Wave Equation for a TEM Transmission Line

Consider a TEM transmission line aligned along the

$z$

axis. The phasor form of the Telegrapher’s Equations (Section 3.5) relate the potential phasor

and the current phasor

to each other and to the lumped-element model equivalent circuit parameters

, and

. These equations are

An obstacle to using these equations is that we require both equations to solve for either the potential or the current. In this section, we reduce these equations to a single equation – a wave equation – that is more convenient to use and provides some additional physical insight.

We begin by differentiating both sides of Equation 3.6.1 with respect to

$z$

, yielding:

Then using Equation 3.6.2 to eliminate

, we obtain

This equation is normally written as follows:

where we have made the substitution:

The principal square root of

is known as the propagation constant:

The propagation constant

(units of m⁻¹) captures the effect of materials, geometry, and frequency in determining the variation in potential and current with distance on a TEM transmission line.

Following essentially the same procedure but beginning with Equation 3.6.2, we obtain

Equations 3.6.5 and 3.6.8 are the wave equations for

and

, respectively.

Note that both

and

satisfy the same linear homogeneous differential equation. This does not mean that

and

are equal. Rather, it means that

and

can differ by no more than a multiplicative constant. Since

is potential and

is current, that constant must be an impedance. This impedance is known as the characteristic impedance and is determined in Section 3.7.

The general solutions to Equations 3.6.5 and 3.6.8 are

where

, and

are complex-valued constants. It is shown in Section 3.8 that Equations 3.6.9 and 3.6.10 represent sinusoidal waves propagating in the

and

directions along the length of the line. The constants may represent sources, loads, or simply discontinuities in the materials and/or geometry of the line. The values of the constants are determined by boundary conditions; i.e., constraints on

and

at some position(s) along the line.

The reader is encouraged to verify that the Equations 3.6.9 and 3.6.10 are in fact solutions to Equations 3.6.5 and 3.6.8, respectively, for any values of the constants