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 CONTENTSContinued
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 fourterminal 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 (429) . . . . . . .
4.6. Network prediction for N' . . . .
4.7. Ladder network for A(s) as given
in (438) with K/H 3 . . . . .
4.8. Ladder network for A(s) as given
in (438) 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 (629) .
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 FIGURESContinued
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 (671) is satisfied . . ... 102
6.6. Regions of the s plane in which
condition (671) 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 (6133) . . . . . 124
6.13. Regions in which the poles of y21 may be
chosen, for YT(s) as given in (6133) . . 125
6.14. Second network realization for YT(s) as
given in (6133) . . . . . . . 126
6.15. Permissible pole locations of z21 for
ZT (s) as given in (6139) . . . . .. 127
6.16. First network realization for ZT(s)
as given in (6139) . . . . . .. 128
6.17. Second network realization for ZT(s)
as given in (6139) . . . . . .. 129
LIST OF FIGURESContinued
Figure Page
6.18. Permissible pole locations of z21
for A(s) as given in (6149) . . . .. 130
6.19. Network realization for A(s)
as given in (6149) . . . . ... 132
6.20. Network realization for A(s)
as given in (6157) . . . . ... 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 twoterminal realization and a fourterminal 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 twoelement
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 twoelementkind 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 fourterminal network shown in Fig. 1.1, terminals
11' are considered as the input, and transmission occurs
from this terminal pair to the output terminals 22'.
1 Il 12 2
E1 t2R2
1' 2'
Figure 1.1 A general fourterminal 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 11' of Fig. 1.1,
three of these transfer functions are defined as
(11)
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 11' of Fig.
1.1, three transfer functions are defined as
(12)
12 E2 E2
Y(s) = G(s)  and G*(s) = 
SE E E
R2O R200 R200
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 (11) it is seen that
12 (E2/R2)
A(s) ZT(s)/R2 (13)
I1 Ii2
If A(s) is specified for a network which is terminated in a
finite nonzero load, an equivalent Z (s) may be easily
T
found from (13). 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 + anlsn ++ als + a0 (14)
T(s) = H = H
Q(s) sm + b ++ bs + b
m1 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 righthand 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) (15)
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 (16)
1 + y22 R2
In selecting the network parameters, the ratio specified in
(14) is inserted in (15) or (16) 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 (15)
(or (16)) 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 twoterminal 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 (15)
and (16), when solved for z22 and y22, have the form
0(s)
z22 21 ( 1 (17)
22 21 HP(s)
and
Q(s)
2= y 1 (18)
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) (19)
and
Q(s)
Y22 1 Ip(s) (110)
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 (17) or (19), or y22 as given in
(18) or (110), is next placed in the form
z22 z' + Z (111)
22 22 s
y y' + Y
22 22 p
(112)
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 (111) or (112) 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 (11) and
(12) 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 (15)
and (16), 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
lefthand 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 righthand 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 lefthand 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 lefthand 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 twoterminal 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 nonminimumresistive 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) (31)
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 (31), z22 has the form
z21
z (R + Z' (s)) 1 (32)
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)) (33)
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 (34)
K + 1
is satisfied. When H is specified, an appropriate value of
K is easily determined from (34). It is seen in (34) 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 (35)
in (33).
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 (34) or (35) 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 (34) or (35) 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 nonode 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
(41)
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
(42)
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 (41) or
(42) 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 twoterminal 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 currentratio 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)
(46)
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
(44)
(45)
realization. z21 should be chosen as a rational function of
the form
N(s) K(szl)(sz2)(sznq+k)
21 K (s) (sr )(sy2)(sy ) k(47)
in which both poles and zeros are real. The transmission
zeros zl, z2,znq+k are chosen according to a set of
rules which will be given and are not necessarily the first
nq+k zeros of T1(s) given in (44). 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 (47).
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) (48)
(spl)(sP2)(spm)
K
(sl)(sy2)(syk) (sk+l)(szq)
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 (48) is substituted in
(45) and (46).
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 twoterminal 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
(49)
[ al ak ak+l am
z2 = K/H ( +  ++ ) + ( ++ ) 
2( + Sy sz SZ
SS1 k k+l m
for ZT(s) or A(s), and
(410)
[ al ak ak+l aq
SK/H a +  ++ ) + ( ++
22 S I STk SZk+l SZq
for A*(s). In (410), 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 (49) or (410), 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 twoterminal 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 +  +  ++ (411)
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'
(412)
and that
a /Bi > H for i 1, 2,, k, (413)
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. (414)
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 (413). This value of H will
determine an acceptable range of K as indicated in (412).
