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A new approach to active RC network synthesis
A NEW APPROACH TO ACTIVE R e NETWORK SYNTHESIS* BY
SANJIT K. MITRA 1 ABSTRACT This paper presents several lemmas which are of importance in the synthesis of one-port and n-porl active RC networks. Syntheses of such networks are viewed from the equivalent active LC network point of view by making use of RC-LC transformation. The first part deals with one-ports. Replacement of the resistance in the Darlington realization of a positive real impedance by an appropriate negative LC impedance results in the odd part of the positive real function at the input. An inverse LC-RC transformation yields the decomposition, partitioning, and consequently the synthesis, of active RC impedances. Admittances are treated by a dual procedure. This approaeh is extended in the second part to the synthesis of active RC immittance matrices of order n. It is shown that in the latter case, the number of active elements needed is at most equal to n. I. INTRODUCTION
Availability of precision, small size, almost drift-free resistors and capacitors having low temperature coefficients, and also extremely reliable small size active elements, has resulted in the broadening of the realm of modern network theory beyond the bounds of passive and bilateral networks. The use of active elements in otherwise passive RC networks (the so-called companion RC networks (1)2) has enabled us to eliminate inductors and crystals (which are undesirable particularly at the low frequencies), and also to realize nonpositive real functions. Furthermore, the use of tantalum films in circuit design has greatly stimulated the interest in the synthesis of active RC networks. Several methods are available for the synthesis of driving-point immittance functions and transfer functions by such networks. The technique followed in most of the synthesis methods can be classified into three major steps: (1) decomposition and partitioning of the given function; (2) identificfition of the parameters of the companion RC networks; and (3) synthesis of the companion RC networks. Although several active RC configurations have been proposed, the major step that has been overlooked is the step of decomposition and partitioning. In a recent paper, Thomas (2) suggested the use of polynomial decomposition in active network synthesis. In this paper we present a more logical and general approach to the problem of decomposition and partitioning, whereby Thomas' result is obtained as a special case. * This work was supported by the National Science Foundation under Grant G-12142. 1 School of Electrical Engineering, Cornell University, Ithaca, N. Y.; formerly, Electronics Research Laboratory, University of California, Berkeley, (-:alif. 2 The boldface numbers in parentheses refer to the references appended to this paper.
~85
I86
SANJIT K.
MITRA
[J.F.I.
In the first part we consider the partitioning of driving-point functions. Starting from the Darlington realization of a positive real impedance, we show that replacing the one-ohm load resistance at the output port by a negative LC impedance (equal in absolute value to the reciprocal of the open circuit output impedance) results in a driving-point impedance at the input port which is the odd part of the original positive real impedance. Dual result is obtained on an admittance basis. An LC-RC transformation on the modified network yields the desired decomposition and partitioning. In the second part we extend the results to the case of active RC n-ports. We show that in this case the n u m b e r of active elements necessary is at most equal to n. II. ONE-PORT ACTIVE RC NETWORK
A. The Odd Part of a Positive Real Function
Darlington has shown that a given positive real function, 3 (1)
F(s) = m, + n, rr/2 +
n2
can always be realized as the input immittance of a lossless two-port 2¢ (henceforth to be called a Darlington two-port) terminated by a one-ohm resistance (Fig. 1) (~). i 2
°
F (s)"w'l ] o ]( LOSSLESS ) I' [
_
~E I.fl.
Fv-
FIG. 1.
IN
/ °
OaF (s)"-l. - ~1--,.--.~ r ~ o,' l ( LOSSLESS) pF-~2, "-T-~
~'22
Darlington realization of a positive real function F ( s ) .
Fro. 2. Realizationof the odd part of a positive real function F(s).
W h e n F(s) is identified as the input impedance, either of two mutually exclusive identifications of the two-port parameters must be made: Case A
~,~ =
Case B
m__!
~,~ =
n2
~22 =
n_L rrl2
/122
--
~22
n2
=
n2
--
(2)
D22 V / D 2 1 D'I 2 - -
n2
n 1n 2
V"n
n 2 --
tn 1 m 2
/T/2
3 The even part of a polynomial will be denoted by the symbol m and the odd part by the symbol n.
Sept., ~,962.1
ACTIVE R C
187
N E T W O R K SYNTttESIS
Now consider the network obtained by terminating the Darlington twoport at the output port by a negative LC one-port impedance 4 - Z t (as shown in Fig. 2) where 1 -g,.
-
(3)
1.. 22
The input impedance of the new network of Fig. 2 for either case is /'-
/7/,)/21
g(,~,)
~
--
Iglln 2
(4)
Equation 4 is recognized to represent the odd part of the positive real function F(s), that is, /N
g(s) = O d F ( s ) = O d m, + n,
(5)
/7/2 q- n 2
The conditions satisfied by a rational function (;(s) =
P(s)/Q(s) to
be
the odd part of a positive real function are (4): (i)
Z'(s) ° _< (L0) ° + 1,
(6) ~
(ii) the poles of G(s) have quadrantal symmetry and the/w-axis poles are simple having real and positive residues. We can summarize the previous results as:
Lemma h A rational function of s, the complex frequency variable, satisfying condition (6) can be realized as the input impedance of an LC two-port terminated by a negative LC one-port impedance. It is clear that dual results hold on the admittance basis.
B. LC-RC Tran.~formation and the Odd Pad Parlilioni,t,,~ In Eq. 4, m~, m 2 are even polynomials and n,, n2 are odd polynomials: and hence we can write m, = a, (s~),
m2 = a~(.~2),
,~, = ,~b, (s~),
~e = sb2(s~) -
(7
Substituting (7) in (4) we obtain
L-
727(,--5--,e;
i
These results can be related to RC networks by means of the L C - R C transformation relationship (5), 4 Unless otherwise stated, by impedance (admittance) we will mean driving-point irnpedancv (admittance). 5 The degree of a polynomial P(s) will be denoted by t'(,~)°.
188
SANJIT K. MITRA
[J.V.I.
T~c(s),__., _1 TRc(s2).
