244
IRE
TRANSACTIONS
ON CIRCUIT
THEORY
September
A New Theory of Cascade Synthesis*
D. G. YOULA,i
SENIOR MEMBER, IRE
Summary-This
paper presents a new result generalizing
Richards’ theorem. It is shown that this result leads to a complete,
simple and unified theory of cascade synthesis which yields the
types A, B, Brune, C and D sections in a direct and natural manner.
The element values of the various sections are obtained in closed
form in terms of three or six indexes. Thus the extraction cycle is
performed once and for all for the whole class of positive-real
functions. Several problems are worked out in detail and a chart is
constructed to facilitate the computations. The formulas are easily
programmed on a digital computer.
I. INTR~DUCTI~N
CCORDING
to Darlington’s
classic theorem,’
any rational, positive-real function Z(p) may
be synthesized as the input impedance of a
cascade of passive, lossless reciprocal 2-ports terminated
in a non-negative resistor.
The idea underlying the method is the following: let
---$$i---,
I $
I
I
I
ZL(Pb+
0
k
I
I
-- -I
I
i
7
LOSSLESS
Fig. l-Four-terminal
, RECIPROCAL
2-PORT
realization of Z,(p).
Many derivations of the Brune, type C and D sections
have appeared in the literature and are extensively
discussed in some of the more recent texts.3-5 However,
most of these derivations rest on separate and distinct
artifices, which though quite ingenious, are not held
r(p) = Z(P) + a--P)
2
together by any single principle. The object of this paper
is to present a new theorem (Theorem 1) which generalizes
denote the even part of Z(p). Any real-frequency zero Richards’ theorem and yields the .three sections in a
of r(p) is removed by a (possibly degenerate) Brune section direct and invariantive manner. The element values of
and the remainder impedance is positive-real and of the various sections are expressed in terms of three or six
lesser degree than the original.’ Similarly, real-axis and indexes. These indexes have the same form, irrespective
complex zeros are removed by type C and D sections, of the positive-real function under consideration! Thus
respectively;
the remainder impedances are again positivethe degree-reducing cycle is performed once and for all.
real and simpler than the originals. The Brune, type C In Fig. 9 a chart has been compiled to summarize the
and D sections, are all passive, lossless, reciprocal four- information in a manner suitable for filter design.
terminal networks. After a finite number of steps, the
The motivation for the approach is to be found in a
remainder is a positive-real function Z,(p) whose even previous publication6 wherein the author demonstrated
part is devoid of zeros in the entire p plane; i.e.,
a natural and straightforward evolution of the Darlington
type
C section by means of two successive applications of
Z,(p) + Zd-P) ~ k
C(I)) =
7
Richards’ theorem.’
2
In order to exhibit the synthesis technique succinctly,
the proof of Theorem 1 is relegated to an appendix.
k being a non-negative
constant. It is easy to show that
However, Section II is self-contained so that a reader
Z,(P) = z&d + k
(1) who wishes to familiarize himself quickly with the synthesis
where Z,(p) is a Foster function. Now (1) permits an procedure will not find himself at any appreciable disobvious interpretation of Z,(p) as a lossless 2-port termi- advantage. Explanatory footnotes are inserted wherever
nated in the resistor k (Fig. 1) and Darlington’s result needed.
is established.
* Received by the PGCT, March 15, 1961; revised manuscript
received, June 15, 1961.
t Microwave Research Inst., Polytechnic Inst. of Brooklyn,
Brooklyn, N. Y.
r S. Darlington, ‘Synthesis of reactance 4-poles,” J. Moth. and
Phys., vol. 18, pp. 257-353; September, 1939.
2 The degree of a rational function is defined as the sum of its
(relatively prime) numerator and denominator degrees. Evidently,
a rational function is a constant if and only if its degree is zero.
3 N. Balabanian, “Network Synthesis,” Prentice-Hall, Inc.,
New York, N. Y.; 1958.
4 E. A. Guillemin, ‘Synthesis of Passive Networks,” John Wiley
and Sons, Inc., New York, N. Y.;1957.
6 D. Tuttle, “Network Synthesis, Vol. 1,” John Wiley and
Son;,$rc.,o~;w pk, .N. Y.; 1958.
Darlmgton Synthesis Via Richards’ Theorem,”
Microwave R&. Inst., Polytech. Inst. Brooklyn, N. Y., Memo
no. 50, PIBMRI-901-61; March 14, 1961.
r P. I. Richards, “General impedance-function theory,” Quart.
Appl. Math., vol. 6, pp. 21-29; 1948.
245
Youla: Theory of Cascade Synthesis
1961
II. THE MAIN
is
THEOREM AND SYNTHESIS
Theorem 1 (the Main theorem) :’ Let Z(p) be any positivereal function (not necessarily rational) of the complex
variable p = u + jw which is neither of the form pL
nor l/pC. Let p, = c,, + jw, be any point in the strict
right half-plane or any finite point on the real-frequency
axis, exclusive of the origin, where Z(p) is analytic. Set
2, = Z(p,,) = R, + jX,, R. and X, denoting the real
and imaginary parts of Z(pO), respectively. Then, if
a(p) = Id
+ I 7-b I*,
r(p ) ~ ad
(3)
C(P) = -19,
(4)
and
(5)
where,
+
0
a--PO)
= 0
2
7
i.e., p, is a zero of the even part of Z(p),
degree W(p) I degree Z(p) - 4, w. # 0,
5 degree Z(p) - 2, w. = 0.
4) If p, = u. > 0 is a real zero of r(p) of at least order
two,
degree W(p) I degree Z(p) - 4.
(2)
= -19,
w
1) positive-real
2) If Z(p) is rational, W(p) is rational and
degree W(p) I: degree Z(p).
3) If Z(p) is rational and
The proof of the above theorem is elementary and is
thoroughly discussed in Appendix I. The purpose of this
section is to show that it leads directly to a unified and
invariantive formulation of the entire theory of cascade
synthesis.
Inverting (10) for Z(p) yields
4dWd
(6)
‘(”
= -c(p)W(p)
- b(d
+ a(p)’
Hence, if Z(p) is rational and p. is chosen to be one of
the zeros of r(p), Z(p) is describable as in (II), W(p)
being rational, positive-real and at least four or two
degrees less than Z(p), depending on whether w. # 0
or w. = 0, u. > 0, respectively.
Eq. (11) shows that Z(p) may be viewed as the input
impedance of a 2-port N,, with polynomial chain matrix
and’
1&o) = mPo),
the function
terminated
* After completing this work, the author discovered that an
essentially equivalent theorem had also been obtained by D.
Hazony of the Case Inst. Tech. and stated without proof in “Synthesis by Cancellation of Zeros,” Rept. No. 9, AF-19(604)-3887.
This report derives the Brune, type C and E sections but not the
type D. In a more recent publication of Hazony entitled, “Zero
cancellation synthesis using impedance operators,” IRE TRANS.
ON CIRCUIT THEORY, vol. CT-8 pp. 114-120, June, 1961, the same
material is presented together with a proof of the theorem. Again
the type D section is not discussed. Moreover, Hazony’s proof of Fig. 2-The
the theorem and formulation of the synthesis procedure are considerably different from that described in this paper.
in W (Fig. 2).
realization of Z(p) corresponding to a single application
of Theorem 1.
