‘ IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AI’-33, NO. 6, JUNE 1985 67 1

Fig. 3. Sum and difference patterns designed to steer main beam to 0, = 112’ and to suppress RFI sources at 0 , = 28” and O2 = 80”. Here, N = 8 and d = w2.

method as that given in [5], it can be shown that this is always the case.

From Fig. 3, it can be clearly seen that both RFI sources are completely cancelled in the sum and difference patterns. In addi- tion, the sidelobes are everywhere below -27 dB in both the sum and difference patterns. In other words, we have been able to nullify the effect of RFI sources completely as well as minimize the sidelobes in this case.

CONCLUSION

A new method of null steering in the difference pattern of a phased array is discussed. Its usefulness stems from its ability to achieve pattern nulling by controlling current amplitudes only. Thus, it is now possible to use phase shifters solely for scanning both the sum and difference patterns, whereas the suppresion of unwanted RFI sources for any scan angle can be independently achieved with the aid of attenuators. In other words, the minimi- zation of noise sources can now be carried out independently of the maximization of the signal. The method is simple and is limited only by the total number of elements used in the array. When the number of jammers is much smaller than half the number of elements, both RFI cancellation and pattern optimi- zation can be achieved.

REFERENCES C. A. Baird and G . G. Russweiler, “Adaptive nulling using digitally controlled phase-shifters,” IEEE Trans. Antennas Propagat., vol. AP- 24, pp. 638-649, Sept. 1976. R. Giusto and P. DeVincenti, “Phase-only optimization for the genera- tion of wide deterministic nulls in the radiation pattern of phased arrays,” IEEE Trans. Antennas Propagat., vol. AP-31, pp. 814-817, Sept. 1983. R. L. Haupt, “Simultaneous nulling in the sum and difference patterns of a monopulse antenna,” IEEE Tram. Antennas Propagat., vol. AP-32, pp. 486-493, May 1984. R. Shore, “Nulling at symmetric pattern locations with phase-only weight control,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 530-533, May 1984. T. B. Vu, “Method of null steering without using phase-shifters,” Proc. Inst. Elect. Eng., vol. 131, Pt. H, pp. 242-246, Aug. 1984. -, “Null steering by controlling current amplitudes only,” in 1984 IEEE Antennas Propagat. SOC. Int. Symp. Dig., pp. 81 1-8 14. R. L. Haupt, “Adaptive nulling in monopulse antennas,” I984 IEEE Antennas Propagat. SOC. Int. Symp. Dig., pp. 819-822.

Hybrid Solutions at Internal Resonances

LOUIS N. MEDGYESI-MITSCHANG, MEMBER, IEEE, AND DAU-SING WANG

AhPucf-The nse of hybrid solutions for integral equation (IE) formalatiom in electromagnetia is illustrated at frequencies where a perfectly conducting scatterer exhibits internal resonances. Hybrid solu- tions, incorporating the Fock theory and physical optics Ansatzes, and the Galerkin representation, are compared with the method of moments (MM) solutions of the electric, magnetic, and combined field formulations at such frequencies. Numerical results are presented for spheres and a right circnler cylinder.

INTRODUCTION

The ease with which complex boundary conditions can be incorporated into integral equation (IE) formulations of Max- well’s equations has made such formulations suitable to handle a wide variety of electromagnetic (EM) radiation and scattering problems. Two formulations, ascribed originally to Maue [ 11, based on the electric and magnetic field integral equations and denoted EFIE and MFIE, respectively, have been used exten- sively. The attractiveness of these formulations is compromised somewhat because their numerical solutions for the exterior radiation or scattering problem are nonunique at frequencies where the associated conductive body exhibits internal (cavity) resonances. This nonuniqueness manifests itself numerically as a noninvertible matrix if, for example, the method of moments (MM) is used. In this discussion, we exclude the case of spurious solutions that arise from inadequate numerical treatment of the IE at surface discontinuities due to sampling and improper re- presentation of the principal value (Cauchy) integrals in the operators.

PREVIOUS INVESTIGATIONS

Alternate IE formulations and solution techniques have been proposed to circumvent the shortcoming of the original EFIE and MFIE formulations. Representative methods include the combined field integral equation (CFIE), the combined source integral equation (CSIE), the augmented boundary condition method (ABC), and the augmented integal equation method (AIE), and the minimum norm solution.

