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A simplified method of synthesizing ladder networks with image-parameter half-sections

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Title:
A simplified method of synthesizing ladder networks with image-parameter half-sections
Creator:
Silber, David, 1922-
Publication Date:
Language:
English
Physical Description:
vii, 79, 1 leaves. : ill. ; 28 cm.

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Subjects / Keywords:
Conceptual lattices ( jstor )
Electric potential ( jstor )
Elliptic functions ( jstor )
Image filters ( jstor )
Insertion loss ( jstor )
Mathematical independent variables ( jstor )
Mathematical maxima ( jstor )
Signals ( jstor )
Sine function ( jstor )
Transcriptional regulatory elements ( jstor )
Electric networks ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 76-78.
General Note:
Manuscript copy.
General Note:
Vita.
Statement of Responsibility:
By David Silber.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ADA5011 ( NOTIS )
13957636 ( OCLC )

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A SIMPLIFIED METHOD

OF SYNTHESIZING LADDER NETWORKS

WITH IMAGE-PARAMETER HALF-SECTIONS










By

DAVID SILBER


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA
August, 1960



































Copyright by


David Silber


1960



















ACKNOWLEDGMENTS


The author would like to express his gratitude to the

members of his supervisory committee for their advice and

helpful criticism. He is especially indebted to Dr. T. S. George

for his supervision and constant guidance, to Dr. W. H. Chen

for his many valuable suggestions and to Dr. H. A. Meyer, Director

of the Statistical Laboratory, for his help in obtaining numer-

ical data by using the University's electronic computer.














TABLE OF CONTENTS


ACKNOWLEDGMENTS ............... .

LIST OF TABLES ............

LIST OF ILLUSTRATIONS .

CHAPTER


I. INTRODUCTION . .

1.1. The Use of Frequency Selective Networks
and Terminology .

1.2. The Image Parameter Method of Network
Synthesis .

1.3. The Insertion Loss Method of Network
Synthesis . .

1.4. Comparison of the Image Parameter and
Insertion Loss Methods .

1.5. Statement of Objectives .

II. THE GENERAL LADDER NETWORK IN TERMS OF ELEMENTARY
BUILDING BLOCKS CONNECTED IN TANDEM .

2.1. The Elementary Building Block .

2.2. Methods of Connecting the Elementary Building
Blocks . .

2.3. A Conventional Low-Pass Ladder in Terms of
Elementary Building Blocks .

2.4. Filters with Incidental Dissipation in Terms
of Elementary Building Blocks .

III. DERIVATION OF THE IMAGE PARAMETERS FOR LADDER
FILTERS HAVING PRESCRIBED INSERTION LOSS
CHARACTERISTICS . .


: : :


Page

ii



vi



1


1


3


5


6

8


10
10


12


15


23



26










Page
3.1. Outline of Approach ... 26

3.2. Insertion Loss of a Two-section Ladder
Filter in Terms of Image Parameters 28

3.3. Tchebycheff and Butterworth Filters Syn-
thesized with Image Parameter Half-sections 33

3.4. The Elliptic Function Filter Synthesized
with Image Parameter Half-sections 44

IV. ANALYSIS OF RESULTS AND CONCLUSIONS 61

4.1. Analysis of Obtained Data 61

4.2. Suggested Modification of Zobel Designs
with the View of Obtaining More Efficient
Filters . 63

4.3. Limits of Improvement of a Zobel Filter 70

4.4. Summary and Conclusions 73

REFERENCES . 76

BIOGRAPHY ......... 79














LIST OF TABLES


TABLE Page

1. CUT-OFF FREQUENCY w. FOR THE BUTTERWORTH
TWO-SECTION FILTER 40


2. IMAGE PARAMETER VALUES FOR THE TCHEBYCHEFF TWO-
SECTION FILTER ................. 45


3. IMAGE PARAMETER VALUES FOR THE ELLIPTIC-FUNCTION
TYPE TWO-SECTION FILTER .. .. 58














LIST OF ILLUSTRATIONS


Figure Page

2.1. The elementary building block 11

2.2. Image impedance of the elementary building block
at terminals 1-0 of Fig. 2.1 11

2.3. Image impedance of the elementary building block
at terminals 2-0 of Fig. 2.1 .. 11

2.4. Tandem connection of the elementary building blocks
giving a non-symmetrical ladder filter 13

2.5. Connection of the elementary building blocks by which
symmetrical and non-symmetrical ladder filters may
be obtained . 13

2.6. A special case of the ladder of Fig. 2.5 16

2.7. Possible locations of elements with arbitrary
values . 20

2.8. Effect of adding proportionately equal amounts of
resistance on the location of poles and zeros of
the insertion voltage-ration function 24

3.1. A two-section ladder filter 29

3.2. Elementary building blocks from which the ladder of
Fig. 3.1 may be obtained .. 29

3.3. Equivalent lattice of the ladder of Fig. 3.1 31

3.4. A Butterworth and Tchebycheff type ladder filter
and its equivalent lattice 31

3.5. Insertion loss characteristics for (a) Butterworth
and (b) Tchebycheff two-section filters .. .. 36

3.6. Cut-off frequency of elementary building blocks
versus maximum attenuation in passband for the
design of filters with Butterworth characteristics 41

3.7. A Butterworth filter in terms of elementary building
blocks with WUc = 1.272 and Lp 3 db 42

vi -











Figure Page

3.8. Image parameters oc rI and r2 versus maximum
peak ripple in passband for the design of filters
with Tchebycheff characteristic 47

3.9. A Tchebycheff filter with peak ripple in passband
L= 1.5 db in terms of elementary building blocks
with wC = 1.0 . 48

3.10. Insertion loss characteristic of an elliptic function
two-section filter . 51

3.11. Minimum stqp-band attenuation L versus selectivity
factor k for the two-section efliptic function
filter . 53

3.12. Variation of minimum attenuation in stop-band with
maximum ripple in passband 53

3.13. Image parameters rl, r2, r3 and r4 for k 0.7 versus
maximum ripple in passband for the design of elliptic
function filters . 60

4.1. Typical insertion loss characteristic in passband
for a Zobel filter . 66

4.2. Typical insertion loss requirement and a possible
insertion loss characteristic which satisfies it. 66

4.3. Two types of insertion loss requirements and
possible filter characteristics 71


vii















CHAPTER I


INTRODUCTION


1.1 The Use of Frequency Selective Networks and Terminology

While transmitting information in the form of electrical

signals from one point to another, we are faced with the problem

of recovering a desired signal from undesired ones and from elec-

trical noise.

When the energies of different signals are in different

frequency regions, or bands (a situation which is often found in

nature, or can be attained by various techniques in the process of

converting the information into an electrical signal at the trans-

mitting points), the separation of one signal from others makes

use of the differences in energy-frequency spectra of different

signals.

This separation is accomplished by placing some device in

the path of the signal (before it reaches the receiving point)

which exhibits frequency selective characteristics, i.e., a device

which is capable of accentuating (or letting pass through without

change) one frequency band, the band containing all or most of the

energy of the desired signal, while suppressing (or attenuating)

another band.

The device usually employed for this purpose consists of

an array of electrical components (resistance, capacitance and

1 -






- 2-


inductance) interconnected in a fashion which produces the desired

frequency selectivity. There are some mechanical devices with

this characteristic which are sometimes used.

A word on terminology: An array of interconnected electri-

cal components is called an "electrical network," or in short a

"network." The manner of interconnection of components, the choice

of their type and number required in order to produce a specified

electrical characteristic (e.g., frequency selectivity), as well as

the mathematical tools employed in network problems, are treated

in network theory.

If a network consists of electrical components, the values

of which are independent of frequency, it is a "linear network";

if the circuit element values are positive constants, it is a

"linear passive network."

A network inserted in a communication path, which usually

consists of two wires, must have two input terminals and two output

terminals, hence falls in the category of "four-terminal networks,"

or, more precisely, "two terminal-pair networks." A four-terminal

network which has frequency discriminating properties is called

an "electric wave filter," or in short, a "filter." Depending on

the region of frequencies a filter favors or attenuates, there are

"low-pass filters" (i.e., these filters pass low frequencies while

attenuating high frequencies), "high-pass filters," "band-pass

filters," "band-elimination filters" and "multiple pass-band

filters" where a filter has several passbands separated by

attenuation bands.







- 3 -


The external characteristics of a linear passive four-

terminal network may be completely described by a set of three

independent parameters, which are in general functions of frequency.

Of the possible sets of parameters, there are some which can be

readily obtained by measurement at the terminals without knowledge

of the internal structure of the network; these are the open-and

short-circuit impedances. A very convenient set of parameters,

the "image parameters," may be obtained as functions of the open-

and short-circuit impedances. The image parameters consist of two

image impedances and an image transfer coefficient.

The external electrical characteristics of a filter in-

serted in a communication path between transmitting point (gen-

erator) and receiving point (load) are the operating characteristics

called "insertion characteristics"; these consist of "insertion

loss" and "insertion phase-shift" characteristics. The first is a

measure of the decrease in amplitude of a sinusoidal voltage passing

through the filter, the second gives the amount of phase shift

(or delay) for a sinusoidal voltage passing from input to output

(i.e., from generator to load) terminals of the filter. Both are,

in general, functions of frequency.


1.2 The Image Parameter Method of Network Synthesis

The problem of designing a filter is to synthesize a net-

work having a prescribed insertion loss characteristic.

There are two distinctly different methods of filter






-4-


design available. The older one, developed by Zobell* from earlier

work of Pupin, Campbell and Foster2, is based on image parameter

theory; the resulting image parameter filter is called a Zobel

filter. The newer method, originated by Darlington,3 is based on

insertion loss theory developed independently by Darlington, Cauer

and Piloty at the same time.

The outstanding feature of the image-parameter design

method is its simplicity. This, coupled with the fairly well

performing filters obtainable by the image parameter design tech-

niques, caused this method to be used almost exclusively in prac-

tical filter design.

The simplicity of this method stems from the "building

block" structure of composite filters. Each "building block," or

"section," is a four-terminal network which can be fully described

by two image impedances Z11 and Z12 and an image transfer coeffi-

cient 91 "= a + Jip, where oe is the image attenuation coeffi-

cient and /31 is the image phase shift coefficient.

A composite filter is formed by connecting sections in

tandem, with image impedances that are equal to each other (matched)

at the terminals of the sections which are joined. The total

transfer coefficient of the composite filter is simply the sum of

the transfer coefficients of individual sections.

If these filters could be terminated at both ends by the

respective image impedances (i.e., matched at the ends) the image


*Raised numbers refer to entries in the References at the
end of this dissertation.







-5-


transfer coefficient would describe their actual (measurable) per-

formance. However, since the image impedances are generally func-

tions of frequency, and since the terminating impedances are usually

constant resistances, the filters cannot, in general, be matched at

the termination. This mismatch causes the quantities of interest,

the insertion loss and insertion phase, to be somewhat different

from the image attenuation and image phase shift.

Moreover, the requirement of matched image impedances at

the points of interconnection of individual sections imposes restric-

tions on the networks obtainable by the image parameter method, as

well as on their performance characteristics (i.e., their insertion

loss and insertion phase characteristic). These restrictions lead

to an inefficient use of network components.


1.3 The Insertion Loss Method of Network Synthesis

The insertion loss method of filter design is characterized

by the fact that filters obtainable by it have an insertion loss

characteristic exactly as prescribed, when inserted between prescribed

terminations (hence the name "insertion loss theory"). It is based

on the treatment of the entire problem of finding a four-terminal

network satisfying given insertion loss requirements. The insertion

parameters, rather than the image parameters of Zobel filters, are

the primary design parameters from which the circuit element values

of the filter are obtained in the last step of a sequence of (often

lengthy) operations. The resulting filter, no matter how compli-

cated, is obtained as an entity, rather than an aggregate of








- 6-


individual building blocks (sections) as in Zobel filters, hence

it does not suffer from the matching restrictions imposed on Zobel

sections and permits the realization of optimum (in the sense of

the number of circuit elements required to meet prescribed charac-

teristics, and the element values) networks.

Another very desirable feature of the insertion loss method

is the correction for incidental dissipation it permits. Though

the range of compensation is limited by the type of insertion loss

function (more specifically, by the distance of the nearest

insertion-loss zero to the j]-axis in complex plane), it permits

taking into account losses in components for a great number of

cases encountered in practice.


1.4 Comparison of the Image Parameter and Insertion Loss Methods

A comparison of the image parameter method and the inser-

tion loss method of network synthesis reveals that the latter is

much more flexible in the results it produces, and much more sophis-

ticated in the overall approach and in the mathematical tools it

employs. As a result of the last, it is taught in most engineering

schools only in graduate curricula, and the majority of practical

design engineers have no opportunity to acquire a working knowledge

of the insertion loss method while in school.

An important economical disadvantage of the insertion loss

method of network synthesis, as compared with the image parameter

method, is the much longer design time required by the insertion







- 7 -


loss techniques, resulting in a more costly filter design than in

the case of image parameter filters.* On the other hand if the

filter designed by the insertion loss method is more economical**

than a corresponding Zobel filter (which is often the case), the

increased cost of design may be offset when it is to be manufac-

tured in large volume.

The situation in the practical filter-design field may be

summarized as follows: Because of the simplicity of the design

procedures and of the acceptable filters the image parameter method

yields, and because of the more analytical background and longer

design time required of the designers using the insertion loss

method, the image parameter method is still used by the majority

of design engineers, though using the insertion loss method may

often result in a better filter.

There have been many contributions in the past to both design

methods;*** a great number of these aimed at bridging the gap exist-

ing between modern theory and practical design methods. The approach

from the image-parameter field is, in general, toward the improvement

of Zobel filters by better matching, e.g., Ref. 6, by reduction of

components in derived terminations, e.g., Ref. 7, or by finding new

structures, e.g., Ref. 8. In the insertion loss field the emphasis

*The design time may be reduced by employing modern means of
computation, e.g., an electronic computer. This, while reducing the
required design time, will not, in general, reduce the overall
design cost.
**An economical filter is one that requires fewer or less
expensive components, or its component values may have higher
tolerances without upsetting the performance, etc.
***For a summary of recent contributions and a good list of
references, see Ref. 5.







-8-


9
is on a more comprehensive presentation of known material, on

simplification of design techniques,10 and on tabulation of

designs.11, 12

This dissertation is concerned with a synthesis method

for ladder networks which employs "building blocks" similar to

the image parameter filter; in fact, the "elementary building

block" used is an image parameter half-section. To form a compo-

site filter these "building blocks" are connected in tandem in a

fashion similar to a Zobel filter, without, however, the funda-

mental restriction of Zobel filters, that of matching image

impedances at the points of interconnection.


1.5. Statement of Objectives

The objectives of this dissertation are two-fold:

1. The introduction and investigation of a new method of

synthesis of ladder networks by which filters can be designed

with a simplicity similar to the simplicity of the image

parameter method of design, yet yielding more general filters

than the Zobel filter of the image parameter method (i.e.,

yielding filters the choice of element values and the perform-

ance characteristics of which do not have the limitations of

Zobel filters).

2. The derivation of relations and determination of

parameter values for use in this synthesis method, by which








-9 -


design engineers familiar only with the image parameter

method will be able to design filters with desirable inser-

tion loss characteristics (Butterworth, Tchebycheff and

elliptic-function type) until now obtainable primarily by

the insertion loss method.

The contents of this dissertation is divided into three

chapters. Chapter II deals with methods of interconnection of

the elementary building blocks. It is shown that any lossless

ladder can be expressed in terms of image impedance half-sections

with one common image parameter. The cut-off frequency is chosen

as the common parameter.

In Chapter III some methods of finding the image param-

eters of the elementary building blocks are investigated. The

image parameters are then determined for two-section ladder filters

having Butterworth, Tchebycheff, and elliptic-function insertion

loss characteristics.

Based on results of Chapter III, a modification of Zobel

filters is introduced in Chapter IV. This modification permits

control of the distortion in passband of Zobel filters; it also

makes the economy of these filters (as far as the performance per

number of elements is concerned) comparable to the economy of

optimum filters.














CHAPTER II


THE GENERAL LADDER NETWORK IN TERMS OF ELEMENTARY
BUILDING BLOCKS CONNECTED IN TANDEM


2.1. The Elementary Building Block

Only the low-pass ladder will be considered here, since

the high-pass, band-pass and band-elimination networks can be ob-

tained from the low-pass ladder by well known frequency trans-

formations.13, 14

The elementary building block (Fig. 2.1) is an image

parameter m-derived low-pass half-section, the circuit element

values of which are determined by the image parameters m, r and c'

using the simple relations from image parameter theory:15

cl = r (2.1)
c
= mr (2.2)

1 m2
c2 -' (2.3)
mr c


mc = 2 _
m = 1 0- ) O m 1 (2.4)

where wc is the cut-off frequency of the half-section, r is the

image impedance at zero frequency, wo is the frequency of infinite

attenuation.

We also have (Fig. 2.2 and Fig. 2.3)


- 10 -







- 11 -


h2



S0l T 0


The elementary building block




real
S\ imaginary (negative)

J -..
I







Image impedance of the elementary building
block at terminals 1--0 of Fig. 2.1.


Imaginary

real (positive) (r
F / i\
i / I \
S/ I \
\ / \ \


negative)


Wc
l-2


c


Image impedance of the elementary building block
at terminals 2--0 of Fig. 2.1.


Zll


Fig. 2.1.




Zi


r .








Fig. 2.2.


Fig. 2.3.


ZI2







- 12 -


r




S2] '2(.



1 ( m2)


Analysis of equations (2.1), (2.2), (2.3) and (2.4) reveals

that the parameters m, r and w> uniquely determine the component
c
values c1, c2 and h (with the restriction that m is a real and posi-

tive constant, 0 t m i 1). On the other hand, given cl, c2 and h,

the design parameters m, r and wc may be found from the following

relations:

2 h
r2 = 2 (2.7)
Cl

m2 = + c2 (2.8)


1W2 1 (2.9)
h2(cl + c2)

Equations (2.7), (2.8) and (2.9) are obtained from equations (2.1),

(2.2) and (2.3) solved for r, m and oo Restricting the design
C
parameters to be positive, it is seen from the last set of equations

that cl, c2 and h2 uniquely determine the design parameters r, m and

W C.


2.2. Methods of Connecting the Elementary Building Blocks

Fig. 2.4 shows one method of connecting the half-sections











- 13 -


-4


r-4

-4 0


r-4

V14







V.4



.M0



0 0


co

WA









V-40


1-4


-4)
GJ-4

0
'30




u00


0 0
u c


0

.0

"4

"0
-4

-A


41
V14






00





,4'





00

V0
C00








- 14 -


into a ladder network; here we connect the even-numbered terminals

with the odd-numbered ones. We note that the even-numbered terminals

have an image impedance given by equation (2.5) (i.e., the image

impedance between the even-numbered terminals and 0, the common, for

one half-section only, is given by (2.5)), and the odd numbered

terminals have an image impedance given by equation (2.6), hence we

connect the terminal pairs having different types of image impedances.

The image impedance at both terminations of the resulting

ladder will not be of the same type; the image impedance at terminals

1-0 is of the type given by equation (2.5), the constant-k mid-shunt

type, whereas, the image impedance at the other termination (at

terminal pair 2n-0) is of the type given by equation (2.6), the

m-derived mid-series type. Therefore the resulting ladder cannot be

symmetrical.*

Another method of connection is shown in Fig. 2.5, where we

connect even-numbered terminals with even-numbered terminals and

odd-numbered terminals with odd-numbered ones, i.e., we connect pairs

of terminals having image impedances of the same type.

