THE EMAGE, PARAMETER METHOD FOR THE
DESIGN CW THE FREQUEWMNSYMMEFRECAL
BANDJASSJ LADDER, FILTERS USING SPECIAL

TYPES OF ELEMENTARY SECTIONS

Thesis EM fie Dwm 0’: pk. D.
MICHEGAN STATE UNEVERSETY

Kudmt Socmintapcera
1965

fl'HESlS

LIBRARY

Michigan Staw
University

 

This is to certify that the
thesis entitled

The Image Parameter Method For The
Design of Frequency-unsymmetrical Band-
pass Ladder Filters Using Special Types of
Elementary Sections

presented by

Kudrat Soe mintapoe ra

has been accepted towards fulfillment
of the requirements for

PhoDo degree in EleCe Engr.

%//7/4&/

ajor professor‘7

Date August 4, 1965

 

0-169

ROW USE Gill

 

ABSTRACT

THE IMAGE PARAMETER METHOD FOR THE DESIGN
OF THE FREQUENCY—UNSYMMETRICAL BAND-PASS LADDER FILTERS
USING SPECIAL TYPES OF ELEMENTARY SECTIONS

by Kudrat Soemintapoera

The design of electrical filters can be accom—
plished by either of two methods, viz., (1) the insertion
parameter method which was developed by Cauer [CA 1] and
Darlington [DA 1] or (2) the image parameter method which
finds its origin in the early works of Campbell [CAM 1]
and Zobel [20 1].

In insertion parameter theory, after special types
of insertion loss requirements are selected (flat loss in
both the pass and stop bands) exact formulas for the char—
acteristic functions of the filter exist. However9 in the
general case, the insertion loss requirement in the block
band is arbitrary and an approximation for the character»
istic function is necessary. Only recently some work
toward this general case has been conducted [EU 2]. The
second part of filter design by the insertion parameter
method is the determination of the network element values.
This necessitates the solution of high-order equations and
the method of zero shiftingo It is known that in these

calculations an abnormal number of digits must be considm

Kudrat Soemintapoera

ered, otherwise the calculated element values are far from
accurate or unrealizable. 0n the other hand, filter
design based on the image parameter method does not ne-
cessitate tedious calculations and the element values
are explicitly given by very simple formulas. Discussions
of the advantages and disadvantages of image parameter
method over that of insertion parameter method can be
found elsewhere [TO 1].

In this thesis some of the work done by Tokad
[T0 1] for the low-pass filters are extended to the
frequency unsymmetric band-pass filters. The contri=
butions of this thesis are

1. Complete characterizations of the elementary
basic sections are developed and the formu-
las for the element values of these sections
are develOped.

2. A systematic design technique is described
for the frequency unsymmetric band-pass
filters.

3. A general approach to the evaluation of
terminating sections is given which utilizes
a network transformation.

4. The image impedance function of a higheru
order terminating section is studied and the

results which are important to the designer

Kudrat Soemintapoera

are shown.
In addition, discussions necessary for completeness in
develOpment of the primary subject material are given

so as to make the thesis self contained.

THE IMAGE PARAMETER METHOD FOR THE DESIGN
OF THE FREQUENCY-UNSYMMETRICAL BAND-PASS LADDER FILTERS
USING SPECIAL TYPES OF ELEMENTARY SECTIONS

by

Kudrat Soemintapoera

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1965

ACKNOWLEDGEMENT

The author is indebted to his thesis advisor,
Dr. Yilmaz Tokad, for his guidance and constant encour-
agement in the preparation of this thesis.

The author wishes to thank his major professor,
Dr. Harry G. Hedges, for his guidance, his encouragement
and his patient advices during the difficult phases of
the author's study.

Thanks also are due to Dr. Joseph A StrelZoff

and Dr. Edward Nordhaus for their encouragements.

11

TABLE OF CONTENTS

CHAPTER I. INTRODUCTION . . . . . . . .

CHAPTER

2.4
2.5

2.6
2.7
2.8

2.9
CHAPTER

3.1
3.2
3.3
3.4
CHAPTER
4.1
4.2
4.3

II. SPECIAL TYPES OF ELEMENTARY SECTIONS
FOR THE FREQUENCY UNSYMMETRIC BAND-
PASS FILTERS. . . . . . . .
Introduction . . . . . . . .
The elementary sections . . . . .
General discussions of the elementary
sections . . . . . . . .
Analysis of the elementary sections .
H—functions and some basic sections
of band-pass filters . . . . . .
Equivalence of the elementary sections
Further equivalence characteristics .
Pole distributions and structure
configurations . . . . . . . .
The impedances o e e , o e e e 0
III. THE FREQUENCY TRANSFORMATION AND

THE TEMPLATE METHOD
Introduction . z. . . . . . .
Template for the low-pass filter . .
Template for the band-pass filter . .

Impedance with normalized frequency .

IV. TERMINATING SECTIONS . . . . .

Introduction . . . . . . . .
The disassociate filter . . . . .

The image parameter ladder terminating
sections . . . . . . . .

iii

11
15

24
26

32

34
36
51

51
51
55
66
69
69
7O

74

CHAPTER V. FILTER DESIGN I - DERIVATION OF
FORMULAS O O O O O O O O O 82

5.1 Introduction . . . . . . . . . 82

5.2 The characterizing function of the
image parameter filters . . . . . . 85

5.3 The chain matrix . . . . . . . . 89

5.4 Current and volta e transmission
factors (M and NT . . . . . . . . 90

5.5 Entries of chain matrix in terms of
the image parameters . . . . . . . 90

5.6 Insertion loss parameters . . . . . 92
5.7 The effective (operating) loss . . . . 95

5.8 Derivation of insertion loss parameters
in terms of image parameters . . . . 101

5.9 Formulas for the operating loss design
technique . . . . . . . . . 108

CHAPTER VI. FILTER DESIGN II - APPROXIMATION
AND DESIGN PROCEDURE . . . . . . 110

6.1 Introduction . . . . . . . . . 110

6.2 Approximation for the attenuation
function of dissymmetrical filters . . 112

6.3 The design procedure . . . . . . . 116

6.4 Some more study of the high order
image impedance me and an . . . . 121

CHAPTER VII. CONCLUSIONS AND FURTHER PROBLEMS . 127
BIBLIOGRAPHY. ' o o o o o o o o o 1 29

APPENDIX. EVALUATION OF ATTENUATION FUNCTION
BY DIGITAL COMPUTER . . . o . . . 134

iv

Chapter I
INTRODUCTION

Although techniques for electrical filter design
are well established, there is still improvement that can
be accomplished in both the image parameter and the inser-
tion parameter methods. In the insertion method, once the
characteristic function is obtained, an exact realization
is available. However if the loss requirement is not
taken as one of a special kind (as is usually done) the
calculation of the characteristic function requires some
approximations. Such approximations are discussed in the
literature [FR1], [EU 2]. Even these approximations can
effectively be done by reducing the problem to the use of
the image transfer function as in the case of a reference
filter [DA 1] or the method described by Fischer [FIS 1].
This, of course, indicates one phase of usefulness of the
image parameter method. In general, it canvbe said that
the image parameter method of filter design is well estab-
lished. In many cases the filter so designed is suffi-
cient for the particular purpose which lead to the design
of the filter. However, certain considerations in terms
of improving this design method may yield a more econom-
ical filter, i.e., a filter with fewer elements. Such

1

considerations can be found elsewhere [BE 2], [TO 1],

[F18 1,2]. Even though the design becomes more involved,
still the simplicity in the calculation of the filter ele-
ment values remains unaltered. However in the insertion
loss parameter method calculation of the element values

is a major problem. Therefore, the image parameter method,
due to some of the simplicities in the design, is still
widely in use.

In this thesis some of the improvements suggested
for the image parameter method [TO 1], which cannot be
used directly for frequency unsymmetric band—pass filters
are considered. A complete study of elementary sections
for this type of band pass filters is given. A technique
for realizing frequency unsymmetric band pass filters
based on the image parameter method is described. Fur-
ther, the prOperties of certain terminating sections are
investigated. A general deve10pment of the derivation of
terminating sections is described. In this derivation
(there is no limitation on the complexity of the terminating
sections as there is in methods previously given by the
other authors [BO 1], [RE 1], [TO 1].

The method of design described in this thesis also
contains the design of crystal ladder filters [SK 1]. In
fact since the branches of the elementary sections are
identical with the electrical circuit representation of

a quartz crystal, a quartz crystal symbol is used in the

branches of these sections. However, crystal filters
require additional conditions on their element values,
therefore the method described in this thesis may not
always lead to a filter whose branches may be replaced
by crystals. This problem is not discussed in this
thesis.

Chapter II

SPECIAL TYPES OF ELEMENTARY SECTIONS FOR THE
FREQUENCY UNSYMMETRIC BAND-PASS FILTERS

2.1 . Introduction.

The image attenuation function of band—pass fil-
ters which can be obtained by a real* frequency transfor—
mation from a low—pass filter attenuation function, has a
geometric symmetry property. Filters of this kind are
generally called frequency symmetric band-pass filters.
The frequency transformation,which is real, reduces the
design of such band-pass filters to the design of low—pass
filters. The low-pass filter can be realized through the
existing several well knowntechniques and the inverse
transformation yields the band4pass filter.

There are band-pass filters whose attenuation
functions cannot be obtained through a real frequency
transformation from the attenuation function of a low—
pass filter. These band—pass filters exhibit non—sym-

metrical attenuation characteristics and therefore they

 

* By the word "real" it is meant that the transformation
function is a positive real function.

4

are, in general, referred to as frequency unsymmetrical
band-pass filters. However, for some special cases, as
Laurent [LA 1] has shown, it is possible to obtain such
characteristics from the low-pass attenuation character-
istics by means of a frequency transformation followed
by a non-constant factor multiplication. This method
yields a band-pass filter section which should be used,
as it is, without any reference as to how it is derived.

In the design of the image parameter filters, the
filters are considered as being composed of cascaded ele-
mentary sections. For this purpose there must be image
impedance matching at the terminal pairs of the cascaded
sections. The elementary sections used in image parameter
filters are, in general, mpderived type sections.
Laurent [LA 1] has described several elementary band—pass
filter sections. One of the elementary sections (called
a zig-zag filter section) is the one that is considered
here in detail.

In this thesis, this elementary section (zig-zag
filter) forms the basis for designing frequency unsym-
metric band-pass filters. The zig-zag filter is a band-
pass ladder network in which the attenuation poles are
created, alternatively in the upper and the lower stOp
bands, by either (1) only series arms or (2) only par-
allel arms or (3) both the series and the parallel arms,

so that the attenuation poles in the upper st0p band are

created by the series arms while those in the lower stOp
band by the parallel arms. The latter type of filter is
used as the ultimate form of the filter designed by the
method developed in this thesis. In general, this type
of filter is frequency unsymmetric.

Economical considerations are also important in
the design of filters. For such reasons it is desirable
to have the least number of inductors and capacitors pos-
sible. For practical reasons, minimum number of inductors
is preferred, For ladder networks, this is achieved if
most of the ladder arms are reactance networks which have
the appearance of the electrical representation of a quartz
crystal.

Watanabe [WA 1] has extended the necessary and
sufficient conditions given by Fujisawa [FU 1] to the
design of frequency unsymmetric band-pass filters based
on the insertion loss method. His method results in a
band_pass filter with minimum capacitors and inductors
without mutual inductance. This network has most of
its ladder branches in the form which could be consid-
ered as the electrical equivalent circuit of a quartz
crystal. Thus, in this case, the zig-zag configuration
appears but partially. In a recent article, using an
insertion loss design technique, Schoeffler [SC 1] has
obtained ladder filters in which all ladder arms are

replaceable by crystals and some capacitors. In this

approach a special form of characteristic function is
produced so that when the synthesis is carried out by
the zero shifting method, a zig-zag type filter is ob-
tained, i.e., most of the arms of the filter are made
of reactance networks which represent a crystal.

An extensive survey [BE 1, CA 1, TO 1, MO 1,
F18 1,2, NO 1, R0 1,2, SA 1, CO 1, BR 1, MA 1, SH 1]
has shown that a complete design procedure of frequency
unsymmetric band-pass filters based on the image para-
meter theory does not exist. The present work is an
attempt to design a frequency unsymmetric band-pass fil-
ter based on the image parameter design technique. Spe-
cial elementary sections, to be used as building blocks
of this filter, will produce a zig—zag filter of the
third type mentioned above. This type of filter has
a minimum number of capacitors and inductors. In certain
cases it is possible to replace some or all ladder arms
by crystals. Therefore, this configuration can also be
utilized in the design of ladder band-pass crystal filters
[SC 1,2]. It has been mentioned above that the elementary
sections used in the design procedure to be described are
of special form. These sections cannot be derived from
other simpler sections as in the case of those derived
from prototype sections by Zobel's m—derivation. For
this reason it is necessary to investigate these elemen-

tary sections separately and establish the necessary

information for the design procedure. The following
sections of this chapter are devoted to the descriptions

of these elementary sections.

2.2 . The elementary sections.

The salient feature of the zig-zag filters to be
considered are that (1) the series arms will produce
poles of the attenuation function only in the upper stOp
band and (2) the parallel arms will produce poles of
the attenuation function only in the lower stOp band.

-The arms of this ladder filter are formed from a react-
ance network which is similar in appearance to that of
the electrical representation of a quartzzcrystal. 0n
the other hand, there are also elementary sections in
which these types of reactances appear only in one of

the components which form the ladder arms. However, these
sections do not form elementary sections for the zig—zag
filter of the third type. All the three types of ele—
mentary sections will however be considered here. Since
reactance network shown in Fig. 2.2.1 resembles closely
the equivalent circuit of a crystal, it will sometimes

be replaced by the crystal symbol. Figure 2.2.2 a, b

and c, represent the three types of elementary sections,
their attenuation curves and image impedance curves, re-

spectively. It is evident from Fig. 2.2.2 that only the

first type or only the second type of section will not
be able to produce the zig-zag filters. This follows
since type E.S.1 has an attenuation pole only in the
lower stOp band, type E.S.2 has an attentuation pole
only in the upper stOp band, while the E.S.Z. type sec-
tion has one attenuation pole in each of the stOp bands.
In this latter section, the upper stOp band attenuation
pole is created by the series arm and the lower stOp
band attenuation pole is created by the parallel arm.
The element values and further prOperties of these sec—
tions will be given later. First it is necessary to give

some general discussions on these elementary sections.

 

.4 _.s.lg|_.

 

FIG. 2.2.1

1O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ZT. Egg Zn1 2T2 :f— Zn2 ZT E22 Zn
F"
% {J 0— fifi’) 0 O
E.S.1 E.S.2 E.S.Z

(a) Elementary sections

/ \ /

 

 

 

 

 

 

E.S.1 E.S.2 E.S.Z
(b) Attenuation Curves

T: :er I: m

 

 

 

— — - — — — - u- — — — - -
- - — — - — - — - — - — -
- — - — — — — — - — —- — u-
— - - — — — — - - — — - -
1
— ~— -— — - - — _ — - nu. —. —

 

 

 

(0) Image Impedance Curves

FIG. 2.2.2

11
2.3 . General discussions of the elementary sections.

In Fig. 2.3.1 the E.S.Z. section is shown in
some detail. As is shown later, once the cut—off fre—
quencies, the poles of the attenuation function and the
constant of one of the image impedances are given, the
section E.S.Z is completely determined.

In the design of image parameter band-pass fil—
ters these sections are connected in cascade as shown in
Fig. 2.3.2 for n = 3. There exists, of course, image
impedance matching between the interconnected terminal
pairs. Note that in Fig. 2.3.2, the series and par-
allel arms of the filter are indicated by the symbol of
a quartz crystal for convenience. The resulting filter
has the form shown in Fig. 2.3.3.

The other types of elementary sections, i.e.,
E.S.1 and E.S.2 are shown in figures 2.3.4-a and
2.3.4-b. Their attenuation poles are on the lower and
upper stOp bands, respectively. Thus, the attenuation
poles of the filters constructed in cascading the E.S.1
sections only are concentrated all in the lower stOp band
and those of the filters constructed from E.S.2 only,
are concentrated in the upper stop band. Once the cut-
off frequencies, the attenuation poles and the constant
of one of its image impedances are given, the E.S.1

and E.S.2 are completely determined.

(a)

w

21
O1

2}

Wk

WK

12

11*“

:Lw—H— I;

 

Yk

 

 

 

FIG. 2.3.1

 

.132 V(s‘2+ w$)(s2+ cog)

 

ZT =

s (saga)
Zn _ Rn (824-00021)

 

 

s \/(s2+ m§)(32+<ng)

upper st0p band attenuation pole

lower stop band attenuation pole

confluence frequency (resonant frequency
for the series arm and anti-resonant
frequency for the parallel arm)

the cut-off angular frequencies

13

 

 

HELL 4’ L ADP “ 0—40.L e

 

 

 

 

 

 

 

 

 

 

 

 

was _J J was was
Zr —> :1 <-—— zn-—> <— zT —-> [:é-Zn
(do: we: 0002[
n n O ' an o ——0
FIG. 2.3.2
LA NFL
e—D U ___.
‘ 09: J was J—
ZT [:5 [:3 Zn
F501 was
FIG. 2.3.3
0001 : full pole 0002 : half pole
0022 : full pole 0°21 : half pole

 

 

 

(a) FIG. 2.34 (b)

2
Zn: = R (52+WO1)
V(sz+ 712)(32+ cog)

 

 

 

 

 

14

. In forming a filter, these different types of
elementary sections can be used provided the image im-
pedance matching exists at the terminal pairs. One
disadvantage of constructing filters this way is that,
once the constant of one of the image impedances is given,
the constants of all the image impedances of the elemen-
tary sections in the filter will automatically be fixed.
This also means that the element values in these sections
are fixed. Therefore, it might not be possible to re-
place the filter branches by the quartz crystals. In
addition, since the impedance level at the other termi-
nal pair is fixed, in general, there is a necessity to
use an ideal transformer at this terminal pair.

