Group Title: synthesis of minimum phase transfer functions by zero sharing
Title: The Synthesis of minimum phase transfer functions by zero sharing
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Title: The Synthesis of minimum phase transfer functions by zero sharing
Physical Description: x, 142 leaves. : illus. ; 28 cm.
Language: English
Creator: Walter, William Austin, 1937-
Publication Date: 1964
Copyright Date: 1964
 Subjects
Subject: Electric networks   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 140-141.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097951
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577383
oclc - 13977417
notis - ADA5078

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THE SYNTHESIS OF MINIMUM PHASE

TRANSFER FUNCTIONS BY ZERO SHARING


















By
WILLIAM AUSTIN WALTER


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











UNIVERSITY OF FLORIDA
December, 1964












ACKNOWLEDGMENTS


The author would like to express his gratitude to the

members of his supervisory committee and to Dr. W. H. Chen

for his supervision and guidance. The author also wishes

to thank the College of Engineering for granting the funds

which made it possible for him to study for the degree of

Doctor of Philosophy.













TABLE OF CONTENTS


ACKNOWLEDGMENTS . . . . . . . . .

LIST OF FIGURES . . . . . . . . .

ABSTRACT . . . . . . . . . .

Chapter

I. INTRODUCTION . . . . . . . .

1.1 The Basic Zero Sharing Procedure . .

II. CONDITIONS OF PHYSICAL REALIZABILITY . .

2.1 Conditions of Physical Realizability
of the Transfer Functions, T(s),
With Ladder Networks . . .....
2.2 Conditions of Physical Realizability
For the Parameters z21 and z22 or
-y21 and y22 With Ladder Network . .

III. TRANSFER FUNCTIONS WHICH ARE POSITIVE
REAL . . . . . . . . . .

IV. TRANSFER FUNCTIONS WITH REAL NEGATIVE
TRANSMISSION ZEROS . . . . . .

4.1 RC or RL Transfer Functions Which
Are Positive Real . . . . .
4.2 A Special Class of RC and RL
Transfer Functions . . . . .
4.3 Synthesis of T1(s) With RC
Networks . . . . . . . .
4.4 Synthesis of the Network N' as
an RC Ladder Network . . . . .
4.5 Illustrations of the Synthesis of
ZT(s), -A(s), or -A*(s) With RC
Networks . . . . . . . .
4.6 Synthesis of T2(s) With RC
Networks . . . . . . . .


Page

ii

v

viii


. 1

S6

S. 10



. 11


S. 12



S. 14


S. 18


S. 18

S. 20

S. 21

S. 29


S. 36

S- 45












TABLE OF CONTENTS--Continued


Chapter Page

4.7 Synthesis of the Network N" . . ... 53

V. TRANSFER FUNCTIONS WITH PURELY
IMAGINARY TRANSMISSION ZEROS . . . .. 54

5.1 Limitations of the Zero Sharing
Technique in the Synthesis of
LC Networks . . . . .... .. 54
5.2 Synthesis of A*(s) or G*(s) With
LC Networks . . . . . . . .. 59

VI. TRANSFER FUNCTIONS WITH COMPLEX CONJUGATE
AND NEGATIVE REAL TRANSMISSION ZEROS .... 71

6.1 Transfer Functions Whose Denominator
Polynomial Is of Second Degree . . .. 72
6.2 The Derivation of Regions In Which
A Positive Real Function, F(s)
May Have Poles ............ . 85
6.3 Transfer Functions Whose Denominator
Polynomial Is of Third Degree . . .. 95
6.4 Synthesis of the Networks N' Or N"
As RC, RL, or RLC Ladders . . . .. 109
6.5 Illustrations of the Synthesis of
RLC Networks . . . . . . ... 123

VII. CONCLUSIONS . . . . . . . . .. 136

LIST OF REFERENCES . . . . . . . .. 140

BIOGRAPHICAL SKETCH . . . . . . . ... 142












LIST OF FIGURES


Figure


1.1. A general four-terminal network . . .

1.2. Network arrangement for a prescribed
transfer function of the set Tl(s) . .

1.3. Network configuration for prescribed
transfer function of the set T2(s) . .

4.1. L network configuration . . . . .

4.2. Standard ladder configuration for
prescribed z21 and z22 . . . . .

4.3. Generalized basic sections (a) A type 1
section (b) A type 2 section . . .

4.4. Basic RC sections (a) Type 1 RC section
(b) Type 2 RC section . . . . .

4.5. Ladder network for ZT(s) as given
in (4-29) . . . . . . .

4.6. Network prediction for N' . . . .

4.7. Ladder network for A(s) as given
in (4-38) with K/H 3 . . . . .

4.8. Ladder network for A(s) as given
in (4-38) with K/H 15/7 . . . .

4.9. L network configuration . . . . .

6.1. The construction of a region in which
poles of F(s) may be selected . . .

6.2. A region in which poles of F(s) may be
selected, for F(s) as given in (6-29) .

6.3. A typical C path with arrows
indicating its negative direction . .


Page

. 1


8


8

S 27


S. 31


S. 31


S. 32


S. 38

S 42


S. 43


S. 46

S. 51


. 82











LIST OF FIGURES--Continued


Figure Page

6.4. A group of C paths on which poles
of F(s) may be placed . . . . ... 93

6.5. Regions of the s plane in which
condition (6-71) is satisfied . . ... 102

6.6. Regions of the s plane in which
condition (6-71) is satisfied . . ... 103

6.7. Generalized basic sections (a) A type 3
section (b) A type 4 section . . ... 110

6.8. Type 1 RLC section . . . . . ... 111

6.9. Type 2 RLC section . . . . . ... 114

6.10. Type 3 basic section with a shunt
resistance R(h) . . . . . . . 117

6.11. Type 4 basic section with a series
resistance R(h) . . . . . . . 121
b
6.12. First network realization for Y (s) as
given in (6-133) . . . . . 124

6.13. Regions in which the poles of -y21 may be
chosen, for YT(s) as given in (6-133) . . 125

6.14. Second network realization for YT(s) as
given in (6-133) . . . . . . . 126

6.15. Permissible pole locations of z21 for
ZT (s) as given in (6-139) . . . . .. 127

6.16. First network realization for ZT(s)
as given in (6-139) . . . . . .. 128

6.17. Second network realization for ZT(s)
as given in (6-139) . . . . . .. 129











LIST OF FIGURES--Continued


Figure Page

6.18. Permissible pole locations of z21
for A(s) as given in (6-149) . . . .. 130

6.19. Network realization for A(s)
as given in (6-149) . . . . ... 132

6.20. Network realization for A(s)
as given in (6-157) . . . . ... 135











Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy


THE SYNTHESIS OF MINIMUM PHASE TRANSFER FUNCTIONS
BY ZERO SHARING


By


William Austin Walter


December, 1964


Chairman: Dr. W. H. Chen
Major Department: Electrical Engineering


A new philosophy is presented for synthesizing ladder

networks from a prescribed minimum phase transfer function,

T(s). Synthesis procedures are developed in which a

sharing of transmission zeros is effected between the net-

work parameters, z21 and z22 (or -y21 and y22), with the

result that the synthesis problem is divided into two parts,

a two-terminal realization and a four-terminal realization.

This division usually reduces the labor involved in the

synthesis and often requires a smaller number of elements

in the final network than are required by other methods.

A wide range of problems exists in which the zero

sharing technique may be advantageously applied. Methods

are described and illustrated for the synthesis of RC, RL,


viii







LC, and RLC networks. For the synthesis of two-element-

kind networks, from transfer functions whose transmission

zeros lie on the negative portion of the a axis or on the

ju axis, the zero sharing approach is straightforward,

regardless of the complexity of the transfer function,

i.e., regardless of the degree of the numerator or denom-

inator polynomial. In the synthesis of transfer functions

which have complex as well as other types of transmission

zeros, the zero sharing approach is applicable to those

transfer functions whose denominator polynomial is of

rather low degree in the complex frequency variable s.

A central portion of the zero sharing approach is

concerned with the choice of the parameters z21 and z22

(or -y21 and y22) from a prescribed transfer function,

T(s) = P(s)/Q(s), of the response/excitation type. It is

shown that this selection is equivalent to removing certain

of the poles of 1/T(s) = Q(s)/P(s) and replacing them with

new poles in such a way that the modified function,

Q(s)/P'(s), has the following three properties:

1) Q(s)/P'(s) is positive real.

2) Q(s)/P'(s) may be broken into a sum of two parts,

each of which is positive real. One part must have for its

poles, all of the poles which were added in forming

Q(s)/P'(s). This part must not have any of the poles of

the original function, Q(s)/P(s). The second part must

have as its poles only the poles of the original function,

Q(s)/P(s).






3) The poles which are selected for the modified

function must not coincide with the zeros of Q(s).

The construction of the modified function Q(s)/P'(s)

is easily accomplished by inspection for those transfer

functions, whose transmission zeros lie on the negative

portion of the a axis or on the jw axis, and whose realiz-

ation is to result in a two-element-kind network. For

those transfer functions which have complex as well as

other types of transmission zeros, the selection of poles

for the modified function, Q(s)/P'(s), is restricted to

certain regions of the s plane. The location and approx-

imate shape of these regions are determined by a procedure

similar to that used in the design of control systems by

the root locus method.











CHAPTER I


INTRODUCTION


Many practical problems encountered in the design of

electrical networks are concerned with the transfer of a

signal from one pair of network terminals to a second pair.

In the four-terminal network shown in Fig. 1.1, terminals

1-1' are considered as the input, and transmission occurs

from this terminal pair to the output terminals 2-2'.



1 Il 12 2


E1 t2R2

1' 2'


Figure 1.1 A general four-terminal network

For a given load the transfer properties of the network may

be expressed in terms of a ratio of input voltage or cur-

rent, E or I to output voltage or current, E or I .

Expressed in the frequency domain, this description has the

form of a ratio of two polynomials in the complex frequency

variable, s o + ju, and is termed a transfer function.

The synthesis problem considered here is one of finding a

ladder network which possesses the properties that are spec-

ified in terms of such a transfer function. The resulting

1








network is to consist of lumped, linear, bilateral, and pas-

sive elements, and is to have no mutual inductances. Reali-

zation procedures will be developed for networks consisting

of RC, RL, LC, and RLC elements.

The network specifications will be given in terms of

one of six transfer functions of the response/excitation

type. For a current input at terminals 1-1' of Fig. 1.1,

three of these transfer functions are defined as

(1-1)
E2 12 I2
Z (s) = A(s) = and A*(s) -
T 1l 1 1
R2/oo R2O R2=O


T (s) will be used when referring to the set, Z (s), -A(s),

or -A*(s). For a voltage input at terminals 1-1' of Fig.

1.1, three transfer functions are defined as

(1-2)
12 E2 E2
Y(s) = G(s) - and G*(s) = -
SE E E
R2O R200 R2-00


T2(s) will be used to denote -YT(s), G(s), or G*(s). For

convenience T(s) will be used when referring to the entire

group of transfer functions, members of either T1(s) or

T2(s). From (1-1) it is seen that


12 (E2/R2)
A(s) -ZT(s)/R2 (1-3)
I1 Ii2









If A(s) is specified for a network which is terminated in a

finite non-zero load, an equivalent Z (s) may be easily
T
found from (1-3). It is sufficient, therefore, to consider

only the synthesis of Z (s). It will be found advantageous,

however, to consider the synthesis of the limiting case,

A*(s), individually. Similarly, it is necessary to consider

only YT(s) and the special case, G*(s).

