Characteristic impedance, SIL, propagation constants

Let the line have series impedance $z = r + j\omega L$ and shunt admittance $y = g + j\omega C$ per unit length.

Characteristic (surge) impedance, $Z_c$

The ratio of voltage to current of a travelling wave on an infinitely long line (or a line terminated in $Z_c$):

$$Z_c = \sqrt{\frac{z}{y}} = \sqrt{\frac{r + j\omega L}{g + j\omega C}}\ \Omega$$

For a lossless line, $Z_c = \sqrt{L/C}$, a pure resistance (about 400 Ω for single-conductor overhead lines, 250–300 Ω for bundled EHV lines, 40–60 Ω for cables). It does not depend on length.

Surge impedance loading (SIL)

The power delivered when the line feeds a resistive load equal to $Z_c$:

$$\text{SIL} = \frac{V_L^2}{Z_c}\ \text{MW}\ (V_L \text{ in kV})$$

At SIL the reactive power generated by the line capacitance equals that absorbed by its inductance, so the voltage is flat along the line.

Propagation constant

$\gamma = \sqrt{zy} = \alpha + j\beta$ per unit length.

Attenuation constant, $\alpha$

The real part of $\gamma$, in nepers per km. It shows how fast the magnitude of a voltage or current wave falls along the line ($e^{-\alpha x}$), because of losses in $r$ and $g$. For a lossless line, $\alpha = 0$.

Phase shift constant, $\beta$

The imaginary part of $\gamma$, in radians per km. It shows how much the phase of the wave changes per unit length. For a lossless line:

$$\beta = \omega\sqrt{LC}, \quad \lambda = \frac{2\pi}{\beta} \approx 6000\ \text{km at 50 Hz}, \quad v = \frac{\omega}{\beta} \approx 3\times10^5\ \text{km/s}$$