--- title: "Conditional stability and unilateral max gain" subject: "RF and Microwave Engineering" chapter: 6 key: "semester_7/rf_and_microwave_engineering:conditional-stability-unilateral-gain" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/conditional-stability-unilateral-gain site: BCT Engineering --- # Conditional stability and unilateral max gain RF and Microwave Engineering · Chapter 6: RF Design Practices - Asked in: 2072 Asoj, 5+5 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/conditional-stability-unilateral-gain - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6 ## Question Justify that a transistor having following S-parameters S11 = 0.894∠−60.6°, S12 = 0.020∠62.4°, S21 = 3.122∠123.6° and S22 = 0.781∠−27.6° is conditionally stable while designing an amplifier. Considering unilateral model calculate maximum gain. ## Short answer K = 0.607 and |Δ| = 0.696, so the device is not unconditionally stable; with |S11| and |S22| below 1 it is conditionally stable. The unilateral maximum gain is 124.5, or 20.95 dB. ## 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) ``` #### Justification of conditional stability ```text S11·S22 = 0.894×0.781∠(−60.6°−27.6°) = 0.6982∠−88.2° = 0.0219 − j0.6979 S12·S21 = 0.020×3.122∠(62.4°+123.6°) = 0.0624∠186° = −0.0621 − j0.0065 Δ = 0.0840 − j0.6913 = 0.6964∠−83.07° K = (1 − 0.7992 − 0.6100 + 0.4850)/(2×0.0624) = 0.0758/0.1249 = 0.607 μ = 0.863 ``` - |Δ| = 0.696 < 1 but **K = 0.607 < 1** (and μ < 1), so the device is **not** unconditionally stable. - Since |S11| < 1 and |S22| < 1, it is stable for a 50 Ω source and load. It is therefore **conditionally stable**: stable for some Γs, ΓL and unstable for others. Stability circles show the unstable zones: ```text CL = 1.363∠46.69°, RL = 0.500 CS = 1.132∠68.46°, RS = 0.199 ``` Both circles cut the unit Smith chart, so parts of the chart (near the top-right edge) are unstable; the stable region is outside the circles. #### Maximum unilateral gain ```text GS,max = 1/(1 − 0.894²) = 4.981 → 6.97 dB G0 = 3.122² = 9.747 → 9.89 dB GL,max = 1/(1 − 0.781²) = 2.564 → 4.09 dB GTU,max = 124.47 → 20.95 dB ``` Check that Γs = S11* and ΓL = S22* are in the stable region: distance of S11* from CS = 0.275 > 0.199 and of S22* from CL = 0.675 > 0.500, so both are stable (|Γin| = 0.912, |Γout| = 0.848). U = 0.557 (error band −3.84 dB to +7.07 dB), so the unilateral value is only an estimate. **Answer: K = 0.607 < 1, |Δ| = 0.696 → conditionally stable; GTU,max = 124.5 ≈ 20.95 dB.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For S11 = 0.894∠−60.6°, S12 = 0.020∠62.4°, S21 = 3.122∠123.6°, S22 = 0.781∠−27.6°, why is the transistor not unconditionally stable? A. |Δ| > 1 B. K = 0.607 < 1 (and μ < 1) C. |S21| < 1 D. |S11| > 1 Answer: B. Although |Δ| = 0.696 < 1, K = 0.607 < 1, so unconditional stability fails. (Key point: Why not unconditionally stable) 2. Why is it called conditionally stable rather than unstable? A. Because K is exactly 1 B. Because |S11| < 1 and |S22| < 1, so it is stable for a 50 Ω source and load but unstable for some other Γs, ΓL C. Because the stability circles lie outside the chart D. Because S12 = 0 Answer: B. It is stable for some terminations (including 50 Ω) and unstable for others, which is conditional stability. (Key point: Why conditionally stable) 3. What is GTU,max for this transistor? A. 9.89 dB B. 6.97 dB C. 20.95 dB (124.5) D. 4.09 dB Answer: C. GTU,max = 4.981 × 9.747 × 2.564 = 124.5, i.e. 6.97 + 9.89 + 4.09 = 20.95 dB. (Key point: Unilateral gain) 4. U = 0.557 with error band −3.84 to +7.07 dB. What does this imply? A. The unilateral value is exact B. The unilateral value is only an estimate because S12 is not negligible C. The device is unstable D. The gain must be 0 dB Answer: B. A large U means feedback through S12 matters, so the unilateral gain is only approximate. 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