a reaction theorem and its application to antenna impedance calculations-pwl

1961

A Reaction Theorem and Its Application to Antenna Impedance Calculations*

J. H. RICHNOKDf-, SENIOR ~ J L Z B E R , IRE

Summary-The reaction theorem is generalized to allow the fields of an antenna in one environment to be employed in calculations of mutual impedance in another environment.

Several expressions for self-impedance and mutual impedance are presented. These are in the form of surface integrals or volume integrals of the field intensities or the current density. It is shown how the fields of an antenna in free space can be useful in calculating the impedance in the presence of scatterers.

INTRODUCTION

R ECIPROCITl- theorems are anlong the most useful tools in field and circuit problems, ranking Ivith the superposition theorem and the equiva-

lence theorems. I t is convenient to classif!. the reciprocity theorems into three types: pure circuit, pure field, and mixed. 'The pure circuit form

V 1 ? 1 1 1 I,'sJa? (1)

developed by Rayleigh for networks of lumped elements n-as extended to antennas by Carson [ l ]. I t applies to a pair of antennas only if each antenna has suitable te rnha ls n-here voltage and current can be defined.

X theorem of the second t>-pe (pure fieldj involving electric- and magnetic-field intensities was derived b>- Lorentz [ l ] in the form of the surface integral

ss :E1 HZ E? H I ) .ds

1 2 ( E 2 H I El Hi).ds, ( 2 )

where surface encloses antenna 1 and encloses antenna 2. (Subscripts indicate the source of the field. For example, El represents the electric-field intensity set up by antenna 1.) Carson also presented a pure- field theorem in the form of a volume integral involving electric-current densit>- and electric-field intensit>-,

where volume V I includes antenna 1 and includes antenna 2. Since terminal currents and voltages are not

received, June 21, 1961. This re-,zarrh was sponsored by the Aero- Received by the PG;\P, FebruarJ- 10, 1961; revised manuscript

nautical Systems Div., --IF S\-stems Command, IYright-Patterson AWB, under Contract AF33(616)-7614.

Columbus. t Antenna Lab., Dept. of Elec. Engrg., The Ohio State Cniv.,

involved, these pure-field theorems apply even when 110

suitable terminals exist. Rumse>- [2] has given t he name "reaction" to the

quantit>- represented by the integrals 11-hich appear i n the reciprocity theorems of Lorentz and Carson and has proposed the synlbol 2:) for the integrals on the left side in (2 j and (3 ) . I11 this terminology, t he reciprocity theorems of Lorentz and Carson state that the reaction of antenna 1 on antenna 2 is equal to the reaction of antenna 2 011 antenna 1.

Kouyoumjian [ 3 ] has developed an expression for the voltage or current induced in one antenna b>- another in terms of the reaction. This reaction theorem states the quantitJ- on the left side in (1) is equal to the nega- tive of the quantity on the left side in ( 2 j 3 that is

This important relation was also pointed out by RumseJ- [2 ] . I t is valid under conditions more general than those required for reciprocity, as will be shown. However, if the conditions for reciprocity are satisfied, this reaction theorem (4) can be combined with a n > - one of the "pure" reciprocity theorems given above to obtain a reciprocity theorem of the "mixed type.'' Circuit quantities (voltage and current) appear on one side of the equation, while field quantities appear on the other side.

In this paper a generalized form of, the reaction theorem is presented, and equations are derived from i t for the mutual impedance of two antennas. Expressions for self-impedance are derived by similar methods.

In most equations from phJ-sics all of the quantities involved in a given equation are understood to relate to a common situation. For example, Sen-ton's second l a w relates the force acting 011 ;L bod!., the mass of the same bod>-, and its acceleration, all at the same instant of time. On the other hand, each of the reciprocity and reaction theorems brings together quantities from different situations into a single equation. In the most familiar versions of these theorems the two situations differ only in that batter!- and an amnleter have traded places i n network. I n the generalized versions of these theorems developed herein some portions of the environment or network are considered to change when the source and observer trade places. -Although this may seem confusing the 111e11tal effort is rewarded in the form of expressions which show how the fields of an antenna in one environment ma>- be emplo>red in calculations of impedance in another environn~ent.

