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Synthesis of active RC multiport networks with grounded ports
Journal of The Franklin Institute DEVOTED
Volume
294,
Number
TO SCIENCE AND THE MECHANIC
ARTS
JVovember
5
1972
fd fdfdfdfdfkjkjkkC Multiport Networks With Grjkjkjkounded Ports by THEODORE
A. BICKART
Electrical and Computer Engineering Department Syracuse University, Syracuse, New York and
DONALD
w.
MELVIN
Electrical Engineering Department University of New Hampshire, Durham, New Hampshire ABSTRACT : A synthesis procedure-easily implemented as a digital computer program-is presented whereby the network function T(s) can be realized as an active RC multiport network with grounded ports. Based on V(s) = T(s) U(s), where T(s) is a q x p matrix of of the complex variable 8, the realization requires a minimum real rational functions number of grounded capacitors-n = degree {T(e)}--and no more than 2(p+n) inverting, grounded voltage amplifiers or p + n differential output, groursded voltage amplifiers. Note: These properties of the realization are desirable if the network is to be fabricated as an integrated circuit.
I. Introduction
Many papers, published in recent years, have dealt with the problem of realizing rational function matrices (1-17). With the exception of Hilberman (13), the results in these papers, (a) deal with restricted classes of rational function matrices, (b) use active elements that are not readily available or (c) require the use of ungrounded capacitors and/or active elements. This paper, which is an extension of work previously reported (17), presents synthesis procedures for any type of rational function matrix (admittance, impedance, voltage gain, etc.). The realized networks require a minimum number of capacitors and use common ground voltage-controlled voltage sources as their active elements. All of the capacitors and active elements share a common ground. Note: The procedures developed here differ significantly from those of Hilberman (13). Let N be a multiport network excited at p of its ports by voltages and/or currents which are elements of the p-vector u(t). Let the responses, the voltages and/or currents of Q of the ports, be elements of the p-vector v(t).
289
Theodore A. Bickart and Donald W. Melvin Note: If N is excited at a port from which a response is derived, then at that port, if the excitation is a voltage (current), the response must be a current (voltage). Let T(s) be a Q x p matrix of real rational functions of the complex variable s such that V(a) = T(s)&%
(1)
where U(s) = LZ’[u(t)] and V(s) = Z[v(t)]. Note: T(s) is said to be a multiport network function. A synthesis procedure is developed by which to realize T(s) as an active RC multiport network with a minimum number of capacitors--n = degree (T(s)} -and with no more than 2(1, + n) inverting, grounded voltage amplifiers or p + n differential output, grounded voltage amplifiers. Furthermore, the capacitors and the amplifiers will share a common ground. The procedure is an extension of that developed by Melvin and Bickart (17). ZZ. Synthesis
Procedure
The procedure, restricted to the realization of any one of the following six cases : (1) a short-circuit admittance matrix, (2) an open-circuit impedance matrix, (3) a short-circuit transfer admittance matrix, (4) a voltage gain matrix, (5) a current gain matrix or (6) an open-circuit transfer impedance matrix, will be developed initially. The unrestricted procedure will follow as an amalgamation of the restricted procedures. By having the method evolve -particular methods to general method-the synthesis technique will not be obscured by the detail of a precipitous generalization. The restricted procedures will be put forth in three phases. In the first phase, state equations needed in the realization of T(s) will be established. In the second phase, short-circuit parameter equations for resistive and capacitive subnetworks will be developed for each of the six oases. In the third phase, realizations of the resistive and capacitive subnetworks will be devised. ZZZ.Restricted
Procedure:
Phase One
Suppose T(s) is not regular at s = co in the extended complex plane 9?*. Let - 01E%‘* be a point of regularity of T(s) on the negative real axis. By invoking the Mobius transformation (18)
with /3> 0, T(s) is transformed
into
defined by (2) is a which is regular at z = 00 in 55’*. The transformation one-to-one mapping of V* onto itself. The inverse transformation is
290
Journal
of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports Now, the results which follow
establish
a synthesis procedure
which will
realize T(z) as an active RC multiport network. In that network each capacitor of admittance cz, should by (4) be replaced by a capacitor and resistor in series, having admittance c/~s/(s + CL).Then the resulting network will realize T(s). Note: This replacement guarantees that if F(s) is realized with n grounded capacitors, then T(s) can be realized with n grounded capacitors. This is a non-trivial observation since one premise was that,: The capacitors will be grounded. * Since degree T(s) = degree{r(z)) (19), the development of the synthesis procedure can now continue, without loss ofgenerality, under the assumption that T(s) is regular at s = co. If it is not, then all that follows is applied to f(x) rather than T(s). Irreducible state equations associated with T(s) are of the form & = Ax+Bu, v = Cx+Du,
I
(5)
where x is an n-vector and {A, B, C, D} is a set of real constant matrices. Note that T(s) = D +C[sI- Al-l B and hence that D = T(W). The set of matrices {A, B, Ct_the uniquely defined D is not included-is not unique. In fact, the family of all such sets of matrices consists of the sets {Q-IA,, Q, Q-l B,, C, Q}, where Q is an arbitrary n x n non-singular real constant matrix and (Ao,BO, C,} is some set in the family. One such set can be obtained by application of the Ho-Kalman algorithm to T(s) -D (20). Therefore, in concluding phase one of the synthesis procedure, it is assumed that a set of irreducible state equations associated with T(s) is known.
IV.
Restricted
Procedure:
Phase
Two
The network that realizes T(s) will be the interconnection of a capacitive subnetwork and a resistive subnetw0rk.t The short-circuit parameter equations for these subnetworks will be related to the state equations, case by case. * Any Mobius transformation s = f(z) can be used provided the following two conditions are satisfied: (i) T( 8) is regular at s =f(co). (ii)f-‘(a) can be realized as E two-terminal network having one or more resistors and just one capacitor which is connected to one of the terminals. t Such a network structure, established in relation to the state equations of a network, is not new. Its dual-resistive and inductive-subnetworks was advanced by Youla in the realization of an open-circuit impedance matrix (23). It has been described in detail by Newcomb (24) and invoked recently by Dewilde et al. (14), and by Melvin and Bickart (17) in establishing synthesis procedures. In the evolution of the synthesis procedure to follow, it is shown that the distinctive realization of the resistive subnetwork arising in the synthesis of a short-circuit admittance matrix by the procedure reported by Melvin and Bickart can be employed in the realization of any multiport network function matrix.
Vol.
294, No. 5, November
1972
291
Theodore A. Bickart and Donald W. Melvin Short-circuit Admittance Matrix By assuming that T(s) is a short-circuit admittance matrix, it is implied that the p-vector u is the vector of network port voltages e, and that v is the p-vector-note the q = p-d corresponding port currents. Consider the network block diagram in Fig. l(a); NR is a (p +n)-port grounded, resistive subnetwork and No is an n-port grounded, capacitive subnetwork. Denote the short-circuit parameter equations of NR as
(‘3) where e, and i, denote the n-vectors of voltages and currents at the ports common to NR and No. The relation imposed by No on e, and i, will be expressed as i, = - Ce,. Substitute
(7)
(7) in (6). The result, with e, = u, is = v and e, = x, yields (5)
(a)
FIG. 1. Network block diagrams.
292
Journal of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports if and only if C is non-singular
and
Open-circuit Impedance Matrix By assuming T(s) is an open-circuit impedance matrix, it is implied that the p-vector u is the vector of network port currents i, and that v is the p-vector-again note that q = p-of corresponding port voltages e,. Consider the network block diagram in Fig. l(b) ; N, and No are specified as in the previous case. The relationship between es, e, and i,--implied by the resistor R in the block diagram-is e, = e,+Ri,,
(9)
where R = diag{r,, r2, . . ., rJ and ri 2 0 for i = 1, . . .,p. Thus there must be a series resistor at each port. Assume G,, is non-regular. Then, in (6), solve for e, and i, in terms of i,Y and e,. Combine that result with (9) ; the expressions achieved are e,- = [R + G;l] i, - G&l Gsce,, i, = G, G&l is -I-[G,, - G, G;l G,,] e,. I
(10)
Substitute (7) in (10). The result, with i, = u, es = v and e, = x, yields (5) if and only if C is non-singular and D = R+G,-,l, - C = G;l G,,, (11)
- CB = G,, G&l, - CA = G, - G,, G$ Get. 1
The elements of R can always be specified such that G,-,’ = D -R is nonsingular; assume such an R is chosen. Then, as previously assumed, G,, is non-singular. Furthermore, (11) yields G=
[D -RI-l [ - CB[D -R-J-l
-[D-RI-lC -CA+CB[D-RI-lC
1 *
(12)
Short-circuit Transfer Admittance Matrix The assumption that T(s) is a short-circuit transfer admittance matrix implies that the p-vector u is a vector of source port voltages, denoted e,, and that the q-vector v is a vector of response currents, denoted i,, with the response ports shorted-that is, with e, = 0. Consider the network block diagram in Fig. l(c) ; NE is a (p + q + %)-port grounded, resistive subnetwork and No is as specified previously. Denote the short-circuit parameter equations of NR as
F;]=[s
Vol. 294,
No. 5, November
1972
5;
$1
E]=G
(13)
293
Theodore A. Bickart and Donald W. Melvin Substitute (7) in (13) and set e, = 0. The result, with e, = u, i,. = v and e, = x, yields (5)-the expression for i, is superfluous-if and only if C is non-singular and G,, D I -CB
G = The unspecified submatrices
GsT G,, G,,
G,, C -CA
(14)
of G are arbitrary.