The transfer specifications in terms of T(s) will be a

ratio of two polynomials in s having the form


P(s) sn + an-lsn- +---+ als + a0 (1-4)
T(s) = H = H
Q(s) sm + b +--+ bs + b
m-1 1 0


Assuming that P(s) and Q(s) have no common factors, the

finite zeros of T(s) occur at the zeros of the numerator

polynomial, P(s), and additional zeros may occur at infinity

for m > n. The zeros of T(s) are termed "transmission

zeros," since at these frequencies no transmission occurs

through the network.

Due to the assumed configuration of the resulting net-

work, only those transfer functions, T(s), whose finite

transmission zeros do not occur in the right-hand half of

the s plane will be considered. Such transfer functions are

commonly referred to as minimum phase functions. T(s) must

possess a number of additional properties in order to be

realizable in a ladder network made up of lumped, linear

elements. Discussion of these properties, however, will be








postponed until Chapter II.

The problem of realizing a ladder network, whose trans-

fer function is specified, is commonly handled by first

finding a pair of network parameters, and from these param-

eters, realizing a network. Use is made of the open circuit

parameters zll, z12, z21, and z22 in the synthesis of the

transfer functions, T (s). It may easily be shown that


z21
Z (s) -RA(s) (1-5)
1 + z22/R2



Use is made of the short circuit parameters yll' y12' y21'

and y22 in the synthesis of the set of transfer functions,

T2(s). It may be shown that

-Y21
-Y (s) G(s)/R2 (1-6)
1 + y22 R2



In selecting the network parameters, the ratio specified in

(1-4) is inserted in (1-5) or (1-6) for the appropriate

transfer function. From the expression which results, z2

and z22 (or -y21 and y22) are each selected as a ratio of

two polynomials in s. In general a great many possible

choices exist, and to ascertain which of these choices

result in a realizable pair of parameters requires a com-

plete description of the conditions of physical realizabil-

ity for the parameters of a ladder network. Such a









description is postponed until Chapter II.

To form a basis for choosing the network parameters,

attention is turned to the transmission zeros of T(s) and

the behavior of the parameters at these zeros. From (1-5)

(or (1-6)) it is evident that at a zero of T1(s) one of the

following two conditions must hold:

1) z21 (or y21) must have a zero which is not a zero of

the denominator, 1 + z22/R2 (or 1 + y22 R2)*

2) z22 (or Y22) must have a pole which is not a pole of

z21 (or Y21).

Two standard methods for choosing the parameters

represent the widely differing approaches which one may

take in making this choice [1l.* In the first approach z21

and z22 are chosen to have the same denominators, which

rules out possibility 2. All transmission zeros are then

the zeros of z21. The second approach assigns a constant

value to z21 with the result that all transmission zeros are

produced as private poles of z22. This method is restricted

in its use to only those transfer functions which are posi-

tive real. However, for this special case it yields a very

simple solution which requires only two-terminal techniques

in the synthesis.

The zero sharing method is primarily concerned with a

new philosophy which may be used in selecting network


*Brackets denote entries in the List of References.








parameters from the given transfer function. An orderly

procedure is developed for utilizing the simplifying proper-

ties of the second approach above while extending its

application to all physically realizable minimum phase

transfer functions.


1.1 The Basic Zero Sharing Procedure

Consider the network arrangement of Fig. 1.1 terminated

in a normalized load, R2 = 1. For such a termination (1-5)

and (1-6), when solved for z22 and y22, have the form


0(s)
z22 21 ( 1 (1-7)
22 21 HP(s)


and

Q(s)
2= y 1 (1-8)
22 21 HP(s)


respectively. For the special open circuit and short

circuit cases, A*(s) and G*(s), z22 and y22 have the form


Q(s)
22 21 HP(s) (1-9)


and

Q(s)
Y22 1 Ip(s) (1-10)


respectively.









The basic zero sharing procedure may be summarized in

the following steps:

1) Determine whether or not Q(s)/P(s) is positive real.

In the event that the positive real property is satisfied,

21 (or -y21) may be set equal to a constant, K, and step 2

may be omitted.

2) In the event that Q(s)/P(s) is not positive real,

choose z21 KN(s)/D(s) (or -y21 KN(s)/D(s)) in such a way

that the product, z21Q(s)/P(s) (or -y210(s)/P(s)) is posi-

tive real and has no zeros on the jo axis. (In the synthe-

sis of A*(s) or G*(s), however, zeros on the ju axis are

permitted.) The degree of the polynomial N(s) and that of

D(s) are to be as low as possible and still meet the desired

conditions on the product z21Q(s)/P(s) (or -y21Q(s)/P(s)).

N(s) is chosen to have certain of the zeros of P(s), as will

be described later in detail. If N(s) has some but not all

of the zeros of P(s) then the product z21Q(s)/P(s) (or

-y21Q(s)/P(s)) must in general satisfy certain restrictions
on the residues at each of its poles in addition to being

positive real. It is this case which represents a sharing

of transmission zeros and is of primary interest.

3) z22 as given in (1-7) or (1-9), or y22 as given in

(1-8) or (1-10), is next placed in the form


z22 z' + Z (1-11)
22 22 s












y y' + Y
22 22 p


(1-12)


respectively.

Here z' (or y' ) is to have the same poles as z21
22 22 21
(or -y ) and both z' (or y' ) and Z (or Y ) are positive
21 22 22 s P
real. The decomposition indicated in (1-11) or (1-12) per-

mits a network realization of the form shown in Fig. 1.2 or

Fig. 1.3, respectively.






With i Zs
parameters
1 z2 = z21 E2 R2
z' z -Z
22 22 s




Fig. 1.2 Network arrangement for a prescribed transfer
function of the set Tl(s)


Fig. 1.3 Network configuration for a prescribed transfer
function of the set T (s)
2^






9


4) The synthesis is completed by realizing the network

N' of Fig. 1.2 (or the network N" of Fig. 1.3) from its

associated parameters z' and z' (or y' and y') in a
21 22 21 22
ladder configuration. The synthesis method which will be

discussed was recently introduced by Chen [1]. This method

provides a well organized and unified approach to synthe-

sizing transfer functions of RC, RL, LC, and RLC ladder

networks.














CHAPTER II


CONDITIONS OF PHYSICAL REALIZABILITY


The six transfer functions, T(s), defined in (1-1) and

(1-2) must satisfy a number of conditions in order that they

represent the transfer properties of a physical network

which is made up of lumped, linear, bilateral, and passive

elements arranged in a ladder configuration. Similarly, the

open circuit and short circuit network parameters must meet

certain conditions in order that they be the associated

parameters of a physical ladder network. From the previous

discussion of the basic zero sharing process and from (1-5)

and (1-6), it is seen that the synthesis of a given transfer

function reduces to the problem of synthesizing a pair of

network parameters, z21 and z22, (or -y21 and y22.) For

this reason, those conditions on the parameters will be con-

sidered which are particularly applicable to the pair of

parameters z and z22 (or -y21 and y 22) The combined
21 22 21 22
realizability conditions on both T(s) and the parameters

will play an important role in the development of the zero

sharing method. First, these conditions aid in establishing

a criterion for selecting the network parameters from T(s),

and second, they are instrumental in determining permissible









decompositions of the parameters z22 or y22 to allow the

sharing of transmission zeros.

Only a summary of the realizability conditions will be

given here. For proofs of these conditions, see the

references [1, 2].


2.1 Conditions of Physical Realizability of the Transfer
Functions, T(s), With Ladder Networks [1]

The six transfer functions, T(s), must satisfy the

following conditions:

1) T(s) is representable as a ratio of two polynomials

in s with real coefficients.

2) The poles of T(s) if not on the ju axis are in the

left-hand half of the s plane, and those on the jw axis are

simple. (Of the six transfer functions, T(s), only A*(s)

and G*(s) may have poles on the ju axis.)

3) For the expressions ZT(s)/R2, -A(s), -R2YT(s), G(s),

-A*(s), and G*(s) the following conditions must be

satisfied: (a) All coefficients in the numerators and denom-

inators of these expressions must be nonnegative. (b) The

numerator coefficients for each of these expressions must be

no greater than the corresponding denominator coefficients.

4) The residues of the transfer functions A*(s) or

G*(s) at any of their imaginary poles, s ju, are

imaginary.

5) The transfer functions A*(s) and G*(s) do not have

a pole at either s 0 or s -oo.









These conditions must hold for a transfer function if

it is to be realizable in a ladder network having no mutual

inductances. Also, those cases of node bridging which pro-

duce transmission zeros in the right-hand half of the s

plane will not be considered. Therefore, T(s) will be a

minimum phase function, and, as a sixth condition, the zeros

of T(s) must be either in the left-hand half of the s plane

or on the ju axis.

Additional restrictions are placed on the transfer

functions when the network is to consist only of two kinds

of elements. RC, RL, and LC network realizations will be

considered separately, and the additional conditions which

apply will be considered along with each individual

synthesis procedure.


2.2 Conditions of Physical Realizability For the Parameters

z21 and z22 or -y21 and y22 With Ladder Networks [1]

The following conditions which the network parameters

must satisfy, are particular applicable to the pairs of

parameters z21 and z22 or y21 and y22:

1) z22 (or y22) and zll (or y11) must be physically

realizable driving point functions.

2) The poles of z21 (or y21), z22 (or y22), and zl

(or y11), if not on the ju axis, are in the left-hand half

of the s plane, and those on the ju axis are simple.

3) In general, the poles of z21 (or y21) are also the








poles of z22 (or y22) and zll (or y11), but z22 or zll (Y22

or y11) may have poles in addition to those of z21 (or Y21).

The poles of z21 (or y21) on the jw axis must always be

poles of z22 (or y22) and zll (or yll).

4) All coefficients in the numerators and denominators

of z21 (or -y21), z22 (or y22), and Z11 (or yll) are nonneg-

ative. The numerator coefficients of z21 (or -y21) are no

greater than the corresponding numerator coefficients of z2

(or y22) or zll (or y11) where z21, z22, and z11 (or -Y21'

Y22 and y11) are placed in a form having the same
denominators.

5) If z22 (or y22) or zll (or yll) is a LC driving

point function, and, therefore, a ratio of odd and even

polynomials in s, then z21 (or -y21) must also be a ratio

of odd and even polynomials.













CHAPTER III


TRANSFER FUNCTIONS WHICH ARE POSITIVE REAL


In the realization of a transfer function, T(s), in

ladder form, it is desirable to first determine whether or

not T(s) is positive real in addition to being physically

realizable as a transfer function. In the event that T(s)

does satisfy the positive real condition, a simple real-

ization may be effected using only two-terminal techniques.

In the interest of completeness, this case will now be

discussed.

For any of the six transfer functions under considera-

tion, if T(s) is positive real then Q(s)/P(s), considered as

an impedance, is a non-minimum-resistive function. This

follows for ZT(s), A(s), Y (s), and G(s) since these func-

tions are not permitted to have poles on the ju axis as

discussed in Chapter II. Although A*(s) and G*(s) may have

poles on the ju axis, the residues at such poles are imagi-

nary as the conditions of physical realizability dictate.

Imaginary residues at these poles,of course, rule out the

possibility of the function being positive real. One may

therefore conclude that if T(s) is positive real then

O(s)/P(s) considered as an impedance is not a minimum

resistive function.








In the following discussion a synthesis procedure is

given for realizing Z (s), A(s), and A*(s). A completely

dual procedure may be followed for YT(s), G(s), and G*(s).

Synthesis may be carried out by first placing Q(s)/P(s) in

the form



Q(s)/P(s) R + Z'(s) (3-1)



where Z'(s) is a minimum resistance driving point impedance

function. When Q(s)/P(s) is an RC or RL driving point

function, or when all transmission zeros lie on the ju axis,

the decomposition of O(s)/P(s) is easily effected by partial

fractioning. If O(s)/P(s) is a general RLC driving point

function, standard methods for the series removal of a

purely resistive element may be used. With Q(s)/P(s)

reduced as in (3-1), z22 has the form


z21
z (R + Z' (s)) -1 (3-2)
22 H


when Z (s) or A(s) is specified, and the network is to be
T
terminated in a normalized resistive load, R2 = 1. z22 has
2 z22
the form


z21
z = --(R + Z'(s)) (3-3)
22 H


for A*(s).








