Max gains for S11 = 0.55∠−150° transistor

The amplifier is designed by (1) checking stability, (2) finding the source and load reflection coefficients for maximum gain in the unilateral (S12 = 0) and bilateral cases, (3) computing the maximum gains, and (4) designing single-stub matching networks.

Given: S11 = 0.55∠−150°, S12 = 0.04∠20°, S21 = 2.82∠180°, S22 = 0.45∠−30° (Z₀ = 50 Ω).

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))

Step 1: Stability

S11·S22 = 0.2475∠180.0° = −0.2475 + j0.0000
S12·S21 = 0.1128∠−160.0° = −0.1060 − j0.0386
Δ = S11·S22 − S12·S21 = −0.1415 + j0.0386
  = 0.1467∠164.7°,   |Δ|² = 0.0215

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.3025 − 0.2025 + 0.0215) / (2 × 0.1128)
  = 0.5165 / 0.2256
  = 2.290

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = 1.436 (> 1, unconditionally stable).

K = 2.290 > 1 and |Δ| = 0.147 < 1, so the transistor is unconditionally stable. Conjugate matching can be used at both ports.

Step 2: Unilateral design (S12 assumed 0)

Maximum gain needs ΓS = S11* = 0.55∠150° and ΓL = S22* = 0.45∠30°.

Unilateral (S12 = 0, ΓS = S11*, ΓL = S22*):
GS = 1/(1 − |S11|²) = 1/(1 − 0.3025)
   = 1.4337  (1.56 dB)
G0 = |S21|² = 2.820² = 7.9524  (9.00 dB)
GL = 1/(1 − |S22|²) = 1/(1 − 0.2025)
   = 1.2539  (0.98 dB)
GTU,max = GS·G0·GL = 14.296  (11.55 dB)

Unilateral figure of merit (to check the error of assuming S12 = 0):

U = |S11·S12·S21·S22| / [(1 − |S11|²)(1 − |S22|²)]
  = 0.0502
1/(1+U)² < GT/GTU < 1/(1−U)²
−0.43 dB < error < +0.45 dB

Step 3: Bilateral design (S12 included)

B1 = 1 + |S11|² − |S22|² − |Δ|² = 1.0785
B2 = 1 + |S22|² − |S11|² − |Δ|² = 0.8785
C1 = S11 − Δ·S22* = 0.4866∠−148.0°
C2 = S22 − Δ·S11* = 0.3728∠−26.7°
ΓS = [B1 − √(B1² − 4|C1|²)] / (2C1) = 0.6307∠148.0°
ΓL = [B2 − √(B2² − 4|C2|²)] / (2C2) = 0.5551∠26.7°
ZS = 50(1 + ΓS)/(1 − ΓS) = 12.2 + j13.6 Ω
ZL = 50(1 + ΓL)/(1 − ΓL) = 109.3 + j78.9 Ω

Bilateral (S12 kept), simultaneous conjugate match:
MAG = Gmax = (|S21|/|S12|)·(K − √(K² − 1))
     = (2.820/0.040) × (2.290 − √(2.290² − 1))
     = 70.500 × 0.2299
     = 16.210  (12.10 dB)

Check: with these ΓS and ΓL, Γin = ΓS* and Γout = ΓL* (simultaneous conjugate match), and the transducer gain G_T works out to 16.210.

Step 4: Matching networks (single open stub + series 50 Ω line)

 50 Ω  --+--[ line dS ]--[ FET ]--[ line dL ]--+--  50 Ω
 source  |                                     |   load
       open                                  open
       stub lS                               stub lL

Each network uses a shunt open stub at the 50 Ω port and a 50 Ω line toward the transistor, chosen so that the transistor sees ΓS (input) and ΓL (output). The stub gives normalised susceptance b with |Γ| = b/√(4 + b²).

Bilateral (ΓS = 0.6307∠148.0°, ΓL = 0.5551∠26.7°):
input : |ΓS| = 0.6307 -> b = 2|Γ|/√(1−|Γ|²) = 1.626
        open stub lS = arctan(b)/2π = 0.162 λ
        series line dS = 0.115 λ
output: |ΓL| = 0.5551 -> b = 1.335
        open stub lL = 0.148 λ
        series line dL = 0.291 λ

Unilateral (ΓS = 0.55∠150°, ΓL = 0.45∠30°):
input : |ΓS| = 0.5500 -> b = 2|Γ|/√(1−|Γ|²) = 1.317
        open stub lS = arctan(b)/2π = 0.147 λ
        series line dS = 0.120 λ
output: |ΓL| = 0.4500 -> b = 1.008
        open stub lL = 0.126 λ
        series line dL = 0.296 λ

(Lengths are in guide wavelengths at the design frequency. The other root, with negative b, gives a second valid solution.)

Step 5: Comparison

The bilateral gain is 0.55 dB higher, because the feedback through S12 adds to the gain when both ports are matched together. The unilateral error bounds (U = 0.0502) are −0.43 to +0.45 dB. The bilateral optimum is slightly above the upper bound because it uses different terminations, but the difference is small, so the unilateral design is a fair first approximation and the bilateral design is the exact one.

Answer: unconditionally stable (K = 2.290, |Δ| = 0.147); GTU,max = 11.55 dB (unilateral); Gmax = 12.10 dB (bilateral).