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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-21, NO. 3, MAY 1974
REFERENCES
[l] M. A. Maidique, “A high-precision monolithic super-beta
operational amplifier,” IEEE J. Solid-State Circuits, vol. SC-7,
pp. 480-487, Dec. 1972.
[2] P. R. Gray, “A 15-W monolithic power operational amplifier,”
IEEE J. Solid-State Circuits, vol. SC-7, pp. 474480, Dec. 1972.
[3] G. Meyer-Brotz and A. Klev. “The common-mode reiection of
transistor differential amplil%rs,” IEEE Trans. Cirwa’t Theory,
vol. CT-13, pp. 171-175, June 1966.
[41 G. Erdi, “Common-mode rejection of monolithic operational
amplifiers,” IEEE J. Solid-State Circuits (Corresp.), vol. SC-5,
pp. 365-367, Dec. 1970.
151 R. J. Widlar, “Design techniques for monolithic operational
amplifiers,” IEEE J. Solid-State Circuits, vol. SC-4, no.
-^ 184-191.
Aug. 1969.
[61 T. M. Frederiksen, W. F. Davis, and D. W. Zobel “A new
current-differencing single-supply operational amplifier,” IEEE J.
Solid-State Circuits, vol. SC-6, pp. 340-347, Dec. 1971.
[71 HA-2600/2602/2605 Operational Amplifier Data Sheet, Harris
Semiconductor Corp., June 1971.
[81 J. E. Solomon, W. R. Davis, and P. L. Lee, “A self compensated
monolithic operational amplifier with low mput current and high
slew rate,” ISSCC Digest Tech. Papers, pp. 14-15, 1969.
191R. W. Russell and J. E. Solomon: “A high-voltage monolithic
operational amplifier,” IEEE J. Solid-State Circuits, vol. SC-6,
pp. 352-357, Dec. 1971.
HOI “A low drift, low noise monolithic operational amplifier for low
level signal processing,” Fairchild Semiconductor, Appl. Brief 136,
July 1969.
1111W. E. Hearn, “Fast slewing monolithic operational amplifier,”
IEEE J. Solid-State Circuits, vol. SC-6, pp. 20-24, Feb. 1971.
[12] H. C. Lin. “Comoarison of invut offset voltage of differential
amplifiers using bipolar transistors and field-elect transistors,”
IEEE J. Solid-State Circuits (Corresp.), vol. SC-5 pp. 126-128,
June 1970.
[1311 B. A. Wooley, S. J. Wong, and D. 0. Pederson, “A computeraided evaluation of the 741 amplifier,” IEEE J. Solid-State
Circuits, vol. SG6, pp. 357-366, Dec. 1971.
[I41 T. J. van Kessel, “An integrated operational amplifier with a
~;ov;l HF behavtour,” ISSCC Digest Tech. Papers, pp. 22-23,
[15] H. Krabbe, “Stable monolithic inverting op amp with 130 V/ps
slew rate,” ZSSCC Digest Tech. Papers, pp. 172-173, 1972.
[16] R. C. Dobkin, “LM 118 op amp slews 70 V/j~s,” National
Semiconductor Linear Brief 17, Aug. 1971.
[17] F. D. Waldhauer, “Analog integrated circuits of large bandwidth,”
in 1963 IEEE Conv. Rec., Part 2, pp. 200-207.
[18] R. J. Apfel and P. R. Gray, “A monolithic fast-settling feedforward operational amplifier using doublet compression techniques,” ISSCC Digest Tech. Papers, pp. 134-155, 1974.
[I91 P. C. Davis and V. R. Saari,. “High slew rate monolithic operational amplifier using compattble complementary PNP’s,” ZSSCC
Digest Tech. Papers, pp. 132-133, 1974.
Maximum Power Transfer in n Ports With
PassiveLoads
M. VIDYASAGAR,
MEMBER, IEEE
Abstracr-The
problem of drawing maximum power from an n port by
suitably terminating it with another passive n port is studied. Methods
are derived for determining sets of passive impedance matrices which are
“optimal” in the sense that they draw maximum power from a given
n port. It is shown that, in all nontrivial cases, if there exists any (passive
or otherwise) optimum-load impedance matrix, then there exist, in fact,
infinitely many passive optimum-load impedance matrices.