(9)
s
For clarity, the subscripts LC and RC have been put in where appropriate. These relations indicate that the equivalent RC network is obtained by replacing each inductance of the LC network by a resistance of equal absolute value. The use of the transformation (9) in (6) and (8) results in the following:
Lemma 2: A rational function of s, Z(s) = N(s)/D(s) satisfying the following conditions: (i) X(s) ° < D(s) °, (10a) (ii) Z(s) has poles and zeros anywhere in the s-plane, with the restriction that the negative real axis poles are simple with real and positive residues, (10b) (iii) Z(oo) > 0, (10c) can always be decomposed and partitioned as (2) Z(s)
(11)
: a (s) b,(s) - a,(s) b (s) a (s) - sb (s)
where [a,(s)/sbi(s)} and {b,(s)/ai(s)l (i = 1, 2; j = 1, 2) are all passive RC impedances. The decomposition (11 ) was originally suggested by Thomas (2), but because of our approach we have obtained an additional restriction (10c) on functions which can be decomposed in the form of (11). Similarly, by duality the following is obtained on the admittance basis:
Lemma 3: A rational function of s, Y(s) = N(s)/D(s), satisfying the following conditions:
(i) N ( s ) ° < D ( s ) ° + 1, (12a) (ii) Y(s) has poles and zeros anywhere in the s-plane, but the negative real axis poles of Y(s)/s must be simple with real and positive residues, (12 b) (iii) Y(0) >__0, (12c) can always be decomposed and partitioned as
r(s)--sL-Fa_2(s)b,(s)
-
a,(s) bz(s) ~
]
(13)
where {a~(s)/bs(s)} and Ib,(s)/sas(s) } (i = 1, 2; j = 1, 2) are all in the form of passive RC admittances. We will designate both the decomposition and the partitioning forms of (11) and (13) as the odd part partitioning. An impedance function satisfy-
ACTIVE RC NETWORK SYNTHESIS
Sept., 1962.]
I8 9
ing condition (10) will be denoted as an oddpart partitionable impedance, and likewise an admittance function satisfying (12) will be denoted as an odd part partitionable admittance. Construction procedures for obtaining the odd part partitioning of a given odd part partitionable impedance Z(s) can be outlined as follows: /,.
1. Obtain the equivalent active LC impedance Z(s) from Z(s) by means of RC-LC transformation (9), 2. Remove thejw-axis poles of Z(s) by partial fraction expansion, go(s)
,x
:
Z(,)
ko -
-
-,
k,
-
--.ks
,+ +
2
I
(14)
3
o
Z(s)--~
N
_ Z L..D,.,
(RG)
0 1I
NIC kzl
o
2' 3' The cascade configuration. Fro. 3.
3. Form an analytic function F,(s)/F2(s ) where F2(s ) is the Hurwitz polynomiM obtained by factoring the left-half plane poles of the remainder function Zm(s). Since there is no restriction on the even part, F 1(s) can always be chosen such that Ft(s)/F2(s ) is positive real and the odd part of F, (s)/F2(s) is equal to Zm(s), that is, ,~,,(s) = O d FI(S) =
F2(s )
Od( m3 + n-2~= m 4 n 3 - - m3/'/4 \m, + n41 m] - n]
(15)
The positive re/d function _F,(s) _
+ - -k0+ F z(s) s
k=s+
k is
, s2 + w
will be called the primary positive real function corresponding to Z(s). 4. Perform an inverse LC-RC transformation on the sum of ,~m(s) and the jw-axis poles of ~(s) to obtain Z(s) =
a4(s) b3(s)
-
a3(s) b4(s) + k +--k° + Z --.k'
(16)
Combining the different terms of (16) we will obtain (11). Similar procedures carl be formulated in the case of an odd part partitionable admittance.
SANJIT K.
I9O
MITRA
[.].F.I.
C. An Application to Cascade M e t h o d of 3ynthesis
A careful study of Section II-A shows the obvious application of the odd part partitioning to the cascade configuration of Kinariwala (6) (Fig. 3). We note that an LC-RC transformation on the network of Fig. 2 results in the network of Fig. 3, where the negative RC impedance -Zt~ has been obtained by terminating a negative impedance converter by a passive RC impedance Z1.. For the case of an odd part partitionable impedance we have the following identifications for the two-port parameters and Zt, : Case A
Case B
Zl, = a l ( s ) l s b z ( s )
z,l = b t ( s ) / a 2 ( s )
zz2 = a2(s)lsbz(s)
z22 = b z ( s ) l a 2 ( s )
Z 212 =
a 1 (s) a z (s) - sb, (s) b 2 (s) sZb~(s)
= b2(s)la2(s)
(17)
sb 1 (s) b z (s) - a 1(s) a 2 (s) Z?2 =
Z,.
=
sa~(s) a2(s)/sb2(s)
which have been obtained using (9) and (7) in (2) and (3). Because of the realizability of the z-parameters given in (2) of the Darlington twoport, the realizability of the parameters of (17) is guaranteed. The mutually exclusive cases, Case A and Case B, are thus determined by the evenness or oddness of the polynomial g'a,(sZ)a2(s 2) - s2b,(s2)b2(s z) (that is, x / m l m z - nln2). Moreover, the usual difficulty of making the numerator ofz~Z2a perfect square applies. However, we are awarded an additional degree of freedom because in starting from the odd part the even part is not uniquely determined. Hence any positive constant may be added to the even part which will often allow us to make the numerator of z ~2a perfect square. Furthermore, Darlington's method of augmentation of the primary positive real function can always be used to guarantee a rational z 12(3). Similarly, in the case of an odd part partitionable admittance we will have: Case A Yu = a l ( s ) / b z ( s )
Case B Yl, = s b l ( s ) / a z ( s )
=
=
(18) y 2 = al(s)a2(s)
- sbl(s)b2(s)
Y?2 = s[sbl(s)b2(s)
b (s) Y,. = sb2(s)/az(s)
- al(s)az(s)l
a (s) Yc = a2(s)/bz(s).
In general, the two-port network realized using the two-port parameters of (17) will contain an ideal transformer. The transformer can be
Sept., ~¢2.1
ACTIVE R C NETWORK SYNTHESIS
I9I
eliminated by the following method. Let the z-parameters of the network N after decomposition and partitioning be zH, zz2 and z~2. We can realize zn satisfying the zeros of z12 by the usual methods (7). The resulting network will be in the form of a ladder if the zeros of z~2 are real. Let this network be designated as N a (Fig. 4a). In general, the open circuit transfer impedance of Na, denoted by z £ , will be different from z~2 by a multiplicative constant, that is,
z,; = f z , 2 .