By direct calculation
4d
d(d - ~W414 = $ + 26~: - dp2 + I po 14, (13)
so that, in general, the determinant 1 tl(p) 1 of the 2 X 2
matrix tl(p) is not the square of a real rational function.
246
IRE
TRANSACTIONS
ON CIRCUIT
Consequently, N, is not always reciproca1.1o (It is most
important to observe that (13) depends only on the point
p. and not on any of the quantities R,, X0 or Z,!)
Two classical exceptions corresponding to p. =jwo(uo = 0)
and to p, = u. > 0 (w. = 0) exist, however. The first
occurs for real-frequency zeros and the second for positive
u-axis zeros of r(p):
ad - bc = (p” + c#,
uo = 0;
(14)
cd - bc = (p” - uy ,
wo = 0.
(15)
We will now prove that when u. = 0, N, is the Brune
section, and when w. = 0, N, is the type C section. The
most instructive way of proceeding is to synthesize N, by
mea.ns of the network depicted in Fig. 3, showing, at the
same time, that the gyrator may be dispensed with in
either of the two cases (14) or (15).
According to the definition of an ideal gyrator,
THEORY
September
V,=
Vb=
ai2,
-ai,
.
---.-L7,
r-i,-
I---
‘2
-_ -.- --
- -I
v”, = cd,,
(16)
LP = primary self-inductance of perfectly coupled transformer.
Ls = secondary self-inductance of perfectly coupled transformer.
M = FtzlL
i;fductance of perfectly coupled transformer:
2
a = the gyraioi’ratio
of the ideal gyrator and is a real number.
v, = -a&.
(17)
Fig. 3-The network realization of the a-port N1 (type E section)
corresponding to a single application of Theorem 1.
From Fig. 3,
v, = (pL,+~)i,+
v, = (PM + &
(PM+;~+a)i,,
- a)& + (6
+ $)i2.
(18)
(19)
More succinctly,
v,
= ZZlil +
ZlZi2,
(20)
2223’2,
(21)
where the meaning of the Z’S is evident. The parameters
A,(p), B,(p), C,(p) and D,(p) of the associated chain
matrix T,(p) are given by
Al(P) = zz = p2j$~+-“,;a
cl(P> = $
= I&L
l+ 1 ,
+ Ls - 2~4 + ,“C)
p2MC - ~CCY+ 1
’
= pzMC -“;Ca
+ 1
(22)
L, i-L,
- 2M +c+
C
Note that because of the perfect
constant, independent of p.
In order that
a(p)
&
(24)
HP)
is the polynomial chain matrix
reciprocal a-port if and only if
A(p) is a
have the polynomial version tl(p) in (12), it is necessary
to make the identifications
(23)
-[ C(P) 1 d(p) 1
coupling,
(26)
(27)
= LPC,
(28)
L, $ Ls - 2M + c&Z,
10More specifically, a 2 X 2 real polynomial matrix
t(p) =
(25)
where
A(P) = ZII.GZ - ~12~21=
Vl = Xl& +
B,(p) = :
p2L,C + 1
D,(P) = ‘E = p2MC _ pea + 1 ’
cm
(30)
and
f&
= L,C.
of a passive, lumped, lossless,
1) a(p) and d(p) are both even or both odd;
2) 6(p) .and c(p) are both even if a(p) IS odd and both odd if
Thus, the capacitance C is explicitly
From (28), (30) and (31)
a(p) is even;
3) ad - bc = P(p), E(p) polynomial and real;.
4) a + b -j- c + d = pv(p”)g(p), where k IS a non-negative
integer, +(~a)
. 1 is an even polynomial and g(p) is a strict Hurwitz
polynomral;
5) pv(p”) divides both a + b and c + d.
I4
1
determined by (30).
Lp=I,=I,r,,
(32)
L,=$,
(33)
247
Youla: Theory of Cascade Synthesis
1961
and
M2 = L,L,
= $ ,
(34)
3
since I, = I;‘.
choices for M:
At this point we see that there are two
M = +
(35)
3
< 0. Inserting this value of M
Choice 1: M = -l/1,
in (29) yields, with the aid of (30), (32) and (33),
a213 = I2 -
2 I, + I, + 2
I3
(36)
1po I2 (1 + IJ2
111,
(37)
I PO I
=- I,I,I,
-
M = -1. -&o)
2
= A-.4cd;
I3
(38)
Eqs. (6)-(8) have been used in going from (37) to (38).
Hence
- u~Z’(u~> < 0
7
co
1
f-J= 2.
uo quo) - u,Z’(u,).
Fig. 4Schematic
of the Darlington type C section to which the
type E section reduces when p, = (ro > 0 is on the positive
u-axis.
(39)
and a negative mutual is possible if and only if w,, = 0,
for otherwise (39) implies an imaginary gyration ratio CL
In short, if p,, lies on the positive a-axis (wO = 0), a choice
of negative mutual M avoids the gyrator and leads to the
familiar Darlington type C section (Fig. 4). In explicit
form the pertinent
parameters are [on the u-axis
R, = Z(U,,)
and
X,/W, = Z’(U,,)]
1
au,> + uJ’(uo)
2ao
Lp=I,T,=
= +z(uO)
(+“.
1
go -quo> - uoZ’(u0) *
Choice 2: M = +1/I,
>
through just as before we get
0. Carrying
a=*--.
(43)
When u0 =
(45)
because both are real.
always leads to physical
= 0, p, is on the realunnecessary. Therefore,
(48)
3
M=+f,
(42)
(49
ff=f---, 2uo
I3
and either value is permissible
In other words, a positive mutual
elements. In particular, when u.
frequency axis and the gyrator is
L”+
(41)
the algebra
(47)
Lp=-j-p
0 and R(w,)
(49)
3
200
I3
=
0 (the Brune
(50)
section),
Ro/“o Z x’(“O) ana
CA.
4u;
cY2=2,
I3
(46)
1
- uOZ’(u~) ,
UO
3
C=&,
(40)
7
au,>+ aoZ'(uo)'
M = -$
the type E section with positive mutual is always realizable
as a passive, lossless, nonreciprocal 4-terminal network
and coincides on the real-frequency axis with (a possibly
degenerate) Brune section (Fig. 5(a)). The element values
of the type E section with positive mutual are
L
P
1
t
QJL!
woX’(wo) - -aJo)
= ~oX’(~o> + -n%)
2wo
’
(51)
(52)
Ls _ 1 box'
- -%41"
2wo woX'(wo>
+ X(u0) '
(53)
j,,f = w~x’(wd - x(wo) > 0
2
2ul
(54)
a = 0.
(55)
248
IRE
--_
TRANSACTIONS
Nl
_
‘2
f&fi-
-1
s
+
,
i
0
-- J
(a)
+ Xbo)7
L = woX’(wo)
2%
THEORY
September
eliminated. This, of course is in agreement with the
former observation that N1 does not always represent a
reciprocal structure. The situation is easily remedied by
applying Theorem 1 once more to W(p) at the same
point p,! Thus, by (II),
“2
I
I
ON CIRCUIT
W(P) =
hoK(p)
-.cI(P)WP) + e(p) ’
(56)
where K(p) is positive-real and e, f, g and h are the polynomials (2)-(5) associated with W(p) :
L-----
e(p) = Jlp2 + I p. 1’
(57)
f(P) = - J*P,
(58)
s(P) = -JJ,P,
(59)
h(p) = J,P’ + I po I’.
