On maximal power transfer problem

On Maximal Power Transfer Problem by F. M. REZAt Concordia University, Canada H3G lM8

Department

of Electrical

Engineering,

Montreal,

Quebec,

ABSTRACT : Given a source vector e E c” applied to a specljied n-port of open-loop impedance Z Oterminated in any n-port load of a family (N). It is desired to determine the optimal load Z which draws the maximum active power. The proposed method of solution of this paper employs operator theoretic concepts of functional analysis and inequalities on positive operators. Among other results, it is shown that the maximal active power transfer across Z, terminated in resistive n-ports is obtained by the network with R = -provided R and Z, are members of a normal family of n-ports. The paper encompasses in fulI, the results of some recent contributions, and the proposed new approach invites further generalizations and applications.

I. Introduction

The central problem of maximal power transfer consists of the following. A linear passive time-invariant n-port No described by an n x n complex open-loop non-singular impedance matrix ZO is given. The network N,, is energized at its ports by a complex voltage n-vector e E C” of a fixed sinusoidal frequency, that is, exp (st). Likewise, a class of linear time-invariant n-ports {N} represented by the open-loop impedance matrix {Z} is specified. The problem is to determine a matching n-port N of the family {N} to be connected in a port-to-port (series) with NO, such that the active power dissipated by N is maximal, relative to the family {N}. In the general case, that is, when the class of {N} is not unduly restrictive, the answer is rather simple. The maximal class is described by its open-loop impedance matrix Z and the unique maximizing current vector i,, E c” as given by Ohm’s law in (1) and (2) where * denotes the adjoint operation e = (Z, + Z,*)i,

(1)

ZiO = Zo*iO.

(2)

Analytic difficulties may arise from singularity of the incurred matrices, reciprocity requirement, or when {N} and {N,} are certain restricted types of n-ports. For instance, {N} may be purely resistive or {N,} purely inductive {N,: Z0 =jXO,X,, > O}. Recently, Calvaer (l), motivated by practical circumstances, considered the maximal power transfer problem for the inductive class of (N,} described above. 7 The author is also affiliated with McGill University.

G The Franklin1nstitute0016-0032/88 $3.00+0.00

335

F. M. Reza

He shows that the corresponding optimal active power transfer networks are purely resistive n-ports with R = X,,. In this case, the difficulty for deriving a simple solution, as in (1) and (2), stems from the fact that for Z,, = jX, the matrix Z, + Z$ in (1) becomes singular and e may not belong to its range. Desoer (2) further clarifies the problem by using the spectral mapping concept ; he arrives at the same optimal solution, that is R = X0 > 0. The more general case of Z0 requires more elaborate work and until the present has not been fully resolved. A comprehensive study of the general maximal power transfer problem using methods of functional analysis is due to Flanders (3). Flanders’s results have enriched the topic ; and his paper, which also contains reference to the work of other contributors, merits further exploration. In this paper a solution to the more general form of the maximal power transfer is derived. We also supplement the previous contributions by emphasizing work on variational techniques and functional analysis. The scope of this paper appears to be broader than those currently employed as it deals with a general load Z0 not restricted to be R0 orjX,. This approach tacitly employs the concept of positive operators. Moreover, the requirement of passivity or reciprocity can be more readily examined in this suggested approach. Functional analysis and convexity methods have the advantage of simplicity and lending themselves to further generalization of similar problems. In the present paper, we first provide a function analytic solution to the known maximal power transfer problem. Then we apply this method for solving a new problem of maximal power transfer; namely, when {N,: Z,} is complex and {N} is purely resistive. The general case has not appeared in the literature and it encompasses the more specialized results of (1) and (2).