In the CFIE approach, an IE is obtained incorporating the boundary condition (BC) on the tangential components of both the electric and magnetic fields. The works of Oshiro and Mitzner [2], Poggio and Miller [3], and Mautz and Hanington [4] are representative. The latter investigation developed the CFIE for a general body of revolution (BOR) and provided extensive nu- merical results. Building on these results and those of Werner and co-workers [5], [6], the CSIE approach was developed for a BOR [7]. The CSIE designation derives from the fact that a combination of electric and magnetic sources (currents) are placed on a conducting body, and the IE is derived from the imposition of BC on the electric fields. In both CFIE and CSIE, an arbitrary weighting constant a is introduced. While the solu-

Manuscript received November 20, 1984; revised January 23, 1985. This work was conducted under the McDonnell Douglas Independent Research and Development Program.

The authors are with McDonnell Douglas Research Laboratories, St. Louis, MO 63166.

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672 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 6, JUNE 1985

tion in principle is independent of the particular choice of LY, numerical results are sensitive to a.

In the ABCmethodl [8], the original BC on the surface is aug mented by an additional BC in the interior of the body at in- terior points not collocated with the nodal surfaces of the reso- nant fields. If a Galerkin expansion procedure is used for the unknown functions in the IE associated with the ABC formula- tion, an overdetermined system of algebraic equations results, solvable by the Moore-Penrose pseudo-inverse. In a variant of t h i s method [9], an Ansatz from the extended boundary condi- tion method (EBCM) [lo] is added to the original IE. This stratagem removes the arbitrariness inherent in the choice of the interior points and obviates the introduction of a weighting constant, but geometry restrictions imposed by EBCM are added. For a detailed critique of the foregoing techniques and a list of extensive references see [ 111 .

In the AIE method [ 111, the EFIS a:d MFIE are auqented with the boundary coylitions V - ( i X H - J ) = 0 and X E ) = 0, respectively, where J is thz surfa2e ( M , ) current density; ti is the unit surface normal; E and H denote the total electric and magnetic fields on aR,. Specific numerical solutions are de- veloped in [ l l] for two- and three-dimensional scatterers. In [12] a minimum norm solution is proposed to solve the EFIE at internal resonances. The procedure is an extension of the well-known Murray method [ 131 for linear operator equations.

In the hybrid formulation discussed here [14], an Ansatz is used to modify the domain of the integrals in the EFIE (or MFIE) associated with a closed body (with possible internal resonances), to that describing effectively an open one (with no resonances). The remainder of the original surface can be visualized as a sheet of current sources.

HYBRID METHOD

Consider a perfectly conducting body with a closed surface aR,, E S. The EFIE formulation is arbitrarily chosen for the subsequent development. Analogous results can be obtained for the MFIE formulation. The tangential scattered and incident fields cancel on the surface, thus

E: = LJ, -+. +

where the integrodifferential operator L is given as

and CP is the free-space Green’s function, i.e., = (1/4rR) exp (-jkR), R is the distance from the source to the field point; V is the surface gradient on the body with respect to the un- primed variables, k = 2nf/c, an$ q = m. The incident field 2’ is given, the current density J is the *own of the problem.

Since the homogeneous equation U = 0 is known to admit nontrivial solutions at a frequency associated with an internal resonance, the inhomogeneous (1) has a nonunique solution at this frequency. A way to circumvent this problem is to modify the domain of integration of the operator L . In the ABC method [8], the integral equation is modified to include additional points internal to the scatterer. In the hybrid method 1141, the domain of the integrodifferential operator is divided into a domain where the Ansatz is applied and in another where it is

I Sometimes the ABC method is termed a “bybrid” method. In this communication, the term “hybrid” is reserved for a different context.

not. For example, assume that the surface of the body is divided into two re ’ons S, and S, havinkthe respective current densi- ties J’, and 1. Then the quantity W in (1) can be rewritten as

+. + + E : = L , J , + L z J 2 (2)

where the subscripts on L denote the domain of integration corresponding to S, and S,. Assume that an Ansatz eith$r from PO and/or the F y k theory can be used to estimate J2 on S2 and further that J , on S1 is governed by the satisfaction of the boundary condition on Sl. Specific examples illustrating the efficacy of this assumption for perfectly conducting bodies are given in [14] and [15] y d for bodies with material dis- continuities in [16]. Thus L 2 J , is a known quantity and (2) is replaced by

L l j l =$- L2&. (3 ) Note the domain of integration in L , and the domain over which Equation (3) is enforced is S,, an open surface. Thus (3) is an integral equation for an open shell and does not support interior resonances.

