THE N-PORT RECEIVING ANTENNA AND ITS ... RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 303* (b) in the receiving situation the incident electromagnetic radiation consists

R908 Philips Res. Repts 30, 302*-315*,1975Issue in honour of C. J. Bouwkamp

THE N-PORT RECEIVING ANTENNAAND ITS EQUIVALENT ELECTRICAL NETWORK

by A. T. DE HOOP *)

Delft University of TechnologyDelft, The Netherlands(Received January 17, 1975)

AbstractA rigorous proof is presented of the commonly accepted theorem thatan N-port receiving antenna is, in several respects, equivalent to anN-port electrical network containing internal sources. Expressions arederived for the quantities that specify the Thévenin representation ofthisnetwork. As a basic tool, the reciprocity theorem that relates the electro-magnetic fields occurring in the transmitting situation to those occurringin the receiving situation, is used. The most general N-port antenna, ifonly linear, passive and time-invariant, is investigated; nonreciprocalones are included as well. The incident radiation consists of an arbitrarilyelliptically polarized plane wave. With the aid ofthe equivalent network,the condition for maximum power transfer from the incident wave toan N-port load is derived.

1. IntroduetionThe electrical properties of a receiving antenna, for example one that is used

in a communication system, are often specified more or less intuitively in termsof an equivalent electrical network with internal sources 1). The parameters ofthe relevant network are commonly accepted to follow from related quantitiesthat characterize the same antenna in the transmitting situation, while thestrengths of the internal sources will depend on the amplitude, phase and stateof polarization of the radiation that is incident upon the antenna in the receivingsituation.The purpose of the present paper is to show how this representation can be

justified rigorously. At the same time, expressions are obtained for the networkparameters involved as well as for the strengths of the internal sources. As aresult, the electrical network that characterizes the properties of the antennain the receiving situation is completely specified. The conditions under whichthe representation is shown to hold are:(a) both in the transmitting and in the receiving situation the antenna is acces-

sible at a finite number of ports at which either the low-frequency voltagesand currents replace the general field concept or the single-mode waveguidedescription (as in a microwave antenna) holds;

*) Dedicated to my friend Dr C. J. Bouwkamp, whose penetrating way of thinking has hada substantial influence on my scientific development.

N-PORT RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 303*

(b) in the receiving situation the incident electromagnetic radiation consists ofa uniform plane wave (with arbitrary amplitude, phase and state of polar-ization);

(c) the antenna system is linear in its electromagnetic behaviour.No further restrictions are imposed. In particular, the materialof which theantenna is made, may be lossy, inhomogeneous and anisotropic; nonreciprocalantenna configurations ale included as well.

The theorem upon which our considerations are based is the reciprocitytheorem that relates, for a single antenna, the electromagnetic fields occurringin the transmitting situation to those occurring in the receiving situation. Thistheorem has been derived by the present author on a previous occasion 2)and is shortly reconsidered here. Subsequently, it is applied to the configurationunder investigation, upon which the desired network representation is obtained.

Once the equivalent network pertaining to the receiving situation is known,several problems related to the further use of the antenna can be solved byemploying pure network methods. Among this is the problem of maximumpower transfer from the incident wave to a load that is connected with theaccessible ports. We show that maximum power transfer occurs if the imped-ance matrix of the load is the complex conjugate (not the Hermitean con-jugate) of the input-impedance matrix of the antenna in the transmitting situa-tion. The load is then "matched" to the antenna.

For recent developments in the network-theoretical aspects of the scatteringproperties of an N-port receiving antenna, we refer to papers by Mautz andHarrington 3) and Harrington and Mautz 4).

Network-theoretical aspects of minimum-scattering antennas are dealt within papers by Kahn and Kurss 5) and Wasylkiwskyj and Kahn 6).

