International Journal of Microwave Science and Technology

Volume 2015 (2015), Article ID 724702, 7 pages

http://dx.doi.org/10.1155/2015/724702

## An Effective Math Model for Eliminating Interior Resonance Problems of EM Scattering

^{1}Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Nanjing 210044, China^{2}Key Laboratory for Aerosol-Cloud-Precipitation of China Meteorological Administration, Nanjing University of Information Science, No. 219, Ningliu Road, Nanjing 210044, China

Received 22 May 2014; Revised 26 November 2014; Accepted 15 December 2014

Academic Editor: Giancarlo Bartolucci

Copyright © 2015 Zhang Yun-feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is well-known that if an -field integral equation or an -field integral equation is applied alone in analysis of EM scattering from a conducting body, the solution to the equation will be either nonunique or unstable at the vicinity of a certain interior frequency. An effective math model is presented here, providing an easy way to deal with this situation. At the interior resonant frequencies, the surface current density is divided into two parts: an induced surface current caused by the incident field and a resonance surface current associated with the interior resonance mode. In this paper, the presented model, based on electric field integral equation and orthogonal modal theory, is used here to filter out resonant mode; therefore, unique and stable solution will be obtained. The proposed method possesses the merits of clarity in concept and simplicity in computation. A good agreement is achieved between the calculated results and those obtained by other methods in both 2D and 3D EM scattering.

#### 1. Introduction

Electric field integral equation (EFIE) and magnetic field integral equation (MFIE) have been widely employed to analyze electromagnetic scattering of conducting bodies [1–3]. However, interior resonance phenomena exist in solving electromagnetic scattering problems with surface integral equations. When working frequencies of conductors are near (or exactly at) to the frequencies associated with interior resonances, the single equation will become highly ill-conditioned (or singular) which makes the solution unstable or nonunique. Also, the interior resonance behavior has significant influence on the late time stability associated with time domain EFIE and MFIE [4, 5].

Several ways of dealing with this numerical problem have been proposed. Nowadays, the popular combined field integral equation (CFIE) technique to overcome this problem is a proper combination of the electric field integral equation and the magnetic field equation [6, 7]. The CFIE technique requires the calculation of both and impedance matrices and it is not suitable for aperture problems. The combined source integral equation (CSIE) [8, 9] technique makes up for aperture structures. Also a technique has been proposed by Mittra and Klein [10], involving application of the generalized boundary condition [11], and consists of additional points in the interior of the conductor and forces the field to be zero at those points. The problem with this technique is the fact that the chosen interior points must be carefully selected so as not to lie on nodal lines, which is not too practical for large bodies of simple shape, for which a slight change in frequency can take us from one resonance to the next, of different modal distribution. There are also some iterative methods reported in the works of Sarkar and Ergül [12, 13] to deal with this situationand they are used to compute the minimum norm solution (which produces the correct scattered fields but not the true tangential fields) and to calculate the LQSR. Another work related to the use of extended integral equations has been presented by Mautz and Harrington [8], which involves application of the boundary element method, with observation points lying on an internal closed surface near the boundary of the scatterer. Unfortunately, the resulting matrix equations are also ill-conditioned since the internal contour can resonate by itself. The authors’ proposed scheme of allowing the internal contour to vary with the wave number seems impractical and in reality does not solve the problem except in some isolated cases of electrically small simple shapes whose resonances are known. More recently, Canning [14] illustrated a matrix algebra technique known as the singular value decomposition (SVD) has been proposed for moment method calculations involving perfect conductors. Such a technique diagonalizes the matrix equation, isolating the resonant contribution, which is then omitted in the calculation. We also refer to the interesting work of Yaghjian, who originally presented his augmented electric or magnetic field equation [15]**, **and more recently Tobin et al. [16] pointed out their drawbacks for an arbitrarily shaped, multiwavelength body and, most importantly, introduced a modification, the so-called dual-surface integral equation, which is applicable to the perfectly conducting bodies and supposedly eliminates all the spurious solutions. Finally, the modal orthogonal characteristics [17, 18] were applied to solve the same problem in two-dimensional EM scattering.

Here, we present an effective method to solve the scattering problem of conductor bodies at or in the neighborhood of the resonant frequencies in both 2D and 3D EM scattering. At the resonant frequencies, Inagaki modes, firstly applied to analyze the antenna array [19], are employed here to be validated away from resonance with a unique and stable solution to the equation. It can both stabilize the numerical calculation and yield to reliable results of the surface current density and exterior field for conductors at interior resonances.

#### 2. Theory

The principle is elaborated as follows. The solution of anill-conditioned system of the equation will consist of (1) the correct physical solution to the problem and (2) the resonant solution, which is in conjunction with Green’s Theorem which produces the nonzero complementary resonant modes. When the orthogonal modes are used to solve the electric field integral equation, the solutions will be divided into the induced modes and the resonant modes corresponding to eigenvalues. We can easily obtain the current density and exterior field from the induced modes.

##### 2.1. Why Is the Solution Unstable?

Consider a PEC scatterer excited by an incident wave and the scatterer defined by the surface , in an impressed electric field (see Figure 1).