(60)
P
Ls _ 1 . boX’(wo)- -w~o)l”
2%
wox’(wo)
+ X(w0)
j/J = WOWWO)- -VW,) > o
7
24
1
c = 2.
wo wox’(wo> - X(wo>’
O--p-O
T w;x’h,,
The coefficients J,, J2, J, and J, are the indexes assigned
to the point p. by W(p). Hence if W, E W(po) =
U, + jV,, the J’s are calculated from (6)-(g), by substituting U, for R,, V, for X0 and W, for 2,.
Part 2 of Theorem 1 guarantees that the degree of K(p)
does not exceed that of W(p). Consequently, K(p) is
also at least two or four degrees less than Z(p) depending
on whether p. is a zero of r(p) lying on or off the u-axis,
respectively. The network equivalent of (56) is shown in
Fig. 6.
(b)
Fig. 6-The realization of W(p) corresponding to a second application
of Theorem 1.
I
0
?) = LIMIT
0
pZ(p)
P*a
Cd)
Fig. 5-(a) Schematic of the Brune section to which the type E
section reduces when po = $10 is on the real-frequency axis and
R(yo) = 0. (b) Degenerate Brune section corresponding to a
pomt PO = ~CQon the real-frequency axis at which .Z(&, = 0.
(c) and (d) The limiting forms assumed by the degenerate section
of Fig. 5(b) when w0 = 0 and o0 = m.
As indicated in the diagram,
nomial chain matrix
UP) = [ziff],
(61)
and since [see (13)l
I tz(p> / = I h(P) I = P” + X6
If G = 0 ad
R(w,) # 0, W(p) = .Z(P)~ and N, degenerates into a pair of wires. iVote that if X(wo) also
vanishes, i.e., Z(jwo) = 0, the network of Fig. 5(a) collapses
into the degenerate Brune section of Fig. 5(b).
In the two limiting cases, o. = 0 and w0 = 00, Fig.
5(b) goes into Figs. 5(c) and 5(d), respectively, and the
familiar types A, B sections are recovered.
In the most general case p. is neither on the u-axis nor
on the real-frequency axis and the gyrator cannot be
N, possesses the poly-
I 4(pP2(p)
I = (pa + %,z,
-
4P2
4)P2 +
I PO I4
+ I PO n2
(62)
(63)
and is the square of the real polynomial,
E(p) = p4 + 26-4 - u2,)p* + I po 14.
(64)
But t(p) = t,(p)&(p) is the polynomial chain matrix of
the over-all 2-port N (the cascade of N, and NJ which,
when terminated
in K(p),
gives rise to the input impedance Z(p) (Fig. 7).
1961
Youla: Theory of Cascade Synthesis
Fig. 7-The
realization of Z(p) corresponding to two applications
of Theorem 1.
”
249
It is easy to show, without evaluating the Vs, l’s and
m’s explicitly, that Z,(p) is compact at all four poles; i.e.,
k,,k,, - lc;, = 0,
Ll,,
Thus, despite the fact that neither N, nor N, are
reciprocal, their cascade N is a passive, lossless, reciprocal
2-port. In the first place, N is reciprocal by construction
and secondly, since N, and N, have already been shown
to be individually passive and lossless (Fig. 3), N is also
passive and lossless. But N has some additional remarkable properties. The (conventional) chain matrix T(p)
of N is given by
where, by direct calculation
(65)
from (2)-(5)
and (57)-(58),
- II, = 0
(76)
and
2
mllmz2 - ml2 = 0.
(77)
For
V4J2 +
= (I,J,
T(p) = ; t,t, = 4
(75)
12JJP2 +
+ I,J,)$
+
I PO I2 (12 + J*)
I po I2 (I3 + 53) ’
where / Z,(p) / is finite at p = 0, and p = m and has
only simple poles at p = &j~,. But from (72)-(74) we
see that 1Z,(p) 1 contains terms of the form
d(P) = U4J4)P4 + [I PO I2 (14 + J4) + I*J,]pZ + I p, j4,
(66)
S(P) = U4J*
+ LJdp”
c”(P) = U3J4 + IIJ3)P3
d(P) = UlJJP4
+
+
I po I2 (I, + J,)p,
+
I po I2 (I, + J,)p,
[I PO I2VI + JI) +
(6%
13J,]p2 + I p0 I4
(69)
and
E(P) = P4 + 2(4
-
U’o)P2 +
1 po y;
(64a)
of course, ci$ - & = E’. The resulting 2 X 2 impedance
matrix.Z,(p)
of N is
!!
ci
T
Z,(p)
=
E
T
“--“_
.
E
d
-2
e
(70)
Clearly, the only poles of Z,(p) occur at p = 0, p = ~0
and at p = =I+,, where
I2(13 + J3)
a’ =
y3J4 + I,J,
Thus, the elements of Z,(p)
fraction expansions
z 11 =
ci
-=
c^
hp
’
(71)
must possess the partial
1
+ 2
and
2pml,
+ -2p +d’
(72)
and the compactness requirements follow. A perusal of
(66)-(69) indicates that N has one more special attribute,
namely
I,, = I,, = I,, = +-y$
3
3
(73)
To summarize, any passive, lossless reciprocal realization of N must possess an impedance matrix Z,(p) with
the structure delineated in (72)-(74) such that
1) Z,(p) is compact at the four poles p = 0, ~0, &GJ, and
2) residue x11 = residue z1a = residue zZZat the pole
p = 0.
Now the Darlington
type D network depicted in
Fig. 8 meets all these specifications!
An application of Kirchoff’s laws together with the
perfect coupling condition L,L, = Mi, enables us to
derive the impedance matrix
IRE
250
+o
TRANsACTIONs
,
:
^__ I--x----
UN
ClhTCUl1’
--------
September
1’HlWliY
Observe that (83)-(91) involve the J’s as well as the
I’s. All that is needed to compute the J’s is W,,, the value
of W(p) at p = po. This is easily found by applying
L’Hospital’s rule to (10):
-
!
w, = -1,
po-uPo)~o - 20%
PoZ’(Po) - x
’
“I
(914
where
I
1
I
x = u,x”+
00
I
1
i---.-.
--..A
Fig. 8-A Darlington type D equivalent of N, the passive, lossless,
reciprocal a-port resulting from two applications of Theorem 1.
where
jw,Ro.
CO
The chart of Fig. 9 is designed to facilitate the synthesis
procedure and to show at a glance the functions of the
different sections. A few words of explanation regarding
the Richards section are in order. Let Z(p) be an arbitrary
positive-real function. Then, Richards’ theorem states
that for any u > 0,
z 1&) =
4P)
-
Pad
-- PZ(P)
-WI
+
u
(92)
is also positive-real. If Z(p) is rational, degree Z,(p) < degree Z(p) and if T(U) = 0, degree Z,(p) I degree Z(p) - 2.
Inverting (92) for Z(p) gives
and
2 = X,/L,.
WC8
Since L,L, = Mf, Z,(p) is also compact at p = m.
Furthermore, at p = 0, residue zll = residue z12 = residue
zZl = S,. Equating the k’s, l’s and m’s in (72)-(74) to
the coefficients appearing in (79)-(82) yields the set of
relations (see Appendix II):
Id,J,
+ I,J,
L1 = I,J,
’
(83)
’
(84)
I*J1
Lz = I,J,
-f I,J,
1
M1 = I,J,
+ I,J,
c;’
+-vzzro,
=
= & = +$
M2
L,
=
-
1 po
)
1
PO
1’
(13
III. EXAMPLES
2
I:;;j4
$--J;o(::
I;
J,)”
5
(87)
‘,
To see how the chart is used, consider the example
203) =P”f9Pf8.
p2 + 2p + 2
JsJ,I,U, - J1)”
I,Jz
=
(85)
A simple analysis now reveals that Z(p) may be realized
by terminating the Richards’ section (shown in Fig. 9) in
Z,(p), an observation already recorded in the literature.”