Voltage Gain Matrix By the assumption that T(s) is a voltage gain matrix, the p-vector u is a vector of source port voltages, denoted e,, and the q-vector v is a vector of response port voltages, denoted e,, with the response ports open-that is, with i, = 0. The network block diagram is shown in Fig. l(c) ; NR and No are as specified in the previous case. Assume G,, is non-singular. Then, in (13) with i, = 0, solve for e, and i, in terms of e, and e,; the expressions derived are e, = - G$ G, e, - Gil G, e,, 4 = [G, - GCTG2 G,I e, + [G, - G,, G2
G,l e,. 1
(15)
Substitute (7) in (15). The result, with e, = u, e, = v and e, = x, yields (5) if and only if C is non-singular and -D
= Gil G,,,
- C = G,-: G,,, (16)
- CB = G,, - G,, G;;l G,,, - CA = G,, - G,, G;;’ G,,. Under the assumption that G,, is non-singular, G=
G,, -G,,D I -CB-G,,D
G G; Gc,
I
.1
(16) yields
G% -G,,C -CA-G,,C
(17)
Except for G,,, which must be non-singular, the unspecified submatrices of G are arbitrary. Note: G,, G,, G,, and G,, depend on G,, and G,,. Current Gain Matrix By assuming T(s) is a current gain matrix, it is inferred that the p-vector u is the vector of source port currents, denoted is, and that the q-vector v is the vector of response port currents, denoted i,, with the response ports shorted. The network block diagram is shown in Fig. l(c) ; NR and No are as specified for the previous two cases. Assume G,, is non-singular. Then, in (13) with e7 = 0, solve for i, and i, in terms of is and e,. When (7) is substituted in the derived expressions, the result with i, = u, i, = v and e, = x, is (5) if and only if C is non-singular
294
Journal of The Franklll Institute
Synthesis of Active RC Multiport Networks with Grounded Ports and G = i
-
G.ss DG, CBG,
Gs, G,, G,,
Gsc C+DG,, - CA - CBG,
(18)
1 .
Except for G,, which must be non-singular, the unspecified submatrices of G are arbitrary. Note: G,, G,,, G,, and G, depend on G, and G,,. Open-circuit
Transfer Impedance Matrix
By the assumption that T(s) is an open-circuit transfer impedance matrix, the p-vector u is a vector of source port currents, denoted i,, and the q-vector v is a vector of response port voltages, denoted e,, with the response ports open. The network block diagram is shown in Fig. I(c) ; N, and No are as specified in the previous three cases. Assume G,, and G,, are non-singular. Then, in (13) with i, = 0, solve for e, and i, in terms of is and e,. When (7) is substituted in the derived relations, the result, with is = u, e, = v and e, = x is (5) if and only if C is non-singular and G
GSS - G,, DII - G,, D]-1 Gss
G;;
- [CB + G,, D] [I - G,, D]-l G,,
G,,
G,, -G,,C-G,,D[I-G,,D]-’
1(19
[Gs, + Gs, Cl
G=
- CA - G,, C - [CB + G,, D] [I - Gs, Dl-l
[
.
Ws,+ Gs,Cl
In addition to the requirement that G,, and G,, be non-singular, G,, must be selected such that [I - G,, D] is non-singular. It is obvious that such G,, exist, since GST= 0 is one such possible choice. The remaining, unspecified submatrices of G-G,, and G,,--are arbitrary. Note: G,,, G,, G,, and Gee depend upon all of the other submatrices of G.
V. Restricted
Procedure:
Phase
Three
Thus far, the n x n capacitance matrix C is only constrained to be nonsingular. Therefore, it can be assumed that C is diagonal, with positive elements on the diagonal. By this assumption, the grounded n-port No contains n grounded capacitors-a characteristic of the realization of T(s) that was to be established-with one capacitor connected to each port terminal, as illustrated by the schematic in Fig. 2. That the realization requires at least n capacitors follows from the fact that C would be singular if No contained fewer than n capacitors. The sub-network Nn in Fig. l(c), consisting of resistors and inverting, grounded voltage amplifiers, is assumed to have the structure shown in Fig. 3, where flR is a 3(p +q+n)-port grounded sub-network of resistors.
VoL294,No. 5,November1972
295
Theodore A. Bickart and Donald W. Melvin The sub-network NR in Figs. l(a) and (b) is obtained by removing the distinct response ports. In that case i?& will be a 2(93+%)-port sub-network. Cl
IO
c2
20
q I ,,C”
no
FIG.
I\
’
2. Capacitive subnetwork Ai,.
t =
t-
%I ‘”
esp+oe,l +o era+-
+ecn
,
&
FIG. 3. Block diagram of N,.
Denote the short-circuit
is i, 4 . lk . lh
296
L % -?+ec1
parameter equations
% [1
of iVRas
e,
=e
ek
eh
Journal
of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports where the diagonal submatrices of c! are symmetric and where [eke;]’ and [i;ijJ’ denote 2(p +q+n)-vectors of voltages and currents at the ports to which the outputs of the amplifiers are connected. Now
(21) where K = diag{k,, . . ., kp+p+n} and I-I = diag(h,, . . ., hp+q+n}. Note: Because the amplifiers are inverting amplifiers, each of the ki and hi is non-positive. Equations (20) and (21) together yield
I iI HK
By comparing
this result with (13) it is found that G = Q+G+K+G_HK,
(23)
where and It will now be shown that i$,, K and H can be determined such that i? is hyperdominant-a necessary and sufficient condition for the 3(p +q +n)port NR to be realizable without internal nodes (21). Select i$, such that it is hyperdominant, with the sum of the elements in a row being strictly positive if the corresponding row of G - Gp has one or more non-zero elements. Let G+,
= P+M,
(24)
where P (M) contains the non-negative (non-positive) elements of G - G9. If G is to be hyperdominant, the elements of [4+ and G__must be non-positive. Lastly, the elements of H and K are non-positive, thus (23) and (24) combine to give
G+K=P
(254
B_HK=M.
Wb)
and
Clearly, by making the 1ki 1and / hi 1sufficiently large the non-zero elements of G-G+ and of G_ can be made as small in magnitude as is necessary to make the first p + q + n rows of G hyperdominant. This is possible because the sum of the elements of a row of Gp is strictly positive if the corresponding row of G - G9 has one or more non-zero elements. If Kt[Ht] denotes the pseudo inverse (22) of K[H], then (25) yields G, = PKt,
Vol. 294, No. 5, November
1972
(26s)
297
Theodore A. Bickart and Donald W. Melvin provided
the consistency
condition
P[I -K+ K] = 0 is satisfied, and
G_ = MK+H+
(26b)
provided the consistency conditions MII - K+ K] = 0 and MK+[I -H+ H] = 0 are satisfied. Note: The consistency conditions can be satisfied. One way of doing so is to select the ki and hi to be strictly negative; then K[H] will be non-singular and K+ = K-l[H+ = H-l]. It then follows that [I-K+K] = 0 and [I-H+ H] = 0. Now, let
be diagonal with the value of each diagonal element selected such that the sum of the elements in each of the last 2(p +q +n) rows of fi is zero. This condition means that there will be no shunt resistors at the ports to which the outputs of the controlled sources are connected. This completes the specification of B.
VI.
Design
Criteria
It has been shown that, for any of the six cases considered, an active RC multiport network realization of T(s) exists. However, it has been established only that 2( p + q + n) amplifiers-the number of diagonal elements k, and h, of K and H-are sufficient. It will now be shown that 2(p + n) amplifiers are sufficient and that, in some cases, not even that many are needed. Consider (25). If j is the index of a zero column of M [both M and P], set hi = 0 [both kj = 0 and hj = 01. With these being the only kj and hj set equal to zero, the consistency conditions for equations (26) are satisfied. This is a consequence of the fact that K+ = diag(kf, . . . . k&+q+n}, where ki = l/4 if ki # 0 and kl = 0 if k, = 0, and that H = diag{h,t, . . . . hA+*+,,}, where hf = l/h, if hi # 0 and hf = 0 if hi = 0. It is evident that the actual number of amplifiers needed is equal to the number of non-zero k, and h,. Now since the number of capacitors in any realization, obtained by the procedure developed herein, is already a minimum, it is reasonable to seek that particular realization requiring a minimum number of amplifiers. Considerable design flexibility is available in the quest for such a minimal realization; for example, the diagonal elements of C can have any positive value, and any nonsingular Q can be used in (A, B, C} = {Q-l A, Q, Q-l B,, C, Q). Bounds on the actual number of amplifiers needed will be established case-by-case. Short-circuit Admittance
Matrix and Open-circuit Impedance
Matrices
When T(s) is a short-circuit admittance or an open-circuit impedance matrix, there are no distinct response ports. Therefore, 2(p +n) is the maximum number of amplifiers needed.
298
Journal of The Franklin Institute
Synthesis of Active RC Multiport Networks with Grounded Ports Short-circuit Transfer Admittance Matrix The subscripts p and m will be used in the following way: Let S be an arbitrary matrix ; then S = S, + S,, where S, (S,) contains the non-negative (non-positive) elements of S. Now, consider G as specified in (14). Let G,, = Dk, G,, = [ - CB]&, and GcT = CL; then Gs, D [ -CB
G =
Dk G,, C;,
[-CBli, C -CA
.
(27)
1
Let G, and G,, be diagonal and let S be symmetric, G,,
Dk G CL
Gp = [ [-%1,
such that
II-CBI:, C s”
(29) 1
is hyperdominant, with the sum of the elements in a row being strictly positive if the corresponding row of G+
0
0
[ [-%],
O 0
=
has one or more non-zero elements.
C, -CA-S
If follows from (29) that
0 P=
D, [-CB],
0
0
0 0
[-CkS],
and M=
0 0 0
I
(29)
1 1
0
0 0 0
0 0 [-CA-S],
(39)
1
(31)
*
Because columns 1 through p of M and p + 1 through p + Q of both M and P are zero, hi = 0 for i = 1,..., p+q and ki = 0 for i =p+l,..., p+q. This implies that p + 2q of the possibly 2(p + q + n) amplifiers that might have been needed are not required. Therefore, no more than pf2n amplifiers are needed to realize T(s) as a short-circuit transfer admittance matrix. Voltage Gain Matrix Consider G as specified in (17). Let G,, = 0, G,, = 0 and G,, = 0 ; then G=
Gss -G,,D i -CB
0 G,, 0
Let G,, be diagonal, let G,, be diagonal symmetric, such that
0 -G,,C -CA
1 .
and non-singular
(32) and let S be
(33)
Vol.294,No.5,November 1972
299
Theodore A. Bickart and Donald W. Melvin satisfies through i=p+l needed. Again, G,, = [ -
all previously imposed properties. It follows that columns p + 1 p + q of both M and P will be zero. Hence, hd = 0 and k, = 0 for , . . .,p + q. As a consequence, no more than 2(p +n) amplifiers are consider G as specified CB]; and G,, = 0 ; then,
in (1’7). Assume
G,, 0 -CB [
G=
0 G,, 0
G%? [ [-:B],
.1
[-CB]:, -G,,C -CA
Let G,, be diagonal, let G,, be diagonal symmetric, such that i$ =
D = 0 and let G,, = 0,
0 G 0”
and non-singular
1
[-CB]:, 0 S
(34) and let S be
(35)
satisfies all previously imposed properties. It follows that columns 1 through p of M and p + 1 through p + q of both M and P are zero. Hence, no more than p + 2n amplifiers are needed. Current Gain Matrix Consider G as specified in (18). Let G,, = 0, G,, = 0 and G,, = 0 ; then, G =
G, DG,, [ -CBG,,
0 G,, 0
0 C -CA
1.
