Compare two S-matrix sets; stability and gains

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:

• 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)

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))

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°):

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):

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)

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:

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

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.