What is the stability factor K and why does it matter when selecting an amplifier for my design?
Amplifier Stability Analysis
Oscillation occurs when the round-trip gain around a feedback loop exceeds unity. In an amplifier, the S12 parameter (reverse transmission) creates a feedback path from output to input. If the source and load impedances create sufficient positive feedback through S12, the amplifier oscillates. The stability factor K quantifies the margin against this condition.
K is calculated from the S-parameters: K = (1 - |S11|² - |S22|² + |Δ|²) / (2|S12||S21|). Both conditions K > 1 AND |Δ| < 1 must be satisfied simultaneously for unconditional stability. The geometric stability factor μ (introduced by Edwards and Sinsky) combines both conditions into a single parameter: μ > 1 guarantees unconditional stability.
Stability must be verified at all frequencies, not just the operating band. Many amplifiers are stable in-band but potentially unstable at low frequencies (where gain is very high) or at frequencies near the transistor's fT. A complete stability analysis checks K from DC to well beyond fT, including frequencies where no S-parameter data may be available.
Δ = S₁₁S₂₂ - S₁₂S₂₁
Unconditionally stable: K > 1 AND |Δ| < 1
μ factor: μ = (1-|S₁₁|²) / (|S₂₂-Δ·S₁₁*|+|S₁₂S₂₁|)
Unconditionally stable: μ > 1
Frequently Asked Questions
What if K < 1 at some frequencies?
Add stabilization: resistive loading at the input or output (reduces gain but increases K), or negative feedback (series or shunt feedback resistors). Place the stabilization elements to affect the unstable frequencies without degrading in-band performance. Common approach: series resistor in the gate/base bias feed for low-frequency stabilization.
Is unconditional stability always needed?
For production amplifiers and gain blocks: yes. For laboratory prototypes with controlled impedance environment: conditional stability may be acceptable if the source and load impedances stay within the stable region. But conditional stability is risky and not recommended for deployed systems.
Does stability change with temperature and bias?
Yes. S-parameters change with temperature and bias point, and K changes accordingly. A marginally stable amplifier at room temperature may oscillate at cold temperature (where gain increases) or at different bias voltages. Stability must be verified across the full operating temperature and bias range.