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Ku-band HEMT amplifier: stability and gains

Short answer

K=1.195, |Δ|=0.488, so unconditionally stable. Bilateral maximum gain GT,max = 46.88 (16.71 dB), unilateral maximum GTU,max = 30.05 (14.78 dB). The unilateral model underestimates by ~1.93 dB due to non-negligible S12.

A microwave amplifier operates in Ku-Band for satellite transmit system uses a High Electron Mobility GaAs MOSFET transistor having following S-parameters at 4 GHz with 50 Ω line impedance as, S11 = 0.72∠−116°, S12 = 0.03∠57°, S21 = 2.60∠76° and S22 = 0.73∠−54°. Check the stability of the given transistor and compare maximum power gain for bilateral and unilateral mode.

Stability is checked with K and |Δ|; the bilateral maximum uses simultaneous conjugate matching, and the unilateral maximum assumes S12 = 0.

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 = 0.72×0.73∠(−116°−54°) = 0.5256∠−170°
        = −0.5176 − j0.0913
S12·S21 = 0.03×2.60∠(57°+76°) = 0.0780∠133°
        = −0.0532 + j0.0570
Δ = −0.4644 − j0.1483 = 0.4875∠−162.29°
K = (1 − 0.5184 − 0.5329 + 0.2377)/(2×0.0780)
  = 0.1864/0.1560 = 1.195

K = 1.195 > 1 and |Δ| = 0.488 < 1 → unconditionally stable (μ = 1.040 > 1).

Bilateral maximum gain

GT,max = (2.60/0.03)(1.195 − √(1.195² − 1))
       = 86.67 × 0.5410 = 46.88 → 16.71 dB
B1 = 0.7478, C1 = 0.3704∠−123.41°
ΓMS = 0.8718∠123.41°
B2 = 0.7768, C2 = 0.3850∠−61.03°
ΓML = 0.8763∠61.03°
GS = 1/(1−|ΓMS|²)            = 4.166 → 6.20 dB
G0 = |S21|²                  = 6.76  → 8.30 dB
GL = (1−|ΓML|²)/|1−S22ΓML|²  = 1.665 → 2.21 dB
GT = 6.20 + 8.30 + 2.21 = 16.71 dB

Unilateral maximum gain

GS,max = 1/(1 − 0.72²) = 2.076 → 3.17 dB
G0     = 6.76                → 8.30 dB
GL,max = 1/(1 − 0.73²) = 2.141 → 3.31 dB
GTU,max = 30.05 → 14.78 dB

Comparison

QuantityBilateralUnilateral
Γs0.872∠123.4°S11* = 0.72∠116°
ΓL0.876∠61.0°S22* = 0.73∠54°
Max gain46.88 (16.71 dB)30.05 (14.78 dB)

Unilateral figure of merit U = 0.72×0.03×2.6×0.73/(0.4816×0.4671) = 0.182, so the error band is −1.45 dB to +1.75 dB. These bounds apply to the gain obtained with Γs = S11* and ΓL = S22* (the actual value there is 36.0, i.e. 15.56 dB). The true bilateral maximum (16.71 dB) is 1.93 dB above the unilateral estimate. Because U is large, the unilateral model is not accurate for this HEMT, and the bilateral design should be used.

Answer: unconditionally stable (K = 1.195, |Δ| = 0.488); GT,max = 16.71 dB (bilateral) vs GTU,max = 14.78 dB (unilateral).

✓ Answer checked by Claude Opus 5.5 (high effort)

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