II. Variational Preliminiaries (a) A fundamental theorem of functional analysis along with some convexity considerations constitutes the central theme of our approach in this paper. It is hoped that introduction of the method described here will open a new path for solving similar power transfer problems for “Hilbert network” in the future. Let A be a linear self-adjoint completely continuous operator defined on a separable complex Hilbert space H. For any complex vector XE H of the domain DA of A, the inner product expression (Ax, x) is a real number. Moreover, the selfadjoint operator A is said to be positive, i.e. A > 0 iff (Ax, x) > 0 for all x E DA. Theorem. (4). Let A > 0, and f EDA be a specified vector. The quadratic (real) functional

Jbl = attains

its minimum

w,

4 - K x) - w-,l

for x0, which is the unique

solution

(3) of

Ax0 =$ Conversely,

336

if x = x0 is the solution

(4)

of (4), then J[xO] < J[x] for all x E DA. Journalofthe Franklm

Institule Pcrgamon Press plc

On Maximal

Power Transfer Problem

(b) Inequalities on positive operators C provide useful tools for the study of variational problems where m and A4 are respectively the smallest and the largest eigenvalue of C, i.e mllx1(2 < (Cx,x> $

[A lesser known inequality inequality is due to Strang all x and y in H, we have

Mllxl12, XEH.

(5)

arising from a generalized form of the Kantorovich (5). Let C > 0, ]ICl] = M and I(C’ /I = m- ‘, then for

l(CX>Y)(X,c-‘Y)l G

(y;;)2 (X,X>(Y,Y>.

This inequality focuses on the interplay between a positive operator Its implementation will be deferred until a subsequent paper.]

III. Unrestricted

Power Transfer Case-A

and its inverse.

Functional Approach

The port-to-port connection of N and N,, under the application of the voltage e is governed by Ohm’s law (1). The active power dissipated by N is a functional of the current vector P[i], and depends on values of [Z], i.e. 2P[i] = (Zi, i) + (i, Zi).

(7)

This active power equals the difference of the total active power provided source and that dissipated by N,,, that is,

by the

2P[i] = (e, i) + (i, e) - (ZOi, i) - (i, ZOi)

(8)

2P[i] = (e, i) + (i, e) - ((Z, + Zai, i).

(9)

This functional depends directly on the choice of the vector i. When Z,+ Z,* > 0, (lossy systems), then according to the stated theorem there is a unique maximizing current vector iO, given by (Z, +Z$)iO = e,

ZiO = Z&

(10)

iO = (Z,+Z$)-‘e

(11)

P[il d P[id,

(12)

Statement I Assuming No to be a linear strictly passive (reciprocal or non-reciprocal) invariant n-port, under the sinusoidal regime of s = jw, we have Z,(jco) = R,(w)+jX,(w). The optimal

current

vector and the maximum

power transfer

i. = (2R&‘e P[i,] = f(R; ‘e, e) 2 P[i]. Vol. 325, No. 3, pp. 335-343, Printed in Great Bntain

time-

(13) are : (14) (15)

1988

337

F. M. Reza

Note that, while for a positive or a negative operator RO, there is only a unique minimizing (maximizing) vector iO, Eq. (2) may have more than one solution in 2. Optimal load does not exist when R0 is an indefinite operator. Power transfer:

optimal and sub-optimal

The above solution for R,, > 0 can be written in another T be the unique square root of RO, i.e. R0 = T2, then

interesting

format.

i0 = ‘,TP2e P[i,] = i(T-‘e,e)

The maximal

power transfer

Let

(16)

= 411T-‘ell

2.

(17)

is P[i,] = 11 TiO I/2.

(18)

Now one may dispense with the voltage e in (9) by expressing terms of the difference i- iO, that is

the active power in

2P[i] = (2T2i,, i) + (i, 2T2i0) - (2T2i, i) P[i] = (T2(i,, -i),

(19)

i)+ (T2i, i,,)

(20)

= (T2(iO-i),i-iO)+(T2(i0-i),iO)+(T2i,i,,) P[i] = (T2i,,, i,,-(T2(i0-i),

(21)

(iO -i)).

