Efficient Computation of Partial Elements in the Full-Wave Surface-PEEC Method

Abstract

The partial element equivalent circuit (PEEC) method provides an electromagnetic model of interconnections and packaging structures in terms of standard circuit elements. The surface-based PEEC (S-PEEC) formulation can reduce the number of unknowns compared to the standard volume-based PEEC (V-PEEC) method. This reduction is of particular use in the case of high-speed circuits and high-switching power electronics, where the bandwidth extends from low frequencies to the GHz range. In this article, the S-PEEC formulation is revised and cast in a matrix form. The main novelty is that the interaction integrals involving the curl of the magnetic and electric vector potentials are computed through the Taylor series expansion of the full-wave Green's function, leading to analytical forms that are rigorously derived. Therefore, the numerical integration is avoided, with a consequent reduction of the computation time. The proposed formulas are studied in terms of the frequency, size of the mesh, and distance between the basis function domains. Three examples are presented, confirming the accuracy of the proposed method compared to the V-PEEC method and surface-based numerical methods from literature.

The Partial Elements Equivalent Circuit Method: The State of the Art

“…However, problems may occur in the numerical implementation due to the high contrast between the conductor and the surrounding dielectric, which makes the resultant linear system highly unbalanced. In [73], the interaction integrals involving the curl of the magnetic and electric vector potentials are computed through the Taylor series expansion of the full-wave Green's function, leading to analytical forms that are rigorously derived, thus reducing the numerical issues. When the skin-effect is well-developed, a valuable alternative option is represented by the use of the surface impedance accounting for the nonuniform distribution of the current density [74], [75].…”

On the rectangular mesh and the decomposition of a Green’s-function-based quadruple integral into elementary integrals

“…Similar to other EM methods for scattering problems, in the S-PEEC method the surface of the object is discretized into rectangular patches. For each patch, the surface electric and magnetic currents and charges are computed by integrating the Green's function of the problem and its curl over the pertinent domain [29]. In particular, the S-PEEC method requires the computation of the integral…”

“…)𝑓 (𝜌) 𝜌 𝑑𝜌 (29) In the following, the above integral 𝐼 𝑝2 for 𝑙 1 > 𝑙 2 will be called as ''𝐼 𝑝2type 2''; consequently, the integral 𝐼 𝑝2 for 𝑙 1 < 𝑙 2 will be called as ''𝐼 𝑝2type 1''. For ''𝐼 𝑝2 -type 2'', the (28) becomes:…”

“…The problem of evaluating the Green's function integrals in triangular patches is also addressed in [28], where the authors exploit a polar-coordinate transformation and a mixed analytic and numerical quadrature that mitigates the singularity and increases the accuracy of the integral computation. For the S-PEEC method with rectangular mesh, some first attempts in this direction have been made in [29,30]. In [29], the Taylor expansion of the exponential term in the Green's function allows a complete analytic formulation of both the self-interaction and the mutual-interaction integrals.…”

“…For the S-PEEC method with rectangular mesh, some first attempts in this direction have been made in [29,30]. In [29], the Taylor expansion of the exponential term in the Green's function allows a complete analytic formulation of both the self-interaction and the mutual-interaction integrals. However, the accuracy depends on the Taylor expansion order, and the method suffers when used for lossy materials in the low-frequency range.…”