INTRODUCTION
T
HE maximum power-transfer
problem for it ports has
been studied by several authors [I], [2]. Consider the
network shown in Fig. 1, and suppose the vector of source
voltages e and the source impedance matrix Z, are fixed.
The problem is to determine a load impedance matrix Z,
which has the property that it draws the maximum possible
power from e.
The best results available to date are due to Desoer [a].
His results are briefly summarized here in the interests of
completeness. In order to obtain a simple solution to this
problem, we treat i as the independent variable instead of
Fig. 1.
2,. Thus, in terms of i, the voltage u is
v = e - Z,i
while the power delivered
(1)
to the load is
P, = 3 Re (u*i) = $(u*i + i*u)
= *(e*i + i*e - i*(Z, + Z,*)i)
(2)
where (*) denotes conjugate
transpose.
Since P, is a
quadratic form in i, it is very straightforward
to characterize
“optimal”
i. The details can be found in [2].
Case I): Z, + Z,* is positive definite. Let
i, = (Z, + Z,*)-‘e.
Manuscript received March 12, 1973; revised August 6, 1973. This
research was supported by the National Research Council of Canada
under Grant A-7790 and by NASA under Grant NGR-05-007-122. This
work was performed at the System Science Department, University of
California? Los. Angeles,
.
^Calif. .1 ne autnor ISon leave from the Department of Electrical Engineering,
Sir George Williams University, Montreal, Canada.
Any Z resulting in this value i, for i is “optimal”
in the
sense that it draws maximum
power. Note that, by
Kirchhoff’s
voltage law, we must also have
Z,i + Z,i = e.
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(4)
328
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, MAY 1974
Thus, from (3) and (4), we get
a single frequency (say 0). Thus, in conditions a), b), and
c), it should be understood that Z and Z, are, in fact,
Z,i, = Z,*i,
(5) Z(jo) and Z,(jw). In the caseof resistivey1ports, of course,
where i, is given by (3), and any Z, satisfying (5) is optimal. conditions a), b), and c) characterizethe maximum powerCase 2): Z, + Z,* is indefinite. No optimal i exists in transfer problem at all frequencies.
this case.
Secondly,for the purposesof this paper, we define a load
Case3~): Z, + Z,* is positive semidefinite,and e belongs impedance matrix Z to be passive if b) holds. Physically,
to the range of Z, + Z,*. In this case,the equation
this condition ensuresthat if Z is driven by a sinusoidal
(vector) current source of frequency w, then the net energy
e = (Z, + Z,*)i
(6) flow is into Z. Now, as o is varied and the maximum
has an infinite number of solutions for i. Any Z, satisfying power-transfer problem is solvedfor all w, then the resulting
optimal Z(jw) will satisfy b) for eacho. However, this does
Z,i = Z,*i
(7) not mean that all poles of Z(s) lie in the closed left halfwhere i is any solution of (6), is optimal.
plane. It does not appear to be possible to overcome this
Case 3b): Z, + Z,* is positive semidefinite, and e does problem in a simple way.
not belong to the range of Z, + Z,*. No optimal i exists in
CONSTRUCTIVEPROCEDURES
this case.
Consider first condition a). The set of all matrices Z in
PROBLEMFORMULATION
C “xn satisfying a) (for fixed i,) can be characterized very
While Desoer’s results are very comprehensive,they do simply. Let IV(&) denote the (n’ - n) dimensional subspace
not include any specialconsiderationsabout the passivity or of C”“” consisting of all matrices that map i, into the zero
otherwise of the load impedancematrices. From a practical vector. Then, the set S of all Z satisfying a) is given by
point of view, it is of interest to draw maximum power from
s = {ZE cnxn: Z = Z,* + Z,, Z, E N(i,)>.
(8)
the given II port using a passiveload, especiallyfrom design
considerations. The results of this paper are addressedto The proof of (8) is elementary. First of all, suppose Z
the maximum power-transfer problem for IZports when the satisfies a); then, Z, = Z - Z,* belongs to N(i,). On the
load impedance matrix is required to be passive, and it is other hand, supposeZ, E IV(&); then if Z = Z,* + Z,, we
further required that the network obtained by inter- have Zi, = Z,*i, + Z,i,, = Zs*io, so that Z satisfies a).
connecting Z, and Z, be solvable. For our purposes, an Thus the set of all matrices satisfying a) is the linear variety
n x n matrix Z, or equivalently an element Z of C” ‘“, is obtained by translating the subspaceN(i,) by the element
consideredto be passiveif Z + Z* is positive semidefinite. Z,X.
