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Two conductors embedded in a dielectric medium
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RLGC Model Waves in Transmission Lines
Voltage and Current Wave Equations (Telegrapher’s Equations)
$$
\frac{\partial V(z, t)}{\partial z} = -L' \frac{\partial I(z, t)}{\partial t} - R' I(z, t) \newline
\frac{\partial I(z, t)}{\partial z} = -C' \frac{\partial V(z, t)}{\partial t} - G' V(z, t)
$$
- These equations describe how the voltage $V(z,t)$ and current $I(z,t)$ change along the length $z$ of the transmission line over time
- Also known as Telegrapher’s Equations
Wave Equations in the Frequency Domain
$$
\frac{\partial^2 \tilde{V}(z)}{\partial z^2} = (\gamma^2) \tilde{V}(z)
$$
- By transforming the time-domain equations to the frequency domain using phasors, the equations simplify and allow analysis of steady-state behaviour
Voltage and Current Solutions
- The solutions to the wave equations describe how voltage and current distribute along the line:
$$
\tilde{V}(z) = V_0^+ e^{-\gamma z} + V_0^- e^{\gamma z}
\newline \tilde{I}(z) = \frac{V_0^+}{Z_0} e^{-\gamma z} - \frac{V_0^-}{Z_0} e^{\gamma z}
$$
- These expressions represent forward and backward traveling waves
- The terms $V_0^+$ and $V_0^-$ correspond to the amplitude of the forward and backward waves
- $Z_0$ is the characteristic impedance of the line
Boundary Conditions and Matching
- Impedance matching and reflection can be analyzed by applying boundary conditions at the load, which determine how much of the signal is reflected back or transmitted