Authorized licensed use limited to: IEEE Xplore. Downloaded on May 10,2010 at 19:00:45 UTC from IEEE Xplore. Restrictions apply.

GENERALIZATION OF THE REACTION THEOREM

The concepts involved in the generalized reaction theorem can be introduced by means of the corresponding network theorem. Consider the two passive networks in Fig. 1. In antenna terminology, network 1 is transmitting and network 2 is receiving. The voltages 1’ and V S l and the current I for this situation are defined in Fig. 1. S o w let network 2 transmit with terminal current 122‘ into a new impedance 2’ as in Fig. 2. The voltage 1” and currents 122’ and I‘ for this situation are defined in Fig. 2. The current generators 111 and I??’ are assumed to have the same frequency.

If the reciprocity theorem applies to network 2 (but not necessarily to network l), it can easilJ- be shown that

V2 ’122 ‘ V I VI’. ( 5 )

Since and 12?’ are independent of Z’, both sides of (5) must also be independent of 2’. This is true even though each of the two terms on the right side are de- pendent on 2’ through 1,’’ and 1’. As an example, suppose network 2 consists of three resistors as in Fig. 3, and let I be 2 amperes and I??’ 1 ampere. Then obvi- ously V is 8 volts and I/’~I is 4 volts. The solutions for three values of are tabulated below:

I’ Ir’ V’I J/TI’

09

a3 0

I

vel

2

I 1

I I Fig. 1-First situation: Network 1 transmits,

network 2 receives.

1‘

Fig. 2-Second situation: Xetmork 2 transmits into a new impedance 2‘.

Fig. .?--In example to illustrate the network version of the reaction theorem.

I n each case the quantity V I - VI’ is equal to four, which is also the value of 1721Iez’ and the [‘reaction of network 1 on network 2.”

This network theorem (5) is not widely used, partly because it is not well known. I t is applicable to transmission-line problems and ladder networks. The quan- t i ty V I - VI’ is independent not only of the impedance Z’, but also of the point along the network or transmission line a t which the theorem is applied.

This network theorem should pave the way to an understanding of the corresponding field theorem, which is presented next.

Consider the situation shown in Fig. 4 in which antenna 1 is transmitting, setting up the field HI) and inducing a voltage V p I at the open-circuited terminals of the receiving antenna (antenna 2). The environment “seen” by antenna 1 may be described b>- the complex permeability p and permittivity E, and i t includes the structure of antenna 2 (with terminals open circuited). The medium need not be homo, ueneous.

In the usual reciprocity-theorem derivations, the roles of the two antennas are reversed a t this point, and antenna 2 is considered to transmit while antenna 1 receives in the same environment ( p , as before. On the contraq-, consider the fields (E?’, Hz‘) of antenna 2 transmitting in a new environment (p ‘ , E‘) which does not necessarily include the struct.ure of antenna 1. (How- ever, let p ’ = p and E‘ within the regular surface enclosing antenna 2.) This situation is depicted in Fig. 5.

Now, as shown in Appendix I , these quantities are related as follows:‘

Fig. 4-First situation: Antenna 1 transmits, antenna 2 receives.

Fig. 5-Second situation: Antenna 2 transmits in a new environment.

are derived in J. H. Richmond, “On the Theory of Scattering by Eq. (6) and many of the other results presented in this paper

Dielectric and Metal Objects,” Ohio State Univ. Res. Foundation, Columbus, Ohio, Antenna Lab. Rept. 786-3; April, 1958.


1961 1iichrn.ond: Reaction Theorem to dnteuna Impedance Calculations 517

Eq. (6) is a generalization of the reaction theorem. The usual time dependence ejwf is understood. It is assumed that all media are linear, the permeabilitJ- and permittivity are independent of time, and antenna 2 is constructed of isotropic media. (The other media are not necessari1)- isotropic.) I t is assumed that the feed system of antenna 2 has a section of periect1)- shielded waveguide or transmission line, and that onlp a single mode exists a t some point on the feed. The antenna terminals are chosen a t such a point, and 12.” represents t he current there when antenna 2 transmits (Fig. 5).