221 may now be chosen as a constant, K, and with the

proper choice of values for the constants, K and H, z22 will

be a physically realizable driving point function. In the

synthesis of ZT(s) or A(s) it is evident from a standpoint

of physical realizability that KR/H must be greater than

one. When it is required that the network be synthesized

within a constant multiplier it is convenient in many cases

to choose a value of K Q 1 and assure that the necessary

condition



H KR (3-4)
K + 1


is satisfied. When H is specified, an appropriate value of

K is easily determined from (3-4). It is seen in (3-4) that

the greatest possible upper bound on H is R. In the reali-

zation of A*(s) it is only necessary to assure that



H R (3-5)



in (3-3).

For ZT(s), A(s), or A*(s) the resulting network has the

form of Fig. 1.2 in which the network N' consists of a

single shunt resistance of K ohms and possibly a series

resistance. When equality occurs in (3-4) or (3-5) the

realization will not require the series resistive element.

For Y (s), G(s), or G*(s) the resulting network has the form

of Fig. 1.3. Here the network N" consists of a single





17


series resistance of 1/K ohms and possibly a shunt resist-

ance. Equality in (3-4) or (3-5) eliminates the shunt

resistive element.













CHAPTER IV


TRANSFER FUNCTIONS WITH REAL NEGATIVE TRANSMISSION ZEROS


The cases which will now be considered are those in

which the resulting network is to consist of only resistive

and capacitive elements, or it is to consist only of resis-

tive elements, and inductive elements with no mutual

coupling. The six transfer functions, T(s), of networks

which are composed of these kinds of elements, arranged in a

ladder configuration with no-node bridging, will possess the

following two properties in addition to the properties of

physical realizability stated in Chapter II:

1) The transmission zeros must lie on the negative por-

tion of the a axis including the points a 0 and o -co.

2) The poles of T(s) must be simple and lie on the neg-

ative portion of the a axis excluding the points a 0 and

0 = 00.


4.1 RC or RL Transfer Functions Which Are Positive Real

As indicated in the discussion of the general procedure

to be followed, it is desirable to first determine whether

or not Q(s)/P(s) is positive real. Any one of the transfer

functions Z (s), -A(s), and -A*(s) for RC networks or

-YT(s), G(s), and G*(s) for RL networks is positive real if

and only if its reciprocal, Q(s)/P(s), is of the form

18









Q (s)
= R(s)
P(s)


(s p1)(s -
(s z )(s -
1


p2)-----(s p)
z )-----(s z )
2 n


where the p. and z. are real with


0 > P > Z1 > p1 > z2 > P2 ---- > Zn > Pn


(4-1)


R(s) = 1


s PO
n(s)
s


Any one of the transfer functions -YT(s), G(s), and G*(s)

for RC networks or ZT(s), -A(s), and A*(s) for RL networks

is positive real if and only if its reciprocal, O(s)/P(s),

is of the form


Q(s) (s pl)(s p2)----(s pn)
R(s)
P(s) (s zl)(s z2)----(s zn)


where the p and z are real with


(4-2)


0 p > pl > >2 > z2 ---- > n > n


0> > PO > z > > z2 > zn > Pn f









for


R(s) 1


or


R(s) (s p ),


respectively. If T(s) satisfies the conditions of (4-1) or

(4-2) a realization may be easily effected by following the

procedure given in Chapter III. If, however, the resulting

network configuration of Fig. 1.2 with a rather complex Zs

and with N' consisting of a single shunt resistor, or the

network of Fig. 1.3 in which N" consists of a single series

resistor is undesirable, various degrees of zero sharing

may be achieved by use of the methods which will be

discussed in Section 4.3.


4.2 A Special Class of RC and RL Transfer Functions [3]

In addition to those transfer functions which are pos-

itive real, there exists another class which may be synthe-

sized using only two-terminal techniques. For RC or RL '

networks, this class consists of those transfer functions,

T(s), whose reciprocal, Q(s)/P(s), has the following

properties:

1) Q(s)/P(s) is of the form


Q(s) (s pl)(s p2)----(s ) (43)
P(s) (s z1)(s z2)-----(s n)


where n < m.









2) The zeros, pi, of Q(s) must be simple and must lie

on the negative portion of the a axis excluding the points

a = 0 and a -oo.

3) The zeros, zj, of P(s) must have no greater multi-

plicity than two and must lie on the negative portion of the

o axis including the points a = 0 and o = -co.

4) If one begins with the critical frequency of

Q(s)/P(s) nearest the origin and divides the entire group of

critical frequencies into adjacent pairs (counting a double

pole as two adjacent poles), then each pair of critical

frequencies must consist of a pole and a zero.

The realization of transfer functions which satisfy

these conditions may be carried out by using the method

which is developed in Section 4.3. An example of the syn-

thesis of such a transfer function is given in Section 4.5.


4.3 Synthesis of T1(s) With RC Networks

Attention is now turned to the problem of synthesizing

an RC network for which Z (s), -A(s), or -A*(s) is speci-

fied. As mentioned previously, T1(s) is used to represent

this set. The only restriction is that T1(s) be physically

realizable with resistive and capacitive elements as

described at the beginning of the present chapter. The

transfer impedance or current-ratio function, T1(s), is

given as a ratio of two polynomials in the form









p(s)
T () H(s)
1 Q(s)


(s zl)(s z2)-----(s zn)
H(s p)( ) -
(s pl)(s p2)-----(p pm)


with poles and

sion for T1(s)

solved for z22


zeros on the -o axis and m L n. The expres-

in terms of the open circuit parameters when

is


Q(s)
z22 21 HP(s)- 1


N(s) Q(s)
z K/H 1
22 D(s) P(s)

0(s)
=K/H ) 1
P'(s)


for T (s) ) ZT(s) or

ation, R2 = 1, or


T1(s) = -A(s) and a normalized termin-


Q(s)
22 = z21 (s)
K P(s)


(4-6)


K/ (s)
P K/H
p'(s)


for Tl(s) A*(s). As suggested in the outline of the

general procedure to be followed, choosing the open circuit

parameters z21 and z22 is the initial step in the network


(4-4)


(4-5)








realization. z21 should be chosen as a rational function of

the form


N(s) K(s-zl)(s-z2)----(s-zn-q+k)
21 K (s) (s-r )(s-y2)----(s-y ) k(47)



in which both poles and zeros are real. The transmission

zeros zl, z2,----zn-q+k are chosen according to a set of

rules which will be given and are not necessarily the first

n-q+k zeros of T1(s) given in (4-4). For ZT(s) or A(s),

q = m, and for A*(s), q may take on either of two values,

q = m or q m + 1, as will become evident in the following

discussion. The choice of poles and zeros of z21 is

dictated by the following considerations:

1) The zeros of z21 are selected from the set of trans-

mission zeros of the given transfer function as indicated

in (4-7).

2) The poles and zeros of z21 must lie on the -a axis

including the point a 0. Poles of z21, however, are not

permitted at s oo.

3) The poles of z21 must not coincide with the zeros

of Q(s).

4) The poles and zeros of z21 must be chosen in such a

way that the product z21Q(s)/P(s) is an RC driving point

function of the form










Q(s) N(s) Q(s) Q(s)
z21 P = K
1 T D(s) P(s) P'(s) (4-8)


(s-pl)(s-P2)--(s-pm)
K
(s-l)(s-y2)--(s-yk) (s-k+l)--(s-zq)


in which the denominator polynomial may or may not have a
simple zero at s = 0. The set of transmission zeros, zk+l,

z k+2---- z consists of those finite transmission zeros
k+2' q
which do not belong to the set, zl, z2,---, zq+k. The

poles and zeros of z21Q(s)/P(s) must alternate on the -a
axis with a pole as the leading singularity. The possible

values of q, q = m for ZT(s) or A(s) and q = m or q m + 1

for A*(s), become evident when (4-8) is substituted in

(4-5) and (4-6).

5) In order to take full advantage of the simplifica-

tion which is possible using this technique one should
assign as few poles and zeros to z21 as possible and still

satisfy condition 4 above. Although such a choice of z2

is not essential, it does assure that two-terminal tech-

niques will be used as much as possible throughout the

remainder of the synthesis.

With z21 chosen according to the considerations above,

one next places the product z210(s)/P(s) in partial-

fractioned form. z22 may then be represented as








(4-9)
[ al ak ak+l am
z2 = K/H ( + -- +-+ --) + (-- +-+ ) -
2( + S-y s-z S-Z
SS-1 -k k+l m


for ZT(s) or A(s), and
(4-10)
[ al ak ak+l aq
SK/H a + -- +--+ --) + (-- +--+
22 S- I S-Tk S-Zk+l S-Zq



for A*(s). In (4-10), a0 = 1 for q m, or a0 = 0 for

q m + 1. At this point it is necessary to determine a

value for K/H in (4-9) or (4-10), and it is well to consider

briefly the significance of this choice. The constant H

determines the overall magnitude of the voltage or current

response for a given voltage or current excitation. The

constant K was introduced as a scale factor of z21 in order

to provide flexibility in synthesizing the networks N' and

N" in Fig. 1.2 and Fig. 1.3 respectively. This flexibility

is desirable since the network N' or N" is to be synthesized

by one of the well known synthesis techniques for ladder

networks which is directed toward synthesizing a given trans-

fer function within a multiplicative constant.

In a particular problem, the constant H may be speci-

fied or it may simply be required that the synthesis be

carried out to within a constant multiplier. When H is

specified or when a high value of H is desired, the permis-

sible range of values for H is of interest. An upper limit









is placed on H by the conditions of physical realizability

for transfer functions as stated in property number 4 of

Chapter II. Additional limits are placed on H by the topo-

logical form of the network realization and considerable

attention has been given to determining these limits for

specific network configurations [4].

Two sets of limits on K and H exist after the zero

sharing step and will now be considered. The first set is

for the special case in which the synthesis is to be car-

ried out entirely on a two-terminal basis. Here, O(s)/P(s)

must satisfy the conditions of section 4.2. After the

application of the zero sharing techniques just discussed,

a pair of open circuit parameters z21 and z22 will result in

which z21 is positive real. z21 may be represented in

partial fractioned form as


S B1 2 Bk I
L K + --- + -- +---+ (4-11)
s 1 s 2 s rk



where SO = 1 if q n or B0 0 if q n. For the reali-

zation of the network N' in an L network as shown in Fig.

4.1 in which Z1 and Z2 are RC driving point impedances, it

is both necessary and sufficient that


K/H 1 > KB,
-- 0'


(4-12)









and that



a /Bi > H for i 1, 2,-----, k, (4-13)



for ZT(s) or A(s) and a resistive termination, R2 = 1.

This realization for a specified A*(s) requires that



ai/Bi > H for i = 0, 1, 2,----, k. (4-14)



r- ----------
-------------------- --------------0
I I
ZI




O II
11

Network N'

Fig. 4.1 L network configuration


When synthesizing ZT(s) or A(s) one may easily select a

suitable value of H from (4-13). This value of H will

determine an acceptable range of K as indicated in (4-12).

K may then be chosen within this range on the basis of

producing desirable element values in the normalized network.

If the value of I is of secondary importance, one may first

make a desirable choice of K and accept further restric-

tions on H. When synthesizing A*(s), H may be selected

within the limits set by (4-14). The value of K is not

restricted and may simply be chosen in such a way as to









produce desirable element values in the normalized

network.

