Plane Wave Propagation

Summary

• Voltage representation in a TL

  $$ \frac{d^2 V(z)}{dz^2} - \gamma^2 V(z) = 0 \newline V(z) = V_0^+ e^{-\gamma z} + V_0^- e^{+\gamma z}

  $$

• Governing equation for EM waves in unbounded space

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

  

• Electric field representation in lossless material

  $$ \tilde{\mathbf{E}}(x, y, z) = \tilde{\mathbf{E}}_0^+ e^{-k \mathbf{R}} \newline k = k_x \hat{x} + k_y \hat{y} + k_z \hat{z} = \omega \sqrt{\mu \epsilon} \, \hat{k}

  $$

• Intrinsic impedance:

  $$ \eta = \sqrt{\frac{\mu}{\epsilon}} \newline \mathbf{H}(x, y, z) = \frac{1}{\eta} \hat{k} \times \mathbf{E}

  $$

• Fundamental quantities

  $$ u_p = \frac{\omega}{k} = \frac{1}{\sqrt{\mu \epsilon}} = \frac{c}{\sqrt{\epsilon_r}} \newline \lambda = \frac{2 \pi}{k}

  $$

Plane wave propagation in the $\hat z$ direction

• Electric field

  $$ \mathbf{E}(x, y, z) = E_{x0} e^{-j k z} \hat{x} $$

• Magnetic field

  $$ \mathbf{H}(x, y, z) = \frac{\sqrt{\epsilon}{\mu}} E_{x0} e^{-j k z} \hat{y} $$

• Wave number

  $$ k = \omega \sqrt{\mu \epsilon} = \frac{\omega}{u_p} $$

• Phase velocity

  $$ u_p = \frac{1}{\sqrt{\mu \epsilon}} $$

• Wavelength

  $$ \lambda = \frac{2 \pi}{k} = \frac{u_p}{f} = \frac{c}{f \sqrt{\epsilon_r}} $$

• Intrinsic impedance

  $$ \eta = \sqrt{\frac{\mu}{\epsilon}} $$

Propagation of a uniform plane wave with a tilted $E$

• Electric field with $E_x$ and $E_y$

  $$ \mathbf{E} = (E_x \hat{x} + E_y \hat{y}) e^{-j k z} $$

• Magnetic field

  $$ \mathbf{H} = \frac{E_0 e^{j \phi_E}}{\eta} \left( -\sin \alpha \hat{x} + \cos \alpha \hat{y} \right) e^{-j k z} $$

• Magnetic field in terms of intrinsic impedance (if the E field is positive, H field is negative?)

  $$ \mathbf{H} = \frac{1}{\eta} \hat{k} \times \mathbf{E} $$