LIMITATIONS ON POWER GA IN
IN FEEDBACK AMPLIFIERS
by
RONALD L.ESLIE 'l'HOMPSON
SUBMITTED IN PARTIAL FULFII..LMENT OF THE
1
RE'~UIREMJ~N rs
FOR rHE DEGREE OF
1
MASTER OF SCIENCE
at the
MASSACHUSETI'S INSTITlYrE OF TECHNOLOGY
August, 1953
•
Signature redacted
Signature of Author ,_,.,.___ . ac:
._. _ . _
Department or'Electrica.l Engineer1ns7Auguet 24, 1953
certified by
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38
. LIMI'[A'rIOJS o:·:
I i~
ro:~ER
GAIN
1-'Er.;"CB.'\CK AMPLIFIERS
by
RONALD LESLIE THOMPSON
Submitted to the Depar tment of Electrical Engineering
on August 24, 1953 in partial fulfillment of the
requirements for the degr_ee of Master of Science.
ABS'l'RAC'f
A method of maximizing the power gain in feedback
amplifiers, subject to a given stability margin, is
investigated. As an example, this method is applied
to a grounded base junction transistor amplifier and
to a grounded emitter junction transistor amplifier.
A second function, the unilateral amplification
factor, U, is also considered. This function is
plotted for the junction transistor amplifier, and
this curve is compared with the gain curves of the
transistor feedback circuit.
Derivation of the equations in this paper is given
in the appendix at the end of chapter 3.
Thesis Supervisor: s. J. Mason
Title: Assistant Professor of Electrical Engineering
-ii-
ACKNOWLEDGMl~ N'rs
•rhe author is indebte d to t.he super visor of this the1,;1 s, Profes sor
s.
J. Mason, for his guidance in the choice of a sub ject which proved
to be very interesting.
I would also like to thank him for the many
discussions we had, in which the general direction of the research was
established.
-111-
TABLE
(I r
CON'l'iN'rS
Page
ABS1'.RACT
.......... .. ...... ..... ... .. ... ........ .. ............... ... ...... u-
ACKNOWLEDGEMENT
Cht1.pter 1.
Chapte r 2.
iii
IUTRODUC'l'ION •.• . . • •.
1 . 1 Gain and StabiliLy
1.2 Purpose of th is Investigation
GAIN AND STABILITY CONSIDERATIONS
2.1 General Repl'esentation of Two-Terminal Pair Device
2.2 Stability
2.3 Gain Maximization
2.4 Unilateral Amplification Factor
............. ....................................
......... ...... ..........................
Chapter 3.
.......... .... ... ... ..
.......... ...... ......
........
... ............
.........
............
..... ................ ............
... ......... ... ..... ..... .. .... ... ....... .... ..........
TRANSISTOR GAIN MAXIMIZA'rION
.. . . . ..
Introduction
Grounded Base Connection
Grounded Emitter Connection
Determination of u
..
3. 1
3.2
3.3
3.4
CONCLUDING REMARKS
... . . . . .
APPENDIX
I.
Grounded Base
II. Grounded Emitter
III. U •••• •• ••
BIBLIOGRAPHY
........ .................... ..........
........ ....... .......................
.......... .. .... ................ .............................
-iv-
l
1
2
3
3
4
5
10
12
12
12
19
25
28
29
29
32
35
LIST OF ILLLJSINJTIONS
Fiirre
Page
1
General Amplifier Circuit .....................................
2
Flow Graph for General Amplifier................................4
3
Reduction of Flow Graph ........................................
4
Vector Diagram Showing Stability Margin .........................
8
5
Network With Arbitrary Feedback ................................
10
6
Transformed Unilateral Network .................................
11
7
Grounded Base Junction Transistor ..............................
12
8
10 Log
9
Grounded Base M, k, and H Curves .....................
01
11
IHI
3
and OH Grounded'Base Junction Transistor ............
0 log (1+r)2 vs r
4
14
15
................................... 17
Maximum Permissible Gain Grounded Base Junction Transistor
Case Two ...................................
.................
18
12
Grounded Emitter Junction Transistor
13
10 Log )Hj and eH Grounded Emitter Junction Transistor .........
21
14
Curves of M, k, and 10 Log
23
15
Maximum Permissible Gain Grounded Emitter Junction Transistor
Case Two ...............
................
24
Unilateral Amplification Factor For Junction Transistor ........
26
-v
-
16
IHI
....
Grounded Emitter
20
LTST OF TABIES
Table
1
Values of ZG and ZL for Grounded Base Amplifier ...............