The incremental area in (6) has a vector direction normal to the surface S p and awa!. from antenna 2.

After using (6) to calculate the voltage induced at the open-circuited terminals of antenna 2 , Thevenin’s theorem may be emploped to determine the terminal voltage for an>- other load impedance.

Perhaps i t should be emphasized here that the vo!t- age in (6) is the induced voltage in t t e original situation shon-n in Fig. 4, m t in the new environment of Fig. 5. If a constant-current generator is employed when antenna 2 transmits, the current JZ2 ’ n.ill be the same regardless of the environment i n which antenna 2 transmits. Then the left-hand side of (6) is independent of the new environment. and so must also be the integral on the right hand, even though the integrand does de- pend on the new environment. (The difference inte- grates to zero.)

Thus, although the field (E2’,H:!’)depends on the environment in which antenna 2 transmits, the integral in (6) is invariant with respect to changes in this environment. Therefore, the environment E ’ ) may be chosen to represent any convenient situation such as

1) free space, 2) the same environment into which antenna 1 trans-

mitted, 3) as perfect]!, cond!lcting metal shell on Se, or

an extension of the structure of antenna 2.

In the last case listed, where the environment is chosen to be an extension of the antenna structure, the generalized reaction theorem shows some similarity tothe induc- tion theorem of Schelkunoff [5].

The transmission formula of Friis [6] and the receiving antenna-sensitivity formula of Levis [5] can be derived from (6). These formulas apply only in the far- field case (where antennas 1 and 2 are far apart), n-hereas (6j does not have this restriction.

The anal!-sis can readi1)- be extended to the case where two or more modes can exist at the terminal surface of antenna 2. I n this case, i t is convenient to use t he set of orthogonal modes of the waveguide or transmission line, letting l,722,L’ and 12?,’ represent the voltage and current for mode when antenna 2 transmits. If T,,721rr and denote the received voltage and current

that the more general form of (6) is

(T~T?l,,r??,t’ I.‘lnV2?,’) n

(Er ’ H1 El H2’) (7)

R’fYTL-.%I. IMPED.%SCE OF TIIT0 .%NTENh-AS

The mutual impedance between antennas 1 and 2 in the environment E ) illustrated in Fig. 4 is

ZZl v?l!I1l Js2(E4 H1 E1 Hi) (8) I11I2.“

where I l l is the terminal current of antenna 1 when it transmits. Eq. (8) follows directly- from the reaction theorem (6).

I t will be recalled that (El, H l ) represents the field of antenna 1 transnlitting in the presence of antenna 2 with its terminals open circuited. \Yith the aid of Thevenin’s theorem it can be shown that a factor must be inserted in front of the integral in (8) in the more general case where (E1, H,) represents the field with an impedance 2 across the terminals of antenna 2. Here and Z,’ represent the input impedance of antenna 2 in-environment E ) and E ’ ) , respectively.

If the conditions for reciprocit!- are satisfied, (8) yields the mutual impedance Z12, as well as T o insure reciprocity, it will suffice if the two antennas are of finite dimensions and a finite distance apart, all media are linear and isotropic, and the medium is homogeneous outside an imaginarl- sphere of finite radius.

In some cases the fields of an extended antenna are known, whereas those of the corresponding truncated antenna are not. Examples include horns and biconical antennas. Eq. (8) permits the use of the known fields of an extended antenna in calculating the mutual impedance for a truncated antenna.

