Bilateral/unilateral gains and design flow chart

The maximum gains are found after checking stability. The unilateral model ignores S12, and the bilateral model includes it (simultaneous conjugate match).

Given: S11 = 0.55∠−150°, S12 = 0.04∠20°, S21 = 2.82∠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 (needed before gain)

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

K = 2.290 > 1 and |Δ| = 0.147 < 1, so the transistor is unconditionally stable. Maximum gains therefore exist.

Unilateral model (S12 = 0)

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)

Bilateral model

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)

The source and load terminations for this gain are ΓS = 0.6307∠148.0° and ΓL = 0.5551∠26.7°.

The bilateral gain is 0.55 dB higher. The unilateral figure of merit U = 0.0502 gives an error range of −0.43 to +0.45 dB, so the unilateral model is a reasonable approximation.

Answer: GTU,max = 11.55 dB (unilateral), Gmax = 12.10 dB (bilateral).

Flow chart of microwave amplifier design

        [ Start: specs f, G, NF ]
                   |
     [ Select transistor, bias; get S ]
                   |
        [ Compute Δ, K (or μ) ]
                   |
          < K>1 and |Δ|<1 ? >
           | yes          | no
           |      [ Draw stability circles ]
           |      [ choose ΓS, ΓL in stable ]
           |      [ region / add loading    ]
           |              |
     < U small (S12≈0)? > |
      | yes        | no   |
 [ΓS=S11*,     [ΓS=ΓMS,   |
  ΓL=S22*]      ΓL=ΓML]   |
      |____________|______|
                   |
   [ Design input/output matching ]
   [ networks (Smith chart, stubs) ]
                   |
   [ Bias network, layout, simulate ]
                   |
           < Specs met? > -- no -> retune
                   | yes
                [ End ]