4.2 Analytical solution methods for the two-dimensional Helmholtz equation

A solution of (4.13) with $n_{port}$ excitation ports, defined between the cover and the bottom plane (Figure 4.2), relates the port voltages $U_{i}(x_{i},y_{i})$ on the ports with index $i$ to the currents $I_{j}(x_{j},y_{j})$ on the ports with index $j$ by the impedance matrix $Z_{ij}$. The coordinates of port i and port j are $(x_{i},y_{i})$ and $(x_{j},y_{j})$ respectively.

$$U_{i}(x_{i},y_{i})=\sum_{j=1}^{n_{port}}(Z_{ij}(x_{i},y_{i},x_{j},y_{j})I_{j}(x_{j},y_{j})) \qquad i=1\ldots n_{port}.$$ (4.17)

An analytical solution for rectangular parallel planes with four open edges depicted in Figure 4.2 was presented by [42]. In this solution, the coefficients of the impedance matrix in (4.17) are

$$Z_{ij}=\frac{j\omega\mu h}{L_{e}W_{e}}\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\left[\frac{L_{mn}N_{mni}N_{mnj}}{{k_{m}^2+k_{n}^2-k^2}} \right],$$ (4.18)

with

$$N_{mni}=\cos(k_{m}x_{i})\cos(k_{n}y_{i})\sinc \left(\frac{k_{m}Wx_{i}}{2}\right)\sinc \left(\frac{k_{n}Wy_{i}}{2}\right),$$ (4.19)

$$N_{mnj}=\cos(k_{m}x_{j})\cos(k_{n}y_{j})\sinc \left(\frac{k_{m}Wx_{j}}{2}\right)\sinc \left(\frac{k_{n}Wy_{j}}{2}\right),$$ (4.20)

$$k_{m}=\frac{m\pi}{L_{e}},\,k_{n}=\frac{n\pi}{W_{e}},\,L_{e}=L+\frac{h}{2},\,W_{e}=W+\frac{h}{2},$$ (4.21)

$$L_{mn}=\begin{cases} 1&\text{, if $m=0\vee n=0$},\\ 2&\text{, ... ...edge (m=0\vee n\neq 0)$},\\ 4&\text{, if $m\neq 0\vee n\neq 0$}, \end{cases}$$ (4.22)

and

$$\sinc (x)=\frac{\sin(x)}{x}.$$ (4.23)

Figure 4.2: Rectangular, parallel metallic planes with four open edges. Equation (4.18) contains the port impedance matrix elements.

In (4.18) $k=\omega/c_{l}$ denotes the wave number, with the speed of light $c_{l}$. $L_{e}$ and $W_{e}$ are the effective plane length and width, respectively. These effective dimensions consider the fringing fields at the cavity edges according to [45]. [43] also provides an analytical solution for equilateral triangular parallel planes with three open edges. Chapter 7 of this work presents an analytical solution for rectangular parallel planes with one open and three closed edges. This is a powerful solution for predesign investigations of the radiated emissions from the slot of a slim cubical enclosure, because discussion of the bias functions of the model provides direct information about the influence of the source position on the emission level.
The analytical method of [43] enables the calculation of parallel plane cavities with fairly arbitrary shapes by connecting rectangular or equilateral triangular parallel-plane segments. This method is illustrated in Figure 4.3.

Figure 4.3: Three rectangular cavities are connected together with interface ports.

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