--- title: "Ku-band HEMT amplifier: stability and gains" subject: "RF and Microwave Engineering" chapter: 6 key: "semester_7/rf_and_microwave_engineering:ku-band-hemt-amplifier" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/ku-band-hemt-amplifier site: BCT Engineering --- # Ku-band HEMT amplifier: stability and gains RF and Microwave Engineering · Chapter 6: RF Design Practices - Asked in: 2077 Chaitra, 5+5 marks - Video: [Microwave Amplifier - RF Stability of Microwave Transistors - Part-2](https://www.youtube.com/watch?v=edQBxxAbpjQ) — Microwave Systems: Theory and Design - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/ku-band-hemt-amplifier - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6 ## Question 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. ## 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. ## Answer Stability is checked with K and |Δ|; the bilateral maximum uses simultaneous conjugate matching, and the unilateral maximum assumes S12 = 0. ```text 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 ```text 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 ```text 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 ```text 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 | Quantity | Bilateral | Unilateral | |---|---|---| | Γs | 0.872∠123.4° | S11* = 0.72∠116° | | ΓL | 0.876∠61.0° | S22* = 0.73∠54° | | Max gain | 46.88 (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).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For K = 1.195 and |Δ| = 0.488, the HEMT is: A. Unconditionally stable B. Potentially unstable C. Unstable D. Stable only for unilateral use Answer: A. K > 1 and |Δ| < 1 give unconditional stability (μ = 1.04 > 1 confirms). (Key point: Stability) 2. How is the bilateral maximum gain obtained? A. (|S21|/|S12|)(K − √(K²−1)) with simultaneous conjugate match B. |S21|² only C. 1/(1−|S11|²) D. K·|S21| Answer: A. GT,max = 86.67 × 0.5410 = 46.88 (16.71 dB), using ΓMS and ΓML. (Key point: Bilateral gain) 3. What is the unilateral maximum gain here? A. 16.71 dB B. 14.78 dB C. 8.30 dB D. 6.20 dB Answer: B. 3.17 + 8.30 + 3.31 = 14.78 dB (30.05). (Key point: Unilateral gain) 4. What does the 1.93 dB difference with U = 0.182 imply? A. Unilateral model is accurate B. Unilateral model is inaccurate; use the bilateral design C. Device is unstable D. Gain is negative Answer: B. U is large, the actual gain exceeds the unilateral estimate by more than the error band, so a bilateral design should be used. 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