Auxiliary Stability Analysis
Understanding Auxiliary Stability Analysis
When a transistor is embedded in a circuit with feedback (which all transistors inherently have via S12), certain combinations of source and load impedance can cause the device to oscillate. This is not a design intent; it is an unwanted parasitic oscillation that corrupts the desired signal, radiates interference, and can destroy the transistor. Stability analysis tells the designer whether the device is safe for all impedances (unconditionally stable) or only safe for certain impedances (conditionally stable).
The analysis operates on the device's two-port S-parameters measured at a specific bias point and frequency. The key insight is that oscillation occurs when the magnitude of the input or output reflection coefficient exceeds unity, meaning the device produces more reflected power than incident power. This is the condition for negative resistance, and it is the mechanism behind every oscillator circuit. The stability tests check whether any passive termination can trigger this condition.
Stability Factor Equations
K = (1 − |S11|² − |S22|² + |Δ|²) / (2|S12||S21|)
Determinant:
Δ = S11S22 − S12S21
Unconditional Stability Condition (Rollett):
K > 1 AND |Δ| < 1
Edwards-Sinsky Mu Parameter:
μ = (1 − |S11|²) / (|S22 − ΔS11*| + |S12S21|)
Unconditional Stability (μ-test):
μ > 1 (single condition, no auxiliary requirement)
The μ parameter is preferred because it combines both Rollett conditions into one test and provides a quantitative measure of stability margin: larger μ means more stable.
Stability Classification
| Condition | K Value | μ Value | Implication | Design Action |
|---|---|---|---|---|
| Unconditionally Stable | K > 1, |Δ| < 1 | μ > 1 | Stable for all passive ZS, ZL | Match freely for gain/noise |
| Conditionally Stable | K < 1 or |Δ| ≥ 1 | μ < 1 | Oscillates for some ZS or ZL | Constrain match to stable region; add resistive stabilization |
| Potentially Unstable | K > 1, |Δ| ≥ 1 | μ < 1 | K-test alone is misleading | Use μ-test; this case is why the auxiliary condition exists |
Frequently Asked Questions
What is the difference between the K-factor and mu stability tests?
Rollett's K-factor requires two simultaneous conditions: K > 1 AND |Δ| < 1. The mu parameter folds both into a single number: μ > 1 guarantees unconditional stability. The mu test is preferred in modern EDA tools like Keysight ADS and AWR Microwave Office because it eliminates the ambiguity of the auxiliary condition. A device can have K > 1 but still oscillate if |Δ| ≥ 1, which is exactly the failure mode the auxiliary condition catches.
What happens when an RF amplifier is conditionally stable?
The device is stable for some source and load impedances but will oscillate for others. Stability circles plotted on the Smith chart delineate the boundary between stable and unstable regions. The designer must ensure the matching networks present impedances within the stable region across the full operating bandwidth. Out-of-band stability is equally critical: matching networks often become reactive at frequencies outside the design band, potentially presenting impedances in the unstable region.
Why must stability be checked at all frequencies, not just the design band?
Transistors have higher gain at lower frequencies due to the typical 6 dB/octave roll-off. A device that is unconditionally stable at 28 GHz may be conditionally stable at 2 GHz where its gain is 20 dB higher. If the bias network or matching network presents a reactive impedance at low frequencies that enters the unstable region, the amplifier will oscillate at that frequency. Series or shunt resistors are commonly added to reduce out-of-band gain and guarantee stability from DC through the device's fT.