If a perfectly conducting metal shell is assumed on 5’2

for environment E ’ ) ! the first term in the integral in (8) will vanish. Letting J 2 = -nXH2’ represent the electric-current densit). induced on the shell by antenna 2, (8) reduces to

1

IllI??’

z21 L 2 J : ! . Elas. (9)

Eq. (9) is particularly convenient in mutual impedance calculations when one of the antennas consists of a waveguide or cavity perforated with holes or slots. Let the perfectly conducting surface S2 coincide with the metal surface of the antenna plus the apertures formed by the holes. Then the integral in (9) vanishes except in the holes, since the tangential components of E1 vanish on the metal. Furthermore, the current J? will

for mode when antenna 1 transmits, it can be shonm in man>- cases be a simple known function, since i t is


the current on the inside malls of the waveguide or cavity when the holes are covered with conductor.

The divergence theorem and Maxwell’s equations can be employed to obtain another expression for mutual impedance from ( X ) ,

-1

I111??1 2 2 1 i,2(J~. E1 K Z . H I ) ~ Z ~ , (10)

where J2 and K2 are the densities of electric- and magnetic-source current, and V2 is the volume within s?. Since magnetic current does not exist in realizable sources, the term Kz.Hl may be omitted.

For an aperture antenna such as a horn or paraboloid, i t is convenient to choose S2 to coincide with the outer surface of the antenna plus the aperture. Then the integrand in (8) vanishes ever>-where on S?, except on the aperture.

Although (8) is suitable for aperture antennas, i t is difficult to apply to cylindrical-wire antennas. The difficult,- arises from the fact that the integrand vanishes everywhere on the antenna surface, except xhere S2 crosses the terminal gap. This problem also exists viith the induced EJ,IF formulas for self- and mutual impedance [4], [SI. The following expression, derived in Appendix I1 is more convenient for c\.lindrical wire antennas:

-1

I11122 z21 L , ( E z X Hli Eli X H2).ds. (11)

Eq. (11) is the same as (8), except that the environment has been chosen to be the same as (p, and the

field of antenna 1 has been replaced by its incident com.ponent.

Eq. (11) is convenient even when S2 coincides with the perfectly conducting metal surfaces of a cylindrical- wire antenna. In this case the first term in the integral vanishes, with a possible exception where S, crosses the terminal gap. The second term, however, does not vanish because it involves the incident field rather than the total field. The surface current J2 on the metal can be introduced in place of n X H B to obtain the following result:

1

I11I2?

ZZ1 L2 JZ (12)

SELF-I Sf PEDANCE

Consider an antenna transmitting in solne en\. won- ment ( p , and let (E, H), V , I , and represent the field, the voltage, the current, and the impedance, respectively. Now let the antenna transmit in a new environment with field (E’, H’), voltage I,” and impedance Z‘. If the current I is adjusted to be the same in both cases, and if the “scattered voltage” V s is defined by

vs 17’ v, (13)

then the impedances are related as follows:

Letting the two environments coincide ( p ’ = p and E) within a surface S which encloses the antenna, a

useful expression for the impedance is

(E’ X H E H’) (15)

The derivation will not be given here, since i t is similar to that in Appendix I.

I t is possible to let the environment (p’, E ’ ) represent an extension of the antenna structure. In this way, (15) and some of the other impedance expressions in this section can be used to calculate the change in impedance associated with an increase or decrease in the length of an antenna. Furthermore, the impedance of a truncated antenna can thereb>- be related to the impedance of the corresponding complete antenna. This procedure would be particularly convenient for antennas of simple geome- trJ- (such as horns and spheroidal antennas) where the impedance and fields of the complete antenna are known.

If the antenna is a waveguide or cavity which is perforated with holes or slots, it mill be convenient to let surface S be the metallic outer surface of the antenna plus the aperture surface of the holes. A perfect!y conducting metal shell ma\- enclose the antenna for environment (p’, in which case the first term in the integral in (15) will vanish. In terms of the current densit\* J’ -n X H‘ induced on the inner surface of the shell, (15) can be written in the form

I t is sufficient to integrate over the holes, since the tangential components of E vanish elsewhere on S. In many cases, Z’ and J’ will be known quantities, namely, the impedance and current density of the cavity or waveguide with no holes. Eq. (16) allows these to be employed in the calculation of the impedance with holes.

