DOCTORAL DISSERTATION SERIES

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UNIVERSITY M/C//'

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PUBLICATION NO.

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SYNTHESIS OP A RESISTANCE - CAPACITANCE FILTER
WHOSE POWER INSERTION RATIO APPROXIMATES A
PRESCRIBED FUNCTION OF FREQUENCY

BY
NOAH HERBERT KRAMER

A THESIS

ubmitted 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
DOCTOR OF PHILOSOPHY

Department of Electrical Engineering
1951

To Dr. J. A. Strelzoff, whose help and
encouragement made this thesis possible

INTRODUCTION
(1)

NETWORK SYNTHESIS.
There are several aspects to the gen­

eral field of Network Syn t h e s i s .
One interesting facet is the Approxi­
mation problem.

The desired characteristic is

usually given in a graphical form.

The problem

requires the construction of a mathematical e x ­
pression to describe a network that possesses
the desired characteristic, within a permissible
approximation.

Elliptic functions, Tschebychef

polynomials, and Butterworth functions often
appear in this type problem.
Another aspect of synthesis is the Real­
ization Problem.

What mathematical or graphical

functions can lead to a physically realizable ne t ­
work?

What geometric forms can this network assume?

Much of the pioneer work in the Realization Problem
has been done by 0. Brune.

(1)

This thesis will deal with both aspects
of Network Synthesis.

1

Brune, 0.

Synthesis of a Finite Two Terminal
Network Whose Driving Point Impedance
is a prescribed function of Frequency.
Journal of Math and P h y s ., vol. 10, ppll91-235

In this thesis,

the response or an ideal

filter Is approximated by a mathematical expres­
sion (Approximation problem).

This expression is

used to realize a symmetrical lattice, composed of
resistances and capacitances

(Realization p r o b l e m ) .

The restriction to these two elements is not
Just an academic choice.

R - C (Resistance-

Capacltance) networks can be designed to give many
responses, and the additional loss that they intro­
duce, due to dissipation in the resistances,
compensated by subsequent amplification.

can be

In many

cases, where only a frequency discriminating cir­
cuit Is desired,

the loss introduced by the resis­

tances is of no importance and can be ignored.

The

exclusion of inductances might introduce economic
and construction advantages.

Certainly the problem

of magnetic pickup is reduced when there are no In­
ductances .
The terminology in Network Synthesis has
become confused,

since Mathematicians,

physicists,

and Engineers are all contributing to the field.
Is advisable to clarify terms at this point.

-2-

It

(2)

POUR TERMINAL NETWORKS
A Pour Terminal Network, or Quadripole, has

a pair or terminals designated as Input, and the
other pair designated as Output.
Associated with the input terminals are the
Input Voltage E^and the Input Current 1^.
The output terminals provide Output voltage
and Output Current I .
In the usual case a generator E with an in­
ternal Generator Impedance Z^ is connected to the
input.

A Load Impedance Z2 is connected to the

output.

This thesis restricts these impedances to

pure resistances R

and R .
1
2
Figure 1 Illustrates the notation that will

be employed.

Since the direction of the currents

is arbitrary,

the notation shown is chosen.

I
Z

I
FOUR TERMINAL
E

E

E
Figure 1

NETWORK
Four Terminal Network with Load.

_ -

3

-

Z

(3)

NETWORK RELATIONS
The relations between the input current and

voltage, and the output current and voltage are
found In any reference on Network Theory - 2,3*4.
I1 - y ll E l * y l2E2

E 1 - zllIi + Z12I2

Eq.1

X2 -= y 21 E 1 4-y22E 2

E q .2

E2 - z 21Z1 + Z2 2 I2

y l l ,y22 are the short circuit driving point admittances.
y12*y21 are the short circuit transfer admittances.
zl l ,z22 are tlrie °Pen circuit driving point impedances.
z12*z21 are

°Pen circuit transfer impedances.

In a passive network y ^2 - y21*

z12 “ Z2 1 *

Gewertz (5) has shown how the four admittances
or impedances uniquely characterize a network, and
has set up realization techniques when these para­
meters are known.

2

Guillemln, Ernst A.
Communication Networks
Vol. II
1935
New York:
John Wiley

3

Guillemin, Ernst A.
Communication Networks
Vol. II
1935
New York:
John Wiley

4

Everitt, W. L.
Communication Engineering
Second Edition
1937
New York
McGraw-Hill

5

Gewertz
Synthesis of a Finite Four Terminal
Network
Jour. Math and P h y s . Vol. 12
1932 - 1933
PP. 1-257

-4-

(4)

TRANSFER IMPEDANCE
Consider the generator impedance and the load

impedance absorbed in the quadripole, as in figure 2
The ratio of the generator voltage to the load cur­
rent is E/l.

This ratio is defined to be the Trans­

fer Impedance

of the quadripole.

New Four
Terminal
Network

Figure 2

T

E,

f

Quadripole for Transfer Impedance

The output voltage of E^ of this quadripole Is zero,
for there Is a short circuit across the new output
terminals.

Also E

*

E1 .

