Max gains for S11 = 0.55∠150° (2071 Bhadra)

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)

Assumptions: Z0 = 50 Ω; the stated S-parameters apply at the design frequency; lossless matching networks.

Stability

S11·S22 = 0.55×0.45∠(150°−30°) = 0.2475∠120°
        = −0.1238 + j0.2143
S12·S21 = 0.04×2.82∠(20°+180°) = 0.1128∠200°
        = −0.1060 − j0.0386
Δ = −0.0178 + j0.2529 = 0.2535∠94.02°
K = (1 − 0.3025 − 0.2025 + 0.0643)/(2×0.1128)
  = 0.5593/0.2256 = 2.479

K = 2.479 > 1 and |Δ| = 0.254 < 1 → unconditionally stable.

Bilateral maximum gain

GT,max = (2.82/0.04)(2.479 − √(2.479² − 1))
       = 70.5 × 0.2106 = 14.85 → 11.72 dB
B1 = 1.0357, C1 = 0.4502∠156.37° → ΓMS = 0.5819∠−156.37°
B2 = 0.8357, C2 = 0.3303∠−19.34° → ΓML = 0.4903∠19.34°
GS = 1.512 (1.80 dB), G0 = 7.952 (9.00 dB),
GL = 1.235 (0.92 dB)

Matching impedances: Zs = 13.75 − j9.70 Ω, ZL = 120.52 + j51.53 Ω.

Unilateral maximum gain

GS,max = 1/(1 − 0.55²) = 1.434 → 1.56 dB
G0     = 2.82² = 7.952          → 9.00 dB
GL,max = 1/(1 − 0.45²) = 1.254 → 0.98 dB
GTU,max = 14.30 → 11.55 dB
U = 0.050 → error −0.43 dB to +0.45 dB

The two differ by only 0.17 dB because |S12| is small (U = 0.05), so the unilateral model is acceptable here.

Answer: GT,max = 14.85 (11.72 dB) bilateral; GTU,max = 14.30 (11.55 dB) unilateral.