2
Values of ZG and ZL for Grounded Emitter ....................
-vi-
19
CHAPTER I
INTRODUCTI'ON
1.1
Gain and Stability
Basic considerations in designing any network are the amount of gain that is
available from the given network configuration and whether the circuit will be
stable when put into operation.
As instability is approached in any network, the
gain approaches infinity, so that some compromise between the two must be reached.
For analysis of a circuit, the network can be represented as a two-terminalpair device.
These devices can be passive circuits, unilateral active circuits,
or non-unilateral active circuits.
It is this latter type circuit that is of
interest.
The non-unilateral active circuit is characterized by the fact that it has
appreciable internal coupling between the output and input terminals of the
amplifier.
This coupling, if it is of the right phase and magnitude, leads to
regenerative feedback and instability of the network.
With the advent of the transistor, interest in non-unilateral active
networks has greatly increased.
Very little has been written on gain maximization
in networks of this type, and what has been written has considered only those.
cases in which stability was not a problem.
With an unconditionally stable
network, the gain was maximized by synthesizing the load and source impedances
on a conjugate image impedance match basis.
Professor S. J. Mason, of the Massachusetts Institute of Technology, has
written a paper1 concerning the problem of power gain maximization in nonunilateral active networks in which there is a stability problem.
Mason, S. J., "Power Gain in Feedback Amplifiers", RLE Report, July 1952.
-l
-
1.
1.2
Purpose of this Investigation
The purpose of this investigation is to see how the methods of analysis
that Professor Mason has put forward are applied, with specific reference to
transistor circuits.
The small signal incremental equivalent circuit for a
transistor will be used in all the discussions of the specific circuit.
-2-
CHAPTER
I
GAIN AND STABILITY CONSIDERATIONS
2.1
General Representation of Two-Terminal-Pair Device
The -two-terminal-pair network that will be considered here can be
represented as shown in Figure 1.
ZG +(L,
The impedances that specify the network
7
~z
12
EZ ZL
Z?
z2 1
N
GENERAL AMPLIFIER CIRCUIT
FIGURE 1.
N are complex quantities.
E
E2
=
1Zi
The equations characterizing such a device are
+ 12 Z 1 2 = EG
11 Z 2 1 + 12 Z2 2 =
G
I2ZL
(2-1)
(2-2)
Because of the active nature of the network N, Z2 1 does not equal Z..
Equations (2-1) and (2-2) can be represented graphically by a signal flow
graph,l as in Figure 2.
1.
Mason, S. J., "On the Logic of Feedback", MIT E.E. Sc. D. Thesis, 1951.
-3-
z_12
I
E____
312
____
Z,,+ Z'
Z2 22+ ZL
FLOW GRAPH FOR GENERAL AMPLIFIER
FIGURE 2.
This graph gives a clear picture of equations (2-1) and (2-2), and lends
itself to an easy solution of them.
2.2
Stability
In any amplifier, the gain is a function of the transfer admittance or
impedance of the network.
By simplifying the graph of Figure 2, the transfer
admittance of the network is plainly shown.
two steps in Figure 3.
This simplification is done in
The resulting solution for the transfer admittance
T
(o.Z11
(~~ %%(Zz~ZL(I
T)
(b)
FIGURE 3.
REDUCTION OF FLOW. GRAPH
-4-.
Zia)
is easily seen from Figure 3(b) to be
(
- 1
)
(2-3)
An examination of equation (2-3) shows that when T = + 1, the gain of the
network becomes infinite and the circuit is unstable.
A closer examination of
T must be made, in order to determine what measures must be taken to avoid the
point +1.
T is defined as the loop transmission or loop gain of the feedback network
and is of the form
Z
(Z1 1
12 Z 21
+ ZG)(Z22 + ZL
(2 -41
Assume for the moment that the circuit parameters are a known function of
frequency.
A Nyquist plot, in the T plane, of T ( j)
as w varies from zero to infinity.
can then be constructed,
If this plot runs too near the critical point,
+ 1, then a change in any of the parameters may sweep T across the critical point.
In order to take care of such a contingency, a stability margin will be imposed.
This margin will be defined as
(2-5)
M>m= IY
In this case M is in an upper bound, which is deterinined by design
considerations, such as allowable tolerances in the system parameters.
larger the value that is assigned to M, the closer T is to +1.
The
This reduces
the limits of design tolerances and increases the probability of instability.
2.3
Gain Maximization
With this stability margin imposed on- the network, gain maximization must
be considered.
