---
title: "Max gains for S11 = 0.55∠−150° transistor"
subject: "RF and Microwave Engineering"
chapter: 6
key: "semester_7/rf_and_microwave_engineering:amplifier-s-0-55-150-gains"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/amplifier-s-0-55-150-gains
site: BCT Engineering
---

# Max gains for S11 = 0.55∠−150° transistor

RF and Microwave Engineering · Chapter 6: RF Design Practices

- Asked in: 2082 Baisakh, 8+4 marks; 2071 Magh, 15 marks (asked 2 times)
- Video: [How to Maximize Power Gain of a RF-Amplifier](https://www.youtube.com/watch?v=iiHJTDVWEfY) — RF & Microwave Engineering
- 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-s-0-55-150-gains
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6

## Question

Using the following S-parameters: S11 = 0.55∠−150°, S12 = 0.04∠20°, S21 = 2.82∠180° and S22 = 0.45∠−30°, design a microwave amplifier for maximum power gain (calculate and compare maximum power gain) for both bilateral and unilateral cases.

## Short answer

Stability: K=2.29>1, |Δ|=0.147<1 – unconditionally stable. Unilateral maximum gain (S12=0): GTU,max=1/(1−|S11|²)·|S21|²·1/(1−|S22|²)=14.3 (11.6 dB). Bilateral maximum gain: MAG=(|S21|/|S12|)(K−√(K²−1))=16.2 (12.1 dB).

## Answer

The amplifier is designed by (1) checking stability, (2) finding the source and load reflection coefficients for maximum gain in the **unilateral** (S12 = 0) and **bilateral** cases, (3) computing the maximum gains, and (4) designing single-stub matching networks.

**Given:** S11 = 0.55∠−150°, S12 = 0.04∠20°, S21 = 2.82∠180°, S22 = 0.45∠−30° (Z₀ = 50 Ω).

```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))
```

#### Step 1: Stability

```text
S11·S22 = 0.2475∠180.0° = −0.2475 + j0.0000
S12·S21 = 0.1128∠−160.0° = −0.1060 − j0.0386
Δ = S11·S22 − S12·S21 = −0.1415 + j0.0386
  = 0.1467∠164.7°,   |Δ|² = 0.0215

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.3025 − 0.2025 + 0.0215) / (2 × 0.1128)
  = 0.5165 / 0.2256
  = 2.290
```

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = **1.436** (> 1, unconditionally stable).

K = 2.290 > 1 and |Δ| = 0.147 < 1, so the transistor is **unconditionally stable**. Conjugate matching can be used at both ports.

#### Step 2: Unilateral design (S12 assumed 0)

Maximum gain needs ΓS = S11* = 0.55∠150° and ΓL = S22* = 0.45∠30°.

```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 = 2.820² = 7.9524  (9.00 dB)
GL = 1/(1 − |S22|²) = 1/(1 − 0.2025)
   = 1.2539  (0.98 dB)
GTU,max = GS·G0·GL = 14.296  (11.55 dB)
```

Unilateral figure of merit (to check the error of assuming S12 = 0):

```text
U = |S11·S12·S21·S22| / [(1 − |S11|²)(1 − |S22|²)]
  = 0.0502
1/(1+U)² < GT/GTU < 1/(1−U)²
−0.43 dB < error < +0.45 dB
```

#### Step 3: Bilateral design (S12 included)

```text
B1 = 1 + |S11|² − |S22|² − |Δ|² = 1.0785
B2 = 1 + |S22|² − |S11|² − |Δ|² = 0.8785
C1 = S11 − Δ·S22* = 0.4866∠−148.0°
C2 = S22 − Δ·S11* = 0.3728∠−26.7°
ΓS = [B1 − √(B1² − 4|C1|²)] / (2C1) = 0.6307∠148.0°
ΓL = [B2 − √(B2² − 4|C2|²)] / (2C2) = 0.5551∠26.7°
ZS = 50(1 + ΓS)/(1 − ΓS) = 12.2 + j13.6 Ω
ZL = 50(1 + ΓL)/(1 − ΓL) = 109.3 + j78.9 Ω
```

```text
Bilateral (S12 kept), simultaneous conjugate match:
MAG = Gmax = (|S21|/|S12|)·(K − √(K² − 1))
     = (2.820/0.040) × (2.290 − √(2.290² − 1))
     = 70.500 × 0.2299
     = 16.210  (12.10 dB)
```

Check: with these ΓS and ΓL, Γin = ΓS* and Γout = ΓL* (simultaneous conjugate match), and the transducer gain G_T works out to 16.210.

