1.1 introduction -...

1.1 Introduction

This chapter is concerned primarily with establishing formulas for the electromagneticfield vectors E and H in terms of all the sources causing these radiating fields, but atpoints far removed from the sources. The collection of sources is called an antennaand the formulas to be derived form the basis for what is generally referred to asantenna pattern analysis and synthesis.

A natural division into two types of antennas will emerge as the analysisdevelops. There are radiators, such as dipoles and helices, on which the current dis-tribution can be hypothesized with good accuracy; for these, one set of formulas willprove useful. But there are other radiators, such as slots and horns, for which anestimation of the actual current distribution is exceedingly difficult, but for which theclose-in fields can be described quite accurately. In such cases it is possible to replacethe actual sources, for purposes of field calculation, with equivalent sources thatproperly terminate the close-in fields. This procedure leads to an alternate set offormulas, useful for antennas of this type.

The chapter begins with a brief review of relevant electromagnetic theory,including an inductive establishment of the retarded potential functions. This is fol-lowed by a rigorous derivation of the Stratton-Chu integrals (based on a vectorGreen's theorem), which give the fields at any point within a volume Kin terms of thesources within Fand the field values on the surfaces S that bound V. This formulationpossesses the virtue that it applies to either type of antenna, or to a hybrid mix of thetwo. Simplifications due to the remoteness of the field point from the antenna willlead to compact integral formulas, from which all the pattern characteristics of thedifferent types of antennas can be deduced.

A general derivation of the reciprocity theorem is presented; the result is used todemonstrate that the transmitting and receiving patterns of an antenna are identical.The concept of directivity of a radiation pattern is introduced and a connection is estab-

3

4 The Far-Field Integrals, Reciprocity, Directivity

lished between the receiving cross section of an antenna and its directivity when trans-mitting. The chapter concludes with a discussion of the polarization of an antennapattern.

A. REVIEW OF RELEVANT ELECTROMAGNETIC THEORY1

It will generally be assumed that the reader of this text is already familiar with elec-tromagnetic theory at the intermediate level and possesses a knowledge of basic trans-mission line analysis (including the use of Smith charts) and of waveguide modalrepresentations. What follows in the next several sections is a brief review of thepertinent field theory, primarily for the purposes of introducing the notation thatwill be adopted and highlighting some useful analogies.2

Throughout this text MKS rationalized units are used; the dimensions of thevarious source and field quantities introduced in the review are listed on the inside ofthe front cover.

1.2 Electrostatics and Magnetostatics in Free Space

A time-independent charge distribution

p(x,y,z) (I-la)

expressed in couloumbs per cubic meter, placed in what is otherwise free space, givesrise to an electrostatic field E(x, y9 z). Similarly, a time-independent current distribu-tion

J(x,y,z) (Lib)

expressed in amperes per square meter, produces a magnetostatic field B(x, y, z). Toheighten the analogies between electrostatics and magnetostatics, it is sometimesuseful to refer to the "reduced" source distributions

P(x, y, z) J(x, y, z) ( 2)

in which eQ is the permittivity of free space and fal is the reciprocal of the perme-ability of free space.

Coulomb's law can be introduced as the experimental postulate for electrostaticsand described by the equations

2The reader who prefers to omit this review should begin with Section 1.7.2The pairing of B with E (and thus of H with D), the use of //Q l, the introduction of reduced

sources, and the parallel numbering of the early equations in this review all serve to emphasize theanalogies that occur between electrostatics and magnetostatics. This is done in the belief that percep-tion of these analogies adds significantly to one's comprehension of the subject. See R. S. Elliott,"Some Useful Analogies in the Teaching of Electromagnetic Theory," IEEE Trans, on Education,E-22 (1979), 7-10. Reprinted with permission.

p(x, y, z) (1.1a)

J(x, y, z) (Lib)

p(x, y, z) J(x, y, z)?o Vol (1.2)

1.2 Electrostatics and Magnetostatics in Free Space 5

F = qE (1.3a)

E(x,y, z) = y 47I€QR3 (1.4a)

in which R is the directed distance from the source point (£, //, £) to the field point(x, y, z), and F is the force on a charge q placed at (x, y, z), due to its interaction withthe source system /?(£, rj, Q.

Similarly, the Biot-Savart law can be introduced as the experimental postulatefor magnetostatics and is represented by the equations

F = qYXB (1.3b)

f J(£,//,0 XKdV{X'y> Z) = I 4jiMolR3 ( l Ab)

One can show by performing the indicated vector operations on (1.4a) that

V X E E E O (1.5a)

V . E = -£ (1.5b)

In like manner, the curl and divergence of (1.4b) yield

V x B - - i (1.5c)Mo1

V . B = 0 (1.5d)

Equations 1.5 are Maxwell's equations for static fields.Integration of (1.5b) and use of the divergence theorem gives Gauss' law, that is,

s 1(£)E • dS = [T~)dV = total reduced charge enclosed (1.6a)

Similarly, integration of (1.5c) and use of Stokes' theorem yields Ampere's circuitallaw:

cB • dl = \~TT) • dS = total reduced current enclosed (1.6b)

In like manner, integration of (1.5a) and (1.5d), followed by the application of Stokes'theorem or the divergence theorem results in the following relations.

j> E-rf /=0 (1.7a)

| B- dS~0 (1.7b)J s

F = qE

E(x,y,z) =JV

• P(Z,t1,OKdV4ne0R

3

(1.3a)

(1.4a)

F = qv X B

B(x, y, z) =JV

'J(£,n,OxRdVAnp^R*

(1.3b)

(1.4b)

V X E = 0

V - E = pf o

(1.5a)

(1.5b)

V X B =

V . B = 0

JMo1 (1.5c)

(1.5d)

tfo.£

'V

s

c(1.7a)

(1.7b)

E-dlssO

B • dS = 0

The Far-Field Integrals, Reciprocity, Directivity

From (1.7a) it can be concluded that E (x, y, z) is a conservative field and that <f E • dlbetween any two points is independent of the path. Equation 1.7b permits the con-clusion that the flux lines of B are everywhere continuous.

Equation 1.4a can be manipulated into the form

E--VO (1.8a)

in which

is the electrostatic potential function. In like manner, Equation 1.4b can be rewrittenin the form

B - V X A (1.8b)

where

A(x,y,z)-^ 4n/iolR ( }

is the magnetostatic vector potential function. One can see that the reduced sources(1.2) play analogous roles in the integrands of the potential functions (1.8a) and(1.8b), as well as in the integrands of the field functions (1.4a) and (1.4b).

There is no compelling reason to introduce either D or H until a discussion ofdielectric and magnetic materials is undertaken, but if one wishes to do it at thisearlier stage, where only primary sources in what is otherwise free space are beingassumed, then it is suggestive to write

Do = 60E (1.10a)

Ho - fa lB (1.10b)

with the subscripts on D and H denoting that the medium is free space. Then it fol-lows logically from (1.5) that

V • Do = /> V X Ho = J (1.11)

and from (1.6) that

<p Do • dS = \ p dV = total charge enclosed (1.12a)

| Ho • dl = f J • dS = total current enclosed (1.12b)

Equations 1.12 are the forms in which one is more apt to find Gauss' law and Ampere'scircuital law expressed. It is apparent from (1.12) that Do and Ho play analogous rolesin the two laws.

E - - V O

O(x, y, z) =ly

p(Z,n,Odv4ne0R

(1.8a)

(1.9a)

B = V X A (1.8b)

(1.9b)A(x, y, z) =IV

mjkQdv4njHolR

Do = 60E (1.10a)

(1.10b)

s

c

Do • dS = p dV = total charge enclosedJy

Ho • dl = \ J • dS = total current encloseds

(1.12a)

(1.12b)

V . Do = /? V x H 0 = J (1.11)

6

1.3 The Introduction of Dielectric, Magnetic, and Conductive Materials 7

When flux maps are introduced, (1.12a) leads to the conclusion that the lines ofDo start on positive charge and end on negative charge. If one chooses to defer theintroduction of D and H until materials are present, a flux map interpretation of(1.6a) includes the idea that the lines of E start on reduced positive charge and end onreduced negative charge.

It has already been noted in connection with equation (1.7b) that the flux linesof B are continuous. Since Ho differs from B only by a multiplicative constant, theflux lines of Ho are also continuous.

1.3 The Introduction of Dielectric, Magnetic,and Conductive Materials

The electrostatic behavior of dielectric materials can be explained quite satisfactorilyby imagining the dielectric to be composed of many dipole moments of the typep = lnqd, in which q is the positive charge of the oppositely charged pair, d is theirseparation, and ln is a unit vector drawn from — q to +q. If P(x, y9 z) is the volumedensity of these elementary dipole moments, one can show3 that their aggregatedeffect is to cause an electrostatic field given by

E(x,y,z)=-V. n p - ^ s 4. r (-Vj-p)<*ri (

with S the dielectric surface and Fits volume. In (1.13a), Vs operates on the sourcepoint and VF operates on the field point.

Similarly, the magnetostatic behavior of magnetic materials can be explained interms of a collection of current loops with magnetic moments of the type m = lnna2l,where na2 is the area of the loop, / is the current, and ln is a unit vector normal tothe plane of the loop in the right-hand sense. If M(x9 y, z) is the volume density ofthese elementary loops, one can show4 that their aggregated effect is to cause a mag-netostatic field given by

ur N T7 w f f M X rfS . f V S X MdV~] n . . . .B( , j , , z) = V X ^ -4^^r^~ + J - 4 ^ ^ T ^ — J 0.13b)

In the more general situation that there is a primary charge distribution p(x, y, z)somewhere in space and secondary (or bound) charge distributions Pn on the dielectricsurface and — V • P throughout its volume, the total electrostatic field is E = Ej +E2, with Ex given by (1.4a) and E2 given by (1.13a). No additional information wouldbe conveyed by using Do = 60E in this situation. However, it is extremely useful5 to

3See, for example, R. S. Elliott, Electromagnetics (New York: McGraw-Hill Book Co., Inc.,1966), pp. 330-37.

4Elliott, Electromagnetics, pp. 404-7.

nbid., pp. 339-40.

t(-Vs^P)dV'4neQKJv

' P . dS ,AneQRs

(1.13a)

M x dSAn^RJ S

B(x,y,z) = WF X f V s X M dV~\

JVAnfJL^R (1.13b)

nbid., pp. 339-40.

1966), pp. 330-37.4Elliott, Electromagnetics, pp. 404-7.

The Far-Field Integrals, Reciprocity, Directivity

generalize the concept of D through the defining relation

D - 6 0 E + P (1.14a)

This insures the desirable feature that

V • D - 60V . Ex + e0V • E2 + V • P

= / > - V - P + V - P (1.15a)

V • D - p

at all points in space (both within and outside the dielectric), thus permitting theassertion that the flux lines of D start and stop on primary charge alone. If there are noprimary charges inside the dielectric, the D lines are continuous there. Outside thedielectric, (1.14a) reduces to D = e0E, which is consistent with (1.10a).

Since V X E = V X E! + V X E2, and since E! and E2 are both expressible asthe gradient of a scalar function, it follows that in this more general situation of pri-mary and secondary charge distributions,

V X E ~ 0 (1.15b)

However, one can see from the defining relation (1.14a) that V X D = V X Pand thus the generalized D, unlike E, may not be an irrotational field everywhere.

Many dielectric materials are linear (or nearly so), in the sense that P = #e60Eholds, where %e is a constant called the dielectric susceptibility. When this can beassumed, Equation 1.14a reduces to

D = 0 +/ t f)60E = 6E (1.16a)

where e is the permittivity of the dielectric medium. The quantity €J€0 = 1 + %e ismore useful and is known as the relative permittivity, or dielectric constant.

Similarly, in the more general situation that there is a primary current distribu-tion J(x, y, z) somewhere in space and secondary (or bound) current distributionsM X ln on the surface of the magnetic material and V X M throughout its volume,the total magnetostatic field is B = I*! + B2, with Bt given by (1.4b) and B2 given by(1.13b). No additional information would be conveyed by using Ho = JUQ lB in thissituation. However, it is extremely useful6 to generalize the concept of H through thedefining relation

H = / / o ' B - M (1.14b)

This insures the desirable feature that

V X H = //o'V X B, + //o V X B2 - V X M= J + V x M - V x M (1.15c)

V X H = J

6op. cit., Elliott, Electromagnetics, pp. 408-10.

8

1.3 The Introduction of Dielectric, Magnetic, and Conductive Materials 9

at all points in space (both within and outside the magnetic material) thus permittingthe assertion that H is irrotational except at points occupied by primary sources.

Since V • B = V • Bt + V • B2, and since Bj and B2 can both be expressed asthe curl of a vector function, it follows that in this more general situation of primaryand secondary current distributions,

V - B - O (1.15d)

However, one can see from the defining relation (Equation 1.14b) that V • H =—V • M, and thus the generalized H, unlike B, may have discontinuous flux lines.

Most magnetic materials are nonlinear, but in the exceptional case that linearitycan be assumed, M is linearly proportional to B and Equation 1.14b reduces to

H = n , B x = — (1.16b)

in which xm *s the magnetic susceptibility and ju is the permeability of the magneticmaterial.

Equations 1.15 are Maxwell's equations for static fields when dielectric andmagnetic materials are present. They are supplemented by Equations 1.14, one ofwhich links E, D, and the secondary sources P, with the other linking B, H, and thesecondary sources M.

The integral forms of (1.15a) and (1.15c) lead to

cb D • dS = primary charge enclosed (1.17a)J s

q> H • dl = primary current enclosed (1.17b)J c

Thus the generalized D and H satisfy Gauss' law and Ampere's circuital law, respec-tively, in terms of the primary sources alone. This is their principal utility. On the otherhand, E and B enter into a calculation of the force on a charge q moving through thefield. In the most general static source situation (primary and secondary charge andcurrent distributions), Equations 1.3, 1.4, and 1.13 combine to give

F-<?(E + v x B) (1.18)

which is the Lorentz force law.

When conductive materials are present and Ohm's law is applicable,

J - < T E (1.19)

at points occupied by the conductor, with a the conductivity of the material.71op. cit., Elliott, Electromagnetics, pp. 473-81.

