What does K > 1 mean for amplifier design?

When K exceeds 1 and |delta| is less than 1, the amplifier is unconditionally stable, meaning it will not oscillate regardless of what passive source or load impedance is presented to its ports. This is the safest design condition because the amplifier will remain stable even if the antenna VSWR changes dramatically (for example, due to nearby objects or ice on the antenna), or if the source impedance varies (different filter characteristics, cable length changes). When K is less than 1 (potentially unstable), certain impedances within the stability circles on the Smith chart will cause oscillation. The designer must then either stabilize the device (using resistive loading to increase K above 1 at all frequencies) or carefully constrain the source and load impedances to remain outside the unstable regions. Resistive stabilization trades gain for stability.

K-factor vs. mu-factor: which is better?

The mu-factor (Edwards-Sinsky) is a single-parameter stability test that eliminates the need for the auxiliary condition (|delta| less than 1 or B1 greater than 0): mu = (1 minus |S11|^2) / (|S22 minus delta*S11*| plus |S12*S21|). Unconditional stability requires only mu greater than 1. The mu-factor has two advantages: it is a single test (no auxiliary condition to check separately), and its magnitude indicates the distance from the nearest instability point on the Smith chart, providing a quantitative measure of stability margin. Higher mu means more stable. K-factor does not provide this geometric interpretation; K = 5 is not necessarily more stable than K = 2 because the auxiliary condition (|delta|) also matters. For these reasons, mu-factor is preferred in modern amplifier design tools. However, K-factor remains widely used in literature and datasheets due to historical precedent.

Amplifier Stability

K-Factor

Q: How do stability circles work?

When K is less than 1 at a particular frequency, not all source and load impedances are safe. Stability circles, plotted on the Smith chart, define the boundary between stable and unstable impedance regions. The source stability circle shows which source impedances cause instability (looking into the output); the load stability circle shows which load impedances cause instability (looking into the input). The circles are defined by center and radius calculated from the S-parameters. Source stability circle: center = (S11 minus delta*S22*) / (|S11|^2 minus |delta|^2), radius = |S12*S21| / ||S11|^2 minus |delta|^2|. The stable region is either inside or outside the circle, determined by checking a known point (typically the center of the Smith chart, Z0). If the amplifier is stable at Z0, then the stable region is the side of the circle containing the center.

/kay fak-ter/ — Rollett Stability Factor

Determines unconditional stability of a two-port amplifier: K = (1−|S₁₁|²−|S₂₂|²+|Δ|²) / (2|S₁₂||S₂₁|), where Δ = S₁₁S₂₂−S₁₂S₂₁. Unconditionally stable when K>1 AND |Δ|<1. If K<1: certain source/load impedances cause oscillation. Stability circles on Smith chart identify unstable regions. Alternative: μ-factor (>1 = stable, no auxiliary condition needed).

Stable: K > 1

Auxiliary: |Δ| < 1

Alt: μ > 1

Understanding K-Factor Stability

Stability analysis is the first step in any amplifier design. Before optimizing for gain, noise figure, or power, the designer must ensure the amplifier will not oscillate at any frequency, with any possible source and load impedance it might encounter. An oscillating amplifier is worse than no amplifier at all, as it generates spurious signals that interfere with all nearby receivers. K-factor analysis provides the mathematical framework for this critical assessment.

The physics behind K-factor is straightforward: an amplifier oscillates when the signal fed back through S₁₂ from output to input arrives in phase with the input and with sufficient amplitude to sustain the loop (Barkhausen criterion). Higher S₁₂ (more reverse leakage) and higher S₂₁ (more gain) increase the oscillation risk. The K-factor combines all four S-parameters into a single metric that captures this risk for all possible termination impedances simultaneously.

Stability Equations

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

μ-factor (Edwards-Sinsky):
μ = (1−|S₁₁|²) /
(|S₂₂−ΔS₁₁*| + |S₁₂S₂₁|)
Stable: μ > 1 (single test)
Higher μ = more stability margin

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

Stability Analysis Summary

Condition	K	μ	Action	Risk
Unconditionally stable	>1, \|Δ\|<1	>1	Any Z_S/Z_L OK	None
Conditionally stable	<1	<1	Check stability circles	Moderate
Potentially unstable	<1 at some f	<1 at some f	Add stabilization R	High
Resistively stabilized	>1 all f	>1 all f	Gain traded for safety	Low
Out-of-band unstable	<1 out-of-band	<1 out-of-band	Lossy stub / R at low f	High (hidden)

Common Questions

Frequently Asked Questions

What does K>1 mean?

Unconditionally stable: no oscillation for ANY passive source/load impedance. Safe even with antenna VSWR changes (ice, nearby objects). K<1: certain impedances within stability circles cause oscillation. Must either stabilize (add series/shunt R, trade gain for K>1) or constrain impedances to safe regions. Check K at ALL frequencies, not just in-band.

How do stability circles work?

When K<1: circles on Smith chart divide stable/unstable impedance regions. Load stability circle: load impedances causing instability. Source stability circle: source impedances causing instability. Calculated from S-parameters (center and radius formulas). Stable region: side of circle containing Z_0 (if stable at Z_0). Design constraint: keep matching within stable region.

K vs. μ-factor?

μ-factor: single test (μ>1), no auxiliary condition needed. Higher μ = more margin (distance from instability on Smith chart). K=5 vs K=2 does NOT tell relative stability (|Δ| also matters). μ preferred in modern tools. K remains in literature/datasheets due to history. Both computed from same S-parameters, equivalent results.

Related Terms

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