If the ladder contains an even number of half-sections, both

image impedances at the terminations will be of the same type, hence

the ladder can be symmetrical. If the number of half-sections is odd,

the image impedances at the terminations are similar to those of

Fig. 2.4. The method of connection shown in Fig. 2.5 results then in


*A 4-terminal network possesses electrical symmetry if the
image impedances at both terminal-pairs are equal, i.e., one terminal
pair cannot be distinguished from the other pair by external
measurements.








- 15 -


a more general ladder than the connection of Fig. 2.4 (in fact,

Fig. 2.4 can be considered as a special case of Fig. 2.5 in which

the components of the second, fourth, etc., section are zero, i.e.,

h4 = h8 h12 ... h2n 0 and c3 c7 '* = c2n-1 = 0) hence

this method of connection will be used in subsequent analysis.

Since every component of this ladder can assume any value depending

on the choice of the image parameters rl, m1, cl' r2, m2' c2'

... rn m Wcn', it is clear that the reverse must hold also, i.e.,

any ladder network having the configuration of Fig. 2.5 can be syn-

thesized with image parameter half-sections by proper choice of the

image parameters ri, mi and wci


2.3. A Conventional Low Pass Ladder in Terms of Elementary
Building Blocks

A particular case of the ladder of Fig. 2.5, one in which

a reduction in the number of components can be realized, would

result if two adjacent parallel resonant circuits in the series

branches of the ladder could be combined into one parallel resonant

circuit. Obviously this can be accomplished only if the resonant

frequencies of the adjacent circuits are equal. The resulting ladders

are shown in Fig. 2.6 (a) and (b), where the components of Fig. 2.5

combine as follows:

ql = Cl (2.10)

q2 h2 + h4 (2.11)
a2 = c2c4 (2.12)
c2 + c4












- 16 -


N 0 ,M 0



-4








I c 10 4





o .







0r 0
41I


c U
-04
eq0




cr V40








- 17 -


a2q2 -1 h2c2 h4c4 (2.13)
2..,

q3 3 c3 + c5 (2.14)


q4 h6 + h8 (2.15)



a4 c6 c8 (2.16)
c6 + c8


a4q 2 = h6c6 = h8c8 (2.17)


In Fig. 2.6 (a) the ladder consists of an odd number of half-

sections (the number of half-sections n odd), hence

(see Fig. 2.5)

qn+l h2n-2 (2.18)


an+1 c2n-2 (2.19)


The ladder of Fig. 2.6 (b) has an even number of half sections

(n-even), hence here (see Fig. 2.5)


n = h2n + h2n-2 (2.20)


a C2n-2 c2n (2.21)
n c2n-2 + c2n


an 2 = h2nc2n = h2n-2c2n-2 (2.22)
n _


qn+1 c2n- (2.23)

The number of image parameters for the ladders of Fig. 2.6








- 18 -


is 3n (three parameters, ri, m, and Wc for each of the n half-

sections). However, due to the conditions of equations (2.13),

(2.17) "' (2.22), some of the parameters will be related by these

equations, resulting in the number of independent image parameters

being equal to the number of components in the ladder.

Consider the problem of expressing a given ladder of the

form of Fig. 2.6 in terms of image parameter half-sections, i.e.,

given a ladder of Fig. 2.6, determine the element values of an

equivalent ladder of Fig. 2.5. One question arises immediately:

Is there a unique equivalent of the ladder of Fig. 2.6 in the form

of one of Fig. 2.5? If not, could there be one specific form of

Fig. 2.5 which is advantageous?

The first question can be answered by noting that there are

3n components in ladder of Fig. 2.5, the values of which are to be

determined from 2n + 1 relations.* Thus there is no unique equiva-

lent of a ladder of Fig. 2.6 in the form of one in Fig. 2.5

The second question can be answered after investigating the

possible equivalent ladders of the form of Fig. 2.5.


*For a ladder of Fig. 2.6 (a) we have three equations of
the form of equation (2.11), (2.12) and (2.13) for each series
branch except the last, i.e., 3(n 1)/2 equations; one equation
for each shunt branch, i.e., (n + 1)/2 equations ((n 1)/2 of the
form of (2.14) and one equation (2.10)), and two relations (2.18)
and (2.19) for the last series branch, giving a total of
3(n 1)/2 + (n + 1)/2 + 2 2n + 1. For Fig. 2.6 (b) we have
3n/2 equations of the form of (2.11), (2.12) and (2.13) for the
series of the form of (2.11), (2.12) and (2.13) for the series
branches, and (n + 2)/2 for the shunt branches, giving a total
of 3n/2 + (n + 2)/2 2n + 1.









- 19 -


In order to obtain a particular ladder equivalent,

3n (2n + 1) n I component values of the equivalent ladder

of Fig. 2.5 will have to be either chosen or specified by addi-

tional n-l equations* (i.e., we have n-l degrees of freedom).

It is clear that the n-I components, the values of which are arbi-

trary, will be located in the ladder in accordance with the location

of the components in Fig. 2.6 which on division produce the n-l

degrees of freedom (e.g., one cannot choose arbitrarily the values

of the first n-i components, starting with cl, c2, h2, c4, h ...

up to the (n-1) th component in Fig. 2.5).

Examination of the ladders in Fig. 2.6 shows that every

branch of the ladder which is divided produces one degree of freedom,

as shown by the symbol "I" in Fig. 2.7 (a). The arrows indicate

where the degrees of freedom originate, hence they also indicate which

of the elements may have arbitrary values.

Fig. 2.7 (b) and (c) show two possible choices of the location

of arbitrary elements: In Fig. 2.7 (b) the first half-section has no

arbitrary elements, the second has one in the series branch and one

in the shunt branch, the third half-section has none, etc. In


*The chosen values of the components, though arbitrary in a
certain range, should not result in a negative element lest some of
the image parameters from which the half-sections can be computed
become imaginary. If instead of a component value a corresponding
image parameter is chosen, a similar limitation applies in addition
to the requirement 0- mi I l(or w < 4).
c









- 20 -


-4

v-]




I I II



4


SI

SII







-4




-4










H-


KHtI













-4










-4


I,.'
C)


44
0
0

0
-4

0
-41

0



-#4




i-4
(U4


,-4


/"









- 21 -


Fig. 2.7 (c) every half-section, except the last one, has one arbi-

trary element. There are a number of combinations of the arrangements

in Fig. 2.7 (b) and (c) possible.*

The arrangement of arbitrary elements in Fig. 2.7 (c) appears

advantageous in that it permits choice of the circuit element value of

one component (or one parameter) in each half-section, except the

last. Hence the resulting ladder, with the exception of one terminat-

ing half-section, can be made to have one arbitrary parameter.**

The values of the n-l arbitrary elements may conveniently

be specified by additional n-l arbitrary equations, in which the

limitations on the chosen values (see footnote* on previous page)

could be included. A simple set of n-1 of such equations could be

Pl m P =2 P3 = P =n (2.24)

where p denotes an image parameter.*** In particular if the same


*Similar reasoning applied to the ladder of Fig. 2.6 (b)
shows that the conclusions reached in analysis of Fig. 2.6 (a)
hold here as well; we may have one arbitrary element in each half
section, except in one of the terminating half-sections.

**Since the component values of a half-section can be ex-
pressed in terms of image parameters r, m and Wc, equation (2.1),
(2.2) and (2.3), the fact that one of the component values can be
chosen arbitrarily means that one of the image parameters can be
chosen arbitrarily.

***We note that from equation (2.24) we have n-1 relations of
the form p m p p P P3,'* Pp, pn, hence the value of p cannot
be chosen but is de ermined from 3n simultaneous equations,
2n + 1 of which are in the form of equation (2.10) to (2.23) with
h's and c's substituted by equation (2.1) to (2.4).









- 22 -


image parameter is chosen for all p's, the resulting ladder will be

very similar to an image parameter Zobel filter1' 15, 16 when p

denotes either r or )c&.*

A convenient choice for p, equation (2.24), is to let it

denote the cut-off frequency a) c, i.e., letting the cut-off frequen-

cies of all n half-sections be equal. This would make the image

parameters m equal for the half-sections with common parallel

resonant frequencies aw, (see equation (2.4)).

The above leads to the following conclusion:

Any conventional ladder filter, Fig. 2.6, can be synthesized

by tandem connected (method of Fig. 2.5) image parameter half-

sections, pairs of which have equal m-values and all of which

have equal cut-off frequencies co .

This is equivalent to stating that:

Any conventional ladder filter can be obtained from an

image impedance Zobel filter by changing the impedance levels

rl, r2, rn of its half-sections.

In Chapter III some methods will be investigated, by which

the required impedance levels ri, the cut-off frequency ojc and the

m-values could be determined so as to obtain a ladder filter having

a prescribed insertion loss.


*A conventional image parameter filter is called a Zobel
filter. It is a filter consisting of connected-in-tandem image-
parameter sections or half sections (prototype or m-derived) with
common cut-off frequencies w and with image impedances (see
equations (2.5) and (2.6)) wh ch match each other at the points of
interconnection. In terms of our elementary building blocks, a
Zobel filter will have jcl= Wc2 ... cn and ri = r2
... = rn.








- 23 -


2.4. Filters with Incidental Dissipation in Terms of Elementary
Building Blocks

One of the major disadvantages of the image parameter

Zobel filter design technique is the fact that dissipation cannot

be taken in account in design (though the effect of dissipation

may be found by analysis).

The modern network theory offers a technique3 which allows

one to synthesize a filter with dissipation producing the same

form of insertion loss curve as one without dissipation, except for

an additional constant (independent of frequency) loss. This tech-

nique, called predistortion, is based on the effect of dissipation

on the location of poles and zeros of the insertion voltage ratio

in the complex frequency plane.

Given the pole and zero distribution of a desired insertion

voltage ratio of a dissipationless ladder filter, e.g., Fig. 2.8

(poles and zeros on solid curve), the addition of positive resistance

to each component (in series with an inductance and in parallel with

a capacitance) in such amounts that would make the Q-factors of all

components equal, would move the zeros and poles in Fig. 2.8 hori-

zontally a distance d 1/Q to the left, as indicated by the poles

and zeros located on the dashed curve. Adding negative resistance

to the dissipationless components in the same amounts as above would

shift the poles and zeros a horizontal distance d to the right, as

indicated by the zeros and poles on the dotted curve* (the dotted

*This horizontal shift is equivalent to transforming the
complex variable s 6' + jco to s + d, where the (+) and (-) signs
correspond to addition of positive or negative resistance, respec-
tively.







- 24 -


Effect of added
positive resistance


I'.


Lossless L's and C's



Effect of added negative resistance


X X


Effect of adding proportionately equal amounts of
resistance on the location of poles and zeros of
the insertion voltage-ratio function


Fig. 2.8.








- 25 -


curve is called the predistorted curve for an amount of dissipation

equal to d).

Suppose that the pole and zero distribution on the dotted

curve is used to synthesize a dissipationless ladder network.* If

now dissipation d is added to each element, the poles and zeros

would move to the left a horizontal distance d and the distribution

on the solid curve (Fig. 2.8) would be obtained, resulting in the

desired insertion loss.

The lossless ladder obtained from the predistorted curve

(dotted curve) can be represented by elementary building blocks

(image parameter half-sections) as outlined in section 2.3. The

image parameters r, m and c c of these building blocks will in gen-

eral not be equal to the image parameters of the building blocks

corresponding to a ladder synthesized from the pole and zero loca-

tions on the solid curve (the pole and zero location of the desired

insertion voltage ratio).

Thus, given a lossless ladder filter in terms of image

parameters r, m and ULc, the effect of predistortion for a certain

amount of dissipation can be obtained by a corresponding (to the

amount of dissipation) change in the image parameters r, m and

W c of the elementary building blocks.


*In practice, predistortion is applied to the zeros only,
since the application to the poles would result in a network func-
tion which is not realizable by a lossless passive ladder without
mutual (inductive) coupling. Thus the transformation of s to s-d
is applied only to the numerator of the insertion voltage ratio
function, and compensation of the effects of dissipation for the
passband only is achieved. Note also that there is an upper limit
of d, which must be smaller than the horizontal distance of the
nearest zero to the jw -axis, for physical realizability with passive
networks.















CHAPTER III


DERIVATION OF THE IMAGE PARAMETERS FOR LADDER FILTERS
HAVING PRESCRIBED INSERTION LOSS CHARACTERISTICS


In this chapter the image parameters will be determined

for the elementary building blocks which, when connected in

tandem, result in ladder filters with Butterworth (maximally

flat) Tchebycheff (equal ripple in passband only), and elliptic

function (equal ripple in passband and in stopband) insertion loss

characteristic. The image parameters will be expressed in terms of

the critical frequencies (poles and zeros) of the insertion loss

function. The investigation will be limited to two-section

symmetrical filters.


3.1. Outline of Approach

The insertion loss L (in db) as a function of normalized

frequency oJ for the above three types of response is given by17

L = 10 log (1 + E2) (3.1)*

where E denotes a polynomial or a ratio of two polynomials in W ,

the form and the coefficients of which determine the type of

response. The zeros and poles of E(w) coincide with the zeros

and poles of the insertion loss function L(w), as can be seen

from equation (3.1). It shall then be attempted to express the

*Equation (3.1) is for symmetrical filters only; it will
be assumed for convenience that the filter is working between
equal resistances of 1 ohm.
26 -








- 27 -


image parameters of the elementary building blocks in terms of

these poles and zeros.


Method I:

One way of accomplishing this could be to find the inser-

tion loss of a ladder consisting of elementary building blocks in

terms of the image parameters of its half-sections. Comparison of

this insertion loss with the insertion loss of Butterworth, Tcheby-

cheff or elliptic-function type would yield a number of equations

which, when solved simultaneously, would give the image parameters m,

r and tc in terms of the poles and zeros of the insertion loss

function.


Method II:

Another method of getting the same results for specific

numerical cases would be to synthesize the ladder from the given

insertion loss function by any suitable method (e.g., Darlington's

method) and then express the ladder in terms of image parameter

half-sections, obtaining a number of equations of the form of

equations (2.10) to (2.23);* substituting for h's and c's equa-

tions (2.1) to (2.4), a set of equations in mi, ri and Wce are

obtained, which could be solved (at least theoretically) simul-

taneously for the image parameters.

The first approach is more general and will be used here.


*It is to be noted here that qi and ai are known.







- 28 -


Some numerical data will be obtained using the second approach.


3.2. Insertion Loss of a Two-Section Ladder Filter in Terms of
Image Parameters

The two section symmetrical ladder which will be used in

subsequent investigation is shown in Fig. 3.1.* This ladder is ob-

tained by tandem connection of image parameter half-sections as

shown in Fig. 3.2.

Comparison of the ladders of Fig. 3.1 and 3.2 yields the

following relations:

q1- l2 1(3.2)
rl wc

m12(rl + r2) (3.3)
q2 oWc



a2 1 m122 (3.4)
r12 Wc(r1 + r2)






4 3(r + r4) (3.6)




m34 c(r3 + r4)


q5 = m34 (3.8)



*The Butterworth and Tchebycheff filter can be considered
as a special case of the ladder of Fig. 3.1 in which a2 w a4 = 0.

































































0V
"4 0 fa4


N U


43
III--





4 4
r -


r U
.r


-HF---
U
413


C4 3
34
"4


-6 -
---II---


^3 u 3,
N U

" N






0)
NN


I3
a |


- 29 -


0 M

V4
"4
.9I
II 44


0
*s






.44
'3
0





*6

00
*I


.94




en



.4
0
V4
0
*0




















00
'0
NN












"4
4
Ak






0

U
0
*s
*g
.4i
60


0
A-
NO
C Ni~

00i




.9.'

fa






- 30 -


where rl, r r and r are the impedance levels of the half-sections
1 2 3 4
1, 2, 3 and 4, respectively; m12 is the common m-yalue of half-

section 1 and 2; m34 is the common m-value of half-section 3 and 4;

Wc is the cut-off frequency common to all half-sections; ql, q ,

q5, a2 and a have dimensions of capacitance; and q2 and q have

dimensions of inductance.

It is much more convenient to find the insertion loss of

a lattice equivalent to the ladder of Fig. 3.1, rather than that

of the ladder directly.

A lattice equivalent of the ladder of Fig. 3.1 is shown in

Fig. 3.3 (a).

The series reactance X and the shunt reactance Xb of the

lattice of Fig. 3.3 (a) are given by


Xa Za WA (3.9)
a J Al A 2o2


Zb 1 A342
Xb j" T (A4 W2 A5) (3.10)


A (3.11)
1I L1

A2 C1 (3.12)


A3 L2C2 (3.13)
A 223 (3.14)

A4 L2C2C3 (3.14)


A5 = C2 + C3


(3.15)







- 31 -


(b)


Equivalent lattice of the ladder of Fig. 3.1.


q2 4 q2


/ "1


A Butterworth and Tchebycheff type ladder filter
and its equivalent lattice


Fig. 3.3.


Fig. 3.4.







- 32 -


The relation between the lattice and the ladder elements

is as follows:

A q q5 (3.16)
q4a4 -- q2a2


A2 q + q2a2A1 (3.17)


A3 + a2 + q4a4 (3.18)
1

A4 A2A3 q2a2q4a4A1 (3.19)

1
AS i (ql + q3 + q5) (3.20)

The insertion loss of the equivalent lattice of Fig. 3.3

is given by equation (3.1), where

Bl(w -w 2B2 + B3)
E W -2 1 (3.21)
U}4B U2B5 + 1

B 2 A2A (3.22)




AIA5 1
B3 AA- (3.24)
A2A4

I
1
4 A l (A2A3 A4) (3.25)


B5 L (A1A3 + A2 A5) (3.26)


The zeros and poles of the insertion loss are given by







- 33 -


the zeros and poles of equation (3.21), i.e.


)(W 4 2B2 + B3) 4 W 2(z1 2 + z 2) + z 1222]

4 2
W4B4- 2 B + 1- + 1 (3.27)
4 5 P12P2 P12 + p22

where z1, z2 and pl, P2 are the frequencies at which the

insertion loss is zero and infinite, respectively. It follows

from equation (3.27)

B2 = z12 + z22 (3.28)


B3 z 2z22 (3.29)


B" p4lp22 (3.31)


Equations (3.27) can be solved for the zeros and poles:

zo 0 (3.32)
1
2 B2 B22 21
z1 2 = 2- + -[ B (3.33)


Po (3.34)

B5 ( B2 4B4) (335)
pl,2 = 2B4 (3.35


3.3. Tchebycheff and Butterworth Filters Synthesized with Image
Parameter Half-Sections

The Butterworth and Tchebycheff-type two-section filter is

shown in Fig. 3.4(a) and its insertion loss characteristic is given

by equation (3.1) with17







- 34 -


E = H6J5 (3.36)

for the Butterworth filter, and

E HT5(o) (3.37)

for the Tchebycheff filter. Here T5 (w) is the Tchebycheff poly-

nomial of order 5, which can be defined by

T5(W) cos (5cos-1 ) (3.38)*


0 5 cdJ 1

The insertion loss zeros of the Butterworth filter, as seen

from equation (3.36) are at C 0; the Tchebycheff polynomial is

zero when

5cos 7T(i + )

i = 0, 1, *** 4

resulting in zero insertion loss at the following frequencies:

zo 0

Z -2 cos2 7T (3.39)

2 cos2 3.
z2 10

*Tn(x) can be defined in polynomial form as Tn(z) = 2n-l Exn
- nxn-2/1!22 + n(n 3)xn-4/2!24 n(n-4)(n-5)xn'6/3!26 + **
where the summation is stopped when the exponents of x are negative.
The form of (3.38) is convenient in that it exhibits the location
of the zeros.