(19)
N I
I
Z($)..,_D.O-~ZlNQ l c~.i KZl2
! ':"'° KZz22
(Q)
G
Z(s)--~
o
Zll
z~2
NIG
KZl2
k: I
(b) Fro. 4.
Most general transformerless cascade configuration.
Moreover,.,~ will have a different z22, denoted z2~- Now we can rewrite the input impedance of the network of Fig. 3 as Q(s) = Z~,
z 2
i2
Z22 -- Zs,
Z,. -
(Kz,2) 2
(20)
K2Z22 -- K 2 Z / .
So our aim now is to get a network .:V whose z-parameters are z ~ , / i z j2 and /f2z22 which is cascaded with a negative impedance converter terminated by a load impedance K2ZI. We have already realized a network .V whose z-parameters are z~, Kz~2 and z/2- Thus we will have to add a series impedance (K2z22 - z]2) as shown in Fig. 4a to get the network .V. In general, we will have
192
SANJIT K. MITRA
K e z = = z 22 ' =
+ h+ +
s + ~rj/
= Z+ -
[.I.F.I. + h2 + ~
\
h.,, ., s + o
Z~
(21)
where Z + and Z ; are passive R C impedances. As a result, the most general cascade configuration for the z-parameter realization without transformers is as shown in Fig. 4b. Similarly, given a set of Yn, Y12, Y22 and YL, we can proceed as before and obtain the most general cascade configuration for y-parameter realization. It should be noted that the numerator of~22 is identical with the numerator of the even part of the primary positive real function. 6 As a result, while constructing the primary positive real function it is possible to insure a ra_ ~ 0
IO ~Ao NIG W=I
10
t
Z (sl ~ o
Flo. 5. Cascade realization of Z(s) = (105s + 80)/(25s2 + 49s + 25). and farads.
Values in ohms
tional Z12, in some cases, by making the numerator of the even part of the primary positive real function a perfect square. This will be clarified from
the following example. Example: Let Z(')
=
N(s) -
-
-
D(s)
105s + 80 25s 2 + 49s + 25
(22)
Z ( s ) is seen to be an odd part partitionable impedance. The Hurwitz polynomial obtained by factoring the left-half plane poles of ,~(s) = s Z ( s 2) is given by F2(s ) = (5s 2 + 5) + s. We will have to identify s Z ( s 2) as O d IF l(s)/Fz(s)], where F l ( s ) / F z ( s ) is the primary positive real function. Let Fl(s) = co + cls + czs 2. Comparing sN(s z) with the numerator of O d [F, ( s ) / F z (s)], we obtain
5q - c2 = 105;
5 q - co = 80.
(23)
The positive real character of F l ( s ) / F 2 ( s ) m a y be insured by making the numerator of Ev [F 1( s ) / F 2 (s)] a perfect square, which implies (5c2 + 5 c 0 - c~)2 = 100c2c0. 6 ~lz = sq2('2) .
(24)
ACTIVE RC NETWORK SYNTttESIS
Sept., 1962.1
193
From (23) and (24) we obtain: c0-- 45; c~ = 25; c2 = 20. Thus we can decompose and partition the given odd part partitionable Z(s) as
Z(s) =
(5s + 5 ) . 2 5 (5S
-
(20s + 45).1
-1- 5 ) 2 - - S"
(25)
1
In this case we have to identify with the two-port parameters of Case A. Hence we obtain from (25) and (17) 20s + 45 Zll
~---
s
5s + 5 ;
Z22
-
-
$
;
1
g,.
5s+5
x/(20s + 45)(5s + 5) - 25s
10s + 15
s
s
Z l 2 ~-
and the resulting network is shown in Fig. 5.
D. The Question of Sensitivity The main problem associated with the active RC network is the sensitivity of the entire network with respect to change in the characteristics of the active element. For RC-negative impedance converter synthesis, the o p t i m u m decomposition of a polynomial P(s) which yields m i n i m u m coefficient sensitivity (8) and m i n i m u m root sensitivity (9) with respect to the conversion ratio of the requisite NIC, is given by P(s) = a2(s) - sb2(s)
(26)
where a(s)/sb(s) and b(s)/a(s) are passive RC impedances. It should be noted from the form of odd part partitioning given in (11) and (13) that the denominators of the odd part partitionable immittances are optimally decomposed. In the cascade synthesis, if transformers are allowed, then the resulting cascade structure is minimum-pole-sensitive. The suggested technique for transformer elimination destroys the minim u m sensitivity feature of the partitioning. In some cases the constant K in (20) m a y be chosen to satisfy the Fialkow-Gerst condition (10) and the RC two-port can be realized without transformers preserving the minimum-pole-sensitivity of the entire network. III. n-PORT ACTIVE RC NETWORK
A. The Odd Part of a Positive Real Matrix It is well known that a given reduced positive real matrix F of order n can be realized as the impedance matrix of an n-port passive network obtained by terminating the (n + 1)'h port of an (n + 1)-port lossless network (henceforth to be called a LeRoy-Bayard network) by a resistance (Fig. 6) (11). Let X denote the open circuit impedance matrix of the LeRoy-Bayard network. Partition X as
194
K. MITRA
SANJIT
[J.F.I.
where U is a symmetric m a t r i x of order n, V is a column vector of order n,
V ' is the transposed of V and W is a matrix of order 1. We can write VV'
F = U
(28)
W+I The even and the odd parts of F are given by EvF & F "
VV' W2 - 1
=
(29)
VV' W W 2- 1
O d F ~ F °a = U
(30)
I O
I
-
I I n 0
I
(LOSSLESS)
I
(LOSSLESSI
0 n
[
C nt
0
#
Fro. 6.
n+l
I od£--~ I
0
F--~I
X
O----I'
n+l
Realization of a reduced positive real matrix F.
Fie. 7.
w Realization of the odd part of a reduced positive real matrix F.
N o w consider the network obtained by terminating the (n + 1)th port of the LeRoy-Bayard network of Fig. 6 by a negative LC impedance of value equal to - l / W , as shown in Fig. 7. The impedance matrix of the modified n-port is VV
F' = U
t
1 W
= U
VV
tW
W2 - 1
(31)
-- --
W
Equation 31 is recognized as the odd part of the positive real matrix F. Thus we have the following lemma, a generalized version of Lemma 1.