Using the hybrid method [14] we apply the Galerkin expan- sion to Jl alone in (3) yielding - -

Z , I , = v, - v, - Y , (4)

where 2, is a matrix embodying t$e EM interactions of the MM represented region spaped by J , ; I , denotes the expansion coefficient array for J , . The right side of (4) is the “hybrid” excitation vector (source), defined as

-+ +. VI = ( W 1 , E:)

and

vl + F1 = ( W l , L 2 J , ) + +

(5) where (+-) denotes the inner product with the testing function array W, over SI. (The choice of the expansion and testing functions is dictated in part by the shape of the body. The exact form of these is immaterial-in the context of the present dis- cussion.) The term Fl + Vl is the nonplane wave excitation arising from $e illuminated and shadowed regions of the body spanned by J z . Detailed expressions for this are given in [14] for a general BOR subject to oblique illumination. For brevity they are not reiterated here. Since the right side of (4) is known, the coefficient array I , of the Galerkin expansion can be de- termined uniquely.

RESULTS The sphere provides an illustrative example of the foregoing

ideas. The lowest internal resonance of a sphere occurs at ku = 2.744. With the Galerkin discretization of the EFIE, the resonance occurs at ku = 2.768, where the condition number (based on the column-sum norm) is largest, Le., the resulting MM system ma- trix is ill-conditioned. In Fig. l , the MM solutions of the EFIE and CFIE (Q = 0.5) are compared with the hybrid solution of the EFIE for this case. (The constant a is defined in terms of CFIE = a EFIE + (1 - a) MFIE.) For the EFIE and CFIE formulations, the conventional Galerkin expansion for the un- knownA currents was represented by 14 triangle $mctions along i(= -8) and by the Fourier terms einQ along 9. In the hybrid analysis, the @-directed current for the part of the body spanned by the fourteenth triangle function is obtained from an Ansatz. The currents, condition numbers, and scattering cross sections

4

6

a

9

I

I IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 6, JUNE 1985 67 3

sphere La= 2.7b3

13 14

e, = 1800 Mie -2.88 - - - EFIE - 12.48 20.2. 57.4' 20.5, -32.0' 1 . 8 ~ 1 0 1 CFlE (U=O.S) -3.07 1.98, -21.4' 2.0. - 110.0~ 50.7

Hybrid -282 1.92, -25.7" 1.92, - 114.4' 454 o,=r EFlE - 12.48 21.3. -115 .T 21.7. 154.3' I . 8 X l V CFIE (a=O.5) -3.07 1.23. -88.6" 1.31. 175.7- 50.7

Hybrid -2.83 1.11, -86.9' 1.21, 179.0' 454

Fig. 1. Comparison of solutions of integral formulations for a resonant sphere, ka = 2.768.

5~

- 10 \

I I I I 1 ' .

0 30 60 90 120 150 I 8 0 (deg)

(b)

Fig. 2. Bistatic cross sections of the resonant sphere in Fig. 1. (a) +$- polarization. (b) @&polarization.

are compared for each of the methods. The Mie solution for the cross section is also given. Comparison of the currents (magnitude and phase) provides a stringent test of the efficacy of the solu- tions. Two cases are depicted in Fig. 1, namely when the Ansatz currents are specified in the shadow region (ei = 180") from the Fock theory and in the illuminated region (ei = 0") with the Ansatz currents given by PO. As expected, the MM solution of the EFIE fails entirely, while the CFIE solution is in close agree- ment with the hybrid solution of the EFIE. The condition number of the hybrid calculation exceeds that of the CFIE, yet consistent answers result. The condition number of the hybrid calculation can be lowered by using additional Ansatz currents in the shadow or illuminated region of the body. The behavior of the bistatic cross sections as a function of scattering angle obtained from the three approaches is depicted in Fig. 2. The corresponding results for the resonance of a sphere at ka = 4.518 are given in Figs. 3 and 4.

Sphere: ka=4.518

J z fmed I

ei = 180" Mie 2.68 EFIE CFlE (a=O.s) 2.66 1.99, n.10 1.98, - 124.00 84 Hybrid 2.44 1.92, 76.9' 1.92, - 123.9' 477

- - - 5.53 123, 39.1' 12.5, -50.9' 2.37 x 1@

e, = 00 EFlE 5.53 113, 38.2' 11.3. - 5 I . Y 2.37x 10' CFIE (a=O.S) 2.66 1.08, 107.7' 1.14. 13.2' 84 Hybrid 244 1.06.Ios.9" 1.11, 13.1' 477

Fig. 3. Comparison of solutions of integral formulations for a resonant sphere, ka = 4.518.