The type E section is capable of extracting any complex
zero of r(p) and leads to a four degree reduction. The type
D section also performs the same operation but leads to
an eight degree reduction if this zero is of order two or
more. The Richards network extracts any u-axis zero of
r(p), yielding a two degree reduction. The type C achieves
the same object but diminishes the degree by four
if the zero is of order two or more. Finally, the Brune
section induces a four degree reduction, provided that
0 < Iwo1 < a.
3
3
2
qp> = flzl(P> + PZ(d,
zm?l + u
-ad
(82)
+ J3) +
L,
(I,
J,
+
=
M;/L,,
I, J3)(13
+
J,)’
’
@)
Then,
(89)
r($)
2
0. =
j
PO I2 (13
13J4 +
+ J,)
I,J3
and
c,’
=
s,
(94)
= wZL3.
(91)
= (P” p”+4
4Y
’
11D. Hazony, “A cascade representation of the Bott-Duffin
cycle,” IRE TRANS. ON CIRCUIT THEORY, vol. CT-5, pp. 144145;
June, 1958.
196i
Youla: Theory of Cascade Synthesis
251
-0
4
8-l”
4
t;
:
t
4
VI
X
d
a
d
= primary self-inductance of upper perfectly coupled transformer.
LP = secondary self-inductance of upper perfectly coupled
transformer.
MI = mutual inductance of upper perfectly coupled transformer.
La = primary self-inductance of lower perfectly coupled transformer.
Fig. g--Chart
L4 = secondary self-incuctance of lower perfectly coupled
transformer.
M2 = mutual inductance of lower perfectly coupled transformer.
2
&a
1
ma=-=L3
L3Ca
depicting the various sections to be used for the extraction of zeros of r(p).
252
IRE
TRANSACTIONS
ON CIRCUIT
September
THEORY
Using the formulas for the element values of the type
and p, = go = 2 is a zero of r(p) of order two. According
to part 4 of Theorem 1, W(p) is at least four degrees less D section (Fig. 9) we find,
than Z(p), i.e., W(p) is a constant. Let us first compute
I, = 0.2
the indexes:
I, = 0.8
Z’(2) = -3
Z(2) = 3;
I, = 1.6
.*. T,(2) = 2;
18(2) = 1
12(2) = 18;
J, = J, = 1,
J, = J, = 0.8
and the parameter values of the type C section are (Fig. 9)
and
1
1
L, = 125/44 h,
LP = m = ijh,
T
L, = 2 = 2h.,
3
M
=
+
=
-lh.,
3
5/44 h,
M, =
25/44 h,
c, =
12 f,
cd,=
c=
Since W(p) is constant,
L, =
3/11,
M, = 136/99 h,
L, = 272/99 h,
it equals its value for p = 0.
L, = 68,/99 h,
C, = 1331/68 f.
This is shown in network terms in Fig. 11.
Fig. IO-Realization of the Z(P) in (94).
Fig. 11-Realization of the Z(P) in (95).
Hence referring to (lo),
W(p) = W(0) = Z(0) = 4fi.
This is shown in network terms in Fig. 10.
A more complicated specimen is
Z(P) =
25~’ + 4p + 1.
p2 + 8p + 1
The zeros of r(p) form a quadruplet
where
(95)
(pO, &, -ppo, -j&)
2
j
po = 5 + 5 = co + &Jo.
Thus,
2, = Z(pJ = 1.5 + jo.5.
Since W(p) is again a constant (a four degree reduction
. occurs),
W(p) = W(0) = Z(0) = 1Q;
.-. R, = 1.5,
i-7, = 1
x, = 0.5
v, = 0.
From a practical point of view, it appears desirable
first to strip Z(p) of all real-frequency zeros and poles;
i.e., to go through the usual Brune preamble. Having
done this, the remaining zeros of transmission are removed by appropriate sections and the remainder is a
non-negative resistor. Naturally the process may be varied
at any stage. The Brune, Richards, type E and type C
sections require, for their determination, only the indexes
I, - I, whereas the type D uses both I’s and J’s. Formulas
(91a) and (91b) supplemented with Z’(pO) and the I’s
suffice and the need for deriving W(p) is completely
obviated (at least for one cycle). To summarize:
1) Apply
the Brune preamble t,o Z(p) to obtain z”(p),
a positive-real function devoid of zeros and poles
on the entire real-frequency axis.
2) Use the chart of Fig. 9 to extract the appropriate
cascade sections corresponding to the zeros of
F(p) = Z(P) + -R-P) .
2
1961
Youla:
Theory of Cascade 8ynthesis
IV. CLOSING REMARKS
By means of Theorem 1 it has been possible to place
Darlington
synthesis on a sound and rational singleprinciple basis and to evolve the various sections in a
natural and direct manner. Moreover, the extraction cycle
has been carried out in abstract terms resulting in closed
formulas for all element values. These values depend
on either three or six indexes. The entire process is now
in a guise suitable for programming on an ordinary gardenvariety digital computer and work in this direction is now
in progress.
In a future paper the author hopes to show that Theorem
1 also leads to a solution of the problem of interpolation
in the right-half p-plane with positive-real functions.
This in turn leads to a simple procedure for the design
of oscillators with prescribed modes and to an alternative
approach to the approximation problem.
APPENDIX
I
The object of this Appendix is to prove Theorem 1.
However, we take this opportunity to develop the meaning
and concept of “indexes” at some leisure, relegating many
of the important side questions to the positions of lemmas.
We begin with a definition.
Dejinition 1: A function Z(p) of the complex variables
p = u + jw is said to be positive-real if
1) it is analytic in the strict right half-plane, Re p > 0;
2) Re Z(p) 2. 0 for Re p > 0;
3) Z(p) is real for all real p in Re p > 0.
Lemma 1 generalizes the notion of the “residue at
infinity” to positive-real functions that are not necessarily
rational.
Lemma 1: Let Z(p) be an arbitrary positive-real function. There exists a non-negative number L 2 0 such
that
-G(P) = Z(P) - PL
is positive-real.
limit Z’(p) = limit z(p> = L
D-m
n-m P
The details are supplied in Valiron.”
Let Z(p) be an arbitrary positive-real
function and set
Z(P) = Rb, 4 + iX(u, 4,
R and X denoting the real and imaginary parts, respectively. On the real-frequency axis, u = 0, p = jw
and Z(jw) = R(o, w) + jX(o, w). Since there is no risk
of confusion, we write this as
Z(j,> = R(u) + ix(u).
(9%
Consider the two quantities
qp> = R(U)4
u
and
X(u 4
cl(P) = *.
Clearly, X(p) is defined for all p in Re p > 0 and p(p)
is defined for all p = u + jw in Re p > 0 for which w # 0.