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 (414). 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 (415)
21 21
and
r al a2 ak ]
z' K/ 1 +  +  ++  1 (416)
22 Sr1l sY2 sk
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
(417)
be satisfied. Equality may occur in (417) 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
(49) or (410) Zs is given by
Z K/H + +  (418)
s s z s z
Zk+1 mz
Finally, the network N' is synthesized either by standard
twoterminal 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 zeroshifting methods, it has two distinct advantages.
First, it is more straightforward in its application than
the zeroshifting 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
zeroproducing 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 szTLk
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) (419)
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) (420)
k b
3) Y* is of the form
k
Mks
Y* =  + Y + 1 (421)
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 (422)
a Mk a zk
4) The impedance looking into the next basic section is
Zk+1 (423)
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= (424)
s z
2) The impedance Z* looking into the remaining portion
k
of the ladder after G (k) is removed is
a
Z* (425)
k (Y G(k))
K a
3) Z* is of the form
k
Nk
Z* =  + Z
k s k + 1 ,
where (426)
Nk (s zk) Z
s zk
The elements of Zb are then
N
1 (k) k
C and R (427)
b Nk b Zk
4) The admittance looking into the next basic section
is
I
Yk + 1 ) Z Nk (428)
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 + (429)
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
(430)
Then
SK/H F (s + l)(s + 4)
22 K/+H )
22  s(s + 3)
(431)
Upon partial fractioning z21Q(s)/P(s), z22 becomes
(432)
The limits on K and H are set by (411) and (412) as
K/H 1 2 K
(433)
2/3 > H
(434)
is chosen equal to its limiting value, that is H 2/3,
H
IK> = 2
1 H
(435)
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
(436)
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
(437)
z' = 2 + 4/s .
21
The resulting network for Z (s) as given in (429) is shown
in Fig. 4.5.
R2/3
R21
Fig. 4.5 Ladder network for ZT(s) as given in (429)
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 (438)
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) (439)
s(s + 4)
For this choice of z21, z22 becomes
22 = K/n [(s+l)(s+3)(s+5)(s+8)(s+12) (440)
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 (441)
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 (442)
For this value of H, K/H must satisfy the condition
K/H > 63/51 .(443)
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 (444)
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 + + (445)
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 + + (446)
22 s s + 4 s(s + 4)
and
(s + 7)(s + 9)47)
z' = =K (447)
21 21 s(s + 4)
Section 2
(2)
Rb
SSection 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 + (448
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 (449)
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 (438)
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
(49). To use this criterion in choosing K/H, it is neces
sary that the inequality
0 < + +  +  < 1 (450)
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 (450) is satisfied,
K/H may be chosen as
(451)
[ 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
(438). The expression indicated in (450) is first eval
uated at each of the transmission zeros to be produced by
z21. From (439) and (441)
F 3+ 1 5
1 + + 4 2 (452)
s s + 4 21
1 1 Is 7
+ 3 + ] (453)
s + 4 15
s = 9
are evaluated. For K/H chosen as
K/H = 15/7 (454)
z22 becomes
(455)
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 (456)
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 (419) through (423) 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 (438)
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)
(457)
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 (458)
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)
(459)
0(s)
K/H
p'(s)
for T2(s) = G*(s). y21 is chosen as
N(s) (s) (sz2)(Sznq+k)
Y21 = K K (460)
21 D(s) (sr )(s2 )(sr )
where q m or q = m 1. Equation (460) indicates that
the number of transmission zeros assigned to Y21 is nq+k.
They are not necessarily the first nq+k zeros of T2(s) as
given in (457). 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
(461)
Q(s) (sPl)(sp2)(sPr)
P(s) (sz )(s)2)(s)k)( szk+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,znq+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
(462)
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
(463)
( k+ls as
s Z s Z
k+l ]
for G*(s). In (462) and (463), al 1 for q m 1,
or a1 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 twoterminal 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 +  + +  (464)
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 (465)
and that
a.