(22)

In this manner, we arrive at a condensed explicit formula for any actual power transfer based on its maximal value, i.e. when Z0 + Z,* is nonsingular, the power transfer for any sub-optimal mode of operation is P[i] = jl Ti,, 11 2- 1)T(i-

iO) 11 2.

(23)

Evidently the maximum of the power transfer occurs when i = iO, that is, when the load is optimal. For any deviation of the current vector i from its optimal value i0 the relative measure of power consumption may be immediately calculated from (23). The term to the right gives a measure of power transfer mismatch or suboptimality when the chosen member of (N) is not the optimal choice. Observe that if Z is the matched load for Z,, then Z* will be the matched load for Z,*.

IV. Maximal

Power Transfer Inductive--Resistive

n-port

In this section, we apply the proposed method for solving inductive-resistive problems. It is assumed that {N,} is purely inductive, that is, Z0 = jX,, with X0 > 0. We wish to determine the optimal resistive load match, that is, Z = R = R*, with R > 0. The active power to be optimized is 338

Journal

ofthc

Franklin Pergiiman

Institute Press plc

On Maximal

Power Transfer Problem

P[i] = [(e, i) + (i, e)]/2

(24)

P[i] = (Ri, i).

(25)

(R+jX,)i

(26)

Using Ohm’s law, = e

P[i] = ((R-jX&‘R(R+jX&‘e,e) (27) P[i] = (Ae, e) where A is a (self-adjoint)

positive

operator

:

A = (R-jX,)yR(R+jX,)y The positivity

= A*.

(28)

of A may be noted for e # 0 by P[i] = (R”2(R+jXo)~‘e,R”2(R+jXo)-1e)

> 0.

(29)

For a specified e, the largest A > 0 would lead to a greater value for P[i]. The situation may be better visualized by requiring the inverse operator A-’ > 0 to attain its smallest possible value, (3) A-’ A-’

= R+X&‘X,,,

X0 > 0

= (R1~2-~XoR-‘~2)(R’~2-R~‘/2XO)+2X0.

(30) (31)

A and A- ’ are positive, independent of the choice of e # 0. The least value of A- ’ occurs when R = X,, > 0 in view of R’/2_X,,R-‘/2 For this optimal

= R’/2_R’/2X,,

= 0.

(32)

load, we have A-’

= 2X0

(33)

G+ l)XOiM = e.

(34)

Statement 2 Maximal power transfer from a linear passive inductive n-port X > 0, terminated in a linear reciprocal passive resistive n-port R > 0 under a sinusoidal voltage at a fixed frequency occurs when R = X0, i.e. P[iM] = $((I,)-‘e,e)

2 P[i].

Note that X0 needs to be self-adjoint but not necessarily symmetric. is slightly more general than the results derived earlier by Culvaer

(35) This statement (1) and Desoer

(2). V. The Resistive Load Problem We search for the maximal attainable active power transfer and the corresponding optimal n-port when Z0 is arbitrary and {IV} consists of the class of all resistive positive definite n-ports. Equation (1) now assumes the form of (36) : Vol. 325, No. 3, pp. 33S343, Printed in Great Britain

‘988

339

F. M. Reza e = (R+ZJi. Assuming

{N,} a time-invariant Z0 +Z;

(36)

and strictly dissipative

= (RO +jX,)

The average active power available

+ (R, -jX,)

network

:

= 2R0 > 0.

to {N} is the functional

(37)

P[i], where

2P[i] = (e, i) + (i, e) - (2R,i, i). According maximum

to the theorem at i,,, where

(38)

of Section II for the unrestricted

case, P[z] has a unique

e = 2R&.

