---
title: "GaAs MESFET at 10 GHz: stability and gains"
subject: "RF and Microwave Engineering"
chapter: 6
key: "semester_7/rf_and_microwave_engineering:gaas-mesfet-10ghz-gains"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/gaas-mesfet-10ghz-gains
site: BCT Engineering
---

# GaAs MESFET at 10 GHz: stability and gains

RF and Microwave Engineering · Chapter 6: RF Design Practices

- Asked in: 2080 Bhadra, 10 marks
- 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/gaas-mesfet-10ghz-gains
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6

## Question

A GaAs MESFET has the following S-parameters measured with a 50Ω resistance at 10 GHz: S11 = 0.55∠−160°, S12 = 0.04∠10°, S21 = 4.82∠180°, S22 = 0.45∠−30°. Determine the stability and compare maximum power gains for unilateral and bilateral modes using supplied formulas.

## Short answer

K=1.306, |Δ|=0.094, so the device is unconditionally stable. Unilateral maximum transducer gain is 16.21 dB; bilateral maximum available gain is 17.49 dB.

## Answer

Stability is checked with the Rollett factor K and |Δ|. Then the maximum gain is compared for the **unilateral** case (S12 = 0) and the **bilateral** case (simultaneous conjugate match).

**Given (10 GHz, 50 Ω):** S11 = 0.55∠−160°, S12 = 0.04∠10°, S21 = 4.82∠180°, S22 = 0.45∠−30°.

```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)]
Gmax (MAG) = (|S21|/|S12|)·(K − √(K² − 1))
```

#### Stability

```text
S11·S22 = 0.2475∠170.0° = −0.2437 + j0.0430
S12·S21 = 0.1928∠−170.0° = −0.1899 − j0.0335
Δ = S11·S22 − S12·S21 = −0.0539 + j0.0765
  = 0.0935∠125.2°,   |Δ|² = 0.0087

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.3025 − 0.2025 + 0.0087) / (2 × 0.1928)
  = 0.5037 / 0.3856
  = 1.306
```

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = **1.147** (> 1, unconditionally stable).

K = 1.306 > 1 and |Δ| = 0.094 < 1, so the MESFET is **unconditionally stable** at 10 GHz.

#### Unilateral mode

```text
Unilateral (S12 = 0, ΓS = S11*, ΓL = S22*):
GS = 1/(1 − |S11|²) = 1/(1 − 0.3025)
   = 1.4337  (1.56 dB)
G0 = |S21|² = 4.820² = 23.2324  (13.66 dB)
GL = 1/(1 − |S22|²) = 1/(1 − 0.2025)
   = 1.2539  (0.98 dB)
GTU,max = GS·G0·GL = 41.766  (16.21 dB)
```

#### Bilateral mode

```text
Bilateral (S12 kept), simultaneous conjugate match:
MAG = Gmax = (|S21|/|S12|)·(K − √(K² − 1))
     = (4.820/0.040) × (1.306 − √(1.306² − 1))
     = 120.500 × 0.4658
     = 56.124  (17.49 dB)
```

Matching terminations: ΓS = 0.7362∠156.7°, ΓL = 0.6831∠25.0°.

#### Error of the unilateral assumption

```text
U = |S11·S12·S21·S22| / [(1 − |S11|²)(1 − |S22|²)]
  = 0.0858
1/(1+U)² < GT/GTU < 1/(1−U)²
−0.71 dB < error < +0.78 dB
```

#### Comparison

| Mode | Max gain | dB |
|---|---|---|
| Unilateral GTU,max | 41.77 | 16.21 dB |
| Bilateral Gmax | 56.12 | 17.49 dB |

The bilateral gain is 1.28 dB higher. This is larger than the upper unilateral error bound (+0.78 dB), because the large S21 = 4.82 makes the feedback through S12 significant (U = 0.0858). So the bilateral design should be used for accurate results.

**Answer: K = 1.306, |Δ| = 0.094 → unconditionally stable; GTU,max = 16.21 dB, Gmax = 17.49 dB.**

## Check yourself

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

1. What are K and |Δ| for the MESFET, and what do they show?
   A. K = 0.61, |Δ| = 0.69, potentially unstable
   B. K = 1.306, |Δ| = 0.094, unconditionally stable
   C. K = 1.306, |Δ| = 1.2, conditionally stable
   D. K = 0.5, |Δ| = 0.094, unstable
   Answer: B. K = 0.5037/0.3856 = 1.306 > 1 and |Δ| = 0.094 < 1, so the device is unconditionally stable. (Key point: Stability result)

2. How is the maximum unilateral gain GTU,max found?
   A. |S21|^2 only
   B. GS·G0·GL = [1/(1−|S11|²)]·|S21|²·[1/(1−|S22|²)]
   C. (|S21|/|S12|)(K − √(K²−1))
   D. |S21|/|S12|
   Answer: B. With S12 = 0 the input match, transistor and output match gains multiply; this gives 41.77 (16.21 dB). (Key point: Unilateral gain)

3. Why can Gmax = (|S21|/|S12|)(K − √(K²−1)) be used here?
   A. Because K > 1, so a simultaneous conjugate match exists
   B. Because S12 = 0
   C. Because |Δ| > 1
   D. Because S21 is real
   Answer: A. For K > 1 the simultaneous conjugate match exists and the MAG formula is real; here Gmax = 56.12 (17.49 dB). (Key point: Bilateral MAG)

4. How do the bilateral and unilateral gains compare and why does it matter?
   A. Equal, since S12 is small
   B. Bilateral is 1.28 dB higher, beyond the unilateral error bound, so use a bilateral design
   C. Unilateral is higher by 1.28 dB
   D. Bilateral is lower because of feedback
   Answer: B. 17.49 − 16.21 = 1.28 dB, above the +0.78 dB bound (U = 0.0858), so the unilateral assumption is not accurate enough. (Key point: Comparison)

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