When these values are inserted into equation
2, there Is obtained

X2 = E ly 21

Zt * e/ I2 “ 1/ y 21

Assuming a passive circuit.
1/ y!2

-5-

Eq. 3

(5)

LOSSES
The most efficient way of coupling the gener­

ator to the load is through an ideal transformer.
The power delivered to the load in that case will
be called P0

.

A quadripole inserted between the

generator and load will transmit a power

to the

load.
The ratio P^/l^
unity.

Is less than or equal to

The equality holds when the quadripole is

an ideal transformer.
It is shown (6 ) that in the ideal case,
Zt ® ^ VZiZ^*
Consider the ratio of powers P^/P0
It is effectively

/I*/* Zt
/T.f 20

.

9 /^|2 - 'Lh
IX*)P o

Divide the fraction by E^E^and there results

£'
|

* llT^RprJ

*

zt
The ratio R. /P0 can be defined -

Power Insertion Ratio *j2 V zlz2 Y t| a

6

Mason, w. P.

Eq* ^

Electromechanical Filters and Wave
Transducers
1st Edition
1942
New York:
D. Van Nostrand. p24

-6-

In the study of four terminal networks,
Insertion Ratio is often considered.

the

This ratio

compares the current that would flow in the load
due to ideal transformer coupling,

to the current

that actually flows when the quadripole is inserted
between generator and load Impedances.

It can be

evaluated, as follows:

This is the Insertion Loss Ratio * .
Bode

(7) prefers to deal with the inverse ratio,
He defines

© - log

A and B are respectively the transfer loss and the
transfer phase.

Bode does not use the absolute values

in the ratio of currents but treats the currents, and
the ratio, as complex numbers.
of a complex number Z )Q

However,

is given by log

the logarithm
|Zi-*■ JO.

Thus, if one takes the reciprocal of the Insertion
Loss Ratio, and then takes the natural logarithm of this
new ratio,

Ibid 3

the Transfer Loss is obtained.

P. 73

-7-

Consider Y t .

Yt =

0 t ^ 3Bt ,

Yt

G t - JBt ,

.

Then

|Yt\ - (0t* ♦

)

Y

This result will be used later in this thesis
in a method suggested by Bode.

7

Bode,
Ibid 7

H. E.

Network Analysis and Feedback Amplifier
Design
19^5 New York:
D. Van Nostrand

P. 230

(6 )

R-C QUADRIPOLES
The essential features of R-C dipoles and

quadripoles have been established by Cauer (8 ),
and only the points of interest to this thesis
are presented h e r e .
The driving-point immittances*of R-C Networks
are rational functions of the complex frequency var­
iable P- J w .

Poles and zeros are simple, real, and

i nterlaced.
The immittances can be expanded into partial
f ractions:
n
, *
Z(p)^a

Caf)

+
P

(<v>
a
P + Pw

K
^
1

Eq. 7

n
,

v

Y(p) = b

<

.p +

b

(uJ
^ p.

J
1

where all the a a n d

b^

b
p +

^

Eq. 8

are positive.

These expansions lead to the two canonic forms
developed by Cauer for R-C networks, figure 3-

8

Cauer, E.
Die Verwirklichung von Wechselstromwiderstanden vorgeschriebener Frequenzahabhangigkeit.
Archlv F. Electrotechnik,
vol. 17 P355
1927

*

Ibid 7 P . 15
-9-

o
(•ai

l/aM i

/WWXA— —J|

(Impedance Form)
Figure 3

(Admittance Form)

Canonic Forms of R-C Networks.

-10-

Quadripole properties are found by insert­
ing the partial fraction expansions, Eq. 7 and
Eq. 8 , into the general equation 1 and 2.
The terms a r e :

z.. = a

n
2/i *

+ 7
p

^ix
p ^ p^

1
r—j
* K

-P

* *>„ + p.

n
xr>

<:v
b /7
P + Qv

n
b/z ■

P

-V b ,2

p . 2
1

btrv
P -►

The very important Residue Theorem0 states:
it CM ^
_
vV)
i
j
a« - 0,
a 2i i 0 ,
aM
.aijt ~(a,^ )
i: 0
0,
for all r .

b j '1i 0 ,

b„<v

.bJt<v

% O

"

as the necessary and sufficient conditions for the
physical realizability of a network.

° Ibid 3

P. 216

-11-

If* the R-C quadripole which meets these con­
ditions Is Inserted between the generator and load,
as In figure 2, the transfer Impedance Is Z t .
The transfer Impedance Zt is then a rational
function of the complex frequency variable, p f with
real coefficients.

Its zeros are the same as any

driving point Impedance seen In any mesh of the n e t ­
work, and since the network Is R-C, the zeros are
real, negative and simple.
Since R, is In series with the Input to the R-C
network,

there can be no zero at p = ^ .

The poles of Zt are produced either by the zeros
of z^2 or by the poles of z //#zJ4 which are not in
There are no restrictions on the zeros of z^2 ,
for It Is not a driving point impedance, so there are
no restrictions on the poles of 2t , other than that
they occur In conjugate pairs when complex.
shows the possible arrangements.
crosses,

zeros by circles.

Figure 4

Poles are denoted by

p-plane

p-plane

o— ©■

Figure 4

Typical Pole and Zero Patterns

The preceding analysis of R-C networks closely
follows the form in a paper by Orchard (9).

