7.2.2 Derivation of an admittance matrix for the consideration of the radiation loss at the cavity field simulation

The pointing vector

$$\vec{S}=\vec{E}\times\vec{H}^{*}$$ (7.35)

is obtained from a multiplication of (7.32) and (7.33). Since this is a multiplication of the two summations, every term in the summation of (7.32) is multiplied with every term in the summation of (7.33), yielding $p^2$ terms,

$$\vec{S}_{cr}=\frac{k^{3}}{16\pi^{2}}\frac{1}{\omega\mu}\sin^{2}(\... ...{L}{p}\right)^{2}U_{c}U_{r}^{*}e^{\left[jk(x_{c}-x_{r})\cos(\vartheta)\right]},$$ (7.36)

which are the self- and mutual pointing vectors of the slot ports with the indexes $c$ and $r$. An integration of these vectors over the sphere is carried out to obtain the total power values of the slot ports with the indexes $c$ and $r$.

$$S_{cr}=\frac{k^{3}}{8\pi^{2}\omega\mu}U_{c}U_{r}^{*}\left(\frac{L... ...heta)e^{\left[jk(x_{c}-x_{r})\cos(\vartheta)\right]}\textrm{d}\vartheta\right\}$$ (7.37)

This pointing power is divided through the voltages $U_{c}$ and $U_{r}^{*}$ on the slot ports $c$ and $r$, respectively, to obtain an admittance matrix element

$$Y_{a\_cr}=\frac{S_{cr}}{U_{c}U_{r}^{*}}=\frac{k^{3}}{8\pi^{2}\ome... ...heta)e^{\left[jk(x_{c}-x_{r})\cos(\vartheta)\right]}\textrm{d}\vartheta\right\}$$ (7.38)

assigned to these slot ports. The index $c$ denotes the column and the index $r$ denotes the row of the admittance matrix element $Y_{a\_cr}$ in the admittance matrix ${\textbf{Y}_{a}}$. Since (7.38) is independent of the voltages at the slot ports, the admittance matrix ${\textbf{Y}_{a}}$ enables the far field pointing power for arbitrary slot port voltage distributions to be calculated. Equation (7.38) was obtained without utilizing the cavity model and is therefore independent of that model. A connection of the admittance network described with ${\textbf{Y}_{a}}$ to $p$ slot ports declared in a cavity model is an analytical application of the domain decomposition approach in Chapter 6, because the admittance matrix introduces the influence of the free space radiation into the cavity model. This enables the correct consideration of the radiation loss in the cavity field calculation.
With this cavity model simulation which considers the radiation loss, the radiated electric far field is calculated from the slot port voltages utilizing (7.32) and the radiated magnetic far field is calculated with (7.33). Equations (7.32) and (7.33) have only one vector component in the spherical coordinate system, defined in Figure 7.3. Equation (7.38) is also much simpler with this coordinate system definition, compared to the commonly used definition. The number of ports $p$ that is necessary to achieve certain accuracy depends on the maximum frequency. A calculation with increased $p$ can be carried out to check whether $p$ is sufficiently high.

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