---
title: "Stability and unilateral gain, S11 = 0.45∠−160°"
subject: "RF and Microwave Engineering"
chapter: 6
key: "semester_7/rf_and_microwave_engineering:amplifier-stability-unilateral-gain"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/amplifier-stability-unilateral-gain
site: BCT Engineering
---

# Stability and unilateral gain, S11 = 0.45∠−160°

RF and Microwave Engineering · Chapter 6: RF Design Practices

- Asked in: 2080 Chaitra, 7+8 marks
- Video: [Microwave Amplifier Stability Introduction](https://www.youtube.com/watch?v=GegxPDDBkHI) — Jordan Edmunds Chetty
- 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-unilateral-gain
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6

## Question

Check the stability and find the unilateral gain of a transistor having following S-parameters: S11 = 0.45∠−160°, S12 = 0.04∠20°, S21 = 3.12∠180° and S22 = 0.35∠−40°.

## Short answer

K=2.745, |Δ|=0.101, so unconditionally stable. Maximum unilateral transducer gain GTU,max = 13.91 (11.43 dB) obtained with Γs=S11*=0.45∠160°, ΓL=S22*=0.35∠40°.

## Answer

Stability is tested with the Rollett factor K and |Δ|; the unilateral gain assumes S12 = 0 and conjugate matching at both ports.

```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 = 0.45×0.35 ∠(−160°−40°) = 0.1575∠160°
        = −0.1480 + j0.0539
S12·S21 = 0.04×3.12 ∠(20°+180°) = 0.1248∠−160°
        = −0.1173 − j0.0427
Δ = (−0.1480+j0.0539) − (−0.1173−j0.0427)
  = −0.0307 + j0.0966 = 0.1013∠107.65°
|Δ| = 0.1013 < 1
K = (1 − 0.2025 − 0.1225 + 0.01027)/(2×0.1248)
  = 0.6853/0.2496 = 2.745 > 1
```

Since **K = 2.745 > 1 and |Δ| = 0.101 < 1**, the transistor is **unconditionally stable** (μ = 1.776 > 1 confirms it). Any passive source and load can be used.

#### Unilateral (maximum) transducer gain

With S12 = 0, Γs = S11* = 0.45∠160° and ΓL = S22* = 0.35∠40°:

```text
GS,max = 1/(1 − 0.45²) = 1/0.7975 = 1.2539 → 0.98 dB
G0     = |S21|² = 3.12² = 9.7344       → 9.88 dB
GL,max = 1/(1 − 0.35²) = 1/0.8775 = 1.1396 → 0.57 dB
GTU,max = 1.2539 × 9.7344 × 1.1396 = 13.91
        = 0.98 + 9.88 + 0.57 = 11.43 dB
```

#### Validity of the unilateral assumption

```text
U = |S11||S12||S21||S22| / [(1−|S11|²)(1−|S22|²)]
  = 0.45×0.04×3.12×0.35 / (0.7975×0.8775) = 0.0281
1/(1+U)² < GT/GTU < 1/(1−U)²  →  −0.24 dB to +0.25 dB
```

The error is only about ±0.25 dB, so the unilateral model is acceptable. For comparison, the exact bilateral maximum is GT,max = 78×(2.745 − √(2.745² − 1)) = 14.71 = 11.68 dB, only 0.24 dB more.

**Answer: K = 2.745, |Δ| = 0.101 → unconditionally stable; GTU,max = 13.91 ≈ 11.43 dB.**

## Check yourself

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

1. Given K = 2.745 and |Δ| = 0.101, the transistor is:
   A. Potentially unstable
   B. Unconditionally stable
   C. Oscillating
   D. Stable only at DC
   Answer: B. K > 1 and |Δ| < 1 satisfy Rollett's condition. (Key point: Stability result)

2. For the unilateral model, what source and load reflection coefficients give maximum gain?
   A. ΓS = S11, ΓL = S22
   B. ΓS = S11*, ΓL = S22*
   C. ΓS = 0, ΓL = 0
   D. ΓS = S22*, ΓL = S11*
   Answer: B. With S12 = 0 each port is conjugate matched independently. (Key point: Unilateral terminations)

3. What is GTU,max?
   A. 13.91 (11.43 dB)
   B. 9.73 (9.88 dB)
   C. 78 (18.9 dB)
   D. 14.71 (11.68 dB)
   Answer: A. 1.2539 × 9.7344 × 1.1396 = 13.91, i.e. 0.98 + 9.88 + 0.57 = 11.43 dB. (Key point: Gain value)

4. Why is the unilateral approximation acceptable here?
   A. U = 0.028 gives only about ±0.25 dB error
   B. S12 is exactly zero
   C. K < 1
   D. Because |S21| is small
   Answer: A. U is small, so GT/GTU is within 1/(1+U)² to 1/(1−U)², about −0.24 to +0.25 dB. (Key point: Validity)

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