(39)

The corresponding ideal unconstrained available average power was given in (15). This ideal situation is physically unattainable for the specified restrictive class of resistive {N} and arbitrary e. In the sequel, we set forth to find the optimal feasible current iM and the corresponding RM > 0 in the sense that for any other attainable i P[i,] 3 P[iM] B P[i] > 0. The average active power employing (36)

dissipated

(40)

by (N} can be equivalently

derived

2P[i] = (Ri, i) + (i, Ri)

(41)

P[i] = ((R+Z,Y)p’R(R+Zo)p’e,e)

(42)

P[i] = (Ae,e). The operator

by

A in (43) is self-adjoint

(43)

and non-negative,

A = (R+Za-‘R(R+Z,,-l

= A*

A = [R1’2(R+Z,,-1]*[R”2(R+Z,,-‘]

(44) > 0

(45)

P[i] = (A “‘e, A “‘e) = I/A “*e 11’3 0. Now we are in a position

to generalize

the previous

(46)

results to prove Statement

3.

Statement 3 The maximal power transfer for a specified n-port of impedance Z0 E {N,} across an arbitrary resistive n-port load R E {IV} under sinusoidal regime occurs for R=JZ,Z,*=&&

(47)

provided (N,} and {N} are normal families. Proof: By normal family it is indicated that all ZO, Z,* and R commute, ZOZ; = Z,*Z,

i.e.

RZO = ZoR.

(48)

The plausibility of this condition will be discussed in Section VI. Thus R being nonsingular and A invertible, it is more convenient to deal in terms of A ‘, i.e. 340

Journal of the

Frankhn Pergamon

Institute Press plc

On Maximal Power Transfer Problem

A-’ = Z,+Zo*+R+Z&‘Z$

=

(4%

(R’12+ZOR-(‘12))(R112+R~(I12)ZOV).

For an arbitrarily specified e, the power inequalities (40) lead to inequality relations between positive operators. Thus, one may verify the validity of the inequality Ap’-Afl > 0, where A, ’ = Z,, + Z$+ Jzozo*+ In view of the commutativity

= Z,+Z$+2Jz,z,*

(51)

= RtZ&‘Z;-22/2uZ,*.

(52)

A-‘-A;’ Using once more the commutativity

assumption,

we find

= R2+Z,,Z$-2Rm

(53)

= (R-JZ,Z,j)(R-JZ,*Z,) Thus, the smallest

(50)

assumption, A,’

(A-‘-A,‘)R

Z,,(ZOZo*) -(r”)Z$.

2 0.

value for A- ’ occurs at R=m=Jm.

This value corresponds ports, and

(54)

to the maximal

power transfer

for the normal

family of n-

iM = (JZoZo*+ZO)-‘e

(55)

P[iM] = (Ae,e) = $((R,+,/mj-‘e,e) P[i,] = d(R;‘e,e)

(56)

$ P[iM] Z P[i].

(57)

Note that in the case of {RO = 0, X,, > 0), (55), (56) yield : R = X0,

i,,, = (2jX&‘e

(58)

P[iM] = j(X;‘e,e). The result derived in (1, 2) is a particular

(59)

case of the broader

Statement

3 above.

VI. Remarks Normal family The normal family {No) considered here, in a way, constitutes extension of the scalar linear reciprocal passive driving-impedance Z,,Z; A normal

n-port may be reciprocal

a non-normal

n-port

Vol. 325, No. 3. pp. 335-343. Printed m Great Britain

= ZgZ,

the most natural functions, i.e.

= R; + X;.

or non-reciprocal,

(60) [like2,,

= (_i

<)].

For

one arrives at :

1988

341

F. M. Reza ZoZ; = R; + X; +j(XoRo Z$Z, The active power dissipated as

- R,Xo)

(61)

= R;+X;-j(XORO-RJ,)

(62)

in an n-port under the voltage vector e may be written

P[i] = Re (Zi, i) = (RZ- ‘e, Zm ‘e) (63) = (Z*-‘RZ-‘e,e). This equation depicts that for the study of power optimization for arbitrary linear systems the question of order of multiplication and commutativity arises naturally. For instance, the commonly used expression for the active power dissipated in a network Z= R+jX, is R(R2+X2)’ in lieu of the operator of Eq. (63). This expression is often used without normality assumption. The correct formulation for the power expression should be written as (63) or (64), i.e. P[i] = ((R+XR-‘X)-‘e,e). For normal R(R2+X2)-‘.