The gain of the network is defined as 'the ratio of some output
-5-
quantity to some input quantity, such as power,. current, or voltage.
The
magnitude of the gain can be either gre3.ter than, less than, or equal to unity.
The
problem here is that of maximizing the power gain in a feedback circuit.
The power gain of a network is defined as the ratio of the power delivered
to the load
PL
(2-6)
RL
122
to the available source power
PS=
k1 2
(2-7)
G
where EG is the peak value of a sine- curve.
The maximum available gain is
therefore
Ga
4 R GRL
(2-8)
2
Combining equations (2-3) and (2-5), the transfer impedance of the network
can be put in the form
12
1
-- = -Z
(2-9)
Substituting this result into equation (2-8), the expression for the maximum
available power gain becomes
Ga = RGRL
(2-10)
2m
Examination of equation (2-10) shows the relationship between gain and
stability margin.
As the margin of stability is decreased (m becomes larger)
the gain is increased.
This is logical, since an oscillator, which is an unstable
feedback circuit, has infinite gain.
The problem then is to maximize this gain
at every frequency by choosing an optimum value of ZG and ZL, while still
maintaining a specified stability margin.
-6-
The problem can be partitioned into
three distinct cases:
and R22 greater than zero and M > m
One
R1
Two
R11 and R22 greater than zero and M = m
Three
R
and/or R22 less than zero and M
=
m
In case one, with M > m, the power gain is maximized at values of ZG and
ZL for which a margin of stability is safely maintained.
For this condition,
a conjugate image impedance match is used for both the generator and load
impedances.
Case three is complicated by possible instability of the amplifier
and the fact that another parameter, N, must be chosen so that
N > n=
R1 1
(2-11)
The choice of N is dictated by design tolerances.
n = N.
For maximum permissible gain
The conjugate image impedance method is not applicable in case three,
and the values of the source and load impedances are based on the value that is
chosen for N.
The method of solution in both these cases is a semi-graphical
one, which will not be considered in this paper.1
A Junction transistor, connected as a grounded base amplifier and as a
grounded emitter amplifier, is considered in this paper.
The input and output
impedances of these devices are positive for all frequencies, and over part of
the frequency spectrum, M > m.
Therefore, case two is of interest.
Again the process of gain maximization is a semi-graphical one, but now
the maximum permissible gain, taking stability into account, will be considered.
The method of gain maximization will be investigat~ed in more detail, but
in order to do this, certain quantities must be defined.
1.
These are
For further information on the design solution for cases one and
three see
Mason, S. J., "Power Gain in Feedback Amplifiers", RLE Report, July 1952.
-7-
H = Z12 Z2 1
f2 (2-12b)
h =
(1+r )2
R2
F
(,+r)2
F =
R
(2-12a)
H
Rl R22
where
(Z
(z=
r =
+ ZG )(Z 2 2 + ZL)
Z 1 2 Z21
(2-13a)
..
Rl
(2-131)
R22
H is a complex number which can be wi-itten as
H
H
LH
(2-14)
With the quantities given in equations (2-12) to (2-14) and the relationship
M
(2-15)
a vector diagram for the determination of ZG and ZL can be drawn.
is shown in Figure 4.
JM
'14
paG-abT
FIGURE 4.
MA:
VECTOR DIAGRAM SHOWING STABILITY MARGIN
-8-
This diagram
In order that a network be considered in case two, E must lie outside
the stability margin curve, M, at some frequency or frequencies.
If, however,
H lies between this M curve and the curve m = oo, conjugate image impedance
matching (case one) can be used for maximizing the gain.
This violates the
given stability margin but the circuit will be stable as long as its parameters
stay constant.
When H lies outside the curve m = oo, then case two must always
be considered, since a conjugate image impedance match will result in an
unstable circuit.
With a chosen value of M, the method of maximization becomes a graphical
one.
A value of r must be chosen so that the construction of Figure 4 can be
made.
From the intersection of f
determined.
with the m = co curve a value of k can be
Using this value of k, and that of r which was previously determined,
equations (2-16) give the optimum values of ZG and ZL at any frequency.
RG = r R
(2-16a)
RL = r R2 2
(2-16b)
XG = (l+r) k
- X11
(2-16c)
XL = (1+r) k R2 2
-X22
(2-16d)
With these values for the generator and load impedances, the maximum
permissible gain becomes
Gp = R1 1 R2 2
Z12
(2-17)
This process of maximization is a long and tedious one, requiring considerable
geometrical construction.
However, if considerable computation is expected,
construction of curves of constant M and constant
k in the 10 log jhj versus eH
plane would be worthwhile.
