Short note: Stability analysis of microwave amplifier

Stability of a microwave amplifier means that it does not oscillate. This needs |Γ_in| < 1 and |Γ_out| < 1, meaning there is no negative resistance at either port, for the source and load terminations used.

Γin  = S11 + S12·S21·ΓL / (1 − S22·ΓL)
Γout = S22 + S12·S21·ΓS / (1 − S11·ΓS)

Types

• Unconditionally stable: |Γ_in| < 1 and |Γ_out| < 1 for every passive source and load (|Γ_S|, |Γ_L| < 1).
• Conditionally (potentially) unstable: stable only for some terminations.

Tests

Δ = S11·S22 − S12·S21
K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
Rollett: K > 1 and |Δ| < 1  -> unconditionally stable
μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|)
μ-test : μ > 1 -> unconditionally stable

Stability circles (|Γ_in| = 1 and |Γ_out| = 1) for a potentially unstable device:

Output (ΓL plane):
 CL = (S22 − Δ·S11*)* / (|S22|² − |Δ|²)
 RL = |S12·S21 / (|S22|² − |Δ|²)|
Input (ΓS plane):
 CS = (S11 − Δ·S22*)* / (|S11|² − |Δ|²)
 RS = |S12·S21 / (|S11|² − |Δ|²)|

If |S11| < 1, the centre of the Γ_L plane (Γ_L = 0) is stable; likewise, if |S22| < 1, the centre of the Γ_S plane is stable. The region on the same side of the circle as the centre is then the stable region. The designer chooses Γ_S and Γ_L in the stable regions, or adds series or shunt resistive loading to stabilise the device.