A study of the pole locations of these elementary
sections shows that the section E.S.Z can be considered
as the cascade connection of E.S.1 and E.S.2. Indeed,
this equivalence exists with the addition of an ideal
transformer at one of the terminal pairs of the cascaded
sections. It is necessary to study this equivalence re-
1ation, because it will help in the determination of the
element values of composite filter which is formed by
cascade connected sections. Before starting a discus-
sion on this equivalence relation, in the next section,
some analytical detail of the different types of elemen-
tary sections is given. The information obtained from

these details of the sections is utilized for the design

15

of the band-pass filter having these sections.

2.4 . Analysis of the elementary sections.

Each of the sections is treated separately in
the following subsections of this section.
2.4.1. Type 1. elementary section (E.S.1)

The network structure of E.S.1 is given in

Fig. 2.4.1.

 

a...“ . .

 

 

or
L1 [—
Z; -—) —__ 6‘—
T1 T Cs Z11:
01
(L T 0
FIG. 2.4.1

The various functions of E.S.1 are given as

 

 

 

follows.
1 Cr 1 s + 1
Z = — 1 + - -2——-2—. (2.4.1)
T‘ or U; 9 s +002
2 2 .
z‘ = 61' 1 .1. (S “”019 (2.4.2)
n: B s 2 2 2 2
/1+_é_cr s+w1)(s+w2)
s
2
w
H _ (s + ) (20403)

A /11.H__Cr \/(8: 2+0.) 5%)

16

where
O1, mg : cut—off angular frequencies
on 2 1
01 =
1767
I4101+1+ 9.1:. 1510's
Cs
2
“’2 = 1 1

L1C1 + L109

The element values of this section can be ex—
pressed in terms of the parameters w01, w1, (oz and

RT as follows:

1

From equation (2.4.4):

2 2
21 _ “’2 - “’01
or '-
“’01
(2.4.5)
2 2

Cr w2,_ ”1
6" = 2 2

8 “’1 " “’01

Let Cr/Cs = K1 and let the constant of the image im-

pedances be RT1 and Rn:, then:

K1 =

1
RT, = U; ’\/1 + K1

Therefore:

Thus, when the critical frequencies and RT:

17

 

 

 

1 T:
w2 w2
2 ' 01
C1 = 2 Cs
“’ 01
1
L1 = 2
(”0101
R _ .12.;
It: — 1 + K1 RT:

 

(2.4.6)

(2.4.6-a)

(or Rn!)

are known all the element values are determined from

equation 2.4.6 and equation 2.4.6-a. '

2.4.2. Type 2.

elementary section

(E.S.2)

The network structure is shown in Fig. 2.4.2.

arm—1 t°2

 

 

 

 

x1 '1
CD

 

 

FIG. 2.4.2

18

The various functions of E.S.2 are given as follows:

ZTQ

“2

where

1 1
_ 1 + .—
Cp V q s

0‘60

 

 

 

 

 

 

From equation 2.4.10 we have,

e°le

2 . 2
C2 __ w21 - 1

'd
‘8

(3
ll
8
'3
I
Nfivnéhl

C s
1 +-5£
q
2 2
1 (s + w1
2’ 2
C (s + w )
1 +'CB 2
q
L202
1 1
E202 + 5 5

1/<s2+ wi><s2+ w§>‘<2.4.7>
(82+ wg1)

(2.4.9)

(2.4.10)

(2.4.11)

19

Let cp/cq = K2, then 2 2
00-0)
2 1-

K =
2 2 2
“’21 " “’2

 

Formulas for determining the element values are:

1
R118 = '6‘ ’\/1+K2
P
— _l_. 4/
Op - RT 1+K2

1
C = -——- ‘V 1 + K
q KZRTa 2 (2.4.12)

 

 

 

w2 2
21 - m1
w
1
1
L2 = 2
”102 ]
K2.
. Rug = 'T_I_Rg (2.4.12-a)

Thus, as in the previous case, the element values and

one of the impedance constants, RTa or R are de-

3113’
termined from Eqs. 2.4.12 and 2.4.12-a, if the critical
frequencies and RTa (or Rug) are given.

2.4.3. Type 3. elementary section (E.S.Z)

The network structure is shown in Fig. 2.4.3.

2O

 

 

 

 

 

 

 

 

 

 

é (1)3: fl
.1 I 1
E[
a L .4—

ZT : 1% w“ —— Cb <3: Zn
01'7”

16— 421

FIG. 2.4.3

The various functions for E.S.Z are given as follows:

The series and parallel arms reactances are:

1 .2. .3
series arm: x1 = '5- '-§-—§-

8.8 8 + (1)21

2 2
llel arm x 1 s + ”01
para : 2=C_—_2-_'2_
bs
s + wO

«10: confluent angular frequency (ofi<wb<w2)

The image parameters are:

22222
.l. [1 +_§E:. A/(s + w1)(s +102) (2.4.13)
03 Cb s(32+ mg1)

 

 

 

 

 

ZT =
2 2
z _1_ ___1____ . (S 1’ “’01) (2.4.14)
3 = Cb c
2 m2

H 1 (S + 0) (2.4.15)

_ c 2 2 2 2 '

1*‘6'3 ’\/(s+w1)(s+w2)

Where

21

 

2
(00 = 1 or
L202
2
“)0: 1+1
L101 L1Gb (2 4 16)
‘”61 = __l_.
L101
2
m21 = 1 1 1
5202 E2Ca

At the cut-off angular frequencies m1 and w2, H
becomes infinite, thus at these frequencies the denom—

inator of H approaches zero. Since

2 ' 2
. s + w .
3:1/_i1_ ‘ o’ ,
x1+X2 1// 2 2 2

1 1 2 2 2 2
62(8 W0) '1' 6:6(8 +0901)(S + (”21)

 

 

 

 

then at s2 = -w§,
Ca (4»? +w§)2 ,
h I_; _ - (-w§ + d12 )(-w? + “2 ) (2.4.17ma)
O1 1 21
and at s2 — —w§,
Ca _ _ ('“2 + ”6)2 !_ . (2.4.17-b)
'6; - (‘“2 + ”61)('“§ 1 ”21)

Then from these two equations, 2.4.17-a and -b, we have

2 2
2 2 2 2

_ (-w1 + too) = _ (-w2 + (no)

2 2 2 2 2 2 2

(‘“H 1*“o1)(‘“H + “21) (‘“2 + ”o1)(‘“2 + “21)

 

22

 

 

 

2 2 2 2 2 2
(”’1 + “’0’ = 2: ('“’1 + “01)(““’1 + “’21) .
2 2 2 2 2 2
(““2 + “’o) (" 2 + “’01’("“’2 1' “’21)

Since only real frequencies are to be considered, in this

expression the negative sign is used which gives

 

 

 

 

 

(2.14.18)
("“2 1 “’21’(“’2 + “’21)
e2 = “’1 + “’2 ("“2 + “21“" 2 + “’21)
o
2 2 2
1 + ("2+“01H‘1 “’21)
(«>3 1... 31>1-w§+ «531)

Substituting this expression into Eq. 2.4.17, we obtain:

2 (2.4.19)
Ca (cog -w2)

Cb - [2/(‘1’2"*’2'“’o1’(“’21"“’2”+ V(”2‘”21’(w21'2’]2

 

 

 

The numerical values of Ca and ob are determined after
the constant RT or R1‘ of the impedances in Eq. 2.4.13
or Eq. 2.4.14 is given. [Note that when one of the above
constants is given the value of the other is fixed auto-

matically. The values of the capacitors Ca and "Cb are

where: (2.4.20)

Ca = T\/ Cb = 'RE 1 + K
63V“

23

Other formulas necessary for the determination of the
element values are developed in the following.

From Eq. 2.4.17:

 

 

2
(402 + <43)
K = 2 2 2 2
(“’1 '“’01’(“’21 " “’1’
2 1/ 2 2 2 2’ 2]
“’0 = i “”1 - “’01)(“’21 -“’1) + “’1

where, since wo>w1, the positive sign is to be used.

From Eq. 2.4.16

 

 

“’2 = “’2“1 + g-
and, then, g I
C1 “2"“21 1/K(“’2-“’21)(“’21-“’2) + (“’2-“21)
2; = “’21 = “’21

Thus C1 is determined in terms of Cb and hence L1

is also found as

 

From the relation given in Eq. 2.4.16

2 2 Cg
a
then
C 1 w2
2 = __ 2
ca( 1 -1)

cal

24

which determines also L2 by the relation I

Further, we have the relation

= .1. ____1__ = K (2.4.21)
R" Cb W' 1" K RT °

In the above analysis, the factor K1, K2 or K appears
for each different section. ,These factors are used later

as the characterizing factor for these sections.

2.5 . H-functions and some basic sections of band-pass

filters.

The H-functions of the sections E.S.1 and
E.S.2 have similar frequencv dependent parts as the
Hefunctions of the sections shown in Figs. 2.5.1 and
2.5.2.
’ For E.S.1, the expression for the H—function is:

k

    

j
2 2 2
(32.. 113) (w?- “’01) (w - .2)

 

(“’3‘ “’51) (“’2’ w?)

The H—function of the section in Fig. 2.5.1 is

31/” ":::\/-(--————"J ”)2) .- (2.5.1)
(w2 -'w2

 

25

Therefore, if one performs an m-derivation Operation
on this section (series m-derivation) a section which
has similar structure and Héfunction to the E.S.1
will be obtained where for the m—parameter the expres-

sion used is

2 2
2__“_’_g“’1"“’01 .
m _ ”1 2

”2‘ ”01

The section in Fig. 2.5.1 is called the basic section
for the band-pass filters.
For the E.S.2 section, the H-function is

 

(s + w 2)

H = 1/11+ K2 lM/(s:1+ w2)

 

 

 

1 2 I
wa-é> <&-%>

(“31' m?) (“g' “2) .

 

in
II

Similarly, from the section shown in Fig. 2.5.2, after
the application of Zobel m-derivation, a structure
having the same circuit configuration and H—function

as that of the section E.S.2 can be obtained. The
H—function of E.S.2 is

 

((02— w?)
(.2—.@

26

 

 

 

 

 

 

 

 

rs { IL 4*

’8 T 2"

¢ 4
FIG. 2.5.1

T 28

L . 43
FIG. 2.5.2

2.6 . Equivalence of the elementary sections.

Comparing the image impedances of E.S.1, E.S.2

and E.S.Z we notice the following:
1. §n8 has the same expression as ZT1’ except
for the constant.’

2. ZTa has the same expression as ZT’ except

for the constant.

27

3. Zm has the same expression as Zn, except
for the constant.

By adjusting the values of the constant in the image
impedances (this can be done by adjusting the element
values) an image impedance matching can be provided for
the cascaded E.S.1 and E.S.2 sections at their inter-
connected terminal pairs. The resulting section will have
the same image impedances as that of E.S.Z except for
the constants (see Figs. 2.5.3 and 2.5.4). However
this cascaded E.S.1 and E.S.2 section and E.S.Z
have identical HEfunctions. Complete equivalence will
be obtained if an ideal transformer is connected at the

end of the cascaded network shown in Fig. 2.5.5.
From Eq. 2.4.12-a and Eq. 2.4.6—a, in Fig.

2.5.3 with Rug = 11,“, we have
Rn = __E1_RT = 31...?! = Li.
2 1+K1 1 1+K1 2 1+K11+K2 T8°

For the network in Fig. 2.5.4, from Eq. 2.4.21 we

have
__ .__.I.<__
R“: " 1 + K RT .
If the constant factors Rm and Rn: are equal, then

the constant factors on the other side of the networks,

i.e., RTa and REP will be identical when an ideal

transformer is connected at the terminals of one of the

28

o
0

 

 

 

 

 

 

 

 

 

 

I F l I _1—
ZT ::> —L' [:3 <::: Z
a fix
. T T 2,
FIG. 2.5.3
A cascade of E.S.1 and E.S.2
o 10'. .,
ZT :E Zn
Qi— 0
FIG. 2.5.4
E.S.Z
o_____%[]¥ n:1
ZT. = ZT :1; Zn = Zn.
,4. .75
FIG. 2.5.5

Equivalent network

29

equivalent networks. Let this transformer be connected
at the terminal pairs of E.S.Z as shown in Fig. 2.5.5.

Then the transformer ratio is given by

 

 

 

2
2 2 2 2

2 __ 311: + ”21"“2 + “’1'“’01 .
n - RTa 002 to: w: on:
21 " 1 2 ‘ 01

It remains to investigate whether the networks in Figs.
2.5.3 and 2.5.5 have equivalent Hefunctions. The
H-function of the network in Fig. 2.5.3 is a composite
Hefunction, i.e., it is related to the H-functionsof
both E.S.1 and E.S.2. Let the H+function of E.S.1
be H1 and that of E.S.2 be H2. Then the H;function
of the composite network in Fig. 2.5.3 is

 

H—H1+H2 .
1 4 H1H2
Since
1
H1 =
1 + K1
and
1 82 + w2
H2: r—— 2 g
1+K2 S+w1
then

(82 + w?) (s2 + mg)

V 1 + K1 + W’1 + K2
H = _
1 __ 2 2 2 2
[1 + V(K1+ 1)(K2+11)] [We + 031MB + 1.12)]

 

 

 

 

 

3O

82Hl(21/1+Km12+111+K1w2)Z(1/1+K1+1+K2)]°
1 + ’V11+K1)(1+K2)
m . «r175

 

 

 

l'¢(s 2 + w1H)(s + w2)

From this final expression and Eq. 2.4.15 we have

w?1/1+K1+w§1/—1__+K—2
W+ W+K2

 

 

I

If 2 2 2 2
1(“1 “ “01)(“21 ' “1)

 

 

 

2 2 2 2 2 2
_ “1 + £02 (“2 ‘ “01)(“21 ' “g1 = mg .
5* 2 2 2 2 ’
1 (“1 ' ”o1)(“21 ' “1)
+

 

2 2 2
(“2 ' “01)(“21 ‘ “2)

On the other hand, since

 

1 + \1(1 + K1)(1 + K2)

«[1 + K;+1/1 + K2

 

 

K1K2
[\l1-1—K1 + V1+K2T
2
(«é-w?)

 

 

 

[:qug‘ ”g1’(“g1w 2’ + IV(“1 ‘ ”01)(“21w $.12

then, from Eq. 2.4.19 it can be seen that this expres-
sion is equal to K. Thus,

31

 

1+ 171+ K1)(1+ K2)

’V1 + K1 + V1 + K2

 

= 1 + K

 

and finally
(82 + mg)

m [v.2 + gm]

H z

 

 

This last expression is exactly the expression of the
HAfunction of the E.S.Z section. Therefore the equiv-
alence of the networks in Fig. 2.5.3 and 2.5.5 is
established.

The last part of the above discussion can be

simplified by the theorem given in the following.

THEOREM:
The H-function of a lossless 2-port network
will not change if the network is augmented by
an ideal transformer connected to one (or both)
of its ports.

PROOF:
The proof is trivial since by definition

so
H = --
Zoc

and the cpen and short circuit impedances Zoc
and Zsc at one of the terminal pairs of this
2-port are either unaltered or both multiplied by

the same turns ratio when ideal transformer is added.

32

Note: The impedance constants on both ports can be
multiplied by a constant factor without changing
the K's of the section as can be seen from the
relation

R = ._E_ .
n 1+K RT
This means that only the element values of the

section are changed.
2.7 . Further equivalence characteristics.

The H-function of the E.S.Z section has the

following form
2 2
1 (s + wo)

1+K NE52+ w$)(s2+ mg)

 

 

 

Note that if K, as well as the critical frequencies,
is not changed the H—function remains unaltered.

Now consider a network consisting of cascade con-
nected sections of Fig. 2.5.3. Let each of these ele-
mentary sections be replaced by its equivalent network,
given in Fig. 2.5.5. The resulting network is shown in
Fig. 2.7.1. The structure to the left of s1 is an
E.S.Z, the structure between s1 and 82, also that
between s2 and s3, can be replaced by E.S.Z's ac—

cording to the above theorem thereby eliminating the

33

 

 

 

 

1
$.19}
1
W}
m

 

 

 

 

 

 

 

II ii III :.
%—. Jr 1 -4 —.—.—- ‘
I
l l

 

 

 

FIG. 2.7.1

 

 

 

 

 

 

 

d——, 1
I' = I II' III' E?
fl———‘ 1

 

 

 

 

 

 

 

 

 

FIG. 2.7.2

transformers. The resulting network will be in the form
as in Fig. 2.7.2, having an ideal transformer. This
transformer has a different transformer ratio as compared
to that last transformer in Fig. 2.7.1, but the H—func—
tions of these networks are identical.

Thus, it can be concluded that any filter consisting
of cascade connected pairs of E.S.1 and E.S.2 can be
replaced by a filter consisting of E.S.Z sectiOns ter-
minated on an ideal transformer at one of its terminal
pairs. Since the H-function will be the same, the calcu-

.1ation of image attenuation and the image phase functions

./-

34

of the complete filter might be easier from one of these

networks as compared to the other.
2.8 . Pole distributions and structure configurations.