H > H for i = 1, 1, 2,, k (466)
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. (467)
When synthesizing Y(s) or ), may be selected in
When synthesizing YT(s) or G(s), H may be selected in
r1 I
Z2
Z1
0 
Network 'N
Fig. 4.9 L network configuration
accordance with (466). A rearrangement of (465),
K/H > 1 (468)
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 (468). 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 (462) 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 (467). 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 (469)
21 21
and
(470)
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 ++  (471)
I k+1 q
The remainder of the procedure consists of synthesizing the
network N" either by standard RC twoterminal 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, Zn2 Z3
Yn1 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 (11) and (12). 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 lefthand 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). (51)
The angle associated with the residue at any one of the
poles, jXi, is
arg ki arg Q(jli) arg (s) (52)
s = ji
The contribution to arg k from the poles is
i
56
ar P(s) i(X + Xi)(X + X2)(j2Xi)
arg
(sJXi) i 2 2
S jXi (Xi + X)2
(53)
Oor 180
For ki to be real, it is then necessary that
arg Q(jXi) = O'or 180 (54)
Q(s) is now represented as
n
O(s) =Zaksk (55)
k0
where a 1. With s Re Q(s) becomes
n
Q(s) a RejkO (56)
k0
In order to meet the condition given in (54), the
imaginary part of Q(s) is set equal to zero
n
0 akRk sin kO (57)
k=0
The only points of interest are those on the ju axis,
for which (57) is satisfied. Therefore, let 9 r/2 in
(57) which gives
n
0 = akRk sin k n/2 (58)
k1
k odd
When n, the number of finite zeros, is odd
(59)
0 Rn(l)(nl)/2 + an2Rn2()(n3)/2 ++ aIR
which has exactly n solutions. When n is even
(510)
0 = a Rnl(_)(n2)/2 + a Rn3(1)(n4)/2++ a R
nI n3 1
which has nI solutions. From (59), (510) 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 n1 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" (511)
In this case the real part of Q(s) is set equal to zero,
which gives
n
0 = akRk sin k t/2 (512)
k0
for 6 n/2. When n, the number of finite zeros is odd
0 nlRnl ()(n1)/2 a n3 (l)(n3)/2
0 a R (1) + a R (1) +
n1 n3
(513)
+ a2R (1) + a0
which has n1 solutions. When n is even
(514)
0 Rn(l)n/2 + a Rn2()(n2)/2 ++ a R2(1) + a0
n2 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 n1 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 twoterminal 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 currentratio function, A*(s),
or transfer voltageratio function, G*(s), is given in the
form
[A*(s) or G*(s)] HP(s)/Q(s)
S J (515)
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) (516)
HP(s)
Y22 = y21Q(s)/HP(s) (517)
It is convenient to choose z21 or Y21 as
z or y H
[21 or H D(s) (518)
H/s(2 +2) ,(+Xnq+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 nq+k are
not necessarily the first nq+k zeros of A*(s) or G*(s) as
given in (515). 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 (518).
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 (519)
(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, nq+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
(520)
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 twoterminal LC synthesis tech
niques are required for this realization. Let z21 (or y21)
as given in (518) be represented in partial fractioned form
as
BO 31s Bks
[z21 or y21] Bis + + 2 + (521)
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. (522)
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) ), (523)
s J^k
then
bk) X X k (if Xk > 0), (524)
b k k k
C(k) = /k
b kk
(if X < 0).
(525)
2) The admittance Y* looking into the remaining portion
k
of the ladder, after ~k) or Cbk) is removed, is
1
Y* (526)
k (k)
Z sL
k b
or
1
Y* (527)
k k
Z /sCb
respectively.
3) Y* has the form
k
Mks
Y* + Y + 1 (528)
k 2 2 k
+ + k
where
I =  k Y*. (529)
ks k
s = Jk
The elements of y(k) are then
a
(k)
L 1/Mk (530)
a
and
(k) 2
C a k/ k (531)
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
(532)
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)
(533)
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)
(534)
(535)
looking into the remaining portion
or L k) is removed, is
ai
1
Z*
k (k)
k sCa
(536)
or
Z* ,(537)
Yk /sLa
respectively.
3) Zk has the form
NkS
Z* + Z (538)
k 2 2 k+l
s +
k
where
2 2
s + \k)
N Z* (539)
k s k s = Jk
(k)
The elements of Zb are then
b
(k)
Cb = 1/Nk (540)
and
(k) 2
Lb = Nkk (541)
4) The admittance looking into the next basic section
is then
Y (542)
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 polezero configurations.
