Typical ABCD values of long lines

For a long line the ABCD constants are given exactly by hyperbolic functions of $\gamma l$. Expanding them in series puts them in terms of the total series impedance $Z = zl$ and total shunt admittance $Y = yl$ (note $\gamma^2l^2 = ZY$ and $Z_c = \sqrt{Z/Y}$).

Exact expressions

$$A = D = \cosh\gamma l, \quad B = Z_c\sinh\gamma l, \quad C = \frac{\sinh\gamma l}{Z_c}$$

Series (in terms of Z and Y)

$$\begin{aligned} A = D &= 1 + \frac{ZY}{2} + \frac{Z^2Y^2}{24} + \frac{Z^3Y^3}{720} + \cdots \\ B &= Z\left(1 + \frac{ZY}{6} + \frac{Z^2Y^2}{120} + \cdots\right) \\ C &= Y\left(1 + \frac{ZY}{6} + \frac{Z^2Y^2}{120} + \cdots\right) \end{aligned}$$

Also, $B = Z\frac{\sinh\gamma l}{\gamma l}$ and $C = Y\frac{\sinh\gamma l}{\gamma l}$. In all cases $AD - BC = 1$.

Typical values

For a well-designed 50 Hz overhead EHV line of 300–500 km ($\gamma l \approx 0.3$–0.5 rad, nearly all imaginary):

Example: a 450 km, 400 kV line with $r = 0.033$ Ω/km, $L = 1.067$ mH/km, $C = 0.0109$ μF/km has $A = 0.886\angle 0.71^\circ$, $B = 145.8\angle 84.6^\circ\ \Omega$ (with $|Z| = 151.6\ \Omega$), $C = 1.48\times10^{-3}\angle 90.2^\circ$ S (with $|Y| = 1.541\times10^{-3}$ S).

Points to note:

• $|A| < 1$ means no-load $V_R = V_S/|A| > V_S$ (Ferranti effect).
• $B$ and $C$ are both slightly smaller than $Z$ and $Y$, because $|\sinh\gamma l/\gamma l| < 1$ when $\gamma l$ is nearly imaginary.
• A short line has $A = 1$, $C = 0$; as length increases $|A|$ falls and $|C|$ rises.