A
A

;__=____.:

‘

____:_:__

57S.

12H

 

A
C

r"; 1‘? '
. arm}:

4 0"
.-

.-
- .

\

-. v.:_..q,. o a
,\

\

.
~.'..A

_ ' 'm'."c
‘39 ..‘:"‘
"Oh.

0
- ‘

own.

-~'..

-"|
\l-

A. O -
"-3616

O}

i .1..-

'xlu.

‘ ‘o
Lanna-1

u‘ |

R

. Lint
l.v.‘
«E

.9

U‘ x;

, «“--‘.——‘

\
‘nl-l.

/‘ \

 

   
  
 
 
  
 
 
  
   
 
 
   

Thlshtoeerflfgthatthe " é

thesis entitled 1 .I

F
" "Feed‘uack Amplifiers Network Analysis" ‘y' ' :
f ' a
I
presented by :

 

John J. LaRue

has been accepted towards fulfillment ‘
of the requirements for ' - ‘ '

M,S. degree big..—

 

U Maia; prof

Date—MERL—

 

 

FEE lBACK A1? PLIF IKE IILT'JORK ANALYSIS

By

John J. LaRue

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

EASTER OF SCIENCE

Department of Electrical Engineering

1954

T\o

\d

\
Ra
\

I "I,
“390

AC K new LED (3- E NT

The author wishes to eXpre

Doctor J. A. :trelzoff for his

development of this thesis.

I” o
u'v
v- o
‘9"
”a:
"' ’«
iv
v a

assistance in the

HHHHHHH HHH
. O O O O O O O
Fuocn~JOWH$7 \ARJF’

[\J PJNNNNNN
O O 0
CD \lokfl-P‘DJNH

Kuhn/eke
0
WNH

4:1:-
0
NH

TABLE or CONTENTS

CHAPTER I

Introduction . . . . . . . . . . . . . . . . . . .
Definitions of Terms . . . . . . . . . . . . . . .
Application of Ohm's Law and Kirchoff's Laws To a
Simple Circuit . . . . . . . . . . . . . . . . .
Mesh Equations . . . . . . . . . . . . . . . . . .
Node Equations . . . . . . . . . . . . . . . . .

.
Constant Current and Constant Voltage Generators . .
Driving Point and Transfer Impedance . . . . . . . .
Driving Point and Transfer Admittance . . . . . . .
Variable Impedance . . . . . . . . . . . . . . . . .
Nesn Equations For a Circui Containing Vacuum
Tubes . . . . . . . . . . . . . . . . . . . . .

CHAPTER II

Elementary Theory of Feedback Circuits . . . . . . .
Return Ratio, Return Difference and Sensitivity . .
Return Ratio and Return Difference . . . . . . . .
Return Difference for a General Reference . . . . .
Return Difference For a Bilateral Element . . . .‘.
Definition of Sensitivity . . . . . . . . . . . . .
Return Difference and Sensitivity in the Case of
Zero Direct Transmission . . . . . . . . . . . . .
General Relationship Between Sensitivity and Return
Difference . . . . . . . . . . . . . . . . . . . .

CRAPTER III

0 O O O O O O O O O O O O O O O O O O O O O O O
Impedance of an Active Circuit . . . . . . . . . .
Examples Of ACtiV'e Impedance 0 e o o o e o o o e o o

Exact Eo"mula for External Gain With Feedback . . .
CHAPTER IV

0 O O O O O C O O O O O O O O 0 O O O O O O O

0
Simplified Computation of WC . . . . . . . . . . .

BIBLIOGRAPHY

Page

\OGJO‘bo NR)

11
18

22

23
26

27
32
33

.2/

q
I

'2
.2

LO
ML
51

56

68

I "1TH CV!) 5. “l 1 'i'IOI‘I

The following paper is based on a portion of the book

Network Analysis and Feedback Amplifier Design b“ H. h. Bode.
y A ) , d

 

Mr. Bode's book may be roughly divided into three sec-
tions. The first section, on which this work is based, con-
sists of basic circuit theory and defining parameters of
feedback amplifiers. The second portion of Bode's book
discusses the tasic properties of the parameters defined in
section one and their use in the design of feedback ampli-
fiers. The third portion of Bode's book deals with Specific
design problems.

This paper is intended to enlarge and illustrate the
theorems of the first section of bode's book by presenting
more detailed derivations and specific examples of some of

the theorems and definitions given in the book.

CHAPTER I

1.1 Introduction

 

The networks considered in this chapter will consist
of linear passive circuit elements and vacuum tubes. The
discussion will be confined to steady state analysis for
simplicity, however, the extension to transient analysis
by the Laplace transformation is immediate and direct.

It is assumed that the reader is familiar with basic
circuit theory, network theorems, and equivalent circuits
for vacuum tubes. The following discussion develops some
of the fundamental concepts of circuit theory in determinant

form for use in later chapters of the paper.

1.2 Definitions of Terms

 

Electric networks are composed of active and passive
elements. Active elements are energy sources such as voltage
or current generators. Passive elements are elements where
energy is stored or dissipated, such as inductors, capaci-
tors and resistors.

The terminals of any element are nodes, i.e. a single
element has two nodes. then two or more terminals are con-
nected together they form a single node.

A number of network elements in series form a branch,

while any continuous closed path forms a loop.

1.3 Application of Ohm's Law and Kirchoff's Laws To a

Simple Circuit

 

Kirchoff's laws may be stated as follows:

1. Voltage law -- The summation of the instantaneous
voltage drops around any closed path is zero.

2. Current law -- The summation of the instantaneous
currents flowing away from any node (flowing to any node) is
zero.

For steady state analysis Kirchoff's laws may be re-
stated, omitting the word "instantaneous."

Ohm's law for steady state:

The voltage drOp across an impedance is equal to the
product of the current flowing through the impedance and the
impedance.

The above three laws in mathematical form are:

{E = o (1-1)
21 2 o . (1-2)
B 2 18 (1-3)

Ohm's law deals with voltage drops or potential differ-
ences between the terminals of any element rather than the
individual potentials at the terminals. Hence, in any net-
work a reference potential must be assumed at some point in
the circuit. A convenient value for the reference potential
is zero.

In Figure 1.1 the resistors in series represent a voltage

dividing network. El: E2, E1, Eu are the potentials at the

terminals indicated. Considering the resistor R1 for example,
by Ohm's law the voltage across R1 is E3 - E2 3 IRl. It is

evident either E3 or E2 may be assumed zero or the reference

 

 

 

 

. Eh
volta e.
S J:“‘
If E2 : O 53
If E3 = 0 -E2 = 1R1
i1 Eh
Fig. 1.1
The correSponding circuits are
£330 Eh
£?=-0
3:
O
o
Ea.
-55
E.
Fig. 1.2 Fig. 1.3

The polarity signs refer to the given reference voltage.

The current law will provide one more equation than is
required for a solution of a problem. In any network contain-
ing n nodes, only n -1 independent current equations may be
written if continuity of current is to hold as stated by the
current law.

The discussion of the number of independent loop equations
is somewhat more involved but may be simplified by writing the
branch equations. Thus, one voltage equation may be written

for each branch of the network. The voltage law may then be

written as the sum of the 11 drops through any given branch,
plus the potential difference between the terminals of the

branch which must be zero.

The following example will illustrate the preceding

 

theory e
R: R3 R!

J
El, ' 1:11 (1 1.1. R1 5 ' EL

 

Fig. l.h

El and E2 represent constant voltage generators. ll, 12,
I3, I“ and IS represent branch currents. Ea, Eb, Ec are the
potentials at the nodes, indicated with respect to an arbi-
trary reference potential.

Choosing Ea = O as the reference for this network, the

voltage equations are

E1 - IlRl -Eb = o (l-t)
I2R2 /'Eb : 0 (1-5)
13R3 -Eb / EC = o (1-6)
IuRu / EC = o (1-7)
EC 4:535 -E2 = o (1-8)
The current equations are
Il / 12 -I3 = o (1-9)
I3 / In -15 = 0 (1-10)

We thus obtain seven equations in seven unknowns.

l.h Mesh Equations

In the preceding equations it should be noted that the

voltages E and Ec can be eliminated from the voltage equations

by use of equations (1-5) and (1-7).

There results three voltage equations:

E1 -IlRl / 12R2 = o (l-ua)
13R3 / 12R2 ~Iufii = o (1-6a)
(l-Ba)

These three equgtions are loop equations as shown in

Figure 1.5.

 

 

 

Two currents may now be eliminated from equations (l-ha),
(l-ba), and (l—Sa) by the use of equations (1-9) and (1-10).

The currents eliminated must be the currents flowing in the

mutual impedance branches, i.e. I2 and In. From (1-9) and

(1-10)

I2 I3 ~Il

Substituting these values of 12

and 1h in (l-ha), (l-éa)

and (l-8a), we obtain
E1 = II (Rl / R2) -I3R2 (l-hb)

o = -Ila2 / I3 (R3 / a2 / Ru) -15a5 (l-bb)

K)

~32 = -1335 / 15 (a5 / Ru) (l-8b)
Equations (l-hb), (l-bb) and (l—8b) are the familiar

mesh equations for the network shown. They may be rewritten

as
31 = Iléll -13213 (l-hb)
o = -IlZ3l / 13333 —15335 (l-bb)

Where Egalis the sum of all passive elements in thex. th
mesh, and Zej represent the sum of all passive elements common
to theeth and J meshes.