In this section possible locations of the atten-
'uation poles of a band-pass filter containing E.S.1,
'E.S.2 or E.S.Z are considered. The following rules
may be observed:

1. The network consisting only of E.S.1, E.S.2
and E.S.Z will be obtained if there are at
most two half poles and the rest of the poles
are full poles.

2. Network consisting only of E.S.1 will be
obtained if besides the condition 1, all
poles are on the lower step band.

3. Network consisting only of E.S.2 will be
obtained if besides the condition 1, all the
poles are on the upper stop band.

4. Network consisting only of E.S.Z will be
obtained if besides the condition 1, there
are equal numbers of poles in the upper as
well as the lower stop band. Full poles are
counted as two.

Figure 2.8.1 is an example of the application of these

rules with six poles.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l I
__ __.__ .J— __ __ ____.,
1:1 1:3 __ [:1 g] E;
0* T o c— T V4)
muggy/1%“ m 1/////////1/1 / // // m //////#1///j//1
l L I'L_ II -
I I l l ‘1 1 ‘5 0“11:11“ 1 1 101—‘0
[:1 El 1;; E 1:1 :1
,d ‘1_‘ ’d n J
Wu m/dzum // 1: Wm; {W
J l I In
”L WDF 1D 1‘ ° 1111 1% °
E :m: 5T3 E? Q
P ,6 c o
WWII/III Mimi/gt“; WWW Willi/@471“;
W JULl/HglLl/Jl
II I 1, IN
° 1 I I1:1| l 1”“)
If] [:3
x half pole FIG. 2.8.1 ® full pole

Possibilities of network configurations

with 6 attenuation poles

36

2.9 . The impedances.

The filters composed only of elementary sections
that have been discussed in detail previously, i.e.,
E.S.1, E.S.2 and E.S.Z have terminal image impedances
of the following forms

 

 

 

 

11.. (.2- 0%.)
Z3040) = E)—
VmZ- «$1.13- .2)
and _
RT VmZ— 2mg— .2)
Z (w) = -——
T ‘” < 2 - 2)
“’21
where
s = 3w
w1’ w2 : angular cut-off frequencies
RT and R1, : constants
m21’ “b1 : critical frequencies that cor-

respond to the attenuation poles.

Using the frequency transformation discussed in Chapter III,
section 3.5, they will have the following form

3-1—1 (ft—710,)
E;-j:i 34—Fi- U1 — izi?

 

 

 

P,
a

(2.9.1)

 

:12 617121 Ti—fi 1/1 471.2
”m 1521—1 3.71 fi-figi

w = Jw1w , .ffi. is the transformed frequency .

 

37

The plot in Fig. 2.9.1 represents these impedances.

They are normalized with respect to the factors

Rn: V 52 '1 for Zn and

a; 3‘7 E01 (2 9 2)
122 3—17.21 for 2T
mm E -1

These two factors still leave RT and Rn free to be
selected. In the design of a filter with E.S.1, E.S.2
and E.S.Z as elementary sections, RT or Rn' is one

of the numbers that should be given before the design is
carried out. It determines the values of the elements.
From the plot of the impedance curves in Fig. 2.9.1 it
is seen that there is only a narrow effective pass band
range, i.e., the range where the impedances are relatively
less fluctuating is a small portion of the entire pass
band. Thus if a wider effective pass band is needed, a
terminating section (T.S.) with higher order image
impedance will be necessary. Note that the extremum point
of these curves are close to one of the cut-off frequen-
cies. This is mainly due to the fact that the form of
these curves is controlled by either .ino1 or :?3:21

and the extremum points in the pass band are closer to
these frequencies which are in the block band. The in—
vestigation of these curves then suggests that if the

image impedance is a function of at least two control

38

 

1O

 
 
   

TLZ(normalized impedance)

'1, II, III represent Z

 

 

 

 

 

 

Zn‘fin 1
51—.
fig
“ o1 ‘ "1'5
1
L. 1
1.0—- _
‘foO1’ pole on
- Power stOp band
_ «Fl—21: pole on
upper stOp band
05——
- 19 2, 3, 4 represent Zn
220%EHJGL211
’ J;ifi= 2
OJ—////’/,,rf””’"flflflfl— fi_\\\\\\\
1 I J 1 l J l L | ‘ EGO 209.1

7

—1 -O.8 —0.2 o 0.2 0.6 +1 JAL

39

frequencies and one of them lies in the lower block band
and the other in the upper block band, then the extremum
point can be pulled towards the center of the pass band
or perhaps the impedance curves will now contain two
maximums or minimums which are located close to the cut-
off frequencies. Indeed, if this is the case then it is
possible to improve the matching requirements by arranging
the location of the controlling frequencies so that the
impedance is relatively less fluctuating in the pass range,
i.e., so that there is a wider effective pass band. Two
types of sections having image impedances with control
frequencies in the upper stop band and lower st0p band,
i.e., of higher order, can be readily obtained from the
E.S.Z. These are sections obtained by series and shunt
m-derivations of E.S.Z. They will be used as the ter-
minating sections. These sections will be studied in
detail in the rest of this section. In order to simplify
the investigation of their image impedances some frequen-
cy transformation will be used.
2.9.1. T.S. made of shunt m-derived E.S.Z.

The structure and the image parameters of this
section are shown in Fig. 2.9.2. In detail the m-

derived impedance is

__1_ W1 («12.01511 Wfl-ufixwéfif
m _ Ca (1-m2)+K w (9?;w5p1)(w2p1’w2)

 

ZT

(2.9.3)

4O

 

 

 

 

 

 

 

 

 

mx:

J 1

1 1
Q_ 0

1 .L
z m x2 C:j'§% z
Tm '12? T n
C} r 0‘

FIG. 2.9.2

Shunt m—derived — E.S.Z section

 

 

 

 

2 2
Ru (9 + “01)
25 = __ 2 2 2 2
s
.V(8 +w0p)(s + wzp)
Z = RTm (s 2+w01) 1/(82 + (:111)(s2 +002)
Tm s (s +1.12 )(s+w2)
S = 3m
-1<1n< 1
wOp’ “2p attenuation poles
2 2
(s + w )
H = O

1/(s2+ 0%)(82-1- mg)

11 : constant

41

and the H—function (H = tanh PI) is

 

 

< 2 2)
H - —-—I-I-l——- (no-w (209.4)
111 + K Wong- (132)03- (1)2)
At the frequencies “’Op1 and w2p1, H200) = 1.

Therefore from the expression for H it follows that

 

 

2 2 2 2 2 2
“’0‘ “021 _ __ (“091+ “WW2" “’0 1)

2 2 - ( 2 2)( 2 )
“’0" c”2p1 “’2p1" “’1 “2p1‘ “’2

01'

 

(«o2 + 002) (1.12- “2 )
Op1 1 2 0p1

2)

(.12

 

1.12 + 002 (1.12 - (1)2)(1112 -
Op1 2p1 2 1 1 2 1

   
   

 

  

 

(2.9.5)

0n the other hand, at the cut-off frequencies (.11 and

<02, H(w) is infinite. Then by a similar discussion

to that in section 2.4.3 we have

 

 

Ca (mg - «€12
3. ” (.3 - «vim? - .3)
2 (2.9.6)
.22 _ (qg - mg)
Cb - (1.122 - 1.131)“); -- 103)

(where 9.92-11) ..
C _

0‘

42

Equating these last two equations the following is obtained.

2 2 2 2
(”0" 2)(“’1" ”011)

 

 

 

 

 

2 2 2 ’2 2 2
.2 _ 9.2.1.1.”;“0‘ “1)(“2‘ ”2L. (2.9.7)
21 fl 2 2 2 2
1 _ (”0" c“’2)(“’1" “01)
( 2 2 2 2
“0" “1)(“2‘ “’01)
We also have
RT _ 1 V1; + K __ ET
m — '5; (1-m‘) ¥IK — (1-m‘) + K
(2.9.8)
where RT = G: 1 + K .

2.9.2. T.S. constructed of series m—derived E.S.Z.

All the formulas from Eq. 2.9.4 to Eq. 2.9.7
above by replacing 0001 by (.121, w0p1 by ”Op? (021”
by w2p2 apply also for this T.S. elementary section.
The structure and the image parameters are shown in

Fig.. 2.9.3. The image impedance in detail has the form

 

 

 

Zn = is (“’2’ “322””322' “2) (2.9.9)
m “’ (.31- .2>1f<.2- .§><..g- .2)
where Rum = Rn[(1‘m:) +3K] and
(2.9.10)

1

__1___.___.
Err-01m

 

43

 

 

 

 

1’

- -- ,,,._.___._______._,._. ._- _. __.._.____0

FIG. 2.9.3

Series m—derived E.S.Z section

 

ET— 4(82+ w§)(82+ (.13)
s

2 2
(s + w21)

 

 

2 2 2 2
Rim (8 + wOD)(s + 002p)
s

(82+ 10:1) «82+- w§)(s2+ 002)

2

(82+ 00(2))

(s2+ w§)(s‘2+ mg)

 

 

Note that for series and shunt m-derived cases

000p > 0001 and

“’21)< m21 .

 

 

44

2.9.3. The frequency transformation.

For the investigation of the impedance curves of
ZTm and an, it is convenient to use a frequency trans-
formation. First, let the expressions for the impedances

be rewritten for ready reference.

‘ (.2_ 2 >
ZTIn RTm% (002 2 (2)1 2 Ww‘g- wfiflwg- 2)
" w0p1)(w2p1" w )

 

 

2 2 2 2
Zn = Rn _1_ (“2112‘ w )(w " ”0p2)

m m w (.31- .2) 1/(.2- .$)<.§- .2)

 

(”01 < m0p< “’1

“’2‘ “’2p< “’21

In these expressions the angular frequencies “01 and
(D21 do not correspond to the attenuation poles for the
corresponding sections. The frequency transformation
‘used here for ZTm will transform (001 to —a5 and
(#21 to +¢> for an. Therefore it will have the form
‘with the following conditions:

(c1)- w1)(w2- 6) + (w- w2)(w1-8)

‘2‘ ‘ «112-91W“ (2.9...)

‘Where 6 is mo, or' w21 depending whether ZTm or

an is being investigated.

 

45

For m = w1 we have JAL = -1

(n =<02 we have JAL = +1
2w1w2 - y(w1+ w2)

w = O we have JAL,= J“L(0)
Y(°°2- 91)

 

(w1+.w2) - 2Y
(W2- ”1)

w = on we have f\. = fl(oo)

 

If 6 = w01 (case ZTm) ,
n(o)>n_(ao) with n.(«»)> 1 .
If a = (.21, n(m)< 11(0) with n(o)< —1.

Since the frequency transformation is a bilinear trans-

formation, so is the inverse transformation

 

 

 

 

fih 2w1w2
031+ (02 — 6
_ J“L + L_ mg — w1 (2 9 13)
w " 5 "001+ng ° '
fl- J- “’2 - “’1 ]

Fig: 2.9.4 shows the transformation from the w-axis to
the JflL—axis. Figs. 2.9.5 and 2.9.6 are the plots

of’the ZTm and an normalized to RTm and Ram

 

 

respectively.
MLEEEW_~ 11131111) fit/.1 ggfip/pjafigj
1’01 Op “1 ‘12 m2p 1’21
lLL/ILJ'II' I£l1’//’I//l ' ’/| ll/IIL/ll/l I Irjry/Il/H/LA
-1 +r +0
FIG. 2.9.4

Frequency transformation

 

46

m%.m :N .mZU.

 

 

 

 

dIW.Q _ _ m _ 0— q _ _

 

 

    

.00w\\um\« v;
4.3 - as» A..R-us\\(A mans»

me.m\.~. \MQFJWN a Qhfidud “ news
NNRKL @ NQhfix ® .uflm\\m§% 9wafix u 0&3 G

0.3 I 2&3Xi08 14.3%:

 

 

 

      

5.5%

n \
SLN N

 

 

 

 

 

 

47

Ad

«Rwfi .N Rubi

 

 

 

a . _ M.Q_ a _ _

q Dd _ q

 

 

G

\mm.mw R53 93:3
kmuémmv +mh§® 9+5?

\N“

 

 

   

u
33

II

 

.l ow.

 

 

ow.

 

 

/

4s

 

 

 

_ n -
a
a
MA 4
_ J
_
o_ _ . _
_
v.0- .
_ a
J
VII

 

®
0\.

 

 

$3 - 3X3...3\.¢e .36

N3 I Rmd
A NEVA dawns Macy

 

 

Sm saw
\mkéw

macs
N
® 8%.»24 e35

.00w\vcmu N 0w

 

 

 

 

 

49

The curves in Figs. 2.9.5 and 2.9.6 represent
the variation of an and ZTm in the pass-band, re-
spectively. From these figures one can observe that the
curves are either flat or have almost a Chebyshev char-
acter over a wider range of the pass band. The latter
type is preferable. Note that these characteristics are
considerably improved as compared to those of elementary
sections. Among these impedance curves one should select
the "best" curve. Since the best image impedance is the
one which causes smallest insertion loss in a given ef-
fective pass—band, then this, of course, implies that
the image impedance must have Chebyshev behavior in this
effective pass-band [CA 1]. In order to obtain best
image impedance one has to locate the critical frequen—
cies of this image impedance in the block bands properly.
To study the effect of the location of the critical fre-
quencies on the form of the image impedance curves which
are already indicated in Riga. 2.9.5 and 2.9.6, a new
frequency scale, J“L, is used. On this .fW.—axis,
either w01 or w21 is transformed to infinity. The
cut-off angular frequencies w1 and ”2 correspond to
11- On the other hand, the critical frequencies .fW_Op
and .fl_2p are chosen so that they satisfy the relation

.IW_Op +_I\_2p = O .
This relation will provide almost a symmetrical character

for the image impedance in the pass band. On the OJ-axis,

 

50

_Fl.= 0 corresponds to an angular frequency which is
located in the vicinity of “a = £(w1 + mg), the arith-
metic means angular frequency. Thus, on the co-axis
these impedance curves are also relatively symmetrical

with respect to w It can be observed that the form

a'
of these curves is almost independent of the location of
w01 or w21, which is the critical frequency of the
impedance. Based on these observations one can locate
”Op and pr’ perhaps by cut and try method on the

computer, to obtain the desired image impedance char-

acteristics.

 

 

Chapter III

THE FREQUENCY TRANSFORMATION
AND THE TEMPLATE METHOD

3.1 . Introduction.

In the design of ideal filters by the image para-
meter theory, as well as by the insertion loss theory,
one of the problems is the determination of the number
and the locations of the attenuation poles. One method
useful in practical filter design involves use of the
template. This method was developed by several authors,
[RU 1, LA 1, SA 2, F0 1], each differing slightly from
the others. The one discussed and used here is the one
set forth by Rumpelt [RU 1], which can be applied to
the low-pass, band-pass or high-pass filters.

In this chapter some techniques of frequency
transformations are considered. The following sections
are devoted to the development of these transformation

formulas and their usage.

3.2 . Template for the low-pass filters.

For this type of filters the normalized frequency

51

 

52

with respect to w1 is
(\ = £L
m,
where w1 is the cut—off angular frequency of the low-

pass filter. The frequency transformation used is

2
JfiL

Y = i 111 (T) (3.2.1)
JAL > 1 .

Since the Hsfunction of the prototype section of the

low-pass filters is given as

 

H(jw) = w (3.2.2)
.2 - .3
then
H(Jl) = n , JIM. (3.2.3)

 

 

V112.»

From Eqs. 3.2.1 and 3.2.3? the following relation is
obtained
H(Y) = eY 0 (30204)

The image attenuation function is given as

 

= 1 H
AI ln ’1 _ H

 

Therefore, substituting Eq. 3.2.4 into this equation

 

the following results (3.2.5)
‘ Y
A = MILLS. = 1n cothl I
I 1 - eY I?

53

On the other hand a simple m-derived section of the

prototype has an Hsfunction of the form'

H(jw) = mm . (3.2.6)

At the attenuation pole w21 this Héfunction has the

value of unity. Therefore we obtain

 

 

 

 

“g1 ‘ “1
m =- (30207)
“21
or, letting JAL21 = w21 ,
“1
2
m = 1/‘(1’21 - 1
fl-21
= e-YQ: o (3.208)?

Thus, in general, for an m-derived section

H(Y) = eY'YQ‘

and

Y-Y21

AI(Y-721) = ln coth __§__

'(3.2.9)

 

 

The total attenuation of a low-pass filter is
Alt = E AI(Y-Yi) (3.2.10)

where Y1 is the attenuation pole.
Thus, the total attenuation curves can be obtained by

plotting the curves represented by Eq. 3.2.9 along

54

the y-axis such that the peaks of these curves occur

at the locations of the attenuation poles on the Yaaxis.
The value of the attenuation at any frequency can be ob-
tained by adding the ordinates of these curves at that
frequency. Repeating this sum for every frequency will
then yield the attenuation curve concerned. In Figs.
3.2.1 and 3.2.2 the template and the total attenuation

curves are shown respectively.

 

 

 

o ‘ >Y
FIG. 3.2.1

Template curve

 

55

 

 

 

 

 

Yas
FIG. 3.2.2

Total attenuation

 

1/////////g////1////J//It”./ (]
m
.flpa

Ull/ ///////n/////////l///1/1m

Yea Ya: °°’Y

 

FIG. 3.2.3

Frequency transformation

3.3 . Template for the band-pass filters.