V • B ~ 0 (1.15d)

B B(1 + Xm)l*0 J*

H = ^ (1.16b)

s

c

D • dS = primary charge enclosed (1.17a)

H • dl = primary current enclosed (1.17b)

F-<?(E + v x B) (1.18)

J - < T E (1.19)

1op. cit., Elliott, Electromagnetics, pp. 473-81.

at points occupied by the conductor, with a the conductivity of the material.7

which is the Lorentz force law.When conductive materials are present and Ohm's law is applicable,

1.4 Time-Varying Fields

If the sources become time-varying, represented by

p(x, y, z, t) coulombs per cubic meter (1.20a)

J(x, y9 z, t) amperes per square meter (1.20b)

and are assumed to exist in otherwise empty space, then Equations 1.5 need to begeneralized. Faraday's EMF law and the continuity equation linking charge andcurrent lead to the result that

dt

€o (1.21)

^o c2 dt

V . B = 0

in which c is the speed of light and E(x, y, z, t) and B(x, y, z, i) are now functions oftime as well as space. Equations 1.21 are Maxwell's equations in their most generalform for primary sources in empty space. If one uses (1.10) and the fact that ju0e0c

2 =1, these equations convert readily to the more familiar set

(1.22)

V • BEEO

If dielectric, magnetic, and conductive materials are present and are representedby time-varying dipole moments, current loops, and drifting electron clouds, respec-tively, if the defining relations in (1.14a) and (1.14b) are extended to apply when thefields and secondary sources are time-varying, and if Ohm's law (1.19) is still valid inthe time-varying case (and all of these are good assumptions in practical situations),then Maxwell's equations become8

V . D = pan <!-23>

V X H = J + ™

V • BEEO

*op. cit., Elliott, Electromagnetics, pp. 393-94, 464, 509.10

VXE=-?dtdBdt

V • D o = p

V X H o = J + ^<?D0

dt

V • BEEEO

(?Bdt

PdD

w(1.23)

V X E

V • D

V X H

V • BEEO

J 4

dBdt

P^ 0

J , 1 dE0O 1 ' C2 dt

V - B E E O

(1.21)

V X B =

V - E

V X E -

8op. cit., Elliott, Electromagnetics, pp. 393-94, 464, 509.

1.5 The Retarded Potential Functions 11

where now D and H have their generalized meanings, as given in the supportingEquations 1.14, and J is linked to E by (1.19) at all points occupied by conductor.

1.5 The Retarded Potential Functions

In antenna problems, one desires to find the field values at a point in terms of all thetime-varying sources that contribute to the fields. This implies an integration of (1.22)or (1.23), a relatively difficult undertaking that will be deferred until Section 1.7. Asimpler but less rigorous approach will be followed in this section, in which E and Bare not found directly, but are found instead through the intermediation of potentialfunctions whose relations to the sources are obtained intuitively.

Let the time-varying sources be given by (1.20) and be assumed to exist in afinite volume V in otherwise empty space. Then Maxwell's equations in the form(1.21) are point relations that connect E(x, y, z, t) and B(x, y, z, t) to the sources.Since V • B = 0, it is permissible to introduce a new vector function A(x, y9 z, i) bythe defining equation

B = V x A (1.24)

Because the divergence of the curl of any vector function is identically zero, it isapparent that (1.24) automatically satisfies (1.21d).

If (1.24) is inserted in (1.21a), one obtains

V X E = - | - ( V X A)<fr (1.25)

V X (E + A) = 0

where the dot over A implies time-differentiation. Since the curl of the gradient ofany scalar function is identically zero, the most general solution to (1.25) results fromthe introduction of a new scalar function cD(x, y, z, t) such that

E--A-V<D (1.26)

Equation 1.26 not only satisfies (1.21a) but, taken in conjunction with (1.24), providesa solution for E and B if the newly introduced functions A and <D can be related tothe sources. This can be done by forcing (1.24) and (1.26) to satisfy the two remainingMaxwell equations, that is, (1.21b) and (1.21c), notably the equations containing thesources.

If (1.24) and (1.26) are used in (1.21), the result is that

V X V X A = * - 1 ( A + V0>)

* i - • ( u 7 )

V(V . A) - V2A = -4r - 4~(A + v ^)Mo c

Equation 1.27 is a hybrid second-order differential equation (hybrid in the sense thatit contains both A and Q>) and as a consequence would be extremely difficult to solve.

d ,

dt

J 1. . - 1 „£Mo c

J 1..-1 r2Mo c

E - - A - V<D

(1.27)


Fortunately, a simplification is possible because, up to this point, only the curl of Ahas been specified, and a vector function is not completely defined until some specifica-tion is also placed on its divergence. It is convenient in this development to choose

(1.28)

for then (1.27) reduces to

(1.29)

Equation 1.29 is an inhomogeneous second-order differential equation in the unknownfunction A, with the negative of the reduced current distribution (which is assumed tobe known) playing the role of driving function. It is variously called the Helmholtzequation or the wave equation, the latter name arising because the solutions to (1.29)away from the sources are waves that travel at the speed of light.

The task remains to insure that (1.24) and (1.26) satisfy the remaining Maxwellequation (1.21b). Substitution gives

V • A + V2$ = — £f o

This is also a hybrid differential equation, but use of (1.28) converts it to

V * * - . * = - A (1.30)C €Q

Thus A and Q> satisfy the same differential equation, the only difference being thedriving function; in (1.30) it is the negative of the reduced charge distribution (whichis assumed to be known) which appears and governs <£.

The development has now reached the point that if (1.29) and (1.30) can besolved for A and 3>, then (1.24) and (1.26) can be used to determine E and B, and thegoal will have been achieved.

A solution of (1.30) can be inferred from the limiting electrostatic case. If thesources cease to vary with time so that p(x, y, z, i) —> p(x, y, z), then (1.25) and (1.30)reduce to

E = - V O (1.31)

V2<D - -P- (1.32)

in which O is now a time-invariant function, that is, 0>(x, y, z, t) —> <D(x, y9 z). But ifone returns to Section 1.2, it can be observed that (1.8a) and (1.31) are identical.Further, if the divergence of (1.8a) is taken and the result is combined with (1.5b),Equation 1.32 is reproduced, and its solution must be (1.9a), namely,

V . A = 6c2

V 2 A - Ac2

JMo1

Pf o

V . A + V2$

C* €o

_6_ p_

p^ 0

Jy ^€0RPit, n, QdvO(x, y, z) = [1.33)

1.6 Poynting's Theorem 13

Thus the limiting (time-invariant) solution to (1.30) is (1.33). How can this be used todeduce the general (time-variant) solution to (1.30)?

It can be argued that a change in the charge density at a source point (£, tj, Qcauses a distrubance which is not immediately felt at a field point (x, y, z), since thatdisturbance, traveling at the speed of light, must take a time interval Rjc to traversethe intervening distance R. Thus if one wishes to find the value of <D at the point(x, y9 z) at the time t, that is, <D(x, y, z, t), one should use the charge densities at thesource points (£, //, Q at the earlier times t — (R/c). This suggests that a solution to(1.30) might be

•*'.?,:.<) = l*i-1-^-RRICUy 0-34)

This is admittedly a highly intuitive argument, and a rigorous solution to this problemwill be presented in the development beginning in Section 1.7. However, if (1.34) isinserted in (1.30), one finds that it is indeed a solution.

By a similar argument it can be inferred that

A(z v z t) - f W>n,Z,t-Rlc)dv mA(z9y,z9t)- ^ 4nju~lR U }

Equations 1.34 and 1.35 are known as retarded potential functions because of the useof retarded time in the integrands. In conformance with the names already given totheir limiting forms in electrostatics and magnetostatics, O is called the electric scalarpotential function and A is called the magnetic vector potential function.

1.6 Poynting's Theorem

One of the most useful theorems in electromagnetics concerns the power balance in atime-varying electromagnetic field. To introduce this theorem, let it be assumed thatthere is a system of impressed sources J' that produces an electromagnetic field E',B1, and that this impressed field causes a response system9 of currents Jr to flow,creating an additional field Er, Br. If all these sources are in otherwise free space, theimpressed and response fields both satisfy Maxwell's equations in the form (1.21).The total current density and field at any point are therefore

J - J1 + Jr

E - E1 + Er

B = B1 + Br

9The decomposition of the total current system into impressed and response current densitiesis arbitrary, but often forms a natural division. For example, the currents that flow in a dipole may beconsidered to be a response to the impressed currents that flow in the generator and transmission linefeeding the dipole.

p(Z,t!,£,t-Rlc)dV4ne0R

<SK.x,y,z.t) =JV

(1.34)

Jtf, tf, C,t-R/c)dV47ZMolR

A(z, v, z, t) =!v

(1.35)

J - J1 + Jr

E - E1 + Er

B = B1 + Br

R/c) dV


If power is being supplied to the field, it must be at the rate10

d3P = - J * • EdV

But from Maxwell's equations (1.22),

J^VxHo-^-J'

so that

cPP = [-E- V X H0 + | r ( y ^ 2 ) + E • Jr] dV (1.36)

Application of the vector identity

V • (E X Ho) = Ho • V X E - E • V X Ho

coupled with the use of (1.22) gives

- E • V X Ho - V • (E X Ho) + Ho • ~at

As a consequence, (1.36) may be rewritten as

<PP = [ | - (yf o£ 2 + y/^-B2) + E • y + V . (E X Ho)] dV (1.37)

This result gives the power balance in a volume element dV. The left side of(1.37) is the instantaneous power being supplied by the impressed sources to dV. Thefactor

Tt&°E2 + WB2)is the time rate of change of density of stored energy.11 The factor E • J r representsthe power density being absorbed from the field by the response current density Jr. If,for example, the response current is flowing in a conductor, this term accounts forohmic loss. Alternatively, if J r is due to freely moving charges, E • J r accounts fortheir change in kinetic energy.

When the law of conservation of energy is invoked, it follows that the termV • (E X Ho) may be interpreted as the volume density of power leaving dV.

This conclusion can be seen from another point of view by integrating (1.37).With the aid of the divergence theorem, one is able to write

P = = : i t \ (~j€°E2 + - J J U ° l B 2 ) d V + f E - J T d v + i E x H o-</S (1.38)10op. cit., Elliott, Electromagnetics, p. 283.lxop. cit., Elliott, Electromagnetics, pp. 193-95, 283-84.

1.6 Ponyting's Theorem 15

The left side of (1.38) represents the entire instantaneous power being supplied by allthe sources. The first integral on the right side of this equation accounts for the timerate of change of the entire stored energy of the field. The second integral stands forthe power being absorbed by the system of response currents. The last integral there-fore represents the entire instantaneous power flow outward across the surface Sbounding the volume V. For this reason, one may define the Poynting vector as

(P = E X H O (1.39)

and place upon it the interpretation that it gives in magnitude and direction theinstantaneous rate of energy flow per unit area at a point. This is Poynting's theorem.

Since the units of E and Ho are volts per meter and amperes per meter, respec-tively, it is seen that the units of (P are watts per square meter.

Cases in which the currents and fields are varying harmonically in time occur sofrequently and have such importance as to deserve special discussion. Expressing allquantities in the form of a complex spatial vector function multiplied by eJcot, such as

E(x, y,z,t) = (R<£ £(x, y, z)ej(ot

one may write

(P = E X H O = #Zeia* + Z*e-j<°1) X (W0ejcot + Kte~Jat)

= %(S X Se? + £* X 3C0) + £(€ X KQeJ2mt + £* X Kte~j2oit) (1.40)

= £(R€(E X Hf) + £<Jte(E X Ho)

The term ^(R€(E X Hf) is independent of time and thus represents the time-averagevalue of <P, giving

<P = £<R*(E X HJ) (1.41)

The term £(R€(E X Ho) contains the factor ej2cot and thus represents the oscillatingportion of Poynting's vector. Therefore <P may be interpreted at a point as consistingof a steady flow of energy density plus a flow which surges back and forth at frequency2co.

Similarly

±€0E2 - i<f0E • E - ifoIK8^"1 + **e~Ja*) • (**** + &*e-Jai)]

= ^ 0 E . E* + i60(R€(E • E)

and

y~Q'B2 = i//o lB • B* + ^ ^ ( R ^ B • B)

The terms ^60E • E* and ^piolB • B* are independent of time and represent the tims-average stored energies; their time derivatives are zero. The terms £eo(R£(E • E) and•J;//o J(R€(B • B) oscillate at a frequency 2co and they represent the variable componentsof the stored energy.

Finally,

E • J r = ^Gte E • J r* + ^(R€ E • J r


Here again, the term ^(ft€ E • J r* represents the time-average power density beingabsorbed by the response currents; the term ffi<£ E • J r oscillates at a frequency 2coand represents the energy density being cyclically absorbed and released by theresponse currents.

With this formulation, Equation 1.38 may be rewritten in two parts. The time-average power balance is seen to be

P = l(R<e f E • Jr*dV + Wte cf E x H f . J S (1.42)

while the time-variable part, oscillating at a frequency 2co, may be written

P(2co) = £ f [^0(R<E • E) + ^ J ' O K B • B)] dVJv (1.43)

+ i(R€ f E • J r dV + Wte (f E X Ho • rfS

Thus, on the time average, the sources supply power only to that component of theresponse currents in phase with the electric field, represented by the first integral in(1.42), and to the net energy flow out of the volume V across the surface S. In addition,the sources may have to furnish energy and take it back at the cyclic rate 2co if theright side of (1.43) is not zero. However, in many practical circumstances, the indi-vidual integrals in (1.43) may not be in phase, but may be adjusted purposely so thatthey cancel each other, thus "matching" the generator.

B. INTEGRAL SOLUTIONS OF MAXWELL'S EQUATIONSIN TERMS OF THE SOURCES

The next four sections and two related appendices are devoted to a rigorous solutionof Maxwell's equations in integral form, giving the fields at any point within a volumeV in terms of the sources within V and the field values on the surfaces S that bound V.One advantage to this development, beyond its rigor, is that the results are in a perfectform to delineate approaches to the two types of antennas mentioned in the introduc-tion, namely those on which the current distribution is known quite well (such asdipoles and helices), and those for which the close-in fields are known quite well(such as slots and horns). Another advantage of the development is that it deliversthe retarded potential functions as an exact consequence of the central results.12

12Some authors, in contradistinction to using the Stratton-Chu formulation (which gives Eand B directly as integrals involving the sources), prefer to present a rigorous proof that the retardedpotential functions A and <I> are given by the integrals shown in (1.34) and (1.35). Then E and B followfrom (1.24) and (1.26). That approach is comparable in complexity to the Stratton-Chu development,and suffers from the ultimate disadvantage of requiring an ad hoc introduction of fictitious magneticsources without rigorous validation. The concept of fictitious magnetic sources arises naturally fromthe Stratton-Chu solution, and their results provide a sound basis for Schelkunoff's equivalenceprinciple. See Section 1.12.