Suppose the two environments coincide everywhere except within a finite region called the “scattering region” enclosed by surface S,. Then an alternative expression for the impedance is

Eq. (17) can be derived from (15) by using the radiation conditions to show that the integral vanishes on an infinite sphere, using Maxwell’s equations to show that V . ( E ’ X H - E X H ’ ) vanishes at each point outside S and Ss, and using the divergence theorem.


1961 Richmond: -1 Reaction Theorem to .Intenna Impedance Calcdations 519

-Antenna impedance can be expressed in terms of a volume integral bp using the divergence theorem, and Maxwell’s equations. The result is

where V represents the antenna region which is enclosed by surface S, J and K are the dmsities of electric and magnetic source current? a!ld ( E ” , H ‘ j i- the scattered field defined b>-

E = E ’ - E 19)

H s H’ H. (20)

-Another expression can be obtained in a similar manner by starting with

Z ‘ = Z - - jw I‘ €)E’. E (p’ p)H’.H]dr, (21)

where I;-, is the scattering region enclosed by surface S,. If environment E ’ ) inclcdes a perfectlJ- conducting

metallic bods- which fills the “~cattering region,” reduces to

where J‘ is the current densits- induced on the conducting surface. The conducting body in the scattering region can be arranged as an extension of a c?-lindrical wire as in Fig. 6. In this \\a\-, (22) expresses the change in impedance associated with a change i n the length of such an antenna. The integral i n (27j has the same form as that appearing in the induced El IF method, but the integrand does not vanish OII the surface of integration.

Fig. 6-Cylindrical wire antennas illustrating an application of (22).

COSCLCSIO~V

The reaction theorem has been generalized to allow the fields of an antenna in one environment to be employed in calculations of mutual impedance in another environment. For example, the kno\m fields of an in-

finitel\- long horn can be used in calculating the mutual impedance between a truncated horn and another antenna. another example, this formulation permits the

current on the inside of a waveguide to be used in calculating the radiation from an antenna consisting of a waveguide with holes or slots.

In some cases i t is advantageous to let the two environments be the same. J f this is done one of the fields involved i n the mutual impedance equation can be replaced b\- the incident field. This avoids the dificulty which arises i n the induced E l I F method, where the integral vanishes over perfectlJ- conducting portions of the antenna.

Several expressions are presented for the self- impedance of an antenna. They permit the kn0n.n fields and impedance of a “complete antenna” to be used in calculating the impedance of the corresponding truncated antenna.

.\PPEKDIS I

GEhXR.\I,IZED I<E.lCTION THECREM

.Antenna 2 ma\. be constructed partlJ- of dielectric and partly of metal, as suggested i n Fig. .Any conductivity in the dielectric portions will be accounted for by letting p and e be complex. llaxn-ell’s equations for the fields within S,. but excluding the “source region” shown in Fig. 7, are

G El jwpH1, (23)

V H I jutE1, (23.)

V Err jwpHz’? (25)

HZ‘ jweE,’. (26)

Fig. ;-.I metal horn with a dielectric lens, .illustrating the surfaces employed in the derivatlon.

( 2 , l i (E?’ H 1 El Hi) (28)

Eq. (28j is simply a definition of reaction; i t differs slightly from the standard form in that the two fields

J s.

I t is assumed that the currents J and K are held constant %-hen which are involved i n the integral are for different en-

the environment is changed from E ) to viro1:nlents. The dixlergence theorem and (27) allow the


IRE TRAXSACTIOSS O N ASTELY~YAS AKD PROPAGATIOS Xovember

surface of integration in (28) to be changed from SI to S,+ST (see Fig. 7) . Furthermore, the integral on the metal surface vanishes if i t is assumed to be perfectly conducting, since E1 and Ez’ are normal to this surface. This leaves only the integration over the terminal surface ST.