A second curve of 10 log (1+r)
versus r would be
2
very helpful for obtaining values of r, since 10 log (l+r) can be obtained
from a plot of 10 log JHf versus OH and the constant M curves.
The inethod of
obtaining r, and the use of these curves, will be explained in Chapter 3.
It must be remembered that although the circuit is in case two for some
frequency range, this is not necessarily true for all frequencies.
Therefore,
when H lies inside the given M curve, the maximization process reverts back to
case one and a conjugate image impedance match.
2. 4
Unilateral Amplification Factor
In the preceding discussion, the feedback in the network was only that
which the network contained as an integral part.
If the network input and
output are coupled through an arbitrary lossless reciprocal network, N, as
shown in Figure 5, any amount or kind of feedback can be introduced into the
FIGURE 5.
network.
NETWORK WITH ARBITRARY FEEDBACK
The problem with the network N is to adjust it so that the overall
power gain is maximized, subject to the same stability margin that was
previously imposed.
With this type of coupling a parameter U, called the unilateral
amplification factor, can be defined.
It is of the form
Z
-Z 2
12
21
(Rl222 - R12 R21)
U=
(2-18)
It can be shown1 that this parameter is invariant under permutations of the
network terminals and under the addition of a lossless coupling network.
Therefore
(2-19)
U(Z) = U(Z')
If
jUf > 1, a network N can be chosen which will make Rll and R2 2 greater
than zero.
Therefore, this network,.N, has operated on the determinant of
the network, Z, in such a manner that a zero has appeared in the 1-2 position.
The resulting network for such a situation is shown in Figure 6.
rI,
Z7
0-
I----4
FIGURE 6.
TRANSFORMED UNILATERAL NETWORK-
For the network that is shown in Figure 6 the gain is
jj2
(2-20)
=U(Z')
G =
4 R
R22
Equation (2-20) shows that under the foregoing conditions, Z1 2
=
0, that U
represents the gain of a two-terminal-pair network, regardless of the way the
terminals are connected.
1. Mason, S.J., "Power Gain and Stability Margin in Feedback Amplifiers",
RLE Group 35 Report, May 1953.
-11-
CHNPI'ER III
TRANSISTOR GAIN
3.1
XIMIZATION
Introduction
In the preceding chapter, a method of determining the maximum permissible
gain in a two-terminal-pair network was discussed.
This method considered
stability margin requirements, in order that instability would not be encountered
due to parameter changes.
In this chapter these methods will be applied to a
junction transistor in both the grounded base and grounded emitter connections.
The function U will then be examined. to see how its value compares with the gain
computed for the unit without coupling.
3.2
Grounded Base Connection
The first circuit to be considered is a grounded base junction transistor
amplifier.
The circuit is shown in Figure 7.
The values chosen for the
ltic
cc
r.
= 20 ohms
Cc = 50 Jgf
rb
= 350 ohms
a
r.
= 1.5x10 6 ohms
FIGURE 7.
=
.95
GROUNDED BASE JUNCTION TRANSISTOR
parameters are the average values obtained from a number of transistors.
The gain of the grounded base amplifier was maximized over the frequency
range for which the network was in case two.
arbitrarily chosen for M.
For this work, a value of 2 was
Other values could have been chosen since the
-12-
consideration here is to show how this method applies to a specific circuit
and the results that can be obtainQd from it.
Before going into the analysis the values of the circuit determinant of
the network should be obtained for the grounded base transistor.
These are in
the form
= r
Z
11
b
+ r
((3-ia)
c
(3-1b)
rb
Z1 2
~arc
Z Z= rb +
Z 2 2 = rb +
22
(3-ic)
c
(3-id)
rc
1+ '7C
where
Tc..
(3-2a)
rccc
(3-2b)
a = j
The first step in the maximization procedure is to obtain a plot of 10 logjH
versus
0
H.
Combining equations (2-12a) and (3-1), and inserting the values
of the circuit parameters, H becomes
)(s,3.12x106-jl.82x,07)(Sal.3x
i.8 (+3.12x106+.82xl0
(s+8.75x106)(s-8.75xl0(')(s+6.25xi06
)
(3-3)
)
H
From equation (3-3), by visualizing a pole-zero plot of H, in the s-plane,
curves of 10 log
in Figure 8.
fHI
and eH versus log w can be drawn.
These curves are shown
From these two plots a curve of 10 log jHj versus eH can be drawn,
as shown in Figure 9.
Before continuing further, the use of Figure
9 will be explained.