9

Orchard, H. J.
The Synthesis of R-C Networks
to have Prescribed Transfer Functions
Proc. I.R.E.
April 1951
p . 428
-13-

(7)

SYNTHESIS
When the generator and load impedance are

absorbed into the quadripole, as in figure 2 ,
the expression for Zt is Zfc = l/y1 2 .
Knowledge of y 12 is not sufficient to uniquely
determine the quadripole0, for y ^

and y22 can be

any values that satisfy the residue conditions.

To

complete the synthesis, many devices can be used.
Darlington (10) uses the Zt and; as additional
conditions,he requires that the quadripole consist
of pure reactances terminated in, at most, one
resistance.

This, of course,

is not acceptable for

R-C n e t w o r k s .
Bode

requires that the quadripole be a con­

stant resistance lattice.

This implies that the

arms of the lattice have an Inverse relationship
to each other.

This could introduce Inductances

and Is not acceptable In R-C networks.
Guillemin (11) has a system that results In a
ladder formation, but requires a generator of zero
internal i m pedance.

°
10
11
00

Ibid 5
Darlington, S.
Synthesis of Reactance Pour Poles.
Jour, of Math. Phys. Vol 18 No. 4 Sept. 1939
pp. 257-353.
Guillemin, E. A.
Synthesis of R-C Networks
Jour, of Math. Phys., Vol 28, p . 22 April 1949
Ibid 7 p.229
-14-

Guillemin*s method also restricts his choice
of functions to the minimum phase shift function.
(All zeros and poles are in the left half of the
p

«

plane.)

This last restriction is due to

his choice of a ladder formation, for &

"Any pas­

sive ladder network is a minimum phase shift
structure.11
Orchard00 has an ingenious device to synthesize
the network.
^11-^22’

Assume a symmetrical network.

Then

Choose y-j^ - y 22 to satisfy the equality

sign in the residue condition (Eq. 10).
Then b ^ s

b 22'°, I

•

For a11 r *

From the expansion of y 12
sible to form y 1 2 , y 22

(Zt )# , it is pos­

making all the residues

of y 12 positive.

"With the specification of

y 2 2 > ^ i 2 > ^21 ’

1S ^ en P ° ssit)l e

synthesize

the network in any suitable form.
Guillemin ° shows that the residue condition
is Identical to the realization condition of a
symmetrical lattice,
in any form,

o

so If the network is realizable

it is always realizable

Ibid 7, p. 243
Ibid 9
Ibid 3, P. 381
-15-

in the lattice form.
other forms,

This does not rule out

but assures the existance of at

least one form,

the lattice structure.

The lattice will have arms:
YB - y ll ' y 12
.E q * 11
12
Y A " y ll
Orchard then extracts equal resistances
from the lattice arms to act as generator and
load resistances.

The result appears as figure

5.
|

Uri

— f+ z *\
4

Figure 5

Extraction of Terminating Resistors

The fact that the generator and load
resistances are equal is a design limitation
of the method.

- 16-

(8 )

APPROXIMATION PROCEDURE
This thesis uses a modification of a device

of Guillemin ° to approximate a frequency response.
F(w) is given graphically over the complete
range --»> <

uj s

«o

A change of variable

ux*

tan ©/2 changes

the range to -it i & s tt t and the F (saO is trans­
formed into a periodic function of

©.

is then approximated by Fourier Series.
transformation, x

the

1 - o*

Another

-= cos © , changes the series

to a polynomial in x , F(x).
ation, x

This F(^)

The last transform­

, returns the approximation to

plane.
The closeness of this approximation is deter­

mined by the nearness of the Fourier Series approx­
imation to the prescribed F(0).

There are many

classes of functions that can be approximated by
Fourier Series in an arbitrarily close manner, and
if the F(®) falls into one of these classes,

the

approximation will be as close as desired.
The mathematical expression that results from
the above manipulation Is not in a form suitable for
R-C networks.
poles.
°

Ibid 11

The denominator would lead to multiple

In this thesis the approximate P ^ )
divided by a function of

will be

that is very close

to unity over the range of ^

.

The choice of

this dividing function will be made to clear
the denominator.
There is liberty in choosing this function,
as long as the choice does not change the final
result to a point that the approximation is not
as close as desired.

- 18-

(19)

APPROXIMATION E X A M P L E . ...LOW PASS FILTER
The graphical representation or the desired

filter is given in figure 6 .
The filter characteristic has been given In
terms of

lYTj*

has been shown to be propor­

tional to the power insertion ratio.

(Eq. 4)

For the sake of simplicity the graph as
been normalized with respect to frequency and m a g ­
nitude.
The first change

^ = tan e/2 results In the

F(e) as shown In figure 6b.
There Is no difficulty in approximating figure
6b, by Fourier Series.

Since F(©) is continuous

and of bounded variation, a Fourier Theorem (12)
states that the series converges uniformly to F(0).
By standard methods the first few terms of
the series are....
F(0

-

.500 - .635 cos©

-.204 cos3© +.1215 cos 50
Eq. 11

Since only a few terms will be used, the above approx­
imation might become negative for some 6 .
allowable for

this is not

|YTix > so a different form is used.

