Compact Quasi-Static PEEC Modeling of Electromagnetic Problems With Finite-Sized Dielectrics

In this article, a compact partial element equivalent circuit (c-PEEC) model is presented for characterizing the electromagnetic (EM) problems with finite-sized piecewise homogeneous dielectrics. Unlike in conventional PEEC models where dielectrics are described by massive subcircuits, the proposed c-PEEC model addresses the composite conductor-dielectric problems via circuit elements resident on conductors only. Novel null-field boundary integral equations (n-BIEs) are formulated based on the surface equivalent principle and quasi-static assumption. The elimination of the magnetic field-related equivalent sources avoids the low-frequency breakdown problem caused by the weak EM coupling at low frequencies and improves the accuracy by inhibiting the numerical errors in discretizing magnetic sources. The computational cost is significantly reduced compared with the conventional models. Furthermore, the concise configuration of the c-PEEC model extends the model-order reduction (MOR) algorithms for fundamental homogeneous models to heterogeneous integration. Three numerical examples including a resonator, a radio frequency (RF) embedded passive circuit, and an interconnection problem, are studied to validate the stability, efficiency, and accuracy of the c-PEEC model. A micro-modeling circuit is obtained from the c-PEEC model to demonstrate its compatibility with the MOR algorithms.

I. Introduction

With the ever and rapid increase of data rate, circuit complexity, and device density in electronic integration and packaging, the signal integrity (SI) and power integrity (PI) of high-speed heterogeneous integrated circuits (ICs) become an imperative and challenging issue especially at the design stage [1]. Numerous methods have been explored in the past decades to address the large-scale electromagnetic (EM) modeling problems, which can be mainly categorized into two groups, the differential equation methods [2], [3] and the integral equation methods [4]. The integral equation methods exhibit superior efficiency for modeling the problems involving conductors in free space as compared to the differential equation methods. Because unknowns from the background space are not required in the integral equation methods, the number of unknowns is thus significantly reduced. However, for heterogeneous problems involving finite-sized dielectrics, additional equivalent sources are needed to describe the dielectrics in the conventional integral equation methods, resulting in exponential increase of the computation effort for electrically large problems. The denser impedance matrix will incur a dramatic hike in the computational overhead. An integral equation method that is capable of modeling EM problems involving piecewise homogenous dielectrics, meanwhile, has less unknowns is highly desirable.