From
these curves the values of r and k, which are needed for the determination of
-13-
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Evaluation of' k is an easy proces
included in Figure
9.
since curves of constant k are
At any frequency, the value of' k is found where the
value of eH at this frequency crosses the given M curve.
Determination of r requires the use of an auxiliary curve, Figure 10.
Figure 9 is a plot in the 10 log
Ihi
versus
the M curve corresponds to a value of 10 log
eH plane, so that any point on
Ihi.
If logarithms are taken of
equation (2-12a), it becomes
10 log
IH
-
10 log
Ih
=
10 log (l+r )2
(34)
The evaluation of equation (3-4) can- be made by taking the difference between
the 10 log
fH
This value
curve and the curve for the given stability margin.
can be used in conjunction with Figure 10 for obtaining r.
With r now determined,relationship (2-17) can be evaluated at every
frequency for which case two applies.
frequency can then be plotted.
A curve of this function, GP, versus log
Substituting parameter values, equation (2-17)
becomes
GP = 16.92 r2
-5)
(s+8.75xl05)(s-8.75xl05)
(a +1.- 3X104) (a -1. 3x104)
Because r is not a known function of s, a break-point method, using the s-plane
poles and zeros of Gp, can not be used for plotting this' curve.
The plot of
Gp, for the frequency range in which the amplifier is in case two, is found in
Figure 11.
At high frequencies, Op falls off very rapidly.
It is here that the s-plane
factors in equation (3-5) become approximately equal and r becomes much less
than one.
Under these conditions, it is probably better to drop the M = 2
-16-
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the gain.
The reason for this is that the H curve now lies within the m = oo
boundary and stability can be maintained by matching in this manner.
The only remaining problem is to evaluate the generator and load impedances
which will give the maximum gain.
With values obtairied from Figure 9 for r
and k, equations (2-16) can be evaluated.
The values of ZG and ZL at two
frequencies are listed in Table 1.
ZGohms
Frequency
cpa
ZL
ohms
Gain
db
1.59x10 3
118+jl46
318xio 3-j349xlo 3
37
8.75x1o 5
362+j952
354-j2698
12
Table 1.
Values of ZG and ZL for Grounded Base Amplifier
In synthesizing the impedances found above, a series R - L connection
can be used for the source impedance and a series R - C connection for the load
impedance, as long as the values can be practically obtained.
3.2
Grounded Emitter
A second connection to be considered is that of a grounded emitter
junction transistor amplifier.
The method of gain maximization is the same
as that for the grounded base amplifier.
shown in Figure 12.
The circuit under consideration is
The values of the circuit parameters are the same as those
for the grounded base connection, as given in Figure 7.
M = 2 was chosen for the stability margin.
-19-
Here again, a value of
U
7-
FIGURE 12.
GROUNDED EMITTER JUNCTION TRANSISTOR
A set of equations similar to equations (3-1) can be written for this
connection.
These are
Z11
rb"
(36a
Z1 2 =r
Z 21 =r
r.
=
r
(3-6b)
+
(3-6c)
arc
l+ TCs
(1-a)rc
(3-6d)
1+ Tcs
where
'c and s have the same meaning as before.
With the use of equation (2-12a), and the 4 preceding equations, H can
be written as a polynomial of s-plane factors, from which Figure 13 was drawn.
H is of the form
(3-7)
)
H H%056(s-6.25x106
Z .056
-~s
l.)(s -1-3x10
x 2 4)
(s+6.6xlo 4 )(s -6.6x10 4
The approximation comes about because H has two. poles and two zeros at high
frequencies (approximately 12 megacycles) which have little effect.
1.
See Appendix II, page 36.
-20-
Combining
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plotted.
described
that be
;ure 14 is the same ascan
The method of use of
H in Figure 14
-
for Figure 9.
From Figure 14 and equation (2-17) a gain curve for the grounded emitter
amplifier can be drawn.
The equation for the gain, with numerical values
substituted is
Op
=
296 r2
(s+7.75xlo7)(s-7.75xlo7)(s+6.6xlo4 )(S-6.6xlO4)
(3-8)
(s+6.25xlo6)(s-6.25xlOb)(s+1.3x,04)(8-1.3xl00)
Again the variation of r with frequency is not a known function of s, and a
point by point method of evaluation is needed.
The curve that was plotted
(Figure 15) was for the frequency range in which the amplifier was in case two.
One of the big differences in this curve from that of the grounded base
gain curve is the short frequency range it covers.