For a network containing n meshestthe equations would

take the form

av

E1 = i, If 1,4,
fll

Ea = , 1'2: £415
’8‘

I I

| I

I L~ , .

En _ i I} its}.
13'

A

As a ru57lt of the stove example we may state the following
theorem:
. l ,
Theoren:1 : In any conductively unLted network,
the number of indeeendent closed meshes is one less
than the difference between the number of branches

and the number of nodes.

 

 

1 Hendrik W. Bode, Network Analysis d :eedback Amplifier
Design D. ’an Nostrand Company, Inc., New lorr. Seventh Printing,

Sept. i051, p. 3.

l

Node Eqpations

 

The node equations or current equations can be developed

from equations (l-b) (1-10) by eliminating the voltage equa—

tions.

This is done by solving equations (l-h) (1-8) for

the currents I1 and If and substituting in equations (1-9)
2

and (1-10).

Performing the indicated Operations, we obtain

L;

1‘ T
.LJC LC
R

Since El and E2 are

terms on one side of the

Ew‘i
n3 R3
E's/32:0
R5 R5

known voltages, we collect the known

equations

EL-= / Eb (1 f l,% l ) - §_
31 R2 R3 R3
E§.4-§h. E (1 1 1
R5"“3’Z ° E3"! w" Is
Using the relation G1 = l_' we obtain
Ri
E235 = —Eb63 / EC (63 / 64 / 65) (l-lOb)

Drawing the

noted that the products E

current generators.

network for the above equations,

it may be

1G1, and E2G5 represent constant

.33.. a

(in 6h

 

 

E‘G, I 5'1

%¥;; (fiSrEacir

Equations (l-9b) and (l-lOb) represent the node equations

Fig. 1.6

for the original circuit. Figure 1.6 is an equivalent circuit
for the original circuit where the constant voltage genera-
tors have been replaced by constant current generators.
The above result leads to the following theorem:
Theorem II2 : In a conductively coupled network, the
number of independent node equations is one less than

the number of nodes.

1.6 Constant Current and Constant Voltage Generatopg

 

The two energy sources in Figure l.h, El and E2, were
referred to as constant voltage generators. In a similar
fashion the energy sources in Figure 1.6, ElGl and E2G2, are
referred to as constant current generators. Although such
devices are physically impossible, they occur quite often
in theoretical analysis. If both types of energy sources
occur in the same circuit, considerable confusion can result.
Consider the case of the terminals of a constant voltage
generator connected to the terminals of a constant current
generator, Figure 1.7a. Although such an extreme base will
never exist practically, it illustrates a very important point.

whit" p. 12.

"

 

10

The voltage across the terminals of a constant current gener-
ator is determined by the voltage generator or, in.general,
the voltage across the terminals of a constant current gener-
ator is determined by the circuit connected to the terminals

of the current generator.

.5 q).

Fig. 1.7a

 

No voltage equations can be written considering the
constant current generator as an independent branch. A
similar situation exists for the constant voltage generator.
The current through a constant voltage generator is deter-
'mined by the external circuit, and no node equation can be
written for it. In the example shown, no equations can be
written. This is obvious since no impedance is shown in the
circuit.

The best way to handle such problems is to replace all
constant current generators by constant voltage generators
for a mesh analysis, and to replace all constant voltage
generators by constant current generators for a node analysis.

‘However, in order to illustrate the point involved, the
basic equations for both the node analysis and the mesh analysis
are presented here for the circuit of Figure l.h. The voltage
generator E2 has been replaced by a constant current generator.

The following rules may be stated for interchange of sources:3

’3 Eu ray—F. Gardne and John L. Barnes Transients in
Linear Sys ems, John kiley and bone, New lorkT—tigh . “ting,
p. 1+3-

 

 

 

ll

1. To replace a constant voltage generator by a con-
stant current, there must be at least one impedance in series
with the constant voltage generator.

2. To replace a constant current generator by a con-
stant voltage generator, there must be at least one impedance

in parallel with the constant current generator.

at. a; K,

 

 

 

— Iawn— a
3"" ,_.. 1 1
' I
E 11“ 1“ #:21h
Fig. 1.7

1.1g Driving Point and Transfer Impedance

th

The driving point impedance of the i mesh may be de-

fined as the ratio of voltage produced by a constant voltage

generator inserted in the ith mesh to the current flowing in

1th

the mesliwith.all other energy sources replaced by their

internal impedance.

In the network shown in Figure 1.8
A ”——
ue. Rt

9 «:62:

 

 

Fig. 1.8
the mesh equations may be written as

E1 = 11 (R1 / R3) -I2H3

o = -IlR3 / 12 (R2 / R3)

l2

 

 

RI +RI ‘R’
The impedance determinant is A =
‘33 81983
2.03%: :5 :R: z 5.4 = A 5f"
' o. Kr.“ ‘ E. (flairflz) R: 4R: u
‘4 ,__

where A andA11 denote the determinant and its cofactor.
The general deveIOpment of the driving point impedance
as the ratio of two determinants follows:

The general mesh equations for an n mesh network are

E1 = 11311 7’ I2312 7’ " " 2‘ Ingln

E1 : I13341.1 / 123.2 7‘ ' " " 7‘ Inzjn

Ilgnl / I2Zn2 % ' ' ' / Inznn

En

Assume a generator BL is placed in thex; th mesh and all

other voltage sources are replaced by their internal impedance.

The current in the4L th'mesh is
.Ait
= T"
0 h A
andZ"=§_S: '5‘ ., :—
u' 1.‘ E7. 4“ 4;; (1-11)

It should be noted that the equation (l—ll) does not
give the driving point impedance when the generator E1 is
connected in series with an impedance which has two or more

mesh currents flowing in that branch. Hence, it is not a

13

defining equation for driving point impedance. For example,
suppose it is desired to compute the driving point impedance
when a generator is placed in series with R3 of Figure 1.8.
The simplest solution for this problem is to rearrange the
meshes so only one current flows in R3. One possible re-

arrangement is shown in Figure 1.9.

R. It;

I. R, a

1
E

 

 

 

Figure 1.9
We then obtain
E = 1132 / 12 (R2 / R3)
0 = 11 (R1 / R2) / I2R2
A , K: ‘K; 14?:
KwK: R:

'8!- 32*?3'
.13". E: 1. “1
K1

 

Ra R: + RI ‘3 " (1?,
53.

822 could be computed, or the driving point impedance

in series with R could be computed from the definition of

3

driving point impedance, using the circuit of Figure 1.8 as

 

 

z: E
11 J2
11 = E (R2 / R3)
A
I2=ER3

‘6

1h

 

- E (R2 {R3 ~EB3 "%L23 3

.A

In general, it will be easier to rearrange the mesh

 

 

currents in a network than to compute a driving point impedance
if there is more than one current flowing throng the branch.
Since in an n mesh network, if the currents are rearranged
properly, it is only necessary to evaluate two determinants,
one of order n and one of order n -1, while if the currents
are not rearranged, it will be necessary to evaluate one
determinant of order n and as many of order n -1 as there
are mesh currents flowing through the branch in question.

The transfer impedance between the4Lth and jth meshes is
defined as the ratio of a voltage E placed in thellth
mesh to the current j flowing in the jth mesh, with all other
voltage sources replaced by their internal impedance.

Consider the network of Figure 1.8.

The transfer impedance from mesh one to mesh two,

312, is then
: '5;

H
N
H
l?

R1R2 ,1 31R} ,1 R23;
. R3

N
H
N

ll

 

A general expression for the transfer impedance from the
A.th to the jth mesh for an n mesh network can be obtained

as fOllO‘ur'd"S :

15

The general mesh equations are

El 3 I1211 / I2812 / ' ' ‘ / Ingln
Ilgél / I22i2 / ' ‘ ‘ / Ingin
Ilznl / IZZnZ / ‘ ' ' / Ingnn

If E1 is placed in the‘L th mesh and all'vcltages replaced

31

En

by their internal impedance

(1-12)

Here again, as in the driving point impedance, the voltage
generator E1 cannot be placed in a branch common to two or
more meshes if the determinant expression is to hold.

A simple example will illustrate the above statement.
Consider the circuit of Figure 1.10 with currents chqfisen as

shown

 

9 to

 

: . : '
R: 1'45 ,- It: 1' Kw
*Kl ; K,
Fig. 1.10

Ia and lb are branch currents. If we attempt to compute
the transfer impedance between branch a and branch b by the

it is not clear just how

 

determinant equation 2 . =
13 zlij

we should apply this equation. Definition Eij is the ratio

of the voltage in mesh 4: to the current in mesh j . Lith

all other voltage sources replaced by their internal impedance.

16

But in the circuit shown, branch a is common to meshes l
and 2 and branch b is common to meshes 2 and 3 .

We note, however, that we can compute the transfer im-
pedance if we define the transfer impedance as the ratio of
the voltage in branch a to the current in branch b with
all other sources of energy replaced by their internal im-
pedance.

This definition is somewhat more general than that given
in terms of mesh voltages and currents, because it is inde-
pendent of the choice of mesh currents. By this definition
the transfer impedance gab is

Zabeiii'
2 3

where E is a voltage generator of zero internal im-
0
pedance inserted between the terminals l ‘ and 12 - 13
o ‘ .3 . '
is the current floszng in branch b 1’19“} the +rrr-rals 22

are shorted. In terms of determinants gab becomes

 

 

3 E .. A
ab " -£ A..+£4.. - Janna: A,,+A.,-A..-4..
A "'4'—

In this particular case we must compute one third order
determinant and four second order determinants in order to
compute the transfer impedance.