0 30 60 90 120 150 180

8 WgJ

(b)

Fig. 4. Bistatic cross sections of the resonant sphere in Fig. 3. (a) 99- polarization. @) BO-polarization.

The computations were repeated for a circular cylinder of radius a and length 2~ with dimensions such that the correspond- ing cylindrical cavity resonated with the TE, , mode. This mode occurs for ka = 2.42. In the numerical discretization of the EFIE, the unstable MM solution occurs at ka = 2.44175. Sub- sequent calculations are given for this value. The comparison of the results from the three IE formulations is given in Fig. 5 . For the EFIE and CFIE formulations, the conventional Galerkin expansion for the unknown currents was used as before, with 15

674 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 6 , JUNE 1985 4 b

b

u//h2(dB) I f l . nrg (J:) IfI. nrg (J? CondlUon No. e, = 1.90- EFIE 2.21 13.1,46.4’ 13.1, -44.0’ I.SXl@ CFIE (a=O.5) 3.38 2.38, -305’ 2.42, -120.7’ 53 Hybrid 3.63 2.51, -30.3’ 2.56, -l20.S0 390

EFIE CFIE (u=0.5) 3.38 0.85. -117.5’ 0.88. 149.0’ Hybrid 3.61 0.84, -127.7’ 0.90, 140.4’ 390

53

e, = 00 2.21 13.7, - 123.3’ 13.7, 146.7’ 1.5X I@

Fig. 5. Comparison of solutions of integral formulations for a circular cylinder for the “ E l l l mode.

triangle functions. In the hybrid analysis, the @-directed current for the region of the cylinder spanned by the fifteenth triangle function is represented by a Fock Ansatz for 0 = 180” and a PO Ansatz for 0 = 0”. Again the hybrid solution is generally in good agreement with the CFIE results. Similar conclusions hold at higher mode resonances.

In conclusion, the hybrid method was applied at frequencies where a scatterer has internal resonances. The examples chosen here were based on the hybrid solution of the EFIE formulation. Analogous results can be established for the MFIE formulation. An advantage of the hybrid method is that it inherently involves smaller matrices than the “pure” MM approach. While the CFIE and other methods cited previously have proven to be effective

totic Techniques in Electromagnetics, R. Mittra, Ed. New York Springer-Verlag, 1975, ch. 5.)

[9] N. Morita, “Resonant Solutions involved in the integral equation approach to scattering from conducting and dielectric cylinders,” EEE

[lo] P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Trans. AntennasPropagat., vol. AP-27, pp. 869-871, 1979.

Proc. IEEE, vol. 53, pp. 805-812, 1965. [ l l ] A. D. Yaghjian, “Augmented electric- and magnetic-field integral

equations,” Radio Sci., vol. 16, pp. 987-1001, 1981. [I21 T. K. Sarkar and S. M. Rao, “A simple technique for solving E-field

integral equations for conducting bodies at internal resonances,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 1250-1254, 1982.

[13] F. J. Murray, “The solution of linear operator equations,” J. Math.

1141 L. N. Medgyesi-Mitschang and D.-S. Wang, “Hybrid solutions for scattering from perfectly conducting bodies of revolution,’’ IEEE Trans. Antennas Propagat., vol. AP-31, pp. 570-583, 1983.

ing from finite circular and elliptic cones,” IEEE Trans. Antennas Propagat., to appear.

I161 L. N. Medgyesi-Mitschang and D.4. Wang, “Hybrid solutions for scattering from large bodies of revolution with material discontinuities and coatings,” IEEE Trans. Antennas Propagat., vol. AP-32, pp.

Phys., V O ~ . 22, pp. 148-157, 1943.

[ l q D . 4 . Wang and L. N. Medgyesi-Mitschang, “Electromagnetic scatter- @

717-723, 1984.

