothered by the
non-linear grid voltage:-plathe cuir'ene
transfer characteristics.
If
two of these tubes were operated in push-pull with a cennr.t&tionjsiihdilar
to that used in this multiplier, with enough cathode feedback to
linearize the grid voltage-cathode current transfer characteristics, it
is quite possible that one could design a multiplier substantially better
than this multiplier.
mwý
- 39 -
BIBLIOGRAPHY
1.
2.
CharleA E.. Cook, "Pulse Compr ession-Key to More Efficient
Radar Transmission", IRE Proc.,
March 1960, p. 310.
J. N. Holmes and J. M. C. Dukes,
"A Speech-Waveform Correla-
tor with Magnetic Tape-Delay and Electronic Multiplication",
Proc. IEE, 101 Part III 225-237, (1954).
3.
John S. Barlowand Robert M. Brown.
System for Brain Potentials ",
4.
5.
MIT, RLE, Technical Report 300.
T. S. Gray, Applied Electronics,
Wiley and Sons, Inc.,
K. W. Goff,
"An Analog Correlator
2nd ed.,
New York, John
1954. Difference Amplifier p. 505.
"Analog Electronic Correlator for Acoustic
Measurements ", Acoustical Society of America, 27, p. 223.
- 40 -
VII.
A.
Appendix
Analysis of a Difference Amplifier
The incremental gain of a difference amplifier can be
computed from the ideal incremental model, Fig. 13.
The
equations for the voltages around the current loops are:
(il
+ i2) Rk +
1 ecl + i
(il + i2) Rk + L Ec2
+
(rpl + RL) = 0
(Al)
(AZ)
i2 (rp + RL) = 0
The grid voltages are:
ecl = esl +(i
(A3)
+ i2) Rk
(A4)
ec2 = es2 + (i 1 +i 2 ) Rk
Combining equation (Al) with equation (A3) and equation (A2) with
equation (A4) and rearranging terms:
(i + i 2) Rk (1 + il) + il (rpl + RL) +l
(il + i 2) Rk (1 + L2 ) + i 2 (rp2 + RL) +
es
2
es
2
=0
(A5)
=0
(A6)
Rearranging terms in equation (A5):
il [(rl1
2
+RL) +Rk ( +1• 1 )
Rk (1 + •i)
+
i l
esl
(A7)
Substituting equation (A7) into equation (A6) and rearranging terms:
ilRk(1 +)
+V.e2
-[i [(rl,+RL)
k
(l+
[Rk (1+~2)+rpZ+R
l Isl] [.+:+
(1L+1
=0
(A8)
-j
1
- 41 -
1i
RL §L ,2
RL
+
e02
rp2
rpi.
-ft 2zec2
- ec
4-c2
+
Es2
si
RK
FIG. 13 IDEAL INCRIMENTAL
DIFFERENCE AMPLIFIER
MODEL OF A
-
-- I
- 42 -
Y
i =
s
es2
Rk (1 +e2e
)+L)2 +RL
Rkkl + l
sl1
Rk (I +I±)+ r
S Rk
R fi1
(A9)
J+ +RL
L
p2
(r)pl
(
+RL)Rk ( 1
+pl) [-Rk(1 +~p)
But the outputs are:
eol =ilRL
(Al10)
eo2 = i2 R L
(Al 1)
Substituting equation (A9) into equation (A10) and rearranging terms:
117 •zL
R es.2
"i
· 2-. +
....L...,1
_
eol =
rp 2
2
+
1
+RL +(
(A12)
__
1 +I
pl
+ RL )
(r p +RL ) (r 2 +RL )
+ p
k
p+
R k (1 + 1
)
or finally
r
(1 F+2 +2Z) R L
k.:.p
+R
. L,
R,k'j'"
eol =
(1 +I
1
)
(rp +RL) +(
+Z)(rpl
.
2
.
+RL) +(rpl +RL) (r
*1
,\
1s
+RL)
(A13)
Rk
and similarly
I -
- 43 -
e o2
(A14)
=-
This result is very interesting and needs some interpretation.
Notice that the denominators in both expressions are the same.
The
gain is proportional to (ýl1 1ý2 + 2 ) in equation (A13) and is proportional
to (ý1 ýl + :L) in equation (A14), therefore t
1
and ý2 can be
substantially different, and the gain for both sides will be the same.
The term:
1+
1
(.A 15:)
1 +-
can be approximated by:
1
i. 2 -
1
(A 16)
++1
which is also very -nearly: equal to: 11..
The term
is also very small compared to 1. This means
pl +L
Rk (-
+
)
that most of the parameters in the difference amplifier can change by
noticeable amou:nts: and:with no noticeable! effect. on the performance of the circuit
B.
Electronic Multipliers that are Commercially .vailable
There are many schemes that have been used for the
multiplication of electrical signals.
This appendix is only a
survey and not an attempt to describe all thermethods that can be used
- 44
to multiply two electrical signals.
1. ..
Time Dimension type multiplier
The time dimension multiplier makes use of the principle:
yT
xy
=
x dt
-
if neither x or y change during the integration.
The x input is
integrated over an interval proportional to y over some set period.
This method is capable of extreme accuracy for low frequency
signals, but is vertially- no good for signals above 10 KC.
Accuracy:
2 per cent - .025 per cent
DC to 100 cycles or 10 KC
Frequency Response:
Price: $750 and up
2.
.
Carrier Type Multiplier
A carrier signal is phase modulated with the x input,
amplitude modulated with the y input, then demodulated with a
syncronous demodulator.
Accuracy: .35 per cent
Frequency Response:
DC to over 5 KC
Price: $1550
3. : .
Diode Quarter Square Multiplier
This unit works on the principle:
xy =
(x +y) 2
4-
(x - y)Z
The squaring function is generated by biasing a number of semiconductor diodes to have break points along a parabola.
- 45 -
Accuracy:
.5 per cent
Frequency Response:
DC - 100 cy
Price: $9875
.
4.
Existing Tabe Quarter Square Multiplier.
The squaring function is generated with the parabolic
characteristics of 6B8 pentodes biased near cutoff.
Accuracy:
Frequency Response:
Price:
.
5.
per cent
not given r5
20 cy - 20 KC
Not commercially available
Hall Effect Device
The x input is a magnetic field, the y input is the current
through the device, and the output is the Hall effect voltage.
The
main difficulty with this device is the problem of establishing a
magnetic field rapidly.
Static Accuracy:
3 per cent
Frequency Response:
Price:
6.
.
DC -
100 cy
not presently available commercially
6AR8 Beam Pentode
The x input controls the current in the tube, the y input
controls the voltage on the deflection plates that deflect the
electrons to one or the other of two plates.
in:the: plate:
cur rt;nt fldwing' thr.ough :.ach .hal .of the tube.
No Information Available
7. '
The output is the difference
5725 Quarter Square Multiplier
Accuracy:
5 per cent
Frequency Response:
DC - 500 KC
Price: not commercially available
1 142
V, .-ZA.
VS 57L
SI-AC-i
-4I.r4 V,4'4
-i
-*1
11),ul OJP-It~
I*f.j
.-//V
4 15o
looM
- I •o
4' lv
Fig.
-"
--
14 Wiring Diagram of Multiplier
47
0
0
p41
Plate 3 Under-chassis View of Multiplier
I
·
--
--
~-~------~~.~---~~.~~1~·
-:~~ ·:-~~:l-iill~·
.,,.... ;*- i . ;.... -, -: ;:- ; ·
..
I
I