If the currents in Figure 1.10 were rearranged as in
Figure 1.11, the mesh definition of transfer impedance could

be used.

17

 

 

.H 1
.1.
U
0-3;";
gs

x. ,

 

Fig. 1.11
In this case a generator E placed between terminals
ll' is an element of the first mesh only and the current

. . I , .
flow1ng between terminals JG? when they are snorted lS simply

12. The desired transfer impedance is then
a -__E;_ =_§____ — 0
12 - ~ 7 - ____.
A

General eXpression can be written for the transfer im-
pedance between any two branches of an n mesh network
regardless of the choice of mesh currents in terms of the
determinants of the network and its cofactors. However, such
expressions would be of little practical value due to the
large number of cofactors which must be computed. It will
always be easier to rearrange the mesh currents and then
compute the transfer impedance.

If the transfer impedance is desired between a branch
common to two or more meshes and another branch eomnon to two
or more mesnes, then the currents or meshes must be rearranged

or a complicated eXpression for the transfer impedance will

result.

18

1.8 Driving Point and Transfer Admittance

 

The driving point admittance for the ith node, is de-

fined as, the ratio of the current of a constant current

generator connected to the ith node, and to the voltage

across the nose with all other current sources reolaeed by

their internal admittance.

The general node equations for an n node network are

Il-

H
[—1.
ll

H
:3
H

m
H.
II

E1Y11 / E2Y12 / - — - / anyln
ElYil 7‘ .132in j! "' -' "' fl EnYin

Elyn.%/ E2Yn2 % ' " ' / EnYnn

generator Ii is placed in the ith

node then

11 An

and Yii _-_- Ii a A

33: .Aii

It should be noted at this point that the determinant

expression for YLL is the same as the determinant expression

for Zii; further, the definition of driving point admittance

is the same

as that for driving point impedance, except that

the terms voltage and current are interchanged.

This is true in general and the symbol A will be used

for either an impedance determinant or an admittance deter-

minant. The name immittance will be used to refer to either

impedance or admittance.

On the

admittance is the ratio of the current at the i

basis of the above, the definition of transfer

th node to the

l9

voltage at the jth node. The determinant expression for Yij
is
Yij = A
T’ii

1.9 Variable Impedance

 

If a variable impedance is placed in the ith mesh

(variable admittance in the ith node), this can be greatly
simplified as follows:

The determinant £3 would have the form

 

 

Z11 ‘12""""”“""z'~
A- 321 322'“" 33¢»
311312/“'/(311/3)”'31n
8n1 2n2 Znn

where 211 represents the sum of all the impedance in the ith
mesh except the variable impedance Z.
The determinant of 2 may be expanded in the following

form by elements of the 1th row.

A g. 2,. 4", 4.251.441 0 ~--- 44!.“ olMfl?---+%im4£n

a!
=- ’E 13’, 4" 'I' 1411.

If we denote the original determinant with the variable
element 8 by A. and let A represent the determinant with Z
removed, then we obtain the expression

A'aAa-EALL
If the letter W is used to represent immittance, then a

general expression for node and mesh equation is

20

l
A :4 /WA11 where W is the variable irmnittance.
As in the case of driving point and transfer immittance,

the variable immittances must be in the ith

mesh (node)
alone, and not a mutual element between two or more meshes
(nodes) in order for the above equations to hold.

The result of the above discussion may be used to sim-
plify the expressions for driving point and transfer immit-
tances.

Let Wij represent a driving point or transfer immittance

and V1. be a variable immittance in the Kth mesh or node.

Then Wijuis given by
Wij g A' : A + WAN
162i ‘AniidvnfiiMK

The variable element W does not appear in A , A'K,

(1-13)

 

Ni and AJfKK . Thus these four quantities may be computed
for a given network and Wij may then be computed for any
value of the variable immittance W by evaluating the right
hand side of equation (1-13).

To illustrate equation (1-13) consider the circuit of
Figure 1.12.
Y'

v. v. Y v.

    

 

 

Fig. 1.12

 

A Bode, Op. cit., p. 11, Equations (1-11) and (1-12).

21

Let Y‘ be the variable element. The node equations are

 

 

I = V1 (Y1 / Y2) - V2Y2
O = -V1Y2 7‘ V2 (Y2 )1 Y3 fl Y4) - VBYLL
The driving point admittance for node 1 is
. 1
2. = A, = A ”‘4” (1-1t)
A» Au+Y3 «u
where YJ’Y; -Y1 o
30 A 2 ‘Ya Yai-Y’i-Y, ’Yf
0 ‘7'; 74 (1'15)
Mo A”: Y' +Y1 -Ya'
~71 Ya st +Y, (1-16)
. Ya.fY,¢-*’ 'Y‘
S. I." -374 ‘YQ
(1-17)
6. Ann: Yz‘Yz'J'! (1-18)

Note that equations (1—15), (1-16), (1-17) and (1-18)
are all constants, that is, they are not functions of the
variable admittance ‘1 . Therefore they need only be com-
puted once and Y11 may be evaluated from equation (l-lu)fbr
any value of Y2. 21f the variable element W appears as a
unilateral coupling element5 such as a vacuum tube (see equa-

tion(1-1o) and(l-ll) equation (1-13) becomes

 

 

wij : AfWAQL (1-11,)
A Atjiwdiifl'
5 Ibid., p. 10, Equations 0-13)and O-lL).

 

22

1.10 Mesh Eqpations For a Circuit Containing Vacuum Tubes

 

The circuit of Figure 1.13 is a portion of an n mesh
network. The current flowing in the grid circuit is Ii, and

the current flowing in the plate circuit is Ij.
‘Ey 4IW;[i K5
3;, “L J.

 

 

 

Fig. 1.13

The equivalent circuit for Figure 1.13 is shown in

h.

4.3
N .- 1 .
I;;:_} gdeK’ K;

Figure 1.1h.

 

Fig. 1.1h
The vacuum tube appears as a voltage generator of voltage

~1’11Rg and internal impedance of r The equation for the

p.
jth mesh may be written as
1.13 -P IiRg = IlZJ-l ,1 - - - ,1 IiZJ-i ,1 --- Inzjn
Equation (1-10) may be rewritten as
1.1L, 0 = 112.31 ,1 - - - ,1 Ii (,0 R8 / 23-1) ---- / Inajn
by tranSposing the term ql’IiRg to the right side of the
equation. Thus an unsymmetrical determinant is formed (zij + Zji).
A similar situation will exist for the node equations.
Examples of both mesh and node equations for circuit containing

vacuum tubes are given in later chapters of this paper.

CHAPTER II

2.1 Elementary Theory of Feedback Circuits

A feedback amplifier consists of a standard amplifier
without feedback, or a}! circuit anda network to feed a
portion of the output of the}! circuit.oack to the input of
the ,u circuit. This is the feedback or; circuit. If the
voltage feedback to the fl circuit is in phase with the input
of the fi' circuit, the feedback is positive or regenerative.
This type of feedback is unstable, and the circuit may break
into oscillations. The type of feedback to be discussed in
this paper is negative, where the voltage feedback to the
input of the fl circuit is 1800 out of phase with the input
voltage to the f’ circuit.

The properties of a negative feedback amplifier are
easily studied, since the behavior of the system is completely
determined by the voltages at the terminals. Consider the
block diagram shown in Figure 2.1. The input voltage to the

the voltages

system 18 denoted by Ein’ the output by Eout’

input to the fl circuit by EO and the feedback voltage by if.

 

 

 

 

 

2h-

The following equations may be written

Eout : [I so (2-2)
sf = Psout (2-3)

Eliminating Bo and Er from the equations KE-l), (2-9),

and (2-3) and solving for the gain of the amplifier,

 

Eout 9
Ein
we obtain
0 -
c - Bout e l’ (2.4+)

t, Ein I-sz
r!

In equation (2-h), G is the logarithmic gain of the

circuit. Since/V represents the gain of the amplifier

without feedback, the following theorem1 may be stated:

Theorem III: Feedback reduces the gain of an ampli-

n ' ".r. 3~,Y!L.:"‘

fier by the factor 1 -/"B . The quantity [”5 is known

as the feedback factor and reprcse ents the transmission

around t1 e loop ([0? loop) frwo1the input of the

 

~ . dud-Ark ‘-

circuit, through the," circuit, through theIB circuit,
and back to the input of the/9 circuit.
In ordinary practice, the product la? is much larger than

unity, and equation ”(Z-L) may be rewritten as

"_ ER _ ' -1. _

From equationp (2-5).weB can conclude that if the product
jf’pl >7 1, the gain of the amplifier varies inversely with

3 and is independent of/" , or the gain is approxinately

OF). ‘j.t., 50619, P. :32.

Proportional to the [9 circuit loss. The error in this con-

clusion, due to the (aparture of I ’79 from unity, will
I-ZF ‘
2 :3

be called the [’5 effect or the/lg error in subsequent dis-
cussion.

in order to show more clearly the indeueudence of the
gain of the amplifier from the ["circuit, we differentiate

ER of equation (2-14-) with respect to /’ keeping fl constant

and obtain
15» : _. f’(’fi)-(l-l’3) .. __ ‘fF-H'WB 2 '
Mr (i-I’B)‘ ‘ u 4*») ’- (“Ww 5) ‘

Dividing by (2-L) and rearranging terms

£2~JP'

" (2 ’4)

Eu» P ’“Ffl

l
cu‘ «115v55»= I-7a; d'L”/’
From equation (2-6) we can state the following theorem3:
Theorem IV: The variation in the final gain character—

stic in decibels, per decibel change in the gain of the

Ho

circuit, is reduced by feedback in the ratio 1 -/’E: l.

arram in Figure 2.1 can be redrawn

a.) “

f—Jo

The basic feedback d
in a single line diagram as shown in Figure 2.2. It should be
noted that all of .he theorems regarding feedback develoned
so far are for the particular case when the output of the .l’

circuit is fedback to the input of the .d’circuit through

 

 

h-’

 

Ibid., p. 33.
3 Ibid., p. 33.

