Transmission Lines and the Smith Chart

At high frequencies, wires become transmission lines: voltages and currents travel as waves, and a mismatched load reflects part of the power.

Line parameters

Per unit length: R, L, G, C.

• Characteristic impedance Z₀ = √((R + jωL)/(G + jωC)); lossless: Z₀ = √(L/C).
• Propagation constant γ = √((R + jωL)(G + jωC)); lossless: β = ω√(LC), v = 1/√(LC).
• Distortionless line: R/L = G/C (Heaviside condition).

Reflection

• Load reflection coefficient Γ_L = (Z_L − Z₀)/(Z_L + Z₀).
• VSWR S = (1 + |Γ|)/(1 − |Γ|); |Γ| = (S − 1)/(S + 1).
• Matched load: Γ = 0, S = 1. Short: Γ = −1. Open: Γ = +1. Both give S = ∞.
• For a resistive load: S = Z_L/Z₀ if Z_L > Z₀, else Z₀/Z_L.
• Return loss = −20 log|Γ| dB; reflected power fraction |Γ|².
• Voltage maxima and minima are λ/4 apart; adjacent maxima are λ/2 apart.

Input impedance (lossless)

Z_in = Z₀ (Z_L + jZ₀ tan βl)/(Z₀ + jZ_L tan βl)

Matching

• Quarter-wave transformer matches resistive loads: Z₀′ = √(Z₀ Z_L).
• Single-stub matching: a short-circuited stub placed where the line admittance is 1 ± jb cancels the susceptance.

Smith chart

A plot of normalised impedance z = Z/Z₀ on the Γ-plane.

• The centre is the matched point (z = 1); the left end is a short, the right end an open.
• Moving along the line traces a circle (constant |Γ| = constant VSWR); one full turn is λ/2.
• Impedance and admittance are diametrically opposite.

Worked example

A 50 Ω line feeds a 200 Ω resistive load.

• Γ = 150/250 = 0.6; VSWR = 4; reflected power = 36%.
• A quarter-wave transformer of 100 Ω matches it.

Practice