Short note: S-parameters and their merits

Scattering (S) parameters describe a linear microwave network in terms of the incident waves (aₙ) and reflected waves (bₙ) at its ports, measured when all other ports are terminated in a matched load Z0. For a two-port:

 b1 = S11·a1 + S12·a2
 b2 = S21·a1 + S22·a2

• S11, S22: input and output reflection coefficients.
• S21: forward transmission (gain or insertion loss); S12: reverse transmission (isolation).
• aₙ = Vₙ⁺/√Z0, bₙ = Vₙ⁻/√Z0, so |aₙ|² and |bₙ|² are proportional to power.

Merits of S-parameters at microwave frequencies:

1. Easy measurement: they need matched terminations, not open or short circuits, which are hard to realise at microwave frequencies.
2. Safe for active devices: transistors stay stable when terminated in 50 Ω, whereas open/short terminations may cause oscillation.
3. Defined for waveguides: voltage and current are not unique in a waveguide, but incident and reflected waves are.
4. Direct physical meaning: |S11| gives return loss and VSWR; |S21|² gives power gain or insertion loss.
5. Measured by VNA with high accuracy, and the reference plane can be shifted mathematically (multiply by e^(−jβl)).
6. Network properties are visible: reciprocal → [S] symmetric; lossless → [S] unitary; matched → Sᵢᵢ = 0.
7. Used in amplifier design: stability factor, gain circles and noise circles are all computed from S-parameters.

Example: a matched 10 dB attenuator has S11 = S22 = 0 and |S21| = |S12| = 10^(−10/20) = 0.316.