26

 

 

 

mm _—— our Pur

 

 

 

IyL
L_____l
r""'1
Lf_J

Fig. 2.2
the fl circuit, and are not valid under other conditions.
For the circuit shown in Figure 2.3, the total gain is given

I
by e8 = f’/’ , and the preceding theorems do not apply.

T177?

INPUT OUT wr

 

2.2 Returngfiatio, Return Difference an Sensitivity

 

The preceding section presented an elementary theory
of feedback amplifiers. From this discussion, there emerged
two main ideas: first, the idea of loop transmission or
return of voltage; and second, the idea of a feedback ampli-
fier being independent of thelf’ circuit. In a normal feed-
back amplifier the }’ circuit is composed of ordinary passive

elements and. vacuum tubes, while the 3 circuit is composed

O

entirely f passive elements. If the ,3 circuit is composed
of passive elements, then the second idea may be restated

fier there is a reduction in the

P-

that in a feedback amol

~

effects of tube variation. In normal circuits these two

 

L Ibid., p. 47.

} .J
[—4
95
<
(I)
U)
0
d
*3
n!
cuf-
(“f—
(7)

ideas are related by cin Je isthemat

term feedback may be aeolied to both.

L .5.

In

tD
c1—
t»
O

xce nal circuits, the correlation between the

down and tie one corresponding to the normal

CL
(5
C4
"3
a
nu
m

t "’V‘ O i

‘

function of feedback

w-

s the idea of loop trapezirsirn or the

product pp . In order to Inc.,-cut any confusion the negative

0
tiff?) T“ tiri {W16

of the loop transmission will be called the rs

 

F3

denoted by the svmb l
J a

O

The quantity 1 “fl? “111 Le given the nape return lif~
/
ferei e and denoted by the symbol 5‘.

 

 

= 1 -/$ = l f T

The concept of reduction in tube variation will be called

[/7

sensitivity and denoted by the s mbol

 

The quantities T, F and S are can loL cue to ~PB, l -fp,
and l - pp respectively. haovever, prose rly def: zed, they are

Inuctirmore gfimuxral timni th6117'.'i 810;:3NIS c'uiiiities and. as ea re-

(1.;
H.
m
0

suit are LUCh LOTS useful in tnsoretical analy

-‘ . In 3. ., w — _1.,- - . -1 i_ V,” ,.
03: IXGtUPH 113 4-0 3:30. 1L6 iJLLIii thiflELIEmCU

 

5 . _ - -. - ~ -* n -- . 1-' 4 ,. - l '3- ~ ‘ ,1 ”A: 2‘» " ~1
Concioer the ClI“ult sheen in rigure 2.L . ror mesh

equation the equivalent circuit appears in Eixure 2.5.

28

 

 

 

 

 

gage T»

 

 

  

 

 

 

 

 

 

 

 

Fig. 2.b
E X B 41,1, ’1
n I'
Fig. 205
[’E, = LCZMsz’zi) "Ia *1 “ 1:11
0 : I1(zy+h*il) ’1‘! if +15. if

0 2 4:3, r1,(i,+a,+ a.)
-P2.1'.= ”-1.315 ’14“, +13) ~125- 1.
(2-7)
0 a «1,1, *1, i,- - I, 2,. + I,(t,-r't.rz,a,)

The mesh equations for the circuit of Figure 2.5 are given

in (2-7). The determinant of (2—7) is given in (2-8).

 

(2-8)
inhale o -?.1 o ‘35;
A: O i‘fi‘fzq O —1 ‘- 15'
-11 O izfigfi’ O O
o ..
“'aV' f’ifi “fhfis i5-
‘11 1g o - a 5. 1,913+},fiy

 

29

If the tube is considered as the variable element in
the circuit, we note W‘ is the quantity/”z,appearing as the
element in the nth row and the third column.

The return ratio, for the element W may be computed

:
by use of the transfer impedance defined in Chapter I.“ By
definition, the transfer impedance from mesh 4 to mesh 3 is
the ratio of a voltage inserted in mesh h to the current
it would cause to flow in mesh 3, with all other voltage
sources replaced by their internal impedance.

Consider the circuit of Figure 2.5 when W = O. This

means that the product,fl Z = O or, in particular, that the

¢
[1 of the tube is zero. This is only a mathematical concept
since the plate resistance and interelectrode impedance re-
main in the circuit. If we then place a constant voltage
generator Eu in mesh h and indicate the determinant with
W SP1. = O by A 0 then, by definition
E E
_E = 2&3 or _E Z I
I z 3
3 Le
The voltage at the grid of the tube is obtained by mul-

tiplying both sides of the above equation by Z ; hence,

' \

r ‘4
.
'51

' ‘Le

The ratio The return ratio may then be obtained

3

54a
n
$53qu

by multiplying by the'fi'of the tube; hence,

 

5 Chapter I, Section 1.7.

\A)

:/12
file
The superscript’ O onlflOhB is unnecessary since the
cofactorA H3 does not contain the element W.
T : WAlm (2-9)
A0

The return difference

 

 

Pal/Tzl/wAm : AO/wAg}
A0 A”

By determinant theory of Chapter I,

F = A -
__.__U__ (2 10)
A .
As a result of (2-10) we may state the following defini—
tion:6

The return difference, or feedback, for any element in
a complete circuit is equal to the ratio of the values assumed
by the circuit determinant, when the Specified element has
its normal valuear=4 of

The equations for the return ratio and return difference
may be deve10ped in terms of the node equations. The equiva-

lent circuit of Figure 2.h is shown in Figure 2.6.

 

 

 

 

Fig. 2.6

 

The node equations for Figure 2.6 are given by equations

(2-11) and the admittance determinant by (2-12).

 

 

5;" ‘1. V'
—: «r— - .1. - J
V&' £-1*—J%:) i; t! '
O : V(?'4 :45.— it) - J11 - .55
V *1 ' 11 ‘13
atr' f‘s z-L~+'-
13 in iv) ‘ (2‘11)
: <15 'r’l/ .1..;.11P_.
'1“; 1. 4 ( "f 1,: 1‘)
a: -_\_/,_ _, V. +\g(._’-+_'. +’_
.l.,.Lm}—L (3 “'cl- 0 - —L-
4: 1'1 i1 it it
0 ##4; O - _'_. - .1...
i‘ *1 i’ i‘ i,
A: - -’_ o -'— 4 J- o 0
i: i: 2. (2—12)
0 -4 1.. _' J. a
‘1‘ fa 1’4£,+1‘
- _'_ l 0 l f ’ |
.- — 0 0-.- —-+—-
z. ‘1 5% it *9

It should be noted that the variable elenmext

as the element in the fourth row

W I Jan-

and

W appears

third column and

The return ratio may be<2omputed in a similar manner as
for the mesh equations. The return ratio is a voltage ratio;
hence, as in the mesh commutation, we introduce a voltage E1+
in place of the voltage generator/p 2g 13 in Figure 2.5.

The constant voltage generator SM is then replaced by a current

generator EE_ . The final circuit is the same as that of
rcp-
Figure 2.6 except that the current generator 7.V3 is replaced
by ELL 0
‘”P

#- =1E“ .-. A
3 (of V3 ALLB

I-.£ulti;_.>lying by}! , we obtain the complete expression as

T = $13 = tA.L1:3 = 7. A h} a W 4Q
”1+ «(A A‘ 4'

As before

F:l/T-l/W_A_%3 =A0/wb = A
A A A6

2.§ Return Difference for a General Reference

 

In Article 2.L an expression for the return difference
for the reference value w = O was developed. This result
can be generalized for any reference value w = k. The gain
of the tube with respect to this reference value k is then

W -K; hence, the return ratio for the reference value k is

TX 3 (W -k)A1!&
A

and the return difference Pk is

 

 

Pk = 1 fl Tk : l % (W ~k) _Au_ : A“ +WA'B'“AH
‘ .A‘ (Y'
Fk = A'uégiwwu-quL : [Ha/Ans : A
‘65 4A“

16
(2-13)
0
If the right hand side of equation (2-13) is multiplied byzAz,

we obtain

‘ ‘_‘A .2: ‘1r‘JH? {51:2—
Fk (ul) - A“ A. A 2A: F‘K) (2-llg)

Stated in words, this result is the theorem:

Theorem V7: The return difference of W for any
reference is equal to the ratio of the return differences,
with zero reference which would be obtained if W assumed
first its normal value, and second, the chosen reference

value.

2.5 Return Difference For a Bilateral Llement
.213... .

developed for a unilateral element. However, it is obvious

The equation for the return difference F =

that the equation is valid mathematically for a bilateral
element. A physical meaning for a bilateral element may be

determined by considering the circuit of Figure 2.8. Let

W : Ri be the internal resistance of the generator E.

