# Transmission Loss

The field strength $\underset{_}{{E}_{t}}$ of the transmitted wave is computed using the transmission factors and the field strength $\underset{_}{{E}_{i}}$ of the incident wave:

(1) $\frac{{E}_{t}}{{E}_{i}}=|\underset{_}{{T}_{12}}|\cdot |\underset{_}{{T}_{21}}|\cdot |\underset{_}{{T}_{2}}|$

$|{T}_{12}|$ and $|\underset{_}{{T}_{21}}|$ are caused by the transition from air to the material and the other way round, whereas $|\underset{_}{{T}_{2}}|$ is caused when the wave penetrates through the material. The values are shown in columns N, O and P in the table. Additionally, column Q shows the product $|\underset{_}{{T}_{12}}|\cdot |\underset{_}{{T}_{21}}|$ , which is the contribution of the losses due to the transition into the material and the transition back to air.

The values are computed as follows1:

(2) $|\underset{_}{{T}_{12}}|=\frac{2}{\sqrt{1+\sqrt{{\epsilon }_{r}^{\text{'}}{}^{2}+{\epsilon }_{r}^{\text{'}\text{'}}{}^{2}}+2\cdot {\left({\epsilon }_{r}^{\text{'}}{}^{2}+{\epsilon }_{r}^{\text{'}\text{'}}{}^{2}\right)}^{\frac{1}{4}}\cdot \mathrm{cos}\left(\frac{1}{2}\mathrm{arctan}\frac{{\epsilon }_{r}^{\text{'}}}{{\epsilon }_{r}^{\text{'}\text{'}}}\right)}}$
(3) $|\underset{_}{{T}_{21}}|=\frac{2\cdot {\left({\epsilon }_{r}^{\text{'}}{}^{2}+{\epsilon }_{r}^{\text{'}\text{'}}{}^{2}\right)}^{\frac{1}{4}}}{\sqrt{1+\sqrt{{\epsilon }_{r}^{\text{'}}{}^{2}+{\epsilon }_{r}^{\text{'}\text{'}}{}^{2}}+2\cdot {\left({\epsilon }_{r}^{\text{'}}{}^{2}+{\epsilon }_{r}^{\text{'}\text{'}}{}^{2}\right)}^{\frac{1}{4}}\cdot \mathrm{cos}\left(\frac{1}{2}\mathrm{arctan}\frac{{\epsilon }_{r}^{\text{'}}}{{\epsilon }_{r}^{\text{'}\text{'}}}\right)}}$
(4) $|\underset{_}{{T}_{2}}|={e}^{-2\pi \cdot f\cdot d\cdot \sqrt{\frac{{\mu }_{0}\cdot {\epsilon }_{0}\cdot {\epsilon }_{r}^{\text{'}}}{2}\left(\sqrt{1+{\left(\frac{{\epsilon }_{r}^{\text{'}\text{'}}}{{\epsilon }_{r}^{\text{'}}}\right)}^{2}-1\right)}}}$

The transmission loss is then computed as follows:

(5) ${L}_{T}=20\cdot \mathrm{log}\left(|\underset{_}{{T}_{12}}|\cdot |\underset{_}{{T}_{21}}|\cdot |\underset{_}{{T}_{2}}|\right)$

1 O. Michler and M. Strey: DAB- Indoor- Versorgung im L- Band. Eine Studie im Auftrag der Sächsischen Landesanstalt für privaten Rundfunk und neue Medien (DAB Indoor Coverage in the L-Band). April 1998