Max gain for S11 = 0.656∠146.7°

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)

Step 1: Δ and K

S11·S22 = 0.656×0.172∠(146.7°−117.1°) = 0.1128∠29.6°
        = 0.0981 + j0.0557
S12·S21 = 0.122×2.3∠(46.1°+44.7°) = 0.2806∠90.8°
        = −0.0039 + j0.2806
Δ = 0.1020 − j0.2248 = 0.2469∠−65.59°
K = (1 − 0.4303 − 0.0296 + 0.0610)/(2×0.2806)
  = 0.6010/0.5612 = 1.071

K = 1.071 > 1 and |Δ| = 0.247 < 1 → unconditionally stable, so simultaneous conjugate matching is possible.

Step 2: Maximum transducer gain (bilateral)

GT,max = (|S21|/|S12|)(K − √(K² − 1))
       = (2.3/0.122)(1.071 − √(1.071² − 1))
       = 18.852 × 0.6876 = 12.96  → 11.13 dB

Step 3: Matching reflection coefficients

B1 = 1 + 0.4303 − 0.0296 − 0.0610 = 1.3398
C1 = S11 − ΔS22* = 0.6612∠150.37°
ΓMS = 0.8504∠−150.37°   (Zs = 4.32 − j13.13 Ω)
B2 = 1 + 0.0296 − 0.4303 − 0.0610 = 0.5383
C2 = S22 − ΔS11* = 0.2467∠−76.27°
ΓML = 0.6548∠76.27°     (ZL = 25.55 + j56.90 Ω)

(The other roots, 1.176 and 1.527 in magnitude, are outside the chart and are rejected.)

Step 4: Gain blocks (check)

GS = 1/(1 − |ΓMS|²)             = 3.613 → 5.58 dB
G0 = |S21|² = 5.29                     → 7.23 dB
GL = (1 − |ΓML|²)/|1 − S22ΓML|² = 0.678 → −1.69 dB
GT = 5.58 + 7.23 − 1.69 = 11.13 dB  ✓

Unilateral comparison

GTU,max = 1/(1−0.4303) × 5.29 × 1/(1−0.0296)
        = 1.755 × 5.29 × 1.030 = 9.57 → 9.81 dB

The unilateral formula underestimates by 1.32 dB because |S12| = 0.122 is large, so the bilateral value is the true maximum.

Answer: Gmax = GT,max = 12.96 ≈ 11.13 dB (with ΓMS = 0.850∠−150.4°, ΓML = 0.655∠76.3°).