7.2.3 Introduction of the radiation loss admittance matrix into the cavity model matrix

The relation of the port voltages to the port currents is given by the impedance matrix

$$\begin{pmatrix} U_{m}\\ U_{s}\\ U_{l}\\ U_{1}\\ \vdots\\... ...ix} I_{m}\\ I_{s}\\ I_{l}\\ I_{1}\\ \vdots\\ I_{p}\\ \end{pmatrix}$$ (7.39)

where indexes $m$, $s$, and $l$ are assigned to a measurement port, a port at the source position of a trace inside the enclosure, and a port at the load position of this trace, respectively. The indices $1$ to $p$ are assigned to the interface ports at the slot of the enclosure. The elements of the impedance matrix are calculated analytically with (7.14) for a slim ( $h\ll\lambda$) rectangular enclosure with a slot on one edge. With $I_{m}=0$ at the voltage measurement port, matrix (7.39) is separated to the slot port matrix equation

$$\begin{pmatrix} U_{1}\\ \vdots\\ U_{p}\\ \end{pmatrix} = ... ... \end{pmatrix} \begin{pmatrix} I_{1}\\ \vdots\\ I_{p}\\ \end{pmatrix}$$ (7.40)

with the matrix notation

$$\textbf{U}_{p}=\textbf{Z}_{ps}\textbf{I}_{s}+\textbf{Z}_{pp}\textbf{I}_{p},$$ (7.41)

and the measurement port matrix equation

$$U_{m}= \begin{pmatrix} Z_{ms}&Z_{ml}\\ \end{pmatrix} \begin{... ... \end{pmatrix} \begin{pmatrix} I_{1}\\ \vdots\\ I_{p}\\ \end{pmatrix}$$ (7.42)

with the matrix notation

$$\textbf{U}_{m}=\textbf{Z}_{ms}\textbf{I}_{s}+\textbf{Z}_{mp}\textbf{I}_{p}.$$ (7.43)

The admittance matrix $\textbf {Y}_{a}$ with the elements from (7.38) relates the voltage vector $\textbf{U}_{p}$ to the the current vector $\textbf{I}_{p}$ at the interface ports with

$$\textbf{U}_{p}=-\textbf{Y}_{a}^{-1}\textbf{I}_{p}.$$ (7.44)

This leads to the final formulation for the voltage on the test port

$$\textbf{U}_{m}=(\textbf{Z}_{ms}-\textbf{Z}_{mp}(\textbf{Z}_{pp}+\textbf{Y}_{a}^{-1})^{-1}\textbf{Z}_{ps}) \textbf{I}_{s},$$ (7.45)

and the voltages on the interface ports

$$\textbf{U}_{p}=\textbf{Y}_{a}^{-1}(\textbf{Z}_{pp}+\textbf{Y}_{a}^{-1})^{-1}\textbf{Z}_{ps} \textbf{I}_{s}.$$ (7.46)

When the radiation loss becomes very low, $\textbf {Y}_{a}$ is almost singular. For a nearly singular $\textbf {Y}_{a}$, (7.45) can be simplified to

$$\textbf{U}_{m}=\textbf{Z}_{ms}\textbf{I}_{s}$$ (7.47)

and (7.46) to

$$\textbf{U}_{p}=\textbf{Z}_{ps}\textbf{I}_{s},$$ (7.48)

to avoid matrix inversion in such a case. Since (7.47) and (7.48) neglect the radiation loss, these equations may only be used at frequencies, where $\textbf {Y}_{a}$ is nearly singular.
The internal enclosure voltages and the slot voltages between the metallic cover and bottom plane are modeled accurately with (7.45), (7.46),  (7.47), and (7.48). With the voltages at the slot (7.32) and (7.33) the free space far field radiation from the enclosure slot can be calculated analytically. This model enables efficient, quantitative investigations in the predesign phase of a device, especially regarding placement decisions. Design rules, derived from the discussion of the analytical cavity model in Section 5.8 and Subsection 7.1.3 are investigated in Chapter 8 regarding their quantitative relevance in the radiated far field.

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