Numerical Evaluation of Complex Resonances of an Elliptic Cylinder

RADHAKRISHNA NAISHADHAM, STUDENT “BER, IEEE, AND L. W O N PEARSON, SENIOR MEMBER, Em

in solving the exterior Dirichlet problem at internal resonances, the hybrid method as demonstrated here provides a viable alter- mined numerical,y by contoar integration of the determinsnt of the Abstracf-Complex natural resonances of an elliptic cylinder are deter-

moment matrix obtained in a homogeneons solution of the electric field nate solution procedure. 4

integral equation (EFIE). These resonances compare well with asymptotic estimates derived from radial eigeofunction expansion for scattering from cylinders.

REFERENCES A. W. Maue, “On the formulation of a general scattering problem by means of an integral equation,” Z. Phys., vol. 126, no. 7, pp. 601-618, 1949. I. INTRODUCTION F. K . Oshiro and K. M. Mitzner, “Digital computer solution of three- dimensional scattering problems,” in Symp. Dig., IEEE Int. Symp. AntennasPropagat., Ann Arbor, MI, Oct. 1967, pp. 257-263. (Also F. K. Oshiro et al., “Calculation of radar cross-section, Part II,” Wright Patterson Air Force Base, OH, AF Tech. Rep. AFAL-TR-7O-21, 1970.) A. J. Poggio and E. K. Miller, “Integral equation solutions of three- dimensional scattering problems,” in Computer Techniquesfor Elec-

J. R. Mautz and R. F. Harrington, “H-field, E-field, and combined field tromagnetics, R. Mittra, Ed. London, Pergamon, 1963, ch. 4.

solutions for conducting bodies of revolution,” Archive Efektronik und Ubertragungstechnik, vol. 32, pp. 157-164, 1978. (See also “Com- puter programs for H-field, E-field, and combined field solutions for bodies of revolution,” 77-215, Rome Air Development Center, Griffiss Air Force Base, N Y , InterimTech.~~Rep. RADC-TR-77-215, June 1977.) H. Brakhage and P. Werner, “Uber das Dirichletsche Aussenraum- problem fur die Helmholtzsche Schwingungsgleichung,” Arch. Math.,

D. Greenspan and P. Werner, “A numerical method for the exterior dirichlet problem for the reduced wave equation,” Arch. Ration. Mech. Anal., vol. 23, no. 4, pp. 288-316, 1966. J. R. Mautz and R. F. Harrington, “A combined-source solution for radiation and scattering from a perfectly conducting body,” IEEE Trans. AntennasPropagat., vol. AP-27, pp. 445454, 1970. (See also report of same title: Dept. Elec. Comp. Eng., Syracuse Univ., Rep. TR- 78-3, Syracuse, N Y , Apr. 1978. Also “Application of the combined- source solution to a conducting body of revolution,” Dept. Elec. C o p . Eng., Syracuse Univ., Rep. TR-78-6, Syracuse, N Y , June 1978.) C. A. Klein and R. Mittra, “An application of the ‘condition number’ concept to the solution of scattering problems in the presence of interior resonant frequencies,” IEEE Trans. Antennas Propagat., vol. AP-23, pp. 431-435, 1975. (See also R. Mittra and C. A. Klein, “Stability and convergence of moment method solutions,” in Numerical and Asymp

VOI. 16, no. 415, pp. 325-329, 1965.

The surface current density on a conducting scatterer is re- presentable in terms of an expansion in complex natural res- ( onances of the object. This expansion is appropriate at inter- mediate and late observation times. At early observation times, a ray formulation involving a representation of the current density in terms of creeping waves is more appropriate [ l ] . The two representatives may be blended systematically into a hybrid expansion, which, in principle, converges for all observation times [ 13.

cylinder excited by a line source, and it is also shown how one may generalize this scheme to smooth convex cylinders of other- wise arbitrary cross section. In this c o r n aication, we report results of implementing the phase integral expression in [ I ] for the complex resonances on an elliptic cylinder of 3:2 aspect ratio. Good estimates of the poles, including the lowest order, are obtained when compared with a method of moments com-

This hybrid scheme is discussed in detail in [ l ] for a circular 1

Manuscript received October 22, 1984; revised January 21, 1985. This work 1 was sponsored by the Office of Naval Research under Contract N00014-81-K- 0256 with the University of Mississippi.

R. Naishadham is with the Department of Electrical Engineering, University of Mississippi, University, MS 38677.

L. W. Pearson was with the Department of Electrical Engineering,

Research Laboratories, P.O. Box 516, St. Louis, MO 63166. University of Mississippi, University, MS. He is now with McDonnell Douglas

0018-926X/85/0600-0674$01.00 0 1985 IEEE