What are K-factor and mu-test?

The Rollett stability factor K is the classical two-port stability criterion: K = (1 minus |S11|^2 minus |S22|^2 plus |Delta|^2) / (2|S12||S21|), where Delta = S11 times S22 minus S12 times S21. The amplifier is unconditionally stable when K greater than 1 AND |Delta| less than 1 (both conditions must be satisfied). The mu-test (Edwards-Sinsky) provides a single, sufficient condition: mu = (1 minus |S11|^2) / (|S22 minus Delta times S11*| plus |S12 times S21|). If mu greater than 1, the device is unconditionally stable. mu greater than 1 is equivalent to K greater than 1 AND |Delta| less than 1, but has the advantage that larger mu indicates greater stability margin (K does not have this property). Both tests must be evaluated at every frequency in the operating range and beyond. A device that is stable in-band may be unstable out-of-band, especially at very low frequencies (where gain is highest) or very high frequencies (where package parasitics create feedback). The stability analysis must cover DC to well beyond f_max.

What are stability circles?

When K is less than 1 (conditionally stable), stability circles on the Smith chart show the boundary between load (or source) impedances that cause oscillation and those that do not. The output stability circle defines the locus of load impedances (Gamma_L) for which the input reflection coefficient equals unity (|Gamma_in| = 1). Inside or outside this circle (determined by testing a known stable point, such as Z0), the device is unstable. Center: c_L = (S22 minus Delta times S11*)* / (|S22|^2 minus |Delta|^2). Radius: r_L = |S12 times S21| / (|S22|^2 minus |Delta|^2). Similarly, input stability circles define source impedances that cause oscillation at the output. The designer must ensure that the actual source and load impedances (including all matching network variations over frequency, temperature, and manufacturing tolerance) stay in the stable region. This is why unconditional stability (K greater than 1) is strongly preferred: it eliminates the need to track impedance variations across all conditions.

How do you stabilize an amplifier?

Resistive loading: add a series or shunt resistor at the input or output to reduce gain and increase K. A shunt resistor at the input (before the matching network) is most common for LNAs. The resistor reduces the maximum available gain but ensures K greater than 1 at all frequencies. Typical LNA stabilization: 50 to 200 ohms shunt to ground at the input. This degrades noise figure by 0.1 to 0.5 dB but guarantees stability. Series feedback: a small resistor (5 to 20 ohms) in the source (FET) or emitter (BJT) lead provides negative feedback that stabilizes the device. This also improves linearity. The emitter degeneration inductance in an LNA serves a similar purpose while also improving noise match. Neutralization: adds a feedback network that cancels the internal feedback (S12) of the device. Used in balanced amplifiers and differential circuits. Effective but narrow-band and sensitive to component tolerances. Out-of-band stabilization: use frequency-selective networks (RC, RL) that provide resistive loading only at frequencies where the device is unstable (usually low frequencies). This preserves in-band gain while ensuring broadband stability.

Amplifier Design

Stability

/stuh-bil-ih-tee/

Will it oscillate? K = (1−|S₁₁|²−|S₂₂|²+|Δ|²)/(2|S₁₂||S₂₁|). Unconditional: K>1 AND |Δ|<1. μ-test: single condition, μ>1. Conditional: K<1, stability circles show safe load region. Stabilize: resistive loading (shunt R), series feedback (emitter/source R), neutralization (cancel S₁₂). Check DC to >f_max. Out-of-band instability = most common failure.

K: >1

μ: >1

|Δ|: <1

Understanding Stability

Stability analysis is the first step in any amplifier design. Before worrying about gain, noise figure, or output power, the designer must ensure the amplifier will not oscillate. An oscillating amplifier is worse than no amplifier at all: it generates spurious signals that interfere with the entire system and can damage components.

The most dangerous instabilities are out-of-band: an amplifier designed for 10 GHz may be perfectly stable at 10 GHz (K>1) but oscillate at 100 MHz where the transistor gain is much higher and package parasitics create feedback paths. Always analyze stability from DC to well beyond the operating frequency.

Stability Equations

Rollett K-factor:
K = (1−|S₁₁|²−|S₂₂|²+|Δ|²)/(2|S₁₂||S₂₁|)
Δ = S₁₁S₂₂−S₁₂S₂₁
Unconditional: K>1 AND |Δ|<1

μ-test (Edwards-Sinsky):
μ = (1−|S₁₁|²)/(|S₂₂−ΔS₁₁*|+|S₁₂S₂₁|)
μ>1: unconditionally stable
Larger μ = more margin

Stability circle (load):
c_L = (S₂₂−ΔS₁₁*)*/(|S₂₂|²−|Δ|²)
r_L = |S₁₂S₂₁|/||S₂₂|²−|Δ|²|

Stabilization Methods

Method	Type	Gain Cost	NF Cost	Best For
Shunt R (input)	Resistive	1-3 dB	0.1-0.5 dB	LNA
Series R (output)	Resistive	0.5-2 dB	None	Driver amp
Source degeneration	Feedback	1-2 dB	~0 (inductive)	LNA
Neutralization	Cancellation	~0 dB	~0 dB	Diff amp
RC low-pass	Frequency sel	~0 in-band	~0 in-band	Out-of-band

Common Questions

Frequently Asked Questions

K and μ?

K = Rollett: needs TWO conditions (K>1 AND |Δ|<1). μ = Edwards-Sinsky: single condition (μ>1), larger = more stable. Both from S-parameters. Check ALL frequencies DC to >f_max. K=0.8 at 100 MHz = will oscillate even if K=5 at 10 GHz. Most dangerous: low-freq (high gain) and high-freq (parasitic feedback).

Stability circles?

K<1 (conditional): circles on Smith chart divide stable/unstable load regions. Center and radius from S-params. Test known point (Z₀) to determine which side is stable. Design must keep all load impedances (including manufacturing spread) in stable region. Unconditional (K>1): entire Smith chart is safe. Strongly preferred.

Stabilize?

Shunt R at input: reduces gain, degrades NF 0.1-0.5 dB. Series source R (5-20Ω): negative feedback, improves linearity too. Source inductance: optimal NF + stability simultaneously (LNA standard). Neutralization: cancels S₁₂, narrowband. RC lowpass: loads only at low freq (out-of-band stabilization).

Related Terms

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Amplifier Design

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