The problem under consideration can be stated as
In order for (8) to be useful, we must be able to determine
follows: suppose e and Z, are given and that we are in the subspaceN(i,) explicitly. This is easily done. First of
Case 1) or Case 3a), so that at least one optimum-load all, note that assumption 2) together with (6), implies that
impedancematrix exists. We wish to determine one or more i, # 0. Thus we can normalize i, to be of norm one, or
matrices Z in C” ’ ” satisfying the following three conditions : equivalently, we can assumethat jli,ll = 1. Let {io,uz, - . *,v,}
be any orthonormal basis for C”. (Such a basis can be
a) Zi, = Z,*i, for some solution i, of (6) (optimality);
easily generatedusing the Gram-Schmidt procedure.) Let
b) Z + Z* is positive semidefinite
(passivity);
T denote the unitary matrix
c) Z + Z, is nonsingular
(solvability).
We remark that, in Case l), Z = Zs* is an impedance
matrix that satisfiesall three conditions, while in Case 3a),
2 = Z,* satisfies a) and b), but not c). In this paper, we
prove much stronger results. We make the following rather
natural assumptions:
1) n 2 2, i.e., the one port caseis excluded;
2) e # 0, i.e., the source voltages are nontrivial.
Subject only to these two assumptions, we show that there
exist, in fact, infinitely many matrices in C”“” satisfying
conditions a), b), and c). Methods of generating such
matrices are also given. Theseresults are particularly useful
in Case3a), when the natural choice Z = Z,* is unsuitable
because it results in an indeterminate network (since
Z, + Z,* is singular).
Some comments are in order regarding the given formulation of the problem. First of all, it should be noted that
we are tackling the problem of maximum power transfer at
*
II :.I
(9)
Then, N(iJ consists of all the matrices of the form TMT*,
where M is any matrix whose first column is identically
zero. Equivalently, IV(&) is the space generated by the
n2 - n matrices
112
0*
77
-
7-.
--
7
v2
3. . .
*
The proof of this last statement is as follows. Let Z, be a
matrix in C” ’ n. Then, the representationof Z, with respect
to the basis{iora2,.. . ,u,} is given by the matrix M = T*Z,T.
Now the condition that Z,i, = 0 simply means that M
must map the vector [lo- . -01’ into the zero vector, or,
equivalently, that the first column of M is identically zero.
Since Z = TMT*, we seethat N(i,,) consists of all matrices
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329
VlLJYASAGAR:POWERTRANSFERIN?ZPORTS
of the form TMT*, wherethe first column of M is identically
zero. Thus we have identified all matrices satisfying a).
Next, consider condition b). Clearly, if 2, and Z, are
passive impedance matrices, then so is Z, + Z,. So if Z,
is any passive element of N(i,), then Z = Z, + Z,* is a
passiveelementof S and, therefore, satisfiesboth conditions
a) and b). (Note that Z,* is passive, since we assumethat
we are in either Case 1) or Case3a).) Thus we seekpassive
elementsof N(i,). Now, supposeZ, = TMT* is an element
of N(i,). Then, it is easy to see that Z, is passive, if and
only if M is passive.One can think of M as the representation of the matrix operator Z, with respect to the basis
{ io,vz, * * - ,v,}. Thus let M be any passivematrix whose first
column is identically zero. (It is clear that there are
infinitely many such matrices. For example, M =
diag {O,CI~,
* * *,a,,}, where Re CL~2 0 for j =’ 2, * . . ,n. In
fact, all such matrices form a convex cone in C”‘“.) Then
Z,* + TMT* satisfiesboth a) and b).
Consider now condition c). Here we are forced to make a
distinction between Cases 1) and 3a). First, suppose
Z, + Z,* is positive definite. We can show very easily
that there exist infinitely many Z E S satisfying a), b),
and c). SupposeM = diag {0,a2; * .,a,} where the aj are
all real and aj 2 0 forj = 2, * * *,n. Then, Z = Z,* + TMT*
satisfiesa) and b). Further, 2 + Z, = Z, + Z,* + TMT*
is the sum of a positive definite matrix and positive semidefinite matrix, whence Z + Z, is itself positive definite,
and therefore nonsingular. Hence, Z satisfiesa), b), and c).
