Stability with flow diagram, S11 = 0.6∠−140°

A transistor is unconditionally stable if |Γin| < 1 and |Γout| < 1 for every passive source and load. This is tested with the Rollett factor K and |Δ| (or the single μ-test), following the flow below.

Flow diagram for stability analysis

      [ Given S11, S12, S21, S22 ]
                  |
      [ Δ = S11·S22 − S12·S21 ]
                  |
 [ K = (1−|S11|²−|S22|²+|Δ|²)/(2|S12S21|) ]
                  |
         < K > 1 and |Δ| < 1 ? >
          | yes            | no
 [ Unconditionally ]  [ Potentially unstable:   ]
 [ stable: any     ]  [ draw input/output       ]
 [ ΓS, ΓL usable   ]  [ stability circles (CS,  ]
          |           [ RS, CL, RL); choose ΓS, ]
          |           [ ΓL in stable region     ]
          |___________________|
                  |
       [ Proceed to gain design ]

Calculation

Given: S11 = 0.6∠−140°, S12 = 0.03∠62°, S21 = 2.40∠54°, S22 = 0.70∠−58°.

S11·S22 = 0.4200∠162.0° = −0.3994 + j0.1298
S12·S21 = 0.0720∠116.0° = −0.0316 + j0.0647
Δ = S11·S22 − S12·S21 = −0.3679 + j0.0651
  = 0.3736∠170.0°,   |Δ|² = 0.1396

K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12·S21|)
  = (1 − 0.3600 − 0.4900 + 0.1396) / (2 × 0.0720)
  = 0.2896 / 0.1440
  = 2.011

Check with the single test μ = (1 − |S11|²) / (|S22 − Δ·S11*| + |S12·S21|) = 1.161 (> 1, unconditionally stable).

Result

• K = 2.011 > 1
• |Δ| = 0.374 < 1
• μ = 1.161 > 1

So the transistor is unconditionally stable: no passive source or load can make it oscillate.

Supporting check with stability circles:

CL = (S22 − Δ·S11*)* / (|S22|² − |Δ|²) = 1.3669∠61.7°
RL = |S12·S21| / ||S22|² − |Δ|²|         = 0.2055
CS = (S11 − Δ·S22*)* / (|S11|² − |Δ|²) = 1.5558∠146.1°
RS = |S12·S21| / ||S11|² − |Δ|²|         = 0.3266

For both circles |C| − R > 1 (1.161 and 1.229), so the circles lie completely outside the Smith chart. The whole chart is therefore the stable region, which agrees with K > 1 and |Δ| < 1.

Since the device is stable, it can be conjugate-matched. For reference, the maximum available gain is Gmax = (|S21|/|S12|)(K − √(K² − 1)) = 21.30 (13.28 dB).

Answer: Δ = 0.374∠170.0°, K = 2.011, |Δ| = 0.374 → unconditionally stable.