21.8: Wave Equation for a Transmission Line
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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 \(I_0^-\).