# Telegrapher's Equation

 This article outlines a derivation of the 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:

.

Differentiating both:

,

where:

,
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:

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:

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.

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:

.

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

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

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.