Definition
Characteristic impedance Z₀ is the ratio of the voltage to the current of a wave travelling in one direction on a transmission line with no reflections (i.e., matched termination). It is a property of the line geometry and dielectric — not of frequency for a lossless TEM line.
Z₀ = √(L′/C′) [lossless line, L′ = series inductance/m, C′ = shunt capacitance/m] Z₀ = √((R′+jωL′)/(G′+jωC′)) [lossy line, frequency-dependent] Phase velocity: vp = 1/√(L′C′) = c/√(εr) [in dielectric] Propagation constant: γ = α + jβ = √((R′+jωL′)(G′+jωC′))
Coaxial Cable Impedance
Z₀_coax = (60/√εr) · ln(D/d) D = outer conductor inner diameter d = inner conductor outer diameter Standard values: 50 Ω coax: D/d ≈ 3.59 (RG58, RG174, SMA, 2.4mm, 3.5mm) 75 Ω coax: D/d ≈ 7.00 (RG59, RG6 — TV, CATV) 93 Ω coax: D/d ≈ 16.2 (RG62 — legacy data networks)
Why 50 Ω?
The 50 Ω standard is a historical compromise between two coax optimums:
- Minimum loss occurs at Z₀ ≈ 77 Ω (for air-dielectric coax — maximum D/d for given outer diameter)
- Maximum power handling occurs at Z₀ ≈ 30 Ω (largest inner conductor)
- 50 Ω ≈ geometric mean of 77 and 30 → best compromise for high-power RF systems
- 75 Ω (minimum loss in dielectric-filled coax) is used in cable TV and video where power is less critical
Mismatch Effects When Z₀ Is Wrong
| Source Z_s | Load Z_L | |Γ| | Power delivered (%) |
|---|---|---|---|
| 50 Ω | 50 Ω | 0.000 | 100% |
| 50 Ω | 75 Ω | 0.200 | 96% |
| 50 Ω | 100 Ω | 0.333 | 89% |
| 50 Ω | 25 Ω | 0.333 | 89% |
| 50 Ω | 5 Ω | 0.818 | 33% |
Reference Impedance in S-Parameter Files
All S-parameters are defined relative to a reference impedance Z₀ (port impedance), typically 50 Ω. Changing Z₀ changes all S-parameter values. The options line of a Touchstone file specifies R [value] — if the VNA and reference impedance don't match, S-parameters must be renormalized:
[S_new] = ([S_old] − Γ₀·[I]) · ([I] − Γ₀·[S_old])⁻¹ Γ₀ = (Z₀_new − Z₀_old) / (Z₀_new + Z₀_old)
RF View: RF View uses 50 Ω reference impedance for all plots. S-parameter data from 50 Ω VNA measurements are displayed directly with no renormalization needed.