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In a lossy medium, waves decay as they propagate due to a complex propagation constant

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Propagation constant: $\gamma = \alpha + j\beta = j \omega \sqrt{\mu \epsilon _c}$, where $\alpha$ is the attenuation constant (Np/m) and $\beta$ is the phase constant (rad/m)

• A wave traveling in the + z direction looks like: $e^{-\gamma z} = e^{-\alpha z}e^{-j\beta z}$

Wave equation (phasor domain)

$$ \nabla^2 \bar{E} - \gamma^2 \bar{E} = 0 $$

Effective Complex Permittivity

$$ \varepsilon_{eff} = \varepsilon' - j\varepsilon'' \quad \text{where } \varepsilon' = \varepsilon,\ \varepsilon'' = \frac{\sigma}{\omega} $$

Solving for $\alpha$ and $\beta$

$$ \alpha = \omega\sqrt{\frac{\mu\varepsilon'}{2}\left[\sqrt{1+\left(\frac{\varepsilon''}{\varepsilon'}\right)^2}-1\right]} \ \text{(Np/m)} \newline \beta = \omega\sqrt{\frac{\mu\varepsilon'}{2}\left[\sqrt{1+\left(\frac{\varepsilon''}{\varepsilon'}\right)^2}+1\right]} \ \text{(rad/m)} $$

Fields propagating along +z

$$ \bar{E}(z) = \hat{x}E_{x0}e^{-\alpha z}e^{-j\beta z} \newline \bar{H}(z) = \hat{y}\frac{E_{x0}}{\eta_c}e^{-\alpha z}e^{-j\beta z} $$

Complex intrinsic impedance

• E and H are no longer in phase since $\eta _c$ is complex

  $$ \eta_c = \sqrt{\frac{\mu}{\varepsilon_{eff}}} = \sqrt{\frac{\mu}{\varepsilon'}}\left(1-j\frac{\varepsilon''}{\varepsilon'}\right)^{-1/2} $$

Skin depth

• Distance for the field magnitude to drop by 1/eL

$$ \delta_s = \frac{1}{\alpha}\ \text{(m)} $$

Two Limiting Cases

Low-loss dielectrics where $\sigma << \omega \epsilon$

$$ \alpha \approx \frac{\sigma}{2}\sqrt{\frac{\mu}{\varepsilon}} \ \text{(Np/m)}, \quad \beta \approx \omega\sqrt{\mu\varepsilon}\ \text{(same as lossless)} \newline \eta_c \approx \sqrt{\frac{\mu}{\varepsilon}}\left(1+j\frac{\sigma}{2\omega\varepsilon}\right) \approx \sqrt{\frac{\mu}{\varepsilon}} $$