Stability and unilateral gain, S11 = 0.45∠−160°

Stability is tested with the Rollett factor K and |Δ|; the unilateral gain assumes S12 = 0 and conjugate matching at both ports.

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 check

S11·S22 = 0.45×0.35 ∠(−160°−40°) = 0.1575∠160°
        = −0.1480 + j0.0539
S12·S21 = 0.04×3.12 ∠(20°+180°) = 0.1248∠−160°
        = −0.1173 − j0.0427
Δ = (−0.1480+j0.0539) − (−0.1173−j0.0427)
  = −0.0307 + j0.0966 = 0.1013∠107.65°
|Δ| = 0.1013 < 1
K = (1 − 0.2025 − 0.1225 + 0.01027)/(2×0.1248)
  = 0.6853/0.2496 = 2.745 > 1

Since K = 2.745 > 1 and |Δ| = 0.101 < 1, the transistor is unconditionally stable (μ = 1.776 > 1 confirms it). Any passive source and load can be used.

Unilateral (maximum) transducer gain

With S12 = 0, Γs = S11* = 0.45∠160° and ΓL = S22* = 0.35∠40°:

GS,max = 1/(1 − 0.45²) = 1/0.7975 = 1.2539 → 0.98 dB
G0     = |S21|² = 3.12² = 9.7344       → 9.88 dB
GL,max = 1/(1 − 0.35²) = 1/0.8775 = 1.1396 → 0.57 dB
GTU,max = 1.2539 × 9.7344 × 1.1396 = 13.91
        = 0.98 + 9.88 + 0.57 = 11.43 dB

Validity of the unilateral assumption

U = |S11||S12||S21||S22| / [(1−|S11|²)(1−|S22|²)]
  = 0.45×0.04×3.12×0.35 / (0.7975×0.8775) = 0.0281
1/(1+U)² < GT/GTU < 1/(1−U)²  →  −0.24 dB to +0.25 dB

The error is only about ±0.25 dB, so the unilateral model is acceptable. For comparison, the exact bilateral maximum is GT,max = 78×(2.745 − √(2.745² − 1)) = 14.71 = 11.68 dB, only 0.24 dB more.

Answer: K = 2.745, |Δ| = 0.101 → unconditionally stable; GTU,max = 13.91 ≈ 11.43 dB.