A transmission line is a structure that guides electromagnetic waves from one point to another. At RF frequencies, ordinary wire connections behave as transmission lines — understanding this is essential for signal integrity and impedance matching.

Telegrapher's Equations

A transmission line is modelled as a ladder network of distributed resistance R, inductance L, capacitance C, and conductance G per unit length. The resulting wave equations are:

Characteristic Impedance

Coax with 50 Ω characteristic impedance: \(Z_0 = (60/\sqrt{\varepsilon_r})\ln(D/d)\).

Reflection Coefficient

When a wave hits a load impedance \(Z_L \neq Z_0\), some power is reflected. The voltage reflection coefficient is:

Return loss \(= -20\log_{10}|\Gamma|\) dB. VSWR \(= (1+|\Gamma|)/(1-|\Gamma|)\).

Standing Waves

The superposition of forward and reflected waves creates a standing wave pattern on the line. The voltage varies with position; the ratio of maximum to minimum voltage is the VSWR. A matched load (Γ = 0) has VSWR = 1 and no standing waves.

Input Impedance

A transmission line of length \(\ell\) terminated in \(Z_L\) presents input impedance:

Special cases: \(\ell = \lambda/4\) → \(Z_{in} = Z_0^2/Z_L\) (quarter-wave transformer); \(\ell = \lambda/2\) → \(Z_{in} = Z_L\) (half-wave section is transparent).