Lemma 4: A given symmetric matrix of order n satisfying the properties of the odd part of a reduced positive real matrix can be realized as the impedance matrix of an n-port network obtained by terminating the (n + 1) 'h port of an (n + 1)-port lossless network by a negative LC impedance of value equal to the reciprocal of the open circuit driving-point impedance of the (n + 1)-port network at its (n + 1)'hport.
Sept., t962.}
ACTIVE
RC
NETWORK
SYNTHESIS
195
Similar statements can be made on the admittance basis.
B. Synthesis of n-Port Active RC Impedance Matrix Let Z be the given impedance matrix of order n, Z = [z,,(s)] .
(32)
Suppose Z is such that upon p e r f o r m i n g a n R C - L C transformation on each of its elements, the transformed matrix Z can be expressed as the odd part of a positive real matrix P (which will be called the primary positive real matrix corresponding to Z), that is,
= [sz,,(s2)] = pod
(33)
A
I f Z is not regular on the imaginary axis, we can write /,,
Z = H + G °~
(34)
where G °d is the odd part of a positive real matrix G which is regular on the imaginary axis and H is a Foster matrix (12) such that the matrices of the residues at the poles are of rank 1. Thus P = H + G.
(35)
Let k be the rank of G% the even part of G. T h e n we can express G °Vas k
G e" = ~
Mi
(36)
i=l
where the matrices M i are each of rank 1 (12). For each MI we can construct a reduced positive real matrix GI such that M,
= G Cv, .
(37)
/%
This implies that we can express Z as k
= H + ~
G °d .
(38)
i=1
As shown in Sec. III-A, each G~d can be realized in the form of Fig. 7. Furthermore, H can be realized by the usual methods (11). Finally a series connection of these networks yields the realization of Z. Next we perform an inverse L C - R C transformation on the n-port =kL, + C network to obtain a realization of the given impedance matrix Z. Since k, the number of negative LC impedances (and hence the number of negative RC impedances)required, is the rank of a (n × n) matrix, k < n, or in other words the n u m b e r of negative impedance converters needed is at most equal to n.7 From these results and the properties of the odd part of a positive real matrix, we have the following lemma which is a generalization of L e m m a 2" 7 Sufficiency of n active elements for the realization of a n a r b i t r a r y n x n m a t r i x of real rational functions was first e s t a b l i s h e d by S a n d b e r g (1.3, 14) following a different approach.
SANJIT K. MITRA
I96
[J.F.I.
Lemma 5: A symmetric (n x n) matrix of real rational functions, Z -
1
D(s)
[n,j(s)]
(39)
satisfying the following conditions,
(i) no(s)° N D(s) °, (ii) Z has poles and zeros anywhere in the complex s-plane with the restriction that the negative real axis poles of Z are simple and the matrices of the residues at the negative real axis poles are positive semi-definite, (iii) Z(oo) is positive semi-definite, can be related to the odd part of a primary positive real matrix P after an RC-LC transformation on each of its elements. Furthermore, Z can be realized as the impedance matrix of an active RC n-port containing k(k < n) negative impedance converters, where k is the rank of per. Similarly, we can obtain a generalization of L e m m a 3 on the admittance basis. It should be noted that as the realization of a LeRoy-Bayard network requires ideal transformers, the n-port active RC network will usually contain ideal transformers. But we have an added advantage, because starting from the odd part of a positive real matrix, the even part is not uniquely determined. So it seems that there exists a possibility of obtaining a transformerless active RC n-port by suitably choosing the even part of the primary positive real matrix. Additional investigation in this direction is suggested. Performing an inverse LC-RC transformation on the right hand side of (31) we can show that the denominator will be in the form o f a Horowitz decomposition (Eq. 26). As a result, the realized n-port active RC networks are minimum-pole-sensitive. IV. CONCLUSION
The decomposition and partitioning of active RC driving-point functions, presented in this paper, are restricted to odd part partitionable impedances which satisfy condition (11) and to odd part partitionable admittances satisfying condition (13). In an actual synthesis it is usually best first to remove any negative real axis poles as RC networks in a manner analogous to the Brune preamble (15). Functions which fail to meet the requirements on either basis m a y sometimes be successfully augmented by the subtraction of a suitable resistance or capacitance. Synthesis of transfer functions can be taken care of by first reducing the problem to the synthesis of driving-point functions by the usual methods (6). A logical extension of the results of the active RC one-ports to the synthesis of active RC n-ports is next presented. In the latter case, the technique is restricted to impedance matrices which satisfy condition (40). Dual conditions for admittance matrices can be easily derived. Further-
Sept., ,962.1
ACTIVE RC
N E T W O R K SYNTHESIS
I97
more, the number of negative RC impedances, and hence the number of active elements, needed is found to be equal to k(k < n), where k is the rank of the even part of a certain primary positive real matrix related to the given impedance matrix by the relation (33). In both the one-port and the n-port cases, the techniques presented hinge upon straightforward algebraic procedures and result in minimumpole-sensitive decomposition and partitioning.
Acknowledgment The author wishes to thank Professor E. S. Kuh for his encouragement and helpful suggestions. Thanks are also due to Messrs. J. D. Patterson and R. A. Rohrer for useful discussions. REFERENCES
(1) S. K. I~IrI'RA,"A Unique Synthesis Method of Transformerless Active RC Networks," .]otR. FRANKLIN INST., Vol. 274, p. 115 (1962). (2) R. E. THOMAS, "Polynomial Decomposition in Active Network Synthesis," IRE Trans. Circuit Theory, Vol. CT-9, pp. 270-274 (1961). (3) E. A. GtrILI.EMm, "Synthesis of Passive Networks," New York, John Wiley & Sons, Inc., 1957, pp. 358-362. (4) D. F. TtvrxH:, "Network Synthesis," Vol. 1, New York, John Wiley & Sons, Inc., 1958, Ch. 8. (5) E. S. KvH and D. O. PEDERSON, "Principles of Circuit Synthesis," New York, McGrawHill Book Co., Inc., 1959, pp. 148-152. (6) B. K. KZNAmWALA, "Synthesis of Active RC Networks," Bell System Tech. J., Vol. 38. pp. 1269-1316 (1959). (7) E. S. KUH AND D. O. PEDERSON,lOC.cil., pp. 160--167. (8) I. M. HOROWlTZ, "Optimization of Negative-lmpedance Methods of Active RC Synthesis," IRE Trans. Circuit Theory, Vol. CT-6, pp. 296-303 (1959). (9) D. A. CALAHAN, "Notes on the Horowitz Optimization Procedure," IRE Trans. (hrcuit Theory, Vol. CT-7, pp. 352-354 (1960). (10) A. FIALKOW AN,) I. GERST, "The Transfer Function of Networks Without Mutual Reactance," Quart Appl. Math., Vol. 12, pp. 117-131 (1954). (11) M. BAYARD, "Th~orie des R~seaux de Kirchhoff-R~gime Sinusoidal et Synth~se," Paris, l~ditions de la Revue d'Optique, 1954, pp. 347-352. (12) P. R. HALMOS, "Finite Dimensional Vector Spaces," New York, D. Van Nostrand Co., 1958, pp. 92 93. (13) 1. W. SANDBERG,"Synthesis of N-Port Active RC Networks," Bell System Tech..7- Vol. 40, pp. 329-348 (1961). (14) I. W. SANnB~RG, "Synthesis of Transformerless Active N-Port Networks," Bell 51vstern Tech. J., Vol. 40, pp. 761-783 (1961). (15) E. A. (;tmIEMIN, IOC.cit., p. 344.
