Stability and gains for S11 = 0.78∠−113°

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)

Stability

S11·S22 = 0.78×0.81∠(−113°−54°) = 0.6318∠−167°
        = −0.6156 − j0.1421
S12·S21 = 0.028×2.60∠(247°+76°) = 0.0728∠323°
        = 0.0581 − j0.0438
Δ = −0.6737 − j0.0983 = 0.6809∠−171.70°
K = (1 − 0.6084 − 0.6561 + 0.4636)/(2×0.0728)
  = 0.1991/0.1456 = 1.367

K = 1.367 > 1 and |Δ| = 0.681 < 1 → unconditionally stable (μ = 1.097 > 1).

Bilateral maximum gain (simultaneous conjugate match)

GT,max = (|S21|/|S12|)(K − √(K² − 1))
       = (2.60/0.028)(1.367 − √(1.367² − 1))
       = 92.86 × 0.4348 = 40.37 → 16.06 dB
B1 = 0.4887, C1 = 0.2347∠−101.90° → ΓMS = 0.7517∠101.90°
B2 = 0.5841, C2 = 0.2840∠−45.19°  → ΓML = 0.7891∠45.19°
GS = 2.299 (3.62 dB)
G0 = 6.76  (8.30 dB)
GL = 2.597 (4.15 dB)   → GT = 16.06 dB

Unilateral maximum gain

GS,max = 1/(1 − 0.78²) = 2.554 → 4.07 dB
G0     = 2.60² = 6.76         → 8.30 dB
GL,max = 1/(1 − 0.81²) = 2.908 → 4.64 dB
GTU,max = 50.20 → 17.01 dB

Comparison

U = 0.78×0.028×2.6×0.81 / (0.3916×0.3439) = 0.342
Error band: −2.55 dB to +3.63 dB

The unilateral formula gives 0.95 dB more than is really possible. Because |S11| and |S22| are large, U = 0.34 and the unilateral model is unreliable; the bilateral value 16.06 dB is the true maximum gain and the design should use ΓMS and ΓML.

Answer: unconditionally stable (K = 1.367, |Δ| = 0.681); GT,max = 16.06 dB (bilateral), GTU,max = 17.01 dB (unilateral estimate).