(36)
Let G,, be diagonal and non-singular, let G,, be diagonal and let S be symmetric, such that Gp in (33) satisfies all previously imposed properties. It easily follows that no more than 2(p + n) amplifiers are needed. Open-circuit Transfer Impedance Matrix Consider G as specified in (19). Let G,, = 0, G,, = 0 and G,, = 0; then,
G=
G,S - G,,DG,, i -CBG,
0
0
G,, 0
- G,,C -CA
I-
(3’)
Let G,, and G,, be diagonal and non-singular and let S be symmetric such that Gp in (33) satisfies all previously imposed properties. It then follows that no more than 2(p + n) amplifiers are needed.
VII.
Unrestricted
Procedure
In treating the unrestricted case, the source ports are divided into four the response ports are divided into two groups groups--+, $22,ss and s,-and
300
Journal of
The Franklin Institute
Synthesis of Active RC Multiport Networks with Grounded Ports -rl and r2. The network is excited by voltages (currents) at the ports in groups s1 and s3 (& and sJ. The response of the network is the voltages (currents) at the ports in groups & and r2 (sl and rl) with the ports in group r2 (rr) open (shorted). The implication is that
u=[g]
and
v=[i],
and that e,, = 0 and i,* = 0. A block diagram of the network is shown in Fig. 4; N, and No are as specified previously. Under relatively minor restrictions on some submatrices of the short-circuit conductance matrix, G, for NE, if3 possible to
4, 0 + i 22 0 +
R +
i
9 0 + Q,
Q2 is4
ic h
+
0 es*
+ Q3
NC
3 0 +
=34
ec
i
3 ert
\; er2
I
FIG. 4. Network block diagram.
solve for isI, es2, ih, era and i, in terms of eS1,is,, e,,, is, and e,. The state equations resulting when the relations i, = - C!hcand e,-, = es,+ Ri,,, where R is diagonal with positive diagonal elements, are incorporated in the derived expressions, can be compared with the state equations in (5) to establish constraints on some of the submatrices of G. The task is conceptually simple, but the resulting expressions are too expansive to present here. By the validity of the restricted synthesis procedures previously established the validity of the procedure just described can be inferred. It is also possible to infer that no more than 2( p + n) amplifiers are needed to realize T(s). However, this bound is an immediate consequence of the more apparent fact that the synthesis procedure can be implemented such that columns other than the first p and the last n of both M and P are zero,
Vol. 294, No. 5, November
1972
301
Theodore A. Bickart and Donald W. Melvin when the short-circuit parameter equations are constrained at the outset to be
e81 e58 e%¶
e54 e,, e
r2
. % The synthesis procedure
and associated results imply Theorem
I.
Theorem I. Any (I x p matrix T(s), of real rationa functions of the complex frequency variable s, can be realized as a multiport network function of an active RC multiport network containing exactly n = degree {T(s)} grounded capacitors and at most 2(r, +n) inverting, grounded voltage amplifiers. In addition, the ports will be grounded. Note: When information about the port variables is available, the bounds on the number of ampli~ers needed may change. Such is the case, for example, when the network function matrix being realized is a short-circuit transfer admittance matrix; recall in that case that 2, + 2n is the bound on the number of amplifiers needed. VIZ& Amplifier
Gain
Selection
The synthesis procedure was established on the observation that amplifier gains exist such that the equations in (25) have solutions corresponding Before turning to to which the rows of [Gp i?$+_a_] are h~erdom~ant. ihustrative examples of the procedure, some criteria for choosing the amplifier gains will be described. Let .X(S) denote the set of indices of the non-zero k&hi) and 9)(A) denote the set of indioes of the rows of P(M) having one or more non-zero elements. Let oi 2 0 denote the sum of the elements in the ith row of QF. Let vi 2 0 (pi > 0) denote the sum of the elements in the ith row of P( -RI). Choose real constants wi such that wi = 0 if i EA% - (9’&), 0 < wa < 1 if i~9’nM and wi = I if iEg--(9’&). Now consider the case k:d = k for i ES and h, = h for i ES’?. It can be shown that if (39a) and (39b) then the rows of /j$,a,fi_] will be hyperdominant. In a particular situation it may be desirable for all the amplifiers to have the same gain. In that case k, = k for i E Y and hi = k for i E%. It follows
302
Journal
of The Frsnklin Institute
Synthesis of Active RC Multiport Networks with Grounded Ports from (39) that k should be chosen to satisfy the inequality - k 2 max
[max cEB (~TJ~~wi)l,l~~r~dul(l-wi)l)~~. II
(40)
Suppose hi = - 1 for i E&. Then, for each i E% the sub-network NR contains a cascade of two amplifiers which, as illustrated in Fig. 5, can be
5. Differential output amplifier.
FIG.
replaced
for VEX.
by a single differential output amplifier. In addition, suppose k, = k Then, it can be shown that k must be chosen such that - k 2 iyU%Urri +r-~i)/c~l.
(41)
It is sometimes possible to select the ki and hi such that the rows of &(4+Q_] are hyperdominant with the sum of the elements in each row being equal to zero. Note: For each row having elements which sum to zero, one less resistor is needed than would otherwise be the case. Now let sp(s,) be the vector having a, o( if i E 9’(ci( 1 - wi) if i E A) and 0 otherwise as its ith element, and let kt(ht) be the vector having k+(ht) as its ith element. Then, if the equations Pkt = -So Wa) and MKtht = -s, are consistent, the non-zero sponding elements of
(42b)
ki and hi will be the reciprocals
of the corre-
kt = -ptsp
(434
ht = - [MKt]t s,.
(43b)
and For reasons given previously, consider the case h, = - 1 for i EX. if the equation [P-M]kt = -[s~+s,] is consistent, elements of
the non-zero
ki will be the reciprocals
(44)
of the corresponding
kt = - [P - M]t [So + s,]. IX.
Then
(45)
Examples
Several features of the synthesis procedure following examples.
Vol. 294, No. 5, November 1972
are illustrated
in the three
303
Theodore A. Bickart and Donald W. Melvin Example 1 The 2 x 2 matrix
T(s) =
s+l
[ ,I1
will be realized as the open-circuit is not regular at s-co, let
S
1
impedance
matrix of a 2-port. Since
T(s)
8 = z/(1-2). Then
4(1-4 W)
l/(1 -4
= [ (-1+22)/(1-z)
2/(1--x)
I
will be realized, after which each capacitor having admittance cz will be replaced by a capacitor and a resistor in series, having admittance cs/(s + 1). The Ho-Kalman algorithm established the state equations which follow: j, = [l]x+[l eg=[
l]i,,
I:]x+[
1:
“l]i,.
The realization will continue with this set of matrices; those associated with equivalent networks will not be sought. Now, let c = [l]. Next, since
D is non-singular, set R = 0 in (12). Then by (12) -1
0
2
-1
1
1 -1
1
-1
G=
-1
1 .
The matrix GP must be selected next. Inspection of G shows that GP cannot be selected such that G-G, has columns of zeros or, next best, non-negative elements, which would thereby reduce the actual number of amplifiers needed to fewer than 2(r, + n) = 6. Set +[
!l
%
Y].
Then, from (24) it follows that 0
P=
304
2 0
0 0 1
0 1 ] 0
and
M=[
-3
‘3
p3].
Journal of The Franklin
Institute
Synthesis of Active RG Multiport Networks with Grounded Ports The realization will be established such that each amplifier has the same gain; therefore (40) will be invoked. Now, i
Oi 71i Pi
1
1
0
2 3
2
3
1
1
3 3 3
.
Set wi = 0 for i = 1 and wi = 4 for i = 2,3. Then by (40) - k > max {[max (0,3,2)],
[max (3,3,6)]f}
= 3.
With (II= - 3, (26) yields e+=[
+
!i
+]
and
i?_=[”
2:
!+I.
It is easily verified that the p + n = 3 rows of [CD i!+ B-1 are hyperdominant. Therefore, 0 composed according to the guidelines previously set forth will be hyperdominant. The realization of T(s), based on the realization of a, is shown in Fig. 6. Note that, as indicated previously, gR has no internal
FIG. 6. Example
1: Realization of T(s).
nodes. Note also that a capacitor and a resistor are in series at each capacitive port of NR. Example 2 The 1 x 1 matrix T(s) = [l/V
+ J(2) s + I)]
will be realized as a voltage gain matrix. Note: The network, once realized, will be a 2-pole Butterworth filter. The Ho-Kalman algorithm establishes
Vol.
294
No.
5, November
1972
305
Theodore A. Bickart and Donald W. Melvin the state equations which follow : 0
ir=
1
-l-J2
[ e, = [l
‘0 lesy I
IL
0] x
[0] e,.
The state equations for equivalent networks will not be sought. Therefore, by (17), this set of matrices {A, B, C, D} together with G,, = [II, G,, = yields
G,, = [Ol,
I ;’ 1
and
Gg, = 10 C=
-
[a 0 1 0
1
1
0
0
-1
0
1
-1
0
0
-1
1
-&
-1
0
1
J2
1
0
0
-1
0
1
-1
0
0
-1
2
-2
-1
0
-a
G=
11, G,, = PI,
II
Set
ilp =
3
t’2
then
P= [
0 0
0 0
have three-the same amplifiers will actually The realization will ki = k for i E X = (3).