1.7 The Stratton-Chu Solution 17

However, the reader who is not interested in delving into the complexities of thisdevelopment, and who is satisfied with the intuitive introduction of the retardedpotential functions given in Section 1.5, may wish to move directly to Section 1.11.This can be done without any loss of continuity.

1.7 The Stratton-Chu Solution

Since Maxwell's equations are linear in free space, no loss in generality results fromassuming that time variations are harmonic and represented by ejcot. The angularfrequency co may be a component of a Fourier series or a Fourier integral, thus bring-ing arbitrary time dependence within the purview of the following analysis. Accord-ingly, if f(x, y, z, t) is any field component or source component, it will be assumed\SasXf(x9y9z9i)=f{x9y9z)e^.

Further, it will be assumed that all of the sources are in what is otherwise freespace. This does not preclude the presence of a dielectric material if it is representedby a P dipole moment distribution, nor the presence of a magnetic material if it isrepresented by an M magnetic moment distribution, nor the presence of a metallicconductor if it is viewed as consisting of a positive ion lattice and an electron cloud,coexisting in free space. With dielectric or magnetic materials present, P = Jb andJm = V X M are the bound current density contributions to the total current densityJ. In the case of the metallic conductor, the electrostatic fields of the lattice and cloudare assumed to cancel each other, thermal motions are assumed to be random with anull sum, and only the oscillatory motion of the electron cloud is germane, making acontribution <JE to the total current density J, with a the conductivity of the metal.All of these assumptions concerning the representation of electrical behavior ofmaterials are valid in the practical realm of the actual materials used to constructmost antennas. For this reason the ensuing analysis has wide applicability.

Maxwell's equations (1.21), for time-harmonic sources in otherwise free space,can be written in the form

V x E = —jcoB

V x B ^ - ^ + ^ EK ' (1-44)

V • BEEO

Since c2ju0€0 = 1, the result if the divergence of the second of these equations is takenis the continuity relation

V - J = -jcop (1.45)

In all five of the above equations, the time factor eJat is suppressed and the fields arecomplex vector functions, as is the current density. The charge density is a complexscalar function.

V • E - P* 0


If the curl of either (1.44a) or (1.44b) is taken and then (1.44b) or (1.44a) is usedto eliminate E or B, one obtains the vector wave equations

V X V X E - /c2E = -j(o(-4rr)

VXVXB-B-VX (-^j)

(1.46)

(1.47)

in which k = co/c is called the propagation constant, for a reason that will emergeshortly. These last two equations can be integrated through use of a technique firstintroduced by Stratton and Chu, and based on a vector formulation of Green'ssecond identity.13

Consider a region V, bounded by the surfaces St • • • SN9 as shown in Figure 1.1.Let F and G be two vector functions of position in this region, each continuous andhaving continuous first and second derivatives everywhere within V and on theboundary surfaces St. Using the vector identity

V . [ A X B ] - B . V X A - A . V X B

FIG. 1.1 Notation for Vector Green's Theorem.

13J. A. Stratton and L. J. Chu, "Diffraction Theory of Electromagnetic Waves," Phys. Rev.,56 (1939), 99-107. Also, see the excellent treatment in S. Silver, Microwave Antenna Theory and Design,MIT Rad. Lab. Series, Vol. 12 (New York: McGraw-Hill Book Co., Inc., 1939), pp. 80-9. The pre-sent development is a reproduction, with permission, of what appears in R. S. Elliott, Electromagnetics(New York: McGraw-Hill Book Co., Inc., 1966), pp. 272-80 and 534-8, and differs from Silver'streatment principally in the nonuse of fictitious magnetic currents and charges.

Si Vi KVi

s2

sNK

VR

1«"

2

P)

1'n

V X V X E - k2E = -jco JUo\

Mo\J

1.7 The Stratton-Chu Solution 19

and letting A = F while B = V X G, one obtains

V - [ F x V x G ] = V x G - V x F — F - V x V x G

If A = G and B = V X F, then

V . [ G x V x F ] = V x F . V x G - G - V x V x F

When the difference in these results is integrated over the volume V, one obtains

f ( F . V x V x G - G . V x V x F ) < / FJv

= [ V . [ G X V X F - F X V X G ] ^Jv

If ln is chosen to be the inward-drawn unit normal vector from any boundary surfaceSt into the volume V, use of the divergence theorem gives

f ( F . V x V x G - G . V x V x F ) ^}V (1.48)

= - I (G X V X F - F X V X G) • ln dS

This result is the vector Green's theorem.Suppose that the fields E and B of (1.46) and (1.47) both meet the conditions

required of the function F in V; let G be the vector Green's function defined by

G = ^ a = ^a (1.49)

in which a is an arbitrary constant vector and R is the distance from an arbitrarypoint P(x, y, z) within V to any point (f, //, f) within V or on St.

As defined by (1.49), G satisfies the conditions of the vector Green's theoremeverywhere except at P. Therefore, one can surround P by a sphere £ of radius 5 andconsider that portion V of V bounded by the surfaces Sx • • • SN, I . Letting E = F,one finds that

f ( E . V s x V s X p - p ' V s x V s x E ) dVJV>

(1.50)= — (pxVsxE-ExVsXp)-U5

in which, since y/ is a function of (x, y, z) as well as (£, //, Q, it is necessary to distin-guish between differentiation with respect to these two sets of variables by subscripting

e-)kR

SX-'-SN

jkR

R a

SI---SN,X


the operators so that

and (1.51)

V — 1 ^ J_ 1 ^ 4 - 1 ^

It is shown in Appendix A that both sides of this equation may be transformedso that a is brought outside the integral signs, with the following result:

a • f (jcoy/^j - f- Vs^) dV - a . f (1. . E) V s ^

= - a • f [;co^(ln X B - (ln X E) X Vs^] dS

Since a is arbitrary, it follows that the integrals on the two sides of the above equationcan be equated, yielding

f ( j coy i - j - Vsv) dV - f [(l. . E) VsV/

+ (1. x E) x VsV/ -ja>y/{K X B)] 5 (1.52)

= f [(I. • E) \ s ¥ + (1. x E) x Vsyf -ja>y(h X B)] dS

where, for convenience, the surface integral over the sphere Z is displayed separately.It is further shown in Appendix A that the right side of (1.52) reaches the limit

—4nE(x, y, z), with (x, y, z) the coordinates of the point P, as X shrinks to zero.Therefore the limiting value of (1.52) is

i f a 5 3 )

+ j- [(1. • E) VsV. + (1. X E) X \s¥ ~ ja^iU X B)] dS/ f c JSI-SN

This important formula gives E at any point in the volume V in terms of the sourceswithin V plus the field values on the surfaces that bound V.

By letting B = F, one may proceed in a similar fashion to deduce a companionformula for B(x, y, z). Alternatively, the curl of (1.53) may be taken and then (1.44a)used to obtain B. By either procedure, one finds that

WJyMo

+ h [ P f (*. X E) + (!" X B) X \s¥ + (1. • B) V^l dSH7lJstSNL C J

(1.54)

d_dx

d

Tyd_dz

daA

dS dt] di

Si •SN,!.

'SI---SN,X

'SI---SN

Vol €o

fl~o Co

'E

1An

E(x,y,z):IV

p

fo\SW - j(OW £

Mo\dV

SI-SN

(1.53)

:(1. • E) VsW + (1. x E) x Vsy/ -jcoy/(U X B)]dS

147T

J^ o "V

Vsy/dV

SI-SN

Vs = l

B(x,y,z)

1.8 Conditions at Infinity 21

Equations 1.53 and 1.54 comprise a solution of Maxwell's equations in terms ofthe time-harmonic charge and current sources within V and the field values on theboundary surfaces S(.

1.8 Conditions at Infinity

Let it now be assumed that the surface SN of Figure 1.1 becomes a large sphere ofradius (R centered at the point P. Initially, (R will be taken great enough to enclose allthe sources J and p of the fields; ultimately (R will be permitted to become infinitelylarge. Under these circumstances, consider the contributions to (1.53) and (1.54) ofthe surface integrals over SN.

If 1& is a unit vector directed outward along the radius of the spherical surfaceSN9 so that 1^ = — 1B, one may write for the appropriate part of (1.54)

hJ pP^ ( l n x E) + ( 1 » x B) x Vs^ + (1° •B) V s ^] d s

= h J [~-$ ( 1 « x E) + (1<*x B) x ^{Jk + i)( 1 \i0~JkR

jk + ~k)r~R~dS (L55)

= ~k\ {~^ ( 1 < R x E )-( J k + i)[{KxlsXB)""(1(a• B ) 1 « ] ) e ~ w d s

Similarly, the appropriate part of (1.53) becomes

~ f [(1. • E) \sy/ + (1, X E) X \sy - j(oy{\u X B)] dSJs" (1.56)

If (R —> oo, since the surface of the sphere increases as (R2, the surface integral in (1.55)will vanish if

lim (RB is finite (1.57)

3 t [ ( l s W xE)-dJ] = 0 (1.58)

Similarly, the surface integral (1.56) will vanish if

lim (RE is finite (1.59)(R-oo

lim CRRla XB) + - 1 = O (1.60)«•-«• L cj

SN

icowc2

(ln X E) + (1. X B) X VSV + (ln . B) V s^ dS

1(R-B 1 < R I

I SN

1An SN

JG>\Cz 1« X E cB B

(R

e-jk&

01rfS

1(R X 1(R X B 1 « - B

lim (RE is finite(R-oo

lim (RB is finite(R-*oo

SN

jo> 1(R X BEc

E5t

e-Jk(KdS

(R1

471


Relations 1.57 through 1.60 are known as the Sommerfeld conditions at infinity.Expressions (1.57) and (1.59) are commonly called the finiteness conditions (End-lichkeit Bedingungen) and Expressions 1.58 and 1.60 are customarily called radiationconditions (Ausstrahlung Bedingungen). The finiteness conditions require that E andB diminish as (R~*, while the radiation conditions require that they bear the relation toeach other found in wave propagation in regions remote from the sources. (SeeSection 1.11.)

It is now possible to demonstrate the extremely important result that realsources, confined to a finite volume, always give rise to fields that satisfy the Som-merfeld conditions. To see this, consider Equations 1.53 and 1.54 when the onlyboundary surface is the large sphere SN, with radius that will be permitted to becomeinfinitely large. It shall be assumed that the real sources J and p are finite and con-fined to a finite volume VQ. With the surface SN becoming an infinite sphere, thevolume V in (1.53) and (1.54) also becomes infinite, but no convergence difficultiesarise with the volume integrals because the sources are all within Vo.

If one borrows from the results of Section 1.6, the fields over SN will consist ofoutgoing waves with power density E x Ho watts per square meter. Since the surfacearea of SN is increasing as (ft2, if there is even the most minute loss in V, the law ofconservation of energy requires that E and Ho diminish more rapidly than (ft"1, andthus Conditions 1.57-1.60 are satisfied. One can then conclude that in an unboundedregion, B(x, y, z) and E(x, y, z) are given solely by the volume integrals that appear in(1.53) and (1.54).

A check on this conclusion for the limiting case of no loss in V may be obtainedthrough an ordering of the terms that comprise the volume integrals. To see this,assume that there are no bounding surfaces except the infinite sphere SNi and that thesurface integrals involving SN in (1.53) and (1.54) are zero. Then, for this situation,Equations 1.53 and 1.54 reduce to

+ k y j ] dv

B(x,y,z) = ±[j^X\sy,dV (1.62)

where the second version of the integrand in (1.61) has been achieved with the aid ofthe continuity equation (1.45). It can now be ascertained whether or not E and B,when computed from (1.61) and (1.62), satisfy Sommerfeld's conditions at infinity.

Let an arbitrary point in Vo be selected as the origin and let r be the vectordrawn from the origin to the field point P(x, y, z); the vector drawn from the sourceelement to P will be labeled R. Then

(J • Vs) yj^w = (J • vs)[iR(> + ±yR

- 1 J d [7 ik 4- l V~ikRl +(ik _U [ V ^ / ^ ' ^ l R , -/^ <?1R\

E(x,^,z)1

4^ r

Pfo

V s ^ - ; c o ^J

Mo*dV - \_

4711

JyJMoJ . Vs V s^

(1.61)

^ / ^ o 1An

-jkR-

p-JkRddR R

e~JkR^/<?1R J*. dlR

/^^' /^ sin 0' (?0'/<7/?

(1.62)

1.8 Conditions at Infinity 23

in which spherical coordinates (r, 6\ 0') centered at P have been used and

i - R

Performing the indicated differentiations, one obtains

( j . vs) vsV, = {(j. wu[^{jk + ) - *] - jr(jk + i)\qi

The functions y/, Vs^, and (J • VS)VS^ are all seen to involve polynomials in thevariable R'1. Retain for the moment only first-order terms; then substitution in (1.61)and (1.62) gives

E(x,y, z) = -L J -^i_[-**(J • 1R)1R + M]^dV (1.63)

B(W) = 4 | ; ^ X 1 R ^ ^ (1.64)

But

R = [(x- Z)2 + (y~ n)2 + (?- 02]1 / 2

- [(r sin 9 cos 0 - ^)2 + (r sin 0 sin 0 - t])2 + (r cos 0 - 02]1 / 2

in which now conventional spherical coordinates (r, 0, 0) centered at the origin havebeen introduced. As P becomes remote, R can be expressed in the rapidly convergingseries

R - r ~ (<J sin 0 cos 0 + t] sin 0 sin 0 + £ cos 0) + O ^ 1 ) (1.65)

Similarly,

R-1 - r~l +0 ( r - 2 ) l imlR - l r

and thus as r becomes very large, Equations 1.63 and 1.64 may be written

E(x,y, z) = f^- J lr X (lr X -±T}eJ**dV+ 0(r-2) (1.66)

B(x,y,z) = re ^ f - ^ x lre^ dV + 0(r"2) (1.67)

in which <£ = ^ sin 0 cos 0 + // sin 6 sin 0 + £ cos 0.If one were to go back and include all the terms in the expressions for V s^ and

(J • VS)VS^, they would alter the results in (1.66) and (1.67) only at the level of 0(r2).Therefore these two expressions for B and E may be taken as exact.