Alternatively, let M be any passive matrix whose first
column is identically zero and consider Z = Z,* + cTMT*,
where c is a positive real number. Then, Z + Z, =
Z, + Z,* + cTMT*. Since Z, + Z,* is nonsingular, so is
Z + Z, for small enough values of c. In fact, [4, p. 5841
Z + Z, is invertible whenever0 I c < 1/(11(Z, + Z,*)-’ 11*
l/MI]). Either way, we see that there exist infinitely many
impedancematrices satisfying conditions a), b), and c),
Now suppose we are in Case 3a), so that Z, + Z,* is
singular. In this case,(6) has infinitely many solutions. Let
i. denote the solution of (6) with minimum norm, and form
the matrix T as before.’ In what follows, we show that,
wheneverM is a block diagonal matrix of the form
positive definite, it follows that y = 0, where x*Px =
PlllXl12 = 0. Sincepi I > 0, lx11 = 0, where x = 0.
Q.E.D.
Lemma 2: Let i, be the solution of (6) with minimum
norm. Then, e*i, > 0.
Proof: Assumption 2) implies that i, # 0, as well as
that Z, + Z,* # 0. Let {w,, . * . ,w,} denotean orthonormal
set of eigenvectors for Z, + Zs*, corresponding to the
eigenvalues {i,J2;~~,&,,0; +*,O}, where ~j > 0 for j =
I,*** ,m I n - 1. Then, since e belongs to the range of
Z, + Zs*, we have
(11)
where the summation in (11) only goesup to m. Now, i, is
given by
Hence,
i,*e = e*i, = i,*( Z, + Z,*)i,
= f
Icj12/Aj > 0.
j=l
(13)
Q.E.D.
Now, to prove our original claim, let
P = T*(Z, + Z,*)T
(14)
and choose Z = Z,* + TMT*, where M is given by (10).
Then, Z + Z, = T(P + M)T*, and Z + Z, is nonsingular, if and only if P + M is nonsingular. However,
note that a) P is positive semidefinite; and b) prl =
i,*(Z, + Z,*)i, > 0, by Lemma 2. Hence, by Lemma] 1,
P + M is positive definite and therefore nonsingular. Thus
we seethat Z satisfiesall three conditions a), b), and c). In
fact, by proceeding as with Case I), one can easily show
that Z = Z,* + T(M + M,)T*, where M is given by (lo),
also satisfiesa), b), and c) wheneverthe first column of M/
is identically zero and the norm of M, is small enough.
Hence, even in Case3a), there exist infinitely many passive
optimal-load impedancematrices.
SUMMARY
In summary, the procedurefor generatingsetsof optimal
passiveimpedancematrices is as follows.
where M, is Hermitian and positive definite, the matrix
1) Obtain the solution of (6) with minimum norm (or the
Z = Z,* + TMT* satisfies conditions a), b), and c). We
unique solution of (6) in caseZ, + Z,* is nonsingular).
prove this by means of two lemmas.
2) Form the orthonormal basis {i0,u2,* * . ,u,} and the
Lemma I: Let P be a Hermitian positive semidefinite
matrix T.
matrix with pr I > 0, and let M be as in (10) et seq. Then,
3) Let M be a matrix of the form (lo), and let M, be
P + M is positive definite.
a matrix whose first column is identically zero. Then,
Proof: Consider the triple product x*(P + M)x. We
Z = Z,* + T(M + cM,,)T* satisfiesconditions a), b), and
wish to show that x*(P + M)x = 0 implies that x = 0.
c) for small enough values of the constant c, including
Partition x as [x1 1y’]‘. Then, x*(P + M)x = x*Px +
c = 0.
y*M,y. SinceP + M is at least positive, x*(P + M)x = 0
implies that x*Px = 0 and y*M,y = 0. Since M, is
ILLUSTRATIVE
EXAMPLES
1 One can easily determine i. using the procedure in [3]. See
specifically Remark (4) on p. 216.
Example 1
Consider the network in Fig. 1, and let
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330
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-21, NO.