2 0 0
“]and*=[iii;i]
three-columns of zeros. This means that only two be needed. be completed with hi = - 1 for i EZ = (3) and with Now, i
)
oi
7ri j.L$
1
0
0
0
2 3
0
0 0
0
4
,,2ep
3
.
1 0
Therefore, by (41) - k>,max [i, 5/(4,/2-
306
5)] = 5/(4J2 - 5)...762.
Journal
of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports With k = - 8, (26) yields
fi+-[”
_Li]andfi-=[ii
i&i].
After composing a according to the guidelines previously developed, a schematic for T(s) can be drawn; it is shown in Fig. 7. A differential output
1
er,ofi_45
h3
32
t
-L
-r
FIG.
7. Example
2: Realization of T(s).
I
amplifier is shown, rather than the cascade of the two amplifiers k, = - 8 and h, = - 1. Note: The 1 mho resistor in series with the response port terminal can be removed since the network realizes T(s) with the response port open. Example 3. The 1 x 2 matrix
T(s) = [(s + $)/(s + 1) (2s - l)/(s + l)] will be realized as the network function v = [ea] and i, = [O]. The Ho-Kalman equations which follow :
of a network for which u = [ire,], algorithm establishes the state
(46)
The unrestricted procedure must be used. The network structure will be that shown in Fig. 4. The short-circuit parameter equations conforming to
Vol.
294,
No.
5, November
1972
307
Theodore A. Bickart and Donald W. Melvin those in (38) are
1[I e1
=
e2
Q
.
e3 e4
Note: e, and e2 are associated with the source ports-groups s3 and s4, respectively-e, is associated with the response port-group r,--and e4 is associated with the capacitive port. The solution for e3 and i, in terms of e,, i, and e4, with i, = 0, is e3 = g& g3, el - 62 g3, g2 i2 - g2 g3, e4, t4 = g,, el + g4, g2 i2 + g4,e4. Set i, = -G4, which implies that C = [l]. Then, comparison with (46), when x = [e,], yields L-92
g31
-sz
932
sa
b41 g42 s31 = I3
= [l
31
21,
and
L--&i
of that result
g341 =
PI?
[g441= PI.
Let gn = 1, g,, = 1 and g,, = ) ; then 1000
Set
Note that &, was chosen such that amplifiers-k, and h,-will be associated with the response port contrary to the practice followed thus far. This choice results in fewer amplifiers-4 vs. 6-than would be the case had GP been set eaual to .
[ 010 01000010 0 01 0 which yields k, and h, both zero.
308
1
Journal of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports The realization will be completed i&=lcforiEZ-=(1,2,3). Now
Therefore,
with hi = - 1 for i E%
i
a,
ri
Pi
1 2
f Q
t B
0 0
3 4
;;i
= (3) and with
by (41) -kamax(*,
with k = -5,
1,$,5) = 5
(26) yields
After composing B, a schematic for T(s) can be drawn ; it is shown in Fig. 8, with a differential output amplifier rather than the cascade of two amplifiers Ic, = - 5 and h, = - 1.
r e,:
+
e4
FIG. 8. Example 3: Realization of T(s).
Vol. 294, No. 5, November 1972
309
Theodore A. Bickart and Donald W. Melvin X.
Concluding
Discussion
It has been shown that any q x p multiport network function T(s) can be realized as an active RC multiport network with the ports grounded. The network will contain a minimum number of capacitors-n = degree(T(s)}and at most 2(p + n) inverting, grounded voltage amplifiers. The fact that all the capacitors and amplifiers are grounded is a distinct advantage if the network is to be fabricated as an integrated circuit. This is an advantage which is not exhibited by a number of active RC synthesis procedures. For example, in the short-circuit admittance matrix synthesis procedures proposed by Sandberg (2), Even (15) and Goldman and Ghausi (12,16) the capacitors will not be, in general, grounded.* Also, of the three ingredients in an active RC network-active component, such as a negative converter or voltage amplifier, resistor and capacitor-the capacitor is the one which most complicates fabrication of the network as an integrated circuit. Therefore, it is also an advantage of the procedure proposed that a minimum number of capacitors are required; such is not the case in many procedures, including those just cited.? It has also been established that if the amplifiers hi have gains of - 1, the cascade of the two amplifiers ki and hi can be replaced by a differential output amplifier. The implication of this is that the condition on the number of amplifiers could read: . . . at most p +n differential output, grounded voltage amplifiers. This is a significant observation, since fabrication of a differential output amplifier is likely to be simpler than the fabrication of two inverting amplifiers. It is conjectured that the realization of T(s) will be relatively insensitive to capacitance variations. This conjecture stems from the following observation: The number of capacitors is minimum, therefore, slight variations in capacitance values will not result in the realized network having more characteristic observable frequencies than the designed network. Even though the values of the characteristic frequencies might change somewhat, it is reasonable to suppose that such changes will have less effect on the realized network properties than the appearance of extraneous characterist,ic frequencies. It is hoped that the continuing investigation of this synthesis procedure will provide a quantitative assessment of the sensitivity of selected network attributes and validation of the conjecture. Note, finally, that the synthesis procedure described herein can be reduced rather easily to a computer program which, given the multiport network
* It should be noted, however, that each of the procedures just singled out do have grounded active components; not 2(p + rz) of them, but just p of them in the first case and 2p of them in the latter two cases. 7 The capacitors are grounded and only a minimum number of them are required in the synthesis procedures evolving from the state equations for a network. That is the case, for example, in the procedures described by Mann and Pike (9) and Dewilde et al. (14). Also, see Newcomb (24).
310
Journal of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports
function, will generate C, K, I-Iand i% Such a program is being prepared in APL, after which, a Fortran version will be developed.
References (1) I. W. Sandberg, “Synthesis of X-port active RC networks”, B.S.T.J., Vol. 40, pp. 329-347, Jan. 1961. (2) I. W. Sandberg, “Synthesis of transformerless active N-port networks”, B.S.T.J., Vol. 40, pp. 761-783, May 1961. (3) D. Hazony and R. D. Joseph, “Transfer matrix synthesis using one amplifier but no inductors”, Eighth Midwest Symp. Circuit Theory, June 1965. (4) J. Barranger, “Voltage transfer function matrix realization using current negative immitance converters”, IEEE Tram., Vol. CT-13, pp. 97-98, March 1966. (5) G. Martinelli, “RC transformerless networks embedding nullors”, Alta l%equenza, Vol. 35, pp. 156-162, Feb. 1966. (6) D. Hazony and R. D. Joseph, “Transfer matrix synthesis with active RC networks”, J. SIAM Appl. Math., Vol. 14, pp. 739-761, July 1966. matrix synthesis with active (7) R. D. Joseph and D. Hilberman, “Immittance networks”, IEEE Trans., Vol. CT-13, p. 324, Sept. 1966. (8) D. Hilberman, “Synthesis ) of rational transfer and admittance matrices with active RC common-ground networks containing unity-gain voltage amplifiers”, IEEE Trans., Vol. CT-15 pp. 431-440, Dec. 1968. (9) B. J. Mann and D. B. Pike, “Minimal reactance realization of N-port active RC networks”, Proc. IEEE, Vol. 56, p. 1099, June 1968. matrix realization using operational (10) S. K. Mitra, “Voltage-transfer-function amplifiers”, Elect. Letters, Vol. 4, No. 20, p. 435, Oct. 1968. (11) B. Bhattacharyya, “Synthesis of an arbitrary transfer function matrix using RC and one-ports and operational amplifiers”, Int. J. Electronics, Vol. 26, pp. 5-16, Jan. 1969. (12) M. Goldman and M. S. Ghausi, “On the realization of rational admittance matrices using voltage-controlled voltage sources and RC one-ports”, IEEE Trans., Vol. CT-16, pp. 544-546, Nov. 1969. “Active network synthesis using amplifiers having any finite (13) D. Hilberman, non-finite gain”, IEEE Trans., Vol. CT-16, pp. 484-489, Nov. 1969. (14) P. Dewilde, L. M. Silverman and R. W. Newcomb, “A passive synthesis for timeinvariant transfer function”, IEEE Trans., Vol. CT-17, pp. 333-338, Aug. 1970. (15) R. K. Even, “Admittance matrix synthesis with RC commonground networks and grounded finite-gain phase inverting voltage amplifiers”, IEEE Trans., Vol. CT-17, pp. 344-351, Aug. 1970. (16) M. Goldman and M. S. Ghausi, “Active RC synthesis of admittance matrix with prescribed RC element constraints”, IEEE Trans., Vol. CT-18, pp. 299-302, March 1971. (17) D. Melvin and T. Bickart, “P-port active RC networks: Short-circuit admittance matrix synthesis with a minimum number of capacitors”, IEEE Trans., Vol. CT-l& No. 6, pp. 687-592, Nov. 1971. (18) E. Hille, “Analytic Function Theory”, Vol. I, pp. 46-58, Ginn, Boston, Mass. (19) R. E. Kahnan, “Irreducible realizations and the degree of a rat,ional matrix”, J. SIAM Appl. Math., Vol. 13, pp. 526-544, June 1965. (20) R. E. Kahnan, P. L. Falb, and M. A. Arbib, “Topics in Mathematical Systems Theory”, pp. 242, 288-294, McGraw-Hill, New York, 1969. (21) P. Slepian and L. Weinberg, “Synthesis applications of paramount and dominant matrices”, Proc. 1958 N.E.C., Vol. 14, pp. 611-630, 1958.
Vol. 294, No. 5, November
1072
311
Theodore A. Biclcart and Donald W. Melvin. (22) H. P. Decell, Jr., “An application of the Cayley-Hamilton theorem to generalized matrix inversion”, SIAM Rev., Vol. 7, pp. 526-528, Oct. 1965. “The synthesis of linear dynamical systems from prescribed (23) D. C. Youla, weighting patterns”, J. SIAM Appl. Math., Vol. 14, pp. 527-549, May 1966. (24) R. W. Newcomb, “Active Integrated Circuit Synthesis”, pp. 80-87, Prentice-Hall, N.J., 1968.