In considering Expressions 1.66 and 1.67 with respect to the Sommerfeld con-ditions, one notices that the terms of 0(r~2) and below satisfy all four conditions and

RR

\vjcoe0

Mo1V

e-JkR

e~JkR


thus concern may be focused on the explicit first-order terms. But

lim rB - & lim e'Jkr f - 4 j X lTejk£ dV (1.68)

and, since the volume integral is a function of the source coordinates and the angulardirection to P, but not of r, this limit is finite. A similar argument establishes thatlim rE is also finite and thus both finiteness conditions are satisfied.

Further,

limrRl, XB) + ^ 1- l { . (1.69)

= lim €^L f \jk\T x - ^ x U ^ U U -A-V** dv

The integrand in (1.69) is identically zero and therefore Condition 1.60 is satisfied.In like manner, Condition 1.58 is also found to be satisfied. This supports the argu-ment that any system of real sources confined to a finite volume Vo gives rise to anelectromagnetic field at infinity that satisfies Sommerfeld's conditions, that the surfaceintegral over an infinite sphere SN gives a null contribution, and that in an unboundedregion the electromagnetic field at any point P, near or remote, is given precisely by(1.61) and (1.62).

Suppose now that parts of the volume Vo are excluded from V by the finite,regular closed surfaces S{ • • • £ , • • • . These surfaces may exclude some of the sourcesfrom V or not, but their presence does not alter the results at infinity. However, nowthe more general expressions in (1.53) and (1.54) apply, and one may conclude bysaying that these expressions are valid even if the volume V is infinite, so long as realsources in a finite volume are assumed. If the volume V is infinite, the surface atinfinity need not be considered.

This solution for E and B, given by Equations 1.53 and 1.54, is in a form that isconvenient for the purpose of drawing a distinction between two types of radiators.Type I antennas will be taken to be those for which the actual current distribution isknown quite well, such as dipoles and helices. Type II antennas will be those thathave actual current distributions which would be difficult to deduce, but which couldbe enclosed by a surface over which the fields are known with reasonable accuracy.These include horns and slots.

For type I antennas, there will be no volume-excluding surfaces and (1.53) and(1.54) will contain only volume integrals. For type II antennas, the volume-excludingsurfaces (usually only one) will be chosen to surround all the actual sources so thatthere are none to be found in the remaining part of space V. Thus for type II antennas,(1.53) and (1.54) will contain only surface integrals. In the developments that followlater in this chapter, it will be seen that it is useful to replace the field values occurringin the integrands of these surface integrals by equivalent sources. Thus for the remain-der of this book, type I radiators will be referred to as actual-source antennas andtype II radiators will be called equivalent-source antennas.

r->oo yMo1 eJk£,

1.9 Field Values in the Excluded Regions

Because of its bearing on the analysis of type II (equivalent-source) antennas, it isimportant to consider the values of the fields E and B at points inside the excludingsurfaces shown in Figure 1.1. In particular, let the field point (x, y, z) lie anywhere inthe volume Vx which has been surrounded by the closed surface Sx. A simple applica-tion of the general results in (1.53) and (1.54) gives

+ i f [(1. • E)VsV. + (1. X E) X \sy/ -jcoy/(h X B)] dSJSz • SN

Hn Jv+Vi Mo

+ Tn \ f ^ ( l n X E) + (1" X B) X V^ + (1" • B ) V s^ l^

Another way to view this situation is to imagine that Vx is the volume regioncomprising the collection of field points and that Sx is the sole surface, performing thefunction of excluding all the rest of space. From this viewpoint, a second applicationof the general results in (1.53) and (1.54) yields

[(1. • E)VsV> + (I. X E) X V s^ -ja^il. X B)] dSJ Si

B(x,y,z) = ±[ - i x V # ^Qt% JVx Mo

-L J [ ^ ( 1 . X E) + (1. X B) X VsV + (In • B)VS ]</S(1.73)

The negative signs in front of the surface integrals in (1.72) and (1.73) are occasionedby the fact that now the normal to the surface S1 is oppositely directed.

If the difference between these two sets of formulas for the fields within Vx isformed, one obtains

+ ^ J [(1. • E)VS^ + (1. X E) X VsW -jawiU X B)] dS

25

E(x,y,z) 1471 W + Vi

Pto

VsV~ jcoy/J

Mo1.\dV

14n SfSN

(1.70)

B(x,y,z)-- 14w.

JVsyrfK

Sf-SN

(1.71)

E(x,^z) 14rc

.£_•

^ 0> - yo>^ J

Mo1/dV

147T

(1.72)

0 147T,

P^o

fsV- jcoy/j

//o\dV

S I - S N

(1.74)


+ A f r^Po. xE| + ( i . « i )x»# t (i. • B)v,J <fs(1.75)

The right sides of (1.74) and (1.75) are seen to be exactly the same as the right sides of(1.53) and (1.54). Therefore one can conclude that if the range of the field point (x, y, z)is unrestricted, when (x, y, z) lies within V, Equations 1.53 and 1.54 will give the truefields E and B. However, when (x, y9 z) lies outside V, Equations 1.53 and 1.54 willgive a null result.

1.10 The Retarded Potential Functions: Reprise

If the volume Vis totally unbounded, Equations 1.53 and 1.54 give

E= fP^dV-jco f j^-.dV (1.76)

B - f JX Vfdv (177)Jv 47T//Q1

Since VF^ = — Vsy, and since J and the limits of integration are functions of (£, tf, Q,but not of (x9 y, z), these integrals may be written in the forms

E = -VP f -P- dV-jco f -r^jdV (1.78)

B = VF X f ^-rdV (1.79)

Therefore it is convenient to introduce two potential functions by the defining

relations

, * * % R dv ( L 8 0 )

<K*, r,0 = / r ^ ^ ^ r f K (1.81)

in which the time factor ejcot has been reinserted and e~JkR/R has been substituted fory/. The function A is called the magnetic vector potential function and <D is called theelectric scalar potential function.

Since k = co/c, one may write

exp [j(cot — kR)] — expf;W* — —Y\

14n

0 JvMo1 \s¥dV

14TT

JI-SW

icow,c2 ln X E !„ X B V s^ l n - B vsy dS

v 4neo

47T//Q1

v47lMo

lv4n€0

Jv4nfio

lK47C^o

Therefore it is convenient to introduce two potential functions by the definingrelations

A(x, y, z, t) J(& w, 0 ^ " * *^

^ iy, g y ^-*«4ne0R

1.11 The Far-Field: Type I Antennas 27

Therefore each current element in the integrand of (1.80) and each charge element inthe integrand of (1.81) makes a contribution to the potential at (x, y, z) at time t whichis in accord with the value it had at the earlier time t — R/c. But this is consistent withthe idea that it takes a time R/c for a disturbance to travel from (£, rj, 0 to (x, y, z).For this reason, (1.80) and (1.81) are often called the retarded potentials.

From (1.78) and (1.79),

E = - V < D - A (1.82)

R = V x A (1.83)

in which the subscripts on the del operators have been dropped, since A and <3> arefunctions only of (x, y, z) and not also of (f, tj9 £).

The differential equations satisfied by A and <D may be deduced by taking thedivergence of (1.82) and the curl of (1.83), which leads to

V»A-i.A=_J (1.84)C JXQ

V23> _ JL$ = _ A (1.85)c e0

These relations are valid whether J and p are harmonic functions of time or moregeneral time functions representable by Fourier integrals. A proof may be found inAppendix B.

All of the results in this section can be seen to be consistent with those obtainedin Section 1.5 by a different line of reasoning.

C. THE FAR-FIELD EXPRESSIONS FOR TYPE I(ACTUAL-SOURCE) ANTENNAS

In antenna problems, one is interested in determining the fields at points remote fromthe sources. This introduces several simplifications in the field/source relations, as canbe seen in the development in the next section.

1.11 The Far-Field : Type I Antennas

The typical situation for an actual-source antenna is suggested by Figure 1.2. Thesources are assumed to be oscillating harmonically with time at an angular frequencyco and to be confined to some finite volume V. There are no source-excluding surfacesSt. For convenience, the origin of coordinates is taken somewhere in V. It is desiredto find E and B at a field point (x, y, z) so remote that R »> max [£2 + rj2 + £2]1/2-Said another way, the maximum dimension of the volume V that contains all thesources is very small compared to the distance from any source point to the fieldpoint.


FIG. 1.2 Notation for Far-Field Analysis.

Because (x, y, z) is outside of V and thus is a source-free point, it follows thatMaxwell's equations (1.44) reduce to

V X E - -jcoB V x B = &E(1.86)

V • E — 0 V - B E E O

As seen either in the development of Section 1.5 or Section 1.10, B can be related tothe time-harmonic current sources by the equation

B - V X A

in which

^ ' - o - I * 6 * ^ "

(1.87)

(1.88)

with k = cole = 2n/X the wave number and R the distance from the source point(£> */> 0 t 0 t n e fi^d point (x, y9 z). From (1.86) and (1.87) it follows that, at source-

:«,*?,?)6/

• 0 -J /

&/

r

(x, y, z)

X

Zi

1.11 The Far-Field : Type i Antennas 29

free points (x, y9 z),

E = (c2ljco) V X V X A (1.89)

For this reason it is not necessary to find <J>. The charge distribution on the antennaneed not be known for far-field calculations; the current distribution will suffice. Theprocedure is reduced to finding A from (1.88) and E and B from (1.89) and (1.87).

The distance between source point and field point is given by

R = K* - O2 + O - n)1 + (z- 02]1/2

= [(r sin 9 cos 0 - £)2 + (r sin 9 sin 0 - rj)2 + (r cos 9 - Q2]172

= [r2 - 2r(£ sin 9 cos 0 + r\ sin 9 sin 0 + C cos 0) + £2 + >/2 + £2]1/2 '

= r — (£ sin 9 cos 0 + ^ sin 9 sin 0 + £ cos 0) + 0(r~J)

in which the last result is obtained via a binomial expansion. If (1.90) is inserted in(1.88) and terms of 0(r~2) are neglected, one obtains the far-field approximation

A(*,y, z, 0 = | ^ J J<& 9, Cy** dV (1.91)

in which

£ = { sin 0 cos 0 + ^ sin 0 sin 0 + C cos 9 (1.92)

The distance £ can be interpreted as the dot product of: (1) the position vectordrawn from the origin to (f, t], (); and (2) a unit vector drawn from the origin toward(x, y, z). The result in (1.91) can be given the interpretation that A(x, y, z, t) is expres-sible as the product of an outgoing spherical wave

0J(oot~kr)

(1.93)47T//01/-

and the directional weighting function

a(9,0) = \v jtf, , c y *fi ^ € (l .94)

The radiated power pattern of the antenna, given by the function (P(0, 0) watts persquare meter can be expressed in terms of this weighting function G(9, 0). To see thisrelation, one can first perform the curl operations indicated by (1.87) and (1.89).When this is done and only the terms in r~l are retained, it is found that14

E =jcolr X (lr X A) = -jcoAT (1.95)

H = M^B = ~ ( ^ ) l r X A = (-L)lr x E (1.96)

14The subscript zero has been dropped on H as a simplification, since it is unambiguouslyclear that the region is free space.

47T//01/-

eJ(oot~kr)


in which 1, is a unit vector in the radial direction and 77 = (/JLO/SO)1/2 = 377 ohms is theimpedance of free space. The transverse part of A is AT = leAe + 10A0. It can beconcluded from a study of (1.95) and (1.96) that the radiated E and H fields areentirely transverse, that E differs from AT only by a multiplicative constant, that H isperpendicular to E, and that

§\ = fl (197)

The complex Poynting vector yields an average power density which can bewritten (see Section 1.6)

<p(09 0) = i-(Re(E X H*)

(1.98)

It is customary to call that part of the radiation pattern associated with E0 the 0-polarized pattern, or the vertically polarized pattern, and to call that part of the radia-tion pattern associated with E^ the ^-polarized pattern, or the horizontally polarizedpattern. From (1.95) and (1.98), it can be seen that these two patterns are given by thefunctions

<Pr,,(0, <j>) = ^ y g L ] | a , ( 0 , <4)|2 (1.99)

<p,,,(^) = ± [ ^ L - ] | a , ( M ) l 2 (1.100)

Often one is interested only in the relative power densities being radiated in differentdirections (9, 0), in which case the factor ^[k2rj/(4nr)2] can be suppressed.

Since the unit vectors in spherical and cartesian coordinates are connected bythe relations

le = lx cos 9 cos 0 + \y cos 9 sin 0 — lz sin 9

1 = —lx sin 0 + 1 cos 0

it follows that the transverse components of (1.94) can be written in the forms

<**(0, 0) = f [cos 9 cos 0 Jx({, r\, 0 + cos 9 sin 0 Jy((, rj, QJv (1.101)-sm9J2({,t1,O]ejk£d£dtidt;

a,(9,0) = j ^ [-sin 0 jxtf9 n, o + cos 0 ./,(£ n, Q]eJk£ d{ dtj dt; (1.102)

These two equations are the key results of this development and form the basis ofpattern analysis and synthesis for actual-source antennas. If one starts with knowncurrent distributions, Q>9 and (£0 can be determined from (1.101) and (1.102) and thenused in (1.99) and (1.100) to deduce the radiation patterns. This is the analysis prob-

HE n (1.97)

i^nryk2rj

_2M aea* + 1

2_«..»•

(1.98)21

- ' .

X4nr)\• k2n

(47rr)2Jk2t]

- sine Jtf,ti,OW**d{dtidC

(1.99)

(1.100)

1.12 The Schelkunoff Equivalence Principle 31

lem. Conversely, if desired patterns are specified, (1.101) and (1.102) become integralequations in the sought-for current distributions. This is the synthesis problem.

The results of this section can be summarized by saying that when one is doingpattern analysis of a type I (actual-source) antenna, the steps to follow are these.

1. Place the known current distribution in (1.101) and (1.102) and determineatf, <t>) and Ct#(0, 0).

2. If the far-field power patterns are desired, use (1.99) and (1.100). Then | ae(9, 0) |2

and | C^(0, 0) |2 will give the vertically and horizontally polarized relative powerpatterns, respectively.

3. If the E and H fields are desired, use (1.95) and (1.96).

For pattern synthesis, (1.101) and (1.102) become integral equations in the unknowncurrent distribution with GLe(9, 0) and Cfc0(0, 0) specified.15

D. THE FAR-FIELD EXPRESSIONS FOR TYPE II(EQUIVALENT-SOURCE) ANTENNAS16

A distinction has already been made between antennas for which the actual sourcedistribution is known to reasonable accuracy and those for which it is not. In thelatter case, it is fortunately often true that the fields adjacent to the antenna are fairlywell known; it is then useful to surround the antenna by surfaces that exclude all thereal sources. If the Stratton-Chu formulation is used, the fields E(x, y, z)ejoit andB(x, y, z)ejcot can then be determined from Equations 1.53 and 1.54 with only surfaceintegrals involved.

