3.6: 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 $\widetilde{V}(z)$ and the current phasor $\widetilde{I}(z)$ to each other and to the lumped-element model equivalent circuit parameters $R'$, $G'$, $C'$, and $L'$. These equations are

$$-\frac{\partial}{\partial z} \widetilde{V}(z) = \left[ R' + j\omega L' \right]~\widetilde{I}(z) \label{m0027_eTelegraphersEquation1p}$$

$$-\frac{\partial}{\partial z} \widetilde{I}(z) = \left[ G' + j\omega C' \right]~\widetilde{V}(z) \label{m0027_eTelegraphersEquation2p}$$

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 $\ref{m0027_eTelegraphersEquation1p}$ with respect to $z$, yielding: $$-\frac{\partial^2}{\partial z^2} \widetilde{V}(z) = \left[ R' + j\omega L' \right]~\frac{\partial}{\partial z} \widetilde{I}(z) \nonumber$$ Then using Equation $\ref{m0027_eTelegraphersEquation2p}$ to eliminate $\widetilde{I}(z)$, we obtain $$-\frac{\partial^2}{\partial z^2} \widetilde{V}(z) = -\left[ R' + j\omega L' \right]\left[ G' + j\omega C' \right]~\widetilde{V}(z) \nonumber$$ This equation is normally written as follows: $$\boxed{ \frac{\partial^2}{\partial z^2} \widetilde{V}(z) -\gamma^2~\widetilde{V}(z) =0 } \label{m0027_eWaveEqnV}$$ where we have made the substitution: $$\gamma^2 = \left( R' + j\omega L' \right)\left( G' + j\omega C' \right) \nonumber$$ The principal square root of $\gamma^2$ is known as the propagation constant: $$\gamma \triangleq \sqrt{\left( R' + j\omega L' \right)\left( G' + j\omega C' \right)} \label{m0027_egamma}$$

The propagation constant $\gamma$ (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 $\ref{m0027_eTelegraphersEquation2p}$, we obtain $$\boxed{ \frac{\partial^2}{\partial z^2} \widetilde{I}(z) -\gamma^2~\widetilde{I}(z) =0 } \label{m0027_eWaveEqnI}$$

Equations $\ref{m0027_eWaveEqnV}$ and $\ref{m0027_eWaveEqnI}$ are the wave equations for $\widetilde{V}(z)$ and $\widetilde{I}(z)$, respectively.

Note that both $\widetilde{V}(z)$ and $\widetilde{I}(z)$ satisfy the same linear homogeneous differential equation. This does not mean that $\widetilde{V}(z)$ and $\widetilde{I}(z)$ are equal. Rather, it means that $\widetilde{V}(z)$ and $\widetilde{I}(z)$ can differ by no more than a multiplicative constant. Since $\widetilde{V}(z)$ is potential and $\widetilde{I}(z)$ 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 $\ref{m0027_eWaveEqnV}$ and $\ref{m0027_eWaveEqnI}$ are $$\widetilde{V}(z) = V_0^+ e^{-\gamma z} + V_0^- e^{+\gamma z} \label{m0027_eV}$$ $$\widetilde{I}(z) = I_0^+ e^{-\gamma z} + I_0^- e^{+\gamma z} \label{m0027_eI}$$ where $V_0^+$, $V_0^-$, $I_0^+$, and $I_0^-$ are complex-valued constants. It is shown in Section 3.8 that Equations $\ref{m0027_eV}$ and $\ref{m0027_eI}$ represent sinusoidal waves propagating in the $+z$ and $-z$ 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 $\widetilde{V}(z)$ and $\widetilde{I}(z)$ at some position(s) along the line.

The reader is encouraged to verify that the Equations $\ref{m0027_eV}$ and $\ref{m0027_eI}$ are in fact solutions to Equations $\ref{m0027_eWaveEqnV}$ and $\ref{m0027_eWaveEqnI}$, respectively, for any values of the constants $V_0^+$, $V_0^-$, $I_0^+$, and $I_0^-$.