1) Let T(s) have the form
H
T(s) = (61)
(s + c jd)(s + c + jd)
where c > 0 and d > 0. z21 (or y21) may be chosen as
1
21 (or y21) = K (62)
where 0 < e 2c. The product z21 Q(s)/P(s) (or y21
Q(s)/P(s) ) then has the form
(63)
z2 Q(s) (or 1 Q(s) K 1 (s+cJd)(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
(64)
S (or y) = K/ + (2ce) + c2+d2(2c)
L22 22)] s [ +e
For A*(s) or G*(s), z22 (or y22) has the form
(65)
(or y ) K/H + (2 ) c2+d2e(2ce)]
[22 (or Y22) K/H (e) s+e
Since z21 (or y21) is a positive real driving point func
tion, a twoterminal 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 (66)
(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 + (67)
121 21Zs + e
where 0 < e < 2c. The product z21Q(s)/P(s) (or y21
Q(s)/P(s) ) then has the form
(68)
(, ) (or Q(s))] K (s+cjd)(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 (64). For A*(s) or
G*(s) specified, z22 (or y22) will have the form given in
(65). Since z21 (or y21) is a positive real driving point
function, a twoterminal 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) (69)
(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. (610)
It will be assumed that T(s) is not positive real and there
fore that the condition given in (610) is not satisfied.
If z21 (or Y21) is chosen as
(s + a jb)(s + a + jb)
[z (or y )] K (611)
21 21 s + e
where 0 s, e 2a and 0 e < 2c, a realization may be easily
effected using only twoterminal techniques. The product
z21Q(s)/P(s) (or y21Q(s)/P(s) ) then has the form given in
(68). If ZT(s), A(s), YT(s), or G(s) is specified, z22
(or y22) will have the form given in (64). If A*(s) or
G*(s) is specified, z22 (or y22) will have the form given in
(65). Here, z21 (or y21) is positive real, and, therefore,
a realization may always be effected using only twoterminal
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) (612)
(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) (613)
is not satisfied. A realization may always be effected by
choosing z21 (or Y21) as
(s + a)(s + b)
[z21 (or y21) ] K + b) (614)
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
(68). In the synthesis of ZT(s), A(s), YT(s), or G(s), z22
(or y22) will have the form given in (64). In the synthe
sis of A*(s) or G*(s), z22 (or y22) will have the form given
in (65). Since z21 (or y21) is positive real, the complete
realization may always be carried out using twoterminal
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 (615)
s+e
The selection of e is made in such a way that the expression
( reb Vc2 + d2)2 2c (e + b) (616)
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
(617)
[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+cjd)(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+cd)(s+c+d) .
22 22 (s+e)(s+b) J
(619)
In the synthesis of A*(s) or G*(s), z22 (or y22) will have
the form
22 (or = K/ (scjd)(/s+c+d)
22 22 1 (s+e)(s+b) I
5) Let T(s) have the form
S (s+aJb)(s+a+jb)
+c)T(s)(s+d)
(s+c)(s+d)
(620)
(621)
(618)
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) (622)
is not satisfied. A realization may always be effected by
choosing z21 (or y21) as
(s+ajb)(s+a+jb)
[z21 (or Y21)] = K (623)
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 (624)
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
(625)
[ cde(c+de)1e
[z (or y )] = K/I s + (c+de) + cde(c1
22 22 s+e
while in the synthesis of A*(s) or G*(s), z22 (or y22) may
be written as
(626)
[z22 (or y22)] = K/H s + (c+de) + cde(cde)
2s+e
Since z21 (or y21) is positive real, a realization using
only twoterminal 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 twoterminal techniques.
Methods have been given for the synthesis of every
physicallyrealizable 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) (627)
(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. (628)
(It is convenient to perform the addition in (628) 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 (628). 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 (627), 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) (629)
(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 (630)
and, second, the radius r is now given by
r = u1 + c + d. (631)
An illustration of this construction is shown in Fig. 6.2,
for F(s) as given in (629), 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 (629)
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 (61), (66),
(69), (612), or (621). 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 twoterminal 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 fourterminal 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 (627) or (629). 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 lefthand half of the a plane and partly
in the righthand 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 lefthand 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
(632)
a > 0 and B 0. The phase function of Q(jw) is given
(633)
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 (634)
F(s) (634)
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 (635)
From (514) the permissible values of w1 are the roots of
the equation
0 R2 + a2 + 2 (636)
which gives
S R a2 + B2
(637)
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 (638), this gives
(638)
exists. This value
(514). For Q (s)
1
0 R2 + cd
0 R + cd
(639)
(640)
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)
(641)
the single condition of physical realizability for a biquad
ratic driving point function, it is necessary that the
condition
[ a2 +2 2 < 4a (642)
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 (642), and then solving for
the roots, Tl and r2, where T1 > T2. The roots, T1 and r2'
are given by
(643)
I ( a+ B2 + 2a) + /[ a 2 + 2a2 (a2 + B2)
and
(644)
2 ( a + 2 + 2a) + a2 B2 + 2a]2 (a2 +B2),
respectively. From (637), yl and r2 become
T1 (1 + 2a) + 1(w1 + 2a)2 2 (645)
1 1 1