An alternate (and equivalent) approach that is rich in physical insight is one inwhich substitute sources are placed on the surfaces enclosing the antenna. Thesesources must be chosen so that they produce the same fields at all points exterior tothe surfaces as the actual antenna does. The next two sections are concerned withdeveloping this alternate approach.

1.12 The Schelkunoff Equivalence Principle

The concept of equivalent or substitute sources is an old and useful idea that can betraced back to C. Huyghens,17 but the development to be presented here is patternedafter S. A. Schelkunoff.18

15Often it is a vexing problem to specify the phase distribution of &e and ct since all that mayreally be desired is some specified \ae(6, 0)| or |a0(0, <j))\. In such cases, one can search for that phasedistribution of <3Le and 0,$ which results in the simplest physically realizable current distribution. Thiscan be a much more formidable synthesis problem.

16Reading the material in Part D of this chapter can be deferred without any loss in continuityuntil Chapter 3 is reached.

17C. Huyghens, Traite de la Lumiere, 1690 (English translation: Chicago: The University ofChicago Press, 1945).

18S. A. Schelkunoff, "Some Equivalence Theorems of Electromagnetics and their Applica-tion to Radiation Problems," Bell System Tech. Jour., 15 (1936), 92-112.


In pursuing this idea, one finds that if the equivalent sources are to reproducefaithfully the external fields, electric sources alone will not suffice. It is necessary tointroduce fictitious magnetic sources. In anticipation of this, consider the situation inwhich real electric sources (p, J) create an electromagnetic field (E1? Bj) and magneticsources (pm, Jm) create an electromagnetic field (E2, B2). The properties of thesefictitious magnetic sources are so chosen that Maxwell's equations are obeyed in theform given below. Away from the sources, no distinction can be made that wouldallow one to determine which type of source had given rise to either field. The twosets of sources and fields satisfy

V x E j - —jcoBi

Mo c

v.El = feo

V • B, = 0The divergence of (1.103e) combined with (1.103h) reveals that V • Jm = —jcopm. Inother words, the manner in which the magnetic sources have been introduced insuresthat the continuity equation applies for magnetic as well as electric sources.

In a development paralleling what is found in Section 1.5, it is useful once againto introduce potential functions, this time by means of the defining relations

B, = V x A (a) E 2 - - V x F (b) (1.104)

As before, A will be called the magnetic vector potential function; by analogy, it isappropriate to call F the electric vector potential function. Equation 1.104a insurescompliance with (1.103d); similarly, (1.104b) is in agreement with (1.103g). Equations1.103a and 1.103f then lead to

V X (Ex +jcoA) EEE 0 V X (B2 + ^ F ) = 0

from which

E, = -jcoA - VO B2 - - - U J W F + V0>m) (1.105)c

with Q> and 0>m called the electric and magnetic scalar potential functions, respectively.If the total fields are E = Et + E2 and B = B! + B2, then

E = - V x F -jcoA - V<D (1.106)

B - V X A - Jj.(jcoF + VOJ (1.107)

Mo cJ , j(oE,

(a)

(b)

(c)

(d)

V X E , = —ir-ywB, (e)Mo

V X B2 = &E2 (f)

(1.103)V • E2 = 0 (g)

V • B2 = £ v (h)Mo

(1.103)cL

_ ja> E2

HZ1J™ -jcoB2

I*VIra

C*J™FB2

c21

V XE,

V X B,=

V - E , ~ -=—* 0

The divergence of (1.103e) combined with (1.103h) reveals that V • Jm = —jcopm. Inother words, the manner in which the magnetic sources have been introduced insuresthat the continuity equation applies for magnetic as well as electric sources.

In a development paralleling what is found in Section 1.5, it is useful once againto introduce potential functions, this time by means of the defining relations

B, = V x A (a) E 2 - - V x F (b) (1.104)

jcoF + V0>m)

V • E2 = 0

(h)

(g)

(0

(e)-jtoB,

^.

V X (E, + 70A) = 0

1

1.12 The Schetkunoff Equivalence Principle 33

Equations 1.103b and 1.103e can next be converted to the forms

V x V x A ^ - i - &(jcoA + V<X>) V x V x F ^ i - ^O'oF + VOJMo c JUQ C

If the divergences of A and F are selected to satisfy

V . A = - ^ * V - F = -&<b

then these hybrid equations reduce to

V2A + /c2A - — i j (a) V2F + k2F - - A - (b) (1.108)Mo Mo

Finally, (1.103c) and (1.103h) transform to

V2<£> + k2® = -JL (a) V2$w + fc2Om - - & s (b) (1.109)

The solutions for A and 0 have already been given (see Section 1.10) and thesolutions for F and Om are obviously similar. If the electric and magnetic sources areconfined to reside in surfaces, then lineal current densities K amperes per meter andKm magnetic amperes per meter replace J and Jm. In like manner, the areal chargedensities ps coulombs per square meter and psm magnetic coulombs per square meterreplace p and pm. The potential functions are then given by

C K(f n r\PJ(o>t-kR) C K (F n r\Pi^t-kR)

«*>*'•» = )™t%R ds ^y^-l^&^R ~ds

W(x, y, z, t) - ^ An^R ab wm{x, y, z, t) - ^ 4n^R as

(1.110)

Suppose one desires to find the values that these surface sources should have inorder to give a specified electromagnetic field external to S but a null field within S.As suggested by Figure 1.3a, let a contour C5 be constructed such that the leg ab isjust outside S and parallel to B tang; the leg cd is parallel to ab and just inside S; bothlegs have infinitesimal lengths dl. Since (1.103b) and (1.103f) combine to giveY x B = (J//Z01) + (jco/c2)E, integration of this result and the application ofStokes' theorem yields

<f B - r f l = f J ^ . d S + 1%1 E - d S (1.111)Jcd Js6Mo c Js6

in which Ss is the membranelike surface stretched over the infinitesimal rectangularcontour C6.

c2cL

Mo1

^0fn

4neQR

injUo'R

4ne0R

4nMo K

c2Mo1

PXL W, Oe^-kR)

ULnX)ej^-kR)dS

-dS

A(x, y, z, t) =Js

$O, y, z, i) =Js

V(x,y,z,t]

®m(x,y,Z,t)

f K.(£, >?, Q^to'-^>

' p,m(.(, n, Oei{c°'-kR)dS

dS

n.no)Js

is'


True fieldsoutside S

(b)

Fig. 1.3 Determination of Equivalent Surface Sources

True fieldsoutside S

As the legs be and da are shrunk toward the limit zero, with ab always outside Sand cd always inside, the electric flux enclosed goes to zero, the current enclosed isKdl, and the line integral in (1.111) gives Bi&ngdl, since there is no contribution frominside. In Figure 1.3a, K emerges from the paper if Btang is in the direction from a to b.One obtains the result that

X B (1.112)

with ln a unit outward-drawn normal vector.Similarly, if (1.103a) and (1.103e) are added and the result integrated, with the

contour taken so that its leg ab is parallel to Etang, one finds that

K m

Mo1In X E (1.113)

Next, imagine that an infinitesimal pillbox has been erected, straddling S asshown in Figure 1.3b. If the view in the figure were to be rotated 90°, one would see an

No fieldsinside S

Rectangularcontour ^

C r yb

Q

adL

In

(a)

S

No fieldsinside S

Edge view ofpillbox volume

c

-J ad

K

Mo1

K

s /

I *

s,.

1.12 The Schelkunoff Equivalence Principle 35

infinitesimal disclike area dS, with the upper surface of the pillbox just outside S andthe lower surface just inside S. Since (1.103c) and (1.103g) combine to give V • E =p/e0, integration plus use of the divergence theorem yields

{ E- dS = f £-dVJ s5 Jvs

€o

in which Ss is the total surface of the pillbox, enclosing the volume V5.Because there is to be no field inside S, as the height of the pillbox is reduced

toward the limit zero, with one pillbox face always on each side of S, in the limitEndS = (ps/e0) dS, with En the normal component of E. Thus

f - = l . . E (1.114)e o

In like manner, if (1.103d) and (1.103h) are combined and this process is repeated,one finds that

^ L - 1 D - B (1.115)Mo1

Schelkunoff's equivalence principle in essence asserts that, if the equivalentsources given by (1.112) through (1.115) are inserted in the potential functions (1.110),and the results are used in (1.106) and (1.107), the calculation of E and B will give thetrue fields at all points external to S and null fields at all points internal to S.

It is not immediately obvious that this should be so, since all that has been doneso far is to choose equivalent sources that would correspond to the situation that thetrue fields exist infinitesimally outside S and that no fields exist infinitesimally inside5, with no obvious indication that this will produce the proper field values at pointsfurther removed from S. However, Schelkunoff's assertion can be affirmed by fol-lowing his suggested procedure. If the factor e3(ot is suppressed, if y/ replaces e~JkR/R,and if equations (1.112) through (1.115) are substituted in (1.110), the result is that

A = ^ J (1- X *)VdS F = ~ 4 £ ( l n X E)y/dS

(1.116)

When these expressions for the potential functions are used in (1.106) and (1.107),and the vector transformations VF^ = — V s^and VF X [^(ln X E)] = VF^ X (ln X E)= (ln X E) X V s^ are employed, one finds that

E(x, y, z) = ± J [(ln -E)VS^ + (1. X E) X Vs^ - jcoy/(lD X B)] dS (1.117)

B(x,y, z) = ± J p ^ ( l n X E) + (1. X B) X Vs^ + (1. • B)VS^] dS (1.118)

Jv8*oJ s5

Mo1

4n

' 4ni An

4TT,

Ps

foIn * E

0fr

dVE - rfS

(1.115)l n ' BPsm

1 I 'dS

rf5c2

dS

dS1

1

A :

(1. • B)yr<Pm

F (In X Ety(In X B)¥<s

's(In • E)p

[(I. -E)VS^ + (1. X E) X Vs</ - y o ^ ( l . X B)] dS (1.117)

7o>y,c2 - (1 . X E) + (1, X B) X \sy/ + (1 . • B)V sy rfS (1.118)

4?rJ,1

4«J^'\_E(x, y, z) =

B(x,y,z)

s

1.


where VF and Vs are the del operators for the field point variables and source pointvariables respectively; they have been defined by Equations 1.51.

These integral solutions for E and B at the field point (x, y, z), in terms of thefield values over the surface S, are seen to be identical to the Stratton-Chu solutions1.53 and 1.54 for the case that all the real sources have been excluded from the exteriorvolume V. Since it has already been shown in Sections 1.7 and 1.9, via a direct integra-tion of Maxwell's equations, that (1.53) and (1.54) give the true fields at all pointsexterior to S, whereas they give a null result at all points interior to S, it follows thatSchelkunoff's equivalence principle has been established.

1.13 The Far Field: Type II Antennas

In a development paralleling what was done in Section 1.11 for actual-source antennas,the potential expressions (1.110) for equivalent-source antennas can be simplified ifthe field point (x, y, z) is remote from all the sources. The details need not be repeated,but the thread of the argument proceeds as follows.

Away from the sources, (1.103b) and (1.103e) give

Ex = 4 - V X B 1 = i - V X V X AJCO JCO

B2 = —i-v XE2 = 4-V x V X FJCO JCO

so that (1.106) and (107) simplify to

E = - V X F + 4 - V x V X A (1.119)jco v J

B = V X A + 4 -V X V X F (1.120)jco v J

As before, one can dispense with the need to know the charge distributions if thefields are only sought at source-free points; knowledge of the current distributions,which determine A and F, is sufficient.

The far-field forms of these vector potential functions can be written as theproduct of the outgoing spherical wave factor (1.93) with the directional weightingfunctions

<*(0,0)= f m,ri,0eJk£dS (1.121)

9(0, 0) - £ KB({, i/, QeJk£ dS (1.122)

When (1.119) and (1.120) are applied to the far-field forms of A and F and only theterms in r~x are retained, the result is

1.13 The Far Field: Type II Antennas 37

E = -jcoA? + jk(h X FT) (1.123)

H = ^ ' B = - ^ ( 1 , X AT) -jcoe0FT = I l r x E (1.124)

with AT and FT the transverse components of the vector potential functions. Onceagain it can be noted that the far field E and H are both transverse to the radial direc-tion and are perpendicular to each other, and that \E/H\ — r\.

In this case of equivalent sources, the complex Poynting vector gives as theaverage power density

<P(0,0) = -j<Rejf.-j(oAT +jk(h x FT)] X [ ( ^ ) ( 1 , X A?) + jme0Ft~

+ ^(Re(ae$* - a#ff?)]

A study of (1.123) reveals that the vertically polarized (Ee) pattern is related to Q.e and3^, whereas the horizontally polarized (E^) pattern is governed by Ct+ and 5> Thusthe component patterns are given by the functions

(Pr,^, 0) = ^J^Ljla^e, 0)|* + ±|SF^, 0)1* - -|(RK«^*)] (1-127)

Once again, the factor %[k2t]l(4nr)2] can be suppressed when only relative levels are ofinterest.

As before, the transverse components of ft and 31 can be obtained by expanding(1.121) and (1.122) into components. This gives

«e(0, <f>) = f [cos 0 cos <f> KM, 17. 0 + cos 0 sin 0 *,(£, V. 0

-siti0KAZ,ri,O]eJkSidS (1.128)

<%(0, 0) - j s [-sin 0 *,(& 7, 0 + cos 0 * # , n, Q]eJk£ dS (1.129)

3=^, 0) = j s [cos 0 cos 0 ^ ( f , j ; , 0 + cos 0 sin 0 KyJ&, V, 0

-smdKzm^,n,O]elkZdS (1.130)

ff,(0, 0) = f [-sin 0 X ^ , 7, C) + cos 0 ^ . ( { , 9> C)]e^£ dS (1.131)

E = -;o)AT+;A:(lr X FT)

H=«0-»B = JG>.

n-(i, x AT) -;cofoFT = i - l r X E1 , v

21<P(0, 0) -(Re\[-ja>AT + jk(l, XF,)] X

W-yofoF*

CfcT -12 i(47ir)2

k2rjM i

c(lr X ffT) X (ir x e$) + l

C* $

(1.125)

= 1 '2A:2M

'2{4nryi[aeaf + a#ay + (5,ffe* + s^*)l

F1

c2-&e(ae7* - a#ffj)+ —a

< P , . ^ , * )

<?ue, 0)

I T A:2/;

2L(4w)2Jl«^ ,0) | 2 + 1

c2 ^ , ^ ) | 2 + -|<Re(a^#*)l2c

(1.126)

(1.127)2c•7(Re(a^f)| a , (M) | 2 + ^|sFe(0,0)|2-

1c 2 1

k2tl(4»r)

12t

«W», 0)

«•(*, 0)

^ , 0)

[cos 0 cos 0 A,(f, r\, Q + cos 0 sin 0 #,(£ >/. 0

dS

dS

dS

dS


These four equations are the key results of this development and form the core ofpattern analysis and synthesis for most equivalent-source antennas.19 If one startswith known equivalent-current distributions, Q,g9 d^ $0, and 9^ can be determinedfrom (1.128) through (1.131) and then used in (1.126) and (1.127) to deduce theradiation patterns. This is the analysis problem. Conversely, if desired patterns arespecified, (1.128) through (1.131) become integral equations in the equivalent currentdistributions that are sought. This is the synthesis problem.