F(©)- B_+.635 cos©-.204 cos 3© +.1215 cos50

Eq. 12

B^ Is chosen large enough to keep F(©) positive.
12 Churchill, R. V. Fourier Series and Boundary Value
Problems.
1941
New York: McGraw-Hill
p .86
-19-

At O = "ir/2, which corresponds to ^-=-1, the
value of F(©)- or

\Yr (©)I2-

- B —

M ake the transformation x ~ cos Q . Then
. 3
^ *■
cos 3© * 4x - 3x# cos 5©
l6x - 20x ■* 5x
The approximation then becomes:
l^x)]1

r ,6 35x 3- .8l6x ♦ .6l2x «.1.944af -2.43x%.6o75x-*

• 1.944x*’- 3 . 2 4 6 / + 1.8545x ♦ B -

Eq. 12

F o r some value of x, and some ^ , the
reaches a minimum.
the lYr (^)i*

|Yr(x)l*

If this minimum is fixed at zero,

will not be qero at

*•»* •« .

This is

the result of using too few terms in the Fourier
Series.
Figure 7 compares the approximation to the
F(©).

The Inclusion of additional cosine terms

would improve the approximation, but increase the
work In the computation.
Themlnlmum Is found by differentiating Eq. 12.
d iYt( X )]*
- 9.720x* - 9 . 7 3 8 x ^ 1 . 8 5 4 5 * 0
Eq. 13
x
This is a quadratic in x
, roots 9 . 7 3 8 * 4.972
197440
The root of interest is x 2 a .245164, x

s. .495

At this value of x, B^jnust be .5823278 to have

CALCULATIONS FOR GRAPHS
©
0
10
20
3°
40
50
60
1°
80
90
100

(Figures 6 & 7)

Cos 0

.635
Cos 0

.204
Cos 3©

.1215
Cos 5©

- .033
Cos 70

1.0
0.9848
0 .9397
0 •866
0.766
0.6428
0.500
0.342
0.1736
0
-.1736

.635
.625
.596
.55
.485
.408
.317
.217
.110
0
-.11

-.204
-.177
-.102
0
. 102
.177
.204
.177
.102
0

.1215
.073
-.0215
-.105
-.144
-.04l6
.0607
.1195
.093
0

-.083
-.0284
.0645
.072
-.0144
-.0816
-.041
.0534
.078
0
.078

■

—

--------- ------------------- --—

From 90 to 180 repeat opposite in sign.

0

3 term
siim

4 term
sum

Corresponding
U)

0
10
20
3°
40
50
60
70
80
90
100
HO
120
130
140
150
160
170
180

.5525
.521
.4725
.445
.473
.5434
.5827
.5135
.305
0
-.305
-.5135
-.5827
-.5434
-.473
-.445
-.4725
-.521
-.5525

.4695
.4926
.5370
.517
.484
.4618
.5417
.5669
.383
0
-.383
-.5669
-.5417
-.4618
-.484
-.517
-.5370
-.4926
-.4695

0
.087
.176
.268
.364
.466
.577
.700
.839
1.0
1.19
1.428
1.732
2.144
2.747
3.732
5.6713
11.430

-21-

The graphs are even functions ofu>
P (w)s F(-to).

.

Then

The graphs could he shown to he

symmetric In oo ahout the origin, hut only posi­
tive to is of interest.
This leads to:
|YT (x)Jx s

1.944x* -3.246x * 1 .8545x 4 .5823278

The next change of variable x, 1 - to*
1 r la*

Eq.l4

gives 1YT («)(*»

1 .944(1-to* )f-3.246 (1 -co* )(lr co1^ ! .8545(1- & )(!**>* Wfrll + q?*)
(1 + w l .- J,
Eq.15

Multiplication and binomial expansion puts the
coefficients in a form expressed in the tables below:
*
CO
10
2
2

CO

1.944
-1
3.246
1
1.8545 -1

5
-1
-3

-10
-2
-2

1

CO

-5
1
3

Cjo
1
-1
1

This gives:
6o’°
-1.944
3.246
-1.8545

CO

coa

19.44
6.492
3.709

-9.720
3.246
5.5635

CO*
9 .720
-3 .246
-5 .5635

-19.44
-6.492
-3.709

CO*
1.944
-3.246
1.8545

The numerator is then formed:
(B„-

.5525)<o°+ (5B«* ♦ .9105)* ♦ (10B„- 29.641 )u>* + • • • •

. . .(1 OB

*►29.641) to* + (5B«» + .9105 )CO*" 4

(B^t.5525)

Wh e n the assigned value for B„ is inserted (B^ *.58 2 3278 )
in the above, there results:

.029827740 + 3.8221394?*-

23.817722 u>*+ 3 5 .464278co*4 2.00113940*+ 1 .1 3 4 8 2 .

- 22-

s

Remove a factor of .0298277 and there is formed:
l^r (u )|* .

.0298277(m'% 128.1 Kg -7 9 8 .5a)*. 11.89.71 u -1 .
. ■ .67.089908 u>L* 38.046

Eq. 16

To get In a form for R-C synthesis, divide the above
fraction by C(co*).
C (Coz) =

(

» .81 )(o3fct .9 0 2 )(a>1^ 1 )(coS 1.02 )(0)1* 1 .2 1 )
i V t

1 )*■

There is a little latitude in the choice of C(u>t),
and the above form is chosen for the ease in subse­
quent calculations.