312
Journal of The Franklin Institute
Volume
294,
Number
TO SCIENCE AND THE MECHANIC
ARTS
JVovember
5
1972
fd fdfdfdfdfkjkjkkC Multiport Networks With Grjkjkjkounded Ports by THEODORE
A. BICKART
Electrical and Computer Engineering Department Syracuse University, Syracuse, New York and
DONALD
w.
MELVIN
Electrical Engineering Department University of New Hampshire, Durham, New Hampshire ABSTRACT : A synthesis procedure-easily implemented as a digital computer program-is presented whereby the network function T(s) can be realized as an active RC multiport network with grounded ports. Based on V(s) = T(s) U(s), where T(s) is a q x p matrix of of the complex variable 8, the realization requires a minimum real rational functions number of grounded capacitors-n = degree {T(e)}--and no more than 2(p+n) inverting, grounded voltage amplifiers or p + n differential output, groursded voltage amplifiers. Note: These properties of the realization are desirable if the network is to be fabricated as an integrated circuit.
I. Introduction
Many papers, published in recent years, have dealt with the problem of realizing rational function matrices (1-17). With the exception of Hilberman (13), the results in these papers, (a) deal with restricted classes of rational function matrices, (b) use active elements that are not readily available or (c) require the use of ungrounded capacitors and/or active elements. This paper, which is an extension of work previously reported (17), presents synthesis procedures for any type of rational function matrix (admittance, impedance, voltage gain, etc.). The realized networks require a minimum number of capacitors and use common ground voltage-controlled voltage sources as their active elements. All of the capacitors and active elements share a common ground. Note: The procedures developed here differ significantly from those of Hilberman (13). Let N be a multiport network excited at p of its ports by voltages and/or currents which are elements of the p-vector u(t). Let the responses, the voltages and/or currents of Q of the ports, be elements of the p-vector v(t).
289
Theodore A. Bickart and Donald W. Melvin Note: If N is excited at a port from which a response is derived, then at that port, if the excitation is a voltage (current), the response must be a current (voltage). Let T(s) be a Q x p matrix of real rational functions of the complex variable s such that V(a) = T(s)&%
(1)
where U(s) = LZ’[u(t)] and V(s) = Z[v(t)]. Note: T(s) is said to be a multiport network function. A synthesis procedure is developed by which to realize T(s) as an active RC multiport network with a minimum number of capacitors--n = degree (T(s)} -and with no more than 2(1, + n) inverting, grounded voltage amplifiers or p + n differential output, grounded voltage amplifiers. Furthermore, the capacitors and the amplifiers will share a common ground. The procedure is an extension of that developed by Melvin and Bickart (17). ZZ. Synthesis
Procedure
The procedure, restricted to the realization of any one of the following six cases : (1) a short-circuit admittance matrix, (2) an open-circuit impedance matrix, (3) a short-circuit transfer admittance matrix, (4) a voltage gain matrix, (5) a current gain matrix or (6) an open-circuit transfer impedance matrix, will be developed initially. The unrestricted procedure will follow as an amalgamation of the restricted procedures. By having the method evolve -particular methods to general method-the synthesis technique will not be obscured by the detail of a precipitous generalization. The restricted procedures will be put forth in three phases. In the first phase, state equations needed in the realization of T(s) will be established. In the second phase, short-circuit parameter equations for resistive and capacitive subnetworks will be developed for each of the six oases. In the third phase, realizations of the resistive and capacitive subnetworks will be devised. ZZZ.Restricted
Procedure:
Phase One
Suppose T(s) is not regular at s = co in the extended complex plane 9?*. Let - 01E%‘* be a point of regularity of T(s) on the negative real axis. By invoking the Mobius transformation (18)
with /3> 0, T(s) is transformed
into
defined by (2) is a which is regular at z = 00 in 55’*. The transformation one-to-one mapping of V* onto itself. The inverse transformation is
290
Journal
of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports Now, the results which follow
establish
a synthesis procedure
which will
realize T(z) as an active RC multiport network. In that network each capacitor of admittance cz, should by (4) be replaced by a capacitor and resistor in series, having admittance c/~s/(s + CL).Then the resulting network will realize T(s). Note: This replacement guarantees that if F(s) is realized with n grounded capacitors, then T(s) can be realized with n grounded capacitors. This is a non-trivial observation since one premise was that,: The capacitors will be grounded. * Since degree T(s) = degree{r(z)) (19), the development of the synthesis procedure can now continue, without loss ofgenerality, under the assumption that T(s) is regular at s = co. If it is not, then all that follows is applied to f(x) rather than T(s). Irreducible state equations associated with T(s) are of the form & = Ax+Bu, v = Cx+Du,
I
(5)
where x is an n-vector and {A, B, C, D} is a set of real constant matrices. Note that T(s) = D +C[sI- Al-l B and hence that D = T(W). The set of matrices {A, B, Ct_the uniquely defined D is not included-is not unique. In fact, the family of all such sets of matrices consists of the sets {Q-IA,, Q, Q-l B,, C, Q}, where Q is an arbitrary n x n non-singular real constant matrix and (Ao,BO, C,} is some set in the family. One such set can be obtained by application of the Ho-Kalman algorithm to T(s) -D (20). Therefore, in concluding phase one of the synthesis procedure, it is assumed that a set of irreducible state equations associated with T(s) is known.
IV.
Restricted
Procedure:
Phase
Two
The network that realizes T(s) will be the interconnection of a capacitive subnetwork and a resistive subnetw0rk.t The short-circuit parameter equations for these subnetworks will be related to the state equations, case by case. * Any Mobius transformation s = f(z) can be used provided the following two conditions are satisfied: (i) T( 8) is regular at s =f(co). (ii)f-‘(a) can be realized as E two-terminal network having one or more resistors and just one capacitor which is connected to one of the terminals. t Such a network structure, established in relation to the state equations of a network, is not new. Its dual-resistive and inductive-subnetworks was advanced by Youla in the realization of an open-circuit impedance matrix (23). It has been described in detail by Newcomb (24) and invoked recently by Dewilde et al. (14), and by Melvin and Bickart (17) in establishing synthesis procedures. In the evolution of the synthesis procedure to follow, it is shown that the distinctive realization of the resistive subnetwork arising in the synthesis of a short-circuit admittance matrix by the procedure reported by Melvin and Bickart can be employed in the realization of any multiport network function matrix.
Vol.
294, No. 5, November
1972
291
Theodore A. Bickart and Donald W. Melvin Short-circuit Admittance Matrix By assuming that T(s) is a short-circuit admittance matrix, it is implied that the p-vector u is the vector of network port voltages e, and that v is the p-vector-note the q = p-d corresponding port currents. Consider the network block diagram in Fig. l(a); NR is a (p +n)-port grounded, resistive subnetwork and No is an n-port grounded, capacitive subnetwork. Denote the short-circuit parameter equations of NR as
(‘3) where e, and i, denote the n-vectors of voltages and currents at the ports common to NR and No. The relation imposed by No on e, and i, will be expressed as i, = - Ce,. Substitute
(7)
(7) in (6). The result, with e, = u, is = v and e, = x, yields (5)
(a)
FIG. 1. Network block diagrams.
292
Journal of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports if and only if C is non-singular
and
Open-circuit Impedance Matrix By assuming T(s) is an open-circuit impedance matrix, it is implied that the p-vector u is the vector of network port currents i, and that v is the p-vector-again note that q = p-of corresponding port voltages e,. Consider the network block diagram in Fig. l(b) ; N, and No are specified as in the previous case. The relationship between es, e, and i,--implied by the resistor R in the block diagram-is e, = e,+Ri,,
(9)
where R = diag{r,, r2, . . ., rJ and ri 2 0 for i = 1, . . .,p. Thus there must be a series resistor at each port. Assume G,, is non-regular. Then, in (6), solve for e, and i, in terms of i,Y and e,. Combine that result with (9) ; the expressions achieved are e,- = [R + G;l] i, - G&l Gsce,, i, = G, G&l is -I-[G,, - G, G;l G,,] e,. I
(10)
Substitute (7) in (10). The result, with i, = u, es = v and e, = x, yields (5) if and only if C is non-singular and D = R+G,-,l, - C = G;l G,,, (11)
- CB = G,, G&l, - CA = G, - G,, G$ Get. 1
The elements of R can always be specified such that G,-,’ = D -R is nonsingular; assume such an R is chosen. Then, as previously assumed, G,, is non-singular. Furthermore, (11) yields G=
[D -RI-l [ - CB[D -R-J-l
-[D-RI-lC -CA+CB[D-RI-lC
1 *
(12)
Short-circuit Transfer Admittance Matrix The assumption that T(s) is a short-circuit transfer admittance matrix implies that the p-vector u is a vector of source port voltages, denoted e,, and that the q-vector v is a vector of response currents, denoted i,, with the response ports shorted-that is, with e, = 0. Consider the network block diagram in Fig. l(c) ; NE is a (p + q + %)-port grounded, resistive subnetwork and No is as specified previously. Denote the short-circuit parameter equations of NR as
F;]=[s
Vol. 294,
No. 5, November
1972
5;
$1
E]=G
(13)
293
Theodore A. Bickart and Donald W. Melvin Substitute (7) in (13) and set e, = 0. The result, with e, = u, i,. = v and e, = x, yields (5)-the expression for i, is superfluous-if and only if C is non-singular and G,, D I -CB
G = The unspecified submatrices
GsT G,, G,,
G,, C -CA
(14)
of G are arbitrary.
