---
title: "Max gains for S11 = 0.55∠150° (2071 Bhadra)"
subject: "RF and Microwave Engineering"
chapter: 6
key: "semester_7/rf_and_microwave_engineering:amplifier-gains-s11-positive-150"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/amplifier-gains-s11-positive-150
site: BCT Engineering
---

# Max gains for S11 = 0.55∠150° (2071 Bhadra)

RF and Microwave Engineering · Chapter 6: RF Design Practices

- Asked in: 2071 Bhadra, 10 marks
- Video: [Microwave Engineering (Operating and Available Power gain )](https://www.youtube.com/watch?v=gklxywCYKPk) — 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-gains-s11-positive-150
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6

## Question

Using the given S-parameters S11 = 0.55∠150°, S12 = 0.04∠20°, S21 = 2.82∠180°, S22 = 0.45∠−30° and required assumptions, calculate maximum gains of this transistor amplifier for bilateral and unilateral modes.

## Short answer

The device is unconditionally stable, K = 2.479 and |Δ| = 0.254. Bilateral maximum gain is 14.85, or 11.72 dB; unilateral maximum gain is 14.30, or 11.55 dB, only 0.17 dB lower.

## 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)
```

Assumptions: Z0 = 50 Ω; the stated S-parameters apply at the design frequency; lossless matching networks.

#### Stability

```text
S11·S22 = 0.55×0.45∠(150°−30°) = 0.2475∠120°
        = −0.1238 + j0.2143
S12·S21 = 0.04×2.82∠(20°+180°) = 0.1128∠200°
        = −0.1060 − j0.0386
Δ = −0.0178 + j0.2529 = 0.2535∠94.02°
K = (1 − 0.3025 − 0.2025 + 0.0643)/(2×0.1128)
  = 0.5593/0.2256 = 2.479
```

**K = 2.479 > 1 and |Δ| = 0.254 < 1 → unconditionally stable.**

#### Bilateral maximum gain

```text
GT,max = (2.82/0.04)(2.479 − √(2.479² − 1))
       = 70.5 × 0.2106 = 14.85 → 11.72 dB
B1 = 1.0357, C1 = 0.4502∠156.37° → ΓMS = 0.5819∠−156.37°
B2 = 0.8357, C2 = 0.3303∠−19.34° → ΓML = 0.4903∠19.34°
GS = 1.512 (1.80 dB), G0 = 7.952 (9.00 dB),
GL = 1.235 (0.92 dB)
```

Matching impedances: Zs = 13.75 − j9.70 Ω, ZL = 120.52 + j51.53 Ω.

#### Unilateral maximum gain

```text
GS,max = 1/(1 − 0.55²) = 1.434 → 1.56 dB
G0     = 2.82² = 7.952          → 9.00 dB
GL,max = 1/(1 − 0.45²) = 1.254 → 0.98 dB
GTU,max = 14.30 → 11.55 dB
U = 0.050 → error −0.43 dB to +0.45 dB
```

| Mode | Gmax | dB |
|---|---|---|
| Bilateral | 14.85 | 11.72 |
| Unilateral | 14.30 | 11.55 |

The two differ by only 0.17 dB because |S12| is small (U = 0.05), so the unilateral model is acceptable here.

**Answer: GT,max = 14.85 (11.72 dB) bilateral; GTU,max = 14.30 (11.55 dB) unilateral.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For S11 = 0.55∠150°, S12 = 0.04∠20°, S21 = 2.82∠180°, S22 = 0.45∠−30°, what is the stability result?
   A. K = 0.5, conditional
   B. K = 2.479 and |Δ| = 0.254, unconditionally stable
   C. |Δ| > 1, unstable
   D. K = 2.479 but unstable
   Answer: B. K > 1 and |Δ| < 1 give unconditional stability, so a simultaneous conjugate match exists. (Key point: Stability)

2. What is GT,max (bilateral)?
   A. 11.55 dB
   B. 9.00 dB
   C. 14.85 dB
   D. 11.72 dB (14.85)
   Answer: D. GT,max = (2.82/0.04)(2.479 − sqrt(2.479^2 − 1)) = 70.5 × 0.2106 = 14.85, i.e. 11.72 dB. (Key point: Bilateral gain)

3. What is GTU,max and how is it built?
   A. GS,max × G0 × GL,max = 1.434 × 7.952 × 1.254 = 14.30 (11.55 dB)
   B. |S21|^2 only = 9.00 dB
   C. 1.56 dB
   D. 14.85 (11.72 dB)
   Answer: A. Multiplying the three blocks gives 14.30, i.e. 1.56 + 9.00 + 0.98 = 11.55 dB. (Key point: Unilateral gain)

4. Why do the bilateral and unilateral gains differ by only 0.17 dB?
   A. Because K is close to 1
   B. Because |S12| is small (U = 0.05), so the unilateral model is acceptable
   C. Because S21 = 1
   D. Because the device is unstable
   Answer: B. A small U means feedback is negligible, so ignoring S12 causes little error. (Key point: Comparing the models)

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