The second set of limits on K and H is for the case in

which the network N' is to be synthesized in a general

ladder configuration of resistive and capacitive elements.

For ZT(s) or A(s), the open circuit parameters for the

network N' are



z' z (4-15)
21 21


and

r al a2 ak ]
z' K/ 1 + --- + -- +---+ -- -1 (4-16)
22 S-r1l s-Y2 s-k



If each parameter is placed in the form of a ratio of two

polynomials with the numerator in expanded form, property 4

for the network parameters may then be applied to determine

limits on H and K. It is generally desirable when synthe-

sizing N' in a general ladder configuration to select a

value for the ratio K/H but to leave the particular values

of K and H unspecified. The value of K/H may be selected

in such a way as to control the spread of element values in

the resulting network. For realizability it is necessary,

but not sufficient, that the condition


K/H > 1


(4-17)









be satisfied. Equality may occur in (4-17) only when

m > n. The manner in which K and H may be determined may

be best explained with the illustrations which will be

given in section 4.5. In synthesizing A*(s) the role of

K/H is not critical and any convenient value of K/H may be

selected.

With the ratio K/H determined one continues the synthe-

sis by realizing Z as an RC driving point impedance. From

(4-9) or (4-10) Zs is given by



Z K/H +- -------+ -- (4-18)
s s z s z
Zk+1 m-z



Finally, the network N' is synthesized either by standard

two-terminal techniques for RC networks or by the methods

to be discussed in section 4.4.


4.4 Synthesis of the Network N' as an RC Ladder Network

In this section a procedure will be described for syn-

thesizing the network N', whose parameters are z' and z'
21 22
As explained earlier, the driving point impedance z22 is

obtained by removing the series impedance Z from z22 and

z' z The method which will be described was intro-
21 21
duced by Chen (1]. Although it produces the same results

as zero-shifting methods, it has two distinct advantages.

First, it is more straightforward in its application than

the zero-shifting methods, and, second, it may be extended









to the realization of transfer functions which have complex

transmission zeros [1]. This extension will be described

in Chapter VI. The underlying principles which will be

considered now are applicable to LC, RC, RL, and certain

RLC synthesis problems.

The basic approach to synthesizing N' consists of the

following two steps. First, a network configuration is

predicted which consists of a number of basic transmission-

zero-producing sections. Second, z' is synthesized as a
22
driving point function having the predicted configuration.

The synthesis of z' is carried out section by section in
22
such a way as to assure that each section produces the

desired transmission zero.

Associated with the network N' is the pair of open-

circuit parameters z21 and z'2. Due to the application of

the zero sharing technique of section 4.3, z' and z' have
21 22
identical denominators. Therefore, all transmission zeros

of the network N' are the zeros of z'1, and may be easily
2i\
determined by inspection. These transmission zeros will

be produced by either shunt or series branches of the

standard ladder configuration of Fig. 4.2. This configu-

ration which is the assumed form of N' will have transmis-

sion zeros which are produced either by a pole of a series

impedance Zk (k 0) or by a pole of a shunt admittance Yi.

In general, each finite nonzero pole of Zk(k / 0) or iY

produces a transmission zero. However, several poles of




















Fig. 4.2 Standard ladder configuration for prescribed
z21 and z22


Zk or Y at s = 0 or s = co may contribute to a simple

transmission zero at s = 0 or s -oo respectively.

The ladder structure of Fig. 4.2 is considered here to

be made up of a predicted configuration of certain building

blocks. These building blocks or generalized basic sections

are shown in Fig. 4.3. Each of the two types of basic

sections consists of a "principal branch" which is shown as

a shaded box and an "auxiliary branch" which appears as a

clear box. In general, the principal branch is used to

produce either one, or a complex conjugate pair of finite,

nonzero value transmission zeros. The auxiliary branch

contributes to a transmission zero at s = 0 or s 00 ,

or it may make no contribution to a transmission zero.

Zb /Zb





o -o o- o
(a) (b)

Fig. 4.3 Generalized basic sections (a) A type 1 section
(b) A type 2 section








Attention is now turned to the problem of synthesizing

N' as a ladder network which is made up of only resistive

and capacitive elements. The first step is to predict a

possible ladder configuration which has the form shown in

Fig. 4.2 and consists of basic RC sections and possibly some

additional RC branches. There are two basic RC sections,

type 1 and type 2, as depicted in Fig. 4.4. The subscript



(k) Rb
Rb I -_-
y(k) r- s-zTLk
a R(k)I Sk Ci
a L G(k)
I- ---- Ga


Zk+l Y* Zk Z* Y
k k k+1 k k
(a) (b)
Fig. 4.4 Basic RC sections (a) Type 1 RC section
(b) Type 2 RC section

or superscript, k, indicates that this is the kth section in

the ladder configuration with the sections numbered consecu-

tively from right to left. In each basic section of Fig. 4.4

a transmission zero is produced at s = z In Fig. 4.4 (a),
(k)k
if Rk) is set equal to zero, the principal branch of this

type 1 RC section contributes to a transmission zero at

s o. In Fig. 4.4 (b), if R(k) is set equal to infinity,
b
the principal branch of this type 2 RC section contributes to

a transmission zero at s 0. Transmission zeros may there-

fore be produced at any point on the negative a axis,

including s 0 and s -00 by the use of type 1 and type 2









RC sections. The prediction step is accomplished as follows:

1) Examine z21 to determine the transmission zeros of

N'.

2) Arrange type 1 and/or type 2 RC sections (and possi-

bly some additional RC branches) in the ladder configuration

of Figure 4.2. (Note that a shunt element appears at the

extreme left, and that ZO may or may not be present as indi-

cated by its presence in dotted lines.)

3) In the arrangement described in 2, the principal

branch of each RC section is assigned one transmission zero

which it is to produce.

4) The auxiliary branch of each RC section makes no

contribution to a transmission zero.

A number of predicted configurations are possible for a

given set of transmission zeros. In this particular situa-

tion two of the possible predictions are (a) a configuration

consisting exclusively of type 2 RC sections with a shunt

resistive branch at the far left, and (b) a configuration

consisting exclusively of type 1 RC sections. It should be

understood that some predictions will be realizable, while

others may be nonrealizable. The appearance of negative

elements will indicate a nonrealizable prediction. A simple

rearrangement of the order of the transmission zeros will

frequently bring a nonrealizable situation into a realizable

form.

The synthesis of the predicted network follows an

orderly procedure in which one basic section is realized at









a time. One begins by synthesizing the first section at the

extreme right (k 1). For this section the impedance
1
Z z' or admittance Y1 is known, and, as a result
22
of prediction, a particular transmission zero has been

assigned to this section. This is sufficient information

to determine the elements of the first section. Upon

removing the elements of this first section from the imped-

ance Z1 or admittance Y1, one obtains the impedance or

admittance looking into the next section, Z2 or Y2, respec-

tively. This process is repeated for the entire predicted

configuration.

The technique for synthesizing a Type 1 RC section is

as follows:

1) Evaluate R(k) where
b


Rb Zk(s) (4-19)
s = z
k


2) The admittance Y*k looking into the remaining por-

tion of the ladder after R(k) is removed is
b


Y* =
k (Z R(k) (4-20)
k b


3) Y* is of the form
k

Mks
Y* = --- + Y + 1 (4-21)
k s z k
k









where



[(s zk) Y|
M = ---- Y*
k k s
S = Zk


The elements of Y are then
a


(k) 1 (k) Mk
R -andC (4-22)
a Mk a -zk



4) The impedance looking into the next basic section is



Zk+1 -(4-23)
Y* Mks
k
s zk



The technique for synthesizing a Type 2 RC section is
(k)
1) Evaluate G(k), where
a

(k) (s)
Ga v= (4-24)
s z



2) The impedance Z* looking into the remaining portion
k
of the ladder after G (k) is removed is
a


Z* (4-25)
k (Y G(k))
K a


3) Z* is of the form
k










Nk
Z* = ---- + Z
k s k + 1 ,


where (4-26)


Nk (s zk) Z
s zk


The elements of Zb are then
N
1 (k) k
C and R (4-27)
b Nk b -Zk


4) The admittance looking into the next basic section

is



I
Yk + 1 ) Z Nk (4-28)

S Z
k



4.5 Illustrations of the Synthesis of ZT(s), -A(s), or

-A*(s) With RC Networks

For the first illustration a transfer function which

belongs to the class described in section 4.2 will be

considered. Let it be required to synthesize


(s + 2)(s + 3)
Z (s) H (s + 2)(s + (4-29)
T (s + 1)(s + 4)


in a network with a normalized load R2 1.








In accordance with the procedure given in section 4.3,

z21 is chosen as


s + 2
z 2 K--
21 s


(4-30)


Then


SK/H F (s + l)(s + 4)
22 K/+H )
22 | s(s + 3)


(4-31)


Upon partial fractioning z21Q(s)/P(s), z22 becomes


(4-32)


The limits on K and H are set by (4-11) and (4-12) as


K/H 1 2 K


(4-33)


2/3 > H


(4-34)


is chosen equal to its limiting value, that is H 2/3,


H
IK> = 2
-1 -H


(4-35)


K may be chosen equal to its limiting value, 2, in which

case z22 and z21 become


If H

then


[ 4/3 2/3
22 /H 1 + --- + -13]
22 1 s s+3









(4-36)


2
2 2 + 4/s + -
122 s + 3


z 2 + 4/s.
21


When the private pole of z22 is removed, the parameters of

the network N' are



z' = 2 + 4/s
22


and
(4-37)


z' = 2 + 4/s .
21


The resulting network for Z (s) as given in (4-29) is shown

in Fig. 4.5.


R-2/3


R2-1


Fig. 4.5 Ladder network for ZT(s) as given in (4-29)


For the second illustration a transfer function will

be considered whose realization requires that the network

N' have the form of a ladder with more than one section.








Let it be required to synthesize


P(s) (s+2)(s+6)(s+7)(s+9)(s+10)
-A(s) H H (4-38)
Q(s) (s+l)(s+3)(s+5)(s+8)(s+12)


in a network which is terminated in a normalized load,

R 1.
2
As explained in section 4.3, one possible choice of

z21 is


21 K (s + 7)(s + 9) (4-39)
s(s + 4)

For this choice of z21, z22 becomes


22 = K/n [(s+l)(s+3)(s+5)(s+8)(s+12) (4-40)
22 1 s(s+2)(s+4)(s+6)(s+10)


The product, z21 O(s)/P(s), is next partial fractioned,

which places z22 in the form


[ 3 45/32 1 15/16 21/32
z22 K/H 1+-+---+-+---+ -1 (4-41)
22 s s+2 s+4 s+6 s+lOJ


At this point it is necessary to select a value of K/IH.

Property 4 for the network parameters, given in Chapter II,

states that the numerator coefficients of z'2 must be greater

than or equal to the corresponding numerator coefficients

of z' Application of this property places the following
21









condition on H.



H 5 12/63 (4-42)



For this value of H, K/H must satisfy the condition



K/H > 63/51 .(4-43)



In selecting a desirable K/H within this wide range of

permissible values, the following three facts are helpful:

1) z2 is given by



z' K/H + 3 + -1 (4-44)
22 s s+4


As K/H decreases from infinity to its lower limit, the zeros

of z' move to the left along the -a axis from their ini-
22
tial positions at the zeros of f(s), where


3 1
f(s) = 1 + + (4-45)
s s+4


The zero movement is covered in detail in reference [5].

2) To prevent the occurrence of a large spread in ele-

ment values in the normalized network one should avoid

selecting a value of K/H which places a zero of z' very near
22
to, but not touching, the transmission zero which is to be

produced by the first section at the right of N'.









