Long line sending end voltage derivation

For a long line (above about 250 km) the resistance, inductance and capacitance are distributed uniformly along the length, so the line is analysed with differential equations rather than lumped circuits. Measure $x$ from the receiving end.

 IS      x <------ measured from receiving end
 o-->--z.dx--+--z.dx--+-- ... --z.dx--+--> IR
             |        |               |
 VS        y.dx     y.dx            y.dx    VR
             |        |               |
 o-----------+--------+-- ... --------+----o
 sending                          receiving

Starting point and solution

For an element $dx$ at distance $x$ from the receiving end, with $z = r + j\omega L$ and $y = g + j\omega C$ per km:

$$\frac{dV}{dx} = zI, \qquad \frac{dI}{dx} = yV$$

Differentiating the first and substituting the second:

$$\frac{d^2V}{dx^2} = zy\,V = \gamma^2V, \qquad \gamma = \sqrt{zy} = \alpha + j\beta$$

The general solution and the current (from $I = \frac{1}{z}\frac{dV}{dx}$) are:

$$\begin{aligned} V &= K_1e^{\gamma x} + K_2e^{-\gamma x} \\ I &= \frac{1}{Z_c}\left(K_1e^{\gamma x} - K_2e^{-\gamma x}\right), \quad Z_c = \sqrt{z/y} \end{aligned}$$

At $x = 0$: $V = V_R$ and $I = I_R$, so $K_1 + K_2 = V_R$ and $K_1 - K_2 = Z_cI_R$:

$$K_1 = \frac{V_R + Z_cI_R}{2}, \qquad K_2 = \frac{V_R - Z_cI_R}{2}$$

So, at any point $x$:

$$V_x = \frac{V_R + Z_cI_R}{2}e^{\gamma x} + \frac{V_R - Z_cI_R}{2}e^{-\gamma x}$$

Sending-end voltage

At the sending end, $x = l$ and $V_x = V_S$:

$$\begin{aligned} V_S &= \frac{V_R + Z_cI_R}{2}e^{\gamma l} + \frac{V_R - Z_cI_R}{2}e^{-\gamma l} \\ &= V_R\,\frac{e^{\gamma l} + e^{-\gamma l}}{2} + Z_cI_R\,\frac{e^{\gamma l} - e^{-\gamma l}}{2} \\ V_S &= V_R\cosh\gamma l + Z_cI_R\sinh\gamma l \end{aligned}$$

Similarly, $I_S = \frac{V_R}{Z_c}\sinh\gamma l + I_R\cosh\gamma l$.

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

• $\gamma = \sqrt{zy} = \alpha + j\beta$ is the propagation constant (per km),
• $Z_c = \sqrt{z/y}$ is the characteristic impedance (Ω),
• $l$ is the line length.

In the exponential form, the first term is the incident wave (travelling from sending to receiving end) and the second the reflected wave. When the line is terminated in $Z_c$, $V_R = Z_cI_R$, the reflected wave is zero and $V_S = V_Re^{\gamma l}$.