2. Description of the configuration

The antenna system under consideration occupies a bounded domain V inspace. Externally, V is bounded by a sufficiently regular closed surface So;internally, V is bounded by a sufficiently regular closed surface Sl' The sur-face Slis considered as the termination of the antenna system, and on it afinite number N of ports is defined through which the antenna system isaccessible from the "interior" (fig. I). Parts of So and S1 may coincide. Theregion V thus introduced allows us to distinguish the antenna system from theenvironment into which it radiates or scatters, as well as from the terminalsat which it is accessible. The cartesian coordinates of a point in space are de-noted by x; y and z; the time variable is denoted by t. The position vector isdenoted by r. The electromagnetic fields occurring in the transmitting situation,as well as those occurring in the receiving, situation.: are assumed to varysinusoidally in time with the same angular frequency w. The complex represen-tation of the field vectors is used, and in the formulas, the complex time factor

i

304* A. T. DE HOOP

environment: ["o.PoI

Fig. 1. Antenna configuration with N accessible ports.

exp (-iwt), common to all field components, is omitted.The antenna consists of a medium, the electromagnetic behaviour of which

is linear and passive, no further restrictions as to its electromagnetic propertiesbeing imposed. The properties of the medium may change abruptly when cross-ing a (bounded) surface, but, across such a surface of discontinuity in properties,the tangential parts of both the electric- and the magnetie-field vector arecontinuous. Other parts of the antenna system may consist of conducting sur-faces. These surfaces are assumed to be electrically perfectly conducting, andon them the tangential part of the electric-field vector vanishes.

The medium outside So is assumed to be linear, homogeneous, isotropic andlossless, with real scalar permittivity 80 and real scalar permeability {-lo; thisincludes the case of free space.

In the following sections, E, H, D and B denote the space- and frequency-dependent complex representations of the electric-field vector, the magnetic-field vector, the electric-flux density and the magnetic-flux density, respectively.All quantities are expressed in terms of SI units. The superscripts Tand Rareused to denote the transmitting and the receiving situation, respectively.

3. The antenna in the transmitting situation

In the transmitting situation (fig. 2) the accessible ports of the antenna arefed by an N-port source. Let InT denote the electric current fed into the nth port,and let VnT denote the voltage across the nth port (n = 1, ... , N). (In thesingle-mode waveguide description, InTand VnT denote the complex amplitudes,in a chosen transverse reference plane, of the transverse parts of the magneticand the electric fields pertaining to the waveguide mode under consideration;the relevant transverse modal functions should be properly normalized.) As aconsequence of the uniqueness theorem of electromagnetic fields, the voltages{VnT} are linearly related to the currents {InT} through the relation

N

VmT = L: Zm.nln InTn=1

(m = 1, ... , N), (1)

where Z",.n In defines the input-impedance matrix ofthe radiating antenna systemI

pin - .i,Re [ ;, V T 1 T*]-2 I..J In na ,m=1

(2)

N-PORT RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 305*

transmitted wave: [E~HTI

environment: ieo.PoI

Fig. 2. Antenna in transmitting situation; the accessible ports are fed by a source.

in the transmitting situation. The time-averaged electromagnetic power pin fedinto the antenna system in the transmitting situation is then given by

where * denotes the complex conjugate. With the aid of (1), eq. (2) can berewritten as

pin = .i, Re[;' ;, Z In 1 HIT]2 f...J I..J m,IJ 111 n

m=1n=1

N N

= t L L (Zm.nIn + Zn.1nIn*) t;T* InT.1n=1 n=1

(3)

In the domain V, between So and Sl' the electromagnetic-field vectors satisfythe source-free electromagnetic-field equations

curl HT + iw DT = 0, (4)

curl ET - ita BT = 0, (5)

and the constitutive equations which express {DT, BT} linearly in terms of{ET, HT}. Owing to the continuity of the tangential parts of ET and HT acrossS 1 we can rewrite the expression for pin as

pin = t Re [{!(ET X HT*) • n ciA1 (6)

where n denotes the unit vector along the outward normal.In the domain outside So, ET and HT satisfy the source-free electromagnetic-

field equationscurl HT + ito 80 ET = 0,

curl ET - io: /ho HT = 0.

