How it works
The Telegrapher's Equations in the phasor domain: dV/dz = −(R+jωL)I and dI/dz = −(G+jωC)V. These yield wave solutions with propagation constant γ = α+jβ = √((R+jωL)(G+jωC)). Characteristic impedance Z0 = √((R+jωL)/(G+jωC)). For a lossless line (R=0, G=0): Z0 = √(L/C), β = ω√(LC), and phase velocity vp = 1/√(LC). For a coaxial line with inner radius a and outer radius b: L = (μ/2π)·ln(b/a) H/m and C = 2πε/ln(b/a) F/m, giving Z0 = (1/2π)·√(μ/ε)·ln(b/a). For air-filled coax with a = 0.455 mm and b = 1.5 mm, Z0 ≈ 50 Ω.
Key points to remember
Phase velocity vp = ω/β = 1/√(LC) = c/√εr for a lossless line, where c = 3×10⁸ m/s. Wavelength on the line λ = 2π/β = vp/f. For a distortionless (but lossy) line, the Heaviside condition R/L = G/C must hold — then α = √(RG) and Z0 = √(L/C), same as lossless. Skin effect at high frequencies makes R increase as √f, a key loss mechanism in GHz-range PCB traces. Input impedance of a lossless line of length l terminated in ZL: Zin = Z0·(ZL+jZ0·tan βl)/(Z0+jZL·tan βl) — this formula generates all the special cases used in Smith Chart analysis.
Exam tip
The examiner always asks you to derive the input impedance of a quarter-wave (λ/4) lossless transmission line terminated in ZL and show Zin = Z0²/ZL — state that βl = π/2, so tan βl → ∞, and simplify the general formula step by step.