Voltage Gain Matrix By the assumption that T(s) is a voltage gain matrix, the p-vector u is a vector of source port voltages, denoted e,, and the q-vector v is a vector of response port voltages, denoted e,, with the response ports open-that is, with i, = 0. The network block diagram is shown in Fig. l(c) ; NR and No are as specified in the previous case. Assume G,, is non-singular. Then, in (13) with i, = 0, solve for e, and i, in terms of e, and e,; the expressions derived are e, = - G$ G, e, - Gil G, e,, 4 = [G, - GCTG2 G,I e, + [G, - G,, G2
G,l e,. 1
(15)
Substitute (7) in (15). The result, with e, = u, e, = v and e, = x, yields (5) if and only if C is non-singular and -D
= Gil G,,,
- C = G,-: G,,, (16)
- CB = G,, - G,, G;;l G,,, - CA = G,, - G,, G;;’ G,,. Under the assumption that G,, is non-singular, G=
G,, -G,,D I -CB-G,,D
G G; Gc,
I
.1
(16) yields
G% -G,,C -CA-G,,C
(17)
Except for G,,, which must be non-singular, the unspecified submatrices of G are arbitrary. Note: G,, G,, G,, and G,, depend on G,, and G,,. Current Gain Matrix By assuming T(s) is a current gain matrix, it is inferred that the p-vector u is the vector of source port currents, denoted is, and that the q-vector v is the vector of response port currents, denoted i,, with the response ports shorted. The network block diagram is shown in Fig. l(c) ; NR and No are as specified for the previous two cases. Assume G,, is non-singular. Then, in (13) with e7 = 0, solve for i, and i, in terms of is and e,. When (7) is substituted in the derived expressions, the result with i, = u, i, = v and e, = x, is (5) if and only if C is non-singular
294
Journal of The Franklll Institute
Synthesis of Active RC Multiport Networks with Grounded Ports and G = i
-
G.ss DG, CBG,
Gs, G,, G,,
Gsc C+DG,, - CA - CBG,
(18)
1 .
Except for G,, which must be non-singular, the unspecified submatrices of G are arbitrary. Note: G,, G,,, G,, and G, depend on G, and G,,. Open-circuit
Transfer Impedance Matrix
By the assumption that T(s) is an open-circuit transfer impedance matrix, the p-vector u is a vector of source port currents, denoted i,, and the q-vector v is a vector of response port voltages, denoted e,, with the response ports open. The network block diagram is shown in Fig. I(c) ; N, and No are as specified in the previous three cases. Assume G,, and G,, are non-singular. Then, in (13) with i, = 0, solve for e, and i, in terms of is and e,. When (7) is substituted in the derived relations, the result, with is = u, e, = v and e, = x is (5) if and only if C is non-singular and G
GSS - G,, DII - G,, D]-1 Gss
G;;
- [CB + G,, D] [I - G,, D]-l G,,
G,,
G,, -G,,C-G,,D[I-G,,D]-’
1(19
[Gs, + Gs, Cl
G=
- CA - G,, C - [CB + G,, D] [I - Gs, Dl-l
[
.
Ws,+ Gs,Cl
In addition to the requirement that G,, and G,, be non-singular, G,, must be selected such that [I - G,, D] is non-singular. It is obvious that such G,, exist, since GST= 0 is one such possible choice. The remaining, unspecified submatrices of G-G,, and G,,--are arbitrary. Note: G,,, G,, G,, and Gee depend upon all of the other submatrices of G.
V. Restricted
Procedure:
Phase
Three
Thus far, the n x n capacitance matrix C is only constrained to be nonsingular. Therefore, it can be assumed that C is diagonal, with positive elements on the diagonal. By this assumption, the grounded n-port No contains n grounded capacitors-a characteristic of the realization of T(s) that was to be established-with one capacitor connected to each port terminal, as illustrated by the schematic in Fig. 2. That the realization requires at least n capacitors follows from the fact that C would be singular if No contained fewer than n capacitors. The sub-network Nn in Fig. l(c), consisting of resistors and inverting, grounded voltage amplifiers, is assumed to have the structure shown in Fig. 3, where flR is a 3(p +q+n)-port grounded sub-network of resistors.
VoL294,No. 5,November1972
295
Theodore A. Bickart and Donald W. Melvin The sub-network NR in Figs. l(a) and (b) is obtained by removing the distinct response ports. In that case i?& will be a 2(93+%)-port sub-network. Cl
IO
c2
20
q I ,,C”
no
FIG.
I\
’
2. Capacitive subnetwork Ai,.
t =
t-
%I ‘”
esp+oe,l +o era+-
+ecn
,
&
FIG. 3. Block diagram of N,.
Denote the short-circuit
is i, 4 . lk . lh
296
L % -?+ec1
parameter equations
% [1
of iVRas
e,
=e
ek
eh
Journal
of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports where the diagonal submatrices of c! are symmetric and where [eke;]’ and [i;ijJ’ denote 2(p +q+n)-vectors of voltages and currents at the ports to which the outputs of the amplifiers are connected. Now
(21) where K = diag{k,, . . ., kp+p+n} and I-I = diag(h,, . . ., hp+q+n}. Note: Because the amplifiers are inverting amplifiers, each of the ki and hi is non-positive. Equations (20) and (21) together yield
I iI HK
By comparing
this result with (13) it is found that G = Q+G+K+G_HK,
(23)
where and It will now be shown that i$,, K and H can be determined such that i? is hyperdominant-a necessary and sufficient condition for the 3(p +q +n)port NR to be realizable without internal nodes (21). Select i$, such that it is hyperdominant, with the sum of the elements in a row being strictly positive if the corresponding row of G - Gp has one or more non-zero elements. Let G+,
= P+M,
(24)
where P (M) contains the non-negative (non-positive) elements of G - G9. If G is to be hyperdominant, the elements of [4+ and G__must be non-positive. Lastly, the elements of H and K are non-positive, thus (23) and (24) combine to give
G+K=P
(254
B_HK=M.
Wb)
and
Clearly, by making the 1ki 1and / hi 1sufficiently large the non-zero elements of G-G+ and of G_ can be made as small in magnitude as is necessary to make the first p + q + n rows of G hyperdominant. This is possible because the sum of the elements of a row of Gp is strictly positive if the corresponding row of G - G9 has one or more non-zero elements. If Kt[Ht] denotes the pseudo inverse (22) of K[H], then (25) yields G, = PKt,
Vol. 294, No. 5, November
1972
(26s)
297
Theodore A. Bickart and Donald W. Melvin provided
the consistency
condition
P[I -K+ K] = 0 is satisfied, and
G_ = MK+H+
(26b)
provided the consistency conditions MII - K+ K] = 0 and MK+[I -H+ H] = 0 are satisfied. Note: The consistency conditions can be satisfied. One way of doing so is to select the ki and hi to be strictly negative; then K[H] will be non-singular and K+ = K-l[H+ = H-l]. It then follows that [I-K+K] = 0 and [I-H+ H] = 0. Now, let
be diagonal with the value of each diagonal element selected such that the sum of the elements in each of the last 2(p +q +n) rows of fi is zero. This condition means that there will be no shunt resistors at the ports to which the outputs of the controlled sources are connected. This completes the specification of B.
VI.
Design
Criteria
It has been shown that, for any of the six cases considered, an active RC multiport network realization of T(s) exists. However, it has been established only that 2( p + q + n) amplifiers-the number of diagonal elements k, and h, of K and H-are sufficient. It will now be shown that 2(p + n) amplifiers are sufficient and that, in some cases, not even that many are needed. Consider (25). If j is the index of a zero column of M [both M and P], set hi = 0 [both kj = 0 and hj = 01. With these being the only kj and hj set equal to zero, the consistency conditions for equations (26) are satisfied. This is a consequence of the fact that K+ = diag(kf, . . . . k&+q+n}, where ki = l/4 if ki # 0 and kl = 0 if k, = 0, and that H = diag{h,t, . . . . hA+*+,,}, where hf = l/h, if hi # 0 and hf = 0 if hi = 0. It is evident that the actual number of amplifiers needed is equal to the number of non-zero k, and h,. Now since the number of capacitors in any realization, obtained by the procedure developed herein, is already a minimum, it is reasonable to seek that particular realization requiring a minimum number of amplifiers. Considerable design flexibility is available in the quest for such a minimal realization; for example, the diagonal elements of C can have any positive value, and any nonsingular Q can be used in (A, B, C} = {Q-l A, Q, Q-l B,, C, Q). Bounds on the actual number of amplifiers needed will be established case-by-case. Short-circuit Admittance
Matrix and Open-circuit Impedance
Matrices
When T(s) is a short-circuit admittance or an open-circuit impedance matrix, there are no distinct response ports. Therefore, 2(p +n) is the maximum number of amplifiers needed.
298
Journal of The Franklin Institute
Synthesis of Active RC Multiport Networks with Grounded Ports Short-circuit Transfer Admittance Matrix The subscripts p and m will be used in the following way: Let S be an arbitrary matrix ; then S = S, + S,, where S, (S,) contains the non-negative (non-positive) elements of S. Now, consider G as specified in (14). Let G,, = Dk, G,, = [ - CB]&, and GcT = CL; then Gs, D [ -CB
G =
Dk G,, C;,
[-CBli, C -CA
.
(27)
1
Let G, and G,, be diagonal and let S be symmetric, G,,
Dk G CL
Gp = [ [-%1,
such that
II-CBI:, C s”
(29) 1
is hyperdominant, with the sum of the elements in a row being strictly positive if the corresponding row of G+
0
0
[ [-%],
O 0
=
has one or more non-zero elements.
C, -CA-S
If follows from (29) that
0 P=
D, [-CB],
0
0
0 0
[-CkS],
and M=
0 0 0
I
(29)
1 1
0
0 0 0
0 0 [-CA-S],
(39)
1
(31)
*
Because columns 1 through p of M and p + 1 through p + Q of both M and P are zero, hi = 0 for i = 1,..., p+q and ki = 0 for i =p+l,..., p+q. This implies that p + 2q of the possibly 2(p + q + n) amplifiers that might have been needed are not required. Therefore, no more than pf2n amplifiers are needed to realize T(s) as a short-circuit transfer admittance matrix. Voltage Gain Matrix Consider G as specified in (17). Let G,, = 0, G,, = 0 and G,, = 0 ; then G=
Gss -G,,D i -CB
0 G,, 0
Let G,, be diagonal, let G,, be diagonal symmetric, such that
0 -G,,C -CA
1 .
and non-singular
(32) and let S be
(33)
Vol.294,No.5,November 1972
299
Theodore A. Bickart and Donald W. Melvin satisfies through i=p+l needed. Again, G,, = [ -
all previously imposed properties. It follows that columns p + 1 p + q of both M and P will be zero. Hence, hd = 0 and k, = 0 for , . . .,p + q. As a consequence, no more than 2(p +n) amplifiers are consider G as specified CB]; and G,, = 0 ; then,
in (1’7). Assume
G,, 0 -CB [
G=
0 G,, 0
G%? [ [-:B],
.1
[-CB]:, -G,,C -CA
Let G,, be diagonal, let G,, be diagonal symmetric, such that i$ =
D = 0 and let G,, = 0,
0 G 0”
and non-singular
1
[-CB]:, 0 S
(34) and let S be
(35)
satisfies all previously imposed properties. It follows that columns 1 through p of M and p + 1 through p + q of both M and P are zero. Hence, no more than p + 2n amplifiers are needed. Current Gain Matrix Consider G as specified in (18). Let G,, = 0, G,, = 0 and G,, = 0 ; then, G =
G, DG,, [ -CBG,,
0 G,, 0
0 C -CA
1.