(7)

(8)

306* A.T.DE HOOP

In addition, the transmitted field satisfies the radiation conditions 7,8,9)

rxHT + (SO/11-0)1/2ET = O(r-2)

rx ET - (fl-o/ SO)1/2 HT = oir: 2)as r- 00, (9)

(10)as r- 00,

in which r(= r/r) denotes the unit vector in the radial direction and r is thedistance from the origin to a point in space. As a consequence of eqs (7)-(10),the following representation holds:

as

in which(12)

.1.0 being the wavelength in the medium outside So. P denotes the point ofobservation with position vector rp. Between eT and hT the following relationsexist:

eT = (#0/SO)1/2 (hTxrp),

hT = (SO/11-0)1/2(rpxeT).

(13)

(14)

These relations, together with eq. (11), are in accordance with eqs (9) and (10).We shall denote eT = eT(rp) as the electric-field and h" = hT(rp) as the magnetic-field amplitude radiation characteristic of the antenna system. For a given N-port antenna, they only depend on the direction of observation and on the wayin which the N accessible ports are fed. Both amplitude radiation characteristicsare transverse with respect to the direction of propagation of the expandingspherical wave generated by the antenna, i.e. rp. eT = 0 and rp. hT = O.Weexhibit the dependence of eT and hT on the way in which the N accessibleports are fed, by writing

N

{eT, hT} = L {enT, hnT} InT,n=1

(15)

The time-averaged electromagnetic power pT radiated by the antenna is given by

pT = t Re [{f (ETxHT*). n dA J. . (16)

where n denotes the unit vector along the direction of the outward normal.Since the medium outside So is lossless, we can replace So in eq. (16) by asphere whose radius is taken to be so large that the representation (11) holds.Then, we can rewrite eq. (16) as

(17)

NNI (8 )1/2pT = L: L: I",T* InT --2 ~ 11 emT*. enT d.Q.

m=1 n=1 32:n; #0 Q

(18)

N-PORT RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 307*

where O denotes the sphere of unit radius. Incidentally, eq. (17) proves thatP" > 0 for any nonidentically vanishing eT. With the aid of (15), eq. (17) canbe rewritten as

Now, for a lossless antenna we have P" = pin. In this case the right-handsides of (3) and (18) should be equal, irrespective of the values of {InT}. Thiscondition leads to the relation

. 1 (80)1/2t rz, nln + z, mln*) = -- - 11 ell,T* • enT d.Q.. . 32:n;2 #0

Q

(19)

A relation similar to (19), has been derived by Van Bladel !").It is known that eT = eT(Tp) and h" = hT(Tp) can be expressed in terms of

the values that the tangential parts of ET and HT admit on So; the relevantexpression for eT is 2)

eT = ikoTpxffn X ET(r) exp (-ikoTp.r) dA +So

-iko {#0/80)1/2 Tp X [TPX{!nXHT(r)eXP(-ikoTP.r)dA 1 (20)

Our proof of the reciprocity relation is in part based on this expression.

4. The antenna in the receiving situation

In the receiving situation a time-harmonic, uniform, plane electromagneticwave is incident upon the antenna system, while the accessible ports are con-nected to an N-port load (fig. 3). As incident field {El, HI} we take

EI = A exp (-i ko ex• r),

Hl = (80/#0)1/2 (Ax ex)exp (-i ko ex. r),(21)

(22)

where A is a constant, complex vector that specifiesthe amplitude and the phaseof the plane wave at the origin, as well as its state of polarization, and-exdenotes the unit vector in the direction of propagation. (We call exthe directionof incidence.) The state of polarization is, in general, elliptic, but is linear ifAxA* = 0 and circular if A. A = O. Since the wave is transverse, we haveex. A. = O. In the domain outside So, the scattered field {ES, HS} is introducedas the difference between the actual (total) field {ER, HR} and the field of theincident wave {EI, Hl}:

(23)

308* A.T.DE HOOP

scattered wave: IE~ HS) Incident wave: IE! H')

environment: (eD'/Io)

Fig. 3. Antenna in receiving situation: a uniform, plane, electromagnetic wave is incidenton it and the accessible ports are connected to a load.

