GaAs MESFET at 10 GHz: stability and gains

Stability is checked with the Rollett factor K and |Δ|. Then the maximum gain is compared for the unilateral case (S12 = 0) and the bilateral case (simultaneous conjugate match).

Given (10 GHz, 50 Ω): S11 = 0.55∠−160°, S12 = 0.04∠10°, S21 = 4.82∠180°, S22 = 0.45∠−30°.

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)]
Gmax (MAG) = (|S21|/|S12|)·(K − √(K² − 1))

Stability

S11·S22 = 0.2475∠170.0° = −0.2437 + j0.0430
S12·S21 = 0.1928∠−170.0° = −0.1899 − j0.0335
Δ = S11·S22 − S12·S21 = −0.0539 + j0.0765
  = 0.0935∠125.2°,   |Δ|² = 0.0087

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.3025 − 0.2025 + 0.0087) / (2 × 0.1928)
  = 0.5037 / 0.3856
  = 1.306

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = 1.147 (> 1, unconditionally stable).

K = 1.306 > 1 and |Δ| = 0.094 < 1, so the MESFET is unconditionally stable at 10 GHz.

Unilateral mode

Unilateral (S12 = 0, ΓS = S11*, ΓL = S22*):
GS = 1/(1 − |S11|²) = 1/(1 − 0.3025)
   = 1.4337  (1.56 dB)
G0 = |S21|² = 4.820² = 23.2324  (13.66 dB)
GL = 1/(1 − |S22|²) = 1/(1 − 0.2025)
   = 1.2539  (0.98 dB)
GTU,max = GS·G0·GL = 41.766  (16.21 dB)

Bilateral mode

Bilateral (S12 kept), simultaneous conjugate match:
MAG = Gmax = (|S21|/|S12|)·(K − √(K² − 1))
     = (4.820/0.040) × (1.306 − √(1.306² − 1))
     = 120.500 × 0.4658
     = 56.124  (17.49 dB)

Matching terminations: ΓS = 0.7362∠156.7°, ΓL = 0.6831∠25.0°.

Error of the unilateral assumption

U = |S11·S12·S21·S22| / [(1 − |S11|²)(1 − |S22|²)]
  = 0.0858
1/(1+U)² < GT/GTU < 1/(1−U)²
−0.71 dB < error < +0.78 dB

Comparison

The bilateral gain is 1.28 dB higher. This is larger than the upper unilateral error bound (+0.78 dB), because the large S21 = 4.82 makes the feedback through S12 significant (U = 0.0858). So the bilateral design should be used for accurate results.

Answer: K = 1.306, |Δ| = 0.094 → unconditionally stable; GTU,max = 16.21 dB, Gmax = 17.49 dB.