3) This shift in the zero locations of z' is identical
22
to the shift which occurs when synthesizing a type 1 RC

section.

With these facts in mind the synthesis is continued by

simply selecting K/H = 2, the next larger integer value

above the limiting value of 63/51, for H 12/63. As the

first step in synthesizing the network N', a network config-

uration is predicted as shown in Fig. 4.6. The realization

of the first basic section at the right of N' is begun by

evaluating Zl(s) -7 z22(s) -7 or Z(s)
Ss -7s = -7 s = -9
z22(s) s Negative values are found for both s = -7
and s -9 which indicates that a negative resistance must

appear in the position of R1l) in Fig. 4.6. Therefore, for

K/H = 2, the network prediction of Fig. 4.6 is nonrealiz-

able. According to remarks 1 and 2 above, a different

choice of K/H may make the prediction of Fig. 4.6 realiz-

able. (See reference [5]).

A second choice, K/H = 3, eliminates the problem just

encountered. For this choice, the open circuit parameters

of N' are

2
9 3 2s + 20s + 36 (4
z' = 2 + + (4-46)
22 s s + 4 s(s + 4)


and

(s + 7)(s + 9)47)
z' = =K (4-47)
21 21 s(s + 4)









-----Section 2


(2)
Rb


S---Section 1---
I I
i R^ bl)
] b


Y* Z Y*
2 2 1


Fig. 4.6 Network prediction for N'


For the network prediction of Fig. 4.6, the synthesis of N'

is carried out as follows:

2
2s2 + 20s + 36
Z1 z2 + (4-48
1 22 s(s +4)


)


R() 2/5
b


8 x 13
2 15


R() 4 x 13
a 25


c() 25
a 4 x 13x9


[s +5/2


(2) 12 x 13
b 5 x 7


R(2) 4 x 13
a 3 x 7


C(2) 3
a 4 x 13


The impedance Zs, whose position in the network is shown in

Fig. 1.2, is given by


Z 9/320 15+ 0 7 (4-49)
s s + 2 s + 6 s + 10








The realization of Z may be carried out by any of the
s
standard synthesis procedures for RC driving point functions.

The resulting network is shown in Fig. 4.7


I R(2 4x13 ,,(I 4x13
{ R a 3x7 a 25


(2) 3 C(1) 25
a 4x13 a 4x13x9

Network N'


Fig. 4.7 Ladder network for A(s) as given in (4-38)
with K/H = 3


The situation frequently occurs in which K/H can be

chosen so that z 2 will possess a zero at one of the trans-

mission zeros which are to be produced by the network N'.

This choice of K/H eliminates the need for the auxiliary

branch in the first basic section at the right of network N'.

The possibility of choosing K/H on this basis depends on the

value of the expression within the first parentheses of

(4-9). To use this criterion in choosing K/H, it is neces-

sary that the inequality










0 < + + -- ---+ -- < 1 (4-50)
s 1 s 2 s J


be satisfied at one or more of the transmission zeros

assigned to z21. Some flexibility in satisfying this condi-

tion does exist, since a great many choices of N(s)/D(s) are

possible, each of which meet the restrictions set by condi-

tions 1 through 5 of section 4.3. However, it is difficult

to select N(s)/D(s) on this basis. If (4-50) is satisfied,

K/H may be chosen as

(4-51)
[ a, a2 ak -1
K/H + -- + -- + +-----+ -
s YI s r2 s k s 1


where z1 is the transmission zero which is to be produced by

the right most section of N'. The remainder of the synthe-

sis is then carried out as described in section 4.4.

The application of the above criterion in choosing K/H

will now be illustrated for the transfer function given in

(4-38). The expression indicated in (4-50) is first eval-

uated at each of the transmission zeros to be produced by

z21. From (4-39) and (4-41)


F 3+ 1 5
1 + + 4- -2 (4-52)
s s + 4 2-1
1 1 Is -7










+ 3 + ] (4-53)
s + 4 15

s = -9


are evaluated. For K/H chosen as



K/H = 15/7 (4-54)



z22 becomes
(4-55)
15 x 45 15 x 15
45/7 7 x 32 15/7 7 x 16 15/32
z 8/7 + + + + + ---
22 s s + 2 s+4 s + 6 s + 10


With the series removal of the impedance 's from z22, z22

becomes


45/7 15/7
z' 8/7 + 45/7+ 15/7 (4-56)
22 s s+4


A network is then predicted having the form given in Fig.

4.6. The realization, however, will not require the resis-

tive element R1). The synthesis procedure given in

equations (4-19) through (4-23) is next carried out. The

resulting network is shown in Fig. 4.8.


4.6 Synthesis of T2(s) With RC Networks


A synthesis procedure will now be developed for the

realization of a network for which Y (s), G(s), or G*(s) is

specified. The only restriction on T2(s) is that it must
2t













(2) 12x13
b 7x7


Network N'


Fig. 4.8 Ladder network for A(s) as given in (4-38)
with K/H 15/7


be a physically realizable transfer function of an RC net-

work, as described at the beginning of this chapter. The

procedure for synthesizing T2(s) is the same as that which

was developed for Tl(s) in section 4.3 except for variations

in the permissible pole and zero locations in choosing the

network parameters. The transfer admittance or voltage-

ratio function is given in the form


P(s)
T (s) H (s)
2 Q(s)
(4-57)

H(s zl)(s z2)----.(s zn)
(s Pl)(s p2)---(s p)
(s pl)(S p2)-....(s -p)









with poles and zeros on the negative a axis and m r n.

From the expression for T2(s) in terms of the short circuit

parameters, y22 is


Q(s)
Y22 -Y21 HP(s)

N(s) O(s)
K/H N(s) (s) 1 (4-58)
D(s) P(s)


0(s)
= K/H ) -1
P'(s)


for T2(s) = YT(s) or T2(s) = G(s) and a normalized termi-

nation, R2 = 1, or


Q(s)
22 = -Y21 HP(s)
(4-59)
0(s)
K/H
p'(s)


for T2(s) = G*(s). -y21 is chosen as


N(s) (s-) (s-z2)----(S-zn-q+k)
-Y21 = K K (4-60)
21 D(s) (s-r )(s-2 )---(s-r )



where q m or q = m 1. Equation (4-60) indicates that

the number of transmission zeros assigned to -Y21 is n-q+k.

They are not necessarily the first n-q+k zeros of T2(s) as








given in (4-57). The poles and zeros of y21 are chosen on

the basis of the following considerations:

1) The zeros of y21 must be selected from the set of

transmission zeros of the given transfer function. Although

they will not be discussed here, certain cases do arise in

which the choice of other zeros is advantageous.

2) The zeros of y21 may lie anywhere on the negative a

axis, including s 0 and s = oo. The poles of y21 must lie

also on the negative a axis including the point s oo but

excluding the point s 0.

3) The poles of y21 must not coincide with the zeros

of Q(s).

4) The poles and zeros of y21 must be chosen in such a

way that the product -y21Q(s)/P(s) is an RC driving point

admittance function of the form

(4-61)
Q(s) (s-Pl)(s-p2)--(s-Pr)
P(s) (s-z )(s-)2)--(s-)k)( s-zk+1)--(-z



in which q = m or q = m 1. The set of transmission zeros,

zk+l, zk+2,---z consists of those finite transmission zeros

which do not belong to the set z1, z2,---zn-q+k. The poles

and zeros of -y21 (s)/P(s) must alternate on the negative a

axis with a zero as the leading singularity. This of course

excludes the possibility of y210(s)/P(s) having a pole at

s 0.

5) One should assign as few poles and zeros as possible









to y21 and still satisfy conditions 1 through 4 above. As
before, various degrees of zero sharing between y21 and y22

may be produced by assigning more transmission zeros to y21

than the minimum required number.

After choosing y21 in accordance with the above consid-

erations one next places the product -y21Q(s)/P(s) in

partial fractioned form. y22 may then be represented as

i ails aks
2 K/H (a s + a + s +----+ ) +
2 -1 0 s r- s Tk

(4-62)

ak+ls as+) -1
s z s Zq
k+l q


for Y (s) or G(s) and



Y22 K/H (a_ ls + a als +---+ ) +
s r s k

(4-63)
( k+ls as
s Z s Z
k+l ]


for G*(s). In (4-62) and (4-63), al 1 for q m 1,

or a-1 0 for q m.

At this point it is necessary to assign a value to K/IH.

As in section 4.3, two different sets of limits on K and H

will be considered. The first is for the more restrictive








case, in which the synthesis of N' is to be carried out

entirely on a two-terminal basis. Here, Q(s)/P(s) must

satisfy the conditions of section 4.2. After the application

of the zero sharing techniques above, a pair of open circuit

parameters, y21 and y22, will result in which y21 is an RC

driving point admittance. y21 may be represented as



- K B s + B + --- + + ---- (4-64)
21 -1 0 s r s r



where -I 1I for n = q + 1, or 8_ = 0 for n I q +1. For

the realization of the network N" in an L network as shown

in Fig. 4.9, in which Z1 and Z2 are RC driving point imped-

ances, it is both necessary and sufficient that



(K/H)a0 1 2 K,0 (4-65)



and that


a.
H > H for i = -1, 1, 2,----, k (4-66)
i

for YT(s) or G(s) with a resistive termination, R2 1.

Such a realization for a specified G*(s) requires that


"i
-> H for i = -1, 0, 1, 2----, k. (4-67)


When synthesizing Y(s) or ), may be selected in
When synthesizing YT(s) or G(s), H may be selected in














r----1- ----I---


Z2


Z1
0 --------------
Network 'N

Fig. 4.9 L network configuration


accordance with (4-66). A rearrangement of (4-65),



K/H > 1 (4-68)
a0 H8O


shows the dependence of K/H on this choice of H. If the

form of the network N" is restricted to that shown in Fig.

4.9, the maximum value of H for this configuration may be

realized by equating H to its upper limit and then choosing

K/H according to (4-68). A second alternative is to choose

K/H in such a way as to reduce the spread in element values

in the resulting network. Examination of (4-62) reveals very

quickly the proper choice of K/H to reduce this spread.

When G*(s) is to be synthesized, H may be selected within

the limits set by (4-67). The value of K is not restricted

and may be chosen so as to produce convenient element

values in the normalized network.

The second set of limits on K and H is applicable when








N" is to be synthesized in a general ladder configuration

composed of resistive and capacitive elements. Property 4

for the network parameters may be applied to y"1 and y2 to
21 a y'i2 to
determine the limits on H and K. Here y" and y" are given
21 22
by


y" = y (4-69)
21 21

and
(4-70)
Y[ als a2s aks
y" = K/H a s + a + ---- + ---- +--+ -- .
22 -1 a0
s I s 2 s yk


When synthesizing YT(s) or G(s), it is usually desirable to

choose a value for K/H but to leave the individual values of

K and H unspecified. The value of K/H may be selected in

such a way as to control the spread of element values in the

resulting network. In the synthesis of G*(s), the value of

K/H is not critical and any convenient value may be chosen.

After a suitable value of K/H has been selected, the

synthesis is continued by realizing Y as an RC driving

point admittance, where

Sak+lS aqs
Y K/H +---+ --- (4-71)
I- k+1 q


The remainder of the procedure consists of synthesizing the

network N" either by standard RC two-terminal techniques or

by the methods to be discussed in section 4.7.









4.7 Synthesis of the Network N"

The method of synthesizing the network N" is essen-

tially the same as that described in section 4.4 for the

network N'. A modification of the predicted ladder config-

uration is necessary due to the fact that the short circuit

parameters y21 and y22 are prescribed for N" whereas the

open circuit parameters were prescribed for the network N'.