A table of values for C(Co*)

shows that the choice of this expression is close
to unity for all oj .
1

CO1

0

c (00l)

.9742

.99952

cox

5

C (A)1)

1.00261

1.5
1.0016

2

31

1.0024

1.00288

.
1.0

The divisions of Eq. 16 by C(co4) leaves as a final result:
lY, (c*)j .

( . 0 2 9 8 2 7 7 X u A 1 2 8 . 14/ - 7 9 8 . 5 0 6 ^ 1 1 8 9 . 710J.V
n —
.H I T T — .£627(
" 1 .Oc!) (
1 T 2 I ) --6 7 .0 8 9 6 8 6 <J" t 3 8 .04595

Eq. 17

This is represented by figure 8 and essentially com­
pletes the approximation problem, except for the ques­
tion of the physical realizability of the trapezoidal
filter.
-23-

V

A P P R O X I M A T I OK/

A

T tR M S

F {*>)

A P P R O X \MAT\OA/

-

3 T E R M S

a

1
2

A/<?

fh

-uj-

5

3

C e M P A P ft erf O p
Ta

r ( u>)

A p p p Q x IAl fiT/pup S

(10)

SYNTHESIS PROCEDURE
The next step is to synthesize a network that

is characterized by equation 17.
The numerator of the expression is a polynomial
of fifth order in to*.
must be found.

The roots of the polynomial

One root is already known for the

|yt (fc*))4, was chosen to have a minimum of zero.

This

m i n i m u m occurred at x » .495 which is u>*=3.04 5 8 7 8 .
An examination of the polynomial shows that
there are two changes of sign.
rules is applied (13)-

One of Descartes

This rule states that there

will be either two or no positive roots for the poly­
nomial.

There Is assuredly one root, and the expres­

sion does not cross the axis in more than one place
on the graph.

Thus there is a double root at

cor -- 3.045878.
To find the rest of the roots, the method of
Lin (14) is employed.

The result is:

l*T (-)i'*
.0298277 ( - 3 . OA59 ) (<**. 134.158 )(to* *. 07375 J- * ■03056783 )
(
{ cjt<■.902 )ecu'*♦ ljlto1* 1 .0 2 )C^*f IT2 1 )
_
Eq. 18

13

Pipes, Louis A.

Applied Mathematics for Engin­

eers and Physicists.
14

p.98, 1946 New York: McGraw

Lin, S. N.

Method of Successive Approximations of
Evaluating the Real and Complex Roots
of Cubic and Higher Order Equations.
Jour. Ma t h and P h y s . Vol. 20. No. 3 Aug. 1941
-24-

The synthesis problem requires the Yr (p) so
since p - J<o, and
made.

cov* -p* , this substitution is then

This leads to:

|YT (p)/z=

Eq. 19.
Equat i o n 6 is applied after the above expression
is b roken down into its simpler factors.

I Y tif )j 's

ki-giiXr**

( p - u .sa z) C p * +.tso7>OB*.nvt></4/)

cp*.9s)C?+/)cp+/-arJCp+i-O

L
\U 7J7J) ( f

y .a v ffj

L

J

t 3. O t / S - y f p I*// &8Z')(f,x-.tSV7/C> JO - . i 7 V 8 W )

L c p - . « ? ; r p - (P+ /Jcp

CP-/-'J

Fo r R-C synthesis only the poles in the left
This restricts the denomin­

ator to Just the positive factors.
There are no restrictions on the zeros of Y^-(p)
so there is m u c h liberty in choosing the numerator
of the expression.
At this stage, G u i l l e m i n ’s method would require
the selection of a numerator that would make Yf a
m i n i m u m phase shift function.

-25-

1

J
Eq. 20

half plane can be used.

J

Lattice synthesis does not required a minimum
phase shift function, but for simplicity in this
example,

and the material following this example,

a m i n i m u m phase shift structure is chosen.
The graph of

| Y T (“Ol^

vs

, in figure 6,

has been normalized with respect to magnitude.
M u l t i p l y the expressions for |YT (di»J*
tor Al

by the fac­

to represent the unnormalized expression.

The significance of this factor becomes apparent
when the full equation for Y T (p) is considered.

Y-r(p)

-

1 Pl

t

2 PoYRjRg
The above is an extension of equation 4.
The m i n i m u m phase shift structure is then:
Yt(p).

.17271 A(P S ,
2 ^ 5 88J0>tl1 52 H p \ .6507P * 1748441
( p J ; § U p t . 9 5 M P + l ) C p + i . o 5 M P t l.lj

Eq. 21

A cutoff frequency (represented by CO* ) could
be introduced here.
of co

This requires the replacement

in equation 18 by Coy/(jt)o.
W h e n that is done the expression for

is found to be

Yr (p)
71 A
.1727:

:

p r *. 1 2 .2327P 6J.1- 1 0 . 7 5 7 P t f ^ 3 9 . 2 2 0 6 p V + 23 .4 8 7 p ^ % 6 .1 6 8 0 3 & r_
5 p V 9 - W 5 p ^
9 87525X?