However, it can be noted
that only over a very small frequency range (9,86 - 19.1 k.c.) does the 10 log HI
curve of Figure 14 lie outside the m = ac curve.
Except for that one
octave in the frequency spectrum, a conjugate image impedance match could be
used, but stability would become a problem at frequencies bordering on this
octave.
This gain curve drops rapidly at high frequencies, resulting from the fact
that r is much less than 1 and the ratio of the vector magnitudes is approaching
unity.
Therefore, at these frequencies, a conjugate image impedance match
should be used in order to try to improve the gain.
However, at low frequencies
the gain is still rising and this would make one other. piece of information of
interest, this being the zero frequency gain of the amplifier.
This gain must
be evaluated on a conjugate image impedance match basis and is 30.9 db.
Although Figure 15 shows the gain to be increasing, it has fallen off somewhat
-22-
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acL
Evaluation of equations (2-16)
.'o frequencies gives the values needed
These values at two frequencies are
-for ZG and ZL to obtain these gains.
listed in Table 2.
Frequency
ZG
ZL'
Gain
cps
ohms
ohms
db
4.46x103
24+j473
30.2x10 3
Table 2.
59+j386
1080+j21,200
34
18.5
832+j5430
Values of ZG and ZL for Grounded Emitter Amplifier
Synthesis of these generator and load impedances calls for a series R
-
L
connection if the element values can be practically obtained.
3.4
Determination of U
In Chapter 2, a unilateral amplification factor, U, was defined.
Since
the value of U is the same regardless of the manner in which the transistor is
connected, one curve suffices for both the grounded base and the grounded
emitter connection.
The numerical value of U is
1
4
(s+9.81x104 )(s-9.81x10
(3-9)
)
U = .845xo14
from which Figure 16 is drawn.
Since U is the gain in a unilateral amplifier, a 'comparison of the curve of
Figure 16 with the gain curves for the grounded base and grounded emitter is
in order.
First look at the grounded base connection.
-25-
Comparison at the lower
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frequencies shows that the grounded base circuit has some
uiilt in negative
feedback, while at higher frequencies, where the gain is dropping, there is
some built in positive feedback.
This u.ans that at the low freque-cyend, a
unilateral grounded base connection realizes the most gain and at high
frequencies, a straight grounded base connection is the best.
However, for the grounded emitter connection, the unilateral connection
realizes more gain over the whole frequency spectrum, indicating that the
grounded emitter transistor has some negative feedback built into it.
The
calculation of the gan of the grounded emitter circuit at zero frequency
showed that even there, this connection could not realize more gain than a
unilateral connection.
-27-
M
M
MW
Concluding Remarks
The method of gain maximization thaL is iudicated in this paper
requires considerable work.
No simple method h&s been arrived at so that
the solution for any network can be reached qtickly.
Values of the generator
and load impedances must be evaluated, frequency by frequency.
However, for
an application of this method, a load and source impedance can be synthesized,
which will maximize the power gain over the frequency range for wkdA
tle
network is to be used.
In determining the function U, which indicates the amount of gain that
can be realized by transforming a feedback network into a unilateral amplifier,
no method has been stated for realizing the transforming network.
Further work
along this line is indicated, since, in some cases, the gain of a network can
be increased by the use of such a coupling network.
-28-
I
Appendix
I
Grounded Base
In Chapter 3, several numerical formulas were given without the actual
mathematics involved in obtaining them.
here.
The mathematics will be outlined
First, the four circuit parameters, Z 1 1 , Z1 2, Z2 1 , and Z22 must be
broken down into their real and imaginary parts.
For Z
very simple, since they are both real impedances at all
Z1 1
rb + r=
Z12 = r
and Z12 this is
frequencies.
-370
(A -l)
=350
(A-2)
However, for the remaining two circuit parameters some work is required.
Z21 is
Z
= rb
21b
+
a0rc
1+ /1a )(1+ frc -S
(A -3)
where a0 is the zero frequency value of a and
a cutoff frequency in radians per second.
Ta is the reciprocal of the
This function can be written as a
product of s-plane poles and zeros or as the sum of its real and imaginary
parts.
Both methods will be considered.
First, Z 2 1 will be written as a quotient of its
pole-zero factors.
Obtaining a common denominator for equation (A-3) and collecting terms of the
same power of a leaves Z21 in the form
Z0
l+ c'
rb'Ta
+ rb(Ta %)s + rb oc(I+TS(
(+ Tas )(1 + 'Ies)
(A J1
Dividing both numerator and denominator by rb Ta Tc, substituting the parameter
values given in Chapter 3, and then factoring the numerator of Z2 1 gives the
relationship
-29-
= 350 (a+3.12xlO6 +jl.82x,0
)(S+3.12x10 -jl.82x107
I
)
Z
~~
)
(s+l.3x104 )(s+6.25xl06
Obtaining Z 2 1 as the sum of its real and imaginary parts requires that the
second term in equation (A-3) be rationalized.