7
-1 +j
-2-j
2
-1 +j
2-j
7 + 2j
-2 .
2-j
10 - 2j
-1
[ -2-j
Then, one can easily verify that Z, + Z,* is positive
definite, and that
Z,=&
I;
i, = ($)[l
e=
[1
- 1 01’.
Z,=$
i, = (+)* [I
1 01’;
vg = [O 0 I]‘.
Then, the matrix T becomes
T = [-ig
2+j
-3+j
[ .i
In this case,one can easily verify that Z, + Z,* is positive
semidefinite and that e belongs to the range of Z, + Z,*.
Furthermore, the solution of (6) with minimum norm is
Choose the vectors v2 and v3 as
v2 = (l/J2)[1
:;;
;].
3, MAY 1974
-1
01’.
Let us choose the same T matrix as in Example 1, and let
M = diag {0,2a$}, where a > 0, fi > 0 (as opposed to
a 2 0, /? 2 0 in Example 1). Then, we seethat any matrix
of the form
Z, = Z, + TMT”
-3-j
Let M = diag {0,2a,B}, where a 2 0, j3 2 0. Then, using
the constructive procedures given in the previous section,
any matrix of the form
2+j
l+j
l+j
5-2j
a a 0
a a 0
I[ 1
-j
+
0
0
P
is an optimal-load impedancematrix and satisfiesconditions
a), b), and c).
Z, = Z,* + TMT*
7
a a 0
-1 -j
-2+j
REFERENCES
-1 -j
-1-2j
2+j
+ a a 01
P. M. Lin, “Determinationof availablepower from resistive
l11
-2+j
2+j
10 + 2j
[ 0 0 P1
multiports,” IEEE Trans. Circuit Theory, vol. CT-19, pp. 385-386,
Tnlv 1973
constitutes an optimum-load impedancematrix and satisfies PI C!:Desoer, “The maximum power-transfer theorem for n ports,”
IEEE Trans. Circuit Theory, vol. CT-201 pp. 328-329, May 1973.
conditions a), b), and c).
I31M. Vidyasagar, “Interative minimizatton in Hilbert space of
quadratic functtonals with nonunique minima,” Proc. 8th Aflerton
Example 2
Co&, pp. 210-218, 1970.
I41N. Dunford and J. T. Schwartz, Linear Operntors. Part I. New
Consider again the network in Fig. 1, and let
York: Interscience, 1959.
=A
1
Synthesisof Biconnected Graphs
F. T. BOESCH, MEMBER,IEEE, AND J. A. M. MCHUGH
Absiruct-In
this investigation, it is necessary to distinguish pseudographs (self-loops and multiple lines allowed), multigraph (no self-loops
but multiple lines are allowed), and graphs (neither self-loop nor multiple
lines allowed). The problem of synthesizing graphs, multigraphs, and
pseudographs having a prescribed degree sequence was solved by Hakimi.
He also determined the conditions under which a connected realization
exists In each of these three cases. The case of biconnected (nonseparable)
realizations of a degree sequence was given by Hakimi for the case of
pseudographs and multigraphs. His methods did not apply to the case of
graphs. The triconnected case was solved by Rao and Rao for graphs;
however, strangely enough, the biconnected case apparently remains
unpublished.
We shall show here that the biconnected case can be handled by a
“surgery” technique similar in spirit to the connected case given by
Hakimi. Several remarks concerning the general n-connected case for
Manuscript received May 8, 1973.
The authors are with Bell Telephone Laboratories, Holmdel, NJ
pseudographs, multigraphs, and graphs will be given. We conclude with
a status report on this subject, which includes the most recent work of
Hakimi, Wang and Kleitman, and Bondy.
I. INTRODUCTION
HE basic notation and terminology follows that of
Harary [l]. The problem considered here is the
characterization of graphical partitions (degreesequences)
of 2-connected graphs. If II = (d,,d,, * * . ,d,,) with di 2
d2 2.e. 2 d,, and p 2 3 is the degreesequenceof some
graph, then when does there exist a 2-connectedgraph with
this degree sequence?A similar question, as well as the
question of the existenceof a connected graph, was solved
by Hakimi [2] for the case of multigraphs. Hakimi’s
T
solution
for l-connected
multigraphs
may be easily modified
to hold true for l-connected graphs. However, his proof for
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