networks

reduces

to

Positivity considerations The approach suggested in this paper may be further explored for problems power transfer. Note that the operator A can also be expressed as

of

A-’

R and

X commute

= (R”2+ZOR.“/2)

and

(64) this

expression

)(R’/2+ZOR-(‘/2))*

(65)

or A-’

=BB*

>o

(66)

where B = R’/2+ZoR-(‘/2)

(67)

or A-’

= (R’/2-ZZoR-

(‘i2))(R’/2-ZZoR-(‘/2))*+Zo+Z~.

The smallest value of A- ’ occurs for normal

network

(68)

at R = dm.

Square roots of a positive operator If T is a self-adjoint operator on a Hilbert space H, then for every vector x E H, the quantity (TX, x) is a real number. Also (T2x, x) = (TX, TX) > 0 implies that T2 is a positive operator. If A is a positive bounded (self-adjoint) linear operator on a complex Hilbert space H, then there exists a bounded self-adjoint linear operator F called a square root of A, that is, F2 = A. If, in addition F 2 0, then F is called the positive square root of A, and is denoted by F = A ‘I2. Every positive bounded self-adjoint linear operator A on H has a positive square root which is unique. For any arbitrary complex operator Z, the operators ZZ* and Z*Z are positivedefinite Hermitian, since for any x E H, we have Journal

342

of the Franklin Pergamon

Institute Press plc

On Maximal

(zz*x, (z*zx,

x) = (z*x, z*x)

Power Transfer

Problem (69)

2 0

x) = (Zx, Zx) 2 0.

(70)

If Z is non-sin ular, then ZZ* and Z*Z are left norm and right norm of Z. The quantities fi ZZ and p Z Z are called respectively left modulus and right modulus of Z. For a normal operator Z the two norms are equal and the two moduli coincide. Also, note that it may be tempting to write the maximum power transfer operator of Eq. (51) as A,’ = t&+&$‘. (71) The inaccuracy of such a representation is evident in view of the fact that ZO, generally a non-normal complex operator does not have a unique square root, unless ZO is a normal positive operator. The statements concerning the minimality of A-’ corresponding to the maximality of A, can be further elucidated by Strang’s inequality (6). In Ref. (3) the same idea is implied and referred to by the duality theorem. Related theorems and recent works

Some energy inequalities for normal and non-normal linear systems pertinent to the foregoing theorem may be found in (5). For some related recent works, see (7) and (8). The suggested Hilbert space format will be more rewarding for the future generalization of this research. Acknowledgement Acknowledgement is made Research Council of Canada.

for the support

of the Natural

Sciences

and Engineering

References (1) A. J. Calvaer, “On the maximum loading of active linear electric’multi-ports”, Proc. IEEE, Vol. 71, pp. 282-283, Feb. 1983. (2) C. A. Desoer, “A maximum power transfer problem”, IEEE Trans. Circuits Syst., Vol. CAS-30, 1, pp. 757-758, Oct. 1983. (3) H. Flanders, “The maximal power transfer theorem for n-ports”, Circuit Theory Appl., Vol. 4, pp. 319-344, 1976. Analysis in Normed Spaces”, (4) I. V. Kantorovich and G. P. Akilov, “Functional Macmillan, New York, 1964. (5) F. M. Reza, “A global energy theorem for linear systems”, J. Franklin Inst., Vol. 313, pp. 97-105, 1982. (6) W. G. Strang, “On the Kantorovich inequality”, Proc. Am. Math. Sot. ZZ,p. 468, 1960. (7) H. Flanders, “Note on maximal power transfer”, IEEE Trans. Circuits Systs., Vol. CAS-32, p. 100, 1985. (8) F. M. Reza, “Two-element type normal n-ports”, J. Franklin Inst., Vol. 323, No. 1, pp. 113-133, 1987.

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