Moreover, if Z(p) # 0,
Rep > 0.
h(P) > 0,
(102)
The domains of definition of X(p) and p(p) are extended
by a limiting process. For example, let p = u. be any
point on the positive u-axis. Then, in a sufficiently small
neighborhood of p = uO,
JXP) = Z(Q) + (P - 4Ud
+ z2 (P - dkak;
(103)
Z’(a,) and the a’s are real. Letting p - u0 = re”,
m
+ C eik+rkak.
k=2
Z(p) = Z(u,) + re’“Z’(u,)
(104)
Since r sin $ = o,
X(U, U) =
(97)
Furthermore
253
wz’(u,,)
+
2
sin k4
r”ak
k=Z
(105)
and
Limit X(u7 u) = Z’(uJ
w
P-no
(98)
uniformly, provided that p tends to infinity in the right
half-plane along any ray arg p = p where 0 < 1p 1 < 7r/2.
Proof: If Z(p) is rational, the lemma is trivial since L
is nothing more than the residue at infinity. However, for
general positive-real functions, the
limit z(jw,
lo-m P
does not necessarily exist and the lemma contributes
essential information. The number L is also known as the
angular derivative of Z(p) at infinity (Caratheodory).
(106)
provided that p + u,, along any ray 4 = constant, 4 # 0,
ir (Fig. 12).
This is written as
ray-limit
P+ro
X(a,
Ld
=
(107)
Z'(uJ.
By definition,
P*(U) = -md,
u > 0.
I2 G. Valiron, “Fonctions Analytiques,”
de France, Paris, France, pp. 79-87; 1954.
(10%
Presses Universitaires
254
IRE
PO”
TRANSACTIONS
September
ON CIRCUL 1’ THEORY
Dejinition 2: Let Z(p) be an arbitrary positive-real
function and p any point in Re p > 0 or any finite point
on the real-frequency axis, where Z(p) is analytic. The
four real numbers
I-
I (pj = VP) - FL(P)
1
VP) + P(P) ’
Fig. 12-Illustration
of the meaning of ray-limit for points on the
positive u-axis and on the finite real-frequency axis.
(119)
Idp)
2 I JxP> I2
= X(p) + P(P) ’
(120)
I,(p)
=
(121)
2
VP> - P(P)
and
Similarly, if p = jw, is any finite point on the realfrequency axis where Z(p) is analytic,
p(jwo) = x(,,>
w. # 0,
,
WO
00 = 0.
= Z’(O),
= -jR’(w)
d(jw)
+ X’(w)
(122)
(109)
are the indexes assigned to the point p by the positive-real
function Z(p). The set of these four numbers is called the
index set relative to Z(p) and is denoted by
(110)
f-UP) = (11,I,, I,, 14).
To discover a meaning for X(j,,) when R(w,) = 0,
we again resort to a limiting process. Thus, suppose
0 I 1 w. 1 < m and R(w,) = 0. Clearly,
-
IdP) = r(P)
It is interesting to see the forms taken by the indexes
at the various points p.
Casel:p=o+jo,a>O,o#O.
R(u
A-Aw) au 4
(111)
Ilcp) = R(u: u) I X(:, w) ’
which implies that
(112)
(125)
because w. must be a zero of R(w) of at least order two.
Consequently, in a sufficiently small neighborhood of
P = PO,
I,(P) = R(u w)
A--
(113)
Z(p) = jX(w3 -I- (p - jwo)X’bo) -I- g2
9 (P - jwojkbk
14(p)
R(a, o) = gX’(w,) -k Re 2 eik%‘bk.
(114)
R(wo) # 0.
= + m,
(116)
= X’bo),
NW,)
R&J
# 0,
= 0.
(126)
W
(127)
K’(P).
Z(u) - u.2%>
I&) = Z(u) + uZ’(u) ’
(128)
I&d
2uZZ(u)
= Z(u) + uZ’(u) ’
(129)
Lb)
= Z(,) ?u2t(u)
(130)
’
I,(u) = I?(u).
Case3:p=
By definition,
X(jw,> = + 03,
=
(115)
provided that p --) jw, along any ray I$ = constant,
7r/2 (Fig. 12). Obviously,
4# -r/2,
ray-limit R(a,
u
D-iwo
X(0 w) ’
Case 2: p = u, u > 0.
Hence,
= X’(wo)
2
U
and
limit ‘+
?)+ioo
(124)
W
U
Z’(jwo) = X’(wo)
(123)
jw,O<
(131)
[WI < a,R(w)
ZO.
(117)
Il(jw)
= 1,
(132)
uw
I&w)
= 0,
033)
&W
= 0,
(134)
In short, X(p) and p(p) have now been assigned a
meaning for all p in Re p > 0 and all finite points on the
real-frequency axis where Z(p) is analytic.
I,(jw) = 1.
(135)
1961
Youla: Theory of Cascade Synthesis
Case4: p = jw,O < 1w 1 < ~0, R(w) = 0.
Il(jw) = w-w.4 - X(4
wX’(w> + X(4
and if p. # po(wo # 0),
’
036)
I&,> = wX’(w) + X(w) ’
(137)
2w
= wX’(w) - X(w) ’
(133)
2wX2(w)
I&)
14(jw) = I;‘(jw).
alternatively,
2 > I -UP01 I
Xo
1
G- I1
Ro
Yip
--~~Po)P,
d@o,PO) = I,(P,)P~ + I PO 1’
are the four indicial polynomials assigned to the point p,
by the positive-real function Z(p).
The importance of the indexes and the indicial polynomials has already been brought out in Theorem 1.
As we will soon see, they possess some vary elegant
and useful properties.
Lemma 2: Let Z(p) be any positive-real function which
is neither of the form pL nor I/PC, L and C non-negative
constants, and let p. be any point in Re p > 0 or any
finite point on the real-frequency axis, exclusive of the
origin, where Z(p) is analytic, Re 2 = 0 and 2 # 0. Then,
(k = 1, 2, 3,4).
(141)
Proof: It suffices to prove that under the prescribed
hypotheses and for the stipulated points p, 0 < X(P) f
CL(P) < a* Now, for any p. in Re p > 0 and any positive-’
real function Z(p), the function
w(p) _ 2oozo
Z(P) + -aPol
(142)
is analytic and bounded by unity in Re p > 0 and vanishes
for p = p,. Thus, by the theorem of the Maximum
Modulus,
I 4-P I =
I 1,
Rep > 0,
(146)
0 < ~cpo>f dpo) < co
040)
C(P, PO) = -UPo)P,
0 < Ik(PO) < O”,
00 z 0.
Eqs. (145) and (146) show that at any point p, in Re p > 0,
4P, PO) = Il(PO)P2 + I PO I27
HP,PO) =
(145)
and
(139)
The functions
and
255
(143)
and (141) follows.
Now let p, = jwo, 0 < I w. ( < 03 be any point on the
real-frequency axis where Z(p) is analytic and R(oo) = 0;
i.e., Z(jwo) = jX(wo) and Z’(jwo) = X’(w3. First, if X(wo)
is also equal to zero, Z(jwo) = 0 and Z-‘(jw,) possesses
a simple pole at p, = jw, with the positive residue
l/X’(w,);
the inequality X’(w,) > ( X(wO)/wo J is trivial.
Hence, we may assume without loss of generality that
R(wo) = 0 and Z(jwo) # 0. Expanding Z(p) in a Taylor
series about the point p. = jwo yields
R(u, wj + jX(u, w) = Z(p) = jX(wo>
+ (P - jwo>X'(wo>
+ z b - jwo>kak.