R; ' 1‘. R1

 

 

 

The circuit equations are
E = 11 W 7! R1 / HO) "IZRl
o g-Ilsl / (a2 ; R3)
The determinant of 8 is then
:3 ‘UV / R1 71 RC "R1
-R1 R2 / R3
The return ratio is

T = W ‘4» = W

'2'.— “—a.
‘6»
. 4’ .
The express1on _____ is the driVing pOint impedance in
I

mesh one if the internal impedance R1 is omitted. Hence, the
return ratio T is tie ratio of the impedance w to the
impedance presented to W by the rest of the circuit.
The return difference A
r = l +W__..Au = A: __3__4-
zy' l!’ 44

4A»
is the ratio of the driving point impedance of the complete
circuit to the driving point impedance when W a O. This
represents a measure of the effectiveness of a generator
with internal impedance in driving its external circuit.' If
* in Figure 2.‘ is a unit voltage, then a voltage drop across
R1 can be thought as the return voltage, and the difference
between E and the return voltage is the return difference.
An exactly analogous situation exists for the node

analysis.

 

2.6 Definition of Sensitivity8
The sensitivity, S, for a given element W is defined as
8:1 g
5 9 where e : Eout (2_15)

 

M W l~ir1
If 9 is a function of W alone, then the partial

derivatives may be replaced by ordinary derivatives.

n

Letting Ein be a unit voltage

9 H
e = Eout

e _
8 d9 _ O'Rout

Substituting these equations in the equation for S,

we obtain

_dWxEM JV

3579? or g: + 7 (2-16)
Thus S is analogous to the quantity 1 ~19'3, in equation (2—6).
The gain of an amplifier may be obtained by dividing the
output immittance by the transfer immittance from the input
to the output; thus, e9 = —:EIZ_.ER where W“ represents the
output immittance. A
If we consider the plate and grid as part of the fourth

and third meshes (nodes) respectively, then by determinant

th eory , .
a An. WK : A“ + W013»

g _. (2—1 )
e A’+W6u WK 7

 

The transmission when W : 0 known as the direct trans-

mission, can be obtained from (2—17) as

 

8 Ibid., p. 32.

8%: AA?" (2-18)

The sensitivity S may be obtained from (2—17) by

applying the definition

.. ‘ n
we
dlbfifi 3;;

from (2-17)
9 = 1.. (A's: 4-way») ”mm: "4» (A‘+WA-aa)

i?— = -. M‘—
JW 411+“ I)” A. +WAQ3

 

 

 

 

Therefore
5 a v _ . [(A'a ewbausflb'e-WAZQ
VV|LiEg1L;;7r-_' '::;~:”‘~;J ‘N’ ‘Ar*‘hbfiiupfiflfl&(darfiibfl
‘3. * at” 0-
S‘ $[(Aaa 4-w4m3)(A eWA ii?)
=—
A’ Ana»; - Ana *4” (2-19)

Equation (2-19) may be simplified by the general formula9

AA‘bga a Aobdcd - And Acb

LO" a“. 58%, cal, 435
About) =AuAnu-Aubna

s a _I_ 4014 (2-20)
- V 438493

 

 

9 himeographed pamphlet 5185, Rep artment of Electrical
Engineering, hichigan State Colle e.

37

2.7 Return Difference and Sensitivity in the Case of Zero

j

Direct Transmission

 

If the transmission is to be zero when t = O, we see
0 1 O ‘ O Q
from equation (2-18) tnat A“ must also be zero. Sucstituting

this result in equation (2-18), we obtain

S ___ WAnvs(4’ {-Wdtn) : “A (2-21)

A. AHIVJ A.
Equation (2-21) is the same as the equation for the
return difference; thus we have the theorem:
The sensitivity and return difference are equal for
any element whose vanishing leads to zero direct trans-

mission through the circuit as a whole.

2.8 General Relationship Between Sensitivity and neturn

Difference

 

According to the theorem V of Article 2.5, he return
difference and sensitivity are equal for a prescribed element
if the transmission through the circuit is zero when the
element is zero.

Q ‘ GO ‘ 0

Since the quantity e represents the transniss1on when
the given element is zero, as might expect the sensitivity

7

of the difference e9 -e90 for the elenent t will be equal

>1“
(D

to the return difference for the element h. Defining t;

quantity e9 l = e9 -e90 we have

,' é’n 43/41:” A’ 2.]
\J c -— __L_
C 2 R A “0- WA” A.

e 6' "W [Mk 1» WAwu)A°‘ A32. (A‘+WAu]
— R A°(A°+WA«)

.. W [ 4.4.Mut WA°Alvn " 4.114.- W473. A93
AWN was.)

“[flu‘u -‘ 4:145;
A°(A' +WA93)

e‘ = 1.. w. + Aw .1. WA m; mam...) - 4.. 4°. ,4. (A°+ WM.)

 

 

. . .
s1 = wJTe-z __ A “M” 4— “ fr (2-22)
aw [v- 5‘34} 4 «4..-»... In

As a result of (2—22) we may state the following heoram:

The sensitivity of the difference eg —eQO, tetween the

normal output an d the -irect transmission for any element w,
is equal to the return diff eleice for L.
Another useful relati n may be derived by OlVlCin: the
return difierence for the element h by the sensitivity for
the element t.

s;— 2'. 29" = _vAl’A’é

_L. 44"- 4.14.
W 4 Ana.

If the ri': ht hand side of the above equation is Y'T‘...‘ltl-ju1_'.',.€d

(1
U
Q

 

by wj_m_a nd the terms rea.rranged, we obtain
Lb.
_If: —w_.£9.‘_3n £215: _ygnquw: _9__
S An N ATRa A'A ‘ AHW'
F ~ 99 —e 0_ 1 _ ego (2-2a)
ES " ”Fifi” - gt“ /

iLquaticn (2-23) is luxnflil in determining ii‘idue difference

10 O»). C0113." EOQG, P. 5&3.

\V J
\3

between P’and & is great enough to be ceisidere co in making
calculations.
Reference Value for u
, , . . 11 my , , .
Lefinition: ine1"eference value oi any element is, that

value which gives zero tr"n¢“1ssion throug h the circu

H-

t as
a whole, when all other elements of the circuit have their
noxi;na 1 values.

we can compute the reference value for L, W by setting
equation (2—17) equal to zero.

Hence, the reference value for h is given by

, .
WO = - Ana. (53-21;)
A3233
If we let Ll reCresent the departure w “ho from the
; , wl _ v p
reference va ue w - u -bo
_ , ‘uv .. O
v; - t1 / “O :- Ll - An. (2-25)
Ann-[3

Substitutin this value in theequation for the gain,

V“
I [KJ
..- l

 

 

 

 

 

 

we obtain a
o ' ‘_ 2941'. AIL‘IB
e9 _ 15.14.“: A;1«_W'A|us VJ
- A' H" 34") r-A J—N‘” VA» ‘ (3'25)
4m“ '” An. '00 .
when Ll - O we note from (2-25) that W : - ‘44» ;
, . Anus
therefore, the ceterminant Will have the value
I . - .
A=A /WA-os - A - AM. 4,;
Au.” AM"
The value of A. contains the first two terms of the de-
nominator of (2-26). Substituting A. in (2- 2(9) vge obtain
e9 _5 W.AI&"$ W3 5. W'Au‘ll (Fl-27)

 

A. 4' W'A” A. ”R

 

313333., p. 61.

Equation (2-27) is an eXpression for the gain,
W - W0 as the reference value of the variable element
) we can compute the

From (2-2
This will be desig-

the circuit.
based on the same reference value of M.
nated as S and known as the relative ,;

 

 

 

 

 

,1 - —'- _'
S - 3—6— : w. 9-!—
)AW' w'
‘ \
w'LAnn _ 4-03 , _ W'A13
w.A|I-‘l‘ A‘N'A'fl A‘ +N‘A’3
31 : A-+th~u_. AttwoAH-HW‘Wdfiu- A. (249)
A.

‘A' ‘4'

From the theorem of (2-9) we may state that the relative
sensitivity for any element t is equal to the return dif-

ference of W for the reference.

The proof of the above theorem is obvious, since the

relative sensitivity is computed on the basis of WC as a

reference value.

CHAPTLI III

a;

This chapter continues the development of the theorv of
feedback network analysis on the basis of the theorems and
definitions developed in the preceding chapters.

3,2 Impedance of an Active Circuit
he definition of driving point impedance Specified that

all energy sources be replaced by their internal impedance

This 18 a passive impedance. The actual input circuit of a
feedback amplifier contains currents due to the vacuum tubes

in the circuit. Thus, the input impedance of a feedback ampli-
fier will be quite different from that obtained by the usual
definition. Because of this reason we shall define an active
Criving point impedance for the nth mesh as the ratio of a
constant voltage generator inserted in the nth mesh to the
current flowing in the nth mesh.

The activ

(0

driving point im‘cdance for the ntn mesh be-
comes

a Z ___JQL___.___ (3-1)1

.600

If we choose any arbitrary impedance, h, eouation 3-1

may be rewritten as

 

 

O
i ’3
a Z fih ‘£%‘”Q§EL”=-£¥Flli£%-x -—-' (3—2)‘
CM
1 92, Cit., Bode, p. 67.
2 Ibid., p. c7.

o
The ratio A maybe considered as the passive impedance
444M“ ‘ m

of the nth mesh with respect to the element R, since R does
not appear in the equation. However, this may not be a pas-
sive impedance in the normal sense of the word, since, if W

1

is considered the impedance of a vacuum tube and if feedoack
O

is still present when W = 0, then A = 20 is a passive
fir-
.AAu»
impedance with reapect to the element W only.
The ratio A is the return difference with ‘e‘.’ = O as a
__1__

A

reference.