The results of this section can be summarized by indicating the procedure fordoing pattern analysis of a type II (equivalent-source) antenna.

1. Surround the antenna with a closed surface S over which the actual fields areknown, at least to a good approximation.

2. Use (1.112) and (1.113) to find the equivalent lineal current densities K(£, //, £)and Km(£, t], 0 on S.

3. Find aT(0, 0) and 3^(0, 0) from (1.128) through (1.131).4. If the component power patterns are needed, use (1.126) and (1.127) to deter-

mine them.5. If the far fields E and B are required, use (1.119) and (1.120).

For pattern synthesis, (1.128) through (1.131) assume the roles of integral equationsin the unknown equivalent-current distributions, with <2T(0,0) and 5^(0,0) specified.20

E. RECIPROCITY, DIRECTIVITY, AND RECEIVING CROSS SECTIONOF AN ANTENNA

This penultimate part of Chapter 1 is concerned with the development of severalconcepts that have proven to be extremely useful in antenna theory. The first of theseis the concept of reciprocity, based on a simple deduction from Maxwell's equations.The second (directivity) is a measure of the ability of any antenna to radiate prefer-entially in some directions relative to others. The last concept (receiving cross section)introduces a measure of the ability of an antenna to "capture" an incoming elec-tromagnetic wave.

1 Occasionally a design problem will be encountered in which the antenna is very long in onedimension and the sources are essentially independent of that dimension. It is then convenient toassume that the problem is two dimensional and use cylindrical coordinate expressions equivalent to(1.128) through (1.131). See Appendix G for the development of these expressions.

20The synthesis problem is actually quite a bit more complicated than this simple statementwould suggest. Often it is only (?r>9(Q, 0) and (Pr^(^, (f>) that are specified. The division into GtT(0, 0)and JFT(0, <t>) is immaterial to the desired result, but it may be critical in terms of physical readabilityof a synthesized antenna. Another difficulty is that the phase of the far-field pattern is seldom specified.This offers the antenna designer an added degree of freedom, but complicates the synthesis problem.One should strive for a phase distribution of the far-field pattern that permits the simplest physicallyrealizable antenna. This can be a formidable undertaking.

1.14 The Reciprocity Theorem

One of the most important and widely used relations in electromagnetic theory is thereciprocity theorem, which will be invoked many times in this text as various subjectsare presented. A derivation of this theorem is based on the idea that either of two setsof sources, (Ja, Jj,, p*, p^) or (Jb, Jb , /?b, pi), can be established in a region, producingthe fields (Ea, Ba) and (Eb, Bb), respectively. It is assumed that the two sets of sourcesoscillate at a common frequency. There may be dielectric, magnetic, and conductivematerials present in which some or all of these sources reside, but if so the electro-magnetic behavior of these materials must be linear. The equivalent situation of freeand bound sources in free space will be used to represent the behavior of the materials,as a consequence of which Maxwell's curl equations in the free space form,

V X Ea = ~-^k -jcoB* V x H a = Ja +jcoDa

"°b (1.132)V x Eb - — ^ -~jcoBh V X Hb = Jb -\-jcoBh

Mo

can be used to connect the fields and current sources for each set. Equations 1.132 area restatement of (1.103) in combined form, with D = 60E and H = pi^B. These curlequations can be dotted as indicated to give

Hb • V X Ea - - / / 0 H b • J t -yayi 0 H b • Ha

Ea • V X Hb = Ea • Jb +jcoe0E* • Eb

J (1.133)

Ha • V X Eb - ~/ / 0H a • Jb -jcojiiH* • Hb

Eb • V X Ha - Eb . Ja +jcoeQEh • Ea

Since

V . (Ea X Hb - Eb X Ha) - Hb • V X Ea - Ea . V X Hb - Ha . V X Eb

+ Eb • V X Ha

it follows from (1.133) that

f (Ea x H b - E b x H 8 ) . J S - f (Eb • Ja - Bb • J -- Ea - Jb

+ B a - Jhm)dV (1.134)

in which integration has been taken over a volume V large enough to contain all thesources of both sets, and in which the divergence theorem has been employed. Equa-tion 1.134 is a statement of the reciprocity theorem for sources in otherwise emptyspace, but with the possibility that some might be bound sources representing thebehavior of linear materials. Several special forms of this reciprocity relation haveproven useful and can be described as follows.

39

V X Ea = ~-^k -ycoBa V x H a = Ja + jajD*

"°b (1.132)V x Eb - — ^ -~jcoBh V X Hb = Jb +y©Db

Mo

ft-*1J i

JtMV

Hb • V X E" = -fioH" ' ^ -j<o/it>W ' Ha

E'.Vxff = E'-J l +j<of0E* • Eb

H« • V X Eb = -fiDH* - Jb -j(on0W - Hb

E" • V X Ha = Eh • Ja +ycoe0Eb • E»

(1.133)

>S(Ea X Hb - Eb X Ha) • dS =

V(Eb • Ja - Bb . Ja - Ea - Jb

+ Ba • Jb) dV (1.134)

+ Eb • V X Ha

V . (Ea X Hb - Eb X Ha) - Hb • V X Ea - Ea . V X Hb - Ha . V X Eb

Since

it follows from (1.133) that

jcoixQW • H b


1. If S is permitted to become a sphere of infinite radius, with the sources con-fined to a finite volume V, the fields at infinity must consist of outgoing sphericalwaves for which Ee = r\H$ and E^ = —Y[He. Under these conditions the surfaceintegral in (1.134) vanishes and one obtains

f (Eb • Ja - Bb • Jl)dV = [ (Ea • Jb - Ba • 3i)dV (1.135)

Equation 1.135 is a principal reduction of the reciprocity theorem, which is used incircuit theory to demonstrate a variety of useful relationships. It will be used in thistext to establish the equality between transmitting and receiving patterns for arbitraryantennas and to develop a basic formula for the mutual impedance between antennaelements.

2. Another important reduction of the reciprocity theorem can be derived byreturning to Equation 1.134 and considering the situation illustrated in Figure 1.4a.

(a) (b)

Fig. 1.4 Geometries for Two Applications of the Reciprocity Theorem

The volume V is enclosed between the surfaces Sx and 52, with S2 completely sur-rounding St. If all the sources are excluded by Sl9 so that none of them lie in V, thenthe right side of (1.134) has a null value. And, if S2 is once again permitted to becomea sphere of infinite radius, the fields at infinity again consist of outgoing sphericalwaves for which Ee = TJH^ and E$ -rjHei and the integral over S2 in (1.134)vanishes. One is left with

f (Ea X Hb - Eb X Ha) • rfS - 0 (1.136)

All sourcesin F1

Vi

V

Si

(a)

^2

All sourcesoutside Vx

Vx

Sx

1.1 5 Equivalence of the Transmitting and Receiving Patterns of an Antenna 41

In (1.136), dS is drawn outward from V, but no change in (1.136) occurs if dS is insteaddrawn outward from Vl9 the volume enclosed by St. Thus (1.136) can be interpretedby saying that if a surface St is constructed to enclose all the sources of both sets in afinite volume Vu then the fields caused by these sources satisfy the relation in (1.136).

Equation 1.136 will be used in Chapter 7 in the establishment of the inducedEMF method for computing the self-impedance of a dipole.

3. A variant on the previous reduction is suggested by Figure 1.4b. The closedsurface Sx excludes all the sources, that is, the volume V1 is source free. Applicationof (1.134) to this situation once again gives (1.136). This result will be used in Chapter3 in the derivation of a formula for the scattering from a waveguide-fed slot.

1.15 Equivalence of the Transmitting and Receiving Patternsof an Antenna

The reciprocity theorem can be used to establish the very important result that thetransmitting and receiving patterns of an antenna are the same. Consider the situationindicated by Figure 1.5, in which two antennas are sufficiently separated so that each

Fig. 1.5 Disposition of Two Antennas in Each Other's Far Field

^Antenna 2

r /

6/

Antenna \ ,

<-0^Y

X

Zi


is in the far-field region of the other. Spherical coordinates are arranged to placeantenna 1 at the origin and antenna 2 at the point (r, 0, 0). Both antennas can be assimple or complicated as one wishes, so long as they are composed of linear materials.It will be assumed that a transmitter is connected to one antenna via a suitabletransmission line and a receiver is connected to the other antenna, also via a suitabletransmission line.21

In accord with the notation used in Section 1.14, let the a-sct of sources occurwhen a transmitter is attached to antenna 1 and a receiver to antenna 2. The &-set ofsources will represent the situation when the positions of transmitter and receiver areinterchanged. The combination of transmitter and receiver used in the ^-situationneed not be the same as in the ^-situation.

It will be assumed that a cross section 1 can be found in the transmission lineconnecting antenna 1 to the transmitter (receiver) at which a single, clean propagatingmode exists, and that similarly a cross section 2 can be found in the transmission lineconnecting antenna 2 to the receiver (transmitter) where a single, clean propagatingmode exists.21

For the tf-situation, let electric and magnetic current sheets be placed at crosssection 1 so that the fields on the antenna side are undisturbed, but so that, with thetransmitter turned off, the fields on the transmitter side have been erased. FromEquations 1.112 and 1.113, these port sources are given by

Ka = ln X Ha Kt, - -juo1 ln X Ea (1.137)

in which ln points along the transmission line toward antenna 1 and Ea and Ha areevaluated in cross section 1. In like manner, let electric and magnetic current sheetsbe placed at cross section 2 so that the fields on the antenna side are not altered, butso that, with the receiver turned off, the fields on the receiver side have been erased.These port sources satisfy

K£ = ln X Ha Kt2 = ~-/iol ln X Ea (1.138)

with ln pointing along the transmission line toward antenna 2, and with Ea and Ha

evaluated in cross section 2.The effective replacement of the transmitter and receiver by equivalent sources

at ports 1 and 2 leaves intact all the a-sources and fields between these cross sections,including the radiation field transmitted by antenna 1 and received by antenna 2.

In precisely the same manner, equivalent electric and magnetic current sheetscan be found which, when placed at cross sections 1 and 2, can serve as proxies forthe transmitter and receiver in the ^-situation.

These two sets of sources and the fields they produce satisfy the reciprocitytheorem in the form of (1.135). The volume V over which the integration is to beperformed must encompass all the original sources between the two cross sectionsplus the equivalent sources in the two cross sections.

21 As a special case of this analysis, the transmission lines may be lumped circuits.

1.1 5 Equivalence of the Transmitting and Receiving Patterns of an Antenna 43

Consider any point between the cross sections that is occupied by sources. Ifthese sources flow in conductive material,

Eb • Ja = Eb • trEa Ea • Jb = Ea - o\Eb (1.139)

with a the conductivity at that point. Similarly, if the material is dielectric,

Eb • Ja = Eb • jcoeQXe^a Ea • Jb = Ea • jcoe0XeEh (1.140)

with Xe the dielectric susceptibility at that point. And if the material is magnetic,

Bb • J i = Bb • jcoMoXnfl* Ba . J^ = Ba • ja>fioXJP (1.141)

in which Xm ls t r i e magnetic susceptibility at that point.In (1.139) through (1.141), the parameters a, Xa a n d Xm c&n be functions of

position, depending on the composition and disposition of the materials that com-prise the two antennas and their feeds, but, with the assumption that all materials arelinear, these parameters are independent of the levels of the fields. Thus, for everysource point between the cross sections, equal contributions are made to the integralson the two sides of (1.135). What remains are the contributions made by the equiva-lent sources in the cross sections.

Equation 1.135 reduces to

f (Eb • Ka - Bb • Ki) dS = f (Ea • Kb - Ba • Kb) dS (1.142)JS1+S2 JS1 + S2

with St and S2 the cross sectional surfaces at ports 1 and 2. By virtue of the set ofrelations of the type (1.137), this can be converted to the form

f (Eb • ln x Ha + Hb . ln x Ea) dSSl+S* (1.143)

f (Ea . ln X Hb + Ha • ln X Eb) dS

It is demonstrated in textbooks dealing with transmission line theory22 that anypropagating mode can be represented by a voltage wave and a current wave, definedso that

E t t l | ( x j , z ) = V(z)g(x,y) (1.144)

ntang(x,y,z) = I(z)h(x,y) (1.145)

with Z the propagation axis and with the functions g(x, y) and h(x, y) characteristic ofthe given mode. The level of these characteristic functions is adjusted so that

f l.-tefrjOx h(x,y)]dS= 1 (1.146)

22See, for example, S. Silver, Microwave Antenna Theory and Design, MIT Rad. Lab. Series,Volume 12 (New York: McGraw-Hill Book Co., Inc., 1939), p. 55.


with S a cross-sectional surface. The functions V(z) and /(z) in (1.144) and (1.145) arecalled the mode voltage and mode current and are given generally by

V{z) = Ae~Jfiz + Be~^z (1.147)

l(z) = YQ(Ae-Jfig - Be"*) (1.148)

with A and B constants to be determined by the boundary conditions, with fi thepropagation constant and YQ the characteristic admittance of the mode.

When this representation is applied to the modes at St and S2, one finds that

E M n X Hf = lZl • Hi X E{ = V\l\\%x • h, X g l

H ! « U E i = lZl • E? X H? = VU\K • gi X h,

H| . ln X Eb2 = 122 . E\ X H | = VUiK ' g2 X h2

Substitution in (1.143) together with use of (1.146) gives

This is a key result of the analysis and can be interpreted as saying that the modevoltages and currents at the two ports satisfy the reciprocity theorem.

