Stability Circle
Mapping the Safe Zone on the Smith Chart
A two-port device (transistor, MMIC) characterized by its S-parameters has two stability circles: one in the source impedance plane (ΓS) and one in the load impedance plane (ΓL). Each circle's center and radius are computed directly from the S-parameters, and the region on one side of the circle represents impedances that will cause oscillation.
If both stability circles lie entirely outside the Smith Chart (|Γ| ≤ 1 region) at a given frequency, the device is unconditionally stable: no passive source or load impedance can trigger oscillation. If either circle intersects the Smith Chart, the device is conditionally stable, and the designer must ensure the matching network avoids the unstable region.
Computing the Circles from S-Parameters
Center: CL = (S22 − Δ·S11*) / (|S22|² − |Δ|²)
Radius: rL = |S12·S21| / (|S22|² − |Δ|²)
Source stability circle (plotted in ΓS plane):
Center: CS = (S11 − Δ·S22*) / (|S11|² − |Δ|²)
Radius: rS = |S12·S21| / (|S11|² − |Δ|²)
Where: Δ = S11·S22 − S12·S21
( * denotes complex conjugate)
Debugging an Oscillating BFP740 LNA at 3.5 GHz
An Infineon BFP740 SiGe HBT biased at VCE = 3 V, IC = 20 mA has these S-parameters at 900 MHz (the parasitic oscillation frequency):
| Parameter | Magnitude | Angle | Note |
|---|---|---|---|
| S11 | 0.62 | −58° | Input match at 900 MHz |
| S21 | 8.4 | 115° | Very high gain at this frequency |
| S12 | 0.045 | 52° | Reverse isolation |
| S22 | 0.38 | −32° | Output match at 900 MHz |
K = (1 − |S11|² − |S22|² + |Δ|²) / (2·|S12·S21|)
K = (1 − 0.384 − 0.144 + 0.072) / (2 × 0.378)
K = 0.544 / 0.756 = 0.72 (conditionally stable)
K < 1 confirms the BFP740 is NOT unconditionally stable at 900 MHz. The stability circles must be plotted to find which impedances trigger oscillation. Adding a 10 Ω series base resistor raises K above 1 at all frequencies below 2 GHz, eliminating the parasitic oscillation while adding only 0.15 dB of noise figure at 3.5 GHz.
K-Factor vs. Mu-Factor: Which Test to Use
| Stability Test | Formula | Unconditionally Stable When | Advantage |
|---|---|---|---|
| Rollett K-factor | K = (1−|S11|²−|S22|²+|Δ|²) / 2|S12·S21| | K > 1 AND |Δ| < 1 | Widely used, in all textbooks |
| Mu-factor (μ) | μ = (1−|S11|²) / (|S22−Δ·S11*| + |S12·S21|) | μ > 1 (single condition) | Single test, indicates stability margin |
| Geometric (B1) | B1 = 1+|S11|²−|S22|²−|Δ|² | B1 > 0 (with K>1) | Auxiliary check for K-factor |
Frequently Asked Questions
What is the difference between K-factor and mu-factor?
The Rollett K-factor needs two conditions: K > 1 AND |Δ| < 1, both satisfied simultaneously. The mu-factor combines these into a single test: μ > 1 guarantees unconditional stability. Mu-factor also indicates the degree of stability margin, which K-factor does not.
Can a conditionally stable transistor be used safely?
Yes, most RF transistors are conditionally stable at some frequencies. The stability circles identify which impedances to avoid. If your matching network stays within the stable region across all frequencies, including out-of-band, the amplifier will not oscillate. Adding stabilization resistors (5 to 15 ohms series at the gate, or shunt at the drain) provides margin against manufacturing tolerances and temperature drift.
Why might an amplifier oscillate out of band but not at the design frequency?
Many transistors have their highest gain below 1 GHz, where they are most likely to be conditionally stable. A matching network optimized for 3.5 GHz presents unpredictable impedances at 900 MHz. If those impedances fall inside the unstable region of the stability circle, the transistor oscillates at 900 MHz despite being perfectly stable at 3.5 GHz. Broadband stabilization resistors prevent this.