Stability and gains for S11 = 0.894∠−60.6°

Stability is tested with K and |Δ|. Then the maximum gain of the unilateral model (S12 = 0) is compared with what is possible in the bilateral model.

Given: S11 = 0.894∠−60.6°, S21 = 3.122∠123.6°, S12 = 0.020∠62.4°, S22 = 0.781∠−27.6°.

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

S11·S22 = 0.6982∠−88.2° = 0.0219 − j0.6979
S12·S21 = 0.0624∠−174.0° = −0.0621 − j0.0065
Δ = S11·S22 − S12·S21 = 0.0840 − j0.6913
  = 0.6964∠−83.1°,   |Δ|² = 0.4850

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.7992 − 0.6100 + 0.4850) / (2 × 0.0624)
  = 0.0758 / 0.1249
  = 0.607

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = 0.863 (< 1, potentially unstable).

K = 0.607 < 1, so the transistor is potentially unstable (|Δ| = 0.696 < 1, but both conditions are needed). Stability circles:

CL = 1.363∠46.7°,  RL = 0.500   (output)
CS = 1.132∠68.5°,  RS = 0.199   (input)

Since |S11|, |S22| < 1, the chart centre is stable, and the unstable regions are the small areas inside these circles near the chart edge.

Unilateral mode

Unilateral (S12 = 0, ΓS = S11*, ΓL = S22*):
GS = 1/(1 − |S11|²) = 1/(1 − 0.7992)
   = 4.9810  (6.97 dB)
G0 = |S21|² = 3.122² = 9.7469  (9.89 dB)
GL = 1/(1 − |S22|²) = 1/(1 − 0.6100)
   = 2.5638  (4.09 dB)
GTU,max = GS·G0·GL = 124.472  (20.95 dB)

Check that these terminations are stable: |S11* − CS| = 0.275 > RS = 0.199 and |S22* − CL| = 0.675 > RL = 0.500. Both points are outside the stability circles, so they are safe.

U = |S11·S12·S21·S22| / [(1 − |S11|²)(1 − |S22|²)]
  = 0.5567
1/(1+U)² < GT/GTU < 1/(1−U)²
−3.84 dB < error < +7.07 dB

U = 0.557 is large (|S11| and |S22| are close to 1), so the unilateral value can be wrong by several dB. With the real S12, these terminations give |Γin| = 0.912, |Γout| = 0.848 and an actual transducer gain G_T = 101.3 (20.06 dB).

Bilateral mode

Since K < 1, the simultaneous conjugate match does not exist (the formula K − √(K² − 1) becomes complex), so Gmax (MAG) is not defined. The upper figure used instead is the maximum stable gain:

MSG = |S21| / |S12| = 3.122 / 0.020 = 156.1
    = 21.93 dB

MSG is the gain obtained when the device is just brought to K = 1 (for example by resistive loading). In practice it is approached but not reached, and a design must keep ΓS and ΓL away from the stability circles.

Comparison

Answer: K = 0.607 < 1 → potentially unstable; GTU,max = 20.95 dB; bilateral MAG undefined, MSG = 21.93 dB.