Eq. 22

- 26 -

One form of a complete problem would specify
the generator and the load resistances.

The required

cutoff frequency would also be given.
For the purpose of demonstration, let:
- R^ - 100 ohms
2

-r 2

Let

.......then

» 200 ohms.

» 1,000,000 rads / sec.

The value of A

cannot be arbitrarily stated

when the impedances and the cutoff frequency are
fixed.

This will become evident as the problem is

solved.
The first step is to expand equation 22 by
partial fractions.
At p

- 0, the residue of the right side of

equation 22 is equal to 6 .2 4 5 9 5 6 9 .

This residue

is subtracted from the expression and partial frac­
tion expansion is continued.

The details of this

expansion are not repeated at this point.
The expanded
y-, p s

1

(p) - y 12 ls:

.17271 A ^6.24596 - 121,418 p
(p

+

L

-895,069 p
u/o

{9

+ U/a

657,443 p

T

* O/0 (p + 1 -0^,)

,560,039 P
...
joJq Kv
•$5u/o )

+

-180,427 p

^p + 1
Eq. 23

-27-

Wh e n all the terms in the previous expressions
are made positive, the equations for y ^
obtained.

«

They are:

y ll * y 2 2 ‘* •1 7271 A] 6.24596 +121,4l8 p
^ I p t»9 cag /
+ 89^069

60*^P

y 22 are

p
■+ 00* )

_ ^

651>44^p
<300 tp

^ 560,039 P
cSt (p +•

„ _ l 8 o A4 27 P

+ I.O 50O0)

60*(p «- l.lOJo)

Eq. 24
The lattice arms can now be formed.
yH
* y !2
11
y 12
y a *.17271 A [l2.491 9+ 1,120,078 p
1,314,886 p
Lu>0 (p ♦. 9$ c00)
6 0 0 \P + I.0 5 <
50«)
“

*

Eq. 25
_

Yb * .17271 A

242,836 p

+

W,lP+-9«»;

1 ,790,136 p

.Mo I P « 4 )

360,854 p

L.

J

Eq. 26
The reciprocals are taken in order to write
the arm impedances.
A

The result is:

p r 64112.4^19

P ( 44.'5532*,/

j

CO,IP . 1-lcJ,)

+

2,5l6>6bo^) «*■ . .

2,455,224^.')

12.454

)

Z_
3 £J.-t- 2.99V»P + -99 »o3 )__ 5 .79005
^ A" %,S^3,B28p£ . 4,775,856^ p . 2 , 3 6 4 , 1 . 2 2 ^ p
Eq. 27

- 28-

These ape driving point impedance expressions
Tor the arms of the lattice.

At this point these

expressions could undergo partial fraction expan­
sion and be synthesized.

This would defeat the

purpose of the problem for the generator and load
resistors would be absorbed in the network.
A standard theorem

4

of lattice networks states

that if an impedance is common to both arms of a
lattice,

it can be removed from the lattice and

placed in series before and after the new lattice.
This theorem is utilized by finding the mini­
m u m value of both arms impedance.

Obviously since

these are R-C arms the minimum impedance occurs
when

f » ••

Z
p -* —

„ __ 5.72 0Q5a__£0* .
A (12.4919
2,515,656)
.

lim Zn
P - . T

then

5.79005 coo
“

A (2,393VBBS)------

Eq. 28

The smaller of these two limits is the first.

Then

a resistance equal in magnitude to the first part
of equation 28 is common to both lattice arms.

This

m i n i m u m resistance can be removed from the lattice.

Ibid 7,

p. 269
-29-

Tills m i n imum r e s istance, designated as Z* ,
Is obviously the generator and load resistance.
If the Z' turns out to be larger than the desired
resistances, only the needed amount need be pulled
from the lattice arms.
In the assigned problem, Z* should equal 100.
Z 1 •> 5.79005 GOo
AU2.'*919ofc+2,5l5,bfc>t>;
F o r the a s s i g n e d o f
of A is

_ 100 ohms

1,000,000,

the derived value

A m 0.003858078

W h e n this value of A is know, it is inform­
ative to solve for the Power Insertion Ratio.
At p = 0, the equation 18 leads to
\*T (p)lp.e

A 1 ( . 0 2 ^ ^ ) ^ . 0 ^ ^ 4 . ^ a i (_ 2 325§Z§al.
-

B u t tYt (p)j*

'

A 2" (.4 1 3 1 8 6 1 7 )
.

4

p 2_____

R-jRg

s

P t,

(40,000)

z A* ( .41318617)

Thus at zero frequency the Power Insertion Ratio is
found to be

PL
P*

_

.24600598

F r o m the graph the ratio of Maximum IYt (^)i*
|Yt (O j*.
is about 1.08/1 so the maximum Power Insertion Ratio
is about

.2 6 5 .

-30-

This is a result that is not evident from
a cursory examination of the problem.

It is now

seen that for a given load and generator resis­
tance, and a specified cutoff frequency, the max­
imum allowable Power Insertion Ratio is automat­
ically determined.

If it had been required to find a

filter with the same load and frequency requirements
as the previous problem, but with a higher Power
Insertion Ratio, some other form of synthesis would
be required.