In the domain outside So, both the incident and the total fields, and hencethe scattered field, satisfy the source-free electromagnetic-field equations

curl HI,s + ico eo El,s = 0,

curl El,s - ico /to HI,s = O.

In addition, the scattered field satisfies the radiation conditions

(24)

(25)

rxHs + (eo//to)1/2 ES = O(r-2)

rxEs - Ctto/eo)1/2 HS = O(r-2) as r~ 00.

(26)

(27)

as r~ 00,

As a consequence of eqs (24)-(27) the following representation holds:

as rp -+ 00. (28)

Between eSand h", relations of the type (13)-(14) exist, and for eS, a representa-tion similar to (20) can be obtained.The time-averaged power P" = PR(a.), received by the antenna system, is

given by(29)

Further, the scattered power P" = PS(a.) is defined as

ps ~ t Re [If (ESxW*) • n dA1Since the medium outside So is lossless, we can, on account of (28), rewrite(30) as

(30)

1 (e )1/2ps = -- ~ 11eS • eS* d.Q.32 :n;2 !-la

{l

(31)

N-PORT RECEIVING ANTENNA AND lTS EQUIVALENT ELECTRICAL NETWORK 309*

Next, we add (29) and (30), subsequently use (23), (21), (22) and the represen-tation of eS similar to (20), and observe that the incident wave would in theabsence of the antenna dissipate no power when travelling in the domain insideSo. This procedure leads to

pR + P" = t Re [(iw fto)-1 A* . eS(-ex)].

Equation (32) is directly related to the "cross-section theorem" in electro-magnetic scattering 11.12).

In the domain V, between So and SI' the electromagnetic-field vectors satisfythe source-free electromagnetic-field equations

curl HR + it» DR = 0,

curl ER - ito BR = 0,

and the constitutive equations which express {DR, BR} linearly in terms of{ER, HR}. The time-averaged electromagnetic power P': = PL(ex) dissipatedin the load is given by

r- = -t Re [[I (ER x HR*) • D dA1 (35)

Owing to the continuity of the tangential parts of ER and HR across SI' wecap express P': also in terms of the electric currents {InR} flowing into theload and the voltages {VnR} across the ports of the load. The result is

Since the electromagnetic properties of the N-port load are assumed to belinear, the voltages {Vn R} are linearly related to the currents {InR} through therelation

N

VmR = L Zm.nL InRn=1

(m = 1, ... , N),

where Zm.n L defines the impedance matrix of the load. With the aid of (37),eq. (36) can be rewritten as

r- - .i, Re [;' ;, Z L 1 R* 1 RJ- "2" LJ LJ m,n m nm=1 n=l

N N

= tL L (Zm.nL + Zn.mL*) ImR* InR.m=1 n=l

Now, for a lossless antenna we have P" = PL. In this case we obtain from(31), (32) and (38)

L- ~_~ _

(32)

(33)

(34)

(36)

(37)

(38)

310* A.T.DE HOOP

NNI (8 )1/2~" "(Z L+Z U) 1 R* 1R+-- ~ If eS. eS*dD=:ij: L...J L...J m,n n,m 111 n 32 2 .m=1 n=1 :Tt!-to n

= t Re [(iw !-tO)-1 A* • eS(-ex)]. (39)

5. The reciprocity relation

The starting point for the derivation of the reciprocity relation is Lorentz'sreciprocity theorem for electromagnetic fields 13). This theorem can be applied,provided that the electromagnetic properties of the medium present in the trans-mitting situation and those of the medium present in the receiving situation areinterrelated in such a way that, at all points in space, the relation

