# Test #4 - Characteristic Far-Fields

The conformity between numerically calculated and the analytically known characteristic far fields can be calculated as

$$\zeta_n = \displaystyle \frac{\max\limits_p \displaystyle \sum\limits_{q=-p}^{p} \Bigg| P_{pq,n}^\mathrm{TE/TM} \Bigg|^2}{\displaystyle \sum\limits_{p,q} \sum\limits_\mathrm{TE/TM}\Bigg| P_{pq,n}^\mathrm{TE/TM} \Bigg|^2}$$

with

$$P_{pq,n}^\mathrm{TE/TM} = \displaystyle \frac{1}{2 Z_0} \int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} \left( \boldsymbol{F}_{pq}^\mathrm{TE/TM} \right)^\ast \cdot \boldsymbol{F}_n \sin \vartheta \,\mathrm{d}\vartheta \,\mathrm{d}\varphi,$$

where $Z_0$ is the free space impedance, $\boldsymbol{F}_n$ is numerically calculated characteristic far field, and $\boldsymbol{F}_{pq}^\mathrm{TE/TM}$ is the analytically known characteristic far field.

#### Results:

Coefficient $\zeta_n$ for spherical shell discretized into 500 triangles (750 RWG functions) at ka = 1/2:

Matlab script generating characteristic far-fields for TM and TE spherical modes is HERE.

For details see IEEE Trans. Antennas and Propagation paper.

/Edited 21. 04. 2017, MC/