After a little algebra,
R(u
4 31 Amu WI = X’(wo) f +
A
W
U
(147)
+ O(r)
for any fixed + in the range -r/2
< 4 < 7r/2, where
p - jwo = re’m. Since Z(p) is positive-real it follows from
(146) that
R(u,w) I Xb, 4 , 0
U
w
for all u > 0. Thus, as is obvious from (147),
R(cT,
X’(wo) f Z?5Yk!d= ray-limit
?-PO
wo
* ZI.kZ.4
U
w.
1
> 0,
-
or
with equality if and only if
X’(w,) 2
w(p) = ejo P2Lz.23
( p+so > ’
8 a real constant.
(144
In this latter case 1 w(jw) I = 1 and this implies, together with (142) that Z(p) is a Foster function bilinear
in p; i.e., Z(p) is either of the form pL or l/pC, L and C
non-negative constants. Excluding these, I s(p) I < 1,
Re p > 0. In particular,
IdPo> I = jZYpo~~/
< 1
1
+$
I
.
Actually, under the prescribed hypotheses, the equality
sign is impossible. Suppose for example that X’(w,) =
X(wo)/wo and w. > 0. Substituting this in the equation
appearing directly above (147) yields
-W
= PLO + g
(p - jwolkak,
(143)
where Lo = X(wo)/wo. Identifying the real and imaginary
parts of the right-hand
side of (148) and setting
IRE
256
TRANSACTIONS
ak = j al, I e”lc we derive
R(u , 4
x(u,
U
wj
_
2
W
1 ak
I rkml
cos
(k+
cos
k=2
+
ok)
cp
_ g j uki rksi”,@++
ek>
2.
(24g)
o.
ON CIRCUIT
THEORY
September
merely necessary to change I 2 I into ] 2 1-l and p into
-p in (152)-(155). This immediately yields (151), Q.E.D.
There is now no difficulty in showing that the indicial
polynomials a(p), b(p), c(p), d(p) associated with Y = 2-l
are given by
a(~, PO) = db> PO),
Choose a $Jin -r/2
< + < r/2 such that
HP, PO) = dP, PO),
cm (24 + 6) < o
cos I$
Now choose r small enough to guarantee that w = w. + r
sin 4 > 0, -a/2 < 4 5 7r/2, and that all terms in (149)
except the one corresponding to Ic = 2 in the first summation are negligible; i.e., to first order,
o -< Nu,
~- 4
U
X(u, 4 _ r I a2 I cos(24 + b> I o
W
cos
Lemma 4 is of an extremely simple nature but is,
nevertheless, the clue to the proof of Theorem 1.
Lemma 4: Let Z(p) be an arbitrary positive-real function and p, = u. + jwo any point in the strict right halfplane. Set 2, = Z(po) = 8, + jX,. Then, if
c$
a) 1 - Il(po) 5 0, there exists a positive-real function
of the form z(p) = r. + pL, such that
Clearly, the only way to avoid a contradiction is to have
I az / = az = 0. But a reference to (149) reveals that with
a, = 0 and r sufficiently small,
R(u
A--= 4
U
W
cos
4
ti(Po) = Z(Po) - 4-PO)
= 0,
wof: 0,
or
r2 1a3 1 cos (34 + e,)
X(u 4
(156)
_c(P,PO)= Npt PO),
d(P, PO)= dP, PO).
wo = 0.
duo) = V(u0) = 0,
f
b) If 1 - II
> 0, there exists a positive-real
and repeating the above argument once again, a3 = 0.
function
x(p)
=
r.
+ X,/p with the above stipulated
By induction, ak = 0, for all k > 2, whence from (148),
properties.
Z(p) = pL,. If X’(wo) = -X(wo)/wo we consider Y(p) =
Z-‘(p) and find that Z(p) = X,/p, S, = -wnX(wo).
Proof: Note first that for w. # 0, 1 - I,(po) ; 0 are
Since both of these cases are explicitly excluded in the
statement of the lemma, X’(wo) > I X(wo)/wo I and we are equivalent to X,/w, $ 0 and for w. = O(p, = a,) to
again led to the two inequalities, 0 < X(jw,) f p(jw,) < 03, Z’(u,) 2 0, respectively. Suppose for the sake of definiteQ.E.D.
ness that w. > 0 and X0 I 0. Then, $(po) = 0 requires
Lemma 3: For any positive-real function Z(p) and any
(157a)
r. = R. + goLo,
p in the domain of definition of the indexes,
Q&9
(150)
= fMP)
Lo = -3.
wo
and
MP)
=
(14,
I,,
I,,
Id,
(151)
in which, of course, $j denotes the complex conjugate of p,
Y = 2-l and &(p) is given by (123).
Proof: Refer to (124)-(127). Observe that at the point
@,w goes into -w and Z(p) goes into Z(p). Hence, X goes
into -X and (150) is obvious by inspection. The relation
(151) is best established by expressing 11, I,, Is, and I, in
trigonometric form. Let p = I p I eimand Z(p) = I 2 I e”.
From (124)-(127),
sin (e - P)
(152)
Ilcpj = sin (e + p) ’
Hence Lo 2 0 and therefore r. > 0, i.e., x(p) = r. + pL,
is positive-real. The element values are expressed most
succinctly in terms of the indexes:
Lo = -1 20 I2(1 -
(153)
sin 28
L(P) = p2 sin (e - 0) ’
I I
(154)
sin (e + P)
14@) = sin (0 - p)’
(155)
(158)
’
2flo
r 0 = -.
I3
(159)
If w. > 0 and X0 > 0 and we choose z(p)
it is found in exactly the same way that
20
I2
(1
I,
r
Since Y = Z-‘, Y = ] Z I-le-” and consequently, to
obtain the indexes assigned to the point p by Y it is
11)
12
&=’
I,(p) = 1
sin (e + p) ’
(157b)
2uo
O=I,I,'
-
11)
’
To +
SO/~,
(160)
(161)
We leave it to the reader to fill in the details for p, on the
u-axis [$(uo) = lC/‘(uo) = 01. The formulas (158)-(161)
are always applicable, irrespective of the location of
p. in Re p > 0. The identifications
(158) and (159)
correspond to 1 - II
5 0 and (160), (161) correspond
to 1 - I,(po) > 0.
257
Youla:’ Theory of Cascade Synthesis
1961
Our next and final lemma is again quite elementary
but useful.
Lemma 5: Let Z(p) be an arbitrary positive-real function and z(p) any positive-real function of the form
Using (161)-(167) we find, after a little simplification,
M-P) + dPMP) + cd-PMP) - dPM--PI
w1(p) = M-P) - dP)lZ(P) + d-PMP) + dPM-PI
(167)
4-P) = To + ZAP),
where r0 > 0 and Z,(p)
Foster function. Then,
is a rational
= (P" + I pa I") -@I + p3Lo+ pWo I p. I2 - 2uoro)
-2p~J(pl
+ p2(ro- 2aoLo) + ToI p. I2
(or meromorphic)
WO
w(p) = Z(P) - d-P)
Z(P) + 4-p>
is a bounded-real scattering coefficient. That is to say,
w(p) is analytic and bounded by unity in Re p > 0 and
real for all p = u > 0.
Proof: Since 2,(-p)
= -Z,(p),
where N(p) and D(p) denote the numerator and denominator of (167) or (168), respectively.