(40)

holds. In the domain outside So this is obviously the case as the constitutiveequations here are simply D = 80 E and B = !-to H, both in the transmittingand in the receiving situation. In the domain V, between So and S l' the situationmay be more complicated. Here, eq. (40) holds without change of propertiesof the medium when the medium is reciprocal. In all other cases, the mediumis nonreciprocal, and the appropriate change in properties has to be madewhen switching from transmission to reception and vice versa. It is noted that,in the general condition (40), nonreciprocal media, including those showing themagnetoelectric effect 14), are included. If (40) is satisfied, Lorentz's theoremstates that

J J (ETxHR - ERxHT). n dA = 0S

(41)

for any sufficiently regular, bounded, closed surface S, provided that the domainbounded by S is free from electromagnetic sources. If(41) is applied to a domainoutside So, we may, on account of eqs (23)-(25), also replace {ER, HR} in(40) and (41) by either {ES, H'} or {El, HI}.

Let Sr denote the sphere with radius r and centre at the origin, where r ischosen so large that Sr completely surrounds So. Since the fields {ET, HT}and {ES, HS} both satisfy the radiation conditions (eqs (9) and (10) and (26)and (27), respectively), we have

lim J J (ETxHS - ES X HT) • n dA = O. (42)r-+ co Sr

Consequently, the application of LOlentz's theorem (41) to the domain boundedinternally by So and externally by Sr> and to the fields {ET, HT} and {ES, HS}leads in the limit r -+ co to

J J (ETxH' - ESxHT). n dA = O.So

(43)

N

L (VII? I",R + V",RImT) = -(iw !l0)-1 A. eT(a).m=l

(48)

N-PORT RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 311 *

Next, we observe that, from (20), (21) and (22), it follows that

11(ETxH'- E'xHT). n dA = (iw !l0)-1 A. eT(a). (44)So

On adding eqs (43) and (44), and using (23), we arrive at

11(ETxHR -ERxHT). n dA = (iw !l0)-1 A. eT(a). (45)So

We proceed with the application of Lorentz's theorem (41) to the domain V,bounded internally by Sl and externally by So and to the fields {ET, HT}and {ER, HR}, which gives

However, on Sl the field description in terms of the N accessible ports holds.Taking into account the direction of n and the chosen directions of the currents{lilT} and {InR}, we can rewrite the left-hand side of eq. (46) as

N

11(ETxHR - ERxHT). n dA = - L (V",T I",R + VmRImT). (47)m=l

Combining eqs (45), (46) and (47), we arrive at the amplitude reciprocity rela-tion (cf. ref. IS)

This relation will serve as the starting point for the derivation of the networkrepresentation of the antenna in the receiving situation.

6. The equivalent network for an N-port receiving antenna

In the reciprocity relation (48) we substitute eq. (1) in the left-hand side anduse (15) in the right-hand side. Rearranging the result we obtain

(49)

This equation should hold irrespective of the values of {I,IIT}. As a con-sequence, it follows that

N

" Z in 1R + V. R - E RL...J n,m n m - m (m = 1, ... , N), (50)n=l

312* A.T.DE HOOP

in which the "equivalent electromotive force" E",R = EmR(et.) is given by

(m = 1, ... , N). (51)

Equation (50) describes the properties of an N-port electrical network withinternal voltage sources (Thévenin representation). The N-port network whosenetwork equations are (50), is the equivalent electrical network for the antennain the receiving situation. Equation (51) shows how E",R depends on the ampli-tude, the phase and the state of polarization of the incident wave and, throughthe geometrical factor emT(et.), on the direction of incidence. Note that theinternal-impedance matrix of the network is the transpose of the input-imped-ance matrix of the antenna in the transmitting situation.

