---
title: "Short note: Stability analysis of microwave amplifier"
subject: "RF and Microwave Engineering"
chapter: 6
key: "semester_7/rf_and_microwave_engineering:amplifier-stability-note"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/amplifier-stability-note
site: BCT Engineering
---

# Short note: Stability analysis of microwave amplifier

RF and Microwave Engineering · Chapter 6: RF Design Practices

- Asked in: 2081 Baisakh, 4 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-note
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6

## Question

Write a short note on stability analysis of a microwave amplifier.

## Answer

**Stability** of a microwave amplifier means that it does not oscillate. This needs |Γ_in| < 1 and |Γ_out| < 1, meaning there is no negative resistance at either port, for the source and load terminations used.

```text
Γin  = S11 + S12·S21·ΓL / (1 − S22·ΓL)
Γout = S22 + S12·S21·ΓS / (1 − S11·ΓS)
```

**Types**
- **Unconditionally stable:** |Γ_in| < 1 and |Γ_out| < 1 for **every** passive source and load (|Γ_S|, |Γ_L| < 1).
- **Conditionally (potentially) unstable:** stable only for some terminations.

**Tests**

```text
Δ = S11·S22 − S12·S21
K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
Rollett: K > 1 and |Δ| < 1  -> unconditionally stable
μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|)
μ-test : μ > 1 -> unconditionally stable
```

**Stability circles** (|Γ_in| = 1 and |Γ_out| = 1) for a potentially unstable device:

```text
Output (ΓL plane):
 CL = (S22 − Δ·S11*)* / (|S22|² − |Δ|²)
 RL = |S12·S21 / (|S22|² − |Δ|²)|
Input (ΓS plane):
 CS = (S11 − Δ·S22*)* / (|S11|² − |Δ|²)
 RS = |S12·S21 / (|S11|² − |Δ|²)|
```

If |S11| < 1, the centre of the Γ_L plane (Γ_L = 0) is stable; likewise, if |S22| < 1, the centre of the Γ_S plane is stable. The region on the same side of the circle as the centre is then the stable region. The designer chooses Γ_S and Γ_L in the stable regions, or adds series or shunt resistive loading to stabilise the device.

## Check yourself

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

1. An amplifier is stable (does not oscillate) when:
   A. |Γin| > 1 and |Γout| > 1
   B. |Γin| < 1 and |Γout| < 1
   C. S21 < 1
   D. S12 = 0 only
   Answer: B. Reflection magnitudes below 1 at both ports mean no negative resistance, so no oscillation. (Key point: Stability condition)

2. Which pair of conditions guarantees unconditional stability?
   A. K < 1 and |Δ| > 1
   B. K > 1 and |Δ| > 1
   C. K < 1 and |Δ| < 1
   D. K > 1 and |Δ| < 1
   Answer: D. Rollett's criterion needs K > 1 together with |Δ| < 1. (Key point: Rollett test)

3. What does 'conditionally (potentially) unstable' mean?
   A. Stable for every passive source and load
   B. Stable only for some source/load terminations
   C. Always oscillates
   D. Stable only at DC
   Answer: B. Some passive terminations give |Γin| or |Γout| > 1, so the device must be used with chosen terminations. (Key point: Conditional stability)

4. For a potentially unstable device with |S11| < 1, how is the stable region found on a stability circle plot?
   A. The side not containing the Smith chart centre
   B. The side containing the Smith chart centre
   C. Always inside the circle
   D. Always outside the Smith chart
   Answer: B. If |S11| < 1 the centre (Γ = 0) is stable, so the region on the same side of the circle as the centre is stable. (Key point: Stability circles)

## Related in RF and Microwave Engineering

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