This modification is shown in Fig. 4.10. The predicted

configuration again consists of type 1 and/or type 2 RC

sections (and possibly some additional RC branches),

however, in this case a series element appears at the

extreme left. Except for this modification, the synthesis







Zn, Zn-2 Z3


Yn-1 Y2 0 Y
Lr-
0



Fig. 4.10 Standard ladder configuration for
prescribed y21 and y22


procedure for the realization of N" is identical to that

given in section 4.4.













CHAPTER V


TRANSFER FUNCTIONS WITH PURELY IMAGINARY TRANSMISSION ZEROS


The general technique of sharing transmission zeros

between z21 and z22 (or -y21 and y22) is based on the fact

that one may replace certain poles of Q(s)/P(s) with poles

which make the modified function, Q(s)/P'(s)= [N(s)/D(s)] x

[Q(s)/P(s)], positive real. The success of this technique

depends on the freedom which exists in making this replace-

ment. It was seen in Chapter IV that for the RC and RL

cases, poles were replaced in such a way that alternation on

the negative a axis would exist between zeros and poles of

the modified function. Also there were certain requirements

at s 0 and s oo Since these were the only restric-

tions on the replacement of poles, considerable freedom was

permitted. In contrast to this freedom which exists for the

RC and RL cases and that which will be found for RLC

networks, the restrictions are so great in the LC case that

the zero sharing technique is of no value in synthesizing

ZT(s), A(s), YT(s), or G(s).


5.1 Limitations of the Zero Sharing Technique in the
Synthesis of LC Networks

It will now be shown that the zero sharing approach








cannot be applied advantageously to the synthesis of Z (s),

A(s), YT(s), or G(s) with LC networks. Consider Q(s)/P(s)

whose reciprocal is ZT(s), or -A(s), or -YT(s), or G(s) as

defined in (1-1) and (1-2). From the conditions of physical

realizability for these transfer functions, it is known that

the zeros of Q(s)/P(s) are restricted to the left-hand half

of the s plane excluding the ju axis. The poles of

Q(s)/P(s) which, of course, represent the zeros of trans-

mission, are to lie on the jw axis. The permissible pole

locations on the ju axis for the modified function,

Q(s)/P'(s), are fixed by the restriction that the residues

of the function at each pole must be real.

In the event that an odd number of finite poles are to

be placed on the jw axis, the denominator will be of the

form


P'(s) s(s jl )(s + jX )(s jX2)(s + j%2)--

(s jXm)(s + jAm). (5-1)



The angle associated with the residue at any one of the

poles, jXi, is


arg ki arg Q(jli) -arg (s) (5-2)

s = ji


The contribution to arg k from the poles is
i






56



ar P(s) i(-X + Xi)(-X + X2)--(j2Xi)--
arg
(s-JXi) i 2 2
S jXi (-Xi + X)2


(5-3)


Oor 180


For ki to be real, it is then necessary that



arg Q(jXi) = O'or 180 (5-4)



Q(s) is now represented as

n
O(s) =Zaksk (5-5)
k-0

where a 1. With s Re Q(s) becomes

n
Q(s) a RejkO (5-6)
k-0

In order to meet the condition given in (5-4), the

imaginary part of Q(s) is set equal to zero

n
0 akRk sin kO (5-7)
k=0

The only points of interest are those on the ju axis,

for which (5-7) is satisfied. Therefore, let 9 r/2 in









(5-7) which gives

n
0 = akRk sin k n/2 (5-8)
k-1
k odd
When n, the number of finite zeros, is odd

(5-9)

0 Rn(-l)(n-l)/2 + an-2Rn-2(-)(n-3)/2 +--+ aIR



which has exactly n solutions. When n is even

(5-10)

0 = a Rn-l(_-)(n-2)/2 + a Rn-3(-1)(n-4)/2+--+ a R
n-I n-3 1


which has n-I solutions. From (5-9), (5-10) and the pre-

ceding development it is seen that if an odd number of

poles are to be placed on the jw axis in such a way as to

make Q(s)/P'(s) positive real, the permissible pole locations

are restricted to a set of n points when n is odd, or to a

set of n-1 points when n is even.

A similar development, when O(s)/P'(s) is to have an

even number of finite poles, shows that



arg P'(jyi) = 90" (5-11)


In this case the real part of Q(s) is set equal to zero,

which gives

n
0 = akRk sin k t/2 (5-12)
k-0








for 6 n/2. When n, the number of finite zeros is odd


0 nlRn-l ()(n-1)/2 a n-3 (l)(n-3)/2
0 a R (-1) + a R (-1) +----
n-1 n-3
(5-13)

+ a2R (-1) + a0


which has n-1 solutions. When n is even

(5-14)
0 Rn(-l)n/2 + a Rn-2(-)(n-2)/2 +----+ a R2(-1) + a0
n-2 2


which has n solutions. If an even number of poles are to be

placed on the ju axis so that O(s)/P'(s) is positive real,

the permissible pole locations are restricted to n-1 points

for n odd, or n points for n even.

These severe restrictions are further strengthened by

the fact that if z22 is a ratio of odd to even or even to

odd polynomials, then z21 must also be a ratio of odd to

even or even to odd polynomials. This condition stipulates

that when P(s) is even, P'(s) must be odd, or that when

P(s) is odd P'(s) must be even. In effect one is forced to

choose precisely the same set of parameters as those which

are chosen in standard LC synthesis methods (1, 5]. That is,

z21 and z22 will have the same set of poles unless cancel-

lation occurs. If such a cancellation is possible, it will

be obvious in the application of the standard method for

choosing z21 and z22 in the LC case.








5.2 Synthesis of A*(s) or G*(s) With LC Networks

Although the zero sharing technique cannot be advanta-

geously applied to the synthesis of ZT(s), A(s), YT(s), or

G(s) with LC networks, the cases involving A*(s) and G*(s)

can be handled very effectively using this method. The

only restrictions placed on A*(s) and G*(s) are that they

be physically realizable with inductive and capacitive ele-

ments. In addition to the conditions of physical realiza-

bility given in Chapter II, this requires that A*(s) or

G*(s) have the following two properties:

1) The transmission zeros must lie on the ju axis

including the points s 0 and s = 00 .

2) The poles of A*(s) or G*(s) must be simple and lie

on the ju axis excluding the points s 0 and s = co .

Before the zero sharing technique as it applies to the

synthesis of LC networks is considered, it should be pointed

out that a special class of LC transfer functions, similar

to that for the RC and RL cases, exists in which the entire

synthesis may be carried out using only two-terminal tech-

niques. This class consists of those transfer functions

A*(s) or G*(s), which are physically realizable with LC net-

works, and whose reciprocal, Q(s)/P(s), has the additional

two properties:

1) The zeros of P(s) must have no greater multiplicity

than two.

2) If one begins with the critical frequency nearest








the origin and divides the group of critical frequencies on

the positive jw axis into adjacent pairs (counting a double

pole of Q(s)/P(s) as two adjacent poles), then each pair of

critical frequencies must consist of a pole and a zero. The

synthesis of this class of transfer functions differs from

the more general case only in the realization of the networks

N' or N" shown in Fig. 2.1 and Fig. 2.2.

A method will now be developed for synthesizing A*(s)

or G*(s). Here, the only restrictions are those of physical

realizability for LC networks as stated at the beginning of

this section. The transfer current-ratio function, A*(s),

or transfer voltage-ratio function, G*(s), is given in the

form



[-A*(s) or G*(s)] -HP(s)/Q(s)
S J (5-15)
2 2 2 2 2 2
(s +-l)(s +-2)---(s +Xn)
= -H
2 2 2 2 2 2
(s +2l)(S +22)---(S +2m)


where poles and zeros lie on the jw axis and m n. The

expression for A*(s) in terms of the open circuit parameters,

or for G*(s) in terms of the short circuit parameters, may

be solved for z22 or y22 respectively. This gives



z22 z21 Q(s) (5-16)
HP(s)








Y22 = -y21Q(s)/HP(s) (5-17)


It is convenient to choose z21 or -Y21 as


z or -y H
[21 or H D(s) (5-18)


H/s(2 +2) ,-(+Xn-q+k)
s 2 2 2 2 (--- 2 2,V -
(s +) 2)--(s


which forms the initial step in the synthesis procedure.
Here, the poles and zeros of z21 (or -y21) lie on the jw
axis. The transmission zeros, jX1, JX'---J n-q+k are

not necessarily the first n-q+k zeros of -A*(s) or G*(s) as
given in (5-15). The following considerations govern the
choice of poles and zeros for z21 (or -y21):

1) The zeros of z21 (or -y21) should be selected from
the set of transmission zeros of the given transfer function
A*(s) (or G*(s)) as indicated in (5-18).
2) The poles and zeros of z21 (or -y21) must lie on

the ju axis including the points s 0 and s oo .

3) The poles of z21 (or -y21) must not coincide with

the zeros of Q(s).

4) The poles and zeros of z21 (or -y21) must be chosen

in such a way that the product z21Q(s)/HP(s) (or -y21

Q(s)/HP(s) ) is an LC driving point function of the form










21 Q(s) or -_ Q(s)]
2 HP(s) 2HP(s)J (5-19)

(s 2 s2 2)-(s 2
1/ 2 2 2 2 2 2 2 2
(s +v )--(s +v )(s +X1 )--(s +1 )
1 k k+1 q

in which the set of transmission zeros j k+l' J k+ 2'

jXq consists of those finite transmission zeros which do not

belong to the set +jXL, jX2,--- n-q+k. Acceptable values

for q are q m and q m 1. The poles and zeros of z21

Q(s)/HP(s) (or y2l Q(s)/IIP(s) ) must alternate on the ju
axis.

5) One should assign as few poles and zeros to z21

(or -Y21) as possible and still satisfy condition 4 above.

With z21 (or -y21) chosen according to the above condi-

tions one next places the product z21 Q(s)/HP(s) (or -y21

Q(s)/HP(s) ) in partial fractioned form. z22 (or y22) may

then be represented as


"a als aks
[z22 or 22] (a s + + -- ---+ )
22 22 1 2 2 2 2
s s +v s +V
1 k
(5-20)
ak+ls aqS
+( +---+ )
2 2 2 2
s + s + X
k+l q

where a 1 1 for q m 1 or a 0 for q m. At this

point, Zs as shown in Fig. 2.1 or Y as shown in Fig. 2.2
P








may be removed from z22 or Y22, respectively. The standard

LC realization for Z or Y may then be performed.

The remaining portion of the realization, the synthesis

of the network N' or N", follows one of two courses. First,

if A*(s) or G*(s) belongs to the special class of LC trans-

fer functions mentioned earlier, then N' or N" may be syn-

thesized in an L network of the form shown in Fig. 4.1 or

Fig. 4.9 respectively. Only two-terminal LC synthesis tech-

niques are required for this realization. Let z21 (or -y21)

as given in (5-18) be represented in partial fractioned form

as


BO 31s Bks
[z21 or -y21] Bis + + ---2 + (5-21)
s s +v s +v
1 k


Then the restrictions on the gain, H, for the L network

configuration may be stated as



a i/Bi 2 H for i = -1, 0, 1, 2,-----k. (5-22)



The second course which may be taken is that of synthesizing

N' or N" in a general LC ladder configuration.

The general approach for synthesizing the networks N'

and N" was described in section 4.4, and details were given

there for the synthesis of RC ladders. In synthesizing LC

ladders, the same general approach is used with respect to

predicting a possible network configuration and synthesizing









22 (or y22) as a driving point impedance, section by

section (1, 6]. Therefore, it is only necessary to intro-

duce the basic LC sections which will make up the ladder,

to indicate the procedure for synthesizing these sections,

and to mention certain factors to be considered in

prediction.

