How it works
The transmission line equations give voltage and current as superposition of forward and reverse travelling waves: V(z) = V₊e^(−γz) + V₋e^(γz). For a lossless line, γ = jβ = j2π/λ. Characteristic impedance Z₀ = √(L/C) where L and C are per-unit-length inductance and capacitance. Reflection coefficient Γ_L = (Z_L − Z₀)/(Z_L + Z₀). Input impedance of a lossless line of length l: Z_in = Z₀[(Z_L + jZ₀tanβl)/(Z₀ + jZ_Ltanβl)]. For a quarter-wave transformer (l = λ/4), Z_in = Z₀²/Z_L — this is the impedance inverter property.
Key points to remember
VSWR = (1 + |Γ|)/(1 − |Γ|); for Γ=0 (matched), VSWR=1; for |Γ|=1 (open or short), VSWR=∞. A short-circuited quarter-wave stub acts as an open circuit at its terminals; an open-circuited quarter-wave stub acts as a short circuit. The Smith chart plots normalised impedance z = Z/Z₀ = r + jx as a point within the unit circle; moving along the chart circumference corresponds to moving along the transmission line. Single-stub matching uses a shunt stub to cancel the susceptance at a point where the normalised conductance is 1. Power delivered to load P = ½|V₊|²(1−|Γ|²)/Z₀.
Exam tip
The examiner always asks you to find VSWR, reflection coefficient, and input impedance for a given load and line length — compute Γ from load and Z₀ first, then use the input impedance formula, not the other way around.