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A two-wire line becomes a transmission line once the wavelength is comparable to the line’s physical length

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Tranmission Modes

TEM

Telegrapher’s equations

$$ \frac{\partial V}{\partial z} = -I(z)(R+j\omega L) \newline \frac{\partial I}{\partial z} = -V(z)(G+j\omega C) $$

Wave equations

$$ \frac{\partial^2 V}{\partial z^2} = (R+j\omega L)(G+j\omega C)\,V(z) \newline \frac{\partial^2 I}{\partial z^2} = (R+j\omega L)(G+j\omega C)\,I(z) $$

Complex propagation constant

$$ \gamma = \sqrt{(R+j\omega L)(G+j\omega C)} = \alpha + j\beta $$

General solutions

$$ V(z) = V_0^+e^{-\gamma z} + V_0^-e^{+\gamma z} \newline I(z) = \frac{1}{Z_0}\left(V_0^+e^{-\gamma z} - V_0^-e^{+\gamma z}\right) $$

Characteristic impedance

$$ Z_0 = \sqrt{\frac{R+j\omega L}{G+j\omega C}} $$

Distortionless line

Historically motivated, choosing R/L = G/C eliminates dispersion, constant attenuatoin and phase velocity for all frequencies

$$ \alpha = \sqrt{RG}, \quad \beta = \omega\sqrt{LC} \implies v_\phi = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}} $$

Terminated lossless line

With $\beta$ purely imaginary (R=G=0)