#### Step 4: Matching networks (single open stub + series 50 Ω line)

```text
 50 Ω  --+--[ line dS ]--[ FET ]--[ line dL ]--+--  50 Ω
 source  |                                     |   load
       open                                  open
       stub lS                               stub lL
```

Each network uses a shunt open stub at the 50 Ω port and a 50 Ω line toward the transistor, chosen so that the transistor sees ΓS (input) and ΓL (output). The stub gives normalised susceptance b with |Γ| = b/√(4 + b²).

```text
Bilateral (ΓS = 0.6307∠148.0°, ΓL = 0.5551∠26.7°):
input : |ΓS| = 0.6307 -> b = 2|Γ|/√(1−|Γ|²) = 1.626
        open stub lS = arctan(b)/2π = 0.162 λ
        series line dS = 0.115 λ
output: |ΓL| = 0.5551 -> b = 1.335
        open stub lL = 0.148 λ
        series line dL = 0.291 λ
```

```text
Unilateral (ΓS = 0.55∠150°, ΓL = 0.45∠30°):
input : |ΓS| = 0.5500 -> b = 2|Γ|/√(1−|Γ|²) = 1.317
        open stub lS = arctan(b)/2π = 0.147 λ
        series line dS = 0.120 λ
output: |ΓL| = 0.4500 -> b = 1.008
        open stub lL = 0.126 λ
        series line dL = 0.296 λ
```

(Lengths are in guide wavelengths at the design frequency. The other root, with negative b, gives a second valid solution.)

#### Step 5: Comparison

| Quantity | Unilateral | Bilateral |
|---|---|---|
| ΓS | 0.55∠150° | 0.6307∠148.0° |
| ΓL | 0.45∠30° | 0.5551∠26.7° |
| Max gain (ratio) | 14.30 | 16.21 |
| Max gain (dB) | 11.55 dB | 12.10 dB |

The bilateral gain is 0.55 dB higher, because the feedback through S12 adds to the gain when both ports are matched together. The unilateral error bounds (U = 0.0502) are −0.43 to +0.45 dB. The bilateral optimum is slightly above the upper bound because it uses different terminations, but the difference is small, so the unilateral design is a fair first approximation and the bilateral design is the exact one.

**Answer: unconditionally stable (K = 2.290, |Δ| = 0.147); GTU,max = 11.55 dB (unilateral); Gmax = 12.10 dB (bilateral).**

## Check yourself

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

1. What conditions show the transistor is unconditionally stable?
   A. K < 1 and |Δ| > 1
   B. K > 1 and |Δ| < 1
   C. |S11| > 1 only
   D. K > 1 and |Δ| > 1
   Answer: B. Rollett's test needs K > 1 and |Δ| < 1. Here K = 2.290 and |Δ| = 0.147, so it is unconditionally stable. (Key point: Stability test)

2. For the unilateral design (S12 = 0), which terminations give maximum gain and what is GTU,max?
   A. ΓS = S11, ΓL = S22; 9.00 dB
   B. ΓS = S11*, ΓL = S22*; 11.55 dB
   C. ΓS = 0, ΓL = 0; 9.00 dB
   D. ΓS = S11*, ΓL = S22*; 12.10 dB
   Answer: B. Conjugate match at each port gives GS·G0·GL = 1.56 + 9.00 + 0.98 = 11.55 dB. (Key point: Unilateral terminations and gain)

3. Which formula gives the bilateral maximum (MAG) and what is the result?
   A. (|S21|/|S12|)(K - sqrt(K^2 - 1)) = 16.21 (12.10 dB)
   B. |S21|^2 = 7.95 (9.00 dB)
   C. (|S21|/|S12|)(K + sqrt(K^2 - 1)) = 16.21
   D. 1/(1 - |S11|^2) = 1.43
   Answer: A. With simultaneous conjugate match, Gmax = 70.5 x 0.2299 = 16.21, i.e. 12.10 dB. (Key point: Bilateral maximum gain)

4. Why is the bilateral gain (12.10 dB) about 0.55 dB higher than the unilateral gain (11.55 dB)?
   A. The unilateral design uses mismatched ports
   B. Feedback through S12 adds to gain when both ports are matched together
   C. S12 is ignored in the bilateral case
   D. The bilateral case has lower loss in the stubs
   Answer: B. S12 is small but not zero, and its feedback adds to the gain when both ports are matched together. The small difference makes the unilateral design a fair first approximation. (Key point: Comparison of the two gains)

## Related in RF and Microwave Engineering

- [Max gains for S11 = 0.55∠150° (2071 Bhadra)](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aamplifier-gains-s11-positive-150) — RF and Microwave Engineering, chapter 6
- [Bilateral/unilateral gains and design flow chart](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aamplifier-gains-flowchart-0-55) — RF and Microwave Engineering, chapter 6
- [Stability and unilateral gain, S11 = 0.45∠−160°](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aamplifier-stability-unilateral-gain) — RF and Microwave Engineering, chapter 6
- [Stability and gains for S21 = 10.11∠91°](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aamplifier-stability-gains-0-64) — RF and Microwave Engineering, chapter 6
- [Stability and gains for S11 = 0.78∠−113°](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aamplifier-stability-gains-0-78) — RF and Microwave Engineering, chapter 6
- [Max gain for S11 = 0.656∠146.7°](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Amax-gain-s11-0-656) — RF and Microwave Engineering, chapter 6