SANJIT K. MITRA 1 ABSTRACT This paper presents several lemmas which are of importance in the synthesis of one-port and n-porl active RC networks. Syntheses of such networks are viewed from the equivalent active LC network point of view by making use of RC-LC transformation. The first part deals with one-ports. Replacement of the resistance in the Darlington realization of a positive real impedance by an appropriate negative LC impedance results in the odd part of the positive real function at the input. An inverse LC-RC transformation yields the decomposition, partitioning, and consequently the synthesis, of active RC impedances. Admittances are treated by a dual procedure. This approaeh is extended in the second part to the synthesis of active RC immittance matrices of order n. It is shown that in the latter case, the number of active elements needed is at most equal to n. I. INTRODUCTION
Availability of precision, small size, almost drift-free resistors and capacitors having low temperature coefficients, and also extremely reliable small size active elements, has resulted in the broadening of the realm of modern network theory beyond the bounds of passive and bilateral networks. The use of active elements in otherwise passive RC networks (the so-called companion RC networks (1)2) has enabled us to eliminate inductors and crystals (which are undesirable particularly at the low frequencies), and also to realize nonpositive real functions. Furthermore, the use of tantalum films in circuit design has greatly stimulated the interest in the synthesis of active RC networks. Several methods are available for the synthesis of driving-point immittance functions and transfer functions by such networks. The technique followed in most of the synthesis methods can be classified into three major steps: (1) decomposition and partitioning of the given function; (2) identificfition of the parameters of the companion RC networks; and (3) synthesis of the companion RC networks. Although several active RC configurations have been proposed, the major step that has been overlooked is the step of decomposition and partitioning. In a recent paper, Thomas (2) suggested the use of polynomial decomposition in active network synthesis. In this paper we present a more logical and general approach to the problem of decomposition and partitioning, whereby Thomas' result is obtained as a special case. * This work was supported by the National Science Foundation under Grant G-12142. 1 School of Electrical Engineering, Cornell University, Ithaca, N. Y.; formerly, Electronics Research Laboratory, University of California, Berkeley, (-:alif. 2 The boldface numbers in parentheses refer to the references appended to this paper.
~85
I86
SANJIT K.
MITRA
[J.F.I.
In the first part we consider the partitioning of driving-point functions. Starting from the Darlington realization of a positive real impedance, we show that replacing the one-ohm load resistance at the output port by a negative LC impedance (equal in absolute value to the reciprocal of the open circuit output impedance) results in a driving-point impedance at the input port which is the odd part of the original positive real impedance. Dual result is obtained on an admittance basis. An LC-RC transformation on the modified network yields the desired decomposition and partitioning. In the second part we extend the results to the case of active RC n-ports. We show that in this case the n u m b e r of active elements necessary is at most equal to n. II. ONE-PORT ACTIVE RC NETWORK
A. The Odd Part of a Positive Real Function
Darlington has shown that a given positive real function, 3 (1)
F(s) = m, + n, rr/2 +
n2
can always be realized as the input immittance of a lossless two-port 2¢ (henceforth to be called a Darlington two-port) terminated by a one-ohm resistance (Fig. 1) (~). i 2
°
F (s)"w'l ] o ]( LOSSLESS ) I' [
_
~E I.fl.
Fv-
FIG. 1.
IN
/ °
OaF (s)"-l. - ~1--,.--.~ r ~ o,' l ( LOSSLESS) pF-~2, "-T-~
~'22
Darlington realization of a positive real function F ( s ) .
Fro. 2. Realizationof the odd part of a positive real function F(s).
W h e n F(s) is identified as the input impedance, either of two mutually exclusive identifications of the two-port parameters must be made: Case A
~,~ =
Case B
m__!
~,~ =
n2
~22 =
n_L rrl2
/122
--
~22
n2
=
n2
--
(2)
D22 V / D 2 1 D'I 2 - -
n2
n 1n 2
V"n
n 2 --
tn 1 m 2
/T/2
3 The even part of a polynomial will be denoted by the symbol m and the odd part by the symbol n.
Sept., ~,962.1
ACTIVE R C
187
N E T W O R K SYNTttESIS
Now consider the network obtained by terminating the Darlington twoport at the output port by a negative LC one-port impedance 4 - Z t (as shown in Fig. 2) where 1 -g,.
-
(3)
1.. 22
The input impedance of the new network of Fig. 2 for either case is /'-
/7/,)/21
g(,~,)
~
--
Iglln 2
(4)
Equation 4 is recognized to represent the odd part of the positive real function F(s), that is, /N
g(s) = O d F ( s ) = O d m, + n,
(5)
/7/2 q- n 2
The conditions satisfied by a rational function (;(s) =
P(s)/Q(s) to
be
the odd part of a positive real function are (4): (i)
Z'(s) ° _< (L0) ° + 1,
(6) ~
(ii) the poles of G(s) have quadrantal symmetry and the/w-axis poles are simple having real and positive residues. We can summarize the previous results as:
Lemma h A rational function of s, the complex frequency variable, satisfying condition (6) can be realized as the input impedance of an LC two-port terminated by a negative LC one-port impedance. It is clear that dual results hold on the admittance basis.