Such a filter as the latter is not

realizable by the method just utilized.
It Is quite possible that there Is no R-C
structure that will satisfy a requirement of a
higher Power Insertion Ratio.

This subject requires

further investigation.
The filter problem can now be completed.
100 ohms, or Z*, is to be subtracted from ZA and Zg.
The modified lattice arms Z^ and ZB * are now com­
puted .
With numerical values the arms are:
Z

.

.1500.76( p* + 2x10 S?

15 .0075bbp*
Zo
*

»

♦ .997x1Q 12, )

* 24.9 8 3 x 1 0 *p «- 12.454

1500.76 (p* » 3x10%***. 2.99xlOa p » .99x10>g)
2.39382p 4 .7 7 5 ^ 5 * 1 0 iP 1- + 2 364 ,122x10* p )
Eq. 30
-31-

W h e n lOO is subtracted from equations 30,
the modified arms are obtained.

_ 1503.22xl0*p + 1.4962577x10 '*______

Z »

15.0075bbpr +

Z_»

2 4 !963x 1 0 Jp + 1 2 .4^4

_ 1 2 6 1 .38ps - 4027.xl0*p *■

+ 4250.86x10/2'p .*.l485.75x
2.5S38£p> +-*7775,U50pi + 2 , ^ 4 , 1 -2 2 x 1 6 ^ ---

B

Eq. 31
The usual methods of partial fraction expan­
sion are then used to realize these driving point
impedances.
F o r the first case the impedance is simpli­
fied by dividing numerator and denominator by

15 .007566 .
V

A

■ 99.7H7*1Q*P,,t

£>

•°997glO

1 ,6 6 4 , 7 4 7 p

.8 2 9 twa

99-7117x10 * p * .0997x10 ,s
— Ip 7176647747 )( p T T G r )
Eq. 32
Equa t i o n 32 is expanded by partial fractions.
The result is:
Z.«
A

(5.98889 )xlO/0
a

( p

V

3.98227x106

0 1 ) --------

P

/ I

. 5 6 4 7 5 0 0 r -

Eq. 33
This driving point impedance is represented by
the figure 8 .

-32-

R^A

» 5.989x10^

ohms

« 2a

"

ohms

C-jA * 16.6975
CgA =* 0.251113

uufd
ufd

Figure 8

>^2/1
j w w l __

\a a a
I— (f--1
Cl A

1—

I
C 2.A

Lattice Arm Z*

A very similar manipulation is done Tor
the other lattice arm.

The numerical calcula­

tions are a bit more complex, because the denom­
inator os the expression is a cubic polynomial.
Luckily, one of the roots of this denom­
inator is at p s 0.

This simplifies the cubic

solution to the solution of a quadratic.
The expression for Zg» is found in equation
31.

D i v i d e both numerator and denominator by

2.39382 and there is obtained as the driving point
impedance of the lattice arm:
2= * . 526 .931pJ
1 6 8 2 .2xlOfc p\. 1 7 7 5 .77x 1 0 '* p+6 2 0 .66x l 0*8
^
p
C p * ■;l,995,674p
875,933x10 * T ------------Eq. 34
There is no great difficulty in obtaining
the roots of the denominator by the quadratic f o r ­
mula.

The roots are:

- 1,084,063;

- 911,010.8;

-33-

0.

The partial fraction expansion, when applied
to Zq * gives the following results
zB * « 6 2 8 .4 5 7
p

^

6 1 ,2 5 3 .1 5 0
p + 1,084,063

„

5 8 3 ,6 3 6
P + 91T 70T 0
Eq. 35

The realization of equation 3 5 , which repre­
sents the lattice arm

—

H V

is shown in figure 9 .

M/V.

...

'

— ^

*

----- 1 1 ------

---- H | ---- 1

C IB

5

LB

Figure 9

Lattice Arm Zg*

The numerical values for the components are:
C nR

■ 0.00159

farads

C 9p

* O.OI63

farads

R 1B

s 56.50

ohms

s 1.713^

farads

Rpr

_.0.6406

ohms

The complete lattice formation is shown in
figure 10.
The quadripole realized in this problem for­
tunately requires the use of components that are
available in a laboratory.
-34-

A less fortunate choice

of cutoff frequency, or of terminal Impedances
could have led to capacitors in the order of
farads, not ^ f a r a d s ;

the result might also

have required m i c rohm resistors.

If that were

to be the case, another method of R-C synthesis
not in the form of a lattice, would be in order
F o r that reason, a high cutoff frequency
was chosen for the demonstration problem.

C»*

CzA

C,B

IB

'33

Lattice Arm A

Lattice Arm B

Ri« ■ 5 . 9 ^ 9 x 1 Ohms

R IB - 56.50

Ohms

R*.*

Rt,® * 0 .6406

Ohms

s

2 .3921

Ohms

Cj*)* 16.698
Cj a s

0.2511

farads

Cj s -

0.00159

4t farads

Cjg =

O.OI63

44 farads

Gse-

1.713^

4f farads

^farads

Terminal Resistors
Rf %

100 Ohms

R£

100 Ohms

=

Figure 10

Final Lattice Structure

-36 -

(11)

REALIZABILITY OF A QUADRIPOLE
A glance at figure 7 shows that the synthe­

sized lattice has a frequency response that merely
approximates the desired trapezoidal pattern.
This is due to the fact that only a few
term of the Fourier Series were used.

