Aperture Theory And The Equivalence Theorem
Adriaan J. Booysen, Member, IEEE Grintek Avitronics, P 0 Box 8492, Centurion, 0046

I. INTRODUCTION
The equivalence theorem [I] provides an elegant method for analysing electromagnetic radiation and scattering from structnres. A specific class of problems is where a source is enclosed within a structure which forms a cavity with an aperture, as depicted in Figure 1. From a radiation point of view, the fields wilhi the cavity are not of any rcal interest, only thc 'external' fields radiated from the apermre. If the field distribution in the aperture is known (even if only approximately),this n priori knowledge can be exploited to limit calculations to the 'external' part of the problem This technique is commonly referred to as 'aperture theory' and is discussed in many antenna textbooks, e.g. [2]. This paper deals with the methods in which the equivalent apermre current densities are incorporated into the surface equivalence theorem. It will be shown that the application of image theory to problems involving radiation hom apertures in half-spaceis incorrect and that the conventional formulation of apelture theory violates a basic condition of the equivalence principle. A new apermre theory formulation is presented. The method is very simple to implement and will be shown to be exact in as far as the exact aperture fields are known.

II. THE SURFACE EQUIVALENCE THEOREM REVISITED
Consider Love's external equivalence as discussed in [3]. Sources 3, and-i radQte in free space (E,, @,)andtogether with equivalent surface current densities ., I and hl , on n ! equiv_alentsurfaces, typically'enclos$g' an eiecuomagnetic scatterer,produce the m e fields E, and H external to-Sand $1 fields 5 and H internal to S. The surface current densities are given by J, = ti x HI , M, = -ii x €,The electric and magnetic fields can be expressed as

,

,

~

E,=E(J,.ii,)+E(S,,ii,) i , i (3,. ii ,) +FIG,. rG J i= i where E (3, M) and H arranged as:

(1)

(2)

(i,fi) denote the radiation integrals.

These equations can be re-

Equations (3) and (4) are commonly referred to as the Magnetic and the Elecmc Field Integral Equation (MFIE and EFIE), respe_ctively, and can be solved by means of the Method of Moments. For perfect conductors M ,= 0 and (3) and (4) reduce to the more familiar forms.

III. FIELDS IN HALF-SPACE
An apermre antenna radiating i the presence of an infMte perfectly conducting half-plane is n depicted in Figure 2(a). In the conventional approach the equivalence theorem is applied IO this problem by regarding the half-plane as one b o u n d 9 of theinfinite s p T e behind (to the left of) the ground plane. Equivalent sources J, = 13,x H and M ,p = -ti, x E :p, ar%placed_in the a p e m to represent the known apcrmre fields, while equivalentcurrent densihes J, and M , are placed on the half-plane to represent the conductor, as shown in Figure 2(b). Since $e internal fields supposedly are zero and th? extemalJmptiaJ elecpic field$ also zero, M , must be zero. To calculate the unknown J,. we set J , J,,, M ,= M .P and M ,= 0 in (3) and solve by means of the MOM. 0-7803-5639-x/99/$10.00 EEE. Q 1999

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Next consider the widely used application of 'image theoxy' to the half-space apenure fields of Figure 2@). The reasoning here (see (21) is that since the equivalent field in the region behind the half-plane is zero, we can place a perfect conductor (electric or magnetic) just behind the ground plane, thereby 'shorting out one of the current densities and doubling the contribution of the other (Figure 2(c)). The use of twice the electric current density, or hvice the magnetic current density, is supposed to yield the exact same field as the combination of electric and magnetic m t densities, everywhere in the region to the right-band side of the ground plane.

As an example,let the aperture& Figure Z(a)represent E - p h i radiation h m a slot or ahorn antenna.WithE,,=E,,,f g d H =HW2,M,,=M,,2andJ,,=J j.Forar-directedM,,, the magnetic field due to M $1 not tend to zero along theyax;:. The magnetic field due to ay-directed J will tend to zero along theyaxis. To a z u e that the magnetic fields given by the three possible combinations (hvice J ,, twice M or the two current densities , combined) are identical is thus incorrect.

, ,

.,

, ,

We next place a co-nductoLin Love's 'null field' region, as shown in Figure 3. The equivalent current densities J,, and M were calculated for the-original problem and can now also be considered as sources. The equivalent current density J, on the surfacc S, of the conductorhas to be solved for in the usual approach. The EFIE of (4) can be written as

,,

We know, however, that the combination of fields on the right-hand side of (5) gives a null field internal t S (Love's equivalence in the original problem). which implies that J, = 0 . In o other words, since the fields internal to Sare zero, placing a conductor in this region will have absolutely no effect In the "shon circuit" approach it is assumed that 'image' currenfs will be induced on the conductor, but this will not actually happen. I h e third consideration is the most irnpottant. Chen (41 presented a mathematical formulation ofthe equivalence theorem which requires that the Green's function and the surface cument ut densities m s be continuous, along with their f i s t and second derivatives, on the equivalent surface S as well as througfroutthe area enclosed by S This means that the reg;on enclosed . by the equivalent currenfs must be homogeneous. This is clearly no the case in conventional aperture theory.

