# 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_nright|\$ 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,varphiright),\$\$
\$\$boldsymbol{J}_{pq}^mathrm{TM} = C_{pq} left( ka right) hat{boldsymbol{r}} times boldsymbol{M}_{pq} left(mathrm{j}_p,a,vartheta,varphiright),\$\$

where

\$\$C_{pq} left( ka right) = frac{k}{gamma_p left( ka right)} displaystylesqrt{frac{Z_0 left(2p+1right)left(p-qright)!}{pi left( 1 + delta_{q0} right) p left(p+1right)left(p+qright)!}},\$\$
\$\$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/