There are two types of band-pass attenuation

 

56

curves. The first is the frequency symmetrical atten—
uation curve and the second one is that of the frequency
unsymmetrical attenuation curve. For the first type of
attenuation curves the deve10pment and the use of the
template method can be reduced to that of the low—pass
filter method. The deve10pment of the second type which
is more general, will be discussed separately.
3.3.1. Frequency symmetrical band-pass filter.

let m1, ”2 be the cut-off frequencies of the
band-pass filter and wm = ([5753. the geometric mean
of these frequencies. The frequency transformation used
for the frequency symmetric band pass filter is

«2- “2.

fl: m . ..
w(w2 _ w1) (3 2 11)

 

Fig. 3.3.1 shows how the w—axis is transformed into
the .fl_-axis by this transformation. It can be observed
from this figure that the .fl.-scale is symmetrical with
respect to .13. = O, which corresponds to mm in the
w-scale. Since the curve of the attenuation is also
symmetrical with respect to a vertical axis passing
through the zero value of the .fW_-axis, if there exists
an attenuation at ma in the upper stOp band, an identi-
cal attenuation will be obtained on the lower stOp band
corresponding to the mirror image of ad. Therefore the
design of the band-pass filter is reduced to the design of
a low—pass filter. Once this lowhpass design is obtained,

57

the desired band-pass filter is then obtained by substi-
tuting Eq. 3.2.11 into the element values formulas.

 

 

£9,218,5',,9:99P, £29119. h 329% 8/1911 W19, .
(:01 um ‘92
Ibo/”3);, g/Egp/ ’1/Jgnd| L 4,12%?!) /§;/9P/ lbfi'pfi
_1 O 1 51.
FIG. 3.3.1

Frequency transformation

[ PASS-B 1/'///////S;D/O/?7IBI’/II ///// I/ / / A
O 1 “7.0.

FIG. 3.3.2

The range which is considered

as low-pass range on the 57L scale

3.3.2. Frequency unsymmetric band-pass filters.
The several methods of frequency transformations

considered here for the frequency unsymmetric filters

58

differ slightly from that of symmetrical case. The
reason it is necessary to have various modifications
is that one can then make a choice as to which form is
more appropriate to apply to a certain problem in order
to treat it in a less complicated manner. These trans-
formations are not only useful in treating the attenuation
characteristic but also usefulfor treating the image imy
pedances. In the pass—band, the consideration of the im-
pedances is particularly essential.

In the following discussions it is necessary to
consider the H—functions of the band-pass filters. They

are of the following forms

 

(s2 + w

H = K
(82 + ”2)

(3.2.12-a)

or

/

(82
K'V( 2 “2) (302012“b)
s +

 

1::
ll

Only these two forms are considered since for all other
forms of the H—functions the treatment can be reduced

to the treatment of the above form of the H-functions.
Note that the frequency dependence part of these Héfunc-
tions is similar to that of the HAfunctions of the basic
sections. Three types of frequency transformations are
considered here. The last two types are frequently used in

the current publications. The Héfunctions are in Eqs.

59

3.2.12-a and -b and the composition of these two, i.e.,

 

 

 

2 2
H 2 1 t (s + “0) (3.2.13)'
V1 + K" 1((sz+ w$)(82+ 00:)
where S = 3” 9
2
m0 : confluent frequency ,

w1, mg : cut-off frequencies ,

K": a constant.

In the next three sections the various frequency trans-
formations that can be used to study the frequency unsym-
metrical filter characteristics are deveIOped.
3.3.2.1. The first method of transformation.

In this method one directly transforms the fre-
quency by not going first through the normalization.
This transformation is only useful for the attenuation

function. The transformation used is

(0)2 - 002)

Y = % 1n -———————2 a (302014)
(‘0 "' 2)

Y0, = 1n :2 (3.2.15)

60

Thus here one does not utilize the .JFL-scale but goes

straight from the (n-scale to the y-scale.

 

 

 

J/l/lx/‘x/A///l//l L,r///// ////4 x w
1 I
/ \\ /
////'////>1 v’///////////V_ AY
-a> O +m 7
Ya)
FIG. 3. 3.3

Frequency axis transformation

The basic section* has poles either at infinity or at

the zero frequency. The thunctions of the sections

for the frequency unsymmetric filters have the same forms

as those in Eqs. 3.2.127a and -b. The first one has a
pole at infinity and the second one has a pole at the
origin. Since the elementary sections of the filters

to be discussed in this thesis have the same HEfunctions ex-
cept fbraconstant factor, the basis for the template

method here will be the H-functions mentioned above.

Thus for the basic sections, the constant of the

 

* The basic section is discussed in Chapter II, section 2.5.

 

61
H4functions are
K = 1 and
K' = 231 . (3.2.16)
“2

Therefore from Eqs. 3.2.14, 3.2.15, 3.2.12—a and -b

we have
H = e+Y or
(3.2.17)
H a e-Yme-Y = e—(Y+1g,)
The attenuation is given by
AI = ln cothlgl or
(3.2.18)
A :-

 

ln coth I“?

l

13.3.2.2. The second method of transformation.

I

This method [BE 2, TE 1] is also developed without
first making a normalization. Thus it is also only useful
for the attenuation consideration. It differs from the
first method in that the transformed frequency is sym-

metric with respect to its origin. The transformation used

is
2 2
Y = % ln w ’ ('01
w? a?
' 2
(3.2.19)
Ya: = iln 22.

 

62

and mm = to 1m 2
0 ——> --Y.»
+00 —-—> W...
w1 ---> .4»
w2 -——-—> .y»
110),, > 0 .

Thus for the basic section we have

 

H = eY'MQ or
(3.2.20)
H e_(Y+Ym) .
Therefore the attenuation function is
AI = ln coth rxfigknl
or (3.2.21)
AI = ln coth Ixflml .
2
Imag. lower upper Imaginary
13‘1- 02992) 2g . ‘ 2239?. Pan freq ency

 

 
       

1

  

w
2 yam

   

FIG. 3.3.4

Frequency transformation
Y - axis

 

63

The above formula can be extended to the cases when the
sections have finite poles. Indeed, suppose there are
attenuation poles at 71 or ”Yi’ then the attenuation

function becomes

 

Y-Y’
AI = 1n coth ‘ 2 1' or
(3.2.22)
'AI = ln coth [IJELE °

It is important to note that the template to be used for
this case will depend on the band width.
3.3.2.3. The third method.

In this method, normalization of the frequency
variable is performed first and then the transformation
follows. The frequency normalization considered here is
also useful in treating the image impedances. The nor-
malization is sometimes called the Goth-transformation.
This transformation has been used by many authors, par-
ticularly by Cauer [CA 1].

a. The normalization (First step transformation)

‘”m
2 2 2
flr-aq +1=w2_m1m +0"In
{12-1 Q2+w1w -0):
112 = JL+a , (n2 = (.12 JL+a

 

64

(I. = 602-001

Sometimes an inverted .fW_-scale is used, i.e.,J;I
and this provides some simplicities. Thus we will use
this .77.-scale in a greater part of the application
of this method:

b. The transformation (second step transformation)

 

fl = .1.
J“L
._ _ 1
Y — tlnifil—{j (3.2.23)
1 = ’d > In m2 =
E ‘2 (1)—1 You
—1 =—E “-9 £111.21]. = _Ym
(1 (02

Thus for the basic sections one has

 

 

H = .V(E'+ 1i _f1.+ 1 or
O. - 1 fl— 1
(3.2.24)
H -.-.

1%:11/311H

Then, substituting Eqs. 3.2.23 into Eqs. 3.2.24

results in (3.2.25)
H = ev—Ym or H = e—(Ywm).

65

Hence,

3>
II

I ln coth 1139391

or (3.2.26)
ln coth Iii-219' .

p
ll

When the sections with finite poles exist then an anal-
ogous method can be used. For example, if there are

poles at Y1 and -Yi’ then

A = 1n coth (3.2.27)

 

YiYil
'_—2__ °

As has been mentioned earlier, the normalization can also
be applied to the image impedances. The following sec-

tion is devoted to this matter. The impedances concerned
are those which appeared in the development of the filters

in an earlier discussion in Chapter II.

-—+ —>
AL/////j[///// PASS “1111/”,1)”. Inl-Lfreq
(A)

 

 

 

/////// ///I P

FIG. 3.3.5

First step and second step transformation

66
3.4.. Impedance with normalized frequency.

The image impedances are of the following form

2

2
z“ = Ro1 (w ' ”01)

w 1/(62- .$)(w2- .3)

 

 

(3.4.1)

 

.221 1/(w2- w$)(w2— mg)
“’ («$1 - we)

N
ll

 

where R01 and R21 are constants.

The following derivations are given to show how
the w variables are replaced by the 1:: variables.
Fig. 3.4.1 shows the scales of w, J”L and 3:1 . Also
it shows the locations of the critical frequencies of
these impedances. Notice the locations of the points
J"LO1 and .IW.21. in -1<J=L<1. 'These points are the
reciprocals of the attenuation poles .ffi;o1 and .FI.21.
It turns out that .f\.01 and .IW_21 _are in the vicinity

of the extremum locations of image impedances Z,[ and ZT’

 

 

 

x I L l x

we: w! w", we we:
4 X l L k I n
.. ..a (1 1
1 11.: (12,
J l - I )l‘ l 41‘ I x J E
-O. Q at -11 no, O n“ l 338! G

FIG. 3.4.1

The derivation of the changing of variables from on
variables to the .75. variables is

 

 

m2 _ (a? = (.92 [flw 1-a] = 0.12.2a (n+1)

 

fl—a +0. m (Ii-a) ( 1 +d)
(as - (02 = (.12 _1_-_1-g_ _ AH: = m2 2a(f\-1)
m 1-a 11:0 m (n-a)(1-a) '

On the II -scale:

Zn = 391 V'1-a2 Vfl-a (£0141)

 

 

 

 

 

“’m f“-01-“ 1/n+a‘ 1L0? .. 7
' (3.4.2)
zT = _R_21 f121"]L Q—a If)? -1“
9.. 17142 Iva (xx-n21)
On the fi-scale:
Zn = 391 1012-1 B-fi (ii-7101)
”m 0-13.01 511—1 :11-f—i2
(3.4.3)
-1 __ .
z. = 3.2.1 .2-1 an. M .
”m 3.27121 E+fi (1121-11)

From inspection of the curves of these impedances it is
seen that these are not satisfactorily suited for ter—
minating impedances. Note also, on the .ffiL-scale the
two points 'fF-O1 and .FI.O1 and also .{\_21 and
j2121 are reciprocals of each other. This fact also

68

shows clearly on Fig. 2.10.1 when w21 or (201 ap-
proaches the cut-off frequencies then the minimum or
maximum points of the image impedance curves in the pass-
band also approach the cut-off frequencies. This suggests
that these extremum points are in the vicinity of _f\.01

or .{\_21 in the pass-band on the.]:L-scale.

Chapter IV
TERMINATING SECTIONS
4.1 . Introduction.

The image impedances of an image parameter filter
are functions of frequencies, whereas the filter is usu4
ally terminated in load impedances at both ends which are
purely resistive. Thus to minimize the loss at least in
the pass band, it is necessary to provide a high order
image impedance at the terminal pairs of the filter, so
that the mismatch at the terminal pairs is somewhat cor-
rected. This can'be done by use of so-called terminating
sections [RE 1]. The whole filter then can be considered
as being composed of the intermediate sections and the
terminating sections. The intermediate section consists
of cascade connected elementary sections, which are-
prototype sections corresponding to the type of the fil-
ter, i.e., low-pass, band-pass or high-pass filters.

The terminating sections, on the other hand, are also
elementary sections or a composition of elementary sec-
tions with a higher ordered impedance at one of its ports.
One method by which terminating sections can be generated

is by the procedures of repeated m-derivations as

69

70

described by Zobel [Z0 1]. This kind of terminating
section, although possessing the desired higher ordered
image impedance, is in general rather cumbersome in
structure, i.e., it contains an unnecessarily high number
of elements and hence is not practical. However, as it
will be shown in this thesis, this does not constitute

a major objection since by using a network transformation,
cascaded Zobel sections can be transformed into the known
economical terminating sections. This approach has the
advantage over the other methods [TO 11 that it can
easily be extended to the general case. These termi-
nating sections suffer from the fact that the poles of
their attenuation function are coincident with the con-
trol frequencies of their image impedances. These sec-
tions then are called "associated sections". It seems
that their use limits the design procedure. The other
types of filters, the so-called "disassociate sections"

will be considered next.
4.2 . The disassociate filter.

The main feature of this filter section is that
the attenuation poles do not coincide with the critical
frequencies of the image impedances. Therefore they can-
not be considered as a collection of matched elementary

sections. However they can be considered as the filters

71

between image parameter filters and insertion loss fil-
ters. The notion of this kind of filters has been in-
vestigated rather thoroughly in recent years by several
authors [00 1, BE 1, R0 1]. The basic idea involved in
these filters comes from the fact that for the lattice
sections or symmetric ladder sections, the image impedance
critical frequencies do not necessarily coincide with the
attenuation poles.

Collins [co 1], in his research directed at
generating new kinds of structures, comes out with dis-
sociate filters produced by using the following proce-
dures:

1. Prescribe the image transmission factor a

of the filter to be designed defined by

Tanh 6 = 53.9.
ZOO

2. Prescribe the image impedance of the filter,
the critical frequencies, none of which coin-
cide with that of the attenuation poles. The

image impedance is:

ZI = Zochc °

3. The frequencies of the unity value of Tanh e
depend on the parameters m of Zobel. The
critical frequencies of the image impedance

ZI depend on the parameters n which are

72

not equal to the parameter m. These para-
meters are related to the critical frequen~
cies of ZI and the relation is identical
in form to that relation between the para-

meters m and the attenuation poles.

4. 2I
20° = tanh 6
ZSC = ZI tanh 6

5. Synthesize Zoo and Zsc by the zero shifting

technique. The resulting filter is the desired

filter. It must have the above open and short

circuit impedances, Zoc and Zsc’
Thus both the image parameter formulations and insertion
loss synthesis are involved in this procedure.

‘Rowland also develOped sections of disassociate
filters [R0 1,2]. However his section has image imped-
ances of m-derived type but the critical frequencies of
the image impedance do not coincide with the attenuation
poles. Later he has shown how his method can be gener-
alized [BO 3].

Belevitch [BE 3] tabulated some of the disasso-
ciate filters in which some sections developed earlier
are included. Some extensions for higher order image
impedances are also given. In the treatment of the dis—

associate filters the above authors have been dealing

73

only with the case of low—pass and high-pass filters.
However as Collins [CO 1] mentioned, his procedure is
also applicable to the band-pass filters. There has not
been any explicit discussion on the problem of frequency
unsymmetric band-pass disassociate filters.

It appears that, although the disassociate fil-
ters provide some savings on the number of elements, they
still require relatively complicated calculations as com-
pared to the associated sections. Thus the use of the,
image parameter theory for the design of filters is still
a simple one. A close comparison.will easily show that,
the disassociated filters will hardly contribute to the
attenuation of the stop band, because they do not possess
as many attenuation poles as the image parameter T.S.
with the same order of image impedance. As a result, we
might be able to reduce the number of intermediate sec-
tions in the image parameter filters, thereby reducing
the number of elements.

The rest of the sections in this chapter are
devoted to discussions on the derivation of the termi—
nating sections, i.e., associated sections. The discus-
sion is kept in general and it is applicable to low-pass,
high-pass and band-pass filters terminating sections.

The resulting sections when applied to the low—
pass filter case agree with the result obtained by Tokad

[TO 1] developed by a different approach.

74
Some network equivalences, which are very impor-
tant for the further development of our sections, will

also be considered in the next following section.

4.3 . The image parameter ladder terminating sections.

 

The terminating section to be considered is a
general one. Some network transformations and the net-
work equivalences are needed for the derivation of these
terminating sections.

4.3.1. The transformation and the equivalent network.

This transformation was also used by Tokad in
his deve10pment of the lowepass terminating sections
[T0 1]. It can be best shown by means of diagrams (Figs.
4.1, 4.2, 4.3). Figure 4.1 shows the original general
half section. Figures 4.2 and 4.3 are the resulting
network on performing the transformation on the original
network of Fig. 4.1.

The real parameter s has values in the interval
O< s< 1. It is obvious from these figures that the re—
sulting networks, i.e., the networks in Figs. 4.2 and
4.3 are equivalent networks. The network of Fig. 4.2
will be designated as T network and that of Fig. 4.3

T
as Tn network.

75

 

 

°——[Z:1.l* o
Z2

c— a fit)
FIG. 4.1

Original section

 

 

 

 

 

 

 

 

 

 

 

s
'——a 22
wx‘_-__> 1-s Ze <f——— wy
U 0
FIG. 4.2
The TT sections
21
c E o
1—sa z
z 9 1
w ——> 4: e———w
5322
n «:6
FIG. 4.3
The Tn section
2
" Z 14-32. z[(1se)+ 22.3-2-
IWX= 2 *5;- .. (40101), WY: 1 1 0(40102)
’ (1-82) +.—— z2
Z1 1+—

76

4.3.2. The derivation of the terminating sections (TS).
For this purpose we use m-derived sections.
The higher the order of the image impedance which is
desired, the more repeated m—derived section has to
be utilized. Terminating sections, having third and
fourth order image impedances, will be derived. Higher
ordered image impedance possessing terminating sections
can be obtained by using more repeated m-derived sections
by the same method. For convenience the m-derived sec-
tions to be used for this purpose are shown in the fol-
lowing diagrams, i.e., Fig. 4.4 and Fig. 4.5. They are
the shunt m-derived and the shunt-series m m'—derived
sections. The procedure for obtaining the terminating
section having a third order image impedance is as follows:
1. Cascade the sections in Fig. 4.4 and Fig.
4.5 such that there is an image impedance
matching at their interconnected terminal
pairs. The resulting network is the network
of Fig. 4.6.
2. Make the rearrangement as in Fig. 4.7. The
parameter s is so chosen that the section

parallel to Z is the Tn: section.