- 35 -


Equation (3.37) can be written, using equations (3.39),

(c22 C T )(W2 C082 37)
E 10 10 (3.40)
1- 2 Co 1 Cos 10)

For both, Butterworth and Tchebycheff, filter types E = H
when 1, giving (see Fig. 3.5 (a) and (b) )

H2 100.11P 1 (3.41)

where L is the maximum loss in passband in db.
P
It can be shown that Tchebycheff and Butterworth two-

section filters possessing electrical symmetry have also geome-

trical symmetry, i.e. (Fig. 3.4 (a) )

q q5

q2 q4 (3.42)

hence the equivalent lattice to the ladder of Fig. 3.4 (a) can be

easily found using Bartlett's bisection theorem.18 The equivalent

lattice is shown in Fig. 3.4 (b).

Comparison of Fig. 3.3 (a) with Fig. 3.4 (b) yields the

following relations:

L1 = L2 q2

Cl C3 z q, (3.43)

C2 (-t

Inserting equations (3.43) into equations (3.11) to (3.15)


gives








- 36 -


L(db)

t


L(db)


-- 60


z1 z2


Insertion loss characteristics for (a) Butterworth
and (b) Tchebycheff two-section filters


Fig. 3.5.







- 37 -


1






A3 2

A qlq2q3
4 2


A5 q + (3.44)


Equations (3.44) inserted in equations (3.22) to (3.26) give


B1 q12q22q3
2

2q + 2q3 q2q3
B2 7 "-
qlq2q3

2ql 2q2 + q3
B3 12q22q3


B4 B5 0 (3.45)

Equations (3.2)to (3.8), with equations (3.42) and with

a2 a = 0 result in 12 = m34 r1 r and r2 r3, giving

1
ql ", c
r c

r + r2
c2 + (3.46)

2 c
3 r2 c






- 38 -


Substitution of equations (3.46) in (3.45) gives


(r1 + r2f
rl12r2 wc5


B c 2r(1 r12 + r )


Wc4rl(1 r1r2)
S= c *1(3.47)
3 r + r

Finally, using equations (3.29) and (3.30) in equations

(3.47) we get


z12 + z 2 W- 2(1 r12 + r
2 c rl+ r2

2 2 c4r(1l rlr2)
z z2 M- + r 2(3.50)


The Butterworth Filter

The insertion loss zeros of the Butterworth filter are at

zero frequency (i.e., at Co 0). Equations (3.50) then become

(with zI = z2 = 0)

0 W c2 1 r 2 + rl
c 1 ri + r2


0 c 4r(1 rr2)51)
r + r

Sol.4ng equations (3.51) simultaneously we obtain
r 1

rI 0.5 + (1.25)21 1.27202

S- 0.78615 (3.52)
r2 r







- 39 -


Comparison of equation (3.21) with equation (3.36) shows

that B1 H, hence from equations (3.41) and (3.47) we get


O.Lp (rI + r2)
10 l 1 =( +2 (3.53)
4 2 10
r1 r2 W*c

From equations (3.51) we get r + r = r 3, which inserted
1 2 1'
in (3.53) with r1r2 1 from (3.52), gives

rl 1.27202
cc H0.2 (100.11P 1)0.1 (3.54)


The cut-off frequency w0 is tabulated for values of L ,

the maximum attenuation in passband, from 0.1 db to 5.0 db in

Table 1, and is plotted versus L in Fig. 3.6.

A two section Butterworth filter in terms of image

parameter half-sections, with L 3 db is shown in Fig. 3.7.*


The Tchebycheff Filter

With the insertion loss zeros of the Tchebycheff filter

given by equations (3.39), equations (3.50) become

2 7T 2 3TF 2 1 2 r
cos -f- + cos -10- c 1 + r


2 7T os2 37T c4rl(1 rlr2)
10 10 r + r2


We also have from equations (3.21) and (3.40)


*For Lp = 3 db, Jc 1.272 since H = 1 in equation (3.54).








-40 -


TABLE 1


CUT-OFF FREQUENCY wc FOR THE BUTTERWORTH TWO-SECTION FILTER

L in db.


Lp wc Lp wc Lp Lc


0.1 1.8526 1.2 1.4263 3.2 1.2612

0.2 1.7265 1.4 1.4011 3.4 1.2503

0.3 1.6560 1.6 1.3792 3.6 1.2399

0.4 1.6071 1.8 1.3597 3.8 1.2300

0.5 1.5698 2.0 1.3421 4.0 1.2205

0.6 1.5396 2.2 1.3261 4.2 1.2113

0.7 1.5143 2.4 1.3113 4.4 1.2025

0.8 1.4925 2.6 1.2976 4.6 1.1939

0.9 1.4732 2.8 1.2847 4.8 1.1857

1.0 1.4560 3.0 1.2726 5.0 1.1776







- 41 -


0.2 0.3 0.5


1.0


2.0 3.0 5.0


Maximum attenuation in passband L in db



Cut-off frequency of elementary building
blocks versus maximum attenuation in
passband for the design of filters with
Butterworth characteristic


1.6





1.4


Fig. 3.6.











- 42 -


M00
* I


.4





41
o c




1 V 1CI


*'W '4







000 0 "
*44













.C.O



o **





Sh






- 43 -


H
Bl' sin2 sin2 :vr
10 10

with which (3.47) becomes

H (rl + r2)2 (3.56)

sin2 3 1 r 2r2

Instead of attempting to solve directly equations (3.55)

and (3.56) for wcj, rl and r it is more convenient to use

equations (3.46) with the following equations derived by Belevitch19
7T
S 2sin 10
g
3TT
2gsin 10
q2 g2 + cos2 3


2 g2 + cos2 )
q3 = (3.57)

g (g2 + cos2 jT


where g is related to H by the transformation

1 sinh 5x
H

g sinh x sinh 5 arc sinh f ) (3.58)


and H is given in terms of L the maximum insertion loss in pass-

band (which is here synonymous with the peak value of ripple in

passband), by equation (3.41).

Equations (3.46) and (3.57) are readily solved for uC ,

r1 and r2, giving







- 44 -


k) 2 A + B
c 4 sin Zt sin 3T
10 10

2 35
2 g sin 3__
r 10
1 (A + B) sin 1
r- 0
r2 1A
B


A 2(g2 + cos2 7L sin j )
10 10


B = g2 + cos 13- (3.59)


A tabulation of W rI and r for values of L from
c 1 2 p
0.1 db to 5 db appears in Table 2.* A plot of COc, rl and r2

from Table 2 versus L is shown in Fig. 3.8, and a Tchebycheff

filter with L = 1.5 db in terms of image parameter half-sections
p
is shown in Fig. 3.9.


3.4. The Elliptic Function Filter Synthesized with Image
Parameter Half-Sections

The elliptic function type (often called equal ripple in

passband and stop-band type) two-section filter which will be used


*The data in Table 2 are computed on the University of
Florida IBM 650 computer from the equations (3.41), (3.58) and (3.59).







- 45 -


TABLE 2


IMAGE PARAMETER VALUES FOR THE TCHEBYCHEFF TWO-SECTION FILTER

Lp in db; r's in ohms



Lp c r r2 r2/rl


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8


1.1724

1.1176

1.0891

1.0704

1.0568

1.0464

1.0379

1.0309

1.0250

1.0199

1.0114

1.0047

0.9926

0.9947

0.9908

0.9874

0.9844

0.9819

0.9796


0.7428

0.6680

0.6197

0.5837

0.5547

0.5304

0.5094

0.4909

0.4743

0.4593

0.4329

0.4102

0.3903

0.3725

0.3565

0.3419

0.3285

0.3161

0.3046


0.8638

0.8262

0.7952

0.7685

0.7448

0.7235

0.7040

0.6859

0.6692

0.6535

0.6247

0.5989

0.5753

0.5537

0.5336

0.5150

0.4975

0.4811

0.4656


1.1613

1.2368

1.2831

1.3166

1.3427

1.3640

1.3820

1.3974

1.4109

1.4228

1.4431

1.4599

1.4741

1.4862

1.4968

1.5061

1.5144

1.5218

1.5284








- 46 -


TABLE 2 -- Continued


L p rl r2 r2/r
2


0.9775

0.9756

0.9740

0.9724

0.9711

0.9698

0.9686

0.9676

0.9666

0.9656

0.9648


0.2939

0.2838

0.2743

0.2653

0.2569

0.2488

0.2412

0.2339

0.2270

0.2203

0.2140


0.4509

0.4370

0.4238

0.4111

0.3991

0.3876

0.3765

0.3660

0.3558

0.3460

0.3366


1.5344

1.5399

1.5449

1.5495

1.5537

1.5575

1.5611

1.5644

1.5675

1.5704

1.5730


3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0










- 47 -


0.9
4 7



Q6 0.7 -..

-s 3r
", 3 I. I-


0 0
S0 ..




03
"^0.3 -.-----.*....



0.2--


01
0.1 0.2 0.5 1.0 2.0

Maximum attenuation in passband L in db
P

Fig. 3.8. Image parameters Jc r1 and r2 versus maximum
ripple in passband for the design of filters
with Tchebycheff characteristics




























.

C4
*







co
* .
O
F^-


-~ if.'

o

F-

I
.0


-48 -


-:1
0






I
o




0
V4
ao
** a




*o


AS


.










-t


.0



Ho
*4



0
A -



a 3



a
4 0)






















d
0.0
0
S8.0


4







44
A.0

00









94








- 49 -


here is shown in Fig. 3.1,* and its insertion loss characteristic,

Fig. 3.10, is given by equation (3.1) with3


2 2 2 2
H (I zW )(1 z2 )
" (1 u2z12)(l &2z2)


(3.60)


The zeros of the insertion loss characteristic are given by
1
2 2K
zi k an ( 2- k)


1
z2 k an ( k)


(3.61)


where an denotes


20, 21
the elliptic sine, defined as


sn(u) sin 0


u = F(k, 0) =/ dO

(1 k2 sin2 e )2


K denotes the complete elliptic integral of the first kind
7T
-7
K K(k) dO
o (1 k2 sin2 9)2


and k is the modulus of the elliptic sine.

The constant H is given by

H 4 (100.1 1)(100.La 1)


(3.62)


(3.63)


(3.64)


*This is the special case of elliptic function filters in
which all the poles are on the jW-axis, hence it can be realized
with a ladder network that does not require mutual coupling.






- 50 -


where L is the maximum distortion in passband, and L is the
P a
minimum attenuation in stopband in decibels (see Fig. 3.10). L ,
22
L and k are related by


La -0 (i 32


Lp 10 logl (1 + H2 2)

5
A k =7 n2 (. k) sn2 3K, k) (3.65)


As can be seen from Fig. 3.10, the modulus k determines

the width of the transition band (k is called the selectivity

parameter),

k = a (3.66)


where u)J is the upper frequency limited of the passband (or the

effective cut-off frequency), and aj is the lower limit of the
a
attenuation band.

When L is small and L large, which is the case for most
p a
filter requirements, the various parameters determining the response

of an elliptic function type 2-section filter are related by the

approximate equation


La = 10 logl0 (10 IL- 1) 50 logl0 q 12.041 (3.67)


where q is the elliptic modular function of k,* La and LP are in


*The function log q is tabulated in most elliptic function
tables, e.g., Ref. 21, pp. 49-51.







- 51 -


L (db)

t


z z2


W p


I
1a
a


Fig. 3.10.


Insertion loss characteristic of an elliptic
function two-section filter







- 52 -


decibels. A plot of La versus k for L 0.1 db is given in Fig. 3.11.

Figure 3.12 shows the dependence of L on L for a given k.
a p
In order to find relations between H, z, z2 of equation

(3.60) and the image parameters aWc, r to r m12, m34, a procedure

similar to one used in the previous section will be followed. The

resulting algebraic manipulations are quite lengthy and will be

omitted here. The following steps are carried out:

1. Equations (3.16) to (3.20) are inserted in equations (3.22)

to (3.26), giving a set of equations, say Set I, relating B's with

q's.

2. Equations (3.2) to (3.8) are inserted in Set I, giving

a set of equations, say Set II, relating B's with the image

parameters.

3. Equate Set II to corresponding equations (3.28) to

(3.33), getting a set of equations, say Set III, relating the

image parameters to the zeros and poles of the insertion loss

function (3.60).

4. Solve equations Set III simultaneously for the image

parameters in terms of zeros and poles of the insertion loss

function.

Step 3 results in five non-linear equations in seven

unknowns (the image parameters); two additional equations are

obtained from the symmetry conditions which the ladder must

satisfy, giving a total of seven independent equations. These

seven equations do not lend themselves readily for use in Step 4,







- 53 -


70


60


50 --


40 -


30 ... .


20 --
0.3 0.4 0.5


0.6 0.7 0.8 0.9 1.0


Selectivity factor k


Fig. 3.11.


-10


0.01


Minimum stop-band attenuation La versus
selectivity factor k for the two-section
elliptic function filter


0.05 0.1


0.5 1.0


5.0


L in db
P


Fig. 3.12.


Variation of minimum attenuation in
stop-band with maximum ripple in
passband


J.o

4.
O 0



0 4j
I 4



. v







- 54 -


hence numerical values for the image parameters will be obtained

using Method II (Chapter III).* Step 4 is carried out for a set

of equations which are close approximations to those obtained in

Step 3.

Following are the equations obtained in Step 3:


12 = m122 (3.68)
wc2


2 1 2 (3.69)
WOc

(m12 m342)(1 + ml2m34D)
H = (3.70)
c5 (rlml2 r4(37

z12 + z 1 + rr4(D + m12m34) (3.71)
2 I1 + ma2m34D

z2z 2 rr4D (3.72)
1 -(3.72)
iJ c4 1 + m12m34D

D rlml2 r4m34(3.73)
r 4m12 rlm34


The equations obtained from symmetry conditions are:


r1r3 = r2r4 (3.74)


*The numerical values appear in Table 3. They were obtained
by analysis of synthesized elliptic function filters in terms of
image parameter half-sections. For this purpose the synthesized
filters published in Ref. 12 were used.







- 55 -


ml2m34 -

m122 m342


r1r4(rl r2)
(r42 rl2)(r1 + r2)


Equation (3.75) can be written as


r2 (r1m2 r 4m34)(r4m12 + rlm34) (3.
rl (r4m12 r1m34)(rlml2 + r4m34)


to which a close approximation turns out to be


r2 r1ml2 r m34 (3.
rl r4ml2 rlm34

with the assumption that*


I L r4ml2 + rlm34 (3.
r1ml2 + r434

With the approximation of equation (3.78) the following

set of approximate relations may be obtained:**


H (m12 + m34)(1 + ml2m34) (3.
rlc5


2 + z2


- uc 2(1 r r2 + l
1 + m12m34


76)


78)


79)


80)


(3.81)


*This assumption is justified by the range of values of the
r's and m's encountered in a great number of filters, as shown by
the numerical data in Table 3. For most of the values computed,
equation (3.79) turns out to be near 0.99.

**In addition to the approximation (3.78), there were also
other approximations used which are implied in (3.78) and (3.79),
e.g., r1I r4)< interest im12 m341<< 1.


(3.75)







- 56 -


z12z22 1 r1( C4(1 rlr3)
rl + r2ml2m34

From (3.68) and (3.69) we have


m12 (1 0 c2zl2)


m34 (1 Ljc2z22)

Using equations (3.83) in (3.80), we may get

(m12 + m34)(1 + m12m34)
rl "wtHum 5b
c

with which r2 may be computed from (3.81)


(3.82)


(3.83)


(3.84)


(3.85)


(3.86)


S2 + z 2
~L22...


Equation (3.82) may be solved for r3, using rl and r2 from (3.85)

and (3.86).


1
r3" 1 -
3


z12z22 1 + r2ml2m34)
r1 Wc4
1 c


(3.87)


Using (3.85) to (3.87) in (3.74), we get


r1r3
r r2


(3.88)


The cut-off frequency W in equations (3.83) to (3.87) may be

computed from the approximation


0.982 + .0 .log1 L
0.982 + 0.01 logo Lp
10 p


(3.89)*


*This equation was derived from the data in Table 3.


r 1 m12m34
2 ri 1 + nl 2m34







- 57 -


If more accurate results are desired, the r's may be

re-evaluated by the following relations, using previously computed

values on the right-hand side of the equations.


(z22 z12) (r + r2m12m34)
r = (3.90)
H Wc3(r1ml2 r4m34)




1 (z12 2+ 22 -1)(1 + r2m12m34 )

r3 2 rl (3.91)

r (l + rlml2m34)
r2



r r3ml2m34 (3.92)
4 (1 -rr) r c 4
2Z 2
z1 z2


With rI, r3 and r4 from above and r2 from (3.74), wcj can be

recomputed from

we 2 R(z2 z12) (3.93)
mlm2

where R is the right-hand side of equation (3.75).*

A tabulation of rI to r4, m12, m34 and jc appears in

Table 3 (see footnote on page 54). A plot of the r's versus L
P

is shown in Fig. 3.13.


*The difference between the recomputed values and the first
values of the r's and w c is an indication of the accuracy of the
results. For greater accuracy, the image parameters may be computed
by an iterative procedure using equations (3.68) to (3.75).








- 58 -


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- 59 -


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- 60 -


0.9





0.8





0.7


0.6





0.5





0.4


0.3


0.05


Fig. 3.13.


0.1 0.2 0.5 1.0 2.0 3.0

Maximum ripple in passband L in db


Image parameters rl, r2, r3 and r4 for k = 0.7
versus maximum ripple in passband for the
design of elliptic function filters














CHAPTER IV


ANALYSIS OF RESULTS AND CONCLUSIONS



4.1. Analysis of Obtained Data

It can be seen from the data in Table 3 that the r's show

very little dependence on the selectivity factor k for quite a

large range of values of k (the maximum change in the r's for a

constant value of ripple in passband is about 5 per cent for

values of k in an interval of about 0.4 to 0.8). If the

Tchebycheff filter is considered as a limiting case of an elliptic

function filter in which k-eO,* i.e., all the poles of which were

moved to infinity without appreciably changing the location of

the zeros, we may use the data in Table 2 for getting the r's

for the lower limit of k 0. It is seen that these r-values

differ little from the corresponding r's (i.e., r's for the same

ripple in passband) of the elliptic-function filter.

The r's do, however, show strong dependence on the amount

of ripple in passband L .
P
Defining mismatch constants as


*This is true only in an approximate sense, when the curve
on which the zeros of the elliptic function insertion power-ratio
are located (in complex frequency phase) approximates an ellipse.
The distribution of the zeros on an ellipse-like curve is found in
most practical cases of elliptic function type ladder filters
without mutual coupling.


- 61 -







- 62 -


u t" "M (4.1)*
t rl rl


uI r2 (4.2)


u2 "r (4.3)
r3


it is seen that as the ripple in passband L increases, u in-

creases, ul decreases, and u2 shows very little dependence on L .

The effect of increasing ut and decreasing u1 may be

visualized in complex frequency plane as moving the zeros of the

insertion voltage ratio toward the j- o* axis, i.e., the effect is

similar to that of predistortion. This suggests that in networks

with incidental dissipation, a compensation for losses similar to

predistortion may be achieved by an increase of ut and a decrease

of ul, in amounts depending on the amount of losses (i.e., Q or d

as defined in 2.4) in the components. A larger change in ut and uI

will be required for components with higher losses.

As seen from equation (3.89), the cut-off frequency wt

varies predominantly with the square root of k, i.e., with the

effective cut-off frequency Wj (see Fig. 3.10).

The m's are functions of Wc and the location of the poles



*r is the terminating resistance, which is equal to 1 ohm
in our analysis.







- 63 -


of the insertion loss function E(&), equation (3.83) and (3.84).*

Since the location of the poles depends on the selectivity parameter

k only, as seen from equation (3.61) and (3.63), and since ?j
c
depends on k predominantly, the m's depend mostly on k and very

little on the amount of ripple in passband L .