Two basic LC sections are shown in Fig. 5.1. The sub-

script or superscript, k, again indicates that this is the

kth section in the predicted configuration, in which the


Y*
()
(a)


~T (k)'
r- b

01 ]c(k)j
b


Yk+l Z* Yk
k+1 k k

(b)
Fig. 5.1 Basic LC sections (a) type 1 LC section
(b) type 2 LC section


sections are numbered from right to left consecutively. In








each basic section of Fig. 5.1, a transmission zero is

produced by the principal branch at s jkk. If, in Fig.

5.1 (a), L(k) is set equal to zero, or if C(k) is set equal
a a
to infinity, the kth section will contribute to a transmis-
sion zero at s oo or s 0, respectively. Similarly, if,

in Fig. 5.1 (b), L(k) is set equal to infinity, or C~k) is
b b
set equal to zero, this kth type 2 section will contribute

to a transmission zero at s 0 or s -oo, respectively.

Note that the auxiliary branches of each section may contrib-

ute either to a transmission zero at a 0 or to a transmis-

sion zero at s o .

In order to synthesize a type 1 LC section, one must

know the impedance looking into that section, Zk, indicated

in Fig. 5.1 (a). One must also know the transmission zeros,

s jkk, or their special cases, s = 0 and s = oo which

are to be produced by this kth section. A knowledge of

these two facts permits the synthesis of a type 1 LC section

as follows:

1) Let


jXk Zk(s) ), (5-23)
s J^k


then

bk) X X k (if Xk > 0), (5-24)
b k k k


C(k) = /k
b kk


(if X < 0).


(5-25)









2) The admittance Y* looking into the remaining portion
k
of the ladder, after ~k) or Cbk) is removed, is

1
Y* (5-26)
k (k)
Z sL
k b

or

1
Y* (5-27)
k k
Z /sCb


respectively.

3) Y* has the form
k

Mks
Y* -+ Y + 1 (5-28)
k 2 2 k
+ + k


where



I = -- k Y*. (5-29)
ks k
s = Jk


The elements of y(k) are then
a


(k)
L 1/Mk (5-30)
a


and

(k) 2
C a k/ k (5-31)
C ak k





67


4) The impedance looking into the next basic section

is then


1
Z =
k+ 1 Mks
Y* -
k 2 2
S+ k
k


(5-32)


The synthesis of a type 2 LC section requires a knowl-

edge of Yk(s) and the transmission zeros which are to be

produced by this section. The procedure for synthesizing

this section is as follows:

1) Let


Bk Yk(S)


(5-33)


then


c(k) = BkAk
a k k





(k)
La 1/(xklBkI)
ak


2) The impedance Z*
k
of the ladder after C(k)
a


(if Bk 2 0)






(if Bk < 0)


(5-34)


(5-35)


looking into the remaining portion

or L k) is removed, is
ai


1
Z*
k (k)
k sCa


(5-36)









or


Z* ,(5-37)

Yk /sLa


respectively.

3) Zk has the form


NkS
Z* + Z (5-38)
k 2 2 k+l
s +
k

where
2 2
s + \k)
N Z* (5-39)
k s k s = Jk

(k)
The elements of Zb are then
b

(k)
Cb = 1/Nk (5-40)


and


(k) 2
Lb = Nkk (5-41)


4) The admittance looking into the next basic section

is then



Y (5-42)
k+l Nks
Z* k
k 2 2
s +\
k








Attention is now turned to those factors which are

important in predicting possible LC network configurations.

When the network parameters z21 and z22 (or y21 and y22) are

chosen by the zero sharing methods above, it is certain that

the parameters for the network N' (or N")will have the same

denominator, and that z' and z' (or y' and y' ) will have
21 22 21 22
no common numerator factors. It is therefore assured that

the zeros of z' (or y' ) are the transmission zeros of the
21 21
network N' (or N"). The procedure for predicting an LC

network may then be stated as follows:

1) Examine z21 (or y21) to determine the transmission

zeros of N' (or N").

2) Arrange type 1 and/or type 2 LC sections (and pos-

sibly some additional LC branches) in the ladder configura-

tion of Fig. 4.2 for prescribed z' and z' or in the
21 22
configuration of Fig. 4.10 for prescribed y' and y'.
21 22
In the arrangement described in 2, the principal

branch of each LC section is assigned a complex conjugate

pair of transmission zeros which it is to produce. In the

event that all transmission zeros are at s 0, s oo, or

a combination of these two, it is necessary that certain

elements in the principal branches take on the limiting

values discussed earlier. The principal branch will then

produce a transmission zero at s 0 or s oo. The auxil-

iary branch of each LC section will contribute to a trans-

mission zero at either s 0 or s = oo

As in the case of RC and RL networks, a number of






70


predicted configurations is possible for a given set of

transmission zeros. Some of these predictions will be

realizable, while others may contain negative circuit ele-

ments and are therefore nonrealizable with passive networks.

If a first prediction is nonrealizable, a rearrangement of

the order of transmission zero assignments may cause the

prediction to become realizable, or it may be necessary to

predict a different configuration of basic sections.













CHAPTER VI


TRANSFER FUNCTIONS WITH COMPLEX CONJUGATE AND

NEGATIVE REAL TRANSMISSION ZEROS


A method will be developed in this chapter for synthe-

sizing minimum phase transfer functions with networks which,

in general, will consist of resistive, capacitive, and

inductive elements. The realizability conditions, which a

transfer function, T(s), must satisfy, are given in Chapter

II. Although these are the only restrictions on the trans-

fer functions, it will be found that the usefulness of the

method under consideration is limited to transfer functions

of low complexity. In fact, only those transfer functions

of third degree or less in numerator and denominator will be

considered in detail. The extension of the zero sharing

method to higher degree transfer functions will be discussed

but has not been placed on a firm basis. That is, in

applying the technique to higher degree transfer functions,

one cannot be sure that full advantage has been taken of the

simplifying effects inherent in the zero sharing process.

The trivial case, in which the denominator of T(s) is a

first degree polynomial with the numerator either a constant

or a first degree polynomial, is always positive real and








therefore realizable with an RC or RL network in the L

configuration of either Fig. 4.1 or Fig. 4.9.


6.1 Transfer Functions Whose Denominator Polynomial Is of
Second Degree

It will be shown in this section that every minimum

phase transfer function, T(s), whose denominator polynomial

is of second degree in s, can be synthesized using only two-

terminal techniques. The resulting network will have the

form indicated in Fig. 1.2 or Fig. 1.3, in which the networks

N' or N" may be realized in the L configuration of Fig. 4.1

or Fig. 4.9 respectively. It is convenient to consider

separately five different forms of T(s) which differ in

their pole-zero configurations.

1) Let T(s) have the form


H
T(s) = (6-1)
(s + c jd)(s + c + jd)


where c > 0 and d > 0. z21 (or -y21) may be chosen as

1
21 (or -y21) = K (6-2)


where 0 < e 2c. The product z21 Q(s)/P(s) (or -y21

Q(s)/P(s) ) then has the form

(6-3)

z2 Q(s) (or -1 Q(s) K 1 (s+c-Jd)(s+c+jd)
L P(s) P(s) s+e 1







For ZT(s), A(), Y(s), or G(s), z22 (or y22) has the form
(6-4)
S (or y) = K/ + (2c-e) + c2+d2(2c-)
L22 22)] s [ +e

For A*(s) or G*(s), z22 (or y22) has the form
(6-5)
(or y ) K/H + (2 ) c2+d2-e(2c-e)]
[22 (or Y22) K/H (-e) s+e

Since z21 (or -y21) is a positive real driving point func-
tion, a two-terminal realization may always be effected. A
very simple sharing of transmission zeros is represented by
the fact that one transmission zero at infinity is produced
by z21 (or -y21), while the other transmission zero at infin-
ity is produced as a private pole of z22 (or y22).
2) Let T(s) be of the form


T(s) = H (6-6)
(s + c jd)(s + c + jd)

where a 0, c > 0, and d > 0. It is assumed that T(s) is
not positive real, and therefore, that a > 2c. One possible
choice of z21 (or -y21) is


[-21 (or -y2)] K + (6-7)
121 21Zs + e

where 0 < e < 2c. The product z21Q(s)/P(s) (or -y21
Q(s)/P(s) ) then has the form








(6-8)
(, ) (or Q(s))] K (s+c-jd)(s+c+jd)
21 P(s) 21 P(s) J s+e

If ZT(s), A(s), YT(s), or G(s) is specified, then z22

(or y22) will have the form given in (6-4). For A*(s) or
G*(s) specified, z22 (or y22) will have the form given in
(6-5). Since z21 (or -y21) is a positive real driving point
function, a two-terminal realization may always be effected.
A sharing of transmission zeros is represented in that the
finite transmission zero at s -a is produced by z21
(or -y21), while a transmission zero at infinity is produced
as a private pole of z22 (or y22).
3) Let T(s) have the form


T(s) = (s + a jb)(s + a + jb) (6-9)
(s + c jd)(s + c + jd)

where a > 0, b > 0, c > 0, and d 0. The single necessary
and sufficient condition for a function of this form to be a
positive real driving point function is given by


S2 + b2- c2 +d2 4 ac. (6-10)

It will be assumed that T(s) is not positive real and there-
fore that the condition given in (6-10) is not satisfied.

If z21 (or -Y21) is chosen as









(s + a jb)(s + a + jb)
[z (or -y )] K (6-11)
21 21 s + e


where 0 s, e 2a and 0 e < 2c, a realization may be easily

effected using only two-terminal techniques. The product

z21Q(s)/P(s) (or -y21Q(s)/P(s) ) then has the form given in

(6-8). If ZT(s), A(s), YT(s), or G(s) is specified, z22

(or y22) will have the form given in (6-4). If A*(s) or

G*(s) is specified, z22 (or y22) will have the form given in

(6-5). Here, z21 (or -y21) is positive real, and, therefore,

a realization may always be effected using only two-terminal

techniques. No sharing of transmission zeros is possible

since both members of the pair of complex conjugate trans-

mission zeros must be assigned to z21 (or -y21).

4) Let T(s) have the form


T(s) (s + a)(a + b) (6-12)
(s + c jd)(s + c + jd)

where a 0, b > 0, c > 0, and d > 0, with a and b distinct.

It will be assumed that T(s) is not a positive real driving

point function and therefore that the condition


Vab V2 + d22 2c (a +b) (6-13)

is not satisfied. A realization may always be effected by

choosing z21 (or -Y21) as









(s + a)(s + b)
[z21 (or -y21) ] K + b) (6-14)


in which 0 e e 2c and 0 o e (a + b). The product z21

Q(s)/P(s) (or -y21Q(s)/P(s) ) then has the form given in

(6-8). In the synthesis of ZT(s), A(s), YT(s), or G(s), z22

(or y22) will have the form given in (6-4). In the synthe-

sis of A*(s) or G*(s), z22 (or y22) will have the form given

in (6-5). Since z21 (or -y21) is positive real, the complete

realization may always be carried out using two-terminal

techniques.

