# 2.3 Analytical and numerical modeling of the EMI effects to classify sources and coupling paths regarding their potential to exceed emission limits

The previously mentioned methods do not provide insight into the coupling process. However, this is required in order to classify sources and geometric elements for the device optimization. A method that does not provide explicit information on the coupling process would need to be very fast, to enable optimizations with a much higher degree of freedom than a method that provides that insight. However, the previously described methods are not that fast. Therefore, researchers concentrated on the modeling of the emission effects, and common mode coupling has been found to be a significant mechanism of electromagnetic emission initiated by sources on a PCB [32][33][34]. The common mode coupling inductance for the current driven mode has been formulated analytically for a trace above a finite ground plane [35][36][37]. Analytical models for the voltage driven mechanism were developed for a PCB with attached cables [38][39]. All formulations consider the PCB and the trace, but not the influence of an enclosure. According to Chapter 1, a metallic plane is parallel to the PCB at an electrically short distance in many applications. This plane has a significant influence on the common mode coupling impedance. Some configurations have been modeled with FDTD simulation tools [40][41].
The two-dimensional Helmholtz equation was utilized to efficiently model the cavity field between the power- and groundplane of a PCB for the purpose of power integrity analysis [42][43]. Traces were introduced into the cavity field model by mode decomposition for signal integrity and power integrity simulations on PCB level [44].
This work develops a model for the field between the ground plane of a PCB and a parallel metallic cover based on this two-dimensional Helmholtz equation. Traces on the PCB are efficiently introduced into the cavity model by an analytical formulation without mode decomposition. The introduction contains explicit information of the common mode coupling. The thereby obtained insight enables the reduction of device optimization only on relevant parameters. The common mode inductance of a trace above a ground plane without a metallic enclosure derived by [36] and [37] depends on a factor d/W (d...trace height above the ground plane, W... ground plane width). It is shown in this work that the same dependence can be obtained from the cavity model and the analytical trace factor. Therefore, the coupling effect, described by the cavity model, is the same as that of a trace above a ground plane. Powerful analytical and numerical methods are presented for the solution of the two-dimensional Helmholtz equation. The external environment of the enclosure can be simulated with another established simulation program, according to a new domain separation method, as mentioned previously. The next section describes an additional application of the proposed cavity model for fast predesign investigations.

C. Poschalko: The Simulation of Emission from Printed Circuit Boards under a Metallic Cover