B. LC-RC Tran.~formation and the Odd Pad Parlilioni,t,,~ In Eq. 4, m~, m 2 are even polynomials and n,, n2 are odd polynomials: and hence we can write m, = a, (s~),
m2 = a~(.~2),
,~, = ,~b, (s~),
~e = sb2(s~) -
(7
Substituting (7) in (4) we obtain
L-
727(,--5--,e;
i
These results can be related to RC networks by means of the L C - R C transformation relationship (5), 4 Unless otherwise stated, by impedance (admittance) we will mean driving-point irnpedancv (admittance). 5 The degree of a polynomial P(s) will be denoted by t'(,~)°.
188
SANJIT K. MITRA
[J.V.I.
T~c(s),__., _1 TRc(s2).
(9)
s
For clarity, the subscripts LC and RC have been put in where appropriate. These relations indicate that the equivalent RC network is obtained by replacing each inductance of the LC network by a resistance of equal absolute value. The use of the transformation (9) in (6) and (8) results in the following:
Lemma 2: A rational function of s, Z(s) = N(s)/D(s) satisfying the following conditions: (i) X(s) ° < D(s) °, (10a) (ii) Z(s) has poles and zeros anywhere in the s-plane, with the restriction that the negative real axis poles are simple with real and positive residues, (10b) (iii) Z(oo) > 0, (10c) can always be decomposed and partitioned as (2) Z(s)
(11)
: a (s) b,(s) - a,(s) b (s) a (s) - sb (s)
where [a,(s)/sbi(s)} and {b,(s)/ai(s)l (i = 1, 2; j = 1, 2) are all passive RC impedances. The decomposition (11 ) was originally suggested by Thomas (2), but because of our approach we have obtained an additional restriction (10c) on functions which can be decomposed in the form of (11). Similarly, by duality the following is obtained on the admittance basis:
Lemma 3: A rational function of s, Y(s) = N(s)/D(s), satisfying the following conditions:
(i) N ( s ) ° < D ( s ) ° + 1, (12a) (ii) Y(s) has poles and zeros anywhere in the s-plane, but the negative real axis poles of Y(s)/s must be simple with real and positive residues, (12 b) (iii) Y(0) >__0, (12c) can always be decomposed and partitioned as
r(s)--sL-Fa_2(s)b,(s)
-
a,(s) bz(s) ~
]
(13)
where {a~(s)/bs(s)} and Ib,(s)/sas(s) } (i = 1, 2; j = 1, 2) are all in the form of passive RC admittances. We will designate both the decomposition and the partitioning forms of (11) and (13) as the odd part partitioning. An impedance function satisfy-
ACTIVE RC NETWORK SYNTHESIS
Sept., 1962.]
I8 9
ing condition (10) will be denoted as an oddpart partitionable impedance, and likewise an admittance function satisfying (12) will be denoted as an odd part partitionable admittance. Construction procedures for obtaining the odd part partitioning of a given odd part partitionable impedance Z(s) can be outlined as follows: /,.
1. Obtain the equivalent active LC impedance Z(s) from Z(s) by means of RC-LC transformation (9), 2. Remove thejw-axis poles of Z(s) by partial fraction expansion, go(s)
,x
:
Z(,)
ko -
-
-,
k,
-
--.ks
,+ +
2
I
(14)
3
o
Z(s)--~
N
_ Z L..D,.,
(RG)
0 1I
NIC kzl
o
2' 3' The cascade configuration. Fro. 3.
3. Form an analytic function F,(s)/F2(s ) where F2(s ) is the Hurwitz polynomiM obtained by factoring the left-half plane poles of the remainder function Zm(s). Since there is no restriction on the even part, F 1(s) can always be chosen such that Ft(s)/F2(s ) is positive real and the odd part of F, (s)/F2(s) is equal to Zm(s), that is, ,~,,(s) = O d FI(S) =
F2(s )
Od( m3 + n-2~= m 4 n 3 - - m3/'/4 \m, + n41 m] - n]
(15)
The positive re/d function _F,(s) _
+ - -k0+ F z(s) s
k=s+
k is
, s2 + w
will be called the primary positive real function corresponding to Z(s). 4. Perform an inverse LC-RC transformation on the sum of ,~m(s) and the jw-axis poles of ~(s) to obtain Z(s) =
a4(s) b3(s)
-
a3(s) b4(s) + k +--k° + Z --.k'
(16)
Combining the different terms of (16) we will obtain (11). Similar procedures carl be formulated in the case of an odd part partitionable admittance.
SANJIT K.
I9O
MITRA
[.].F.I.
C. An Application to Cascade M e t h o d of 3ynthesis
A careful study of Section II-A shows the obvious application of the odd part partitioning to the cascade configuration of Kinariwala (6) (Fig. 3). We note that an LC-RC transformation on the network of Fig. 2 results in the network of Fig. 3, where the negative RC impedance -Zt~ has been obtained by terminating a negative impedance converter by a passive RC impedance Z1.. For the case of an odd part partitionable impedance we have the following identifications for the two-port parameters and Zt, : Case A
Case B
Zl, = a l ( s ) l s b z ( s )
z,l = b t ( s ) / a 2 ( s )
zz2 = a2(s)lsbz(s)
z22 = b z ( s ) l a 2 ( s )
Z 212 =
a 1 (s) a z (s) - sb, (s) b 2 (s) sZb~(s)
= b2(s)la2(s)
(17)
sb 1 (s) b z (s) - a 1(s) a 2 (s) Z?2 =
Z,.
=
sa~(s) a2(s)/sb2(s)
which have been obtained using (9) and (7) in (2) and (3). Because of the realizability of the z-parameters given in (2) of the Darlington twoport, the realizability of the parameters of (17) is guaranteed. The mutually exclusive cases, Case A and Case B, are thus determined by the evenness or oddness of the polynomial g'a,(sZ)a2(s 2) - s2b,(s2)b2(s z) (that is, x / m l m z - nln2). Moreover, the usual difficulty of making the numerator ofz~Z2a perfect square applies. However, we are awarded an additional degree of freedom because in starting from the odd part the even part is not uniquely determined. Hence any positive constant may be added to the even part which will often allow us to make the numerator of z ~2a perfect square. Furthermore, Darlington's method of augmentation of the primary positive real function can always be used to guarantee a rational z 12(3). Similarly, in the case of an odd part partitionable admittance we will have: Case A Yu = a l ( s ) / b z ( s )
Case B Yl, = s b l ( s ) / a z ( s )
=
=
(18) y 2 = al(s)a2(s)
- sbl(s)b2(s)
Y?2 = s[sbl(s)b2(s)
b (s) Y,. = sb2(s)/az(s)
- al(s)az(s)l
a (s) Yc = a2(s)/bz(s).