A realization may frequently be effected in which the

transmission zeros are shared. In order to produce this

sharing, the numerator polynomial, N(s), of z21 (or -y21)

should be chosen as either s + a or a + b. Without loss of

generality s -a is chosen as the zero of N(s). z21 (or

-y21) is assigned a simple pole at s -e, which gives


s + a
[z21 (or -y21)] K a (6-15)
s+e


The selection of e is made in such a way that the expression


( reb- Vc2 + d2)2 2c (e + b) (6-16)


is satisfied. It will be found convenient to assure that

e satisfies the restriction,










[e (e b)]2 > (c e)2


for e > b, or


(6-17)


[b (b e)]2 > (c b)2 + d2


for b > e. The product z 21(s)/P(s) (or -y Q(s)/P(s) )
then has the form


[z Q(s) (or y (s)) =
21 P(s) 21 P(s) J


(s+c-jd)(s+c+Jd)
(s+e)(s+b)


In the synthesis of ZT(s), A(s), YT(s) or G(s), z22 (or y22)
will have the form


[z22 (or y22)] K/H (s+c-d)(s+c+d) .
22 22 (s+e)(s+b) J


(6-19)


In the synthesis of A*(s) or G*(s), z22 (or y22) will have
the form


22 (or = K/ (s-c-jd)(/s+c+d)
22 22 1 (s+e)(s+b) I


5) Let T(s) have the form


S (s+a-Jb)(s+a+jb)
+c)T(s)(s+d)
(s+c)(s+d)


(6-20)


(6-21)


(6-18)








where a > 0, b > 0, c > 0, and d > 0 with c and d distinct.
It will be assumed that T(s) is not a positive real driving
point function, and therefore, that the condition


( -d -a2 + b2) < 2a(c + d) (6-22)


is not satisfied. A realization may always be effected by
choosing z21 (or -y21) as


(s+a-jb)(s+a+jb)
[z21 (or -Y21)] = K (6-23)
s+e

in which 0 < e < 2a and 0 < e < (c+d). The product z21

Q(s)/P(s) (or -y21Q(s)/P(s) ) then has the form


QF Q(s) Q (s)) K (s+c)(s+d)
Lz2 (or -ys K (6-24)
21 P(s) -Y21 P(s) J s+e

In the synthesis of ZT(s), A(s), YT(s), or G(s), z22
(or y22) will have the form
(6-25)
[ cd-e(c+d-e)1e
[z (or y )] = K/I s + (c+d-e) + cd-e(c1
22 22 s+e

while in the synthesis of A*(s) or G*(s), z22 (or y22) may
be written as
(6-26)

[z22 (or y22)] = K/H s + (c+d-e) + cd-e(cd-e)
2s+e








Since z21 (or -y21) is positive real, a realization using

only two-terminal techniques is possible. No sharing of

transmission zeros is possible in this case since both

members of the complex conjugate pair of transmission zeros

must be assigned to z21 (or -Y21).

The five cases considered above include all physically-

realizable transfer functions, whose denominator is of sec-

ond degree, which were not included in the synthesis methods

given in Chapter IV and Chapter V. The transfer functions

considered in Chapters IV and V, which have denominators of

second degree are found to belong to the special classes,

which were defined in section 4.2 and section 5.1, respec-

tively. It may, therefore, be concluded that all transfer

functions which have denominators of second degree may be

synthesized using only two-terminal techniques.

Methods have been given for the synthesis of every

physically-realizable minimum phase transfer function whose

denominator is of second degree. However, only those

choices of z21 (or -y 21) have been considered in which the

zeros of D(s) are assigned to points on the negative o axis

or on the ju axis. The possibility of choosing a complex

conjugate pair of poles for z21 (or -y21) has not been con-

sidered. As previously described, one should assign as few

poles and zeros to z21 (or -y21) as possible and still

assure that the product z21Q(s)/P(s) (or -y21Q(s)/P(s) ) is

positive real. In general, when more poles and zeros than









the minimum required number are assigned to z21 (or -y 21),

an excessive number of circuit elements is required in the

realization, Since the synthesis of all transfer functions,

T(s), whose denominator is of second degree can be carried

out with z21(or -y21) having a single pole on the negative o

axis, it would be expected that the assignment of a pair of

complex poles to z2 (or -y 21) will cause an excessive number
21 21
of circuit elements to be necessary in the realization.

Although this is true, consideration of the complex conjugate

pole assignment is important from a theoretical standpoint,

and will be found useful in the synthesis of transfer func-

tions whose denominator polynomial is of third degree or

higher in s.

A procedure will now be given for synthesizing any of

the five forms of T(s) given above, in which one has the

freedom to choose the poles of z21 (or -y21) and z22 (or

y22) as a complex conjugate pair.
Attention is first turned to a preliminary subject, the

problem of constructing a positive real function, F(s), whose

poles are to appear in complex conjugate pairs. With the

form of F(s) given as



F(s) = (s + c jd)(s + c + jd) (6-27)
(s + e jf)(s + e + jf)


the problem may be stated as follows.

If the zero locations, s = -c jd, are known, what








values may be chosen for e and f so that F(s) will be posi-

tive real? Permissible values for e and f are determined by

a simple geometric construction which is explained in the

following four steps: (A proof of the validity of this

procedure is given in section 6.2.)

1) On a piece of graph paper construct a rectangular

coordinate system to represent the s plane. With a compass,

mark the point where a circle, whose center is at the origin

and which passes through the point -c +jd, intersects the

positive jw axis. (This intersection is, of course, at

s j 4c2 + d2.) Call this point jucl

2) From the specified value of c, and u1 as just deter-

mined, calculate r, where r is given by



r 2c + 1. (6-28)



(It is convenient to perform the addition in (6-28) by sum-

ming lengths on the jw axis.) With a compass mark the

points at which a circle, whose radius is r and whose center

is at s = jul, intersects the real axis. Call the point at

the intersection of this circle with the negative o axis

-x0, and the point at the intersection with the positive

a axis xg.

3) Construct a circle, C1, whose center is at s -x0,

and whose radius is r, as given in (6-28). Construct a

second circle, C2, whose center is at s x0, and whose








radius is also r. This construction is illustrated in Fig.

6.1 for c 1 and d 2.


Fig. 6.1 The construction of a region in which
poles of F(s) may be selected.


4) A pair of complex conjugate poles may be placed

anywhere within or on the closed region whose interior is to

the left and outside C2, and inside C1. The shaded area in

Fig. 6.1 indicates such a region. For this assignment of

poles, F(s), as given in (6-27), is a positive real driving

point function.

The case will now be considered in which the zeros of








F(s) lie on the negative portion of the a axis. F(s) is to

have the form


F(s) (s + c)(s + d)
F(s) (6-29)
(s + e jf)(s + e + jf)


The zeros at s = -c and s -d are considered to be previ-

ously specified, and e and f are to be chosen so that F(s)

will be positive real. A region in which poles of F(s) may

be selected can be determined by following steps 1 through

5 with two modifications. First, the point, jul, is now

given by



ju j e (6-30)


and, second, the radius r is now given by



r = u1 + c + d. (6-31)


An illustration of this construction is shown in Fig. 6.2,

for F(s) as given in (6-29), with c 1 and d 2.

If a pair of complex conjugate poles is chosen for

F(s) within one of the regions just defined, it is certain

that F(s) will be positive real. Although such a choice is

sufficient to assure that F(s) is positive real, it is not

necessary that the poles be chosen within these regions.





































Fig. 6.2 A region in which poles of F(s) may be selected, fpr
F(s) as given in (6-29)


Larger regions have been shown to exist. However, these

regions are much more difficult to construct [7].

The above procedures for choosing the poles of a posi-

tive real function may be used in synthesizing any one of

the five forms of T(s) given in equations (6-1), (6-6),

(6-9), (6-12), or (6-21). After choosing an acceptable pair

of poles for z21 (or -Y21), the synthesis is carried out by

first, selecting a value of K/H, and second, synthesizing

the network N' or N". If z21(or -y21) is chosen as a








positive real driving point function, the synthesis may be

carried out on a two-terminal basis, with the resulting net-

work having an L configuration. For such a realization, K/H

should be chosen in such a way that z' z' (or y' y' )
22 21 22 21
is a positive real driving point function. If, however, a

general four-terminal realization of N' or N" is to be

effected, then in the selection of K/H, it is only neces-

sary that property 4 for the network parameters, as given

in Chapter II, be satisfied.

Illustrations will be given in section 6.5 for those

transfer functions, T(s), whose denominator polynomial is of

second degree in s.


6.2 The Derivation of Regions In Which A Positive Real

Function, F(s), May Have Poles

The validity of the construction procedure, given in

steps 1, 2, 3, and 4 of section 6.1, will now be established.

F(s) is to be a biquadratic driving point function of the

form given in (6-27) or (6-29). For specified zero loca-

tions, if the poles of F(s) are chosen to lie within a

certain region of the s plane, it will be assured that F(s)

is positive real. A derivation of these regions will be

given in the following eight steps:

1) Definition of a C path [7]: Consider a circle which

lies partly in the left-hand half of the a plane and partly

in the right-hand half of the s plane, and whose center is

on the real axis. A C path is, by definition, that portion









of the circle which lies in the closed left-hand half of the

s plane. Also by definition, the negative direction on a C

path always points to the left. Fig. 6.3 shows a typical C

path with its negative direction indicated.


Fig. 6.3 A typical C path with arrows indicating its
negative direction


2) The phase function of a quadratic: Let Q(s) be a

real quadratic given by


2 2
Q(s) 2 a + 2as + a + 2


with

by


(6-32)


a > 0 and B 0. The phase function of Q(jw) is given


(6-33)


-1 2au
(w) tan- 2
Q 2 2 2
a +8 -


3) A property of the phase function as the zeros of

Q(s) are moved along a C path [7]: Let the zeros of Q(s)








move in the negative direction on a C path that intersects

the ju axis at s ju (WI > 0). Then the phase function

SQ(w) of Q(jw) decreases monotonically for every fixed w in
the interval wl < w < oo and increases monotonically for

every fixed w in the interval 0 < w < u This has been

shown by Steiglitz and Zemanian [7].

4) A property of a biquadratic driving point function,

F(s), whose poles are restricted to lie on the ju axis: Let

F(s) have the form


Q1(s) 92 + 2as + a2 + 2 (6-34)
F(s) (6-34)
Q2(s) 2 + 2
s + W1



If values of a and B (a > 0, B > 0) are specified, only one

permissible value of wl exists for which F(s) is positive

real. A proof of this fact using the general procedure

given in Chapter V is as follows: Q1 () has the form


Q1(s) = s2 + 2as + a2 + B2 (6-35)


From (5-14) the permissible values of w1 are the roots of

the equation


0 -R2 + a2 + 2 (6-36)


which gives










S- R a2 + B2


(6-37)


Similarly, if F(s) has two distinct zeros on the negative a

axis at s -c and s -d, then F(s) has the form


s2 + (c + d) s + cd
2 2
s +WJ
1


Again, only one permissible value of w'
1
of w' may be found by applying equation
1
as given in (6-38), this gives


(6-38)


exists. This value

(5-14). For Q (s)
1


0 -R2 + cd
0 -R + cd


(6-39)


(6-40)


S- R -cd .
1


5) A property of a biquadratic driving point function,

F(s), which has a double pole on the negative portion of the

a axis: First, let F(s) have the form


Q1(*) 52 2 2
F(s) 8s2 + 2as +a2 B2
F(s)
Q2(s) (g + T)2


where values of a and B (a > 0, B >0) are specified. From


F(s)
Q2(s)


(6-41)








the single condition of physical realizability for a biquad-

ratic driving point function, it is necessary that the

condition


[ a2 +2 2 < 4a (6-42)


be satisfied [8]. This inequality determines a segment of

the negative a axis on which a double pole of F(s) may be

placed, with the result that F(s) is a positive real driving

point function. The end points of this segment are deter-

mined by assuming equality in (6-42), and then solving for

the roots, Tl and r2, where T1 > T2. The roots, T1 and r2'

are given by

(6-43)

I ( a+ B2 + 2a) + /[ a 2 + 2a2 (a2 + B2)


and

(6-44)

2 ( a + 2 + 2a) + a2 B2 + 2a]2 (a2 +B2),


respectively. From (6-37), yl and r2 become



T1 (1 + 2a) + 1(w1 + 2a)2 2 (6-45)
1 1 1




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