Wave Propagation in Lossy Dielectric
1. Medium Definition
A lossy dielectric is a medium in which an EM wave loses power as it propagates owing to an imperfect dielectric (a partially conducting medium with conductivity $\sigma \neq 0$).
For a linear, isotropic, homogeneous lossy dielectric medium that is charge-free ($\rho_v = 0$):
Maxwell's Equations (Time factor $e^{j\omega t}$ suppressed):
$$\nabla \cdot \mathbf{E}_s = 0 \quad \text{--- (1)}$$
$$\nabla \cdot \mathbf{H}_s = 0 \quad \text{--- (2)}$$
$$\nabla \times \mathbf{E}_s = -j\omega\mu \mathbf{H}_s \quad \text{--- (3)}$$
$$\nabla \times \mathbf{H}_s = (\sigma + j\omega\varepsilon)\mathbf{E}_s \quad \text{--- (4)}$$
2. Wave Equations & Propagation Constant
Taking the curl on both sides of equation (3) and applying the vector identity $\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2\mathbf{A}$:
$$\nabla^2 \mathbf{E}_s - \gamma^2 \mathbf{E}_s = 0 \quad \text{--- (6)}$$
$$\nabla^2 \mathbf{H}_s - \gamma^2 \mathbf{H}_s = 0 \quad \text{--- (8)}$$
Where the complex propagation constant $\gamma$ is:
$$\gamma^2 = j\omega\mu(\sigma + j\omega\varepsilon) \quad \text{--- (7)}$$
$$\gamma = \alpha + j\beta \quad \text{--- (9)}$$
Constants Derivation:
Solving for real and imaginary parts yields the phase and attenuation constants:
$$\text{Phase Constant } (\beta) = \omega \sqrt{\frac{\mu\varepsilon}{2} \left[ \sqrt{1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2} + 1 \right]} \text{ rad/m} \quad \text{--- (14)}$$
$$\text{Attenuation Constant } (\alpha) = \omega \sqrt{\frac{\mu\varepsilon}{2} \left[ \sqrt{1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2} - 1 \right]} \text{ Np/m} \quad \text{--- (15)}$$
3. Electric & Magnetic Field Solutions
Assuming propagation along $+a_z$ with an $a_x$-polarized electric field:
$$\mathbf{E}(z,t) = E_0 e^{-\alpha z} \cos(\omega t - \beta z) \hat{\mathbf{a}}_x \quad \text{--- Instantaneous E-field}$$
$$\mathbf{H}(z,t) = \frac{E_0}{|\eta|} e^{-\alpha z} \cos(\omega t - \beta z - \theta_\eta) \hat{\mathbf{a}}_y \quad \text{--- Instantaneous H-field}$$
Intrinsic Impedance ($\eta$):
$$\eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\varepsilon}} = |\eta| e^{j\theta_\eta} = \frac{\sqrt{\mu/\varepsilon}}{\left[1 + \left(\frac{\sigma}{\omega\varepsilon}\right)^2\right]^{1/4}} e^{j\theta_\eta}$$
$$\tan 2\theta_\eta = \frac{\sigma}{\omega\varepsilon} \quad (0^\circ \le \theta_\eta \le 45^\circ)$$
4. Loss Tangent & Key Parameters
The ratio of conduction current density to displacement current density determines how lossy the medium is:
$$\text{Loss Tangent: } \tan \theta = \frac{\sigma}{\omega\varepsilon} = \frac{\varepsilon''}{\varepsilon'} \quad \text{--- (21)}$$
$$\text{Complex Permittivity: } \varepsilon_c = \varepsilon' - j\varepsilon'' = \varepsilon \left(1 - j\frac{\sigma}{\omega\varepsilon}\right) \quad \text{--- (19, 20)}$$
$$\text{Wave Velocity: } v = \frac{\omega}{\beta}, \quad \text{Wavelength: } \lambda = \frac{2\pi}{\beta}$$