One application of eqs (50) and (51) is the experimental technique for deter-mining the electric-field amplitude radiation characteristics {emT(cx)} of a givenN-port antenna, in the transmitting situation, by using the antenna as a receivingantenna and measuring its reaction on an incident plane wave. A simple pro-cedure performing this, runs as follows:(a) a specific value of et. and two different values of A, corresponding to two

independent states of polarization, are chosen (the values of A can be deter-mined experimentally by removing the antenna and measuring E' at theorigin of the coordinate system, the latter point serving as a reference pointfor the phase of the fields);

(b) all ports are left open, i.e. InR = 0 for all n, and VmR is measured for all m(in this case V,,,R equals E",R as (50) shows);

(c) from eq. (51) we calculate the two complex components of emT at theselected value of et. (note that both A and em

T are transverse with respectto et.);

(d) new values of et. and A are selected, and the procedure is repeated untilenough values of {emT(et.)} have been obtained.

7. The condition for maximum power transfer from the incident wave to the load

Let now an N-port load be connected to the accessible ports. Then, the rela-tions (37)hold. For any given load, substitution of (37) in (50) leads to a systemof N linear, algebraic equations, from which the currents {InR} can be deter-mined. Once this has been done, the power P': dissipated in the load can becalculated from (38).

One ofthe problems associated with the use of a receiving antenna is to designsuch a load that, for a given antenna system, maximum power is transferredfrom the incident wave to the load. On account of the relations (37) and (50),this is a pure network-theoretical problem, and it has a unique solution. Theanswer is that the impedance matrix of the load should be the Hermitean con-jugate of the internal-impedance matrix of the electric network under consider-ation. For reference, a proof of this statement is included in an appendix. As

Z L = Z In*m,n m,n (m,n = 1, ... , N). (52)

N-PORT RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 313*

the internal-impedance matrix of the equivalent network for the antenna in thereceiving situation is the transpose of the input-impedance matrix of the antennain the transmitting situation, maximum power is dissipated in the load if

Hence, for maximum power transfer, the impedance matrix of the load shouldbe equal to the complex conjugate (not the Hermitean conjugate) of the input-impedance matrix of the antenna in the transmitting situation. If (52)holds, theload is called "matched" to the antenna. Apparently, the condition for matchingis independent of the direction of incidence ex.

Appendix

The maximum-power-transfer theorem for an N-port network with prescribedinternal sourcesWe consider a linear, passive, time-invariant electric network with prescribed

internal sources and N accessible ports (N ~ 1). The network is operating inthe sinusoidal steady state. The equations corresponding to the Thévenin repre-sentation of the network are given by

N

LZm.nln + Vm= Emn=1

(m = 1, ... , N), (A.I)

where {In} are the currents flowing out of the ports, {Vm} are the voltagesacross the ports, Zm.n defines the internal-impedance matrix and {Em} are theequivalent electromotive forces ofthe internal sources. The time-averaged powerP': dissipated in an N-port load is given by

pL = t Re [mt1Vm Im*1 (A.2)

We now define the "optimum state" ofthe network as the one that maximizesPL. In the optimum state, let {VmoPt} be the voltages, {InoPt} the currents andpL.opt the power dissipated in the load. Then .

pL.opt = t Re [mt1VmoPt ImoPt*1 (A.3)

Let us now consider arbitrary (not necessarily small) variations *) {(W;n} and{Mn} around. {VmoPt} and {InoPt}, respectively, and take

(m = 1, ... , N) (A.4)and

(n = 1, ... , N). (A.5)

*) While preparing the manuscript, the author's attention was called to a paper by Desoer 16),where a similar line of reasoning is followed and where a more sophisticated treatment isgiven.

314* A. T. DE HOOP

While "varying the voltages and the currents, we keep {ZIII.n} and {Em} fixed.Consequently, {t5Vm} and {bIn} are interrelated through

N

LZrn.n st, + t5Vrn= 0n=1

(m = 1, ... , N), (A.6)

as follows from (A.I). Substitution of (A.3), (A.4) and (A.5) in (A.2) yields

pL = pL.OPt+ t Re L~5t5Vn,IrnoPt* + Vmopt MIII*+ t5Vrn bIlII*) 1 (A.7)

Sincet Re (t5v,n ImoPt*) = t Re (t5Vrn* ImoPt), (A.8)

the first-order terms on the right-hand side of (A.7) cancel if

Re [m~ 1(t5Vm* Imopt + Vmopt st;*)] = O. (A.9)

