--- title: "Stability and gains for S11 = 0.78∠−113°" subject: "RF and Microwave Engineering" chapter: 6 key: "semester_7/rf_and_microwave_engineering:amplifier-stability-gains-0-78" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/amplifier-stability-gains-0-78 site: BCT Engineering --- # Stability and gains for S11 = 0.78∠−113° RF and Microwave Engineering · Chapter 6: RF Design Practices - Asked in: 2074 Magh, 12 marks - Video: [Microwave Engineering (Stability)](https://www.youtube.com/watch?v=xmpxNnYmxRg) — Taghreed Al -Erani - 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/amplifier-stability-gains-0-78 - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6 ## Question Given S-parameters for microwave transistor amplifier: S11 = 0.78∠−113°, S12 = 0.028∠247°, S21 = 2.60∠76°, S22 = 0.81∠−54°. Determine the stability and compare maximum power gain for unilateral and bilateral modes using supplied formulas. ## Short answer Compute Δ and K; K=1.367, |Δ|=0.681 so unconditionally stable. Bilateral simultaneous conjugate match gives GT,max=16.06 dB, unilateral estimate 17.01 dB; use bilateral values. ## Answer ```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 ```text S11·S22 = 0.78×0.81∠(−113°−54°) = 0.6318∠−167° = −0.6156 − j0.1421 S12·S21 = 0.028×2.60∠(247°+76°) = 0.0728∠323° = 0.0581 − j0.0438 Δ = −0.6737 − j0.0983 = 0.6809∠−171.70° K = (1 − 0.6084 − 0.6561 + 0.4636)/(2×0.0728) = 0.1991/0.1456 = 1.367 ``` **K = 1.367 > 1 and |Δ| = 0.681 < 1 → unconditionally stable** (μ = 1.097 > 1). #### Bilateral maximum gain (simultaneous conjugate match) ```text GT,max = (|S21|/|S12|)(K − √(K² − 1)) = (2.60/0.028)(1.367 − √(1.367² − 1)) = 92.86 × 0.4348 = 40.37 → 16.06 dB B1 = 0.4887, C1 = 0.2347∠−101.90° → ΓMS = 0.7517∠101.90° B2 = 0.5841, C2 = 0.2840∠−45.19° → ΓML = 0.7891∠45.19° GS = 2.299 (3.62 dB) G0 = 6.76 (8.30 dB) GL = 2.597 (4.15 dB) → GT = 16.06 dB ``` #### Unilateral maximum gain ```text GS,max = 1/(1 − 0.78²) = 2.554 → 4.07 dB G0 = 2.60² = 6.76 → 8.30 dB GL,max = 1/(1 − 0.81²) = 2.908 → 4.64 dB GTU,max = 50.20 → 17.01 dB ``` #### Comparison | | Bilateral | Unilateral | |---|---|---| | Γs | 0.752∠101.9° | 0.78∠113° | | ΓL | 0.789∠45.2° | 0.81∠54° | | Gmax | 40.37 (16.06 dB) | 50.20 (17.01 dB) | ```text U = 0.78×0.028×2.6×0.81 / (0.3916×0.3439) = 0.342 Error band: −2.55 dB to +3.63 dB ``` The unilateral formula gives 0.95 dB more than is really possible. Because |S11| and |S22| are large, U = 0.34 and the unilateral model is unreliable; the bilateral value **16.06 dB** is the true maximum gain and the design should use ΓMS and ΓML. **Answer: unconditionally stable (K = 1.367, |Δ| = 0.681); GT,max = 16.06 dB (bilateral), GTU,max = 17.01 dB (unilateral estimate).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For S11 = 0.78∠−113°, S12 = 0.028∠247°, S21 = 2.60∠76°, S22 = 0.81∠−54°, what is found? A. K = 0.5, conditionally stable B. |Δ| > 1, unstable C. K = 1.367 and |Δ| = 0.681, unconditionally stable D. K = 1.097 and unstable Answer: C. K = 1.367 > 1 and |Δ| = 0.681 < 1, so the transistor is unconditionally stable. (Key point: Stability result) 2. What is the bilateral maximum gain and how is it found? A. 16.06 dB from (|S21|/|S12|)(K − sqrt(K^2 − 1)) B. 17.01 dB from 1/(1−|S11|^2)·|S21|^2/(1−|S22|^2) C. 8.30 dB from |S21|^2 D. 20.95 dB from the MSG Answer: A. GT,max = 92.86 × 0.4348 = 40.37, i.e. 16.06 dB, using simultaneous conjugate match. (Key point: Bilateral gain) 3. How is the unilateral maximum gain GTU,max composed? A. GS,max + GL,max only B. G0 = |S21|^2 only C. 1/(1−|S21|^2) times |S12| D. GS,max × G0 × GL,max = 2.554 × 6.76 × 2.908 (17.01 dB) Answer: D. GTU,max = [1/(1−|S11|^2)]|S21|^2[1/(1−|S22|^2)], which is 4.07 + 8.30 + 4.64 = 17.01 dB. (Key point: Unilateral gain) 4. The unilateral estimate (17.01 dB) exceeds the true bilateral maximum (16.06 dB). Why should the bilateral value be used? A. Because U = 0.34 is large (|S11|, |S22| are large), so ignoring S12 is unreliable B. Because the device is unstable C. Because unilateral design needs K < 1 D. Because S21 is below 1 Answer: A. With U = 0.34 the error band is −2.55 to +3.63 dB, so the unilateral model is not reliable; the bilateral value is the real maximum. 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