4.2. Suggested Modification of Zobel Designs with the View of
Obtaining More Efficient** Filters

The method of designing a Zobel filter to satisfy given

insertion loss requirements has been appropriately called the cut-

and-try method. One reason for this is the fact that the performance

of a filter (i.e., its attenuation and phase-shift when inserted

between generator and load) is only approximately given by its image

attenuation and phase. This approximation is fairly good in the

attenuation band, but is very poor in the pass-band.

The labor required to design a Zobel filter that approx-

imately meets given requirements is usually very small. A commonly

employed method of obtaining a filter which satisfies given require-

ments may be described in the following general steps:

1. Design a Zobel filter which approximately satisfies

insertion loss requirements.


*In these equations the zeros z, and z2, rather than the
poles, appear. However, the poles are located inversely with
respect to the zeros in the elliptic function filter, as seen from
equation (3.60).

**The term "efficient" is used here in the sense of
high contribution to filtering action for each component.







- 64 -


2. Build a model filter according to design of Step 1.

3. Test the model of Step 2, establishing the difference

between its performance and the requirements.

4. Dictated by the results of Step 3, make corrective

changes in the design of Step 1, aiming at eliminating the

difference between the filter performance and the requirements.

5. Repeat Steps 2, 3 and 4 until the aim of Step 4 is

reached.

It should be pointed out here that the cost (in terms of

effort or dollars) of arriving at a satisfactory design using the

above method is usually smaller than the cost of a design carried

out by modern network-design techniques (i.e., by insertion parameter

synthesis, leading to a satisfactory design in a more direct way).*

This may partially account for the fact that Zobel designs are still

so widely employed, although the modern techniques (in existence

now for about twenty years) are known to produce better, often more

economical, filters.

It is in general easier to correct the performance of a

filter (Step 4) in attenuation band than in passband. In fact,

unwanted distortion in passband may sometimes be eliminated only by

adding components to the filter (e.g., m-derived terminal half-

sections), or by using an additional network (an attenuation equalizer).

This is due to the pattern of distortion in passband inherent in


*This is particularly so if the designer has reasonable
ability to predict the effects of changes in design (i.e., changes
of Step 4).







- 65 -


Zobel filters, as shown in Fig. 4.1.

The insertion loss in passband of Zobel filters with equal

image impedances Z, at both terminations, inserted between equal

resistances Rt, is given by24


L 10 log10 + Rt) sin2 BJ
10 4 Rt ZI

where B is the total phase shift through the filter. The envelope

of the insertion loss in passband is


L -'20 log R)
env 10 2 Ri Z ]


Since the image impedance ZI is continuous in passband and

becomes zero (or infinite) at the cut-off frequency Wc in a manner

indicated by Figs. 2.2 and 2.3, termination of the filter by a

constant resistance Rt results in an insertion loss envelope in pass-

band which increases near the cut-off frequency )c as indicated

in Fig. 4.1.

Most filter requirements allow a certain amount of maximum

distortion in passband. Fig. 4.2 shows a typical insertion loss

requirement; any insertion loss curve laying in the shaded area would

satisfy the requirements. The curve shown may be considered typical

for an efficient* one-section filter.


*In general, the number of components needed in a filter in-
creases with increased selectivity k wUp/Wa, with increased min-
imum loss in attenuation band La and Lh, and with decreased maximum
distortion in passband Lp. The curve shown utilizes the limits of
the requirements in a fashion which could lead to a filter with
fewest components.







* 66 -


Lenvelope


- Maximum distortion
in passband


C1. (2 c


Fig. 4.1.


Typical insertion loss characteristic in
passband for a Zobel filter


_____ -4---


p p Ca


Fig. 4.2,


-IN W


Typical insertion loss requirement (straight
lines) and a possible insertion loss characteristic
which satisfies it


Jh








- 67 -


In case of the Tchebycheff and elliptic function filters,

utilization of the maximum allowable distortion in passband, i.e.,

designing a filter that has this maximum distortion in passband,

leads to meeting the requirements with a filter having a minimum

number of elements. Since the Tchebycheff and elliptic function

filters can be designed to produce a predetermined amount of maximum

distortion in passband (in contrast to the Zobel filter, in design

stage of which there is no provision for obtaining a certain pre-

determined amount of distortion in passband), these filters can be

designed to satisfy given requirements with a minimum number of

components.*

In case of the Zobel filter, the amount of distortion in

passband can be controlled by mismatch at the filter terminals,

i.e., by choice of the value for Rt, within a relatively narrow

range only. In general, a Zobel design is termed successful if the

filter satisfies the requirements in attenuation band, and turns

out to have passband distortion which is below (or equal) the speci-

fied maximum allowable distortion.


*Tuttle25 showed in a comparison of the performance of
Butterworth, inverted Tchebycheff, elliptic-function, constant-k
and m-derived Zobel filters that the Zobel filter was inferior to
the elliptic function type only. However, a comparison of a
constant-k Zobel filter with a Tchebycheff filter should prove
that the latter has superior performance for the same number of
elements.







- 68 -


If the requirement calls for a very small distortion in

passband (in the 0.01 db. range), the usual way to obtain it with

a Zobel design is by the use of m-derived terminating half-sections,

the image impedance of which (see Fig. 2.3) has less variation

over an appreciable portion of the passband than the prototype

(constant-k) image impedance, Fig. 2.2. This then results in

smaller passband distortion due to better matching at the termina-

tions, the price for which is an increased number of elements (above

the number necessary to produce a given selectivity and discrimina-

tion) required in the m-derived terminations.*

If the allowable distortion in passband is high (in the

I db. range), the Zobel design can take little advantage of it

since at best the maximum distortion will be reached at one point

in the passband, as indicated in Fig. 4.1.**

From this it may be concluded that in order to make a

Zobel filter more efficient (comparable to a Tchebycheff or

elliptic function filter), some means of

1. Controlling the amount of distortion in passband

2. Achieving equal (or approximately equal) maximum ripple

in passband should be included in the design procedure.


*In some cases additional attenuation equalizing networks
are used instead of, or for severe requirements, with the m-derived
terminations.

**An efficient (in terms of the number of components)
design results when the maximum allowable distortion in passband
is reached a maximum possible number of times3 as in the case of
the Tchebycheff and elliptic function filters.







- 69 -


It was shown in Article 4.1 that for the elliptic function

filter the values of the r's are determined mainly by the ripple

in passband, whereas all the other parameters ( w and the m's)
C
depend mainly on the selectivity k. This means that the main

effect of a change in the r's will be a change in passband distor-

tion. There is a certain relation between the changes in the

individual r's which results in an equal ripple of a certain mag-

nitude. These changes depend only little on the position of the

poles (i.e., Wc and m) in the attenuation band.

From the above reasoning the following may be concluded:

Given a Zobel filter designed with any image parameters

( r and m's), the amount of distortion in passband may be

controlled by changes in the impedance levels (i.e., in the

r's) of its half-sections. In particular, for a Zobel filter

consisting of m-derived sections, an approximately equal ripple

in passband of magnitude L will be obtained if the values of

the r's are made equal to the corresponding r's of the elliptic-

function type filter having the same amount of ripple in pass-

band and approximately the same selectivity k; for a Zobel

filter consisting of constant-k sections, an approximation to

equal ripples of magnitude L will be achieved by letting the
p
r's of the Zobel filter assume the values of corresponding r's

of the Tchebycheff filter with a passband distortion of L .
P
Since, as it was pointed out in Article 4.1 (p. 61), the






- 70 -


r-values of a Tchebycheff filter having a certain amount of pass-

band ripple L differ only slightly from corresponding r-values of

an elliptic-function type filter having the same passband ripple L ,

it appears that:

Making the impedance levels (the r's) of the half-sections

of any Zobel filter (i.e., consisting of m-derived or constant-k

sections and having any selectivity) equal to the corresponding

r's of a Tchebycheff filter with passband ripples of magnitude

L will result in the insertion loss of the thusly modified
P
Zobel filter to have approximately equal passband ripples of

the same magnitude L .
P

4.3. Limits of Improvement of a Zobel Filter

There are many ways by which one may carry out the details

of a Zobel filter design as outlined in Step 1, Article 4.2. These

will in general depend to some extent on the type of the insertion

loss requirements. Since this subject is well covered in litera-

ture,15' 16, 26 it will not be repeated here.

For the purpose of this article it will be assumed that

Step 1 has been carried out for two types of insertion loss require-

ments shown by the straight lines in Fig. 4.3 (a) and (b),* and

that the resulting Zobel filters have an insertion loss as shown

by the solid curves in Fig. 4.3.

If now the half-sections of these filters are mismatched as


*These two types of requirements are among the most commonly
encountered in practice.







- 71 -


WJp Ja


CJOp (b


cb


h


Two types of insertion loss requirements and
possible filter characteristics


Lb



L -1


Fig. 4.3.


-- W








- 72 -


outlined in Article 4.2, an insertion loss indicated by the dotted

curves will be obtained. The question arises, how much of an im-

provement in performance of these filters may be expected from this

mismatch?

In the case of the filter in Fig. 4.3 (a), the answer comes

from the fact that the new insertion loss (dashed curve) can be at

best as good as the insertion loss of an elliptic function type

filter with passband ripple L and selectivity k j / w The
P P a
minimum insertion loss in attenuation band Lae of such a filter is

given by equation (3.65) or (3.67), or can be found from the curve

in Fig. 3.11. Hence, at best, the improved performance of a modi-

fied Zobel filter will be an increase of the minimum insertion loss

in stop-band by*

6L = La L (4.4)
ae a

In the case of the filter in Fig. 4.3 (b), it is convenient
26
to use a transformation which will express the performance of the

filter in terms of a performance as in Fig. 4.3 (a). It turns out

that the minimum attenuation Lbe in the frequency range wob to W h

is approximately the same as the minimum attenuation Lbe for


*If the Zobel filter (before modification) had equal minima
of insertion loss in passband, as indicated by the solid curves in
Fig. 4.3, the minima will not, in general, be equal after the modi-
fication, i.e., after mismatching the half-sections. For most cases
occurring in practice, however, the difference in the minima will
be small and can easily be corrected by a small change in the
m-values, the effect of which on the passband ripple is negligible.







- 73 -


a filter as Fig. 4.3 (a) with a selectivity of




k 6 Wh (4.5)*

W 1 h

Lbe may be determined in the same manner as Lae for the previous

case, giving the maximum possible improvement as


ALb = Lbe Lb (4.6)


4.4. Summary and Conclusions

A method of synthesis of ladder networks was investigated

with which conventional ladder networks may be obtained by proper

choice of parameters of elementary building blocks.


*The frequency scale in Fig. 4.3 (b) is transformed to an
m-scale by the relation m (I WJ 2/w 2)1/2, hence m z 0 cor-
responds to Oj m 1 corresponds to W oo, mb and mh correspond
to Lob and WJh, respectively. Multiplication of the m-scale by a
factor I/mh introduces a new scale, say m' scale, in which m' 0
corresponds to wop, m' I1 corresponds to Wuh, and mb' = mb/mh
corresponds to c An inverse transformation of m, i.e.,
CJ' = c p/(I m, )1/2, results in a frequency scale 6w' which is
infinite at Wh, giving an insertion loss requirement with equal
minimum from cb' (corresponding to dob) to Lw' oo. The selec-
tivity of the transformed requirement is k wJ '/Lb ', giving
(4.5) upon substitution of w p uj and W b p /(l b)1/2







- 74 -


These parameters were determined for three types of two-

section filters having desirable insertion loss characteristics.

As a result of analysis of the obtained parameter values,

a method for improving Zobel filters was suggested.

The design of a two-section Butterworth filter can be

quite simply accomplished with image parameter half-sections having

a constant amount of impedance mismatch given by equation (3.52) and

a cut-off frequency Wc given by equation (3.54) or Fig. 3.6. The

impedance levels of the half-sections, rI and r are constant,

i.e., independent of any design parameter; the cut-off frequency

cc is a function of the maximum distortion in passband only, as

given by equation (3.54).*

The design of two-section Tchebycheff filters may be as

easily performed using equation (3.58) and (3.59), or the graphs

in Fig. 3.8.**


*The ctt-off frequency W- is not a critical design parameter
for constant -k type low-pass ana high-pass filter types, i.e., it
has no effect on the shape of the response curve. A change in the
cut-off frequency amounts to a linear transformation of the frequency
scale of the insertion loss versus frequency curve.

**Butterworth and Tchebycheff filters with lossless elements
can be designed with not much effort using already developed formulae
for element values (i.e., without going through the actual synthesis
from the insertion loss function). The engineering value of our
approach is in the simplifications resulting from its application to
lossynetworks, for which no explicit formulae for circuit element
values are available at the present.








- 75 -


Computation of the element values by slide rule gives suffi-

cient accuracy for commonly encountered tolerances in manufacturing

of the components.*

A two-section filter of the elliptic-function type can be

designed using either the equations (3.83) to (3.89),** or the

graphs in Fig. 3.13 with equations (3.83), (3.84), (3.89) and

(3.74); use of the graphs is simpler but more restricted in the

range of application.


*A manufacturing tolerance in component values of + 1.0 per
cent to + 2 per cent is quite common for capacitors in the range of
approximately 100 A/f to 0.2 /ff and for inductors in the range
of approximately 1 mh to 50 hy.

**The values of the zeros, equation (3.61), can be found from
many elliptic function tables, e.g., Ref. 21; they are tabulated
for use in elliptic-function filter design in Ref. 23.














REFERENCES


1. 0. J. Zobel, "Theory and Design of Uniform and Composite Electric
Wave Filters," B.S.T.J., 2 (1923) pp. 1-46, and "Transmission
Characteristics of Electric Wave Filters," B.S.T.J., 3 (1924),
pp. 567-620.

2. M. I. Pupin,"Wave Propagation Over Non-Uniform Cables and Long
Distance Air Lines," Trans. A.I.E.E., 17,(1900), pp. 445-507.

G. A. Campbell, "On Loaded Lines in Telephonic Transmission,"
Phil. Mag., 5,(1903), pp. 313-330, and "Physical Theory of
the Electric Wave Filter," B.S.T.J., 1 (1922), pp. 1-32.

G. A. Campbell and R. M. Foster, "Maximum Output Networks for
Telephone Substation and Repeater Circuits," Trans. A.I.E.E.
vol. 39 (1920), pp. 231-280.

3. S. Darlington, "Synthesis of Reactance 4 Poles," J. Math.
Phys., vol. 18 (1939), pp. 257-353.

4. W. Cauer, Theorie der linearen Wechselstromschaltungen, Akademische
Verlag, Berlin, Germany (1954), 2nd ed.

H. Piloty, "Weichenfilter," Zeitschrift fUr Telegraphen und
Fernsprechtechnik, vol. 28 (1939), pp. 291-298, 333-344, and
"Wellenfilter, Insbesondere symmetrische und antimetrische mit
vorgeschriebenem Betriebsverhalten," Zeitschrift fUr Tele-
graphen und Fernsprechtechnik, vol. 28, No. 10 (1939)
pp. 363-375.

5. V. Belevitch, "Recent Developments in Filter Theory," IRE
Transactions on Circuit Theory, vol. CT-5 (Dec. 1958)
pp. 236-252.

6. T. Laurent, "Allgemeine physicalische ZusammenhMnge bei
Filterketten," Arch. d. El. U. B. 12, H.1 (Jan. 1958) pp. 1-8,
and "Echostatanpassung, eiee neue Methode zur Anpassung von
Spiegelparameterfiltern," Arch. d. El. U. B. 13, H. 3
(March 1959) pp. 132-140.

7. R. 0. Rowlands, "Composite Ladder Filters," Wireless Engineer,
vol. 29 (Feb. 1952) pp. 51.

8. J. E. Colin, "Two-Branch Filter Structures with Three Cut-Off
Frequencies," Cables & Transmission, vol. 11 (July 1957)
pp. 179-217.
76 -







- 77 -


9. W. Saraga, "Insertion Parameter Filters," TMC Technical J.,
vol. 2 (March 1951), pp. 25-36.

10. A. J. Grossman, "Synthesis of Tshebycheff Parameter Symmetrical
Filters," Proc. IRE, vol. 45 (April 1957) pp. 454-473.

10. E. Green, Amplitude-Frequency Characteristics of Ladder Networks,
Marconi's Wireless Telegraph Co., Essex, England; 1954.

11. J. K. Skwirzynski and J. Zdunek, "Design Data for Symmetrical
Darlington Filters," Proc. IRE, vol. 104, pt. c (Sept. 1957)
pp. 366-380.

12. S. D. Bedrosin, E. L. Luke, and H. N. Putchi, "On the Tabulation
of Insertion Loss Low-Pass Chain Matrix Coefficient and
Network Element Values," Proc. Natl. Electronics Conf., vol. 11,
(1955) pp. 697-717.

13. T. Laurent, Vierpoltherie und Frequenztransformation, Springer-
Verlag, Berlin, 1956.

14. H. W. Bode, Network Analysis and Feedback Amplifier Design,
D. van Nostrand Company, Inc., New York, 1945.

15. T. E. Shea, Transmission Networks and Wave Filters, D. van
Nostrand Company, Inc., New York, 1929.

16. F. Scowen, Introduction to Theory and Design of Electric Wave
Filters, Chapman & Hall, Ltd., London 1950.

17. W.H. Chen, Elements of Electrical Analysis and Synthesis,
McGraw Hill Co. In press.

18. A. C. Bartlett, The Theory of Electrical Artificial Lines and
Filters, John Wiley & Sons, New York, 1931.

19. V. Belevitch, "Tchebycheff Filters and Amplifier Networks,"
Wireless Engineer, vol. 29 (April 1952) p. 106.

20. L. M. Milne Thomson, Die elliptischen Funktionen von Jacobi,
Springer Verlag, Berlin, 1931.

21. E. Jahnke and F. Emde, Tables of Functions, Dover Publications,
New York, 1945.

22. W. N. Tuttle, "Design of Two-Section Symmetrical Zobel Filters
for Tchebycheff Insertion Loss," Proc. IRE, vol. 47 (Jan.
1959), pp. 29-36.






78 -




23. E. Glowatzki, "Sechsstellige Tafel der Cauer-Parameter,"
Abhandlungen der Bayerischen Akademie fer Wissenschaften,
Neue Folge, Heft 67 (1955).

24. W. Saraga, "Insertion Loss and Insertion Phase-Shift of Multi-
section Zobel Filters with Equal Image Impedances,"
P.O. Elec. Eng. J., vol. 39 (Jan. 1947), pp. 167-172.

25. W. N. Tuttle, "Applied Circuit Theory," IRE Trans. on Circuit
Theory, vol. CT 4, (June 1957) pp. 29-32.

26. J. H. Mole, Filter Design Data, John Wiley and Sons, Inc.,
New York, 1952.







- 79 -


BIOGRAPHY


David Silber was born on October 2, 1922, in Lodz, Poland.

His undergraduate studies were pursued at the O.V.M. Polytechnikum

in Munich, Germany, from which he received the degree of Electrical

Engineer in June, 1950.

After arrival in the United States in 1951, he pursued

graduate studies at the University of Cincinnati, Evening College

and Summer School, while employed by the Keleket X-Ray Corporation

in Covington, Kentucky. Since 1954 he has been employed by the

Communication Accessories Company in Kansas City, Missouri, taking

leave in 1956 to enter the University of Florida for further graduate

studies.

At the University of Florida he held a graduate fellowship

for two years and taught electrical engineering for one year. He

received the degree of Master of Science in Engineering in

August, 1957.










This dissertation was prepared under the direction of the

chairman of the candidate's supervisory committee and has been

approved by all members of that committee. It was submitted to the

Dean of the College of Engineering and to the Graduate Council, and

was approved as partial fulfillment of the requirements for the

degree of Doctor of Philosophy.