Y
3. The result of replacing TI‘by TT in item 2, is given

in Fig. 4.8. This is the desired ladder ter-
minating section.

It should be noted that a minimum mumber of ladder arms

77

mz,

 

 

 

1:1
1:35“ |

ZTm 32 Z
m

 

FIG. 4.4

Shunt derived section

 

 

 

 

 

m 1-m'a 1-m"
m'1-m

m 1 2
mm‘ 9

 

FIG. 4.5

Shunt-series mm' derived

(1Hm')mz,

0:: 1+ ' i, 0
mg m 222
-—1:}——~

 

 

-m

 

 

 

1-m'9 2;
T“: r

 

 

FIG. 4.6

78

 

 

 

 

 

 

 

 

 

 

 

 

Zn
0 —E_.L o
1-s2
88 ZO.
'ng' z E] é————— z
Y n
Zn ' ——->- 2.2
mm s2
(1 “0
FIG. 4.7
a CZ:
___{fg_"__ ‘———{::F--
0—— o
ng ng
_——{:]F-—i 1———£:j————
323 fza
Zn --9 Zn
mm'
c4. as
FIG. 4.8
Terminating section
a = (1-m'8)m b = m 1-m'2
1-m9
c - mm'(1+m') d = mm'S1+m')
1-m m'
e = f =

5L4

79

will be obtained if the rearrangement by equivalences,
as shown in Fig. 4.7, is always started from the side
of the network with the higher ordered image impedance.
If a higher ordered image impedance is required
at one of the terminal pairs, higher order derived sec—
tions will be used. Thus, for example, if an mm'm"-
derived section is utilized the terminating section of
fourth order image impedance will be obtained as shown

in Fig. 4090

 

{—7 F——{:3--* F__{:j_——_
az, oz, 1 ez,

 

 

 

 

 

 

 

 

 

 

 

 

 

0—— ~42
1323 (129 f2 3
mm'm" :1 Za Z ] Z
m( 1-—mB m‘i 25 mm'(1-%E"5) mm'm"
G 3
FIG. 4.9
a = mg 1—m'9)
1—nf9m" 9
b = m 1am“
1-m m" 1-m
c = mm'1+m' 1-m."8
1 m' m" 1+m'm"
d = mm' 1+m' 1-m"9
1+m'm" 1-m m 1-m m

 

(D
II
1‘?
8.5.
as.
r
a

f = (1+m'm1%(1+m')mm'm"

fi+m'm" ) [1+m(m'm

80

The other type of terminating sections can also
be derived in a similar manner using the transformed net-
work in Fig. 4.3. These terminating sections are given
in Figs. 4.10 and 4.11. It can be shown that these
are the dual to those in Figs. 4.4, 4.8 and 4.9.

Since our purpose is to investigate frequency
unsymmetric band-pass filters, the T.S. derived from
the basic sections and E.S.Z will be as shown in

Figs. 4.12, 4.13, and 4.14, respectively.

 

 

 

mza mm' 22
l 1- 9 21 1—m9m'9 z,
m 1..m'2 mm' (1+m' )
Z —> {.—
Tmm' 1 1 1 1 ZT
is Tin“ 2° E W 2*
¢ r 15
FIG. 4.10

m(1-m’3m"8)za mm'(1I-_+m‘m'a) z, mm'm" z;

 

 

 

 

 

zF== L-J ‘ ‘fi__J
%Z2 '3' Z: [5'15 28
211 - z '1' [5'1" z <: ZT
mm'm" b 1 c 23 f 1
0 so

 

FIG. 4.11

 

 

 

[I

ll

 

———

 

 

 

81

l l

j

I

 

 

J_

.L

“T

 

 

 

L11—

 

 

FIG. 4.12

ll

1 l

1
ll

 

 

FIG. 4.13

 

 

 

 

{D}

 

D

l
I

 

 

 

 

 

 

I
l

U

1‘
l

 

 

 

 

 

»——[1+— ——1£11.—«
~111—

 

 

FIG. 4.14

Chapter V

FILTER DESIGN I
DERIVATION OF FORMULAS

5.1 . Introduction.

For the general design of electrical filters,
two different procedures are available. One of them
which was established earlier is the image parameter
method [20 1]. The other is the insertion loss method
[DA 1, CA 1]. The image parameter method utilizes the
image parameters of the filters, i.e., the image imped-
ance (Z1) and the image transmission factor (PI).

On the other hand, in the insertion loss method these
parameters are in general, the insertion transmission
factor (P8) and the driving point impedance (Zd).

For this latter method, some authors prefer to use the
reflection factor |p| and the characteristic function 9 .

Under certain conditions, the two parameters Ps
and PI become identical. This condition is obtained
if image impedance matching at all terminals of the fil-
ter is provided. This can be clearly seen from the for-
mula derived by Zobel for the insertion loss in terms of
the image parameters [TO 1]. Both of the above methods

. 82

83

have their own advantages and disadvantages. In the image
parameter method, due to the fact that simpler calcula—
tions are required, realization can be achieved in a
shorter time. However the resulting filter might COD!
tain more elements than is actually needed. The inser-
tion loss method on the other hand generally contains a
smaller number of elements. However, a more complicated
method of calculation is required which implies the use
of electronic calculators in this design. Using the
image parameter method for filter design we can immedi-
ately obtain the elementary sections completely which
are the building block of the filter.

There often arise occasions in which the filters
which are designed by the insertion loss method will have
the same number of elements as those designed by the
image parameter method. The only difference between
these two filters is that there is an improvement in the
electrical prOperties of the filters designed by the in-
sertion loss method, which are not really required. In
such cases, indeed due to its simplicity, the image para-
meter method is preferable.

Another design technique which has been established
is the reference filter design method. In this design
technique both the image parameter and the insertion loss
parameters are involved. The synthesis is mainly carried

out by the insertion loss method, and the image parameter

84

is used for finding the locations of the attenuation
poles and also to determine the characteristic function.
This can be seen from the following formula of the char—
acteristic function for the filter

m = E) sinh (PI)
where 43 is a constant, c is the characteristic func-
tion, PI is the transmission factor of an image para-
meter filter (reference filter) which has nothing to do
with the actual filter, except that it has identical
attenuation pole locations with this filter. The advan-
tage of the reference filter method over the image para—
meter method is that it provides a flat loss in the pass
band, i.e., the Chebyshev type of attenuation character-
istics. However, the calculation of the filter elements
is by no means as easy as that of the filter designed by
the image parameter method.

It is than desirable at this point to establish
some formula which will furnish the relationship between
the insertion loss properties of the filters and its
image parameters. When such a relationship is established,
then from the given insertion loss requirements, the image
parameters of the filter can be obtained so that the fil—
ter can be designed by means of the image parameter meth-
od. Based on these parameters, some exact design proce-
dures for low-pass filters exist and can be found else-

where [TO 1]. Fisher [FIS 1] has used an approximation

85

formula and carried out an insertion loss design for
symmetrical and antimetrical band-pass filters Uti-
lizing the image parameter. However the design was
completed by insertion loss synthesis. Some fundamental
discussions on the attenuation and phase functions, es-
pecially for symmetric and antimetric .filters, are
given by Belevitch [BE 1,2].

The present work is the extension of the method
presented by Tokad [TO 1] to the design of frequency
unsymmetric band-pass filters, especially those having
dissymmetrical characteristics. In the following sec-
tions of this chapter the important features of the in-
sertion loss and image parameter methods as well as the
tOpics pertinent to the deve10pment of the desired for-

mulas are presented.

25.2 . The characterizing function of the image parameter

filters.

The salient features of the image parameters,
i.e., the image impedances and the transmission factor
(21:, Z19 and PI) are considered first.
5.2.1. Input and output image impedances, Z1, and ZI,‘
For convenience, normalized values of image im-
pedances are used. The normalization being with respect

to the terminating resistors. The normalized impedances

86

are indicated by the symbols z1 and z2 and their
properties are as follows:
(i) in the pass band they are real and a func—
tion of the angular frequency w .
(ii) in the st0p band they are purely imaginary
and a function of w .
(iii) for symmetrical filters, 21 = 22.
(iv) for antimetrical filters, Z122 = con-
stant, usually taken as unity.
5.2.2. The transmission factor, PI'
PI = AI + jBI
where- AI is the image attenuation or loss function.
AI is identically zero in the pass band and non—negative
in the stOp band.for all types of lossless filters, i.e.,
all elements are lossless. BI is the image phase func-
tion. It has distinct features for different types of
filters. The following are the pr0perties of BI and
AI in detail.
5.2.2.1. In the pass band.
AI is zero. BI is monotonic increasing as a
function of frequency. The properties are
(i) For frequency unsymmetric band—pass filters.
1. Symmetrical types
-m:n§_BIg_n:rt (5.2.1)

2. Antimetrical filter types

(5.2.2)
—(mn + 3/2) {BIQ (nn + J1/2)

87

where m and n are the number of the
attenuation pole locations in the lower
and upper st0p bands respectively.
(ii) For lowhpass filters.
1. Symmetrical types
ogBIgn (5.2.3)
2. Antimetrical types
03 BIS. (nn + Tl:/2) (5.2.4)
where n is the number of the locations
of the attenuation poles. The properties
of the frequency symmetrical band-pass
filters are implicitly covered by the low—
pass filters.
(iii) For dissymmetrical filters we have the pos-
sibilities of either one of the cases in
(ii) and (1).
5.2.2.2. In the stOp band.
1A1) 0. (AI = O at cut-off frequencies.) BI is
a constant except at the pole locations where it jumps
down by It or It/2 depending whether there is a full
pole or a half pole. The properties are
(i) For frequency unsymmetric band-pass filters.
1. Symmetrical types
I. In the lower stOp band
BI: -(m—11)n,..., -(m—2):n,...,

-(m-1)fl. -mIt (5.2.5)

(11)

where:

3.

p

v

88

II. In the upper stop band
BI = neg (n—1)n,..., (ndv)n
(5.2.6)
Antimetrical types
I. In the lower st0p band
BI = 0, -[(m-p,)n + n/2],...,
-[(m-1)n + n/2]. -[mn+ 11/2]
(5.2.?)
II. In the upper stop band
BI == nfl+n/2, (n—1):t + ”/2,...,
(n-v)n + 1!/2, 0 (5.2.8)
For dissymmetrical filters BI could
be given as in 1 or 2.
low-pass filters.
Symmetrical types

BI = nn, (n-1)n,..., (n-v)n
5.2.9)

Antimetrical types
BI = nJ + “/2, (n~1)n + J1V2,...,
(n— )n + “V2, 0 (5.2.10)
For dissymmetrical types BI could

be given as in 1 or 2.

"O, 000, m

0, 0.0, n o

89

5.3 . The chain matrix.

In order to obtain a formula for the insertion
loss function in terms of the image parameters, the chain
matrix of the 2-port filter network is utilized. It re-
lates the terminal currents and voltages of the filter.
From the diagram in Fig. 5.3.1, we have the following
relationships

V1

AV2 + 312 = AVZ + eve/32 (5 3 1)

I1 CV2 + D12 == 019:2 + DI2

where A, B, C, D are the elements of the chain matrix.
If R1 = ZI1 and R2 = 219’ i.e., matching at the termi-
nations, then V1/I1 = 21, and V2/12 = ZIa and from

Eq. 5.3.1 we have

_ AZ + B
Z _ Is
I, 7—35
(5.3.2)
Z = DZI‘ + B
I” 021: + D
From these equations we obtain
zI 21 = 3/0
' ' (5.3.3)
ZI,/ZI. = A/D .
Hence
Z = ’BA
(5.3.4)

Z1 8 = BD

90

 

 

 

 

 

R
1 I I
5 z 1 A B z 2 R
. I I 2
Y ‘ c D a
FIG. 5.3.1

4 T.N. Reactance network

5.4 . Current and voltage transmission factors (M and N).

These functions are used by Cauer [CA 1] in treating
the insertion loss filter design technique. From Eq.

5.3.1 we have

v1 sz + 312 = AV2 + BV2/R2

Therefore.
I /I = CR + D = M
1 2 2 (5.4.1)
The driving point impedance is
Z = V1/I1 = R2 (N/M). (50402)

5.5 . Entries of chain matrix in terms of the image

parameters.

From Eq. 5.4.2, we have

ZOO = A/C (R2 = w)

91

230 = B/D (R = O) . (5.5.1)

Then the H—function is given by

H = lips; = ‘IBC . (5.5.2)
280 If

From this equation, since H = tanh PI, we have

 

ePI = 1 + H = (BC + VAD
(1 .. H 11m-
9391 = (cosh P]; + sinh PI)
(cosh PI - sinh PI) °
Therefore
’VAD = cosh PI
(5.5.3)
‘VBC = sinh PI
Since
AD - B0 = —sinh2P + cosh2P = 1,

I I
passivity of the network is implied.

From' Eqs. 5.5.3 and 5.3.3, the following expressions

are immediate:

ZI.
C = 1 sinh PI
VZI, ZI,a
D = 213 cosh PI ,

92
5.6 . Insertion loss parameters.

Figures 5.6.1 and 5.6.2 are to be used as
an aid in deriving an expression for the insertion func-
tion. It is assumed that the reactance 2-port network
is terminated in two resistances, R1 and R2, and
driven by the voltage driver E.
5.6.1. The insertion transmission function.

In Fig. 5.6.1, the power delivered to the load

through the 2-ports is given by
. 2 2
[Nll = IIZRZI = 1.2. 0 (5.6.1)
R2

The power delivered directly to the load as can be seen

from Fig. 5.6.2, is

2
111.1 = 11321121 = |_Vé_| . (5.6.2)
R2

The insertion transmission factor is defined as

_ N
Ps ‘ 5 In N% (5.6.3—a)
= fi-ln.l§g‘ + j arg Nd
N1 Fl
P3 = A8 '1’ 3B8 0
Thus
A =

511115?" = ln I‘2l = Inn;

93

 

 

 

 

B3 = arg .32 = arg I: , (5.6.3-b)
I2 V2
where
As = attenuation function
B8 = phase function .
R
{—11 \ I1 . 12.
g l___J I 7

 

 

 

 

 

 

 

FIG. 5.6.1

 

 

 

 

 

E
FIG. 5.6.2
R1 I1
—3 ,
E f; v; R1
4

 

 

 

FIG. 5.6.3

94

5.6.2. The echo-loss (return-loss).

This loss function is related to the power re-
flected back to the driver. Thus, it is the power de-
livered directly to the load minus that delivered through
the 2-port, i.e.,

[Nel = [Nd] - lNll- (5.6.4)

The echo transmission factor is defined as

Pe

N = N N
ilnfid ilnlfigl+jarg[_dx

e Ne
= Ae + ,jBeo
Thus, the echo—loss is given by

A8 = 1n & o (5.605)
Ne

 

5.6.3. The characteristic function.
From the relation in Eq. 5.6.4 we have

lNdl = lNll + lNel
1 = N1 « Ne
ta [ta
1 — e‘ZAS + e‘ZAe .
Therefore
e2A8 = 1 + e-2A6
e-ZAS ‘

The characteristic function is defined as

 

 

2 —2Aew
= 6
[ml I —2As . (5.6.6)
e
Thus A = t In [1 + lml2] .

8

95
5.7 . The effectiveLoperating) loss.

The definition of insertion transmission factor
given in Eq. 5.6.3-a can be modified if Fig. 5.6.2
is replaced by Fig. 5.6.3. This will yield a new trans—

mission factor, P which is called the effective or

0’
Operating transmission function. Ps and Po will be
identical if the terminating resistances are identical.
The advantage of using P0 in the design is that it will
avoid the occurrence of negative losses in the pass band.
The possibility that negative loss occurs is evident from
the definition of P3 in Eq. 5.6.3. When an ideal
transformer at one port is used, as will be apparent
later, the form of the formula is also simplified.
5.7.1. Effective transmission factor.

Using the diagram in Fig. 5.6.3, the maximum

available power is

2
[mm] = [1112,]. (5.7.1)
The power delivered to the load, from Fig. 5.6.1, is
2
[N1] = [12122]. (5.7.2)

The transmission factor is defined as

2
Po = i In Nm - i lnIEl +'§ ln'El
If 2
l 2 12
(5.7.3)
P0 = A0 + 3B0 o

96

Thus the attenuation is:

A0 = i-ln.El + ln‘El
R2 I2
and the phase is: (5.7.4)

B = arg( I1) .
o ‘T_
2

The characteristic function is also defined here as before

-2A
2 e e

-2Ao . (5.7.5)

lwl "

 

e

where Ae is the effective return loss (echo loss).

5.7.2. The echo loss.

The echo power here is defined as the total maxi-
mum available power minus the power delivered to the load.
Thus, in the definition of the characteristic function
above, we have the situation that a fraction e-ZAO of
the total power is delivered to the load and another

fraction, e-2Ae, is reflected, hence

1 = emzA0 + 6"2Ae

(a relation due to Feldtkeller).

