5.1 Calculation of Kcouple with mode decomposition

The introduction of wires into a rectangular enclosure by a multi-mode analogous transmission line theory has been presented by [49] and [50]. The introduction of traces to the parallel-plane cavity field formulations in Chapter 4 can be achieved by mode decomposition [44]. The cover plane return current is identified as the parasitic common mode current depicted in Figure 5.3.

\includegraphics[width=13cm,viewport=110 560 510
Figure 5.3: Identification of the cover plane return current as the parasitic common mode current.

The conductor with number 1 in Figure 5.3 is the trace, the ground plane is assigned to number 2 and the cover plane to number 3. The partial capacitances between these conductors are indexed accordingly. A source which drives the traces against the ground plane will excite both, the differential mode and the common mode currents. The partial capacitance between the cover and the ground plane $ C_{23}$ is high, due to the large extent of these planes. A source current that drives the trace is divided by the capacitances $ C_{12}$ and $ C_{13}$. Therefore, the excitation of the cavity field $ I$ in (4.13) by the trace is expressed by the trace currents multiplied by the coupling factor

$\displaystyle K_{couple}=K_{couple\_md}=\frac{C_{13}}{(C_{13}+C_{12})}$ (5.7)

To extract the $ 1/2n_{co}(n_{co}-1)$ partial capacitances between $ n_{co}$ conductors, the Laplace equation

$\displaystyle \vec{\nabla}(\epsilon\vec{\nabla}\varphi)$   with$\displaystyle \qquad\begin{cases}
 \varphi=V&\text{, on conductor boundaries}\\...
...vec{\nabla}\varphi =0& \text{, on open boundaries with no charges}
 \end{cases}$ (5.8)

for the electrostatic potential $ \varphi$ has to be solved for $ 1/2\cdot
n_{co}(n_{co}-1)$ different voltage distributions [71], [74]. The surface normal vector at the boundary is $ \vec{n}$. The Smart Analysis Program (SAP), a FEM based interconnect simulation software from [75], performs this partial capacitance extraction automatically. SAP is also capable of automated resistance and inductance extraction of interconnects.
Figure 5.4 depicts the difference in the electrostatic potential distribution between a trace with and without cover plane.

\includegraphics[height=4 cm,viewport=100 80 380
250,clip]{pics/PotentialTrace_under_Plane.eps} \includegraphics[height=4 cm,viewport=100 80 380
(a) Field with cover plane. (b) Field without cover plane.
Figure 5.4: Electrostatic potential with and without the metallic cover plane (qualitative diagram).

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