In general, the two-port network realized using the two-port parameters of (17) will contain an ideal transformer. The transformer can be
Sept., ~¢2.1
ACTIVE R C NETWORK SYNTHESIS
I9I
eliminated by the following method. Let the z-parameters of the network N after decomposition and partitioning be zH, zz2 and z~2. We can realize zn satisfying the zeros of z12 by the usual methods (7). The resulting network will be in the form of a ladder if the zeros of z~2 are real. Let this network be designated as N a (Fig. 4a). In general, the open circuit transfer impedance of Na, denoted by z £ , will be different from z~2 by a multiplicative constant, that is,
z,; = f z , 2 .
(19)
N I
I
Z($)..,_D.O-~ZlNQ l c~.i KZl2
! ':"'° KZz22
(Q)
G
Z(s)--~
o
Zll
z~2
NIG
KZl2
k: I
(b) Fro. 4.
Most general transformerless cascade configuration.
Moreover,.,~ will have a different z22, denoted z2~- Now we can rewrite the input impedance of the network of Fig. 3 as Q(s) = Z~,
z 2
i2
Z22 -- Zs,
Z,. -
(Kz,2) 2
(20)
K2Z22 -- K 2 Z / .
So our aim now is to get a network .:V whose z-parameters are z ~ , / i z j2 and /f2z22 which is cascaded with a negative impedance converter terminated by a load impedance K2ZI. We have already realized a network .V whose z-parameters are z~, Kz~2 and z/2- Thus we will have to add a series impedance (K2z22 - z]2) as shown in Fig. 4a to get the network .V. In general, we will have
192
SANJIT K. MITRA
K e z = = z 22 ' =
+ h+ +
s + ~rj/
= Z+ -
[.I.F.I. + h2 + ~
\
h.,, ., s + o
Z~
(21)
where Z + and Z ; are passive R C impedances. As a result, the most general cascade configuration for the z-parameter realization without transformers is as shown in Fig. 4b. Similarly, given a set of Yn, Y12, Y22 and YL, we can proceed as before and obtain the most general cascade configuration for y-parameter realization. It should be noted that the numerator of~22 is identical with the numerator of the even part of the primary positive real function. 6 As a result, while constructing the primary positive real function it is possible to insure a ra_ ~ 0
IO ~Ao NIG W=I
10
t
Z (sl ~ o
Flo. 5. Cascade realization of Z(s) = (105s + 80)/(25s2 + 49s + 25). and farads.
Values in ohms
tional Z12, in some cases, by making the numerator of the even part of the primary positive real function a perfect square. This will be clarified from
the following example. Example: Let Z(')
=
N(s) -
-
-
D(s)
105s + 80 25s 2 + 49s + 25
(22)
Z ( s ) is seen to be an odd part partitionable impedance. The Hurwitz polynomial obtained by factoring the left-half plane poles of ,~(s) = s Z ( s 2) is given by F2(s ) = (5s 2 + 5) + s. We will have to identify s Z ( s 2) as O d IF l(s)/Fz(s)], where F l ( s ) / F z ( s ) is the primary positive real function. Let Fl(s) = co + cls + czs 2. Comparing sN(s z) with the numerator of O d [F, ( s ) / F z (s)], we obtain
5q - c2 = 105;
5 q - co = 80.
(23)
The positive real character of F l ( s ) / F 2 ( s ) m a y be insured by making the numerator of Ev [F 1( s ) / F 2 (s)] a perfect square, which implies (5c2 + 5 c 0 - c~)2 = 100c2c0. 6 ~lz = sq2('2) .
(24)
ACTIVE RC NETWORK SYNTttESIS
Sept., 1962.1
193
From (23) and (24) we obtain: c0-- 45; c~ = 25; c2 = 20. Thus we can decompose and partition the given odd part partitionable Z(s) as
Z(s) =
(5s + 5 ) . 2 5 (5S
-
(20s + 45).1
-1- 5 ) 2 - - S"
(25)
1
In this case we have to identify with the two-port parameters of Case A. Hence we obtain from (25) and (17) 20s + 45 Zll
~---
s
5s + 5 ;
Z22
-
-
$
;
1
g,.
5s+5
x/(20s + 45)(5s + 5) - 25s
10s + 15
s
s
Z l 2 ~-
and the resulting network is shown in Fig. 5.
D. The Question of Sensitivity The main problem associated with the active RC network is the sensitivity of the entire network with respect to change in the characteristics of the active element. For RC-negative impedance converter synthesis, the o p t i m u m decomposition of a polynomial P(s) which yields m i n i m u m coefficient sensitivity (8) and m i n i m u m root sensitivity (9) with respect to the conversion ratio of the requisite NIC, is given by P(s) = a2(s) - sb2(s)
(26)
where a(s)/sb(s) and b(s)/a(s) are passive RC impedances. It should be noted from the form of odd part partitioning given in (11) and (13) that the denominators of the odd part partitionable immittances are optimally decomposed. In the cascade synthesis, if transformers are allowed, then the resulting cascade structure is minimum-pole-sensitive. The suggested technique for transformer elimination destroys the minim u m sensitivity feature of the partitioning. In some cases the constant K in (20) m a y be chosen to satisfy the Fialkow-Gerst condition (10) and the RC two-port can be realized without transformers preserving the minimum-pole-sensitivity of the entire network. III. n-PORT ACTIVE RC NETWORK
A. The Odd Part of a Positive Real Matrix It is well known that a given reduced positive real matrix F of order n can be realized as the impedance matrix of an n-port passive network obtained by terminating the (n + 1)'h port of an (n + 1)-port lossless network (henceforth to be called a LeRoy-Bayard network) by a resistance (Fig. 6) (11). Let X denote the open circuit impedance matrix of the LeRoy-Bayard network. Partition X as
194
K. MITRA
SANJIT
[J.F.I.
where U is a symmetric m a t r i x of order n, V is a column vector of order n,
V ' is the transposed of V and W is a matrix of order 1. We can write VV'
F = U
(28)
W+I The even and the odd parts of F are given by EvF & F "
VV' W2 - 1
=
(29)
VV' W W 2- 1
O d F ~ F °a = U
(30)
I O
I
-
I I n 0
I
(LOSSLESS)
I
(LOSSLESSI
0 n
[
C nt
0
#
Fro. 6.
n+l
I od£--~ I
0
F--~I
X
O----I'
n+l
Realization of a reduced positive real matrix F.
