3.6: Wave Equation for a TEM 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 ˜V(z) and the current phasor ˜I(z) to each other and to the lumped-element model equivalent circuit parameters R′, G′, C′, and L′. These equations are
−∂∂z˜V(z)=[R′+jωL′] ˜I(z)
−∂∂z˜I(z)=[G′+jωC′] ˜V(z)
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 ??? with respect to z, yielding: −∂2∂z2˜V(z)=[R′+jωL′] ∂∂z˜I(z)
The propagation constant γ (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 ???, we obtain ∂2∂z2˜I(z)−γ2 ˜I(z)=0
Equations ??? and ??? are the wave equations for ˜V(z) and ˜I(z), respectively.
Note that both ˜V(z) and ˜I(z) satisfy the same linear homogeneous differential equation. This does not mean that ˜V(z) and ˜I(z) are equal. Rather, it means that ˜V(z) and ˜I(z) can differ by no more than a multiplicative constant. Since ˜V(z) is potential and ˜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 ??? and ??? are ˜V(z)=V+0e−γz+V−0e+γz
The reader is encouraged to verify that the Equations ??? and ??? are in fact solutions to Equations ??? and ???, respectively, for any values of the constants V+0, V−0, I+0, and I−0.