The ratio 404* can be considered as the return difference
llama

with respect to W :hen the self—impedance of the nth mesh
is made infinite, or the terminals between which E is measured
are open.

Denoting the passive impedance by 80, the normal feedback
by F (O) -- (since this is the return ratio when the terminals
across which 2 is measured are short circuited) and the return
difference with 8 open circuited by F («3), (3-2) may be

written as

Z : go %—%£;%~ (3‘3)3

In a similar manner we can obtain an eXpression for the

driving point admittance

Y : YO Ff.) (3-1+)’+
r(¢°)

 

7
7.

C‘ C”

3.19;” p.
& Ibid., p.

 

142

If, in equation (3-1) we let W represent the transimped-

ance of a tube and give it a definite value W1, we obtain

Au; A _, A‘ , I
gwl- x4213... _xflx__ ~if__"’(°’(

Amnn Ann 4' A’» A’ Aw.” Fly ( 6°)
A’nn

 

3-5)

Making the same comp utation for a different value of

' “9 h“: W2, we obtain

 

 

we "42'; A’ .32.? 5E... ‘ H.300) (3‘6)
4'1":
The ratio of the impedances for the two values of W is
gal 2 FW. (0) PM (‘0) (3’7)
2'51; Fur, (o) F “f ( °°’ '

If W1 is considered as the normal Operating value of w
and W2 is a prescribed reference, the following theorem may
be stated:

Theorem VI;5 The ratio of the impedances seen at any
point of a network when a given element W is assigned
two different values, is equal to the ratio of the return
differences for W when the terminals between which the
impedance is measured are first short-circuited and then

open—circuited; if the return differences are computed

by letting the first value of W be the operating value,

and the second be the reference.

 

’5 Ibid., p. 68}

143

The relation between feedback and impedance can be stated

in another way. Suppose an arbitrary impedance Z is added
n

in series with the nth.mesh and let.6 1 represent the deter-

. . . a
minant Wlth 8p included and A represent the impedance

L

when an O. The return difference for the circuit containing

an is—

A . A-fzudnn (3-8)

F w-—- -
A” A’+ In. 4.1:".

Where the symbols A anddnn represent the circuit deter-
minant and its cofactor when the circuit is in the normal con-
dition, that is, Zn is not present.

Choose an such that F = 0
g :._ £3 , (3-9)

n
Abnnv

By the definition of active impedance, equation (3-9)

 

may be written as

an 2 .A : —zn (3—10)
Am»
As a result of (3-10) we can state the following theorem:

6

Theorem VII: The impedance seen in any mesh is the
negative of the impedance whose insertion in that mesh
would give zero return difference for an arbitrarily

chosen element in the circuit.

 

6 lbid., p. b9.

n

333_ Examples of Active Impedance

Consider the circuit of figure 3.1.

 

 

 

 

 

 

 

 

Fig. 3.1

If we assume there is no grid current flowing and that
T1 and T2 are ideal transformers, the input circuit to the

left of AB may be redrawn as follows.

L. L: A

L:
Eu;

 

Fig. 3.2

- 'r. T?
where L - n, /‘m

By applying Thevinen's theorem to the circuit to the

left of the points AB, we obtain the circuit

 

Eu!

 

where Ein = j Ein W L

 

N
ll

030’?

8,; §“Ll)(JWL3 ) )/ ij2

Assuming T2 is an ideal transformer, the circuit to the

right of CD may be redrawn as shown in Figure 3-h-

C
D
Fig. 3.4
where 2L = 1:. and a is the turn's ratio of T .
“1?“ 2

a
Replacing the vacuum tubes in Figure 3.1 by their equi-

valent circuits, we obtain the circuit of Figure 3. S.
“P

 

 

 

 

 

 

 

 

 

 

 

5’, “PE,
+ -
Eu! '
- E.

 

 

 

 

 

 

146

The mesh equations for the circuit_of Figure 3.5 are

‘f” E" :- 1.0m * 1‘.)— I: 3-"?

0 ‘3 ~I’, ¥4+I,(£,+Z;+h)
~PZE73= I3(Af,+i’) -I., 2:,

O a {[31, +I,(1'g+ig+}n)
-}’3 E”: 15(hf3*i1'i3)" Is i 3

O '3 ~15- }: "’ I¢(§,v‘13+13)

In addition the following equations may be written:

Combining these equations we obtain

-hE’g 3 I'o‘" {‘i‘) -11!“ i. I‘ F‘io
O 2 ”I014 *Ia(iu*2r*z‘)

z 13%}, +I:(Ap.+7-’1)-I61'1
2.". ~13}, +I1(%I+i1*i")

14h?" + IsOoru— ?;+1,)- 1"},

o
o
o
a «13-52, 4- Ids. r1,+‘1.)

The determinant of the preceding mesh equation is

1+7

 

 

In -111- - O O O I": I,
‘21: 23: O O o O
o #31; 2 u "tut 0 o
A cal
*0 O o “2., a.“ O o
0 o O f, 1,, 7 s: " i u
o O o O “1"- 1“
where Zij = gji except where indicated.

The active impedance between the terminals AAl, or the
active driving point impedance 266 may be computed in several
ways. The first method will be by applying formula (3—3) of

Article 3.2 and using the determinants.

p.

gee = go 2 (2;

’11

where Z0 is the passive impedance of the 6th mesh. F (O)
is the return difference with the terminals AAl short circuited
or the circuit in the normal condition, and F ('9) is the re-
turn differences with the terminals AAl open circuited.

The element W in this example will be taken as P. 810.

By determinate theory

a = A.
o, 66 T
2., db; 0 o 0 M2,
-1” 1” O o o a (3-11)

. 0 I31; “I” ~iu O a

—A-.- a: o O - in "tn" 0 O
A“ o o a o 1,, 4:“
a 0 0 0 ~ i“- 1»

 

 

LB

2” ’ in, O O 0
-;2l *2:- o O o
o
A“ 2 o A k 2,, ‘2” o
' a ‘iq: 1 u ¥3f
o O O 0 " it:

 

 

EXpanding the numerator and denominator of (3-11) by a
Laplace development of the second and third columns
I in “I u. 0 flfl; I
"*3 is: O O

 

 

 

i 33 " 124' 0 0 19-,- -Zu (3_12)
A. - 47:» in i o a «1:»- *4
Eu ~ 2;; * 2:1 2:- ~11; 0
‘iv: 321" '1’): '13: 0
o o 2 5’:
Again eXpanoing
2» ‘ in. 255 ~1uI *3: 15‘

 

 

 

 

 

A ’12: 7-1; " its in ‘3 b! *4
3 - .

A “ l in “its
”22: in.

is:
2,, (3-13)

 

Substituting values from the mesh equations in equation (3-13),

we obtain 'flfwiQ-ris ~23 l
.. ~3‘ A ' 5! +1. i
20, 66 ' ‘ ’ , I If ’-
“’31'21 +13

 

(3-1u)
-a /3 ,la -21
- l 3 3 el— . P

.nfg /I31,/ ‘33
It should be noted that the E0 given by (3-lL) is not

the value of SO we would expect from an examination of the

original circuit in Figure 3.1.7 The reason for the discrepancy

 

I Bode, p. 57.

1:9

appears very clearly in the equivalent circuit of Figure 3.5.

Lith the W of he third tube C, there is still a complete
path for the current I5 and, thus, we should expect a term

. . p' . . a l a
involv1ng at to appear in the eXpreSSion for 5AA or e64.

If it is assumed that the output impedance is large
compared with the 3 circuit impedance, then equation (3-lh)

becomes

0, as “ Zl / 22 / 23 (3-15)

A similar difficulty may exist in the input circuit.

Z

n order that 80 be the passive inpedance, as used by Lode,
it is necessary to make the additional assumption that input

impedance be large compared to the 5 circuit.

 

 

 

F (O) = 27 ‘4 1*P3210.éA_:1
12a "2nu O ‘0 1%2;
~15, ‘11; o «o o
O Pair in 0 0 -(3-l€:)
° ° 'iwz 0’ °
19:! 2 ‘0 o 0 "1ur 15‘
‘A. 2% “IWL O ‘3 " ’7iu
'i'g. in o o o o
0 5*: in " ‘2" 0 o
O 0 wk“ 2,, a o
‘9 4O ‘0 f3lh¢ ihufila
o 0 0 o - hr 2“

 

 

By a Laplace development (3-16) becomes

 

' o MZ'|(-iac)0’a ERGO-Ln)

” "}H§ in;

‘ 2.. -;.:| 2.. 4.1“!“ 4..
“in 1" ‘12. “2:; ‘1’»- z

: I": 2. genre 2 5 Z 13
( 133 in ‘ Z;)( 15 z..-L.)(h:1...~ 25:)

(3-17)
yJIIOPl 2: 21; P3 Zcha

«" = ’4' ——§ fir ‘ '2...
I (O ) (}33}«'zfl)(ia zat”zls)(}sy iuy'zg '2 18 )

Equation (3-18) gives the complete exoression for L (O)
or the return difference when the circuit is in

the normal
condition.

