Summary of Equations

Telegrapher’s Equations

These equations describe how voltage and current propagate along a TL

$$ \frac{\partial I(z, t)}{\partial z} = -G' V(z, t) - C' \frac{\partial V(z, t)}{\partial t} \newline\frac{\partial V(z, t)}{\partial z} = -R' I(z, t) - L' \frac{\partial I(z, t)}{\partial t} $$

• Any term with $’$ indicates this is the per-unit-length quantity

Complex Propagation Constant

$$ \gamma = \alpha + j\beta = \sqrt{(R' + j\omega L')(G' + j\omega C')} $$

• $\alpha$: attenuation constant, describes how signal amplitude decreases
• $\beta$: phase constant, describes phase change per unit length

General Solution to the Wave Equation

$$ \tilde{V}(z) = V_0^+ e^{-\gamma z} + V_0^- e^{\gamma z} \newline \tilde{I}(z) = \frac{V_0^+}{Z_0} e^{-\gamma z} - \frac{V_0^-}{Z_0} e^{\gamma z} $$

• $V^+_0$ and $V^-_0$ are the forward and backward voltage waves
• $Z_0$ is the characteristic impedance of the line
• Used to describe complete behaviour of the voltage and current on the line, showing both forward and backward traveling waves

Characteristic Impedance ($Z_0$)

• For a general lossy line:

  $$ Z_0 = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}} $$

• For a lossless line $R’ = 0, G’ = 0)$:

  $$ Z_0 = \sqrt{\frac{L'}{C'}} $$

Phase Velocity

• Speed at which phase of the wave propagates along the line

  $$ v_p = \frac{\omega}{\beta} = \frac{\lambda}{T} $$

Wave Behaviour and Attenuation