Next, let Zx t be the impedance of antenna 1 referenced at port 1, and let Z22 bethe impedance of antenna 2 referenced at port 2. Then

V\ = 1\ZXX Vb2 = lb

2Z22 (1.150)

Further, let ZRl be the impedance of the receiver transformed to port 1 in the b-situation, and let Z^2 be the impedance of the receiver transformed to port 2 in the^-situation. Then

V\ = -l\ZRl V\ = -Ia2ZR2 (1 .151)

When (1.150) and (1.151) are placed in (1.149), one finds that

The transformed receiver impedances are obviously independent of the direction (9, (j>)from antenna 1 to antenna 2 and, since the two antennas are in far fields of the other,so too are the driving point impedances Zx x and Z22. Thus

v vb

7f = *7> (1.153)

with K = [1 + {ZR2/Z21)]/[[ + (ZRt/Zn)], a constant.

V(z) = Ae--"" + Be~Jfi!

l(z) = Y0(Ae-Jfz - Be1"*)

(1.147)

(1.148)

EJ • 1. X H| = 1,, • H! X E\ = V\l\K • h, X g l

HJ . 1. X E* = 1,, • E? X H? = V\l\\%l - g, X h,

H| • in x E5 = K • E5 x Hi = vUiK • g* x h2

2 V'A = L v\n (1.149)

V\ = l\Z,x Vl = nz12 (1.150)

V\ = -I\ZRX V\ = -l\Zn (1 .151)

ft' + ® ] - * + @l)]ill \ z , , / J /fl_ \ z 2 2 / j^ r , , /zRI\i Kir, , /z,,\'i i -

n i\v°x -- KVI (1.153)

(1.152)

(1.151)

1.15 Equivalence of the Transmitting and Receiving Patterns of an Antenna 45

If Vf is held fixed and /£ is measured as a function of (0, 0) while antenna 2 ismoved along some programmed path on the spherical surface of radius r, the trans-mitting field pattern of antenna 1 is recorded. Reciprocally, if V\ is held fixed and /?is measured as a function of (0, 0) while antenna 2 is moved along the same pro-grammed path, the receiving field pattern of antenna 1 is recorded. But Equation1.153 leads to the conclusion that

n<*.*)= (^3)/f(M) (1.154)

In words, the normalized transmitting field pattern and the normalized receivingfield pattern of any antenna are identical.

Some features of this proof are worth noting. No specification of the size, shape,or type of either antenna was necessary, nor were there any restrictions on the typesof transmission lines feeding the two antennas, except that each should exhibit asingle, clear propagating mode at the chosen ports. The materials of which theantennas and their feeds were composed were arbitrary except that they needed to belinear. It was not necessary for either antenna to be matched to its transmission line,nor was there any requirement that the transmitter or receiver be matched to eithertransmission line. Also, there was no restriction on the orientation of antenna 2 as itmoved along its programmed path. It could be continuously reoriented to measureEe(0, 0), or £0(0, 0), or E2(Q, 0), or some arbitrarily shifting polarization. All that isneeded is for antenna 2 to replicate its orientation at each point along the path after ithas shifted from receive to transmit. One can conclude from this that the proof is verygeneral.

Equation 1.154 establishes the equivalence of the transmitting and receivingfield patterns of any antenna. A simple extension shows that this equivalence appliesto the power patterns as well. If (1.154) is multiplied by its complex conjugate, theresult can be used to deduce that

!l/?(M)l^, = ( y 1|/!(M)I2*« (1.155)

The quantities | / | | 2 RR1I2 and | / f]2 RRl/2 that appear in (1.155) are the powersabsorbed in the receiver when antenna 1 is transmitting and receiving, respectively.Since each is linearly proportional to the power density of the waves passing thereceiving antenna, it is proper to infer that they are measures of the transmitting andreceiving power patterns of antenna 1. With K' = (RRJRR2)\ KVIJV1|2, one can write

(Pre%e, 0 ) - x:'<P"(0,0) (1.156)

Care must be taken in interpreting (1.156). For example, if antenna 2 is linearlypolarized and always oriented as it moves along its programmed path, in order toreceive or transmit only 0-polarized waves, then (I.156) becomes

<PJ"(0, 0) = tf'<PJ'(0, 0) (1.157)

2KVlV

2

21

|/f(M)|2*JH (R*i\RR1>

v?/KVfr


from which one can conclude that the normalized 0-polarized component of thepower pattern of antenna 1 is the same for receive and transmit. Similarly, if antenna 2is linearly polarized but aligned to receive or transmit only 0-polarized waves, (1.156)reduces to

6>;"(0, 0 ) - K'<P{{0, <t>) (1.158)

and once again equivalence is demonstrated in the component power patterns forantenna 1. And if, for example, antenna 1 is linearly polarized with only an Ee electricfield, then (1.158) gives a null result, as it should. If antenna 1 does not radiate an E+field, it cannot detect an incoming E^ field.

The sum of Equations 1.157 and 1.158 shows that the total power patterns areequivalent:

<%UM) = * ' ^ U M ) (1-159)

Acceptance of the conclusion that the normalized total power patterns of anyantenna are the same for transmit and receive, and thus that one need not determineboth, still leaves a measurement difficulty that should be noted. This concerns the factthat not any antenna can be chosen to play the role of antenna 2, make one traverseof the programmed path, and at each point in the path be oriented so that the receivedpowers in (1.155) coincide with the power densities in (1.159). This will occur only ifantenna 2 \s polarization-matched to antenna 1. For example, if antenna 1 is circularlypolarized, antenna 2 must be circularly polarized in the proper screw sense in order tohave the received powers in the a- and -situations that can be interpreted as the totalradiated and received power patterns of antenna 1.

However, if one is content to use as antenna 2 a linearly polarized antenna,make two traverses of the programmed path, one with 0-orientation and the otherwith 0-orientation, and keep transmit power and receiver sensitivity stable, then theseparate measurements give the component power patterns. Their sum gives the totalpower pattern, and Equations 1.157 through 1.159 indicate that it does not matterwhether the measurements are made with antenna 1 transmitting and antenna 2receiving, or vice versa.

1.16 Directivity and Gain

Often a principal goal in antenna design is to establish a specified radiation pattern(P(0, 0) watts per square meter through a suitable arrangement of sources. Thespecified pattern frequently embodies the intent to enhance the radiation in certaindirections and suppress it in others. A useful measure of this is the directivity, whichis simply the radiated power density in the direction (0, </)) divided by the radiatedpower density averaged over all directions; that is,

IK0,*) = , , , , . ^ ' ^ d-160)(I/4«r2) 9(0', <t>')r2 sin 9' dO' d<j>'

Jo Jo

1.16 Directivity and Gain 47

Equation 1.160 contains the implications that the origin for spherical coordinates hasbeen chosen somewhere in the immediate vicinity of the antenna, and that powerdensities are being evaluated on the surface of a sphere whose radius r is large enoughto ensure being in the far field of the antenna.

If the radiation intensity is defined by

P(09 </>) = r2(P(0, 0) (1.161)

then, since (P(0, <f>) is measured in watts per square meter, it follows that P(9, (f>) ismeasured in watts per steradian. Substitution in (1.160) gives the equivalent expres-sion

D ( M ) - r f 2 a 4 7 * m 0 ) (1.162)P(9\ f ) sin 9r dO'dp

Jo Jo

The value D{9, </>) is a pure numeric. It will have a value less than unity indirections in which radiation has been suppressed, and a value exceeding unity wherethe radiation has been enhanced. If (0O, 0O) is the direction in which the radiationintensity is greatest, then D has its largest value at (0O, 0O) and D(90, 0O) is the peakdirectivity.

In characterizing an antenna, one must be careful to distinguish between direc-tivity and gain. Directivity is used to compare the radiation intensity in a given direc-tion to the average radiation intensity and thus pays no heed to the power losses inthe materials comprising the antenna. Gain includes these losses, and the definition ofgain is therefore

in which Paccis the total power accepted by the antenna from the transmitter, measuredin watts. The denominator of (1.163) is the value, in watts per square meter, thatthe radiated power density would have if all the power accepted by the antenna wereradiated isotropically. Since the power accepted is greater than the actual powerradiated, the denominator of (1.163) is larger than the denominator of (1.160), and,as a consequence, G(9, 0) < D(9, 0).

Most antennas are constructed of linear materials; in this case, one may arguethat

Pace = KL in CK(P(9\ 0 > * sin 9' d0§ dp (1.164)Jo Jo

with KL a pure real constant that has a value somewhat greater than unity. When thisis so, Equation 1.163 becomes

G(0, 0) = M - i ) (1.165)

P(9, 0) = r><?(9, 0) (1.161)

w.e,*) = 4nP(d,6){" [ P(6',P)wiO'd9'dpJo Jo

(1.162)

G(9, $)" P«J4nr*__ mt) (1.163)

P =•*- ace Jo JQ

*(P(0',fy2sin0V0'#' (1.164)

G(M) = ^KL

D(0,<t>) (1.165)


The gain and directivity differ by a multiplicative factor that is independent of direc-tion. In particular, the peak gain occurs in the same direction (0O, 0O) as the peakdirectivity.

Often, gain and directivity are expressed in decibels (dB). From (1.165),

logloG(0, 0) = logIO/)(0, 0) - logl0KL (1.166)

The gain in any direction is seen to be 10 \og10KL decibels below the directivity in thatdirection; 10 log1QKL thus represents the power losses in the materials forming theantenna.

For some applications it is useful to introduce the concept of partial directivityand partial gain. As an example of how this is done, a return to Equations 1.98through 1.100 or 1.125 through 1.127 helps to recall that

<W 0) = <Pr.e(0, 0) + &rM 0) (1-167)

If this relation is inserted in Equation 1.160, it can be seen that it is possible to write

D(69 0) - Z>'(0, 0) + D"(09 0) (1.168)

in which

D'(6,4>) = r . rz*'^0'® (1-169)(l/4w2) 9(0', 0>* sin 9' dB' # '

Ja Joand

D"(d, <f>) = r ( . 2 , 5 * ( M ) (1-170)(1/47U-2) &(B', 0 > 2 sin 6' dB' d<j>'

Jo Jo

are the partial directivities associated with the -component and 0-component pat-terns, respectively. Similar definitions follow readily for the partial gains.

An example of the utility of this concept would be when an antenna is to bedesigned to give peak radiation at an angle (0O, 0O), but all the radiation should be0-polarized; any 0-polarized radiation is unwanted, but for practical reasons somemay be unavoidable. In such a circumstance it is the peak partial directivity Df(9Q, 0O)that is a pertinent measure, not the peak total directivity D(0O, 0O).

The division of the total power pattern into components can be done in otherways than the 0/0 partition indicated above. For example, the decomposition couldequally well be into right-handed and left-handed circularly polarized componentpower patterns. In that case one could identify right-handed and left-handed partialdirectivities and gains.

1.17 Receiving Cross Section

A receiving antenna will absorb energy from an incident plane wave and feed it via atransmission line to its terminating impedance. A useful measure of its ability to dothis results from introducing the concept of the absorption cross section of the antenna

1.17 Receiving Cross Section 49

or, as it is more commonly known, its equivalent receiving cross-sectional area. If S isthe power density of the incoming plane wave in watts per square meter and Pr is theabsorbed power in watts, then the equation

Pr(0,4>) = SAr(0,<l>) (1.171)

serves to define the receiving cross section, in square meters, as a function of the angleof arrival of the incoming signal. In order to have Ar(9, <f>) be a maximum measure ofthe capture property of the antenna, it is customary to assume that the incoming planewave is polarization matched to the antenna, and that the antenna is terminated by amatched receiver. With these assumptions, Equations 1.155 and 1.159 are applicableand one can write

SA,(O,<I>) = y |/?(M)l2**i = Pr*M4>) = K'WoUO^) (1172)

An integration of (1.172) gives

±j" j2nAr(0', 0') sin 9' dO1 d<f>'= ^ ^ £ (?ZUO\ 0> 2 sin 9' dO* d# (1.173)

If the ratio of (1.172) to (1.173) is taken, one obtains

^M> = D(e^) (1174)Ar

in which D(0, 0) is the directivity of antenna 1 when it is transmitting, as given by(1.160). Then Ar is the average receiving cross section of antenna 1, defined by

Ar = ^*^Ar{e\V)<fa0'd0ldV (L175)

It is a remarkable fact that the average receiving cross section Ar is the same forall lossless antennas that are polarization matched. This can be demonstrated asfollows.

Consider again the situation of two antennas, depicted as in Figure 1.5, withantenna 1 transmitting and antenna 2 receiving in the a-situation and the reverseoccurring in the Z?-situation. To obtain maximum power transfer, assume that in thea-situation the transmitter attached to antenna 1 has an internal emf Vg and an internalimpedance that has been adjusted to equal Zfx, with Zx x the driving point impedanceof antenna 1. Similarly, in the -situation, let the transmitter attached to antenna 2have an internal emf Vg and an internal impedance Z?2, with Z2 2 the driving pointimpedance of antenna 2.

In the a-situation, /? = Vgl2Rlx and the power delivered to antenna 1 is^l/fl2/?,! = \Vg\

2j%Rlx. If the losses in the antenna can be neglected, all of this

21

Anr1K'

4nS

Ar1

47T .

(1.172)sAxe, 0) =

*n fin

'o JoJo Jo

*n r>2n

Ar(0',<t)')sm9'd6'd(l>'*n ?ln

0 JO

A,(O, <t>)Ar

'-D(0,4>) (1174)


power is radiated, with an average density in watts per square meter given by

4nr2

When antenna 2 is located at the point (r, 9, 0), the power density in the wave arrivingfrom antenna 1 is given in watts per square meter by

g(M) = |K'|2,ff*"fl.(M)

in which Dx(0, 0) is the directivity of antenna 1. If use is made of (1.171) with S =(P(0, 0), it can be argued that the power absorbed by the receiver attached to antenna 2is given by

Pr = ^'^"Dtf, 4>)Art2(0, 0) watts

with Ar2(6, 0) the receiving cross section of antenna 2. Use of (1.174) converts this to

Pr = lVtU;*2

RllD1(0, 0)/)2(0, 0K > 2 (1.176)

The power absorbed is also given by (l/2)(Re I%[?ZR2 but, with a matched receiver,ZR2 = Zf2, and thus

When (1.176) and (1.177) are combined, the result can be written in the form

DM, 4>)D2(0, <l))Ar>2 = 16nr>M^2R" (1.178)

If this analysis is repeated for the fe-situation one finds that

^ i (MMM,i = i6^a|/f|a^y2^ (1.179)

The currents and voltages in the two situations are related generally by Equation1.152. In the circumstance being considered here, V? = TfZtl = (Vg/2R1 i)Zll9 V\ =/ | Z 2 2 = (VJ2R22)Z229 ZRl = Zfu and ZR2 = Z?2; as a consequence of this, (1.152)reduces to

/f = /f (1.180)

Hence, upon comparing (1.178) and (1.179), one can see that

A r A =Ar>2 (1.181)

4nr2W.WSRti

4nr2\vgmRu

4nr2W.V&Rn

4nr2\vK\mRlx

\vg\2

2 ^2 ^ 1 1 ^ 2 2

\Vg\*\nm±Sti

(1.181)A T t \ — A r > 2

(1.177)Pr=t\H\2Rl2

i />.(MM,.2(M) watts

(1.178)

1.17 Receiving Cross Section 51

Since antennas 1 and 2 are completely arbitrary (except that they must be polariza-tion-matched), Equation 1.181 is a general result.

