Power Wave Definition
S-parameters use power waves (a, b) rather than voltage/current. Power waves are defined so that |aᵢ|² = power incident at port i and |bᵢ|² = power leaving port i:
aᵢ = (Vᵢ + Z₀·Iᵢ) / (2·√Re{Z₀}) [incident wave]
bᵢ = (Vᵢ − Z₀*·Iᵢ) / (2·√Re{Z₀}) [reflected wave]
[b] = [S][a] → Sᵢⱼ = bᵢ/aⱼ (all aₖ=0, k≠j)
Properties of the S-Matrix
| Network Type | S-Matrix Property | Condition |
|---|---|---|
| Reciprocal (passive) | [S] = [S]ᵀ (symmetric) | Sᵢⱼ = Sⱼᵢ for all i,j |
| Lossless | [S]†[S] = [I] (unitary) | Columns (and rows) are orthonormal |
| Matched | Sᵢᵢ = 0 for all i | All diagonal elements zero |
| Unilateral | S₁₂ = 0 | No reverse transmission |
Converting Between Parameter Sets
S → Z: [Z] = (I + S)·(I − S)⁻¹ · Z₀
Z → S: [S] = (Z − Z₀·I)·(Z + Z₀·I)⁻¹
S → Y: [Y] = (I − S)·(I + S)⁻¹ / Z₀
S → ABCD (2-port):
A = (1+S₁₁)(1−S₂₂) + S₁₂S₂₁) / 2S₂₁
B = Z₀·((1+S₁₁)(1+S₂₂) − S₁₂S₂₁) / 2S₂₁
...
S-Matrix Renormalization
S-parameters depend on the reference impedance Z₀. To convert from Z₀=50Ω to Z₀=75Ω:
Γ₀ = (Z₀_new − Z₀_old)/(Z₀_new + Z₀_old) [S_new] = ([S_old] − Γ₀[I])·([I] − Γ₀[S_old])⁻¹
RF View: All RF View analysis uses 50 Ω reference impedance consistent with standard VNA measurements. The SNP Converter can process files with non-standard reference impedances when specified in the Touchstone options line (R [value]).