# Analytic Solution of Characteristic Modes for Spherical Shell

### Characteristic numbers

Characteristic numbers of spherical shell of radius a made of PEC are:

$$\lambda_n^\mathrm{TE} = - \frac{\mathrm{y}_n \left(ka \right)}{\mathrm{j}_n \left( ka \right)},$$
$$\lambda_n^\mathrm{TM} = - \frac{\left( n + 1 \right) \mathrm{y}_n \left(ka \right) - ka \, \mathrm{y}_{n+1} \left(ka \right)}{ \left( n + 1 \right) \mathrm{j}_n \left( ka \right) - ka \, \mathrm{j}_{n+1} \left( ka \right)},$$
where $\mathrm{j}_p$  and $\mathrm{y}_p$ are the spherical Bessel (Neumann) functions of $p$-th order.

Characteristic angles (only re-scaling of characteristic numbers):

$$\delta_n^\mathrm{TE/TM} =180 \left( 1 - \frac{1}{\pi} \mathrm{atan} \left( \lambda_n^\mathrm{TE/TM} \right) \right).$$

Characteristic numbers calculated as $10 \, \text{log}_{10} \left|\lambda_n\right|$ for three different normalized frequencies ka:

### Characteristic currents

Characteristic currents (normalized with respect to unitary radiated power) are:
$$\boldsymbol{J}_{pq}^\mathrm{TE} = C_{pq} \left( ka \right) \hat{\boldsymbol{r}} \times \boldsymbol{N}_{pq} \left(\mathrm{j}_p,a,\vartheta,\varphi\right),$$
$$\boldsymbol{J}_{pq}^\mathrm{TM} = C_{pq} \left( ka \right) \hat{\boldsymbol{r}} \times \boldsymbol{M}_{pq} \left(\mathrm{j}_p,a,\vartheta,\varphi\right),$$

where

$$C_{pq} \left( ka \right) = \frac{k}{\gamma_p \left( ka \right)} \displaystyle\sqrt{\frac{Z_0 \left(2p+1\right)\left(p-q\right)!}{\pi \left( 1 + \delta_{q0} \right) p \left(p+1\right)\left(p+q\right)!}},$$
$$\gamma_p \left( ka \right) = Z_0 ka \, \mathrm{j}_p \left( ka \right) \frac{\partial \left( ka \, \mathrm{j}_p \left( ka \right) \right)}{\partial ka},$$
$Z_0 = \sqrt{\mu / \epsilon}$ being impedance of vacuum, functions $\boldsymbol{M}_{pq}$ and $\boldsymbol{N}_{pq}$ being defined in Stratton.

The aggregated index $\boldsymbol{J}_n = \left\{ \boldsymbol{J}_{pq}^\mathrm{TE}, \boldsymbol{J}_{pq}^\mathrm{TM} \right\}$ is used thoroughly on this web page.

Characteristic numbers $\lambda_n$, first six TM and TE modes:

Characteristic angles $\delta_n$, first six TM and TE modes:

/Edited 17. 07. 2017, MC/