Letting subscript denote tangential-field components on the terminal plane, the voltage and current are defined as follows when antenna 2 transmits 191:

E2t’ VzZ’e (29) and

The vector-mode functions are related by

h = n X e , (3

and they are normalized bs; letting

A T e . e d s I. (32)

The voltage and current induced at antenna 2 when antenna transmits are defined in a similar manner by

Elt V2le (33)

and

Hlt Iplh. (34)

Eqs. (29)-(34) can be,used to reduce (28) to

(2, Vs?’Iz1 v211221. (35)

The received current 1 2 1 is zero, since the terminals are assumed to be open circuited when receiving. There- fore, (35) becomes

V 2 1 I 2 2 1 (2, (36)

This leads directly to the “reaction theorem,” ( 6 ) .

APPENDIX I I

h’IUTCAL IMPED.4NCE IN TERMS OF THE INCIDENT FIELD

Let the field (E?, H?) be the field of antenna 2 when transmitting with current I?? in the environment E ) ,

which includes the structure df antenna l .3 The field of antenna 1 in this same environment is resolved into incident and scattered components by means of the following equations:

El Eli ElS (37)

3 To be more specific, let (E*, H?) be the field of antenna 2 in the presence of antenna 1, the generator across the terminals of antenna 1 being replaced by its internal impedance.

and

HI HIi HIS. (38)

To complete the definition of these fields, l\:Iaxwell’s equations for the incident and scattered fields are given:

V H I S jwrEls (41)

V X ElS jwPHlSJ outside SI.

Eq. (11) can be derived bq- starting with (8) and using (35) and (38) to split the integral into one involving the incident field and another involving the scattered field. The integral of EZXHlS- ElS XH2 on sphere vanishes as the radius goes to infinity in view of the radiation conditions. (Antenna 2 is assumed to be of finite dimensions, and the medium outside a finite sphere is assumed to be free space.) The integral on the infinite sphere can be expressed via the divergence theorem as a volume integral of V (E2 HIS ElS XHz). h1axLvell’s equations can be used to show that this integrand vanishes a t each point outside S2. I t follows that the integral of E? HIs ElS H2 on S? is zero, and (1 1) is established.

ACKSOWLEDGMEXT

Discussions with Dr. R. G. Kouyoumjian of The Ohio State University have been most helpful and are grate- fully acknowledged. Many of the ideas presented in this paper were stimulated by Prof. V. H. Rumsey through a course he preseated in the 1950’s at The Ohio State University.

REFEREWES [ l ] J. R. Carson, “Reciprocal theorems in radio communication,’’

H. Rumsev. “Reaction concem in elecmomaenetic theorv.” PROC. IRE, vol. 17, pp. 952-956; June, 1929.

L .

Phys. Rex, vd.’94, pp. 1483-1491’; June 15, 1954; < I

[3] R. G. Kouyoumjian, “The Calculation of the Echo Areas of Perfectly Conducting Objects by the Variational Method,” An- tenna Lab., Ohio State Univ. Res. Foundation, Columbus,

A. B. Bronyl l and R. E. Beam, “Theory and Application of Rept. No. 444-13, pp. 29, 80-84; 1953.

Microu-a\Tes, XIcGraw-Hill Book Co., Inc., Sew York, N. Y., pp. 424431; 1917.

[5] S. A. Schelkunoff and H. Friis, “Antennas: Theory and Practice,” John il-iley and Sons, Inc., New York, N. Y., p. 516; 1952.

[6] H. T. Friis, note on a simple transmission formula,” PROC IRE, 34, pp. 254-256; May, 1946. C. Levis. “ho te on Receiving. Antenna Sensitivitv.” 4ntenna Lab., Ohio State Unit-. Reg Foundation, Columbus, Ohio, Rept. So. 486-29; 1954.

[8] P. S. Carter, “Circuit relations in radiating systems and applica- tions to antenna problems,” PROC. IRE, vol. 20, pp. 1004-1041; June, 1932.

Co., Inc., S e w York, N. Y., Sec. 1-2; 1951. [9] hIarcuvitz, Ed., “n’aveguide Handbook,” McGran-Hill Book


a reaction theorem and its application to antenna impedance calculations-pwl

Documents