To study the relationship between A9 and the termi-
nating impedances R1 and Z (or 21), consider Figs.
5.7.1, 5.7.2 and 5.7.3. In general, it is sufficient
to consider only Fig. 5.7.2 and Fig. 5.7.3. In Fig.
5.7.2, Z represents the load impedance. The circuit in
Fig. 5.7.3 contains the same driver E, but instead of

Z there is a driver V representing the

39

97

 

 

 

 

 

a
l__J '
O
E ., v1 [1 zI
FIG. 5.7.1
R1 I1

 

 

 

 

 

 

 

R1 I1
r—fl \
l__fi I
A R1
1 1’ “3
FIG. 5.7.3

echo, and resistor R1. V8 is selected in such a way

that the current I1 and the voltage V1 in Fig. 5.7.3
are identical to those in Fig. 5.7.2. Then the following

98

relationship can be written

 

 

E = E - ve
R1 + Z 2R1
or
Ve = Z "' R .
2 RE
Hence
2
lel = ——E
2R1
v 2 2 2
lNe' = e = Z - R .
2R1 ‘Z + R E‘ 4R1
Thus
Ae = {rinlfigl = lnlz + R11 . (5.7.6)
Ne ‘Z — R1
Let

IZ+R1 = 1

Z-R1l m

where [PI is referred to as the reflection factor.

The echo loss is important for the filter design by the
insertion loss method in the pass band. In the remaining
parts of this section, discussion is devoted to the study
of the echo loss, in the pass band, for dissymmetrical
filters. Substituting Eq. 5.4.2 into Eq. 5.7.6 we
obtain

Ae ln M + N

M - N
'CR2 + B/R2 + D + A
Gig—B/Rg-t-D—A

 

1n

99

As = 4;. ln (az1 + [1/a]zg)2coszBI + (1/a + az1zz)sin2BI

 

(az1 - [1/a]zz)2cos2]31 + (1/a az1zz)sinZBI

where a = 1/Eh'
R2

Ae = % 1n (a2z1 + 222)2 + (1 - 222)(1 - a4z12)sinzBI
(azz1 - 222)2 e (11- 222)(1 --a4z12)sin2BI

 

(5.7.7)
In order to determine a bound on Ae function let

Eq. 5.7.7 be written as

 

Ae zek ln ~(a221+22)2 + {(1+a221zg)2 - (a221+zg)2}sinzBI ,
(a221-22)2 + {(1-a22122)2 - (a221-22)2}sin231
(5.7.8)
From Eq. 5.7.8 it can be seen that if

(a2z1 + 22)2 z (1 + a221z2)2
then also
2

1(a221 - z2)2 z (1 - a 2122)2.

Since, in the pass band, Z1, 22, and a are positive,
then

(a221 + 22)2 > (8221 - 22)2
(1 + a22122)2 > (1 - a22122)2 .

The value of the numerator expression in Eq. 5.7.8 is

always between (azz1 + 22)2 and (1 + a22122)2 and the

100

value of the denominator is between (3221 - 22)2 and

(1 - a2

2122)2. The curve corresponding to the denomina-
tor will always be below that corresponding to the nu-

merator. In Fig. 5.7.4 the curves are sketched.

 

 

 

 

FIG. 5.7.4

Curves 1 and 2 correspond to the numerator of Eq.
5.7.8 when sin2BI is O and 1, respectively.

Curves 3 and 4 correspond to the denominator of Eq.-
5.7.8 when sinzBI is O and 1, respectively.

Curve 5 is the echo-loss, Ae.

Curve I corresponds to the numerator of Eq. 5.7.8.

Curve II corresponds to the denominator of Eq. 5.7.8.

101

5.8 . Derivation of insertion loss parameters in terms

of image parameters.

Referring to Figs. 5.6.1 and 5.6.2, consider
Eq. 5.6.3:

P8 = ln I2' .
I2

This relation can be put into the following form

P8 = 1n<_I_'g £1): 1n<£g + 1n(fl) .
2 1 I1 I2

(5.8.1)

From Figs. 5.6.1 and 5.6.2 we also have

IQ = E
R1+R2
I1 = E .
R1+Z
Equation 5.4.1 gives
_I_1_ = CR2+D = M,
I2

Substituting these expressions into Eq. 5.8.1 we obtain

Pa =-. ln R1 + Z + 1n M . (5.8.2)
R1 + R2

When Eq. 5.4.2 for Z is substituted into Eq. 5.8.2,

it gives

 

Ps = 11,121+ 1112(N/M)M =ln 1"$111“ 157112 .
R1 + R2 (121-+122)

Using the espressions for N and M we have

102

rs = 1n R132 + 1n[:QR1R§c +1’El D +,«Eg A.+ B
R + R R R .
1 2 2 1 ‘VR1R2

Substituting Eq. 5.5.4 into this equation and making

the following normalizations,

21 = :11
1
(5.8.3)
22 = 213 ’
R2

the desired result is obtained as

P8 = ln 2 MR1R2 + ln[; + Z1Z2 sinh PI + Z1 + Z2 cosh PI] .

R1 + R2 z z2 2:92 z
1 1 2 (5.8.4)
Since P8 = As + st the derivations of the attenuation

function AS and the phase function. B5 for special types
of structures, i.e., symmetrical, antisymmetrical or dis—
symmetrical filters are cbnsidered next.
5.8.1. Symmetrical filters.
For this filter since, by definition,
21 = 22 = z ,
then
= In ’(5755- + 1n 2[}osh‘PI + 1 + 22 sinh P£]
R1 + R2 22 '
a) In the pass band
AI = 0
PI = 331

Z1, Z2 are real, thus 2 is real .

105

Then

to
m
I

_ 1n 2 (R 1R2 + 1n [cos BI + j 1 + 22 sin BI]

RT— + R2 22
= ln 2 R1R2 + % ln [cos2BI + (1 +2 2 222sin2BI]
R1 + R2 422

+ j are tan [1 + 22 tan BI]
2z

Therefore, we have:

ln 2 R132 + 6 ln [00823]: + (1 + 2222 13111231]

(5.8.5)

As

Bs

arc tan, 1 + 22 tan BI .
22

b) In the stop band.

2 is purely imaginary

z = jx

PI: AI+jkn, (k=0,+1, ...)
and

cosh PI =.i cosh AI

sinh PI = + sinh AI

where the upper or lower signs must be used simultaneously.

Therefore ‘ 2
P8 = ln 231131112 + 1b In [costhI + 1 - x2 sinhZAI]
R1 + R2 2x

+jarctan [_1-x2tanhAI]
2x

104

As = ln 2 R132 + % 1n [oosthI + 1 - x? 2sinh2Ai]
R1 + R2 . 2x
(5.8.6)
B8 = arc tan[:-21 - xx2 tanh AI .

5.8.2. Antimetric filters.
For this filter ZI:ZIa = R1R2. Thus,using
the same normalization as in Eq. 5.8.3 we have
2221 = 1
21 = 1/22 = 2 .

The transmission factor is then

P = ln 2 R122 + 1n [sinh P + 1 + 22 cosh P 1'
s -——————- I --- I
R1 + R2 22

a) In the pass band

AI = 0
PI = 3B1
and
sinh PI = sinh jBI = 3 sin BI
cosh PI = cosh jBI = 3 cos BI .
Then

P = 1n 2‘VR1RZ + ln 3 sin B + 1 + 22 cos B '
s —————-—- I ~-—-—-- I
R1 + R2 22

= ln 2 R1Rg + % ln[sinzBI + 1 + 22)2 cos2B;]
R1 + R2 22

+ j are tanh 22 tan BI .
[1 + 22 ]

Therefore,

105

 

As = % ln sin2BI + £1 + 2212 coszBi] + 1n 2 R1R2
(5 8 7)
B3 z are tan 22 tan Bi] .
1 + 22

b) In the block band

2 = jx
PI.= AI+ (kit‘i'n/Z), k=0, :19...
sinhPI= isinh (A1». 3 “/2 ) =.t:1 001.:thI
cosh PI = '1 cosh (AI + j 372 ) =.i j sinh AI .
Then:
P8 = ln 2 R1R2 + ln [3 cosh AI + 1 — x2 sinh A£]
R1 + R2 21
= 1n 21111112 + a} 1n E03112“ + (1 — x2 2s1nh2AI
R1 + R2 -21
+ j are tan ' 2x coth AI
1 - x2
and
A8 = 1n 2 R1R2 + 4} ln [:cosh2AI + 1 - 2xx___2_)2 sinhZAI]
R1 + R2
(5.8.8)
B8 = arc tan 2x coth AI] .
1 - x

4.8.3. Dissymmetrical filters.

The transmission factor is

106

P8 = 1n 2 R122 + 1n.[1+-21Z2 sinh PI + Z1 + 22 cosh Pi] .

2 V2122

R1 + 32

2 V2122
a) In the pass band

21, z2 are real

Therefore,

P8 = 1n 2;nyRz + lnljz1 + 22 cos BI + 3 1"'21952 sin Bi]

R1 + R2 2 2122
and thus
A8 =

R1 + R2 42122
B z:

are tan 1 + 2122 tan BI
21 + 22

b) In the stop band

2

2122

Z1 = JX1
22 = 3x2
PI = AI + jkn or
PI = AI + J(kn + F/Z),
(1) PI = AI + jkfl , k = o, 11,.32, ...
P3 = 1n 2 R132
R1 + R2

2 2
1n 2 R132 + & 1n [:1+Z122) sinzBI + (21+22) cosZB?]
42122

(5.8.9)

k=0, i1, 000

+ 1n[1 ‘ x1x2 sinh A1 + x1 + x2 cosh At]

ZJ‘Vx1x2

2

X1X2

107

AS = ln 2 R132
R1 + R2

2 2 .
+ i In [(1-1122) sinthI + (21+X2) cosh2A{]

4X1X2 4X1X2
B8 = arc tan ._ 1 - X112 tanh AI .
X1 + X2
(ii) PI = A1 + j(kn + 3/2)
P8 = ln 23(3132
R1 + R2

+ 1n 1"I122 cosh AI — 3 x1 + x2 sinh AI

24x1x2 2 Vx1x2

A = ln 2 R132
R1+R2

+.§ ln[(1"‘x1x2)2 cosh2AI + (x1+12) 2sinthJE]

4x1x2 4X1x2
BS = arc tan _X1 + x2 tanh AI
1- X1X2

(5.8.10)

As was mentioned earlier the operating loss will be the
same as the insertion loss if the terminating resistors
at both terminal pairs are identical. Thus, in the 0p-
erating loss, we have the same formula as in the case of
insertion loss, except that the term containing R1, R2
disappears. Thus, we have a more convenient set of for-
mulas if Operating loss formulas are used. This is what

will be done in the following sections and if insertion

108

loss is required then the term
ln 2fR1Rz
R1+R2
will be added to the formula. In the next section the

formulas for the operating loss will be presented.
5.9 . Formulas for the operating loss design technique.

The operating transmission factor is

Po = ln [1 + 2122 sinh PI + z1 + z2 cosh PI

2 V2122 2 V2122
(5.9.1)
5.9.1. For symmetrical filters.
a) In the pass band
A - i In cosh2B S1 222 inh2B
o — I + +22 3 I
42
(5.9.2)
Bo z are tan[_1_____2 + 22 tan BI].
b) In the stOp band
A0 = é'ln [costhI + 1 - x2 2 sinhzal]
4x2
(5.9-3)
B0 = arc tan ‘_ 1 - x2 tanh AI .
2x
5.9.2. Antimetric filters.

a) In the pass band

B0

109

% 1n [sinZBI + g1 + 2222 coseBI]
4

2

(5.9.4)
= arc tan 22 tan BI .
1 + 22
block band
= % ln.[cosh2AI + {13-2x222 sinheAIi]
4x ' (5.9.5)

arc tan.[' 2x coth AI .
1 - xfi

5.9.3. Dissymmetrical filters.

a) In the

A0

pass band

 

 

2 1 2'
% ln[}1 + 2122) sinZBI + (Z1 + Z2) c08231:]

42122 42122
1 (50906)
are tan + Z1Z2 tan BI .
Z1 + 22

b) In the block band

or

2 2
g ln (1-x112) sinthI + (11+12) costhI
4X112 4x1x2

1 (50907-8)
arc tan ‘_ ' x112 tanh AI ,
X1 + X2

_ 2 2
Ԥ In (1-x1x2) cosh2AI + (X1+12) sinthI
4X1X2 4X1X2

(5.9.7—b)
arc tan ‘_ x1 + x2 tanh AI .
1 - X1X2

Chapter VI

FILTER DESIGN II
APPROXIMATION AND DESIGN PROCEDURE

6.1 . Introduction.

The filter design procedure considered in this
chapter is based on the image parameter method. It is
assumed that the insertion (effective loss) requirements
of the filter are specified. The synthesis of the filter
network is carried out using the image parameter method.
Some exact design procedures for the filter utilizing the
image parameter method and approximate techniques for sym-
metrical and antimetrical. filters with the insertion
loss method are already develOped [TO 1, F18 1]. In this
thesis the work is mainly devoted to the.design of frequen-
cy unsymmetric band—pass filters, especially those having
dissymmetric configurations. The realization procedure
is carried out by the image parameter method. The ele-
mentary sections used in this type of filter cannot be ob—
tained as those of frequency symmetrical filters by the
frequency transformation from a low-pass filter. The
technique deve10ped by Laurent, to generate elementary
sections for general filters, can be utilized. However

110

111

these generated sections must be used as is without re-
ferring to how they are generated [LA 1]. There are other
techniques available to generate a set of elementary sec-
tions [BR 1, SH 1, MA 1, NO 1, BO 1, CO 1, SA 1]. These
elementary sections should be used as is. Since each ele-
mentary section is considered as independent, its prOp—
erties must be investigated separately. The necessary
information required for the design can be obtained from
analytical investigations.

One method which seems to be less complicated
than others, hence preferable, is to develop the elemen~
tary sections from the basic sections by m-derivation
[NO 1, B0 1, see also Chapter II, section 2.5]. One
other factor to be considered before deve10ping relations
for the elementary section is the fact that these sections
should contain a minimum number of inductors [SA 1, WA 1]
The elementary sections which are suited to the discus-
sion of this thesis are those E.S.1, E.S.2, and E.S.Z
which are presented in Chapter II.

The approximation of the loss functions for sym-
metric and antisymmetric filters are given by the formulas

AeZAI - 1n 2 Nepers for stOp band
A332 ln {z 22122 Nepers for pass band

where A8 is the insertion attenuation

AI is the image attenuation

112

z is the normalized image impedance .

The approximation has to satisfy the overall requirements,
1.8.,

(1) in the pass band Asg’As min

(2) in the stOp band AsZ.As max .

An improvement on the approximation in the stop band for
symmetrical and antisymmetrical filters [FIS 1] is

2
A0 é AI + 1n [z2-+ 1] - ln 2 Nepers
2z
where A is the effective attenuation

I
All 3 Nepers .

6.2 . Approximation for the attenuation function of dis—

symmetrical filters.

In the pass band Belevitch [BE 1] has made an
extensive discussion especially for the low-pass filters,
frequency symmetrical and unsymmetrical band-pass fil-
ters of symmetrical and antimetrical types. However
for the dissymmetrical case he considered only epecial
types of filters. Here formulas for completely dissym-
metrical filters will be established. From Eq. 5.9.6-a

and -b, in the pass band, we have

113

‘F 2 2
e2A° = (1+z122) sin2BI + (21+Z2) cos2BI
_ 42122 42122

 

2 2 2
e2A0 [_(1+z122) + ((z1+z2) _ (1+z122) )coszBI .

 

 

L. 4z1z2 4z1z2 4z1z2
2
Let (1 + Z122) = B
42122
2
(21 + 22) = C
42122 .

Consider now the value of the function e2A0 at a fixed
frequency. If C - B<1O then, at this frequency, e2A°
will have a maximum when coszBI = O. This maximum is
B. The minimum at this frequency will occur when

cos2BI = 1. This minimum is C. The converse is true

when B — C<<O. Thus, the curve of A0 will always
lie between the two curves of % 1n (”2122)2 and

 

 

 

    
 

42122
2
% ln (21+22) as is shown in Fig. 6.2.1.
42122
A when cos2B = 1 A0 when coszBI = O
ALN O I
or or % ln (1+Z122)2
% 1n (z1+22)2 /,.AZ1Z2
‘ ‘-4z122 ’
/ \\

 

 
 

 

 

FIG. 6.2.1

114

In the block band, from Eqs. 5.9.7-a and -b, it is
evident that
2A 2 2
e 0 = } (1-X1x2) sinhZAI + (X1+X2) cosh2AI
X1X2 X1X2

or
2 2
e2A° = & (14112) cosh2AI + (X1112) sinhZAI .
x1x2 x1x2
If AIZ.3 Neper the following approximations can be
made

sinh AI: cosh AI: eAI/2

V e2AI- 1 2 6A1 0

Thus, the above formulas of eZAO, using these approx-

imations give

2 2
32Ao zi. l’(1-x1x2) eZAI + (x1+x2) BZAI
x1x2 4 x1x2 4

2 2
A0 :A1 + a} 1n(1+x1)(1+x2)_ 1n 2
4X1XZ

Hence,

2 2
AI on — it ln(1+x1)(1+12)+ ln 2 . (6.2.1)
4X1x2

( 2 2
If 2 ln 1+x1 )(1+12 ) is equal to In 2 then A0 = A1.
4X1X2

 

This will occur when |z1| = |22| = 1, which is the case
when an image impedance matching exists. On the other

hand if

115

2 2
f(x1,x2) = i In (1+x1 )(1+x2 ))>ln 2 ,
4X1 X2
then AI<1AO. Therefore it is necessary to investigate
the behavior of the function f(x1,x2) to check if the
condition ln 2<if(x1,x2) is possible.