(36)
Let G,, be diagonal and non-singular, let G,, be diagonal and let S be symmetric, such that Gp in (33) satisfies all previously imposed properties. It easily follows that no more than 2(p + n) amplifiers are needed. Open-circuit Transfer Impedance Matrix Consider G as specified in (19). Let G,, = 0, G,, = 0 and G,, = 0; then,
G=
G,S - G,,DG,, i -CBG,
0
0
G,, 0
- G,,C -CA
I-
(3’)
Let G,, and G,, be diagonal and non-singular and let S be symmetric such that Gp in (33) satisfies all previously imposed properties. It then follows that no more than 2(p + n) amplifiers are needed.
VII.
Unrestricted
Procedure
In treating the unrestricted case, the source ports are divided into four the response ports are divided into two groups groups--+, $22,ss and s,-and
300
Journal of
The Franklin Institute
Synthesis of Active RC Multiport Networks with Grounded Ports -rl and r2. The network is excited by voltages (currents) at the ports in groups s1 and s3 (& and sJ. The response of the network is the voltages (currents) at the ports in groups & and r2 (sl and rl) with the ports in group r2 (rr) open (shorted). The implication is that
u=[g]
and
v=[i],
and that e,, = 0 and i,* = 0. A block diagram of the network is shown in Fig. 4; N, and No are as specified previously. Under relatively minor restrictions on some submatrices of the short-circuit conductance matrix, G, for NE, if3 possible to
4, 0 + i 22 0 +
R +
i
9 0 + Q,
Q2 is4
ic h
+
0 es*
+ Q3
NC
3 0 +
=34
ec
i
3 ert
\; er2
I
FIG. 4. Network block diagram.
solve for isI, es2, ih, era and i, in terms of eS1,is,, e,,, is, and e,. The state equations resulting when the relations i, = - C!hcand e,-, = es,+ Ri,,, where R is diagonal with positive diagonal elements, are incorporated in the derived expressions, can be compared with the state equations in (5) to establish constraints on some of the submatrices of G. The task is conceptually simple, but the resulting expressions are too expansive to present here. By the validity of the restricted synthesis procedures previously established the validity of the procedure just described can be inferred. It is also possible to infer that no more than 2( p + n) amplifiers are needed to realize T(s). However, this bound is an immediate consequence of the more apparent fact that the synthesis procedure can be implemented such that columns other than the first p and the last n of both M and P are zero,
Vol. 294, No. 5, November
1972
301
Theodore A. Bickart and Donald W. Melvin when the short-circuit parameter equations are constrained at the outset to be
e81 e58 e%¶
e54 e,, e
r2
. % The synthesis procedure
and associated results imply Theorem
I.
Theorem I. Any (I x p matrix T(s), of real rationa functions of the complex frequency variable s, can be realized as a multiport network function of an active RC multiport network containing exactly n = degree {T(s)} grounded capacitors and at most 2(r, +n) inverting, grounded voltage amplifiers. In addition, the ports will be grounded. Note: When information about the port variables is available, the bounds on the number of ampli~ers needed may change. Such is the case, for example, when the network function matrix being realized is a short-circuit transfer admittance matrix; recall in that case that 2, + 2n is the bound on the number of amplifiers needed. VIZ& Amplifier
Gain
Selection
The synthesis procedure was established on the observation that amplifier gains exist such that the equations in (25) have solutions corresponding Before turning to to which the rows of [Gp i?$+_a_] are h~erdom~ant. ihustrative examples of the procedure, some criteria for choosing the amplifier gains will be described. Let .X(S) denote the set of indices of the non-zero k&hi) and 9)(A) denote the set of indioes of the rows of P(M) having one or more non-zero elements. Let oi 2 0 denote the sum of the elements in the ith row of QF. Let vi 2 0 (pi > 0) denote the sum of the elements in the ith row of P( -RI). Choose real constants wi such that wi = 0 if i EA% - (9’&), 0 < wa < 1 if i~9’nM and wi = I if iEg--(9’&). Now consider the case k:d = k for i ES and h, = h for i ES’?. It can be shown that if (39a) and (39b) then the rows of /j$,a,fi_] will be hyperdominant. In a particular situation it may be desirable for all the amplifiers to have the same gain. In that case k, = k for i E Y and hi = k for i E%. It follows
302
Journal
of The Frsnklin Institute
Synthesis of Active RC Multiport Networks with Grounded Ports from (39) that k should be chosen to satisfy the inequality - k 2 max
[max cEB (~TJ~~wi)l,l~~r~dul(l-wi)l)~~. II
(40)
Suppose hi = - 1 for i E&. Then, for each i E% the sub-network NR contains a cascade of two amplifiers which, as illustrated in Fig. 5, can be
5. Differential output amplifier.
FIG.
replaced
for VEX.
by a single differential output amplifier. In addition, suppose k, = k Then, it can be shown that k must be chosen such that - k 2 iyU%Urri +r-~i)/c~l.
(41)
It is sometimes possible to select the ki and hi such that the rows of &(4+Q_] are hyperdominant with the sum of the elements in each row being equal to zero. Note: For each row having elements which sum to zero, one less resistor is needed than would otherwise be the case. Now let sp(s,) be the vector having a, o( if i E 9’(ci( 1 - wi) if i E A) and 0 otherwise as its ith element, and let kt(ht) be the vector having k+(ht) as its ith element. Then, if the equations Pkt = -So Wa) and MKtht = -s, are consistent, the non-zero sponding elements of
(42b)
ki and hi will be the reciprocals
of the corre-
kt = -ptsp
(434
ht = - [MKt]t s,.
(43b)
and For reasons given previously, consider the case h, = - 1 for i EX. if the equation [P-M]kt = -[s~+s,] is consistent, elements of
the non-zero
ki will be the reciprocals
(44)
of the corresponding
kt = - [P - M]t [So + s,]. IX.
Then
(45)
Examples
Several features of the synthesis procedure following examples.
Vol. 294, No. 5, November 1972
are illustrated
in the three
303
Theodore A. Bickart and Donald W. Melvin Example 1 The 2 x 2 matrix
T(s) =
s+l
[ ,I1
will be realized as the open-circuit is not regular at s-co, let
S
1
impedance
matrix of a 2-port. Since
T(s)
8 = z/(1-2). Then
4(1-4 W)
l/(1 -4
= [ (-1+22)/(1-z)
2/(1--x)
I
will be realized, after which each capacitor having admittance cz will be replaced by a capacitor and a resistor in series, having admittance cs/(s + 1). The Ho-Kalman algorithm established the state equations which follow: j, = [l]x+[l eg=[
l]i,,
I:]x+[
1:
“l]i,.
The realization will continue with this set of matrices; those associated with equivalent networks will not be sought. Now, let c = [l]. Next, since
D is non-singular, set R = 0 in (12). Then by (12) -1
0
2
-1
1
1 -1
1
-1
G=
-1
1 .
The matrix GP must be selected next. Inspection of G shows that GP cannot be selected such that G-G, has columns of zeros or, next best, non-negative elements, which would thereby reduce the actual number of amplifiers needed to fewer than 2(r, + n) = 6. Set +[
!l
%
Y].
Then, from (24) it follows that 0
P=
304
2 0
0 0 1
0 1 ] 0
and
M=[
-3
‘3
p3].
Journal of The Franklin
Institute
Synthesis of Active RG Multiport Networks with Grounded Ports The realization will be established such that each amplifier has the same gain; therefore (40) will be invoked. Now, i
Oi 71i Pi
1
1
0
2 3
2
3
1
1
3 3 3
.
Set wi = 0 for i = 1 and wi = 4 for i = 2,3. Then by (40) - k > max {[max (0,3,2)],
[max (3,3,6)]f}
= 3.
With (II= - 3, (26) yields e+=[
+
!i
+]
and
i?_=[”
2:
!+I.
It is easily verified that the p + n = 3 rows of [CD i!+ B-1 are hyperdominant. Therefore, 0 composed according to the guidelines previously set forth will be hyperdominant. The realization of T(s), based on the realization of a, is shown in Fig. 6. Note that, as indicated previously, gR has no internal
FIG. 6. Example
1: Realization of T(s).
nodes. Note also that a capacitor and a resistor are in series at each capacitive port of NR. Example 2 The 1 x 1 matrix T(s) = [l/V
+ J(2) s + I)]
will be realized as a voltage gain matrix. Note: The network, once realized, will be a 2-pole Butterworth filter. The Ho-Kalman algorithm establishes
Vol.
294
No.
5, November
1972
305
Theodore A. Bickart and Donald W. Melvin the state equations which follow : 0
ir=
1
-l-J2
[ e, = [l
‘0 lesy I
IL
0] x
[0] e,.
The state equations for equivalent networks will not be sought. Therefore, by (17), this set of matrices {A, B, C, D} together with G,, = [II, G,, = yields
G,, = [Ol,
I ;’ 1
and
Gg, = 10 C=
-
[a 0 1 0
1
1
0
0
-1
0
1
-1
0
0
-1
1
-&
-1
0
1
J2
1
0
0
-1
0
1
-1
0
0
-1
2
-2
-1
0
-a
G=
11, G,, = PI,
II
Set
ilp =
3
t’2
then
P= [
0 0
0 0
have three-the same amplifiers will actually The realization will ki = k for i E X = (3).
