Conditional stability and unilateral max gain

Δ   = S11·S22 − S12·S21
K   = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
Unconditionally stable if K > 1 and |Δ| < 1
GTU,max = 1/(1−|S11|²) · |S21|² · 1/(1−|S22|²)
GT,max  = (|S21|/|S12|)·(K − √(K² − 1))
B1 = 1 + |S11|² − |S22|² − |Δ|²,  C1 = S11 − Δ·S22*
B2 = 1 + |S22|² − |S11|² − |Δ|²,  C2 = S22 − Δ·S11*
ΓMS = [B1 − √(B1² − 4|C1|²)] / (2C1)
ΓML = [B2 − √(B2² − 4|C2|²)] / (2C2)

Justification of conditional stability

S11·S22 = 0.894×0.781∠(−60.6°−27.6°) = 0.6982∠−88.2°
        = 0.0219 − j0.6979
S12·S21 = 0.020×3.122∠(62.4°+123.6°) = 0.0624∠186°
        = −0.0621 − j0.0065
Δ = 0.0840 − j0.6913 = 0.6964∠−83.07°
K = (1 − 0.7992 − 0.6100 + 0.4850)/(2×0.0624)
  = 0.0758/0.1249 = 0.607
μ = 0.863

• |Δ| = 0.696 < 1 but K = 0.607 < 1 (and μ < 1), so the device is not unconditionally stable.
• Since |S11| < 1 and |S22| < 1, it is stable for a 50 Ω source and load. It is therefore conditionally stable: stable for some Γs, ΓL and unstable for others.

Stability circles show the unstable zones:

CL = 1.363∠46.69°,  RL = 0.500
CS = 1.132∠68.46°,  RS = 0.199

Both circles cut the unit Smith chart, so parts of the chart (near the top-right edge) are unstable; the stable region is outside the circles.

Maximum unilateral gain

GS,max = 1/(1 − 0.894²) = 4.981 → 6.97 dB
G0     = 3.122² = 9.747          → 9.89 dB
GL,max = 1/(1 − 0.781²) = 2.564 → 4.09 dB
GTU,max = 124.47 → 20.95 dB

Check that Γs = S11* and ΓL = S22* are in the stable region: distance of S11* from CS = 0.275 > 0.199 and of S22* from CL = 0.675 > 0.500, so both are stable (|Γin| = 0.912, |Γout| = 0.848).

U = 0.557 (error band −3.84 dB to +7.07 dB), so the unilateral value is only an estimate.

Answer: K = 0.607 < 1, |Δ| = 0.696 → conditionally stable; GTU,max = 124.5 ≈ 20.95 dB.