# Telegrapher's Equation

 This article outlines a derivation of Oliver Heaviside's Telegrapher's Equation and application to solution of steady state transmission line problems.

# Introduction

A transmission line can be represented as an infinite series of cascaded identical two port networks each representing an infinitely small section of the transmission line. The small networks represent:

• the distributed resistance of the conductors is represented by a series resistance per unit length R
• the distributed inductance is represented by a series inductance per unit length L
• the capacitance between the conductors is represented by a shunt capacitance per unit length C
• the conductance of the dielectric material separating the two conductors is represented by a conductance per unit length G

Fig 1 shows the small networks.

R,L,G, and C may be frequency dependent. In practical transmission lines at HF and above the following assumptions are often appropriately used:

• inductance per unit length as constant (due partially to skin effect and a fully effective outer conductor));
• capacitance per unit length as constant;
• resistance per unit length is subject to skin effect and is proportional to the square root of frequency;
• conductance per unit length is due to dielectric loss and is proportional to frequency.

## Derivation

The line voltage V(x) and the current I(x) where x is displacement can be expressed in the frequency domain as:

$\frac{\partial V(x)}{\partial x}=-(R+j \omega L)I(x)$

$\frac{\partial I(x)}{\partial x}=-(G+j \omega C)V(x)$

Differentiating both:

$\frac{\partial^2 V(x)}{\partial x^2}=\gamma^2V(x)$

$\frac{\partial^2 I(x)}{\partial x^2}=\gamma^2 I(x)$

where:

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

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

γ is the transmission line complex propagation constant, and Z0 is a complex value known as the characteristic impedance of the line.

A solution for V(x) and I(x) is:

$V(x)=V_f e^{\gamma x}+V_r e^{- \gamma x}$

$I(x)=I_f e^{\gamma x}-I_r e^{- \gamma x}$

where x is the displacement from the load, negative towards the source, and Vf, Vr, If and Ir are forward and reflected voltages and currents respectively at the load end of the line.

# Application

The above expressions can be rewritten as:

$V(x)=V_f( e^{\gamma x}+ \Gamma e^{- \gamma x})$

$I(x)=I_f( e^{\gamma x}- \Gamma e^{- \gamma x})$

where Γ is the complex reflection coefficient at the load.

Transmission line behaviour is described by these equations and the boundary conditions imposed by the load.

$\Gamma=\frac{Z_l-Z_0}{Z_l+Z_0}$

$\Gamma(x)=\Gamma \frac{e^{\gamma x}}{e^{- \gamma x}}$

These equations fully describe the behaviour of a transmission line with a given load impedance.

From these, the relationships for rho; and VSWR can be developed:

$\rho=|\Gamma|$

$VSWR=\frac{1+\rho}{1-\rho}$

We can write $$Z_l \text{ in terms of } Z_0 \text{ and } \Gamma$$:

$Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}$

Input impedance Zin of line of length l can be calculated from the load impedance:

$Z_{in}=Z_0 \frac{Z_l+Z_0 tanh(\gamma l)}{Z_0+Z_l tanh(\gamma l)}$

In the special case of a lossless line, input impedance Zin of line of length l can be calculated from the load impedance:

$Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}$

The following relationships exist for the two port network equivalent of a transmission line:

$V_1=V_2 cosh(\gamma l)+I_2 Z_0 sinh(\gamma l)$

$I_1=\frac{V_2}{Z_0} sinh(\gamma l)+I_2 cosh(\gamma l)$

where V1 and I1 are the voltage and current at the input port, and V2 and I2 are the voltage and current at the output port.