2 0 0
“]and*=[iii;i]
three-columns of zeros. This means that only two be needed. be completed with hi = - 1 for i EZ = (3) and with Now, i
)
oi
7ri j.L$
1
0
0
0
2 3
0
0 0
0
4
,,2ep
3
.
1 0
Therefore, by (41) - k>,max [i, 5/(4,/2-
306
5)] = 5/(4J2 - 5)...762.
Journal
of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports With k = - 8, (26) yields
fi+-[”
_Li]andfi-=[ii
i&i].
After composing a according to the guidelines previously developed, a schematic for T(s) can be drawn; it is shown in Fig. 7. A differential output
1
er,ofi_45
h3
32
t
-L
-r
FIG.
7. Example
2: Realization of T(s).
I
amplifier is shown, rather than the cascade of the two amplifiers k, = - 8 and h, = - 1. Note: The 1 mho resistor in series with the response port terminal can be removed since the network realizes T(s) with the response port open. Example 3. The 1 x 2 matrix
T(s) = [(s + $)/(s + 1) (2s - l)/(s + l)] will be realized as the network function v = [ea] and i, = [O]. The Ho-Kalman equations which follow :
of a network for which u = [ire,], algorithm establishes the state
(46)
The unrestricted procedure must be used. The network structure will be that shown in Fig. 4. The short-circuit parameter equations conforming to
Vol.
294,
No.
5, November
1972
307
Theodore A. Bickart and Donald W. Melvin those in (38) are
1[I e1
=
e2
Q
.
e3 e4
Note: e, and e2 are associated with the source ports-groups s3 and s4, respectively-e, is associated with the response port-group r,--and e4 is associated with the capacitive port. The solution for e3 and i, in terms of e,, i, and e4, with i, = 0, is e3 = g& g3, el - 62 g3, g2 i2 - g2 g3, e4, t4 = g,, el + g4, g2 i2 + g4,e4. Set i, = -G4, which implies that C = [l]. Then, comparison with (46), when x = [e,], yields L-92
g31
-sz
932
sa
b41 g42 s31 = I3
= [l
31
21,
and
L--&i
of that result
g341 =
PI?
[g441= PI.
Let gn = 1, g,, = 1 and g,, = ) ; then 1000
Set
Note that &, was chosen such that amplifiers-k, and h,-will be associated with the response port contrary to the practice followed thus far. This choice results in fewer amplifiers-4 vs. 6-than would be the case had GP been set eaual to .
[ 010 01000010 0 01 0 which yields k, and h, both zero.
308
1
Journal of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports The realization will be completed i&=lcforiEZ-=(1,2,3). Now
Therefore,
with hi = - 1 for i E%
i
a,
ri
Pi
1 2
f Q
t B
0 0
3 4
;;i
= (3) and with
by (41) -kamax(*,
with k = -5,
1,$,5) = 5
(26) yields
After composing B, a schematic for T(s) can be drawn ; it is shown in Fig. 8, with a differential output amplifier rather than the cascade of two amplifiers Ic, = - 5 and h, = - 1.
r e,:
+
e4
FIG. 8. Example 3: Realization of T(s).
Vol. 294, No. 5, November 1972
309
Theodore A. Bickart and Donald W. Melvin X.
Concluding
Discussion
It has been shown that any q x p multiport network function T(s) can be realized as an active RC multiport network with the ports grounded. The network will contain a minimum number of capacitors-n = degree(T(s)}and at most 2(p + n) inverting, grounded voltage amplifiers. The fact that all the capacitors and amplifiers are grounded is a distinct advantage if the network is to be fabricated as an integrated circuit. This is an advantage which is not exhibited by a number of active RC synthesis procedures. For example, in the short-circuit admittance matrix synthesis procedures proposed by Sandberg (2), Even (15) and Goldman and Ghausi (12,16) the capacitors will not be, in general, grounded.* Also, of the three ingredients in an active RC network-active component, such as a negative converter or voltage amplifier, resistor and capacitor-the capacitor is the one which most complicates fabrication of the network as an integrated circuit. Therefore, it is also an advantage of the procedure proposed that a minimum number of capacitors are required; such is not the case in many procedures, including those just cited.? It has also been established that if the amplifiers hi have gains of - 1, the cascade of the two amplifiers ki and hi can be replaced by a differential output amplifier. The implication of this is that the condition on the number of amplifiers could read: . . . at most p +n differential output, grounded voltage amplifiers. This is a significant observation, since fabrication of a differential output amplifier is likely to be simpler than the fabrication of two inverting amplifiers. It is conjectured that the realization of T(s) will be relatively insensitive to capacitance variations. This conjecture stems from the following observation: The number of capacitors is minimum, therefore, slight variations in capacitance values will not result in the realized network having more characteristic observable frequencies than the designed network. Even though the values of the characteristic frequencies might change somewhat, it is reasonable to suppose that such changes will have less effect on the realized network properties than the appearance of extraneous characterist,ic frequencies. It is hoped that the continuing investigation of this synthesis procedure will provide a quantitative assessment of the sensitivity of selected network attributes and validation of the conjecture. Note, finally, that the synthesis procedure described herein can be reduced rather easily to a computer program which, given the multiport network
* It should be noted, however, that each of the procedures just singled out do have grounded active components; not 2(p + rz) of them, but just p of them in the first case and 2p of them in the latter two cases. 7 The capacitors are grounded and only a minimum number of them are required in the synthesis procedures evolving from the state equations for a network. That is the case, for example, in the procedures described by Mann and Pike (9) and Dewilde et al. (14). Also, see Newcomb (24).
310
Journal of The Franklin
Institute
Synthesis of Active RC Multiport Networks with Grounded Ports
function, will generate C, K, I-Iand i% Such a program is being prepared in APL, after which, a Fortran version will be developed.
References (1) I. W. Sandberg, “Synthesis of X-port active RC networks”, B.S.T.J., Vol. 40, pp. 329-347, Jan. 1961. (2) I. W. Sandberg, “Synthesis of transformerless active N-port networks”, B.S.T.J., Vol. 40, pp. 761-783, May 1961. (3) D. Hazony and R. D. Joseph, “Transfer matrix synthesis using one amplifier but no inductors”, Eighth Midwest Symp. Circuit Theory, June 1965. (4) J. Barranger, “Voltage transfer function matrix realization using current negative immitance converters”, IEEE Tram., Vol. CT-13, pp. 97-98, March 1966. (5) G. Martinelli, “RC transformerless networks embedding nullors”, Alta l%equenza, Vol. 35, pp. 156-162, Feb. 1966. (6) D. Hazony and R. D. Joseph, “Transfer matrix synthesis with active RC networks”, J. SIAM Appl. Math., Vol. 14, pp. 739-761, July 1966. matrix synthesis with active (7) R. D. Joseph and D. Hilberman, “Immittance networks”, IEEE Trans., Vol. CT-13, p. 324, Sept. 1966. (8) D. Hilberman, “Synthesis ) of rational transfer and admittance matrices with active RC common-ground networks containing unity-gain voltage amplifiers”, IEEE Trans., Vol. CT-15 pp. 431-440, Dec. 1968. (9) B. J. Mann and D. B. Pike, “Minimal reactance realization of N-port active RC networks”, Proc. IEEE, Vol. 56, p. 1099, June 1968. matrix realization using operational (10) S. K. Mitra, “Voltage-transfer-function amplifiers”, Elect. Letters, Vol. 4, No. 20, p. 435, Oct. 1968. (11) B. Bhattacharyya, “Synthesis of an arbitrary transfer function matrix using RC and one-ports and operational amplifiers”, Int. J. Electronics, Vol. 26, pp. 5-16, Jan. 1969. (12) M. Goldman and M. S. Ghausi, “On the realization of rational admittance matrices using voltage-controlled voltage sources and RC one-ports”, IEEE Trans., Vol. CT-16, pp. 544-546, Nov. 1969. “Active network synthesis using amplifiers having any finite (13) D. Hilberman, non-finite gain”, IEEE Trans., Vol. CT-16, pp. 484-489, Nov. 1969. (14) P. Dewilde, L. M. Silverman and R. W. Newcomb, “A passive synthesis for timeinvariant transfer function”, IEEE Trans., Vol. CT-17, pp. 333-338, Aug. 1970. (15) R. K. Even, “Admittance matrix synthesis with RC commonground networks and grounded finite-gain phase inverting voltage amplifiers”, IEEE Trans., Vol. CT-17, pp. 344-351, Aug. 1970. (16) M. Goldman and M. S. Ghausi, “Active RC synthesis of admittance matrix with prescribed RC element constraints”, IEEE Trans., Vol. CT-18, pp. 299-302, March 1971. (17) D. Melvin and T. Bickart, “P-port active RC networks: Short-circuit admittance matrix synthesis with a minimum number of capacitors”, IEEE Trans., Vol. CT-l& No. 6, pp. 687-592, Nov. 1971. (18) E. Hille, “Analytic Function Theory”, Vol. I, pp. 46-58, Ginn, Boston, Mass. (19) R. E. Kahnan, “Irreducible realizations and the degree of a rat,ional matrix”, J. SIAM Appl. Math., Vol. 13, pp. 526-544, June 1965. (20) R. E. Kahnan, P. L. Falb, and M. A. Arbib, “Topics in Mathematical Systems Theory”, pp. 242, 288-294, McGraw-Hill, New York, 1969. (21) P. Slepian and L. Weinberg, “Synthesis applications of paramount and dominant matrices”, Proc. 1958 N.E.C., Vol. 14, pp. 611-630, 1958.
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Theodore A. Biclcart and Donald W. Melvin. (22) H. P. Decell, Jr., “An application of the Cayley-Hamilton theorem to generalized matrix inversion”, SIAM Rev., Vol. 7, pp. 526-528, Oct. 1965. “The synthesis of linear dynamical systems from prescribed (23) D. C. Youla, weighting patterns”, J. SIAM Appl. Math., Vol. 14, pp. 527-549, May 1966. (24) R. W. Newcomb, “Active Integrated Circuit Synthesis”, pp. 80-87, Prentice-Hall, N.J., 1968.
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