---
title: "Compare two S-matrix sets; stability and gains"
subject: "RF and Microwave Engineering"
chapter: 6
key: "semester_7/rf_and_microwave_engineering:amplifier-two-s-matrix-sets"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/amplifier-two-s-matrix-sets
site: BCT Engineering
---

# Compare two S-matrix sets; stability and gains

RF and Microwave Engineering · Chapter 6: RF Design Practices

- Asked in: 2082 Bhadra, 4+12 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/amplifier-two-s-matrix-sets
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6

## Question

Find the basic differences of the systems having following sets of S-Matrix. a) S11 = 0.65∠146°, S12 = 0.12∠46°, S21 = 2.30∠44°, S22 = 0.17∠−172° b) S11 = 0.30∠−134°, S12 = 0.03∠46°, S21 = 2.20∠44°, S22 = 0.10∠−172°. Using the given S-Parameters given in (a), determine the stability and compare maximum power gains for unilateral and bilateral modes using supplied formulas.

## Short answer

System (a) has larger S11 (0.65 vs 0.30) and S12 (0.12 vs 0..03), making it less unilateral and closer to instability; system (b) is nearly unilateral and very stable. For (a): K=1.202, |Δ|=0.339, unconditionally stable. Unilateral max gain =9.433 (9.75 dB), bilateral simultaneously conjugate matched gain =10.249 (10.11 dB).

## Answer

#### Basic differences between the two systems

Both are two-port amplifying devices with nearly the same forward gain (|S21| = 2.30 and 2.20, same angle 44°) and the same S22 angle. They differ mainly in input match and reverse isolation:

| Point | Set (a) | Set (b) |
|---|---|---|
| Input reflection, mag S11 | 0.65 (poorly matched) | 0.30 (well matched) |
| Output reflection, mag S22 | 0.17 | 0.10 |
| Reverse transmission, mag S12 | 0.12 (−18.4 dB) | 0.03 (−30.5 dB) |
| Forward gain, mag S21 squared | 7.23 dB | 6.85 dB |
| Stability factor K | 1.202 | 6.834 |
| Magnitude of Δ | 0.339 | 0.045 |
| Unilateral figure U | 0.0544 | 0.0022 |
| Nature | Bilateral, closer to instability | Almost unilateral, very stable |

- System (a) has strong **feedback** (large S12) and a large input mismatch. Matching adds more gain, but the design is more sensitive and closer to the stability limit (K near 1).
- System (b) is nearly **unilateral** (U ≈ 0.0022), so its input and output can be designed separately with little error. It is strongly stable (K ≫ 1).

#### Stability of system (a)

```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))
```

```text
S11·S22 = 0.1105∠−26.0° = 0.0993 − j0.0484
S12·S21 = 0.2760∠90.0° = 0.0000 + j0.2760
Δ = S11·S22 − S12·S21 = 0.0993 − j0.3244
  = 0.3393∠−73.0°,   |Δ|² = 0.1151

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.4225 − 0.0289 + 0.1151) / (2 × 0.2760)
  = 0.6637 / 0.5520
  = 1.202
```

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = **1.317** (> 1, unconditionally stable).

K = 1.202 > 1 and |Δ| = 0.339 < 1, so system (a) is **unconditionally stable** (though with a small margin).

#### Maximum gains of system (a)

**Unilateral mode** (ΓS = S11* = 0.65∠−146°, ΓL = S22* = 0.17∠172°):

```text
Unilateral (S12 = 0, ΓS = S11*, ΓL = S22*):
GS = 1/(1 − |S11|²) = 1/(1 − 0.4225)
   = 1.7316  (2.38 dB)
G0 = |S21|² = 2.300² = 5.2900  (7.23 dB)
GL = 1/(1 − |S22|²) = 1/(1 − 0.0289)
   = 1.0298  (0.13 dB)
GTU,max = GS·G0·GL = 9.433  (9.75 dB)
```