The value of the constant Ar for linearly polarized antennas can be deduced asfollows: Let antenna 1 be completely arbitrary and located at the origin, as shown inFigure 1.5, except that it is linearly polarized and has been oriented to transmit an Efield that has only a 0-component. Antenna 2 is a single current element of length dl,located at the point (r, 9, 0), and oriented parallel to 10 so that the two antennas arepolarization matched.

In the ^-situation, let antenna I be transmitting with antenna 2 absent. In the^-situation, the current element \Jt\ dl (antenna 2) is present and radiating, andantenna 1 is receiving. Port 2 is taken to be the 0-directed line segment of length dllocated at (r, 0, 0). In this case the reciprocity relation (1.149) becomes

V \l\ -f- V 2*2 — y 1M

which can be rewritten in the form

V\I\ - VUi = Eae(r, 0, 0)/} dl (1.182)

Since /f = Kf/Zn when antenna 1 is transmitting, and /? = -~V\\ZRX = —Vf/ZTiwhen antenna 1 is receiving, (1.182) assumes the form

Vf V\Zx± + f * = -Mn^VtVi = EUidl (1.183)

When (1.183) is multiplied by its complex conjugate, the result is

^ i | K f ^ f | a = |£g|2|/Jrf/|2 (1-184)— i

| Z n

In the ^-situation, antenna 1 accepts an amount of power given by

i ] \va\2

racc — ~Y\1\\ ^ 1 1 — T | 7 2 ^M 12 | Z U

from the transmitter and, if losses in antenna 1 are neglected, all of this power isradiated. The power density at (r, 0, <f>) in the a-situation is, therefore,

1 | £ j ( r , f l , 0 ) | 2 _ I ^ l 2 R

D^4>) (] 1 8 5)

When | E% |2 is eliminated from (1.184) and (1.185), one obtains the result that

\V»\2 = 1\ Z^/ZR]' 0.(0, 0) 1K dl\z (1-186)

^1 \Z\ 1 I \ 1 IZ _4_ y* JD

11 -f ZJ\\ £f\ 1 1

2 n |Z M | * " 4 ^1 | £ g ( r , 0 , 0 ) | 2 ^ |KfP „ /? , (g ,0)

« n

IZ i . l 44/?,? KfK?|2 = | £ g | 2 | / J < « l 2

l\=-V\\Zn = -V\fflx

(1.183)

V \ l \ "f- V 2 * 2 — ^ 1 7 1

Kf/? - V\I\ = E&r, 6, m dl (1.182)

\6xr2nlz^i'/Ru| K f | 2 : U/),(0, 0)1/£<//p (1.186)

ajb ra1b WbTa


In the ^-situation, the receiver attached to antenna 1 absorbs the power

i'^-^I^^TJIS*11 (L187)

since ZR1 = ZJV This absorbed power can also be expressed in terms of the powerdensity in the waves radiated by the current element and the receiving cross section ofantenna 1. From Equations 1.99 and 1.101 the maximum23 power density radiated bya single current element is

and thus the power absorbed by antenna 1 is also given by

Pabs = \-^y2\ndl\2AU^<l>) (1-189)

If (1.187) and (1.189) are combined and the result solved for | V\ |2, further combinationwith (1.186) gives

^ . i ( M ) = ^ J > i ( M ) (1-190)

and thus the universal value of the average receiving cross section for linearly polarizedantennas is X2/4n.

Equation 1.190 is an extremely useful result. It permits computation of theoptimum power level in a receiver which is attached to an antenna of peak directivity£(0o> 0o) when the power density in the incoming signal is known. This value isdiminished slightly by the losses in the antenna. It is also diminished by the multipli-cative factor (1 — |F|2) when the receiver and the antenna are mismatched, with Fthe reflection coefficient.24

F. POLARIZATION

This concluding section of Chapter 1 is concerned with characterizing the polarizationof an electromagnetic field far from the sources which produce it. Such characteriza-tion is important in many practical applications. Prominent examples include thefollowing. (1) For purposes of optimizing propagation through a selective medium(such as the ionosphere), or optimizing back-scattering off a target, it may be desirableto specify the polarization the wave should have. This places a constraint on thedesign of the transmitting antenna. (2) When a sum pattern is required to have a

23It is the maximum value that should be used since the current element is oriented so thatits maximum power density is directed at antenna 1.

24See, for example, Silver, Microwave Antenna Theory, and Design, pp. 51-53.

*abs12

®r.e1 k2ti2 (4nr)2 \IUI\2 (1.188)

* i i

<t>)

1.18 Polarization of the Electric Field 53

specified polarization and low side lobes, it is important to check that the antennabeing proposed does not produce a cross-polarized pattern at a height which exceedsthe desired side lobe level. This possibility exists, for example, with parabolic reflectorantennas. (3) The polarization of an incoming wave may have to be accepted, whichplaces a constraint on the design of an antenna that will receive this wave optimally.(4) The polarization of an incoming wave may be unpredictable, in which case it maybe desirable to design a receiving antenna which will respond equally to all polariza-tions. To be equipped to deal with these and similar problems, it is important to beable to describe the polarization of an electromagnetic wave unambiguously.

1.18 Polarization of the Electric Field

It has been shown in Sections 1.11 and 1.13 that the far field of a transmitting antennacan be viewed as the product of an outgoing spherical wave and a complex directionalweighting function. For the electric field (which is conventionally used as the vehiclefor describing polarization), this complex directional weighting function is given by(1.95) for type I antennas and by (1.123) for type II antennas. In either case, at a farfield point (r, 9, (/>), the electric field can be represented by

E = (1,E, + l,E,y«" (1.191)

when time-harmonic sources are used in the transmitting antenna.The functions Ee(r, 9, 0) and E^r, 9, <f>) that appear in (1.191) are, in general,

complex. If this is recognized by the notation

Ee = E'e + jE'i Et = E'++ jE'i (1.192)

then it can be appreciated that what is really meant by (1.191) is that

E(r, 9, 0, 0 = <Re[l,(E'$ + jE'£) + HE', + JE'W* m)

= \6{E'e cos cot — E'e sin cot) + \^{E^ cos cot — E'l sin cot)

with E'0i E'e\ E'^ R'l all real functions of r, 9, and 0.Equation 1.193 can be rewritten in the form

E = \eA cos (cot + a) + \^B cos (cot + ft) (1.194)

in which

A = V(E'e)2 + (E'e')1 B - J(E',Y + (E'tfF» f> (1.195)

a = arctan §f fi = arctan ^be L4>

With no loss in generality, the origin of time can be selected so that a = 0 (that is,E'l = 0). Then (1.194) becomes

E = y c o s © r + \^B cos (cot + fi) (1.196)

E = (\eE6 + l,E,y«' (1.191)

E$ = E'9+jE'l E, = E',+jEf; (1.192)

(1.193)E(r, 9, 0, 0 = (Re[ld(E'e + jE'£) + 1,(^ + / W

= \e(E'e cos cot — E'e sin cot) + \^{E^ cos cot — E'l sin cot)

E = \eA cos (cot + a) + l^B cos (cot + {!) (1.194)

E = \9A cos cot + \^B cos (cot + fi) (1.196)

E'e

17"^ea = arctan

A = V(E'e)2 + {E'lY B = v W 2 + (E';y

0 = arctan ^J7"

E*

(1.195)

in which

]ejcot

+ p)

54 The Far-Field Integrals, Reciprocity. Directivity

purpose of identifying its polarization.

(a) LINEAR POLARIZATION If B = 0, the electromagnetic wave is said to belinearly polarized in the ^-direction. Similarly, if A = 0, the wave is 0-polarized. Butmore generally, if /? = 0 but A, B ^ 0, the 9 and $ components of the electric field arein phase. The polarization is then tilted, but it is still linear, as can be seen from thetime plots of Figure 1.6a. Therefore the most general example of linear polarizationoccurs when E9 and E$ are in phase.

(b) CIRCULAR POLARIZATION If A - B and fi = -90° , Equation 1.196becomes

E = A(le cos cot + 1* sin cot) (1.197)

In this case the magnitude of E is constant with time. The angle that E makes with thele direction is cot and this angle changes linearly with time. The locus of the tip of E isa circle, as indicated in Figure 1.6b. For this reason, the field is said to be circularlypolarized.

The sequence in Figure 1.6b is drawn as though the observer were lookingtoward the transmitting antenna from afar, along a longitudinal line in the (0, 0)direction. The progression of E with time is seen to be counterclockwise, which is thedirection of rotation a right-hand screw would have if it were being turned to progressin the direction of propagation. For this reason, (1.197) is said to represent a right-handed circularly polarized wave. If one were to write

E = A(U cos cot - 1 sin cot) (1.198)

so that £0 leads Ee by 90°, instead of lagging by 90° as in (1.197), then a left-handedcircularly polarized wave would be described.

(c) ELLIPTICAL POLARIZATION The most general case of (1.196) occurs whenA ^ B, 0 ^ 0. The magnitude of E is given by

|E(0l = [A2 cos2 cot + B2 cos2 (cot + fi)]l/2 (1.199)

If the time derivative of this function is set equal to zero, the extrema of | E(t) | can beidentified. They occur at angles cot = 8 governed by

t a n 2 J = - * a g ? 2 ^ (1.200)A2 + B2 cos 20 v '

If 8X is the angle in the first quadrant which satisfies (1.200), then 82 = 8{ + n/2 alsosatisfies (1.200).

Substitution of the angles 8i and 82 in (1.196) reveals both the direction andmagnitude of each of the two extrema of E(t). The two directions are at right anglesto each other and form the principal axes of the locus. It is left as an exercise to show

E = A(1B cos cot + 1* sin cot) 1.197)

A2 + B2 cos 20B2 sin 2jgtan 2 8 = (1.200:

|E(OI = [A2 cos2 cot + B2 cos2 (atf + j8)]1/2 (1.199)PW1

If /I - B and fl = -90° , Equation 1.196

POLARIZATION

POLARIZATION

1.18 Polarization of the Electric Field 55

Fig. 1.6 Phasor Plots of E Versus Time for Electromagnetic Waves of Various Polariza-tions

that this locus is an ellipse.25 Its semimajor and semiminor diameters can also befound by substituting 5X and S2 in (1.199). This gives

|E| =A2 + B2 (B\ sin_2£T/2

2 + V2/sin2«5j(1.201)

25The components Ee and E^ that occur in (1.196) are analogous to the voltages applied to thetwo sets of deflecting plates in an oscilloscope in order to create a Lissajou figure on the screen.

(a) Linear polarization

h

(b) Circular polarization

(c) Elliptical polarization

ojt = 0 OOt =7T

4

1.

cot7T

2OJt =

_ 3?r

4C0r = 7T

sin 2diJ

sin2^- 1/2


A typical plot of (1.196) is shown in Figure 1.6c. As in the case of circularpolarization, the direction of rotation of E can be either clockwise (left-handed ellip-tical polarization) or counterclockwise (right-handed elliptical polarization). This isdetermined by whether the phase angle /? is lead or lag.

REFERENCES

COLLIN, R. E. and F. J. ZUCKER, Antenna Theory, Part I (New York: McGraw-Hill BookCo., Inc., 1969).

ELLIOTT, R. S., Electromagnetics (New York: McGraw-Hill Book Co., Inc., 1966).ELLIOTT, R. S., "The Theory of Antenna Arrays," Chapter 1 in Volume II of Microwave

Scanning Antennas, ed. R. C. Hansen (New York: Academic Press, 1966).SILVER, S., Microwave Antenna Theory and Design, MIT Rad. Lab. Series, Volume 12 (New

York: McGraw-Hill Book Co., Inc., 1939).

PROBLEMS

1.1 Complete the Stratton-Chu derivation by letting F = B and repeating the analysis thatwas used in Section 1.7 to obtain B, thus establishing Equation 1.54 of the text.

1.2 Alternatively, take the curl of Equation 1.53 to find —jcoB and in this manner verifyEquation 1.54 of the text.

1.3 Use the expression for the curl of a vector in spherical coordinates and begin withEquation 1.91 in the form

ej(cot-kr)

Then use (1.10b), (1.87), and (1.89) to deduce that, in the far-field

E = ~jca AT

•—•-(I)thus confirming (1.95) and (1.96).

1.4 Demonstrate the validity of equations (1.113) and (1.115) in the text.

1.5 Use equivalent-source Equations 1.112 through 1.115 in the retarded potential functions(1.110) and show in detail that the results agree with the surface integrals in the Stratton-Chu formulation for E(JC, y, z) and B(x, y9 z).

1.6 Begin with the far-field expressions (1.123) and (1.124) and show that the power radiatedhas a density given by (1.125).

1.7 Enumerate the theorems in circuit analysis that can be proven with the aid of the reci-procity relation (1.134). Sketch the proof of each.

1.8 An antenna A, when transmitting, radiates a circularly polarized field in the direction(9, <f>), which is right-handed. If antenna A is receiving an elliptically polarized electro-

\q>(E

E = ~jCO A T

H = l r X

4njLLolr

gj(cot-kr'.

jGtf,*)A(x, v , z,t)=-A(x, y, z, t)

Problems 57

magnetic wave, incident from the direction (0, 0), state the conditions of ellipticity whichwill maximize the received signal. State those which will minimize it.

1.9 Show that |E(7)|, as given generally by Equation 1.199, has as its locus an ellipse withaxes that occur at angles 8\ and S2 = Sx + iz/2, with these angles satisfying (1.200).What is the ellipticity ratio?

1.1 introduction -...

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