August 13, 1960


Dean, ollegeof Enginerin



Dean, Graduate School


SUPERVISORY COMMITTEE:


Chairman


~G'.









(C -KV E&^^
































UNIVERSITY OF FLORIDA
1 II II 11111 Il III III
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PAGE 1

A SIMPLIFIED METHOD OF SYNTHESIZING LADDER NETWORKS WITH IMAGE-PARAMETER HALF-SECTIONS By DAVID SILB E R A DISSERTATION PRES ENTE D TO THE GRADUATE COUNOL OF THE UNIVERS I TY OF FLORIDA I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEG R EE OF DOCTOR O F PHILOSOPHY UNIVERSITY OF FLORIDA A u gus t 196 0

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Copyright by David Silber 1960

PAGE 3

ACKNOWLEDGMENTS The author would like to express his gratitude to the members of his supervisory committee for their advice and helpful criticism. He is especially indebted to Dr. T. s. George for his supervision and constant guidance, to Dr. w H, Chen for his many valuable suggestions and to Dr. H. A. Meyer, Director of the Statistical Laboratory, for his help in obtaining numerical data by using the University's electronic computer. ii

PAGE 4

TABLE OF CONTENTS Page ACKNOWLEDGMENTS LIST OF TABLES ii V LIST OF ILLUSTRATIONS CHAPTER vi I. INTRODUCTION oooooo 1 1.1. The Use of Frequency Selective Networks and Terminolog y . . l 1.2. The Image Paramete r Method of Network Synthesis . . . . . . . 3 1. 3. The Insertion Loss Method of Network Synthesis . . . . . 5 1.4. Comparison of the Image Parameter and Insertion Loss Methods 6 1.5. Statement of Objectives 8 II. THE GENERAL LADDER NETWORK IN TERMS OF ELEMENTARY III. BUILDING BLOCKS CONNECTED IN TANDEM. 10 2 .1. The Elementary Building Block 10 2.2. Methods of Conn ecting the Elementary Building Blocks . . 12 2.3. A Conventional Low-Pass Ladder in Terms of Elementary Building Blocks 15 2.4. Filters with Incidental Dissipation in Terms of Elementary Building Blocks. 23 DERIVATION OF THE IMAGE PARAMETERS FOR LADDER FILTERS HAVING PRESCRIBED INSERTION LOSS CHARACTERISTICS iii 26

PAGE 5

3 .1. Outline of Approach 3.2. Insertion Loss of a Two-section Ladder Filter in Terms of Image Parameters Page 26 28 3 .3. Tchebycheff and Butterworth Filters Synthesized with Image Parameter Half-sections 3 3 3.4. The Elliptic Function Filter Synthesized with Image Parameter Half-sections 44 IV ANALYSIS OF RESULTS AND CONCLUSIONS 4.1. Analysis of Obtained Data 4.2. Su ggested Modification of Zobel Designs with the View o f Obtaining More Efficient Filters 61 61 63 4.3. Limits of Improvement of a Zobel Filter 70 4.4. REFERENCES BIOGRAPHY Summary and Conclusions . iv 7 3 76 79

PAGE 6

I I I ( I 1 I TABLE 1. LIST OF TABLES CUT-OFF FREQUENCY We FOR THE BUTTERWORTH TWO-SECTION FILTER ....... 2. IMAGE PARAMETER VALUES FOR THE TCHEBYCHEFF TWOPage 40 SECTION FILTER . . . 45 3 IMAGE PARAMETER VALUES FOR THE ELLIPTIC-FUNCTION TYPE TWO-SECTION FILTER ... .... 58 V

PAGE 7

Figure 2.1. LIST O F ILLUSTRATIONS The elementary building block 0 2.2. Image impedance of the elementary building block Page ll at terminals 1 0 of Fig. 2.1 . 11 2 .3. Image impedance of the elementary building block at terminals 2-0 of Fig. 2 .l. . . 11 2.4. Tandem connection of the elementary building blocks giving a non-symmetrical ladder filter ... 13 2.5. Connection of the elementary building blocks by which symmetrical and non-symmetrical ladder filters may be obtained . . . 13 2.6. A special case of the ladder of Fig. 2 5 2.7. Possible locations of elements with arbitrary values. 2.8. Effect of adding proportionately equal amounts of resistance on the location of poles and zeros o f 16 ,20 the insertion voltage-ration function 24 3.1. A two -section ladder filter 29 3.2. Elementary building blocks from which the ladder of Fig. 3.1 may be obtained. . . 29 3.3. Equivalent lattice of the ladder of Fig. 3Gl 3 1 3 .4. A Butterworth and Tchebycheff type ladder filter and its equivalent lattice 31 3.5. Insertion loss characteristics for (a) Butterworth and (b) Tchebycheff two -section filters 3 6 3.6. Cut-off frequency of elementary building blocks versus maximum attenuation in pass band for the design of filters with Butterworth characteristics 41 3 .7. A Butterworth filter in terms of elementary building blocks with wc = 1.272-and LP= 3 db . 42 -vi -

PAGE 8

Figure 3.8. Image parameters we, r 1 and r 2 peak ripple in passband for the with Tchebycheff characteristic versus maximum design of filters Page 47 3.9. A Tchebycheff filter with peak ripple in passband 1;> = 1.5 db in terms of elementary building blocks With WC = 1. 0 o , o , , , 48 3 .10, Insertion loss characteristic of an elliptic function two-section filter . . 51 3.11. Minimum stq>band attenuation L versus selectivity factor k for the two-section efliptic function filter o o o 53 3.12. Variation of mini11U1m attenuation in stop-band with maxiimllll ripple in passband 53 3.13. Image parameters r 1 r 2 r 3 and r 4 fork~ 0.7 versus maximum ripple in passb~nd for the design of elliptic function filters ..... 60 4.1. Typical insertion loss characteristic in passband for a Zobel filter 66 4.2. Typical insertion loss requirement and a possible insertion loss characteristic which satisfies it. 66 4.3. Two types of insertion loss requirements and possible filter characteristics . . 71 vii

PAGE 9

CHAPTER I INTRODUCTION 1.1 The Use of Frequency Selective Networks and Terminology While transmitting information in the form of electrical signals from one point to another, we are faced with the problem of recovering a desired signal from undesired ones and from electrical noise. When the energies of different signals are in different frequency regions, or bands (a situation which is often found in nature, or can be attained by various techniques in the process of converting the information into an electrical signal at the transmitting points), the separation of one signal from others makes use of the differences in energy-frequency spectra of different signals. This separation is accomplished by placing some device in the path of the signal (before it reaches the receiving point) which exhibits frequency selective characteristics, i.e., a device which is capable of accentuating (or letting pass through without change) one frequency band, the band containing all or most of the energy of the desired signal, while suppressing (or attenuating) another band The device usually employed for this purpose consists of an array of electrical components (resistance, capacitance and -1 -

PAGE 10

-2 -inductance) interconnected in a fashion which produces the desired frequency selectivity. There are some mechanical devices with this characteristic which are sometimes used. A word on terminology: An array of interconnected electrical components is called an "electrical network," or in short a "network." The manner of interconnection of components, the choice of their type and number required in order to produce a specified electrical characteristic (e.g., frequency selectivity), as well as the mathematical tools employed in network problems, are treated in network theory. If a network consists of electrical components, the values of which are independent of frequency, it is a "linear network"; if the circuit element values are positive constants, it is a "linear passive network. 11 A network inserted in a communication path, which usually consists of two wires, must have two input terminals and two output terminals, hence falls in the category of "four-terminal networks," or, more precisely, "two terminal-pair networks.11 A four-terminal network which has frequency discriminating properties is called an "electric wave filter," or in short, a "filter." Depending on the region of frequencies a filter favors or attenuates, there are "low-pass filters (i.e., these filters pass low frequencies while attenuating high frequencies), "high-pass filters," "band-pass filters," "band-elimination filters" and ''multiple pass-band filters" where a filter has several passbands separated by attenuation bands.

PAGE 11

-3 -The external characteristics of a linear passive fourterminal network may be completely described by a set of three independent parameters, which are in general functions of frequency. Of the possible sets o f parameters, there are some which can be readily obtained by measurement at the terminals without knowledge of the internal structure of the network; these are the open-and short-circuit impedances. A very convenient set of parameters, the "image parameters," may be obtained as functions of the openand short-circuit impedances. The image parameters consist of two image impedances and an image transfer coefficient. The external electrical characteristics of a filter inserted in a communication path between transmitting point (generator) and receiving point (load) are the operating characteristics called "insertion characteristics"; these consist of "insertion loss" and "insertion phase-shift11 characteristics. The first is a measure of the decrease in amplitude of a sinusoidal voltage passing through the filter, the second gives the amount of phase shift (or delay) for a sinusoidal voltag e passing from input to output (i.e., from generator to load) terminals of the filter. Both are, in general, functions of frequency. l.2 The Image Parameter Method of Network Synthesis The problem of designing a filter is to synthesize a net work having a prescribed insertion loss characteristic. There are two distinctly different methods of filter

PAGE 12

-4 -design available. The older one, developed by Zobel1* from earlier work of Pupin, Campbell and Foster2 is based on image parameter theory; the resulting image parameter filter is called a Zobel filter. The newer method, originated by Darlington,3 is based on insertion loss theory developed independently by Darlington, Cauer and Piloty4 at the same time. The outstanding feature of the image-parameter design method is its simplicity. This, coupled with the fairly well performing filters obtainable by the image parameter design techniques, caused this method to be used almost exclusively in practical filter design. The simplicity of this method stems from the "building block" structure of composite filters. Each "building block," or "section," is a four-terminal network which can be fully described by two image impedances z11 and z12 and an image transfer coefficient 8 1 ot.. 1 + Jfix, where 'I is the image attenuation coefficient and ;31 is the image phase shift coefficient. A composite filter is formed by connecting sections in tandem, with image impedances that are.equal to each other (matched) at the terminals of the sections which are joined. The total transfer coefficient of the composite filter is simply the sum of the transfer coefficients of individual sections. If these filters could be terminated at both ends by the respective image impedances (i.e., matched at the endsh the image *Raised numbers refer to entries in the References at the end of this dissertation.

PAGE 13

5 transfer coefficient would describe their actual (measurable) performance, However, since the image impedances are generally functions of frequency, and since the terminating impedances are usually constant resistances, the filters cannot, in general, be matched at the termination. This mismatch causes the quantities of interest, the insertion loss and insertion phase, to be somewhat different from the image attenuation and image phase shift. Moreover, the requirement of matched image impedances at the points of interconnection of individual sections imposes restrictions on the networks obtainable by the image parameter method, as well as on their performance characteristics (i.e., their insertion loss and insertion phase characteristic). These restrictions lead to an inefficient use of network components. 1.3 The Insertion Loss Method of Network Synthesis The insertion loss method of filter design is characterized by the fact that filters obtainable by it have an insertion loss characteristic exactly as prescribed, when inserted between prescribed terminations (hence the name "insertion loss theory"). It is based on the treatment of the entire problem of finding a four-terminal network satisfying given insertion loss requirements. The insertion parameters, rather than the image parameters of Zobel filters, are the primary design parameters from which the circuit element values of the filter are obtained in the last step of a sequence of (often lengthy) operations. The resulting filter, no matter how complicated, is obtained as an entity, rather than an aggregate of

PAGE 14

-6 -individual building blocks (sections) as in Zobel filters, hence it does not suffer from the matching restrictions imposed on Zobel sections and permits the realization of optimum (in the sense of the number of circuit elements required to meet prescribed characteristics, and the element values) networks. Another very desirable feature of the insertion loss method is the correction for incidental dissipation it permits. Though the range of compensation is limited by the type of insertion loss function (more specifically, by the distance of the nearest insertion-loss zero to the jwaxis in complex plane), it permits taking into account losses in components for a great number of cases encountered in practice. 1.4 Comparison of the Image Parameter and Insertion Loss Methods A comparison of the image parameter method and the insertion loss method of network synthesis reveals that the latter is much more flexible in the results it produces, and much more sophisticated in the overall approach and in the mathematical tools it employs. As a result of the last, it is taught in most engineering schools only in graduate curricula, and the majority of practical design engineers have no opportunity to acquire a working knowledge of the insertton loss method while in school. An important economical disadvantage of the insertion loss method of network synthesis, as compared with the image parameter method, is the much longer design time required by the insertion

PAGE 15

-7 -loss techniques, resulting in a more costly filter design than in the case of image parameter filters.* On the other hand if the filter designed by the insertion loss method is more economical** than a corresponding Zobel filter (which is often the case), the increased cost of design may be offset when it is to be manufactured in large volume. The situation in the practical filter-design field may be summarized as follows: Because of the simplicity of the design procedures and of the acceptable filters the image parameter method yields, and because of the more analytical background and longer design time required of the designers using the insertion loss method, the image parameter method is still.used by the majority of design engineers, though using the insertion loss method may often result in a better filter. There have been many contributions in the past to both design methods;*** a great n b e r of these aimed at bridging the gap existing between modern theory and practical design methods. The approach from the image-parameter field is, in general, toward the improvement of Zobel filters by better matching, e.g., Ref, 6, by reduction of components in derived terminations, e.g., Ref. 7, or by finding new structures, e.g., Ref. 8. In the insertion loss field the emphasis *The design time may be reduced by employing modern means of computation, e.g., an electronic computer. This, while reducing the required design time, will not, in general, reduce the overall design cost. **An economical filter is one that requires fewer or less expensive components, or its component values may have higher tolerances without upsetting the performance, etc. ***For a sumnary of recent contributions and a good list of references, see Ref. 5.

PAGE 16

-8 -9 is on a more comprehensive presentation of known material, on 10 simplification of design techniques, and on tabulation of 11 12 designs. This dissertation is concerned with a synthesis method for ladder networks which employs "building blocks" similar to the image parameter filter; in fact, the "elementary building block" used is an image parameter half-section. To form a composite filter these "building blocks" are connected in tandem in a fashion similar to a Zobel filter, without, however, the fundamental restriction of Zobel filters, that of matching image impedances at the points of interconnection. 1.5. Statement of Objectives The objectives of this dissertation are two-fold: 1 The introduction and investigation of a new method of synthesis of ladder networks by which filters can be designed with a simplicity similar to the simplicity of the image parameter method of design, yet yielding more general filters than the Zobel filter of the image parameter method (i.e., yielding filters the choice of element values and the performance characteristics of which do not have the limitations of Zobel filters) 2. The derivation of relations and determination of parameter values for use in this synthesis method, by which

PAGE 17

9 design engineers familiar only with the image parameter method will be able to design filters with desirable insertion loss characteristics (Butterworth, Tchebycheff and elliptic-function type) until now obtainable primarily by the insertion loss method. The contents of this dissertation is divided into three chapters. Chapter II deals with methods of interconnection of the elementary building blocks. It is shown that any lossless ladder can be expressed in terms of image impedance half-sections with one common image pars.meter. The cut-off frequency is chosen as the common parameter. In Chapter III some methods of finding the image parameters of the elementary building blocks are investigated. The ima ge parameters are then determined for two-section ladder filters having Butterworth, Tchebycheff, and elliptic-function insertion loss characteristics. Based on results o f Chapter III, a modification of Zobel filters is introduced in Chapter IV. This modification permits control of the distortion in passband of Zobel filters; it also makes the economy of these filters (as far as the performance per number of elements is concerned) comparable to the economy of optimum filters.

PAGE 18

CHAPTER II THE GENERAL LADDER NETWORK IN TERMS OF ELEMENTARY BUILDING BLOCKS CONNECTED IN TANDEM 2.1. The Elementary Building Block Only the lowpass ladder will be considered here, since the high-pass, band-pass and band-elimination networks can be obtained from the low-pass ladder by well known frequency transformations.13, 14 The elementary building block (Fig. 2.1) is an image parameter m-derived low-pass half-section, the circuit element values of which are determined by the image parameters m, r and w C using the simple relations from image parameter theory: m cl = r w C mr hz-= -w e 1 m2 c2::, mr w c Q-;m~l 15 (2 .1) (2. 2) (2. 3) (2. 4) where w is the cutoff frequency of the half-section, r is the C image impedance at zero frequency, w0o is the frequency of infinite attenuation. We also have (Fig. 2.2 and Fig. 2.3) -10 -

PAGE 19

11 h2 l o I 00 0 2 Zi2 zll cl I c2 --0 0 0 0 Fig. 2.1. The elementary building block r I I \ I imaginary (negative) --w Fig 2.2. Image impedance of the elementary building block at terminals 1--o of Fig. 2.1. t l real (positive) I / / / / / / / Imaginary 1 / :\r (negative) / I \ / \ " ' ' Fig 2.3. Image impedance of the elementary building block at terminals 2--o of Fig. 2.1.

PAGE 20

-12 -r (2. 5) (2.6) Analysis of equations (2.1), (2.2), (2.3) and (2.4) reveals that the parameters m, r and c.,., uniquely determine the component C values c 1 c2 and h (with the restriction that mis a real and posi-tive constant, 0 m 1). O n the other hand, given c 1 c2 and h, the design parameters m, r and w c may be found from the following relations: 2 =~ (2. 7) r cl m2 = c1 (2.8) cl + Cz w2= l (2.9) hz (cl + c2) Equations (2.7), (2.8) and (2,9 ) are obtained from equations (2.1), (2.2) and (2.3) solved for r, m and w C Restricting the design parameters to be positive, it is seen from the last set of equations that c 1 c 2 and h 2 uniquely determine the design parameters r, m and 2.2. Methods of Connecting the Elementary Building Blocks Fig. 2 4 shows one method of connecting the half-sections

PAGE 21

h2 l C 0 I cl 2 0 2 hl h4 h 2n l ...........J -~!'tQ2n c ....._ c2 ...1. c c2 0 l o J : I 4 o--o Jc2n-l n o 0 n number of half-sections Fig. 2.4. Tandem connection of the elementary building blocks giving a non-symmetrical ladder filter h6 ha h2n-2 4 3 5 6 8 7 2n-3 rYY't2n-2 2n h 2n 2n-l c4 ....L c ..L C6 C8 J_ _L C2n-2 <>-0 I o 3 o I cs o-o I :~-. 0 Ic2n-3 0 c2n 0 L2n-l n = number of half-sections Fig. 2.5. Connection of the elementary building blocks by which symmetrical and non-symmetrical ladder filters may be obtained .... w

PAGE 22

-14 into a ladder network; here we connect the even-numbered terminals with the odd-numbered ones, We note that the even-numbered terminals have an image impedance given by equation (2.5) (i.e., the image impedance between the even-numbered terminals and O, the common, for one half-section only, is given by (2.5)), and the odd numbered terminals have an image impedance given by equation (2,6), hence we connect the terminal pairs having different types of image impedances, The image impedance at both terminations of the resulting ladder will not be of the same type; the image impedance at terminals 1-0 is of the type given by equation (2.5), the constant-k mid-shunt type, whereas, the image impedance at the other termination (at terminal pair 2n-O) is of the type given by equation (2.6), the m-derived mid-series type. Therefore the resulting ladder cannot be symmetrical. Another method of connection is shown in Fig. 2.5, where we connect even-numbered terminals with even-numbered terminals and odd-numbered terminals with odd-numbered ones, i.e., we connect pairs of terminals having image impedances of the same type. I f the ladder contains an even number of half-sections, both image impedances at the terminations will be of the same type, hence the ladder can be symmetrical. If the number of half-sections is odd, the image impedances at the terminations are similar to those of Fig. 2.4. The method of connection shown in Fig. 2.5 results then in *A 4-terminal network possesses electrical symmetry if the image impedances at both terminal-pairs are equal, i.e., one terminal pair cannot be distinguished from the other pair by external measurements.