Eqs. (161a), (161b), (166) and (167) directly show that
d--P”) = d--150) = 0,
Npo) = N&J,) = @a) = @a) = 0,
Z(P) + Z,(P) - To
w(p) = Z(p) + Z,(p) + f-0
wo# 0
(169)
w0 = 0.
(170)
and
and is therefore the reflection coefficient of the positivereal function Z(p) + Z,(p) normalized to the positive
number r,,, Q.E.D.
Now that these preliminary considerations are out of
the way, we may turn our attention to the proof of
Theorem 1.
N(u,) = N’(u,)
Proof of Theorem 1: Let Z(p) be an arbitrary positivereal function and p, = u,, + jw, any point in the strict
right half-plane. Set 2, = Z(p,,) = R, + jX, and assume,
to begin with, that w0 > 0, X, < 0. According to Lemma
4, there exists a positive-real impedance z(p) = r,, + pL,
such that the function #(p) = Z(p) - 2(-p) satisfies
either
N-PO) = DC-PO) = s(poM--Pa) + 4-~0)1
= -dPoMPo) - 4-PO)1 = 0.
tiCPa) = iGo) = 0,
W” > 0,
(161a)
or
$(a”) = ~‘(a,) = 0,
W” = 0.
(16lb)
By Lemma 5,
w(p)
=
Z(P) - d-P)
Z(P) + Z(P)
(162)
in a passive scattering coefficient. Since the numerator of
w(p) is divisible by the quadratic factor (p - p,J(p - j&J,
it follows from the Maximum Modulus Theorem that
the function s(p) defined by
(P - POXP - 15”)
w(p) = (p + P”)(P + PO)s(p)
coefficient.
(163)
Consequently,
(164)
is positive-real.
Let
065)
where
g(p) = P2 + 2@“P + I PO 12.
= D(uo) = D’(uo) = 0,
Thus, N(p) and D(p) both are always divisible by the
factor (p - p,,)(p - j%J = g(-p).
Now suppose that
Z(p) is meromorphic and p, is a zero of its even part;
i.e., Z(p,,) + Z(-pO) = 0. From (167),
w3
(171)
Hence if w0 # 0, both N(p) and D(p) are divisible by
(P + PO)(P + PO) = g(p). If W” = 0,
N(-a,)
= D(-ua,)
= 0
(172)
and therefore, w,, # 0 and Z(pO) + Z(-pO) = 0 imply
the divisibility
of numerator and denominator by g(p)
while w,, = 0 and Z(pO) + Z( -pO) = 0 imply their
divisibility by the linear factor (p + uO). One more point
is of paramount importance. Suppose that u,, is a zero of
r(p) of a least order two. Then, Z’(a,) - Z'(-a,)
=
0.
Differentiating
N(p) and D(p) and setting p = -u,,
yields with the aid of the relation Z'(U,)
=
Z'(
-uo),
N’(-uo)
= D’(-a,)
But by construction
= Z’(u,) + x’(-u,,).
[see (161b)],
!b’(U”) = $ [Z(p) - x(-p)lp=oO = Z’(u,) + Z’(--0)
= b(PMP)
is also a passive scattering
that
= 0,
whence, p = - u,, is also a zero of N(p) and D(p) of at
least order two. Thus, in this case, N(p) and D(p) are
both divisible by the quadratic factor (p + a,,)‘.
To summarize:
1) N(p) and D(p) always possess the common quadratic factor g( -p).
2) They possess the common quadratic factor g(p)
when r(p,J = 0 and w0 # 0.
3) They possess the common linear factor (p + uo)
when w0 = 0 and u. is a first-order zero of r(p) and
the common quadratic factor (p + uJ2 when u,, is a
zero of r(p) of at least order two.
258
IRE
TRANSACTIONS
We are now in a position to bring the proof to a quick
conclusion by simplifying
(168) in such a way as to
remove the p3 term in N(p). It is easily seen by inspection
that the angular derivative (or residue for rational 2)
at infinity L, of W,(p) is
L, =
L + Lo
To - 2U”(L” + L) 2 O;
it is non-negative because W,(p) is known to be positivereal (L, is the angular derivative of Z(p) at infinity).
Thus, if
then
0 I E I Ll,
(176)
Using (174) and (168), we get
=
dPP(P> + b(P)
~PPYP>+ d(p)
a(p) = r. ~“~~,,,
U” 0
2U”P
(177)
(179)
k+2
k-k3
and in each case the apparent increase in degree is four
units. However, a decrease of four units is always brought
about by the cancellation of the common quadratic
factor g(-p)
from the numerator and denominator of
W(p) and so degree W(p) I degree Z(p). Furthermore,
if Z(po) + Z( -pa) = 0 an additional diminution in degree
of either four or two units occurs because of the cancellation of either the second-order factor g(p) or the linear
factor (p + uo), depending on whether w. # 0 or w. = 0,
respectively. Thus, if pa is a zero of the even-part of Z(p),
degree W(p) 5 degree Z(p) - 4, w. # 0,
d(p) = (1 - y)p’
+ I p, 1’.
uw
Eqs. (6)-(g) are readily obtained by substituting
(157a) and (157b) in (178)-(181). We now assert that
the numerator and denominator of W(p) also have the
divisibility
properties l), 2) and 3) enumerated in the
summary appearing above (172). The proof is immediate
because by (176),
= r.
As mentioned previously,
whenever u. is a zero of r(p) of at least order two.
All that remains is the removal of the restrictions
w. 2 0, X0 I 0. First, if w. < 0, the theorem is applied
at the point p,. Since Q,(@,) = QZ(po), the polynomials
a, b, c and d remain unchanged. Second, if X0 > 0, we
apply the theorem to Y(p) = Z-‘(p). We already know
that the polynomials a, b, c and d associated with Y(p)
are determined from (156). Hence, the positive-real
function
(1W
and
N(P)
- PE D(P).
D(P)
At this stage, it is possible to compare the degrees of
W(p) and Z(p) when the latter is rational. Note first that
a(p) and d(p) are of degree two and b(p) and c(p) are
of degree one. Since Z(p) is positive-real and rational,
its numerator and denominator degrees cannot differ by
more than unity. Symbolically,
zag,
k-F3
k-l-2’
degree W(p) I degree Z(p) - 4
(173)
+ I P” I22
= --20-~1,(To+ *),
W(P)
k-k2
W(P) =:k+2’
degree W(p) I degree Z(p) - 2, w. = 0.
where
Nd
September
and
W(p) = ToWl(P) - 7-034
J/jqp)
THEORY
Referring to (177) we see that the corresponding sequence
for W(p) is
(175)
and from Lemma 1,
is also positive-real.
ON CIRCUIT
-k+l
k
’
-’
k+l
k
has all the properties enumerated
Theorem 1. Consequently, so does
in the statement
of
Now let the point p, approach any finite point on the
real-frequency axis, exclusive of the origin, along an
arbitrary ray not parallel to the axis. At those points
where Z(p) is analytic the ray-limits of all indexes are
perfectly well-defined by (132)-(138); and since the limit
of a sequence of positive-real functions is obviously a
positive-real function, we conclude that W(p), as defined
by (176), is also positive-real for p, on the jw-axis.
By using (136)-(138), it is found by straightforward
calculation that
1) forp,
= jwo,O < Iwo] < Q,
(UZ + b),-i,,
= 0 = (cZ + d)l)‘jWO.
2) If, in addition, r(jwo) = 0,
(aZ + b)imjwO = (cZ + d)L=jw. = 0.
