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# 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: $-\frac{\partial^2}{\partial z^2} \widetilde{V}(z) = \left[ R' + j\omega L' \right]~\frac{\partial}{\partial z} \widetilde{I}(z)$ Then using Equation \ref{m0027_eTelegraphersEquation2p} to eliminate $$\widetilde{I}(z)$$, we obtain $-\frac{\partial^2}{\partial z^2} \widetilde{V}(z) = -\left[ R' + j\omega L' \right]\left[ G' + j\omega C' \right]~\widetilde{V}(z)$ 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: $\gamma^2 = \left( R' + j\omega L' \right)\left( G' + j\omega C' \right)$ 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 $$I_0^-$$.