Stability of transistor at 6 GHz

Stability is first tested with the Rollett factor K and |Δ| (and μ). Since the device turns out to be potentially unstable, the input and output stability circles are found to show which source and load terminations must be avoided.

Given (6 GHz): S11 = 0.894∠−60.6°, S12 = 0.020∠62.4°, S21 = 3.122∠123.6°, S22 = 0.781∠−27.6°.

Step 1: Rollett test

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 (although |Δ| = 0.696 < 1), and μ = 0.863 < 1. So the transistor is potentially (conditionally) unstable at 6 GHz. Some passive source or load impedances will make it oscillate.

Step 2: Stability circles

Output (load) stability circle, |Γin| = 1:
CL = (S22 − Δ·S11*)* / (|S22|² − |Δ|²)
   = 0.1702∠46.7° / (0.6100 − 0.4850)
   = 1.3626∠46.7°
RL = |S12·S21| / ||S22|² − |Δ|²|
   = 0.0624 / 0.1249 = 0.500

Input (source) stability circle, |Γout| = 1:
CS = (S11 − Δ·S22*)* / (|S11|² − |Δ|²)
   = 0.3556∠68.5° / (0.7992 − 0.4850)
   = 1.1317∠68.5°
RS = |S12·S21| / ||S11|² − |Δ|²|
   = 0.0624 / 0.3142 = 0.199

Step 3: Stable regions

• Load plane: |CL| = 1.363 > RL = 0.500, so the circle does not enclose the chart centre. Since |S11| = 0.894 < 1, ΓL = 0 is stable. So the region outside the output stability circle (inside the Smith chart) is stable, and the part of the chart inside the circle (around 47°, near the edge) is unstable.
• Source plane: |CS| = 1.132 > RS = 0.199, and |S22| = 0.781 < 1, so ΓS = 0 is stable. The small area inside the input stability circle (around 68°, near the edge of the chart) is unstable.

Conclusion

The transistor is conditionally stable. A design must keep Γ_S outside the circle (C_S = 1.132∠68.5°, R_S = 0.199) and Γ_L outside the circle (C_L = 1.363∠46.7°, R_L = 0.500), or add resistive loading to stabilise it. For example, the conjugate points ΓS = S11* = 0.894∠60.6° and ΓL = S22* = 0.781∠27.6° lie at distances 0.275 and 0.675 from the centres, which is more than R_S and R_L, so they are in the stable regions.

Answer: Δ = 0.696∠−83.1°, K = 0.607 < 1 → potentially unstable; CL = 1.363∠46.7°, RL = 0.500; CS = 1.132∠68.5°, RS = 0.199.