On account of (A.6), this can be rewritten as

Re [;' (-;, Z * 1 opt+ V, oPt) M *]= 0i..J LJ n,UI n m 111 •111=1 n=1

(A.IO)

Equation (A.IO) holds for arbitrary complex {bIm*}, provided thatN

V, opt= "Z *1optUI .l...J n,m n

n=1

(m = 1, ... , N). (A.U)

Introducing the impedance matrix of the load throughN

VIII =L ZIII.nL Inn=1

(m = 1, ... , N), (A.I2)

eq. (A.1l) implies that in the optimum state we have

Z L.opt = Z *m.n n,UI (m, n = 1, ... , N). (A.13)

Equation (A.13) states that the optimum N-port load has an impedance matrixwhich is the Hermitean conjugate of the internal-impedance matrix of the net-work. It now remains to be shown that the condition (A.9) maximizes PL. Tothis aim we observe that if, apart from the sources, the given N-port networkis passive, we have

t Re [m~1 ElbIlII* ZIII.n bIn] ~ 0

for any sequence {bIn}. Using (A.6), (A.I4) and (A.9) in (A.7) we arrive at

(A.I4)

pL ::::;;;pL.opt (A.I5)

(A.l6)

N-PORT RECEIVING ANTENNA AND ITS EQUIVALENT ELECTRICAL NETWORK 315*

for any sequence {c5In}. This result completes the proof. If the given N-portnetwork is dissipative, the left-hand side of (A.14) is positive for any non-identically vanishing sequence {c5In} and hence P': < pL,oPt for any non-identically vanishing sequence {c5In}.The time-averaged power PI that is dissipated internally in the network is

given by

In the optimum state this reduces, on account of (A.13), to

(A17)

The time-averaged power P, that is delivered by the sources is given by

(A18)

Using (Al), (A.2) and (A.16), this becomes

r. =PI +pL. (A19)

In the optimum state we therefore have, by virtue of (A.17) and (A.19)

(A20)

REFERENCES1) R. E. Collin and F. J. Zucker, Antenna theory, Part 1, McGraw-Hill Book Company,

New York, 1969, Chapter 4.2) A. T. de Hoop, Appl, sci. Res. 19, 90-96, 1968.3) J. R. Mautz and R. F. Harrington, IEEE Trans. Ant. Prop. AP-21, 188-199, 1973.4) R. F. Harrington and J. R. Mautz, IEEE Trans. Ant. Prop. AP-22, 184-190, 1974.5) W. K. Kahn and H. Kurss, IEEE Trans. Ant. Prop. AP-13, 671-675, 1965.6) W. Wasylkiwskyj and W. K. Kahn, IEEE Trans. Ant. Prop. AP-18, 204-216,1970.7) S. Silver, Microwave antenna theory and design, McGraw-Hill Book Company, New

York, 1949,p. 85.8) C. M ü l ler, Foundations of the mathematical theory of electromagnetic waves, Springer-

Verlag, Berlin, 1969, p. 136.9) J. van Bladel, Electromagnetic fields, McGraw-Hill Book Company, New York, 1964,

p.204.10) J. van Bladel, Arch. elektro Uebertr. 20, 447-450, 1966.11) A. T. de Hoop, Appl. sci. Res. B7, 463-469, 1959. ,12) J. van Bladel, Electromagneticfields, McGraw-Hill Book Company, NewYork, 1964,

p.258.13) R. E. Collin and F. J. Zucker, Antenna theory, Part 1, McGraw-Hill Book Company,

New York, 1969, p. 24.14) T. H. 0'Dell, The electrodynamics of magneto-electric media, North-Holland Publishing

Company, Amsterdam, 1970, p. 22.15) A. T. de Hoop and G. de Jong, Proc. lEE 121,1051-1056, 1974.16) C. A. Desoer, IEEE Trans. Circuit Theory CT-20, 328-330, 1973.