PAGE 23

15 -a more general ladder than the connection of Fig. 2.4 (in fact, Fig. 2.4 can be considered as a special case of Fig. 2.5 in which the components of the second, fourth, etc., section are zero, i .e., h4 =ham hl2 m m hzn = 0 and c3 = c7 a = czn-1 = 0) hence this method of connection will be used in subsequent analysis. Since every component of this ladder can assume any value depending on the choice of the image parameters r1 tni, wcl, r2 m 2 wc2 rn, m, w it is clear that the reverse must hold also, i.e., n en any ladder network having the configuration of Fig. 2.5 can be syn-thesized with image parameter half-sections by proper choice of the image parameters r i, mi and w ci. 2.3. A Conventional Low Pass Ladder in Terms of Elementary Building Blocks A p a rtic u l a r case of the ladder of Fig. 2.5, one in which a reduction in the number of components can be realized, would result if two adjacent parallel resonant circuits in the series branches of the ladder could be combined into one parallel resonant circuit. Obviously this can be accomplished only if the resonant frequencies of the adjacent circuits are equal. The resulting ladders are shown in Fig. 2.6 (a) and {b), where the components of Fig. 2.5 combine as follows: ql ::;:: cl (2.10) q2 = h2 + h4 (2 .11) a 2 = c2c4 (2 .12) c2 + C4

PAGE 24

q2 q4 qn+l 1 ----~2 ql a2 Iq3 84 I I : I qn an+l 0 0 0 n = an odd number (a) q2 q4 qn ,_. 1 ---I CJ 0 2 I q10 T a2 a4 Tq3 I ~5-J qn-1 an I 4:+1 0 0 n = an even number (b) Fig. 2.6. A special case of the ladder of Fig. 2.5

PAGE 25

-17 a2q2 = l = hzcz = h4c4 w2o0 2 (2.13) q3 a: c3 + cs (2 .14) q4 ,.. h6 + ha (2 .15) a = c6c8 4 c6 + ca (2 16) l = h6c6 = hacs a4q4 = W4 z oC) (2 .17) In Fig. 2.6 (a) the ladder consists of an odd number of halfsections (the number of half-sections= n odd), hence (see Fig 2.5) q
PAGE 26

-18 is 3n (three parameters, rl.., mi and w for each of then halfci sections). However, due to the conditions of equations (2 13 ) (2.17) (2.22), some of the parameters will be related by these equations, resulting in the number of independent image parameters being equal to the number of components in the ladder. Consider the problem of expressing a given ladder of the form of Fig. 2 6 in terms of image parameter half-sections, i e given a ladder of Fig. 2.6, determine the element values of an equivalent ladder of Fig. 2.5. One question arises immediately: Is there a unique equivalent of the ladder of Fig. 2.6 in the form of one of Fig. 2.5? If not, could there be one specific form of Fig. 2.5 which is advantageous? The first question can be answered by noting that there are 3n components in ladder of Fig. 2.5, the values of which are to be determined from 2n + 1 relations.* Thus there is no unique equivalent of a ladder of Fig. 2.6 in the form of one in Fig. 2.5 The second question can be answered after investigating the possible equivalent ladders of the form of Fig. 2.5. F o r a l adder of Fig. 2.6 (a) we have three equations of the form of equation (2.11), (2.12) and (2.13) for each series branch except the last, i.e. 3(n 1)/2 equations; one equation for each shunt branch, i.e., (n + 1)/2 equations n -1)/2 of the form of (2.14) and one equation (2.10, and two relations (2.18) and (2 19) for the last series branch, giving a total of 3(n -1)/2 + (n + 1)/2 + 2 = 2n + 1 For Fig 2.6 (b) we have 3n/2 equations of the form of (2.11), (2 12) and (2.13) for the series of the form of (2.11), (2 12) and (2.13) for the series branches, and (n + 2)/2 for the shunt branches, giving a total of 3n/2 + (n + 2)/2 = 2n + l.

PAGE 27

19 -In order to obtain a particular ladder equivalent, 3n (2n + 1) = n -1 component values of the equivalent ladder of Fig. 2.5 will have to be either chosen or specified by additional n-1 equations* (i. e we have n-1 degrees of freedom). It is clear that the n-1 components, the values of which are arbitrary, will be located in the ladder in accordance with the location of the components in Fig 2.6 which on division produce the n-1 degrees of freedom (e.g., one cannot choose arbitrarily the values of the first n-1 components, starting with c 1 c2, h 2 c 4 h 4 ... up to the (n-1) -th component in Fig. 2.5). Examination of the ladders in Fig. 2.6 shows that every branch of the ladder which is divided produces one degree of freedom, as shown by the symbol "l" in Fig 2. 7 (a) The arrows indicate where the degrees of freedom originate, hence they also indicate which of the elements may have arbitrary values. Fig. 2.7 (b) and (c) show two possible choices of the location of arbitrary elements: In Fig 2 7 (b) the first half-section has no arbitrary elements, the second has one in the series branch and one in the shunt branch, the third half-section has none, etc. In *The chosen values of the components, though arbitrary in a certain range, should not result in a negative element lest some of the ima g e parameters from which the half-sections can be computed become ima ginary. If instead of a component value a corresponding image parameter is chosen, a similar limitation applies in addition to the requirement O-=m 1 (or w <-w...,).,_ C

PAGE 28

(a) 1 1 rO----cJ-r1 rCJ--r--ri OJ I T T L _____ _l I o (b) 1 1 y0----cJ-r rCJ---CJ--r 1---1 OJ l I I l I L -I o (c) Fig. 2.7. Possible locations of elements with arbitrary values N 0

PAGE 29

21 -Fig. 2.7 (c) every half-section, except the last one, has one arbitrary element. There are a number of combinations of the arrangements in Fig. 2.7 (b) and (c) possible.* The arrangement of arbitrary elements in Fig. 2.7 (c) appears advantageous in that it permits choice of the circuit element value of one component (or one parameter) in each half-section, except the last. Hence the resulting ladder, with the exception of one terminating half-section, can be made to have one arbitrary parameter.** The values of the n-1 arbitrary elements may conveniently be specified by additional n-1 arbitrary equations, in which the limitations on the chosen values (see footnote* on previous page) could be included. A simple set of n-1 of such equations could be (2.24) where p denotes an image parameter.*** In particular if the same *Similar reasoning applied to the ladder of Fig. 2.6 (b) shows that the conclusions reached in a n alysis of Fig. 2.6 (a) hold here as well; we may have one arbitrary element in each half section, except in one of the terminating half-sections. **Since the component values of a half-section can be expressed in terms of image parameters r, m and ~c' equation (2.1), (2.2) and (2.3), the fact that one of the component values can be chosen arbitrarily means that one of the image parameters can be chosen arbitrarily. ***We note that from equation (2.24) we have n-1 relations of the form p 1 = p 2 p 1 = p 3 ... p 1 = Pn, hence the value of p cannot be chosen out is determined from 3n simultaneous equations, 2n + l of which are in the form of equation (2.10) to (2.23) with h's and e's substituted by equation (2.1) to (2 .4).

PAGE 30

22 -image parameter is chosen for all p's, the resulting ladder will be 1 15 16 very similar to an image parameter Zobel fLlter' when p denotes either r or we.* A convenient choice for p, equation (2.24), is to let it denote the cut-off frequency lJ i.e., letting the cut-off frequen-c cies of all n half-sections be equal. This would make the image parameters m equal for the half-sections with common parallel resonant frequencies w ( see equation (2. 4)) The above leads t o the following conclusion: Any conventional ladder filter, Fig. 2.6, can be synthesized by tandem connected (method of Fig. 2,5) image parameter halfsections, pairs of which have equal mvalues and all of which have equal cut-off frequencies Lv. C This is equivalent to stating that: Any conventional ladder filter can be obtained from an image impedance Zobel filter by changing the impedance levels r1 r2 rn of its half-sections. In Chapter III some methods will be investigated, by which the required impedance levels r., the cut-off frequency w and the L C m-values could be determined so as to obtain a ladder filter having a prescribed insertion loss. *A conventional image parameter filter is called a Zobel filter. It is a filter consisting of connected-in-tandem imageparameter sections or half sections (prototype or m-derived) with common cut-off frequencies w and with image impedances (see equations (2 5 ) and (2.6)) whtch match each other at the points of interconnection. In terms of our elementary building blocks, a Zobel filter will have l
PAGE 31

-23 -2.4. Filters with Incidental Dissipation in Terms of Elementary Building Blocks One of the major disadvantages of the image parameter Zobel filter design technique is the fact that dissipation cannot be taken in account in design (though the effect of dissipation may be found by analysis). 3 The modern network theory offers a technique which allows one to synthesize a filter with dissipation producing the same form of insertion loss curve as one without dissipation, except for an additional constant (independent of frequency) loss. This technique, called predistortion, is based on the effect of dissipation on the location of poles and zeros of the insertion voltage ratio in the complex frequency plane. Given the pole and zero distribution of a desired insertion voltag e ratio of a dissipationless ladder filter, e.g. Fig. 2.8 (poles and zeros on solid curve), the addition of positive resistance to each component (in series with an inductance and in parallel with a capacitance) in such amounts that would make the Q-factors of all components equal, would move the zeros and poles in Fig. 2.8 horizontally a distanced l/Q to the left, as indicated by the poles and zeros located on the dashed curve. Adding negative resistance to the dissipationless components in the same amounts as above would shift the poles and zeros a horizontal distanced to the right, as indicated by the zeros and poles on the dotted curve* (the dotted *This horizontal shift is equivalent to transforming the complex variables= 6' + j W to s d, where the(+ ) and(-) signs correspond to addition of positive or negative resistance, respectively.

PAGE 32

Effect of added positive resistance -jd I 'f ;w t ,c I I X I I I I I I I i 1 1 I : t + 24 -Lossless L's and C's of added negative resistance Fig. 2.8. Effect of adding proportionately equal amounts of resistance on the location of poles and zeros of the insertion voltage-ratio function

PAGE 33

25 -curve is called the predistorted curve for an amount of dissipation equal to d}. Suppose that the pole and zero distribution on the dotted curve is used to synthesize a dissipa~ionless ladder network.* If now dissipation dis added to each element, the poles and zeros would move to the left a horizontal distanced and the distribution on the solid curve (Fig. 2.8) would be obtained, resulting in the desired insertion loss. The lossless ladder obtained from the predistorted curve (dotted curve) can be represented by elementary building blocks (image parameter half-sections) as outlined in section 2.3. The image parameters r, m and w of these building blocks will in gen-e eral not be equal to the image parameters of the building blocks corresponding to a ladder synthesized from the pole and zero locations on the solid curve (the pole and zero location of the desired insertion voltage ratio). Thus, given a lossless ladder filter in terms of image parameters r, m and we, the effect of predistortion for a certain amount of dissipation can be obtained by a corresponding (to the amount of dissipation) change in the image parameters r, m and w of the elementary building blocks, C *In practice, predistortion is applied to the zeros only, since the application to the poles would result in a network function which is not realizable by a lossless passive ladder without mutual (inductive) coupling. Thus the transformation of s to s-d is applied only to the numerator of the insertion voltage ratio function, and compensation of the effects of dissipation for the passband only is achieved. Note also that there is an upper limit of d, which must be smaller than the horizontal distance of ~e nearest zero to the jw -axis, for physical realizability with passive networks.

PAGE 34

CHAPTER III DERIVATION OF THE IMAGE PARAMETERS FOR LADDER FILTERS HAVING PRESCRIBED INSERTION LOSS CHARACTERISTICS In this chapter the image parameters will be determined for the elementary building blocks which, when connected in tandem, result in ladder filters with Butterworth (maximally flat) Tchebycheff (equal ripple in passband only), and elliptic function (equal ripple in passband and in stopband) insertion loss characteristic. The image parameters will be expressed in terms of the critical frequencies (poles and zeros) of the insertion loss function. The investigation will be limited to two-section sy,mnetrical filters. 3.1. Outline of Approach The insertion loss L (in db) as a function of normalized frequency W for the above three types of r esponse is given by17 L = 10 log (1 + E2) (3.1)* where E denotes a polynomial or a ratio of two polynomials in w, the form and the coefficients of which determine the type of response. The zeros and poles of E ( w ) coincide with the zeros and poles of the insertion loss function L( w), as can be seen from equation (3.1). It shall then be attempted to express the *Equation (3.1) is for synn:netrical filters only; it will be assumed for convenience that the filter is working between equal resistances of l ohm. .. 26

PAGE 35

27 image parameters of the elementary building blocks in terms of these poles and zeros. Method I: One way of accomplishing this could be to find the insertion loss of a ladder consisting of elementary building blocks in terms of the image parameters of its half-sections. Comparison of this insertion loss with the insertion loss of Butterworth, Tchebycheff or elliptic-function type would yield a number of equations which, when solved simultaneously, would give the image parameters m, rand wc in terms of the poles and zeros of the insertion loss function. Method II: Another method of getting the same results for specific numerical cases would be to synthesize the ladder from the given insertion loss function by any suitable method (e.g., Darlington's method} and then express the ladder in terms of image parameter half-sections, obtaining a number of equations of the form of equations (2.10) to (2.23);* substituting for h's and e's equations (2 .1) to (2 .4), a set of equations in m 1 ri and w c are obtained, which could be solved (at least theoretically) simultaneously for the image parameters. The first approach is more general and will be used here. *It is to be noted here that qi and a 1 are known

PAGE 36

28 -Some numerical data will be obtained using the second approach. 3.2. Insertion Loss of a Two-Section Ladder Filter in Terms of Image Parameters The two section synunetrical ladder which will be used in subsequent investigation is shown in Fig. 3 .1.* This ladder is obtained by tandem connection of ima g e param eter half-sections as shown in Fig. 3 2 Comparison of the ladders of Fig. 3.1 and 3.2 yields the following relations: ql m12 r1 w e l (3. 2) m12(r1 + r2) q2 ... W e (3.3) (3. 4) (3.5) (3. 6) (3. 7) (3. 8) *The Butterworth and Tch ebycheff filte r can be considered as a special case of the ladder of Fig. 3.1 in which a 2 a 4 = O.

PAGE 37

q2 q4 1 2 041 I a2 Jq3 a4 J qso 0 Fig. 3 l. A two section ladder filter N \0 ml2rl ml2r2 -m34r3 ~4r4 w C WC We We I 0 Q -0 0 I 0 I m12 _L l ml2 2 1 m 12 2 m12 m34 1 m342 1 m342 m34 rl We T "'12r1 We m12r2 we I r2 WC r3 W e I m34r3 W c m34r4 W e I r4 We 0 0 ,. 0 0 0 l 2 3 4 Fig 3.2. Elementary building blocks from which the ladder of Fig. 3.1 may be obtained.

PAGE 38

-30 -where r 1 r, r and r are the impedance levels of the half-sections 2 3 4 1, 2, 3 and 4, respectively; m12 is the common m-value of half-section land 2; m34 is the common m-value of half-section 3 and 4; We is the cut-off frequency common to all half-sections; q1 q3 q5 a2 and a4 have dimensions of capacitance; and q2 and q4 have dimensions of inductance. It is much more convenient to find the insertion loss of a lattice equivalent to the ladder of Fig. 3.1, rather than that of the ladder directly. A lattice equivalent of the ladder of Fig. 3.1 is shown in Fig. 3 3 (a). The series reactance X8 and the shunt reactance of the lattice of Fig. 3.3 (a) are given by A _l 1 L1 A 3 s LC 2 2 (3.9) (3 .10) (3 .11) (3 .12) (3 .13) (3 .14) (3.15)

PAGE 39

/ / / -31 / / z a / / / / ~o--~?~--------~---0 o-----
PAGE 40

-32 The relation between the lattice and the ladder elements is as follows: (3 .16) (3 .17) (3.18) (3.19) 1 A5 = 2 ( q1 + q3 + q5) (3.20) The insertion loss of the equivalent lattice of Fig. 3.3 is given by equation (3.1), where Bl ( w 4 2 w B2 + BJ) E == 4 w 2 B 5 + l (3.21) uJ B4 A2A4 (3. 22) Bl .,.--Al Bz ... A1A4 + AzA5 -A3 (3 23) A2A4 B = A 1 A 5 -l (3. 24) 3 A2A4 1 B4 = A1 (AzA3 -A4) (3.25) l (A1A3 + Az A5) (3. 26) B = -5 A1 The zeros and poles of the insertion loss are given by

PAGE 41

-33 4 the zeros and poles of equation (3.21), i.e. 4 2 B3) "" [w 4 w 2(z 2 z 2) + z 2z 2 ] W ( W w B2 + -+ l 2 l 2 w4 B 2 w 4 w 2 + l (3. 27) 4 w BS + l = P12P2 pl2 + P22 where z 1 z2 and p1 P z are the frequencies at which the insertion loss is zero and infinite, respectively. It follows from equation (3.27) (3.28) B .. z 2z 2 3 1 2 (3. 29) (3. 31) Equations (3.27) can be solved for the zeros and poles: z 0 ... 0 (3.32) l z/2 B2 + [ B t Bli "', 2 (3. 33) Po = 0() (3. 34) B22 -1 B5 4B 4)Z P1~2 = 2B 4 (3",35) 3.3. Tchebycheff and Butterworth Filters Synthesized with Image Parameter Half-Sections The Butterworth and Tchebycheff-type two-section filter is shown in Fig. 3,4(a) and its insertion loss characteristic is given by equation (3.1) w1thl7

PAGE 42

-34 -E = Hw5 (3.36) for the Butterworth filter, and (3. 3 7) for the Tchebycheff filter. Here T 5 ( w ) is the Tchebycheff polynomial of order 5, which can be defined by Ts ( w) = cos -1 (Scos w ) (3. 38)* The insertion loss zeros of the Butterworth filter, as seen from equation (3 36) are at w = O; the Tchebycheff polynomial is zero when a TI (i + ) i ; O, 1, 4 resulting in zero insertion loss at the following frequencies: z12 a cos2 TT 10 3 IT 10 (3. 39) n-1 [ *Tn(x) can be defined in polynomial form as Tn(z) 2 xn nxn2/1!22 + n(n 3)xn-412:24 n(n-4)(n-5)x0 -6/3!26 + J where the summation is stopped when the exponents of x are negative. The form of (3.38) is convenient in that it exhibits the location of the zeros.