259
Youla: Theory of Cascade Synthesis
1961
Thus, for pa = jwo, the numerator and denominator of
W(p) are always divisible by the quadratic factor (p” + wi)
and it is always true that degree W(p) 5 degree Z(p).
When p, = jw, is a zero of r(p), the numerator and denominator of W(p) are divisible by the biquadratic factor
(p” + w$” and degree W(p) < degree Z(p) - 4. This
terminates the proof of Theorem 1, Q.E.D.
In retrospect, one can easily find several ways of
simplifying the proof. However the method we have
presented has the advantage of motivation. At no point
in the demonstration was it ever found necessary to leave
the domain of positive-real functions. Interestingly enough,
(163) may be viewed as a generalized form of Schwartz’s
lemma which, when applied to a bounded-real scattering
coefficient, yields another simpler bounded-real function.
The usual Schwartz lemma preserves the bounded but not
the real character of the function.
and
m 22 = g$L,.
Note that (187)) (192)) (193) and the perfect coupling
condition, L,L, = Mi, automatically imply (191). For,
invoking (77))
m
11
-
-m,,
2
mz2
x = I,J, + I,J,,
(194
y = I, + J3.
(195)
Then
w~JP!?-!b)
&A.$
Y
L,=I,J,,
L2 = k22 = I, J, + I,J,
(184)
’
L,=L&.
(185)
and
w:M, =
0
of La, L,, C, and M,.
w:(“,
+
M2)
= -ui(x
jPo I”(13 + Ja)
I,J, + I,Js
’
=
I3
J,
+
1,
+ I,J,
J,
(187)
’
(188)
... M 2 = -d(x
+ y)” - wZ(x -
+ Y)" + 4x
2.
XY2 I P" I
y)“.
Y)"
(200)
But
089)
,
and
xi-y=
%
x-y=
-%
(Uo + RoIJ
Gw
and
,c& + w;(~2 + Lo) = 1 P” I2 (I1 + J1) + ITaJ2.
I,J, + I,J,
Eqs. (188)-(190)
equalities
Y
xy” 1 pa I2 M, = - I pa I2 (x - ~1” - 4&y
2(w: - a;)
+
X2
X
Thus,
L ,w; + so + s, = 1PO I2(1, + Jd + IsJ3
‘6
2(1 p, I2 - 2uZ) l14t!&Ly.
W@
All that remains is the determination
Continuing the identification,
I,J,
(199)
X
1
+ I, J, ’
~
L3
(198)
2
I,J,
2
so
w,=--=
(197)
M, =$
(183)
L1 = kll = I, J, + I, J3 ’
&-’ = fJ* = y$.
W-3
X
I,J,
M1 = k12 = I,J,
= -1 &.
2
Consider (189)) and for the sake of brevity, set
The object of this Appendix is to derive (83)-(91) for
the element values of the Darlington type D section.
A straightforward
comparison of (72)-(74) with (79)(82) yields the relations
and
1
= W2M2
LL-2 = -wiL,
2L,
2
II
APPENDIX
(193)
result, respectively,
(190)
from the residue
S
ml1 = 22 ’
(191)
1
w:M,
2
(192)
m 12
--
-
(V” + XOIJ,
(202)
where
J,“I: I Wo + ZoI, I2
M2 = - I PO I2 V,J,
+ I,J,>(I,
+ J,)”
In passing from (200) to (201)-(203))
the identity
’
‘*
(203)
we have used
260
IRE
1 Z” I2 I,I,
TRANSACTIONS
= I,.
ON CIRCUIT
(204)
In similar fashion,
xy2 1 PO I2 -L = xyl3J2
-
I pa I2 6~ - I,Y)(x
-
J,Y). (205)
However, [use (201) and (202) again],
(x - IIY)(X
so that
- Jd
= - J,J,I,(I,
J,J,I,V, L4 = ( pa Ia ?+
J,) + (I,J,
+ I, J,)(I,
- Jl)“,
Jl)”
+ J,)“’
(206)
(207)
THEORY
September
Consequently, L3 is found from the formula L, = Mi/L4
and S, from the formula S, = wzL3.
The only equation which has not been used is (188).
Nevertheless the remark directly below (190) renders
(188) superfluous. Since the indexes are all non-negative,
L,, L,, L,, L,, C, and C, are also non-negative and we
have succeeded in proving that two successive applications of Theorem 1 or, stated differently, two type E
sections in cascade are equivalent to a Type D section,
Q.E.D.
Synthesis of Active RC Networks*
J. M. SIPRESSt,
procedures are presented which establish
Summary-Synthesis
that one grounded J-terminal negative-impedance converter,
embedded in an unbalanced grounded RC structure, is suflicient
to realize
1) any driving-point function,
2) any two of the four short-circuit admittance parameters of a
two-port network, and
3) certain sets of n short-circuit admittance parameters of an
(n + 1)-terminal network,
where each of the parameters is specified as the ratio of any two
polynominals in the complex-frequency variable, with real coefficients.
Furthermore, the required RC networks can always be made to
take the form of grounded ladder-type structures, some of which, in
particular cases, reduce to two-terminal admittances.
I. INTRODUCTION
CTIVE network synthesis is concerned with the
realization of networks in which active elements
A
are used to shift the poles and/or zeros of passive
immittance functions so as to realize a desired immittance
function. Consequently, networks which are not realizable
with passive elements only, such as nonpositive-real
driving-point
functions, or networks from which inductance or capacitance has been eliminated, may be
synthesized. Active RC networks, i.e., networks containing only resistors, capacitors and active elements,
seem particularly attractive at lower frequencies, where
magnetic elements become rather large and expensive.
Several techniques have been proposed for the active
MEMBER,
IRE
of any voltage-transfer function’ with only one 4-terminal
controlled source of the amplifier type.2 Linvill presented
a technique demonstrating that an RC network containing one 4-terminal negative-impedance converter (NIC),3
could be used to realize any immittance-transfer
function.?
Yanigasawa, Sandberg and Myers, each independently
advanced a method of using a S-terminal NIC to synthesize any voltage- or current-transfer function.“-‘”
Subsequently, interest shifted to the realization of
driving-point functions by active RC techniques. Sandberg
showed that any driving-point function could be realized
with one 4-terminal controlled source whose gain is
allowed to approach infinity9 or with one 4-terminal
NIC.” Kinariwala advanced a technique demonstrating
that one 4-terminal controlled source of the amplifier
type embedded in an RC network is suitable for the
1 In this paper “any function” is defined to mean any function
that can be expressed as the ratio of any two polynomials in the
complex frequency variable, p = c + Jo, with real coefficients.
2 R. L. Dietzold, “Frequency discriminative electric transducer,”
U. S. Patent No. 2,549,065; April 17, 1951.
3An NIC can be defined as a two-port network for which the
input impedance at either port is porportional to the negative
of the impedance connected to the other port.4-6
4 J. L. Merrill, “Theory of the negative impedance converter,”
Bell System T’ech. J., vol. 36, pp. 88-109; January, 1951.
5 J. G. Linvill, “Transrstor negative impedance converters,”
PROC. IRE, vol. 41, pp. 725-729; June, 1953.
6 A. I. Larky, “Negative impedance converters,” IRE TRANS. ON
CIRCUIT THEORY, vol. CT-4, pp. 124-131; September, 1957.
Linvill
CC77 J.
rn” G.nr-~
1 :n,,“RC active filters,” PROC. IRE, vol. 42, pp.