Doing this, and then collecting
even power terms of s as the real part of Z21 and odd power terms of s as the
imaginary part of Z21 results in
Z21= R2 1 + J X 2 1
aorc(1+ Ta cs2 )
A
ac
a
Ta+
a T)s
(1- I CS Al . -ls
(.
[ a )(1.'
c I aa
P 0
(A-6
0
)
rb + -.
Substituting in equation (A-6) gives
2+8.3x10 1 0
(3-3 9xlol2)(s2-l.69xl1M)
(A-7a)
)
R21 = 350 + 1.19 x 1017 (s2
and
5
7.63 x 1023
(s2-39x1012 )(s2
-.
(A -7b)
1.69x108
)
X '= -
A similar procedure must be followed for Z2 2 , except only the real and
imaginary parts need to be obtained.
Z2 2 is
r
(A -8)
1++c
-
Z22 = rb +
Rationalizing this gives
rb +
T
R2 2
350 -
2.7x10l4
(s2-1.69xlob)
X22
2x10 1 0
Z22
-r
A -9)
Substituting
(
A -10a)
82-1.69xl0
A-o0b)
5
-30-
Because of the form in which H was obtained, it will be necessary to
determine R2 2 as a quotient of factors.
Evaluating equation (A-10a) in this
manner gives
(s+8.75xl 5 )(s-8.75xl05)
R22 = 350
(A-11)
)
(s+1. 3xlO )(s -1.3xl04
Substituting the values for Z1 1 ) Zl2, Z2 1, and Z22 in equation (2-12a) results
in equation (3-3) for H.
A similar substitution in equation (2-17), with M = 2,
gives equation (3-5) for the maximum permissible gain.
Evaluation of the generator and load impedances was done at two frequencies.
The first was w = 10
k
=
0.3.
radians per second.
At this frequency r = 0.32 and
From equation (2-16) optimum values of ZG and ZL at this frequency are
ZG = r R
+ j (l+r) k R 11
=.32 x 370 + J (1.32)(.3)370
= 118 + j 146
(A -12)
and
ZL = r R 2 2 + j [(1+r) k R22
-
X22
= .32x995x103 + j [1.32x3x995xlo 3
-
7.44x105]
=318 x 103 - J 349 x 10 3
(A-13)
The second frequency at which the generator and load impedances are
calculated is to = 5.5x106 radians per second.
Optimum values of ZG and ZL at this frequency are
ZG = r Rl
1
+ j (1+r) k R11
=.98x370 + J (1.98) 1.3 370
362 + j 952
and
-31-
(A-14
)
k = 1.3.
At this frequency r = 0.98 and
ZL= r
+ j [(L+r) k R 2 2
22
=.96x3 66 +
=
II
' 22]
- 3.64x10 3 j
l1.98xl.3x36C
(A-15)
354 - j 2698
Grounded Emitter
An analysis similar to that for the grounded base connection will be done
for the grounded emitter connection.
be evaluated.
As in the grounded base case Z
and Z12 are re&l impedances
frequencies and are
Zi = rb + rc = 370
(A-16
Z1 2 = r
(A-17)
)
at all
First, the four circuit parameters will
= 20
Again the remaining two parameters are complex impedances.
Z
= r. 21. era (1+T
(l+
First look at
(A-18)
aC
)
This function will be evaluated both as a quotient of .its s-plane factors and
as the sum of its real and imaginary parts.
Consider first the quotient of factors.
Obtaining a common denominator
and collecting terms of the same power of s, Z2 1 becomes
Zr
a Ic 2 + r6( Ta+ V)s + re-aorc
(1+ rcs)(l+ Ta)
(A-19)
Substitution of the circuit parameter values in equation (A-19) and
obtaining the roots of the polynomials results in
Z
7 )(s-7.4x107 )
= 20(s+8x10
2~ (s-1.3xl04)(s+6.75xl06)
(A-20)
Next the real and imaginary parts of Z21 must be obtained.