F (0) may also be written as

 

F (O) = 1 4‘33- (3-19)
PC“)?- A—iawzli- PJI'IoALQS‘V
. A.“ A“ A'u
I. ’2'!)- ° 0
_u 65.. Zn. 0 o a.
Ausv’ F.- o ’32, 2” o o
c’ 9 41¢) "

 

 

2. an. o o o
. I ‘qu 7taz.‘° a '0
A“.Tr 0 [’12,- 1” 'i” O
.. O O fin 1'" °

 

 

 

o O O 0 if?
Therefore, F‘(°°) is given by
F (‘39) 3’1!” in (‘) :l (3'20)
.
5‘

The comwlete expression for the active impedance of EAA

is given by the product of (3-1L), (3-15) and (3-20).

3-
626 [ f 1* 3 ’I' 114,] (‘ PF)

Under the condition that the output impedance is much

greater than the fl circuit impedance, (3-21) reduces to

2% = (21 ,1 22 ; 23m +3) (3-22)

U
o

The active impedance 866 is, therefore, smaller than

(3 r-v

iv im nee e

d-
H4
P!
(D
’U
(I)

o, 66'

The preceding calculations could have been simplified,
somewhat. If the assumption that the input impedance and
output impedance are much greater than the ,0 circuit im-
pedance, the passive impedance 20’ 66 can be computed immedi-

ately as

”0,66 ' éi / Z2 / 53
F (O) is the normal return difference and must be computed.
('9) = lxfiIB when the terminals AAl are Open. In this

condition the product [*5 3 0, therefore

gab : (g1 / 22 / 23)(1 efimB)

jgh Etact Formula for External Gain iith feedback

 

In Chapter II, it was stated that the gain of an ampli-
9 1 ‘ 1 8 '
fier was reduced by tne mlnt of feedback. This statement

seems simple enough, but in attempting to apply it, it is not
clear just what the gcin before 1eedback is. ince trere

is energy lost in the B circuit, it is not clear if the B
circuit should be considered in computing the gain before feed-
back. In addition, the circuits containing appreciable direct
transmission, the question arises as to whether we should in-

clude the direct transmission as part of the gain.

 

8 See Article 2.1, p.

The last problem presents a point of deoarture. we

shall define the gain before feedback or fractionated gain

K.

as follows:

e°"= F (.9 - e90) 9 (3-23)
It should be noted that in defining eQF, the direct

transmission term is subtracted from the final gain e9 and.

is not considered as a gain term in the definition.
Equation (3-23) may be rewritten as

e0 - e90 u 1 e9? (3-2h)‘

_
T"
L1 ‘

.L

From equations (2-17) and (2-18) we obtain

90 -e00 =(AA2' _ AZZL)WK (3"«z-5)

3 A0114 W411” .. A’n WR‘
( A. + VA 13 A. .
3 Act“ 4- A.VA [1’3 ' AOA.II ‘ VA71 A u

A (A6 4’WAUJ) WR

.,_._ yM‘Ams ~A°u A»
M A’HVA .3) WK

The last equation of (3-25) may be simplified by the
general determinant relation of Chapter II, and the expression

becomes

e9 ~€GO : ‘WéiAu wR (3-26))

A A‘
. A.
Multiplying the right side of (3-2b) by we obtain

90 -890 Z .. WA—ll All TR A (3-27)

 

 

9 92. Cit., Bode, p. 81.

 

53

Comparing equation (3~2h) and (3—27) we find
86F: - W Al) A01. (3-28)
lY' (V

The formula for the fractionated gain (3-28) can be

IV?

 

considered as the product of three factors 4:3 which repre-
O

sents the transmission from the input circuit to the grid

of the element W, 1341 hh which represents the transmis-
(3'

sion from the plate of the tube to the output circuit, and n

the transmittance of the tube in question. The first two

terms A" and 44; WE: contain the B circuit impedance.
(5‘ 1A'

An example of the calculation of the fractionated gain
is furnished by the circuit of Figure 3.1. The actual cal-
culation will be based on the equivalent circuit of Figure
3.5. The element w‘ will again be taken as the third tube

.
and 1: 2.0.3310.

In the original derivation of gain in Chapter II, the
subscripts l,_2, L, and 3 referred to the input, output,
plate and grid of the circuit and tube under discussion re-
Spectively. In the equivalent circuit of Figure 3.5, the
subscript 1 refers to the input circuit, the subscript h refers
to the grid of the third tube and the subscript 5 refers to
both the plate of the third tube and to the output circuit.
With this change in notation, (3-28) becomes

ees = 4‘ 3310 A" A.“ a. (3-29)
(A‘)’
Substituting in (3-29), we obtain

 

 

 

 

 

 

 

 

’11: In. 0 O 0 1,, 0 O 0 hi.
6‘ ' o In. 1.. o o l ,1“ a o o o ,
C’filbfp 0 a 1w 0 ° (77, o E” ”131 0 0 i"
0 0 hi» {53'2“ a a )3}, '1‘, -1«
0 o 0 “za- 2“ 0 “in in.
17:7 {7-3: -le 2. -z.. a” —- :al (3 30>
"ill 3 211 ‘i 2. 131. 1‘; a“
‘2), 2”- O 1" O I”;
0 Fair‘in If” “Izl win 0 0
=~f32m. O O 2" O ’3 ‘0 “in. in:
”231' 1 33 " i3! 2’. " I"; 1
-:: -if’ '12: 1::

 

: P1210 lat/’12;- 7:411:31": in is: in I" ‘1-
(1917:“~2u)(luzcv°z”)t(z ‘W-i

‘The third expression in equation (3—30) represents the
complete expression for the fractionated gain of the circuit
of Figure 3.1.

The circuit of Figure 3.1 presents another problem in
the use of the determinant formulas. Consider the expression

. . 0 . .
for the gain e in equation (3-17)

0_ An.
Changing the subscripts to agree with Figure 3.1, equation (3-17)

.
becomes 60 = —A--” —-—WK- Aunt-W034,- (3-31)
I! (S"’\Vllur

A.az+ WAn-H
A. *WAIn

e WR = wn (3-17)

The second order cofactor £3 lSLS seems to represent
an unsymmetrical determinant or to have no meaning. The
apparent discrepancy does not exist; however, for an exam-
ination of the original determinant, it is noted that the
element W does not appear in the cofactmq'ZSlS. Hence,
the last term on the right of equation (3-31) does not exist,
and the correct expression for the gain e9 is simply

g A An

= ._£L_ :- a-
e A M A'+ IVA.” (J 32)

The direct transmission gain equation (3-18) becomes
39 : A"
A.

In many cases an unsymmetrical array will appear and

“VR

 

seem to indicate an error in the determinant theory. However,
all of these cases will be similar to equation (3-31), and
the term containing the unsymmetrical array actually does

not exist and may be dropped from the equation.

C ILAPTLR IV

“.1
This chapter will be devoted to the comoutation of HO.
Since many of the calculations involving feedback amplifiers

can be carried out easier in terms of W1 and 81

and these
quantities in turn depend on W0, it is of some importance to

be able to compute ho easily.

Q.2 Simplified Computation of We
If the amplifier in question belongs to a class indi-

cated by Figure #01, 4,2, u,3 and h.u, the computation of We
1

can be simplified.

 

Fig. M-1 Fig. h-Z

   

Fig. u.3 I Fig. Uou

 

l 00. Cit., bode, p. 89.

By definition W0 is the reference value of any element
which.gives zero transmission throughthe circuit as a whole
when all other elements have their normal values.

When the circuit is in the reference condition, the
output is zero; thus, the total voltage across the load or
the current through the load must be zero. This demands
that the voltage or current supplied to the load by the tube
must cancel the voltage or current supplied to the load by
the rest of the circuit. This calculation might still be
difficult, for in order to evaluate the voltage or current
supplied to load by the tube, one must take into account
the feedback of the circuit. However, if the feedback path
can be interrupted, the complete calculation can be simpli-
fied.

Consider the case of circuit such as that in Figure b.l.
Since there is no voltage in the load, it follows that there
is no voltage between the terminals AAl when the tube is in
'the reference condition. Hence, in order to compute NO,
the terminals AAl may be shorted provided that we define the
reference condition as that which supplies zero current
through the short circuit. Eith the terminals AAl short—
circuited, the feedback path is interrupted, and W0 may be
computed as a transmittance between the grid and the plate
with the tube dead. The tube must have this transmittance
in order that the current be zero through the short circuit.

The circuit of Figure L.S provides an example of this

type of calculation. he shall first conpute Lb by requiring

58

that the gain be zero and then by the simplified method

f—Ei—I _

. . 2
discussed above.

 

 

 

 

 

 

 

Fig. LIME
The equivalent circuit on the node basis for the circuit

of Figure b.S is shown by Figure h.6.

 

 

 

 

 

 

Ya ‘ V1
IvfiY. f ‘3‘“ “p? fig
Fig. h.b

The node equations are

(ll-1)
- ngl 3 -VlY3 / V2 (Yu / Y5 f l_)
“P’
where W is the transconductance of the tube 54». ab may be

computed by requiring that V2 be zero. Therefore

Y1 7‘ Y2 7‘ Y3 EY.
"ID-Y3 0

II
0

(h-B)

 

2 Ibido, p. 900

 

59

Hence, EYl (gm -Y3) - 0. Since Y1 f 0, W0 : gmo = Y3.

By the simplified method of computation the admittances
Yu and Y5 are shorted. W0 is then the ratio of the current
through Y3 divided by voltage between the grid and cathode.
By inspection this is

W : V1Y3

° : Y (L-.>
“ii—1" 3 )4

The value of WC obtained by the two methods is identical.