The inclusion

of additional terms would bring the response of the
filter mor e like the desired characteristic, but
the inclusion of more terms would require more com­
plex calculation and require the addition of ad d i ­
tional circuit components to the lattice.
Since the desire F (9) is a continuous func­
tion, it was stated before,

that the desired res­

ponse could be approximated arbitrarily close.

A

F o u rier Theorem® states that if F (9) satisfies
these conditions:

F(0) * F(©+2 tt) for all 9, and

F(9) is sectionally continuous in interval

(—/r, tt),

then the Fourier Series converges to the value
£^F ( d * ) t P ( O' )] at every point where F(0)
has a left and right hand derivative.
The trapedoidal pattern fulfills these r e ­
quirements^ so an infinite number of terms could
result in a theoretics expression that exactly
realizes the desired response.
Ibid (f2) p. 70
-37-

(12)

E X T E N S I O N TO THEORY OF EQUALIZERS.
Consider the system shown In block diagram

form in figure 11.

In this system, a signal passes

through a Low Pass Filter.

Some of this signal

is utilized in the filtered form, and the rest of
the signal is isolated by a vacuum tube and sent
on to an Equalizer.

Since the filtered signal

passed through a Low Pass Filter,

the Equalizer

must be a Hig h Pass Filter.
A system such as this can be handled by
the methods of this thesis, for the problem of the
interaction between the High Pass and the Low Pass
F i l t e r is solved by the use of the Isolation A mpli­
fier.
The system requirement might be that the
product of the Power Insertion Ratios of the High
and Low Pass Filters be a constant; or the require­
m en t might be that the sum of the ratios be a con­
stant,

for all to .
The characteristic required for the High Pass

Filter can then be transformed Into a Fourier
Polynomial and the synthesis performed.

-38-

SIGNAL

LOW PASS FILTER
FILTERED SIGNAL

*

ISOLATION AMPLIFIER

EQUALIZING HIGH PASS
FILTER

|

Figure 11

4

— EQUALIZED SIGNAL

Block System using Equalization

-39-

(13)

HIGH PASS FILTER SYNTHESIS
It Is desired to synthesize a High Pass

Filter such that the sum of the Power Insertion
Ratio of this filter and the filter previously
designed in this thesis be a constant for all.
The Power Insertion Ratio characteristic
desired for this filter is shown in figure 12.
The Fourier representation for this filter
is readily obtainable from the Fourier represen­
tation of the Low Pass Filter,

equation 36.

The expression for the High Pass Filter is
obtained when the Low Pass characteristic is sub­
tracted from its maximum value.
Thus,
unity,

if equation 18 were subtracted from

the result would be:

.0 2 9 8 2 7 7 (<ol -3 .0459)l (to1* 1 3 4 .1 5 8 ) (Co+07375

1-

9 0 2 ) <£,»; i

'

y

c

<

o

v

.03056783

)---------------

E q . 36
This equation would be a starting point for
the synthesis procedure.

The remainder of the

problem would be solved in the same manner as the
solution for the Low Pass Filter.

-40-

4

Figure

IP

Fourier Representation of High Pass
Filter

-41-

(14)

CONCLUSION
If* It Is desired to synthesize an R-C

quadripole,

to have a Power Insertion Ratio,

that approximates a given characteristic,

there

are sufficiency theorems that will permit the
synthesis.
If there exists a series of transforma­
tions that can put the desired characteristic in
a periodic form, and if this periodic form is, at
least,

sectionally continuous with left and right

hand derivatives,

it is possible to approximate

the desired form by a Fourier Series technique.
As long as the derived expression for the
y,a is realizable,

it is realizable as a lattice.

If the expression is also of “minimum phase shift"
it is also realizable as a ladder network.
There are certain restrictions on the rela­
tions between terminating impedances,

cutoff fre­

quency, and m a x i m u m Power Insertion Ratio,
should be further investigated.

-42-

that

Literature Cited

1

Brune, O.

Jour. Math, and P h y s . Vol 10, No. 3

2

Guillemin, E.

Communication Networks

Vol. I

3

Guillemin, E.

Communication Networks

Vol.

4

Everitt, W.L.

Communication Engineering

5

Gewertz

6

Mason, W. P.

Electromechanical Filters and
W a v e Transducers.
Ed. 1

7

Bode, H. E.

Network Analysis and Feedback
Amplifier Design.

8

Cauer, E.

9

Orchard, H. J.

P r o c . I.R.E.

10

Darlinton, S.

Jour. Math, and Phys.

Vol. 18

11

Guillemin, E.

Jour. Math, and Phys.

Vol. 28

II

Ed.

2

Jour. Math, and Phys. Vol 12. 1932

Archlv f

12

Churchill, R. V.

13

Pipes, L. A.

14

Lin, S. N.

Electrotechnik,

1927

April 1951

Fourier Series and Boundary
Value Problems
1941

Appxied Mathematics for
eers and Physicists
Jour. Math, and Phys.

-43-

Engin­

Vol. 20 No. 3