1500 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 6, JUNE 2012 Ef ficient Low-Frequency Breakdown Free Full-Wave PEEC Modeling Based on Geometrical Optics DCIM Zhi-Yuan Zong, Member, IEEE, Wen Wu, Senior Member, IEEE, Feng Ling, Senior Member, IEEE, Ji Chen, Senior Member, IEEE, and Da-Gang Fang, Fellow, IEEE Abstract—The partial element equivalent circuit (PEEC) mod- eling based on geometrical optics discrete complex image method (GODCIM) is proposed. In GODCIM, the spectral domain dyadic Green’s functions for layered media (DGFLM) are expanded into a geometrical optics series. The coefficient of each term is related to the Green’s function for a half-space problem and then is expanded into an exponential series with frequency-independent parameters, which avoids the repeated computations of the complex images at each frequency brought by the conventional discrete complex image method (DCIM) and facilitates its application to the PEEC method to solve the layer medium problems. To address the low- frequency breakdown issue in PEEC modeling, a capacitor-opened (CO) model is introduced, which leads to fewer unknowns com- paring with modified nodal analysis (MNA) model. The calcula- tion errors between CO and MNA model are verified to be negli- gible. Easy switching between CO model and modified loop anal- ysis (MLA) model is realized by a switching criterion based on Kalman filter which leads to a full spectrum simulation from dc to high frequencies. Numerical results ranged from Hz to 30 GHz are given to validate the proposed method. Index Terms—Discrete complex image method (DCIM), Green’s functions, geometrical optics discrete image method (GODCIM), low-frequency breakdown, partial element equivalent circuit (PEEC). I. INTRODUCTION T HE partial element equivalent circuit (PEEC) method is equivalent to the traditional Method of Moment (MoM) by selecting the same pulse function for basis and testing functions, but it differs in that actual circuit elements are used. Through introducing the partial capacitances, or coefficients of potential, and partial inductances, the PEEC approach can generate the electromagnetic models of various structures in terms of lumped element circuit network, which can be easily managed by general-purpose circuit simulators like SPICE [1], [2]. In addition, the PEEC method can be employed in both frequency and time domains [3], [4]. All of these merits make the PEEC method popular within the fields of electromagnetic Manuscript received October 02, 2011; revised February 12, 2012; accepted February 16, 2012. Date of publication April 05, 2012; date of current version May 25, 2012. Z.-Y. Zong, W. Wu, F. Ling, and D.-G. Fang are with the Ministerial Key Laboratory of JGMT, Nanjing University of Science and Technology, 210094, China (e-mail: zongzhiyuan@sina.com). J. Chen is with the Department of Electrical and Computer Engineering, Uni- versity of Houston, Houston, TX77204 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2012.2189238 compatibility, electrical interconnect analysis, and signal in- tegrity [5], [6]. Many variants of PEEC models have been devised for appli- cations after its introduction by Ruehli during the 1970s [7], [8], such as the PEEC models including incident fields or scattering fields [9], dielectrics [10], the retarded PEEC models (rPEEC) [11], and macromodeling PEEC [12]. However, all these PEEC models used only simple Green’s functions for the free or half space [13]. This means that the dielectric has to be discretized using a 3-D volume grids, which leads to a large full matrix system. In this case, the computational costs are higher than those using differential methods. Recently, the dyadic Green’s functions for layered media (DGFLM) were introduced into the PEEC method to solve the strip-line structures in layered media [14]. With the aid of the Green’s functions for layered media, the proposed DGFLM-PEEC method requires the discretization only on con- ductors. The method greatly improves the efficiency compared with the method requiring the discretization of both conductor and dielectric. It can also be applied to anisotropic piecewise ho- mogeneous layered media, However, the paper only addressed strip-line structures and only the quasi-dynamic model is given for time domain [13] due to the difficulty in obtaining the Green’s functions in the spatial domain for arbitrary multilayer medium. New progresses have been presented in [15] and [16] where the discrete complex image method (DCIM) was used in PEEC analysis for planar circuits. DCIM was adopted to avoid the time-consuming and complicated numerical computation of the Sommerfeld integrals (SIs) that is used in calculating spatial Green’s functions [17]–[21]. However, in the DCIM-PEEC approach, the complex coefficients of the complex images are frequency dependent; therefore, the discrete complex images must be determined for each frequency. For wideband analysis or for time-domain analysis through FFT, it can be time con- suming. In this paper, we borrow the idea of the geometrical optics discrete complex image method (GODCIM) first devel- oped for time-domain Green’s function derivation [22], [23] and use it in our PEEC modeling. In GODCIM, the spectral Green’s functions for multilayered media are decomposed into a series by using the principle of geometrical optics [24], [25]. Each term in the series is correspondent to the Green’s function for the half-space problem and then may be expanded into exponential series with frequency-independent parameters. Consequently, this series is available to arbitrary frequency, and the expansion is needed only once. As a special DCIM, GODCIM introduced into PEEC modeling cannot only enhance the computation efficiency but also share the recent progress 0018-9480/$31.00 © 2012 IEEE