Fie. 7.
w Realization of the odd part of a reduced positive real matrix F.
N o w consider the network obtained by terminating the (n + 1)th port of the LeRoy-Bayard network of Fig. 6 by a negative LC impedance of value equal to - l / W , as shown in Fig. 7. The impedance matrix of the modified n-port is VV
F' = U
t
1 W
= U
VV
tW
W2 - 1
(31)
-- --
W
Equation 31 is recognized as the odd part of the positive real matrix F. Thus we have the following lemma, a generalized version of Lemma 1.
Lemma 4: A given symmetric matrix of order n satisfying the properties of the odd part of a reduced positive real matrix can be realized as the impedance matrix of an n-port network obtained by terminating the (n + 1) 'h port of an (n + 1)-port lossless network by a negative LC impedance of value equal to the reciprocal of the open circuit driving-point impedance of the (n + 1)-port network at its (n + 1)'hport.
Sept., t962.}
ACTIVE
RC
NETWORK
SYNTHESIS
195
Similar statements can be made on the admittance basis.
B. Synthesis of n-Port Active RC Impedance Matrix Let Z be the given impedance matrix of order n, Z = [z,,(s)] .
(32)
Suppose Z is such that upon p e r f o r m i n g a n R C - L C transformation on each of its elements, the transformed matrix Z can be expressed as the odd part of a positive real matrix P (which will be called the primary positive real matrix corresponding to Z), that is,
= [sz,,(s2)] = pod
(33)
A
I f Z is not regular on the imaginary axis, we can write /,,
Z = H + G °~
(34)
where G °d is the odd part of a positive real matrix G which is regular on the imaginary axis and H is a Foster matrix (12) such that the matrices of the residues at the poles are of rank 1. Thus P = H + G.
(35)
Let k be the rank of G% the even part of G. T h e n we can express G °Vas k
G e" = ~
Mi
(36)
i=l
where the matrices M i are each of rank 1 (12). For each MI we can construct a reduced positive real matrix GI such that M,
= G Cv, .
(37)
/%
This implies that we can express Z as k
= H + ~
G °d .
(38)
i=1
As shown in Sec. III-A, each G~d can be realized in the form of Fig. 7. Furthermore, H can be realized by the usual methods (11). Finally a series connection of these networks yields the realization of Z. Next we perform an inverse L C - R C transformation on the n-port =kL, + C network to obtain a realization of the given impedance matrix Z. Since k, the number of negative LC impedances (and hence the number of negative RC impedances)required, is the rank of a (n × n) matrix, k < n, or in other words the n u m b e r of negative impedance converters needed is at most equal to n.7 From these results and the properties of the odd part of a positive real matrix, we have the following lemma which is a generalization of L e m m a 2" 7 Sufficiency of n active elements for the realization of a n a r b i t r a r y n x n m a t r i x of real rational functions was first e s t a b l i s h e d by S a n d b e r g (1.3, 14) following a different approach.
SANJIT K. MITRA
I96
[J.F.I.
Lemma 5: A symmetric (n x n) matrix of real rational functions, Z -
1
D(s)
[n,j(s)]
(39)
satisfying the following conditions,
(i) no(s)° N D(s) °, (ii) Z has poles and zeros anywhere in the complex s-plane with the restriction that the negative real axis poles of Z are simple and the matrices of the residues at the negative real axis poles are positive semi-definite, (iii) Z(oo) is positive semi-definite, can be related to the odd part of a primary positive real matrix P after an RC-LC transformation on each of its elements. Furthermore, Z can be realized as the impedance matrix of an active RC n-port containing k(k < n) negative impedance converters, where k is the rank of per. Similarly, we can obtain a generalization of L e m m a 3 on the admittance basis. It should be noted that as the realization of a LeRoy-Bayard network requires ideal transformers, the n-port active RC network will usually contain ideal transformers. But we have an added advantage, because starting from the odd part of a positive real matrix, the even part is not uniquely determined. So it seems that there exists a possibility of obtaining a transformerless active RC n-port by suitably choosing the even part of the primary positive real matrix. Additional investigation in this direction is suggested. Performing an inverse LC-RC transformation on the right hand side of (31) we can show that the denominator will be in the form o f a Horowitz decomposition (Eq. 26). As a result, the realized n-port active RC networks are minimum-pole-sensitive. IV. CONCLUSION
The decomposition and partitioning of active RC driving-point functions, presented in this paper, are restricted to odd part partitionable impedances which satisfy condition (11) and to odd part partitionable admittances satisfying condition (13). In an actual synthesis it is usually best first to remove any negative real axis poles as RC networks in a manner analogous to the Brune preamble (15). Functions which fail to meet the requirements on either basis m a y sometimes be successfully augmented by the subtraction of a suitable resistance or capacitance. Synthesis of transfer functions can be taken care of by first reducing the problem to the synthesis of driving-point functions by the usual methods (6). A logical extension of the results of the active RC one-ports to the synthesis of active RC n-ports is next presented. In the latter case, the technique is restricted to impedance matrices which satisfy condition (40). Dual conditions for admittance matrices can be easily derived. Further-
Sept., ,962.1
ACTIVE RC
N E T W O R K SYNTHESIS
I97
more, the number of negative RC impedances, and hence the number of active elements, needed is found to be equal to k(k < n), where k is the rank of the even part of a certain primary positive real matrix related to the given impedance matrix by the relation (33). In both the one-port and the n-port cases, the techniques presented hinge upon straightforward algebraic procedures and result in minimumpole-sensitive decomposition and partitioning.
Acknowledgment The author wishes to thank Professor E. S. Kuh for his encouragement and helpful suggestions. Thanks are also due to Messrs. J. D. Patterson and R. A. Rohrer for useful discussions. REFERENCES
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