Since

2 2
f(X1,X2) = i In (14-11 )(1+X2 )

 

 

 

 

 

 

4X1 12
then
2
Br = (1 + x22) (x1 - 1)
2
3x1 2x2 x1
2 ’ '
a r = (1 + 122) 1
2 3
3x1 x2 x1
bf = (1 + x12) (x22 - 1)
2
3x2 2x1 x2
62: = (1 + ‘12) 1
""2 "‘3
3x2 x1 x2
622 = x12 ‘ 1 x22 ' 1
2 2
5x15x2 x1 x2
8x1
bf = o ———9 x2 = 1

116

x2=1
fx,x, = 2 >0
,/:1—1
xg-1
fxaxa = 2 >0 0
1/41=1
xg=1
Thus we will have an extremum at (x1,x2) = (1,1) be- '

cause DD’O. However this extremum is a minimum

(f > O, f > O). From this discussion it is seen

xxx: xexa
that f(x1,x2) can be made as large as we desire. This,
of course, implies that, according to Eq. 6.1.2, AI

can be made as small as possible, thus a minimum number

of elements will be required in the filter. However, this
possibility is limited by the available types of imped-

ances, z1 and 22.

6.3 . The design procedure.

As the starting points for the design for the
insertion loss filters, the following requirements are
given:

I. Effective pass band and the required effec-

tive return loss in this range, i.e.,

117

A or Ipl = e 9

e .

II. Effective stOp band and the corresponding
attenuation requirements.

III. The requirements on the imput and output

impedances in the stOp band.

For the image parameter filters we have the following
requirements:
If The interval in which the image impedances
are real.

I

II and III’ as in the II and III above.

Consider the function

2 2
f<x19X2) -- tlnilifll+tlnflifal
2x1 2x2 °

The following curve in Fig. 6.2.2 is the curve

of % ln §1+x22 , where x) O. This curve is to be used
2x

as an aid in determining the attenuation curve in the stop

band. The curve is symmetric with reSpect to x = 1. Thus
only the part of this curve for x< 1 is plotted in Fig.

6.2.2.

118

2.00

1.75

1.50

1.25

1.00

 

 

J A l

l l l l l
.01 .02 .04 .06 .2 .4 .6 .8 1.0

x-1o X "9

FIG. 6.2.2

k ln §1+x22 Nepers
2x

The design procedure consists of the considera-

tions of the following items: I’, II’, 111’.

If Gives the cut-off frequencies.

/

II. Gives the lower bound on the attenuation

or the poles of attenuation.

119

III{ Gives the types of image impedances.

Additional requirements are: filter is lossless and
contains a minimum number of inductors. The procedure
is described briefly in the following:
(1) Plot the y-scale.'
(ii) Plot the impedances on the Yescale, then
with the aid of Fig. 6.2.2, plot
ln f(x1,x2) on the yhscale.
(iii) Substract the curve in (ii) from the
required attenuation curve.
(iv) The remaining curve is usually, after
adding ln 2 to it, the requirement

for the intermediate section.

By using the template we can find the poles of attenua-
tion, as is shown in Fig. 6.2.4. In Fig. 6.2.3 the
plots of A0 min and in f(x1,x2) are shown. In Fig.
6.2.4 the template curves which will approximate AI

are illustrated, i.e., the image attenuation of the inter—
mediate section to be designed. Since the terminating
image impedances are supposed to be given in advance,

this means that the conditions in the pass band are

completely satisfied.

120

 

 

 

 

 

 

 

 

 

    

 

 

 

,/ A Nepers
“aft
.fl
/: Upper stop band ӣ9229
2 2
Aomin lower stop band 1‘ %1n(1+x1)(1+x2)
2 2 C ‘ 4X1XQ
%1n(1+X1)(1+X2) , /’
4X1X2
\\\\ //’/’ \\\
\\ /// AI
g \/
\
'FIG.6.2.3
1A Nepers
' Template .
i curve
i x /”
\ /
Q J \/
\Y Yes You Yes Ya:

FIG. 6.2.4

121

_§;4 . Some more study of the high order image impedance

"Z'Tiil and ermo

 

In Chapter II, general discussions of these
types of impedances are given. In this section these
impedances are studied to obtain some details useful
in the filter design.

Figures 6.3.1 and 6.3.2 illustrate the plot
of the image impedances me and an, which are nor-
malized with reapect to the constants RTm and an°
Note that the frequency axis is the J”L-axis described
in Chapter II rather than the actual (n-axis. These
curves are calculated with the aid of digital computer.
In these figures, curve (1) represents the image imped-
ance with critical frequencies (attenuation poles),
mop and wgp which are relatively close to the cut-off
frequencies. In the image impedance corresponding to
curve (2), the critical frequencies are taken far from
the cutaoff frequencies, and finally curve (3) corre—
sponds to the situation that the critical frequencies
are taken still farther from the cut-off frequencies.
The effect of the location of the critical frequencies
of the image impedance on the impedance curve is evident
from these examples. It is to be noted that the shape
of these curves depends also on the band width. In fil-

ter design, using these types of impedances9 one should

122

plot Z1 or Z2 and choose the best curve fitting the‘
requirements.

The following is a table of the quantities which
is useful in the design.' All of the quantities are cal-
culated for a fixed band width of (22—18) = 4 k rad/sec

and only the best curves are included in the table where:

JfiLmin,1JfiLmax,.J”Le are extremum frequencies
D1 : maximum

D2 : minimum

JfiLZ — JAL1 : effective band width

Amax : expected attenuation in the pass-band .

Table I
Z1 = ZTm/Rmm (refer to Fig. 6.3.1)
w01 1 2 5 10 k rad/sec
J“Lop1 -1.10 —1.10 -1.10 -1.10
.!\_2p1 1.10 1.10 1.10 1.10
nmin 0.090 0.050 0.090 0.120
Joimax -O.895 —0.890 -0.965 -O.965
, JfiLe 0.875 0.990 0.870 0.885
JAL1 -O.9650 -0.970 -0.965 -0.965
JALQ 0.945 0.945 0.915 0.925
D1 0.282 0.283 0.269 0.215
02 0.208 0.283 0.192 0.152

Amax 0.0275 0.0190 0.0195 0.02 Nepers

123

Table II

(refer to Fig. 6.3.2)

“21 35

JoLOpg —1.10
47L2p2 1.10
JflLmax —0.110
JflXmin 0°895
17L. —0.875
J“L1 —0.925
J"12 0.960

D1 5.88 10‘3
02 4.16 10‘3
Amax 0.019

49
-1.10
1.10
-0.105
0.990
-0.930
-0.950
0.975
2.42 10"3
1.61 10’3
0.02N

30 k rad/sec
—1.10

1.10
-0.150

0.885
—O.87O
—0.920

0.965

9.9 10’3

6.9 10"3

0.02N Nepers

In.general the image impedances are normalized with

respect to the terminating resistance, RL° Thus, the
normalized impedance used in the design is
ZTm _ Z ZTm RTm z RTm
RL I RTm RI: 1 RI.
or
E12. = Z — ..Z—J-IE Ell Z ELISE °
RL II R m RL 2 RL
Let
max min 2 max min 2
ZTm ZTm = RI: (OI‘ Zflm an RL )0

124

Then Rum/BL (or RTm/RL) is equal to 1/V575E .
Thus, RTm = R 45755 (Rum = 85/957553 which can
also be used to determine the value of RTm or Rum .
Note that the expected attenuation in the pass band,

A is the only required condition for the determina—

max 9

tion of the image impedance if the filter is symmetric.

125

N .06 .03.
Sid.

 

 

. “.9— _ . . . ”QT.

0
1
q

 

——.-—___l

 

 

 

 

 

 

 

 

 

00on 8‘03
toxunxcutw uflmuxu MN$M~

 

3.0:» 398$. \Wmfixko 103
00 1 H d All .03 ‘
in
A... .. ..328w3...3 3 EN

 

M231u3yma.3143&\(an31.3x .\I 1 ELM

 

 

 

 

126

.WHU

Rem-Q ffi‘h .

 

 

.e

— d d 0— ~— . J - “J' .

 

 

 

 

 

 

 

 

 

 

. tokaxcwvvw 3.590 was
nuautoshvuua \mu..w.20

~ fiN
8+“.«qx 3

 

2. -...xs-.e>§- we e 1..
A... .. 333.3 -.e

 

MU\OO\ M Nmflg

    

£55m

 

EhN

 

I. So.

36.

one.

 

Chapter VII

CONCLUSIONS AND FURTHER PROBLEMS

Complete characterizations of elementary band—
pass filter sections have been developed as well as
formulas for the values of the elements of these sec-
tions. The elementary sections discussed in this the-
sis are of Special types (see Chapter II). However,
the deve10pment can also be applied to other types of
sections. The reason that only the Special type of
sections are considered here is that a filter made out
of these sections is an economical filter, i.e., it
contains a minimum number of elements. A systematic
design technique is described using an approximation
formula for the attenuation function, which takes into
consideration the effect of image impedance. The effects
of image impedances are generally omitted in the earlier
approximation formulas for the attenuation function.

The study of higher order image impedances by
using a frequency transformation technique is discussed
and the selection of the location of the critical fre-
quencies of the impedance function is considered. For
the determination of the location of these critical
frequencies a digital computer program has been employed.

127

128

After using a frequency transformation on the image im—
pedance expression, through a trial and error method
these frequencies are located to give the "best" image
impedance function. However, an analytical approach,
perhaps utilizing elliptic functions, could be used.
Such an approach is not considered in this thesis, but
rather is left as a further problem.

In this thesis only lossless band-pass filters
are considered. In the ease of incidental losses, as
is known for the low-pass case [T0 1], as long as the
losses are assumed to be uniformly distributed, a simple
computer program can be written to take into account the
effect of losses. In this thesis, the program written
and used for the calculation of the insertion loss func-
tion can easily be extended to the lossy case. However,
since the main objective in this thesis for the designing
of "zig-zag" type of band-pass filter is to describe an
exact design procedure, such an additional program

coveringthe lossy case is not written.

[813 1,2]

[BE 3]

[80 1]

[BR 1]

[CA 1]

[CAM 1]

[001]

BIBLIOGRAPHY

Belevitch, V., "Elements in the design of
Conventional filters", Electrical Communi-
cations, Vol. 26, March, 1949, pp 84—90 and
June 1949, p 180.

, "Recent deve10pments in filter
theory", IRE Transaction on Circuit Theory,
Vol. CT—S, December, 1954. PP 236—240.

Bode, H.W., "A general theory of Electric
Wave Filters", J. Math. Phys., Vol. XIII,
1934. PP 275-362.

Brandt, R.V., "Ein einheitliches System der
Dimensionierung von Bandpfissen nach Zobel

und Laurent", Frequenz, Band 7, Juni 1953,
Nr 6, pp 167-180.

Cauer, W., "Theorie der linearen Wechselstrom
schaltungen" (book), Akademie Verlag Berlin,
1954, 2nd edition.

Campbell, G.A., "Physical Theory of the
Electric Wave Filters", B.S.T.J., Vol. 1,

pp 1-32, November 1922.

Collins, J.E., "Les Filtres dissocies Passe—
bas et Passe-haut", Cables et Transmission,
198 Annee, Janvier 1965, No 1, pp 9-19.

129

[DA 1]

[F18 1]

[FIS 2]

[FU 1]

[FR 1]

[FU 2]

[LA 1]

130

Darlington, 3., "Synthesis of reactance
four poles", J. Math. and Phys., Vol. 28,
September 1939. Pp 257-353.
Fischer, B.J., "Uber elektrischer Wellen—
filter mit vorgegebenen Betriebseigenschaften",
A.E.U., Band 14, Juli 1960, Heft 7.

"Ein Beitrag zur Berechnung elek-
trischer Wellenfilter", A.E.U., Band 17,
Heft 6, Juni 1963, Heft 7, Juli 1963.
Fujisawa, T., "Realizability theorem for
Mid-Series or Mid-Shunt Low-Pass Ladders
Without Mutual Inductance", IRE. Transc.
on Circuit Theory, Vol. CT-2, December 1955, pp
236-252.
Fromagoet, A. and LaLande, M.A., "Utilisa-
tion d’une methode degabarit pour le calcul
pratigue des filters", Ann. Télécomm. ,
Septembre 1950, pp 277-290.

, "Theory and Procedure for optimaza—
tion of low-pass attenuation characteristics",
IEEE Transact. 0f Circuit Theory, Vol. CT-11,
No 4, December 1964.

Laurent, T., "Vierpoletheorie und Frequenz-
transformation" (book), Springer—Verlag,

Berlin/Gottingen, Heidelberg, 1956.

[MA

[M0

[N0

[RE

[R0

[R0

[R0

[RU

1]

1]

1]

1]

1]

2]

1]

O.

131

Mason, W.P., "Electromechanical transducers
and Wave Filters" (book), D. von Nostrand
Company, Inc., New York, Second edition 1958.
Mole, J.H., "Filter design data for Communi-
cation Engineers" (book), John Wiley & Sons,
Inc., New York, 1952.

Nonnemacher, W., "Uber den Aufbau von Band-
passen aus Stamgliederen", Frequenz, Vol.16,
1957.

Reed, M.B., "Electric Network Synthesis"
(book), Prentice Hall, Englewood Cliffs,
N.J., 1955.

Rowlands, R.0., "Composite ladder filters",
Wireless Engrs., Vol. 24, January 1952, No 340,
PP 50-55.

"Double derived Terminations", Wire-
less Engr., Vol. 23, pp 52-56, February 1946
and pp 292-295, November 1946.

"Impedance transformations in Four
element Band-Pass Filters", Proc. Inst.
Radio Engrs., Vol. 37, November 1949, pp
1337-1340.

Rumpelt, Z., "Schablonverfahren fur den
Entwurf elektrischer Wellenfilter aus der
Grundlage der Wellenparameter", T.F.T.,

August 1942, Vol. 31, pp 203—210.

[SA

[SA

[80

[80

[SH

[SK

[TO

[WA

1]

2]

1]

2]

1]

1]

1]

132

Saraga, W., "Minimum inductor or capacitor
filters", Wireless Engr., Vol. 30, No 5,
May 1953.

, and Fosgate, L., "New Graphical meth-
ods for analysis and design", Wireless Engr.,
January, 1952, Vol. 24. PP 50-55.

Schoeffler, J., "On the existence of Crystal
ladder filters", Proc, First Allerton Con—
ference on Circuit Theory, November 1963.

, "A solution to the approximation and
realization problems for Crystal Ladder fil-
ters", 1964 IEEE International Convention
record Part 1, March 23-26.

Shea, T.E., "Transmission Network and Wave
filters" (book), Von Nostrand Company, Inc.,
New York, 1929.

Sykes, R.A., "A new approach to the design
of high frequency crystal filters", IRE
National Conv. Rec., Part 2, March 24-27, 1958.
Tokad, Y., "Some improvement of the image
parameter methods of the design of L—C fil-
ters", Doctoral thesis at Michigan State
University, 1959.

Watanabe, H., "Synthesis of Band-Pass Ladder
Network", IRE Transact. on Circuit Theory,

Vol. CT—5, September 1958, No 3.

133

[Z0 1] : Zobel, 0.J., "Theory and design of uniform
and composite electric Wave Filters", B.S.T.J.

Vol. 2, No 1, January 1923, pp1-.46

APPENDIX

EVALUATION OF ATTENUATION FUNCTION
BY DIGITAL COMPUTER

The following is an example of the evaluation
of attenuation function using a digital computer. The
filter considered in this example is of the form shown
in Fig. A—1. The ideal transformer of turn ratio
1xn is used to make both the image impedances of this
filter identical.

40: ~40—
101 .3

2T we}: ‘T‘woz we:
m wept

 

 

 

 

 

 

 

 

 

01—

[11+

(00 P-t
we?!

 

 

+1

The following table gives the explanation of the symbols
used in the computer program. The program is also
included after the table and it is used for the filter

in Fig. A—1 with the following parameter values.

1001 10 kc/sec

30 kc/sec
134

“’21

135

—1.1

“0p1 = “(“0p1)
92p1 = 9(w2p1) = 1.1
Amax a max. attenuation in the pass band
0.02 Nepers
(see also Table I on page 122 for case w01 = 10)
The result is included in the following and the sketch
of the attenuation function in both stop bands and pass

band are given in Fig. A-2.

