Voltages and currents vary along a long line

In a long line the resistance, inductance and capacitance are distributed uniformly along the whole length. Each small element carries a different current and has a different voltage, so $V$ and $I$ are functions of distance $x$.

Starting point

Let per-unit-length series impedance be $z = r + j\omega L$ (Ω/km) and shunt admittance $y = g + j\omega C$ (S/km). Measure $x$ from the receiving end. Take an element of length $dx$ at distance $x$:

 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

• Voltage rise across the element (series drop): $dV = I\,z\,dx$
• Current change across the element (shunt current): $dI = V\,y\,dx$

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

Solving

Differentiating the first equation and substituting the second:

$$\frac{d^2V}{dx^2} = z\frac{dI}{dx} = zy\,V = \gamma^2 V, \qquad \gamma = \sqrt{zy}$$

The general solution is $V = A_1 e^{\gamma x} + A_2 e^{-\gamma x}$, and from $I = \frac{1}{z}\frac{dV}{dx}$:

$$I = \frac{1}{Z_c}\left(A_1 e^{\gamma x} - A_2 e^{-\gamma x}\right), \qquad Z_c = \sqrt{z/y}$$

At $x = 0$: $V = V_R$, $I = I_R$, giving $A_1 = \frac{V_R + Z_c I_R}{2}$, $A_2 = \frac{V_R - Z_c I_R}{2}$.

$$\begin{aligned} V(x) &= \frac{V_R + Z_c I_R}{2}e^{\gamma x} + \frac{V_R - Z_c I_R}{2}e^{-\gamma x} \\ &= V_R\cosh\gamma x + Z_c I_R\sinh\gamma x \\ I(x) &= \frac{V_R}{Z_c}\sinh\gamma x + I_R\cosh\gamma x \end{aligned}$$

Why V and I differ from point to point

• $\cosh\gamma x$ and $\sinh\gamma x$ change in both magnitude and angle as $x$ changes, so $V(x)$ and $I(x)$ are different at every point.
• With $\gamma = \alpha + j\beta$, the first term $e^{\alpha x}e^{j\beta x}$ is an incident wave that grows in size and advances in phase towards the sending end; the second is a reflected wave. Their sum varies along the line.
• Only when the line is terminated in $Z_c$ ($V_R = Z_c I_R$) does the reflected wave vanish; even then the magnitude changes as $e^{\alpha x}$ and the phase as $\beta x$.
• At $x = l$ these give the sending-end values: $V_S = V_R\cosh\gamma l + Z_c I_R\sinh\gamma l$, $I_S = \frac{V_R}{Z_c}\sinh\gamma l + I_R\cosh\gamma l$.

So for a long line voltage and current are different at every point along it, unlike the short-line model, where the current stays the same along the whole line.