**Bilateral mode** (simultaneous conjugate match):

```text
Bilateral (S12 kept), simultaneous conjugate match:
MAG = Gmax = (|S21|/|S12|)·(K − √(K² − 1))
     = (2.300/0.120) × (1.202 − √(1.202² − 1))
     = 19.167 × 0.5347
     = 10.249  (10.11 dB)
```

```text
B1 = 1 + |S11|² − |S22|² − |Δ|² = 1.2785
B2 = 1 + |S22|² − |S11|² − |Δ|² = 0.4913
C1 = S11 − Δ·S22* = 0.6121∠150.0°
C2 = S22 − Δ·S11* = 0.1624∠−88.9°
ΓS = [B1 − √(B1² − 4|C1|²)] / (2C1) = 0.7433∠−150.0°
ΓL = [B2 − √(B2² − 4|C2|²)] / (2C2) = 0.3778∠88.9°
ZS = 50(1 + ΓS)/(1 − ΓS) = 7.9 − j13.1 Ω
ZL = 50(1 + ΓL)/(1 − ΓL) = 38.0 + j33.5 Ω
```

**Error from the unilateral assumption:**

```text
U = |S11·S12·S21·S22| / [(1 − |S11|²)(1 − |S22|²)]
  = 0.0544
1/(1+U)² < GT/GTU < 1/(1−U)²
−0.46 dB < error < +0.49 dB
```

#### Comparison

| Mode | Max gain | dB |
|---|---|---|
| Unilateral GTU,max | 9.433 | 9.75 dB |
| Bilateral Gmax | 10.249 | 10.11 dB |

The bilateral gain is 0.36 dB higher, because the relatively large S12 = 0.12 gives useful feedback when both ports are matched together. Most of the gain from matching comes from the input side (G_S = 2.38 dB), since |S11| is large.

(For system (b), for comparison: GTU,max = 7.30 dB and Gmax = 7.32 dB, which are almost equal because S12 is very small.)

**Answer (a): K = 1.202, |Δ| = 0.339 → unconditionally stable; GTU,max = 9.75 dB, Gmax (bilateral) = 10.11 dB.**

## Check yourself

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

1. What is the main difference between set (a) and set (b)?
   A. (a) has much less forward gain
   B. (a) has a poor input match and strong feedback (large S12); (b) is nearly unilateral and very stable
   C. (b) has more reverse transmission
   D. They differ only in the S21 angle
   Answer: B. Set (a) has |S11| = 0.65 and |S12| = 0.12 (K = 1.202), whereas set (b) has |S12| = 0.03 and U = 0.0022, so it is almost unilateral. (Key point: Difference between sets)

2. What is the stability result for set (a)?
   A. K = 1.202, |Δ| = 0.339: unconditionally stable with a small margin
   B. K = 0.8: unstable
   C. K = 6.8: very stable
   D. |Δ| > 1: unstable
   Answer: A. K > 1 and |Δ| < 1, so it is unconditionally stable, though K near 1 gives little margin. (Key point: Stability of (a))

3. For set (a), what are the unilateral and bilateral maximum gains?
   A. GTU,max = 10.11 dB; Gmax = 9.75 dB
   B. GTU,max = 9.75 dB; Gmax = 10.11 dB
   C. Both are 7.23 dB
   D. GTU,max = 7.30 dB; Gmax = 7.32 dB
   Answer: B. The unilateral value is 9.433 (9.75 dB) and the bilateral value is 10.249 (10.11 dB), 0.36 dB higher. (Key point: Gain comparison)

4. What does the unilateral figure of merit U tell you?
   A. The stability of the device
   B. The error band in dB from assuming S12 = 0
   C. The bandwidth
   D. The noise figure
   Answer: B. U bounds GT/GTU between 1/(1+U)^2 and 1/(1-U)^2. For (a), U = 0.0544 gives about -0.46 to +0.49 dB. (Key point: Unilateral figure of merit)

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