PAGE 43

-35 Equation (3 37) can be written, using equations (3.39), Hw(w2 -cos2 1T )(w2 cos2 i~) E=--------1_0 __________ ( l 2 cos TT 10 ) (1 -cos2 i~ ) (3.40) For both, Butterworth and Tchebycheff, filter types E: H when W 1, giving (see Fig. 3.5 (a) and (b) ) a2 = 10 1, -1 (3. 41) where L is the maximum loss in passband in db. p It can be shown that Tchebycheff and Butterworth two-section filters possessing electrical symnetry have also geome-trical symmetry, i.e. (Fig. 3.4 (a) ) (3. 42) hence the equivalent lattice to the ladder of Fig 3.4 (a) can be easily found using Bartlett's bisection theorem.18 The equivalent lattice is shown in Fig. 3.4 (b). Comparison of Fig 3.3 (a) with Fig. 3.4 (b) yields the following relations: (3. 43) Inserting equations (3.43) into equations (3.11) to (3.15) gives

PAGE 44

L p L(db) L(db) (a) (b) I 11 1 36 -w 6J Fig 3.5. Insertion loss characteristics for (a) Butterworth and (b) Tchebycheff two -section filters

PAGE 45

l Al .. q2 -37 -A q + q3 5 l 2 (3. 44) Equations (3.44) inserted in equations (3.22) to (3.26) give 2ql 2q2 + q3 ql 2q/q3 (3.45) Equations (3.2)to (3.8), with equations (3.42) ~nd with ql .. 1 rl W C q ... r1 + r2 (3. 46) 2 W e q3 Cl 2 r 2 We

PAGE 46

.. )8 -Substitution of equations (3.46) in (3.45) gives (r1 _-+: r} B ::a----1 r 2r w 5 l 2 C B .., W 2 (1 r 2 + rl ) 2 C l rl + r2 B = WC 4rl (l rlr2) 3 r + r2 l (3. 47) Finally, using equations (3.29 ) and (3.30) in equations (3. 47) we get ) (3. 50) The Butterworth Filter The insertion loss zeros of the Butterworth filter are at zero frequency (i.e., at w 0). Equations (3.50) then become 2 ( 2 r1 \ 0 : WC l -r + --) 1 r1 + r2 Sol.tug equations (3.51) simultaneously we obtain 1 r1 m [o.s + (1.25) J2 = =-l._ = o. 78615 rz rl 1.27202 (3.51) (3.52)

PAGE 47

39 Comparison of equation (3.21) with equation (3.36) shows that B1 H, hence from equations (3.41) and (3.47) we get O.lL 10 -p -1 m 4 (r1 + r2) 4 2 10 r1 r2 Wc (3. 53) From equations (3.51) we get r + r = r 1 ~ which inserted l 2 in (3.53) with r1 r 2 = l from (3.52), gives 1.27202 (3. 54) The cut-off frequency w c is tabulated for values of LP, the maximum attenuation in passband, from 0.1 db to 5.0 db in Table 1, and is plotted versus L in Fig. 3.6. p A two section Butterworth filter in terms of image parameter half-sections, with L = 3 db is shown in Fig. 3.7.* p The Tchebycheff Filter With the insertion loss zeros of the Tchebycheff filter given by equations (3.39), equations (3.50) become 2 7T 2 3 Tf w 2 ( 1 2 + r1 ) cos To+ cos .. -r 10 C l rl + r2 4 2 rr 2 3 7T We rl(l r1 r2) (3.55) cos cos = 10 10 rl + r2 We also have from equations (3.21) and (3. 40) *For Lp =3 db, W = 1.272 since H c 1 in equation (3.54). C

PAGE 48

40 -TABLE 1 CUT-OFF FREQUENCY We FOR THE BUTTERWORTH TWO-SECTION FILTER LP in db. W e LP u.) C L W e p 0.1 1.8526 1.2 l. 4263 3.2 1.2612 0.2 l. 7265 1.4 l. 4011 3.4 1.2503 0.3 l. 6560 1.6 1.3792 3.6 1.2399 0 4 1.6071 1.8 l. 3597 3.8 1.2300 0 5 1.5698 2 0 l. 3421 4.0 l. 2205 0.6 1.5396 2.2 1 3261 4.2 l. 2113 0.7 1.5143 2 4 l. 3113 4.4 1.2025 0.8 l. 4925 2.6 l. 2976 4.6 1.1939 0.9 1.4732 2.8 l. 2847 4.8 1.1857 1.0 1 4560 3.0 1.2726 5.0 1.1776

PAGE 49

. u Cl) (/) ....... a, C: ell 'Q ell 1-4 c:: u 3 ;:,,.. u C: Cl) ::, c:r Cl) 4-t 4-t 4-t 0 I .1,.1 ::, (J .. 41 .. 2.0 1.8 "' 1.6 1------I ~ I'-' 1.4 "-1.2 1.0 0.1 0.2 0.3 0 5 1.0 2.0 3 0 Maximum attenuation in passband LP in db Fig 3.6. Cut-off frequency of elementary building blocks versus maximum attenuation in passband for the design o f filters with Butterworth characteristic ' 5.0

PAGE 50

C : r "" h .. 1.0 I 0 618 () l.272 0 618 0.618 LOI -c1 0.786 0.786 h in henrys; c in farads; r in ohms l.O ::r I 1.272 Fig. 3.7. A Butterworth filter in terms of elementary building blocks with w = l.272 and L 3 db C p N

PAGE 51

with which sin 2 43 H B = 1 sin2 TT 10 sin2 (3. 47) becomes H {rt+ rz}2 2 = 2 5 rr 3 TT sin rl r2 WC 10 10 3TT 10 (3. 56) Instead of attempting to solve directly equati9ns (3.55) and (3.56) for w r 1 and r it is more convenient to use C 2 equations (3.46) with the following equations derived by Belevitch19 TT .,. 2sin 10 ql g 3 TT 2gsin 10 q "" + cos2 3 TT 2 2 g 10 2 (g2 cos2 3 lT ) + 10 q3 = g ( l + cos2 ~) where g is related to H by the transformation .!. c sinh Sx H g =sinh x = sinh ( arc sinh J (3 57) (3.58) and His given in terms of L, the maximum insertion loss in passp band (which is here synonymous with the peak value of ripple in passband), by equation (3.41). Equations (3 46) and (3.57) are readily solved for Wc, r 1 and r 2 giving

PAGE 52

44 -2 WC A+ B = ----------4 sin 21.. sin 3 TT 10 10 = 3 TT 10 -----( A + B ) sin .. r1A rz B .LL 10 T\ 10 B = g2 + cos2 3 TT 10 sin 2l '\ 10 ) (3. 59 ) A tabulation of we' r 1 and r 2 for values of LP from 0.l db to 5 db appears in Table 2.* A plot of We, r 1 and r 2 from Table 2 versus L is shown in Fig. 3.8, and a Tchebycheff p filter with L = 1 5 db in terms of image parameter half-sections p is shown in Fig. 3.9. 3.4. The Elliptic Function Filter Synthesized with Image Parameter Half-Sections The elliptic function type (often called equal ripple in passband and stop-band type) two -section filter which will be used *The data in Table 2 are computed on the University of Florida IBM 650 computer from the equations (3.41) (3.58) and (3.59)

PAGE 53

45 TABLE 2 IMAGE PARAMETER VALUES FOR THE TCHEBYCHEFF TWO-SECTION FILTER LP in db; r's in ohms WC rl r2 r/r1 0.1 1.1724 0.7428 0 8638 1.1613 0.2 1.1176 0.6680 0.8262 1.2368 0.3 1.0891 0.6197 0,7952 1. 2831 0.4 1.0704 0.5837 0. 7685 1.3166 0.5 1.0568 0 5547 0. 7448 1.3427 0.6 1 0464 0,5304 0.7235 1.3640 0 7 1.0379 0 5094 o. 7040 1.3820 0.8 1.0309 0.4909 0.6859 1.3974 0.9 1 0250 0.4743 0.6692 1.4109 1.0 1.0199 0.4593 0.6535 1. 4228 1.2 1.0114 0.4329 0.6247 1. 4431 1.4 1.0047 0.4102 0.5989 1. 4599 1.6 0.9926 0.3903 0.5753 1. 4741 1.8 0.9947 0.3725 0.5537 1. 4862 2.0 0.9908 0.3565 0.5336 1.4968 2.2 0.9874 0.3419 0.5150 1.5061 2.4 0 9844 0.3285 0.4975 1.5144 2 6 0,9819 0.3161 0.4811 1 5218 2 8 0.9796 0.3046 0.4656 l.5284

PAGE 54

46 TABLE 2 --Continued L WC rl r r/r1 p 2 3.0 0.9775 0.2939 0.4509 l.5344 3 2 0 9756 0,2838 0.4370 1 5399 3.4 0.9740 0.2743 0.4238 1 5449 3.6 o. 9724 0,2653 0. 4111 1.5495 3 8 0.9711 0.2569 0.3991 1. 5537 4.0 0 9698 0.2488 0.3876 1.5575 4.2 o. 9686 0 2412 0.3765 1. 5611 4.4 0 9676 0.2339 0.3660 1.5644 4.6 0.9666 0.2270 0.3558 1.5675 4,8 0 9656 0.2203 0.3460 1.5704 5.0 0 .9648 0 2140 0.3366 1.5730

PAGE 55

47 L ., -l.2 l.l LO ...._ ------W e --:____ r---_ '--I --I ------I I ........... K I I "-. -~""I I --'---'1 ---I "" "' ,-1 "' ""' M 03 0 2 O.l 0.1 0.2 0.5 l.O 2.0 Maxinrum attenuation in passband L in db p Fig. 3.8. Image parameters we, r1 and r2 versus maximum ripple in passband for the desig n of filters with Tchebycheff characteristics 5.0

PAGE 56

h ... 0.4 0.585 0.585 0.4 c~ 0 T -c r .. 0.4 Fig. 3o 9. T T T 0 0.585 0.585 0.4 h in henrys; c in farads; r in ohms A Tchebycheff filter with peak ripple in passband L 1.5 db in terms of elementary building blocks with w l~O C 0:,

PAGE 57

-49 -here is shown in Fig. 3.1,* and its insertion loss characteristic, Fig. 3.10, is given by equation (3.1) with3 E = 2 2 2 2 H w ( l0 z1 ) ( w z2 ) 2 2 2 2 (1 lJ z1 )(1 -W z2 ) (3.60) The zeros of the insertion loss characteristic are given by 1 z1 = k2 an ( 2K 5 k) l ( 4K Zz =-k"'Z an k) -5 (3. 61) 20, 21 where an denotes the elliptic sine, defined as sn(u) sin QI u = F(k, ~) = J d8 0 (3 0 62) (l k 2 sin2 8 )2 K deno tes the complete elliptic integral of the first kind 7f K "' K(k) = J-,: d9 (3.63) I 0 (1 k 2 sin2 8)2 a nd k is the modulus of the elliptic sine. The constant His given by 4 O lL O o lL H =(10 '--P -1)(10 8 -1) (3. 64) This is the special c ase of elliptic function filters in which all the poles are on the jW-axis, hence it can be realized with a ladder network that does not require mutual coupling.

PAGE 58

50 -where LP is the maximum distortion in passband, and La is the minimum attenuation in stopband in decibels (see Fig. 3.10)., LP, 22 L and k are related by a 2 L 10 log (l + !:L) a 10 62 2 2 LP = 10 loglO (l + H 6 ) 5 "2" 2 K 2 3K b. k sn (5, k) sn rs, k) (3. 65) As can be seen from Fig. 3.10, the modulus k determines the width of the transition band (k is c alled the selectivity parameter), (3. 66) where w is the upper frequency limite of the passband (or the p effective cut-off frequency), and w is the lower limit of the a attenuation band. When L is small and L large, which is the case for most p a filter requirements, the variou s parameters determining the response of an elliptic function type 2-section filter are related by the approximate equation 0. lLp La= 10 log 10 (10 -l) 50 log10 q -12.041 (3 67) where q is the elliptic modular function of k,* La and, are in *The function log q is tabulated in most elliptic function tables, e.g., Ref. 21, pp. 49-51.

PAGE 59

-51 -L (db) I U I I -:_ ~---------I I LP I f -z1 Z2 I 1 l z2 z l w p s: 7/k l lJ = a 7/k Fig. 3.10. Insertion loss characteristic of an elliptic function two-section filter w

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-52 -decibel s o A plot of L versus k for L a 0.1 db is given in Fig. 3 11. a p Figure 3.12 shows the dependence of La on LP for a given k. In order to find relations between H, z1 z2 of equation (3.60) and the image parameters we' r 1 to r 4 n,_2 m34, a procedure similar to one used in the previous section will be followed. The resulting algebraic manipulations are quite lengthy and will be omitted here. The following steps are carried out: 1. Equations (3.16) to (3 20) are inserted in equations (3.22) to (3 .26), giving a set of equations, say Set I, relating B's with q's. 2. Equations (3.2) to (3.8) are inserted in Set I, giving a set of equations, say Set II, relating B's with the image parameters. 3. Equate Set II to corresponding equations (3.28) to (3 .33), getting a set of equations, say Set III, relating the image parameters to the zeros and poles of the insertion loss function (3.60). 4. Solve equations Set III sinrultaneously for the image parameters in terms of zeros and poles of theinsertion loss function. Step 3 results in five non-linear equations in seven unknowns (the image parameters); two additional equations are obtained from the symmetry conditions which the ladder must satisfy, giving a total of seven independent equations. These seven equations do not lend themselves readily for use in Step 4,

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53 .. 80 70 0.3 0 4 0.5 0.6 0.7 0.8 0.9 1.0 Selectivity factor k Fig. 3 .11. Minimum stop~band attenuation La versus selectivity factor k for the two-section elliptic function filter 20 -10 0.01 0.05 0.1 0.5 1.0 L in db p 5.0 Fig. 3 .12. Variation of minimum attenuation in stopband with maximum ripple in passband

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.. 54 -hence numerical values for the image parameters will be obtained using Method II (Chapter III). Step 4 is carried out for a set of equations which are close approximations to those obtained in Step 3. Following are the equations obtained in Step 3: H = z 2 2 == 1 m342 w 2 C (ml22 ~42)(1 + m12m34D) w c5(r1m12 r41_3L) l r1r4o l + m1zm34D The equations obtained from symmetry conditions are: (3.68) (3.69) (3. 70) (3. 71) (3. 72) (3.73) (3. 74) *The numerical values appear in Table 3. They were obtained by analysis of synthesized elliptic function filters in terms of image parameter half-sections. For this purpose the synthesized filters published in Ref. 12 were used.

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55 Equation (3.75) can be written as r2 = (rl~2 r4m34)(r4m12 + rlm34) rl (r4m12 r1m34)(r1m12 + r4m34) to which a close approximation turns out to be r2 rlml2 r4m34 r1 r4m12 r1m34 with the assumption that* l r4m12 + r1m34 rlm12 + r4m34 (3.75) (3. 76) (3.78) (3. 79) With the approximation of equation (3.78) the following set of approximate relations may be obtained:** H (m12 + m34)(l + m12m34) rl wc5 2 2 z + z 1:11 l 2 (3 0 80) (3.81) *This assumption is justified by the range of values of the r's and m's encountered in a great number of filters, as shown by the numerical data in Table 3. For most of the values computed, equation (3.79) turns out to be near 0.99. **In addition to the approximation (3.78), there were also other approximations used which are implied in (3.78) and (3.79), e.g., lr1 r41 <
PAGE 64

z 2z 2 1 2 56 -,:: rl We 4(1 r1 r3) r1 + r 2ml 2m34 From (3.68) and (3.69) we have Using equations (3.83) in (3 .80), we may get with which r 2 may be computed from (3.81) (3 82) (3. 83) (3. 84) (3.85) (3. 86) Equation (3 82) may be solved for r 3 using r1 and r 2 from (3.85) and (3 .86). 1 [ z12z22(r1 + r2m12m34) ] r3 =--;--1 1 4 rl W e (3. 87) Using (3 85) to (3.87) in (3.74), we get (3.88) The cut-off frequency w in equations (3.83) to (3.87) may be C computed from the approximation 1 2.. w k c 0,982 + 0.01 loglO LP *This equation was derived from the data in Table 3. (3.89)*

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-57 -If more accurate results are desired, the r's may be re-evaluated by the following relations, using previously computed values on the right-hand side of the equations. (3. 90) 1 z1 2 + z 2 ( 2 -1) (l + r2m12m34 ) w 2 r1 C r3 = (3.91) r 1 (1 + r1m12m34 r ) 2 rm m 3 12 34 (3.92) With r1 r 3 and r 4 from above and r2 from (3.74), Wc can be recomputed from (3.93) where R is the right-hand side of equation (3.75).* A tabulation of r 1 to r 4 m 12 m 34 and wc appears in Table 3 (see footnote on page 54). A plot of the r's versus L p is shown in Fig. 3.13. *The differen,ce between the recomputed values and the first values of the r's and w is an indication of the accuracy of the C results. For greater accuracy, the image parameters may be computed by an iterative procedure using equations (3.68) to (3.75).

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TABLE 3 IMAGE PARAMETER VALUES FOR THE ELLIPTIC-FUNCTION TYPE TWO-SECTION FILTER L8 in db, r's in ohms La k r1 r2 r 3 r 4 m12 m34 We LP = 0.1 d b 35 0 764 0 744 0.823 0 810 0.7 00 0 818 0.552 0.983 40 0.707 o. 746 0 836 0.8 22 0.734 0.854 0.637 0.952 45 0.649 0 748 0.843 0.832 o. 738 0.881 0.703 0.919 50 0 591 0.752 0.750 0 973 0 745 0 903 0.759 0.884 55 0.537 0 745 0 852 0 .844 0 738 0. 922 0 .804 0 847 \/1 00 60 0 488 0 740 0.856 0.849 0.734 0 937 0 840 0.809 65 0.442 0.739 0.858 0 853 o. 734 0.950 0.871 o. 771 70 0.398 0 741 0.860 0 856 0 737 0.960 0.896 0.735 = 0.5 db 35 0.843 0.530 0.739 0.693 0.495 0.791 0.526 0 .954 40 0.787 0 532 0.745 0.707 0 503 0.831 0.601 0 927 45 0.731 0 539 o. 749 0. 718 0.518 0.864 0.674 0.895 50 0.675 0.544 0.747 o 719 0 531 0.891 0.733 0 860 55 0.618 0 557 0.757 o 716 0.556 0 912 o. 784 0.824 60 0 561 0.550 0 746 0.733 0.540 0 929 0.824 0 .789 65 0 510 0.550 0. 745 0.735 0 542 0.943 0.857 0 753 70 0.462 0 555 0 753 0. 744 0 548 0 9 5 4 0.883 o. 720

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TABLE 3 -Continued La k r1 r2 r3 r4 m m lJJc L = 1.0 db p 35 0.874 0.426 0.661 0.598 0.388 0. 772 0.502 0.949 40 0.822 0.430 0.660 0.613 0.400 0.815 0.580 0. 923 45 0.803 0.443 0.670 0.633 0.419 0.850 0.649 0.895 50 o. 711 0.447 0.66 0.640 0.429 0.881 0. 715 0.859 55 0.654 0.449 0.661 0.641 0.435 0.904 o. 768 0.824 60 0.597 0.451 0.657 0.642 0 441 0.923 0.812 0. 788 65 0.543 0.452 0.653 0.642 0.443 0.938 0.847 0.752 70 0.491 0.452 0.655 0.650 0.443 0.951 0.875 o. 718 \JI

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(7l ] 0 i:: """' .._jM ,-c, i:: ell "" M N M ,-4 M (/) M (1) .., (1) ffl M C\I 0. (1) 00 m 0.9 0.8 0.7 0 6 0.5 0.4 0.3 0.05 -60 -"" r 4 \ 0.1 0 2 0.5 1.0 Maximum ripple in passband L in db p \ \ 2.0 3.0 Fig. 3.13. Image parameters r 1 r2 r3 and r 4 fork= 0.7 versus maximum ripple in passband for the design of elliptic function filters

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CHAPTER IV ANALYSIS OF RESULTS AND CONCLUSIONS 4.1. Analysis of Obtained Data It can be seen from the data in Table 3 that the r's show very little dependence on the selectivity factor k for quite a large range of values of k (the maximum change in the r's for a constant value of ripple in passband is about 5 per cent for values of kin an interval of about 0 4 to 0.8). If the Tchebycheff filter is considered as a limiting case of an elliptic function filter in which k -O,* i .e., all the poles of which were moved to infinity without appreciably changing the location of the zeros, we may use the data in Table 2 for getting the r's for the lower limit of k = O. It is seen that these r-values differ little from the corresponding r's ( i .e., r s for the same ripple in passband) of the elliptic-function filter. The r's do, however, show strong dependence on the amount of ripple in passband L. p Defining mismatch constants as *This is true only in an approximate sense, when the curve on which the zeros of the elliptic function insertion power-ratio are located (in complex frequency phase) approximates an ellipse. The distribution of the zeros on an ellipse-like curve is found in most practical cases of elliptic function type ladder filters without mutual coupling. 61 -

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62 rt 1 u ... =t r1 r1 (4.1)* r1 ul -r2 (4.2) u2 .. r3 (4.3) it is seen that as the ripple in passband LP increases, ut increases, u 1 decreases, and u 2 shows very little dependence on LP. The effect of increasing ut and decreasing u 1 may be visualized in complex frequency plane as moving the zeros of the insertion voltage ratio toward the j-w axis, i.e., the effect is similar to that of predistortion. This suggests that in networks with incidental dissipation, a compensation for losses similar to predistortion may be achieved by an increase of Ut and a decrease of u 1 in amounts depending on the amount of losses (i.e., Q or d as defined in 2.4) in the components. A larger change in ut and u 1 will be required for components with higher losses. As seen from equation (3.89), the cut-off frequency wc varies predominantly with the square root of k, i.e., with the effective cut-off frequency WP (see Fig. 3 .10). The m's are functions of wc and the location of the poles *r is the terminating resistance, which is equal to 1 ohm t in our analysis.

