Digital Wave Formulation of Quasi-Static Partial Element Equivalent Circuit Method Piero Belforte Independent Researcher Via G. C. Cavalli 28bis, 10138 Turin, Italy Email: piero.belforte@gmail.com Luigi Lombardi, Daniele Romano and Giulio Antonini Dipartimento di Ingegneria Industriale e dell’Informazione e di Economia Universit` a degli Studi dell’Aquila Via G. Gronchi 18, 67100, L’Aquila, Italy, Email: giulio.antonini@univaq.it Abstract—This paper presents a digital wave formulation (DWF) of the quasi-static Partial Element Equivalent Circuit formulation. Through the use of a pertinent change of variables and the choice of a specific implementation of PEEC cell elements in the Digital Wave domain, the standard PEEC model is transformed into and solved as a wave digital network. The example reported shows the accuracy and the significant speed- up of the proposed DWF-based PEEC solver when compared to the standard Spice solution. I. I NTRODUCTION One well-known approach for generating accurate circuit models for 3-D electromagnetic structures is the partial el- ement equivalent circuit (PEEC) approach [1]. The PEEC method is based on the mixed potential integral equation (MPIE) and the continuity equation. It provides a circuit interpretation of the electric field integral equation (EFIE) and continuity equation [2] in terms of partial elements, namely resistances, partial inductances and coefficients of potential. Hence, the resulting equivalent circuit can be directly embedded in a circuit environment like SPICE-like circuit solvers [3] and the entire problem can be described by means of the circuit theory and solved in both the time and frequency domain. Time domain solutions are especially advantageous and the unique possibility, if there are nonlinearities in the circuit environment. Over the years, several improvements of the PEEC method have been performed allowing to handle complex problems involving both circuits and electromagnetic fields [1], [4]–[14]. Introduced originally by Fettweis [15], wave digital filters (WDFs) and scattering methods have been widely used for numerical simulations [16] and applied to numerical methods quite popular in the community of applied electromagnetics. In particular, they have been applied to transmission lines and their basic idea is strictly related to the the Transmission Line Matrix (TLM) method [17], [18]. Starting from the mid 70’s several generations of circuit simulation tools based on DWF have been developed and applied mainly for the state-of-art signal and power integrity applications in the field of high- speed digital systems [19]–[21]. Despite its generality, the digital wave formulation (DWF) has been mainly applied to differential methods, where the interactions are local leading to sparse matrices. Aim of this work is to apply DWF to integral-equation based techniques. Among the others, the PEEC method, especially for its capability to translate an elec- tromagnetic problem into a circuit form, is the best candidate to be combined with the DWF method. The paper is organized as follows. Section II briefly sum- marizes the PEEC method. The wave digital elements are introduced in Section III in both the scalar and multidi- mensional case. The resulting PEEC-digital wave simulator (PEEC-DWS) is briefly described in Section IV. A numerical example is presented in Section V where the PEEC-DWS method is compared with a standard Spice solver in terms of accuracy and cpu-time requirements. The conclusions are drawn in Section VI. II. BASIC PEEC FORMULATION FOR CONDUCTIVE MATERIALS The standard PEEC method is based on the equation of the electric field and the continuity equation. Using a pertinent spatial discretization of currents flowing in conductors and di- electrics and charges located on their surfaces, the application of the Galerkin’s process allows to identify topological entities, namely nodes and branches, and equivalent circuit elements. Figure 1 shows an example of PEEC circuit where the electric field coupling is represented in terms of current controlled current sources (CCCS) and the magnetic field coupling in terms of current controlled voltage sources (CCVS). The enforcement of Kirchoff Voltage Law (KVL) and Kirchoff Current Law (KCL) leads to the following set of ordinary differential equations in the MNA form: C x()  = −Gx()+ Ku() (1) where C = [ 0 L  C  0 ] (2a) G = [ A R G  −A  ] (2b) K = [ ℐ 0 0 ℐ ] (2c) x() = [() i()]  (2d) u() = [ −v  () i  () ] . (2e) where A is the connectivity matrix, L  is the partial in- ductance matrix, C  = P −1 is the short circuit capacitance matrix, R the resistance matrix, G  is the lumped memoryless elements matrix.