IV. DERIVATION OF AN EXACT APERTURE THEORY FORMULATION
Consider again the cavity example of Figure I . Our aim is to exploit our knowledge of the a p e m fields to exclude the 'internaP part of thc s m c p e from-the calculations. With this in xH, mind, we place equivalent a p e m e current densities J,, =i,P and M,, = -6, x E in the aperture, as indicated in Figure 4(a). The presence of these additional sources will. however. yield incorrect results. To correct-t we place another set of current densities -J,v and -M ,~ an infiitesimal distance 6 from J,, and M ,, on the side of the cavity. Since these sources , are finite and infiitely close to each other (but not coincident), the total radiated field fLom them will still be zero. We can now follow the same reasoning and place ahibitrary sources J,,, and M,,, together with the oppositely directed cancelling current densities, within the infiitely narrow gap. This is depicted in Figure 4(b).

, ,

, Since 2 and k can have arbitrary values, wc_canplace an elstric conductor immediately behind g e apesources and k and let J,,, and h,* represent the eurrent densities that would be induced on t h z conductor. This will result in M,, -0 and from an external point of view the a p s e can indeed be 'closed'. The current density J,,, can then be viewed a a con@uaKon of J,, . ?his i equivalent to placing an infinitely thm conductor between the s s sources J,,, M,, and the oppositely directed cancelling sources, as shown in Figure 4(c). As long as we integrate along all the paths (intemally as well), the correct radiated field will be obtained, since the sources in thc a p c m e cancel. This selection of surface current densities
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.,

will also ensure tbat is, continuous everywhere as required. is

The geomehy of thisproblem is identicalto that of Figure 1 in which case the equivalent fields inside the object are zero (Love's equivalence). The geomehy of fie intern$ problem in Figure 4 is also of thisnature, with a line source and current densities -J,# and -M iniiitely close to the conductor that covm the apcrIure. This ipis tbat in both cases a nui! field will mle be produced in thc region which is not of interest, effectively 'isolating' the external and internal equivalence problems from each other. The E-plane (HJ apcrture geomehy shown i Figure 1 serves as a good example to n demonstrate the new method. To simulate a constant E-field distribution, source J, was selected to be an elecic line source of uniform amplitude. The calculations were performed at 3 GHz with the dimensions in Figure 1 given by CD=40nun, DE=235mm, AB=lOmm, FG=210mm, x=lOmm and the length of the uniform line source 8 m The far field radiation pattern as calculated for the original problem of Figure 1 and by the new method are shown in Figure 5 and are virmally indistinguishable. The pattern yielded by the conventional formulation is also shown and is clearly inaccurate. This erroneous result confirms that the conventional formulation is mathematically unsound.

VI. SUMMARY
It was shown that conventional techniques for applying aperture theory to the equivalence theorem are mathematically incorrect as far as the free space equivalence theorem is concerned. n e accuracy of these techniques varics from fair to totally unreliable, depending on the s e o m e q of the problem. A rigorous method for incorporating aperture theory into the equivalence theorcm was subxquently presented. The new technique requires knowledge of both the electric and magnetic fields in the aperture, but these can usually be derived quite easily to good approximation. The technique is easy to implement and yields excellent correlation between theory and measurement.

REFERENCES
1.

A.E.H. Love, "The integration of the equations of propagation of electic waves". Phil. Trans. Roy. Soc. London, Ser. A, vol. 197, pp.1-45, 1901. R.F. Harrington, ~me-Ha~onicElecnonrngneric Fields, New York McGraw-Hill. 1961, pp. 106 113.

2.
3.

-

C.A. Balanis, Advanced Engineering Elecrromagnelics, New York John Wiley 8: Sons,1989, pp. 329 334.

-

4.

K.Chen, "A MathematicalFormulation of the Equivalence Principle", I€€€. Microwave Theory Tech., vol. MTT-37, pp.1576-1581, October 1989.

Trans.

Figure I . A structure wilh an aperture (AF)

Figure 3. Conductor in the 'null field' region.

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(a)

(b)

(C)

(a) Original problem (b) Love's equivalence (c) Electric current density removed

Figure 2. Apemre radiating in half-space

t
Figure 4. Derivation of new formulation

Figure 5. Radiation patterns for Figure 1.

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