Symbol used in

 

136

Table III

Meaning

Symbol used

 

 

 

 

 

the program in the text

201(1) I attenuation poles in the (.001
upper and the lower st0p -

P02(I) band (I = number of E.S.Z. w21

‘ sections or its m derived) “

PP1 transformed critical fre- 90p1
quencies in impedance in-

PP2 vestigation (PP1 = -PP2) 92p1

21(1) attenuation poles in the no,

/ fi-scale

P2(I) §21

GAP1(I) attenuation poles in the Y01
y-scale (logarithmic ‘

GAP2(I) scale) Y21

H(I) the Hqunction of E.S.Z. Hi
of its m—derived section

2 _
Hf e mi (6 -1) $992512)
(5456(1))2 (9241)

-S(I) confluence frequenc of 50(1)
each E.S.Z. in the -scale
§0(1)=§2 119591-15502119EE1"

, 2
47:13:51-1 Alton—1
20(1) constant of the Héfunction m1
2 2
m1: 1 (L92p1W90p1-1“3'90p1W9221-1
1’—-2
ATI total attenuation AIt

‘-

137

(Table III continued)

Symbol used in

the

AI

BIT

BI

X1

A01

A03

DEL

OS
0B
A0
OM

1

program

Meaning

...“ ..-... .. ...—....

 

‘ image attenuation of each

section

AI = ln 1+H
1-H

total image phase
BIt = 213I

image phase

BI = arc tan [2H1]
normalized impedance
x1 -—- ZTm/RTmW

STOP band attenuation
(exact)

STOP band attenuation
(apprOX)

0.5 log f(x1,x2)

logarithmic frequency
scale

transformed frequency for

:impedance investigation

angular frequency

pass band attenuation

transformed angular
frequency

Symbol used
in the text

AI

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140

* 041917 SOEMINTAPOERA

10

11

12

13

100

15
16
18

17
21

PROGRAM DESIGN

ODIMENSION POI(2)0P02(2)0P1(2)0PZ(2)0GAPI(2)0GAP2(2)0S(2)0
120(210H12’0HA(2)0AG(2)0AI(Z)0BI(2)

PP18-1.100

PPZSI .100

P01(2):109000

POI(1)=PPO(POI(2)0PPI)
POZ(1)RPPO(POIIZI0PPZI

P0212):300000

DO 2 I3102

P11I)=POLE(POI(I))

PRINT 30 (PI(I)0I81021

FORMAT (1H003X07HPOLEIB 0F80303X0F803l
DO 4 18102

P211)=POLE(POZ(I)1

PRINT 50 (PZIII0ISI02)

FORMAT (1H003X07HPOLEZ= 0F80303X0F803)
DO 6 I8102
GAPI(I)=005iLOGF((PI(I)+10)/(P1(I)-10)1
PRINT 70 (GAPI(II0I8102)

FORMAT (IHO03X06HGAPI3 0F80303X0F803)
DO B 18102
GAP2(II'D-SiLOGFC(P2(I)+lo)/(PZIII-Iol)
PRINT 90 (GAP21I10131021

FORMAT (IHO03X06HGA923 0F80303X0F803)
DO 10 1'102

S‘II3COL(PI(II0PZII)I

PRINT 110 (5(I1018102)

FORMAT (IHO03XOIZHCONFLUENCES 03X05150803X051508)
DO 12 I310?

ZO(I)8CONS(PI(I’0PECIII

PRINT 130 (ZOCI9018102)

FORMAT'11H003X0IZHM =03X0F150803X0F1508)
PRINT 100

FORflAT (1HO03X025HSTOP BAND ATTEN IN NEPERS)
ON8-9025

DO 14 K'I068
IFIK-33) 15015016
GO T0 17

IF (K-34) 15018015
0N30M+2025

GO TO 17

IF (K-I) 20021020
PRINT 22

ZZOFORMAT (1H006X02H0M0IOX03HGAM016X03HATI018X03HA01018X.3HAO3.

20

25
24

115X0IIHLOGF(XI0X2))

CONTINUE

ATI'O

DO 23 18102
HCI)'RATHCZOCIIOSCIIOOM)
HACII‘fiBSFIHII’)

IF (HAfII-Io) 3‘02502‘

GO TO 23
AG(II'ABSF((Io+HA(III/Ilo-HACIIII
IF (AGCIII 23023035

141

38 IF (AGIII-100**8) 39039023

39 AICI)*20*LOGF(AG(III

23 ATIBATI+AIII1
STIIISINH1ATII)**2
OBSABSF($ORTF((180*2201*(100+OH1/(100-OM111
XI‘ZIMCOB0POICZI0POIII10P02(1I)
IF (XI) 27028027

28 GO TO 14

27 CSABSFIXII
IF (C-lo) 29030029
IF (C‘100**6) 29029030

30 GO TO 14

29 CONTINUE
GICCIC+XI**2)/(20*X1II**2
DIABSF110+G*ST1)

IF (DI 40040041

40 GO TO 14

41 A013=005*LOGF(DI
TaABSFISORTFCG’I
IF (T) 42042043

42 GO TO 14

43 ZSLOGFCT)
6.005*LOGF(ABSF(10M+101l¢0fl~1011I
AO3IATI+Z-LOGF(201
PRINT 310 0MOGOATIOAOIOAO30Z

31 FORMAT (3X0F8020513X02150811

14 on-OM+O.25
PRINT 110

110 FORMAT (1H003X021HPASS BAND ATTENUATIONI
PRINT 34

34 FURNAT (1H008X02HXI0IBX02HAO018X03HBIT012X02H0H012X03HDEL)
DEL8-1.OOO
DO 19 L31064
IF (L‘Io) 490‘9050

49 GO TO 60

50 IF (L-55) 70070049

60 DEL'DEL+0001
GO TO 80

7O DEL'DEL+0004
GO TO 80

80 OS'PPOIPOICZIODELI
OH'POLE(OSI
BIT'O
DO 35 I'102
HIII'R‘TH‘ZOCII05‘IIOOMI
BICII.2¢*(ATANF(20*H(III)

35 BIT'BIT+BIIII
5T2ICSINF(BITI)**2
X1=ZIH1050POI(210POI(I)0POZ(I))
02'1110-X1**2)/(20*X1))**2
A03005*LOGF(ABSF(10+02*ST2I1
PRINT 360X10AO0BIT0OH0DEL

36 FORNAT (3(3X0EI505)03X0F80303X0F803I

19 CONTINUE -

142

STOP

END

FUNCTION COL(AOBI
DIMENSION A13I0BI3I
RISORTF(A*A-IOI
OISORTF(B*B-10)
COL'(A*O+B*R)/(O+RI
RETURN

END

FUNCTION CONSIAOBI
DIMENSION A13IOBK3)
C3100
R-(C*AI*SQRTF(B*B-10)
O'CC-BI*SORTF(A*A-Io)
RBB-A

S‘SORTFCCiC-IOI
CONS3(G+R)/(R*SI
RETURN

END
FUNCTIONMRATHCA0B0CI
DIMENSION A13)0B(3)
DSIOQ

GBSORTFCD*D-Io)

TnD-B

PBC-B
OBSORTFIABSFICic-IOII
RATH“A§G§RI/CT*Q)
RETURN

END

FUNCTION ZIN(P¢GOUOSI
TORI'IIo/RI*CP*R‘O*O)/I(9*P-U*U)*(5*S-R*RII
TOR23$ORTF1ABSFC(9*R-180*180I*(220*220-R*P)I)
01800215

02300152
COIIlolsoRTFKD1*02)
ZIM'TORI*TOR2*COI
RETURN

END

FUNCTION ROLE(OI
GSSORTFCIaoi22oI
A8001
TASA*(O*0+G*GI/IQ*Q-G%GI
ROLEFIo/TA

RETURN

END

FUNCTION RPOCAOBI
PEN=B+¢40.-2.*la.*22o/A3/4.
REH'B-C4Oo-20*A1/40
PDQ-AlPEN/PEM

RETURN

END

END

143

STOP BAND ATTENUATION IN NEPERS

OH
-9025
'9000
-8075
-8050
"8025
-8000
'7075
-7050
-7025
“7000
-6075
-6050
-6025
“6000
-5075
-5050
-5025
-5000
“0075
-0050
-0025
-4000
-3075
-3050
-3025
-3000
-2075
'2050
-2025
-2000
’1075
-1050
-1025

1025
1050
1075
2000
2025
2050
2075
3000
3025
3050
4000
4025
4050
0075
5000
5025
5050
5075
6000
6025

GA!

“010853225E+OO
’011157178E+OO
*01I478722E+OO

011819439E+OO
‘0IZIBIIOQE+OO
-0I2565721E+OO
'0I2975560E+OO
’013‘I3I99E+OO
‘0I3881587E+OO
“010384106E+OO
’014924649E+OO
-015507746E+OO
'0I6138670E+00
”0168236I2E+00
-017569394E+OO
-018386239E+00
‘0I928312‘E+00
”020273255E+OO
‘021372201E+OO
“022599256E+00
”02397865‘E+OO
-025541281E+OO
”027327I85E+00
‘029389333E+00
-031799‘38E+00
’03‘657359E+O°
’038107003E+OO
“042369893E+OO
‘0‘7775572E+OO
-050930610E+00
'06‘96‘!‘9E+00
‘060471996E+00
-010986123E+OI

010986IZBE+01

0804716965+00.

060960109E+00
050930614E+00
0QT775572E+00
042364393E+00
03BIO7003E+OO
030657359E+00
031799938E+OO
029309333E+00
02554128IE+00
023978650E+00
022599256E+00
02I372201E+00
020273255E+00
019283124E+OO
018386239E+00
017569394E+00
016823612E+00
016I38670E+OO

AT!
012153058E+02
012239682E+02
012337574E+02
012449155E+02
012577634E+02
0127273BIE+02
012904549E+02
013118161E+02
0I3382188E+02
013719900E+02
014174373E+02
010839844E+02
0159989555+02
020222259E+O2
016282573E+02
014626395E+02
013636026E+02
012396565E+02
012286595E+02
01I753083E+02
011267581E+02
0108125922+02
010376121E+02
099991111E+01
095240619E+01
090941730E+01
086527426E+01
081926852E+01
077061754E+01
071847347E+01
066215437E+01
060274069E+01
0559III23E+01
072523363E+01
071555523E+01
0771672142+01
083098361E+OI
089980807E+01
096632370E+OI
0I0364844E+02
0III3BOS3E+02
012050583E+02
013282581E+02
016207BIBE+02
0I4039082E+02
013163435E+02
01260336OE+02
012289783E+02
012031465E+02
011533969E+02
013678126E+02
011552237E+02
011008683E+02

A01
012032159E+02
012291305E+02
012547024E+02
012809294E+02
013085837E+02
013388769E+02
013726091E+02
014113623E+02
01¢573289E+02
015140750E+02
015881228E+02
016936590E+02
018731231E+02
025121373E+02
019266597E+02
016836987E+02
015406330E+02
0143526IIE+02
0134940655+02
012751526E+02
0120827585+02
0114620152+02
010871897E+02
010299546E+02
0973‘6209E+OI
091681059E+01
08591544QE+01
079965588E+01
073747732E+01
067189042E+01
060285976E+01
05341783OE+01
050112407E+01
071047596E+OI
065006564E+01
070268563E+OI
077054453E+01
080210205E+01
0915931882+01
099344433E+01
010780384E+02
011765822E+02
013061211E+02
016114935E+02
014007BIIE+02
0I3I92399E+02
01273147OE+02
012436238E+02
012235745E+02
01209584IE+02
011997651E+02
011929784E+02
011884955E+02

A03
012032159E+02
012291305E+02
012547024E+02
012809294E+02
013086837E+02
013388769E+02
01372609IE+02
014113623E+02
014573289E+02
0151407SOE+02
015881228E+02
01693659OE+02
0187312315+02
025121373E+02
019266597E+02
016836987E+02
015406330E+02
014352611E+02
013094065E+02
012751526E+02
012082758E+02
011462015E+02
010871897E+02
010299546E+02
097346209E+01
091631059E+01
085915444E+01
079965589E+01
073747732E+01
06718904IE+01
060285965E+01
053417774E+01
050112324E+01
07104759BE+01
065006555E+02
070268561E+01
077054453E+01
08¢210205E+01
091593188E+01
09934443SE+01
0I0780348E+02
011765822E+02
0130612IIE+02
016I14935E+02
014OO7BIIE+02
013192399E+02
012731470E+02
012436238E+02
012235745E+02
012095841E+02
011997651E+02
0119297845+02
0I188‘955E+02

6050 015507706E+00
6075 014920649E+00
7000 014384104E+00
7025 013881587E+00
7050 013413199E+00
7075 012975560E+00
8000 012555721E+00
802$ 0I2I81104E+00
8050 011819039E+00
8075 011078722E+OO
9000 011157178E+00
9025 010853225E+00
9050 010565455E+00
9075 010292603E+00
PASS BAND ATTENUATION IN
X1 A0
067256E+00 015919E-01
087333E+00 0334I4E’02
098897E+00 03II49E-04
010622E+01 0II‘20E-02
011103E+01 0398085-02
0II‘21E+01 071442E-02
01163IE+01 099590E-02
0II763E+01 012150E¢01
0IIB40E+01 013651E-OI
011877E+01 014501E-01
0II794E+0I 013285E-01
0II505E+01 039334E-02
011250E+01 048403E-02
010954E+01 0211285-02
010625E+01 069647E-03
010410E+OI 0144‘2E‘03
010170E+0I 099766E-05
099512E+00 013073E-06
097530E+00 099010E’06
095736E+00 0‘3064E-04
094114E+00 024592E-03
092650E+00 075248E-03
091332E+00 0167675-02
090148E+00 030576E‘02
089089E+00 048310E-02
038105E+00 063230E-02
087311E+00 087651E‘02
086579E+00 010332E’01
085945E+00 OIIZOIE-OI
085405E+00 OIIIZZE-OI
080955E+00 0999‘9E‘02
084594E+00 079361E-02
084318E+00 053053E-02
08¢I28E+00 026703E-02
084023E+00 07I3IIE-O3
084004E+OO 047371E-06
080073E+00 081‘22E-03
080230E+00 029902E-02
080080E+00 05934IE'02

144

011362254E+02
011289252E+02
011226971E+02
011173381E+02
011126930E+02
011086407E+02
011050855E+02
011019507E+02
010991742E+02
010967OSIE+02
010945013E+02
010925278E+02
010907553E+02
010891589E+02
NERERS
BIT
-058256E+01
~056363E+01
~0509IIE+01
-053687E+01
~052609E+01
~051635E+01
~050738E+01
-049904E+01
~049II9E+01
-048376E+01
-0457I5E+01
-043396E+OI
-04I3OOE+01
-039360E+01
-037532E+01
~035787E+OI
-034104E+01
*032463E+01
-030853E+01
-029260E+01
-027675E+01
-026089E+OI
-024‘92E+01
-022878E+01
~021238E+01
-0I9566E+01
~017855E+01
-016099E+01
~014292E+01
~012428E+OI
~0105052+01
-085182E+00
“060672E+00
-043524E+00
-021765E+00
055562E-02
023366E+00
046570E+00
070077E+00

011858321E+02
011846602E+02
011847579E+02
0118598135+02
011882479E+02
0119153OOE+02
0119585585+02
01201319BE+02
012081077E+02
012165502E+02
012272414E+02
012413356E+02
012614687E+02
012961136E+02

OM DEL
'0993 ‘0990
“0985 ‘0980
-0978 “097°
‘0971 1 '0960
-0963 “0950
‘0956 '0940
‘0948 -0930
“0941 -0920
-0933 ‘0910
‘0926 -0900
“0895 -0860
‘0865 -0820
'0834 -0780
-0802 -0740
‘0771 -0700
‘0739 “0660
-0706 -0620
’0673 “0580
-0640 -0540
‘0607 -0500
’0573 -0460
“0538 ‘0420
-0503 -0380
“0‘65 -0390
’0433 -0300
*0397 -0260
‘0360 ‘0220
“0323 '0I80
‘oZBO -0140
-0248 ‘0100
-.210 -0060
“0171 -0020
‘0132 0020
-0092 0060
“0052 0100
-0012 0140

0029 0180
0071 0220
0113 0260

0118583215+02
011846602E+02
011847579E+02
011859813E+02
011882479E+02
011915300E+02
011958558E+02
01201319BE+02
012081077E+02
0121655025+02
0122724I4E+02
012413356E+02
012614687E+02
012961136E+02

084826E+00
085272E+OO
085825E+00
086491E+00
087278E+OO
088195E+00
089253E+00
090460E+00
091829E+00
093365E+00
09507OE+OO
096926E+OO
098878E+00
010078E+01
010224E+01
010226E+01
010176E+OI
0100928+01
099614E+00
097652E+00
094761E+00
090500E+00
084115E+00
074149E+00
057OO3E+OO

088024E-02
010782E-01
011352E-01
010431E’01
083613E-02
057581E-02
032843E-02
0143I7E-02
038675E-03
023730E~04
022847E-04
065044E-04
021663E“04
017872E-04
020803E-03
024743E*03
015267E-03

-041875E-04

071882E-05
025796E‘03
012176E-02
036901E-02
091383E-02
020132E-01
041003E-01

145

093762E+00
011752E+01
014126E+01
016490E+01
018838E+01
021170E+01
023486E+01
025792E+01
028096E+01
030411E+01
032756E+01
035155E+01
037642E+01
040265E+01
043101E+01
046283E+01
047159E+01
048081E+01
049056E+01
050097E+01
051224E+01
052464E+01
053866E+0

055520E+01
057668E+01

0156
0199
0243
0288
.332
0378
.424
0471
0518
0566
0615
0664
0714
0764
0815
0867
0880
0893
0907
0920
0933
0946
0960
0973
0987

0300
0340
0380
0420
0460
0500
0540
0580
0620
0660
0700
0740
0780
0820
0860
0900
0910
0920
0930
0940
0950
0960
0970
0980
0990

”If!
RI!
81118
HHS
VHS
T"
al

I]

3 [11111111711]

[WNW]