A similar situation occurs in the circuit of Figure h.2.
However, if the terminals B, B" are shorted, the feedback path
is not interrupted. This may be accomplished by opening the
terminals B B" and requiring that the voltage between these
terminals be zero in the reference conditions.

An example of.this type of calculation is furnished by

the circuit of Figure 14.7.3

:3 ‘3

Fig. u.7

As in the previous example, we shall compute W0 first by

'0
0‘

 

requiring that the gain be zero. The equivalent circuit on

the mesh basis is shown by Figure L.8.

 

3 Ibid., p. 90.

60

 

 

 

" 99' _
a «We a
a 2' ' Q

 

The mesh equations are

0 =,u 2211 .122L; 7‘ I3 (Pp )1 Eli)

W in the above ecuation is 2 ~ thus we shall comoute
i 2) i

a value W0 = P082 which will make the current 13 = 0.
Thus
: 0
Z Z
2 Pp / u
and
W0 = Poge : 33 (IT) 7! EL) (14-7)
ml
Ll

This is in contrast to the value of WC given by Bode’s
equation (6--’~)).l‘L

W0 may also be computed by the node equations. The
equivalent circuit of Figure L.7 on the node basis is piven

by Figure 4.9.

 

Li Ibido, p. (XL).

61

 

 

 

 

‘M VE— g E?
3.
[fj ¢’.WT~Q 4?
EE" 7 w; ’ 3:.

 

 

Fis . 14.9

The node equations are

it; : wvrl (:1- 7! 22) "'V»)
=1 r51 a2 a; (1,-8)
-sm ”1 - v3)= v2 (1 74 l 7&1.) - v3 (3.; 7‘ 3;)

rp Ir a; 1‘1) =1;

“vagrant-i,- ;~)-I6 (if? i, l”: ( '2". *éfié’, +#+r-)

The determinant of (L-B) is

 

 

21(3- / 3:) o ’ El. "’
£51 n2 fa"
an '(i 2‘ a. .-/- l) - (l -/— l -/— an) (ta-9)
Pp EL gr rp Eu
r 2
-.1_-£*'1 -(l2‘l) Mil/.1. /l/em>
82 rp an 82 £3 Eh rp

gm -(l / l / gm) (L-lO)
Pp Eu
- wag/gm) (.1. r’l /_l_ 21;. mm)
8’3 1!: g I")
L 3 L_ 1
sm(l 7‘; 7‘}. /_1_ r’sx)=(;l_ ’Uu)(_1_;/_l_r’sh)
82 53 EA 1p :2 Pp nu

62

P w \
W0 ._._ 5mg g 63 (Pp 71 2:1,)- (ii-ll)

 

F7 f'
a h

,3 ’ u
L e
a ti (--11) ' ° ‘t‘a t t a * .. et'o (6-10 5
aqua on . 18 in con res 0 Boos s equa ion _ .
W is essentially a function of the tube parameter/U o
0
or gmo.

If the equation

gmo = 33 (rp / EL)

Eggu rp

is multiplied by rpza, we obtain

(rp / it) ' (f—12>

F0 2 ‘3 m
a“

To

The value of ’10 obtained in equation (u-l ) is the

same as that in equativn (h-7).

The value of W

0 may be computed by the Simplified method

discussed previously. The conditioh that the voltage across

\ l o o ' a
BB be zero in Figure L.8 requires that the voltage across

be equal and opposite to tha across 8 . W

, , under these
r .2 0

EL
conditions is the ratio of the voltage across EA to the cur-

rent in the grid circuit. Assuming a curre.t 11 the voltage

 

across 23 is I, 83.
V83 : Ilg3 (u-l3)
The voltage across EL is
V L “III 89 X g!
is 1 ‘- 1* (14-11;)

rp /fgu
Equating (L-lB) and (h-lh) and changing the sign we obtain
1123 = .é;}132 K-gt
rp fwgh (L'lj)

 

 

f‘,

5. Ibid., p. ‘7 I

0‘
U)

0
Thus

wb =1 p082 = ”3 (Sp / gt) (t—lé)
or

p0 = 23 (radii). we

 

5‘?
aamu

This agrees with the value of ’10 given by equations
(h-Y) and (L-lZ).

On the node basis, Figure 4.9, the calculation is simi-
lar to that of the previous method except the W0 = the ratio
of the current in an to the voltage across the grid and
cathode of the tube. Thus, as before, ssume a current I1
in 82. The voltage across Z0 is

.2

V3? 2 1123 (u’lg)

The voltage across St is

Vin - gm I32 X Pp g#__ (u_19)
rp 7 Eu
Equating h-lB and the negative of h—l9, we obtain

IlZEPP ZN
m..- ._. .‘1’...
Pp /~Eu

Thus we = gm is

1123 3 gmo

W0 3 gmo = 33 (Pp % Eb) (L~20)
Pp gégu

 

61+

This is the same value as obtained from equation (h-ll).

The circuits of Figures h.3 and h.h differ slightly from
those just discussed.

Consider the circuit of Figure h.3. As in the previous
cases when the tube is in the reference condition, there is
no voltage or current in the load. However, we cannot short
circuit the plate and cathode as in Figure h.l since this
condition would still leave the feedback path open. he can
make the output hero by shorting the grid to the cathode
and interrupting the feedback path. This can be done by
supposing a voltage generator of zero internal impedance is
connected between the grid and cathode. The reference value
W6 may then be computed as the ratio of a current generator
in the plate circuit to the voltage in the grid circuit when
both sources are adjusted to produce the same current in the
load with the tube dead.

This type of calculation can be illustrated by the cir-
cuit of Figure h.5 and its equivalent circuit, Figure h.6.

If a voltage generator is placed between the grid and cathode

of Iigure 4.6, a voltage V will appear in the load equal to

 

 

v2 = 1: Y3 (l / Yu 2‘ Y5) 1 (14,21)
rpr v x 7:1./ Yu /Y5)
3";[137TLL'T7FYSJ rp

A current generator I placed across the load will
cause a voltape VL across the load equal to

V=I(rp 121-17 /——1:)
L . /IY3 1h 15 (h—QQ)

65

Equating (h-Zl) and (h-EZ), we obtain

W0 : 3; : Y3 (14")3)

The netWork of Figure b.u, may be analyzed in a similar
fashion except that in this case the feedback path is inter—
rUpted by connecting a current generator of infinite internal
impedance in series with the grid. W0 is then equal to the
ratio of a constant voltage generator connected in the plate
circuit to the current generator in the grid circuit when
both are adjusted to produce the same output with the tube
dead.

An example is furnished by the circuit of Figure h.7
and its equivalent circuit Figure L.8. The equi*alent cir-
cuit of Figure h.8 with a constant current generator in the

grid circuit 18 shown in Figure h.lO.

 

 

 

 

 

A? 7 Vb
E z”
I“’ Wt
2:
Fig. h.lO

The constant current generator I has an infinite
internal impedance and therefore interrupts the feedback
path. By superposition we can compute the voltage across

‘f'

35 due the constant current generator l and the constant

voltage generator E. W0 will then be the ratio of E
' I

66

The node equations for the circuit of Figure u.lO are

l
I = V .i / rEZM _ V l ('_2 )
8" Z3 rp 7 24 b( Ezrp L h
I’p ; 21L,
0 .—

— lr-v 7! VT 1 l
rp ; Zn

The voltage across 85 is

Vb = I (rp / 8h)

 

 

 

 

a
m L- (11-25)
1 1 1
33:5 fa rpzu / a rpgu
rp Eu 5 rs ab

= I (rp / a“) 2325
rpEL / rpZS / EH25 / rpZ3 / Z3ZH

 

The mesh equations for Figure h.ll are

- Il (rp / an) - Izau (u-aé)

['11
l

0
II

- Ilgh / I2 (83 / Eu / 25)
The voltage across 25 may be computed as

vb = Bag 55
r933 7 rain 7 PPS; /Z3Zu / 2&3; (h-ET)

 

 

Equating equations (h-Zb) and (h-EB) we obtain
(h-ZS)

67

Thus w is
o

 

3 (Pp / at ‘ (L-aa)
g

Equation (u-30) gives the same result for no as obtained
before. However, in this case, the computation by the simpli—
fied method is more tedious than simply equating the,:ain
to zero as in equation (b-lO). Thus in using the simplified

method on simple circuits, a certain amount of judgment must

be used in selecting the method of calculation.

I”LIO”WAL’E

 

Bode, Hendrik W. Network Analysis a1d beedbech Amplifier
Desig;n. D. Van Nostrand Con1pany, riew lork, I925.

Cruft Electronics Sta ff. “lect roqic_ Ci rcu 'ts_\nd Tub.
KcGraw-Hill Bock CompanL New lork, l9L7.
Gardner, Murray F. and Ba nes, Join L. Transient. in Lines

 

C‘
SESEEE§° John Riley and Sons, New Iorh, 19L2.

Guillemin, E. A. Communica ion Networks, Vol. 1. John
Wiley and Sons, New lork, l9h9.

Guillemin, E. A. Co :runica tion Networks, Vol. 2. John

4. -. .-_

Riley and Sons, new lork, l9h0.

 

, J
m
F—J

he
(0
F.
[D
O

 

Guillemin, E A. The Hathematics of'Circuit A:
93

John hi 16 and Sons, flew lorh, 951.

t

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MICHIGAN STATE U

IVERSITY LIB

N RARIES
3

015 6532

3 1293 0