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63 -of the insertion loss function E(w), equation (3.83) and (3.84). Since the location of the poles depends on the selectivity parameter k only, as seen from equation (3.61) and (3.63), and since w C depends on k predominantly, the m's depend mostly on k and very little on the amount of ripple in passband LP. 4.2. Suggested Modification of Zobel Designs with the View of Obtaining More Efficient** Filters The method of designing a Zobel filter to satisfy given insertion loss requirements has been appropriately called the cutand-try method One reason for this is the fact that the performance of a filter (i.e., its attenuation and phase-shift when inserted between generator and load) is only approximately given by its image attenuation and phase. This approximation is fairly good in the attenuation band, but is very poor in the pass-band. The labor required to design a Zobel filter that approximately meets given requirements is usually very small. A commonly employed method of obtaining a filter which satisfies given requirements may be described in the following general steps: l. Design a Zobel filter which approximately satisfies insertion loss requirements. *In these equations the zeros z and z 2 rather than the poles, appear. However, the poles are located inversely with respect to the zeros in the elliptic function filter, as seen from equation (3 .60). **The term "efficient" is used here in the sense of high contribution to filtering action for each component.

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64 2. Build a model filter according to design of Step 1. 3. Test the model of Step 2, establishing the difference between its performance and the requirements. 4. Dictated by the results of Step 3, make corrective changes in the design of Step 1, aiming at eliminating the difference between the filter performance and the requirements. S. Repeat Steps 2, 3 and 4 until the aim of Step 4 is reached. It should be pointed out here that the cost (in terms of effort or dollars) of arriving at a satisfactory design using the above method is usually smaller than the cost of a design carried out by modern network-design techniques (i.e., by insertion parameter synthesis, leading to a satisfactory design in a more direct way).* This may partially account for the fact that Zobel designs are still so widely employed, although the modern techniques (in existence now for about twenty years) are known to produce better, often more economical, filters. It is in general easier to correct the performance of a filter (Step 4) in attenuation band than in passband. In fact, unwanted distortion in passband may sometimes be eliminated only by adding components to the filter (e.g., m-derived terminal halfsections), or by using an additional network (an attenuation equalizer). This is due to the pattern of distortion in passband inherent in *This is particularly so if the designer has reasonable ability to predict the effects of changes in design (i.e. changes of Step 4),

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65 -Zobel filters, as shown in Fig. 4.1. The insertion loss in passband of Zobel filters with equal image impedances z1 at both terminations, inserted between equal resistances Rt, is given by24 where Bis the total phase shift through the filter. The envelope of the insertion loss in passband is L 20 log [1 ( ZI Rt)] env 10 2 Rt z1 Since the image impedance z1 is continuous in passband and becomes zero (or infinite) at the cut-off frequency w in a manner C indicated by Figs. 2.2 and 2.3, termination of the filter by a constant resistance Rt results in an insertion loss envelope in passband which increases near the cut-off frequency W as indicated C in Fig. 4. l. Most filter requirements allow a certain amount of maximum distortion in passband. Fig. 4.2 shows a typical insertion loss requirement; any insertion loss curve laying in the shaded area would satisfy the requirements. The curve shown may be considered typical for an efficient* one-section filter. *In general, the number of components needed in a filter increases with increased selectivity k a wp/Wa, with increased minimum loss in attenuation band L8 and Lh, and with decreased maximum distortion in passband~-The curve shown utilizes the limits of the requirements in a fashion which could lead to a filter with fewest components.

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66 -L t Lenvelope Maximum distortion in pauband Fig. 4 .1. Typical inaertion loss characteristic in passband for a Zobel filter L t i -------w Wp W a Fig. 4.2 Typical insertion loss requirement (straight lines) and a possible insertion loss characteristic which satisfies it

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67 In case of the Tchebycheff and elliptic function filters, utilization of the maximum allowable distortion in passband, i.e., designing a filter that has this maximum distortion in passband, leads to meeting the requirements with a filter having a minimum number of elements. Since the Tchebycheff and elliptic function filters can be designed to produce a predetermined amount of maximum distortion in passband (in contrast to the Zobel filter, in design stage of which there is no provision for obtaining a certain predetermined amount of distortion in passband), these filters can be designed to satisfy given requirements with a minimum number of components.* In case of the Zobel filter, the amount of distortion in passband can be controlled by mismatch at the filter terminals, i.e., by choice of the value for Rt, within a relatively narrow range only. In general, a Zobel design is termed successful if the filter satisfies the requirements in attenuation band, and turns out to have passband distortion which is below (or equal) the specified maximum allowable distortion. *Tuttle25 showed in a comparison of the performance of Butterworth, inverted Tchebycheff, elliptic-function, constant-k and m-derive d Zobel filters that the Zobel filter was inferior to the elliptic function type only. However, a comparison of a constant-k Zobel filter with a Tchebycheff filter should prove that the latter has superior perfonnance for the same number of elements.

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68 -If the requirement calls for a very small distortion in passband (in the 0.01 db. range), the usual way to obtain it with a Zobel design is by the use of m-derived terminating half-sections, the image impedance of which (see Fig. 2.3) has less variation over an appreciable portion of the passband than the prototype (constant-k) image impedance, Figo 2.2. This then results in smaller passband distortion due to better matching at the terminations, the price for which is an increased number of elements (above the number necessary to produce a given selectivity and discrimination) required in them-derived terminations.* If the allowable distortion in passband is high (in the l db. range), the Zobel design can take little advantage of it since at best the maximum distortion will be reached at one point in the passband, as indicated in Fig. 4.1.** From this it may be concluded that in order to make a Zobel filter more efficient (comparable to a Tchebycheff or elliptic function filter), some means of 1 Controlling the amount of distortion in passband 2 Achieving equal (or approximately equal) maximum ripple in passband should be included in the design procedureo *In some cases additional attenuation equalizing networks are used instead of, or for severe requirements, with them-derived terminations. **An efficient (in terms of the number of components) design results when the maximum allowable distortion in passband is reached a maximum possible number o f times3 as in the case of the Tchebycheff and elliptic function filters.

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-69 -It was shown in Article 4.1 that for the elliptic function filter the values of the r's are determined mainly by the ripple in passband, whereas all the other parameters (w and the m s ) C depend mainly on the selectivity k. This means that the main effect of a change in the r's will be a change in passband distortion. There is a certain relation between the changes in the individual r's which results in an equal ripple of a certain magnitude. These changes depend only little on the position of the poles (i.e., W c and m) in the attenuation band From the above reasoning the following may be concluded: Given a Zobel filter designed with any image parameters (W, r and m's), the amount of distortion in passband may be C 0 controlled by changes in the impedance levels (i.e., in the r's) of its half-sections. In particular, for a Zobel filter consisting of m-derived sections, an approximately equal ripple in passband of magnitude, will be obtained if the values of the r's are made equal to the corresponding r's of the ellipticfunction type filter having the same amount of ripple in passband and approximately the same selectivity k; for a Zobel filte r consisting of constant-k sections, an approximation to equal ripples of magnitude L will be achieved by letting the p r's of the Zobel filter assume the values of corresponding r's o f the Tchebycheff filter with a passband distortion of L. p Since, as it was pointed out in Article 4.1 (p. 61), the

PAGE 78

70 r~values of a Tchebycheff filter having a certain amount of passband ripple L differ only slightly from corresponding r-values of p an elliptic-function type filter having the same passband ripple L, p it appears that: Making the impedance levels (the r's) of the half-sections of any Zobel filter (i.e. consisting of m-derived or constantk sections and having any selectivity) equal to the corresponding r's of a Tchebycheff filter with passband ripples of magnitude L, will result in the insertion loss of the thusly modified p Zobel filter to have approximately equal passband ripples of the same magnitude L. p 4 .3. Limits of Improvement of a Zobel Filter There are many ways by which one may carry out the details of a Zobel filter design as outlined in Step 1, Article 4.2. These will in general depend to some extent on the type of the insertion loss requirements. Since this subject is well covered in litera ture,15 16, 26 it will not be repeated here. For the purpose of this article it will be assumed that Step 1 has been carried out for two types of insertion loss requirements shown by the straight lines in Fig. 4.3 (a) and (b),* and that the resulting Zobel filters have an insertion loss as shown by the solid curves in Fig. 4 .3. If now the half-sections of these filters are mismatched as *These two types of requirements are among the most commonly encountered in practice.

PAGE 79

L a L p -71 -(a) (b) Fig. 4.3. Two types of insertion loss requirements and possible filter characteristics L1 L -w ----w

PAGE 80

-72 -outlined in Article 4, 2 an insertion loss indicated by the dotted curves will be obtained. The question arises, how much of an im provement in performance of these filters may be expected from this mismatch? In the case of the filter in Fig. 4.3 (a), the answer comes from the fact that the new insertion loss (dashed curve) can be at best as good as the insertion loss of an elliptic function type filter with passband ripple L and selectivity k = w I w The p p a minimum insertion loss in attenuation. band Lae of such a filter is given by equation (3.65) or (3.67), or can be found from the curve in Fig. 3.11. Hence, at best, the improved performance of a modified Zobel filter will be an increase of the minimum insertion loss in stop-band bY* L = L L ae a (4. 4) In the case of the filter in Fig. 4.3 (b), it is convenient 26 to use a transformation which will express the performance of the filter in terms of a performance as in Fig. 4.3 (a). It turns out that the minimum attenuation Lbe in the frequency range wb to wh is approximately the same as the minimum attenuation Lbe for *If the Zobel filter (before modification) had equal minima of insertion loss ln passband, as indicated by the solid curves in Fig. 4.3, the minima will not, in general, be equal after the modification, i.e., after mismatching the half-sections. For most cases occurring in practice, however, the difference in the minima will be small and can easily be corrected by a small change in the ro-values, the effect of which on the passband ripple is neglibible.

PAGE 81

-73 a filter as Fig. 4.3 (a) with a selectivity of l ( ~/ u.J 2 1-(i!J 2 (4.5)* L may be determined in the same manner as L for the previous be ae case, giving the maximum possible improvement as (4.6) 4.4. Sulllnary and Conclusions A method of synthesis of ladder networks was investigated with which conventional ladder networks may be obtained by proper choice of parameters of elementary building blocks. *The frequency scale in Fig. 4.3 (b) is transformed to an m-scale by the relation m .,. (1 w 2/w 2 ) l/2 hence m s O cor-p responds to w m "" 1 corresponds to W a oo m and m. correspond p b h to wb and u.J h> respectively. Multiplication of the m-scale by a factor 1/mh introduces a new scale, say m' scale, in which m' 0 corresponds to wp, m' = 1 corresponds to Wh' and~' "" mb/~ corresponds to W~ An inverse transformation of m! i.e., w' = W /(1 -m' ) 112 results in a frequency scale w which is p infinite at wh' giving an insertion loss requirement with equal minimum from wb' (corresponding to uib ) to w -= oo. The selectivity of the transformed requirement is k = wp'/wb', giving (4.5) upon substitution of wp' =-wp and wb' = wp/(l ~1 )112

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74 -These parameters were determined for three types of twosection filters having desirable insertion loss characteristics. As a result of analysis of the obtained parameter values, a method for improving Zobel filters was suggested The design of a two-section Butterworth filter can be quite simply accomplished with image parameter half-sections having a constant amount of impedance mismatch given by equation (3.52) and a cut-off frequency wc given by equation (3.54) or Fig. 3.6. The impedance levels of the half-sections, r 1 and r 2 are constant, i.e., independent of any design parameter; the cut-off frequency w is a function of the maximum distortion in passband only, as C given by equation (3.54).* The design of two-section Tchebycheff filters may be as easily performed using equation (3.58) and (3.59), or the graphs in Fig. 3.8.** *The ctt-off frequency w is not a critical design parameter for constant k type low-pass an~ high-pass filter types, i.e., it has no effect on the shape of the response curve. A change in the cut-off frequency amounts to a linear transfonnation of the frequency scale of the insertion loss versus frequency curve. **Butterworth and Tchebycheff filters with lossless elements can be designed with not much effort using already developed formulae for element values (i.e., without going through the actual synthesis from the insertion loss function) The engineering value of our ap proach is in the simplifications resulting from its application to lossynetworks, for which no explicit formulae for circuit element values are available at the present.

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-75 Computation of the element values by slide rule gives sufficient accuracy for commonly encountered tolerances in manufacturing of the components A two-section filter of the elliptic-function type can be designed using either the equations (3.83) to (3.89),** or the graphs in Fig. 3.13 with equations (3.83), (3.84), (3.89) and (3.74); use of the graphs ~s simpler but more restricted in the range of application. *A manufacturing tolerance in component values of 1 0 per cent to 2 per cent is quite common for capacitors in the range of approximately 100 tU./1<. f to O. 2 f and for inductors in the range of approximately 1 mh to 50 hy. **The values of the zeros, equation (3.61), can be found from many elliptic function tables, e g., Ref. 21; they are tabulated for use i n elliptic-function filter design in Ref. 23.

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REFERENCES l. O J. Zobel, "Theory and Desig n of Uniform and Composite Electric Wave Filters," B .S.T.J., "2 (1923) pp. 1-46, and "Transmission Characteristics of Electric Wave Filters,11 B.S.T.J. 3 (1924), pp. 567-620 2. M I. Pupin,''Wave Propagation Over Non-Uniform Cables and Long Distance Air Lines," Trans. A.I.E.E., 17,(1900), pp. 445-507. G. A. Campbell, 110n Loaded Lines in Telephonic Transmission, 1 1 Phil. Mag., 5, (1903), pp. 313-330, and 11Physical Theory of the Electric Wave Filter," B.S.T.J., 1 (1922), pp. 1-32. G. A Campbell and R. M. Foster, "Maximum Output Networks for Telephone Substation and Repeater Circuits, 11 Trans. A. I.E.E. vol. 39 (1920), pp. 231-280. 3 S. Darlington, 11Synthesis of Reactance 4 Poles,'' J. Math. Phys., vol. 18 (1939), pp 257-353 4 w. Cauer, Theorie der linearen Wechselstromschaltungen, Akademische Verlag, Berlin, Germany (1954), 2nd ed. H Piloty, ''Weichenfil ter, 11 Zei tschrift fUr Telegraphen und F ernsprechtechnik, vol. 28 (1939), pp 291-298, 333-344, and ''Wellenfilter, Insbesondere symmetrische und antimetrische mit vorgeschriebenem Betriebsverhalten," Zeitschrift fUr Telegraphen und Fernsprechtechnik, vol. 28, No. 10 (1939) pp. 363-375. 5. v Belevitch, "Recent Developments in Filter Theory," IRE Transactions on Circuit Theory, vol. CT-5 (Dec. 1958) pp. 236-252 6. T Laurent, "Allgemeine physicalische Zusammenhttnge bei Filterketten," Arch d. El. ii. B. 12, H.l (Jan. 1958) pp. 1-8, and 11Echostatanpassung eiee neue Methode zur Anpassung von Spiegelparameterfiltern," Arch. d. El. ii. B. 13, H. 3 (March 1959) pp. 132-140. 7 R O Rowlands, 11Composite Ladder Filters," Wireless Engineer, vol. 29 (Feb 1952) pp. 51. 8 J E Colin, "Two-Branch Filter Structures with Three Cut-Off Frequencies, 11 Cables & Transmission, vol. 11 (July 1957) pp. 179-217 76 -

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-77 -9. W. Sa raga, "Insertion Parameter Filters," TMC Technical J. vol. 2 (March 1951), pp. 25-36. 10. A. J Gros s man, "Synthesis of Tshebycheff Parameter Symmetrical Filters," Proc. IRE, vol. 45 (April 1957) pp 454-473. 10 E o Green, Amplitude-Frequency Characteristics of Ladder Networks, Marconi's Wireless Telegraph Co., Essex, England; 1954. 11. J, K. Skwirzynski and J. Zdunek, "Design Data for Symmetrical Darlington Filters," Proc. IRE, vol. 104, pt. c (Sept. 1957) pp. 366-380. 12. S. D. Bedrosin, E. L. Luke, and H. N. Putchi, "On the Tabulation of Insertion Loss Low-Pass Chain Matrix Coefficient and Network Element Values,11 Proc. Natl. Electronics Conf., vol. 11, (1955) pp. 697-717. 13. T. Laurent, Vierpoltherie und Frequenztransformation, SpringerVerlag, Berlin, 1956. 14. H W. Bode, Network Analysis and Feedback Amplifier Design, D. van Nostrand Company, Inc., New York, 1945. 15. T. E. Shea, Transmission Networks and Wave Filters, D. van Nostrand Company, Inc., New York, 192 9 16. F. Scowen, Introduction to Theory and Design of Electric Wave Filters, Chapman & Hall, Ltd., London 1950. 17 W.H, Chen, Elements of Electrical Analysis and Synthesis, McGraw Hill Co. In press. 18. A. C. Bartlett, The Theory of Electrical Artificial Lines and Filters, John Wiley & Sons, New York, 1931. 19. v Belevitch, "Tchebycheff Filters and Amplifier Networks," Wireless Engineer, vol. 29 (April 1952) p. 106. 20. L. M Milne Thomson, Die elliptischen Funktionen von Jacobi, Springer Verlag, Berlin, 1931. 21. E Jahnke and F. Emde, Tables of Functions, Dover Publications, New York, 1 945. 22. w. N Tuttle, "Design of Two-Section Symmetrical Zobel Filters for Tchebycheff Insertion Loss," Proc. IRE, vol. 47 (Jan. 1 9 5 9), PP o 2 9-36.

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78 -23 E Glowatzki, "Sechsstellige Tafel der Cauer-Parameter," Abhandlungen der Bayerischen Akademie fer Wissenschaften, Neue Folge, Heft 67 (1955), 24. w. Saraga, "Insertion Loss and Insertion Phase-Shift of Multisection Zobel Filters with Equal Image Impedances," P O Elec. Eng. J vol. 39 (Jan. 1947), pp 167-172. 25. W. N, Tuttle, "Applied Circuit Theory," IRE Trans. on Circuit Theory, vol. CT -4, (June 1957) pp. 29-32. 26. J H. Mole, Filter Design Data, John Wiley and Sons, Inc., New York, 1952.

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-79 -BIOGRAPHY David Silber was born on October 2, 1922, in Lodz, Polando His undergraduate studies were pursued at the O.V.M. Polytechnikum in Munich, Germany, from which he received the degree of Electrical Engineer in June, 1950. After arrival in the United States in 1951, he pursued graduate studies at the Uni versi ty of Cincinnati, Evening College and Swmner School, while employed by the Keleket X-Ra y Corporation i n Covington, Kentucky. Since 1954 he has been employed by the Communication Accessories Company in Kansas City, Missouri, taking leave in 1956 to enter the University of Florida for further graduate studies. At the University of Florida he held a graduate fellowship for two years and taught electrical engineering for one year. He received the degree of Master of Science in En gineering in August, 1957.

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 13, 1960 Dean, Colle Dean, Graduate School SUPERVISORY COMMITTEE: ;zsA Chairman =G). f ~~
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