Z21 = R2 1 + j X
2
1
a r (1+ 7
= r22
s2 )
2
(1- or2)(1:2s2)
-32-
aorc( a+ 1)s
+ (II
0
(1_ 7 2
a(A-21)
_ 722
Numerical substitution in equation (A-21) gives
2
21 = 20 - 1.19x1017
(6 -39x12
10
(A-22a)
6 9x1)
)( -1.,
and
x
s
(S2 -39xl01-2)(82-j.69x18-)--
7.63x102 3
=
21
(A -22b)
It will be obtained
.
The only remaining parameter to be evaluated is Z
22
as the sum of its real and imaginary parts.
R22 + j X 2 2
Z22
ao
+
\c
+
r
(A-23)
1 + -cs
er
2-
84 - [r 6 (eT2+'?')+ Tar
s 2 +rg -r (1-a
aIar)
(A-24a)
-
=
)
This equation can be broken down into its real and imaginary components.
(1- TC
0
X2 2
Tcs3+rcap(
Tr
(
Q1-(l
-;
a2(-2a+
2 2
. (A-24b)
Substituting in equations (A-24) and factoring gives
20 (8+7.75x107)(s-7.75xl07)(s+6.6x0o4)(s-6.6x
(s.-1 3 x,04)(s+1. 3 x04)(s-6.25xo6.)(s+6.25xlO 6
10)
(A-25a)
)
=
s(s+j6.l7xl06)(s -j6.17xl0 6
)
X22 = 2x10 1 0
(s-1.3xl04)(s+1. 3 x,04)(s-6.25xlO6)(s+6.2 5 xlO6)
Introduction of relations (A-16 ), (A-17),
(A -25b)
(A-20), and (A-24a) into
equation (2-12a) results in equation (A-2 6 ) for H.
)
H = .056 (s+8xl07)(s-7.4xl07)(s-6.25xlo6)(s-l.3xlo 4
(s+7.75x107 )(s -7.75xl07)(s+6.6x1o4)(s-6.6xi0)
(A-26)
The factors which occur at the large values of s (about a a 7.5x107 ) have
little
or no effect on the phase and gain of H at the frequencies of interest.
-33-
Therefore, for computation purposes, these factors may be nrglected.
This is
the approximation that is referred to in connection with equation (3-9).
A similar substitution of equation (A-24 a) in equation (2-17), with M = 2, results
in equation (3-5).
The generator and load impedances were evaluated at two frequencies.
The
first was o = 2.8x104 radians per second, at which frequency r = .065 and
k = 1.2.
Equations (2-16) then gives for ZG and ZL
ZG = r R1 1 + j (l+r) k R 1 1
=
.065x370 + j (1.065)(1.2) 370
(A-27)
= 24 + j 47 3
and
r R2 2 +
ZL
(l+r) k R2 2
-
X22
=
.06 5x16 ,6 00 + j 1.o65xl.2x16,600-201
=
1080 + j 21,200
(A-28)
The second frequency is o = l.9x10 5 radians per second for which r = 0.16
and k
=
0.9.
This leads to the following source and load impedances
ZG = r RH1
+ j (l+r) k RB1
= .16 x370 + J 1-16x9x370
=
59
+
j 386
(A-29)
and
ZL = r R 2 2 + j [(l+r) k R
2 2 - X2 21
= .16x5200 + j
[1.16x.9x5200
= 832 + j 5430
- 0.5]
(A-30)
-34-
III
U
The function U, which is the same for the grounded base and the grounded
emitter connections, was determined by using the circuit values for the
grounded emitter connection.
For the function U, as defined in relationship
(2-18)
Z
21
-
Z - = r
12
e
-
a
1+ 'sc
-
=- arc
(A-31)
1+ 7c5
and
R R
11 22
2R
12 21
-Frao
Ta
Lb 0r a
r
+ (r+r)
ls
2
al
+ (rb+rc)rG +rbrc(1-ao)-
(A-32)
Substitution of the transistor parameters in these two relations, followed
by subsequent division of equation (A-31) by equation (A-32) times 4, results in
equation (3-9).
-35-
-1
BIBLIOGRAPHY
1.
Llewelyn, F. B., "Some Fundamental Properties of Transmission Systems",
IRE Proc. 40, March 1952, pp. 271-283.
2.
Mason, S. J., "On the Logic of Feedback", MIT E.E. ScD. Thesis, 1951.
3.
Mason, S. J., "Power Gain in Feedback Amplifiers", RLE report, July 1952.
4.
Mason, S. J., "Power Gain and Stability Margin in Feedback Amplifiers",
RLE group 35 report, May 1953.
5.
Roberts, Shepard, "Conjugate Image Impedance", IRE Proc. 34, April 1946,
pp. 198P-203P.
-36-