Amplifier Stability: How to Read the Rollett K Factor
An amplifier is unconditionally stable at a frequency when K is greater than 1 and |delta| is less than 1. With S12 = 0.1 and S21 = 2, K is 1.256 and the amplifier passes; with S12 = 0.15 and S21 = 4, K falls to 0.519 and it does not.
For practitioners
What stability means for an amplifier
An amplifier is stable when it does not oscillate on its own. It should only amplify the signal you give it. An unstable amplifier can turn some of its output back into input, in phase, and then it makes its own signal. The result is a spur, a damaged load, or an amplifier that no longer meets its gain and linearity targets.
You can judge this from the S-parameters of the device. The two-port data already contains the feedback path (S12), the gain (S21), and how well each port is matched (S11 and S22). The Rollett K factor condenses these into one number per frequency.
The delta term
The first quantity to compute is the determinant of the S-parameter matrix:
With measured data the S-parameters are complex, so delta is a complex number. Use its magnitude |delta| in the tests below. Delta has two parts. The product S11 * S22 comes from the two port reflections. The product S12 * S21 is the loop formed by the forward gain and the reverse leakage.
The Rollett K factor
The Rollett stability factor is:
Read the formula in two halves.
- Numerator: 1 minus the squared input and output reflection magnitudes, plus |delta| squared. Well-matched ports (small |S11| and |S22|) make the numerator larger.
- Denominator: twice the magnitude of S12 * S21, the loop gain of the internal feedback path. A larger loop makes K smaller.
A small S12 means little feedback from output to input, which makes a high K easier to reach. This is why devices with good reverse isolation are easier to stabilize, and why high-gain devices with leaky feedback paths are harder.
The two conditions for unconditional stability
An amplifier is unconditionally stable at a frequency when both of these hold:
- K is greater than 1.
- |delta| is less than 1.
When both are true, the amplifier is stable for any source and load. No passive termination you connect can make it oscillate at that frequency. Both conditions are required. K above 1 alone is not enough, and |delta| below 1 alone is not enough.
If either one fails, the amplifier is potentially unstable. That does not mean it oscillates. It means that some source and load impedances can make it oscillate. You then have two choices: keep the terminations away from the unstable region, or add loss or feedback to push K above 1.
Worked example: one stable and one unstable device
Take two devices with all-real S-parameters, to keep the arithmetic visible. Both have S11 = 0.5 and S22 = 0.5.
Device A: S12 = 0.1 and S21 = 2.
Delta = 0.5 * 0.5 - 0.1 * 2 = 0.25 - 0.20 = 0.05.
The numerator of K is 1 - 0.25 - 0.25 + 0.0025 = 0.5025. The denominator is 2 * 0.2 = 0.4.
K = 0.5025 / 0.4 = 1.256.
K is above 1 and |delta| = 0.05 is below 1, so device A is unconditionally stable.
Device B: S12 = 0.15 and S21 = 4.
Delta = 0.25 - 0.15 * 4 = 0.25 - 0.60 = -0.35.
The numerator is 1 - 0.25 - 0.25 + 0.1225 = 0.6225. The denominator is 2 * 0.6 = 1.2.
K = 0.6225 / 1.2 = 0.519.
|delta| = 0.35 is below 1, but K = 0.519 is below 1, so device B is potentially unstable for some source and load.
Compare the two. The reflections are the same. What changed is the loop S12 * S21, which grew from 0.2 to 0.6. More reverse leakage and more gain mean a bigger feedback loop, and the denominator of K grew with it. Device B also shows why you must check both conditions: |delta| alone would have passed it.
A device that passes at one frequency says nothing about the others. Compute delta and K at every frequency point in the S-parameter file, and look at the whole curve, not one value.
Why stability is checked over a wide frequency range
Stability has to be checked at all frequencies where the transistor has gain, not only in the working band. An amplifier can oscillate out of band. Your match and filtering may be tuned for the working band, while at some other frequency the terminations look quite different and the device still has enough gain to oscillate.
In practice this means you need S-parameter data that extends well beyond the band of use. Plot K and |delta| against frequency. Wherever K drops below 1 or |delta| rises above 1 and the device still has gain, you have a potential problem. Load the data with a viewer or script first; the page on opening an S2P file shows how. Reflection terms such as S11 and S22 are explained further in return loss and insertion loss.
In plain words · for students and beginners
Amplifier stability in plain words
An everyday analogy
Think of a microphone held near its own loudspeaker. A small sound goes in, comes out louder, travels through the air back to the microphone, and goes round again. If each trip around makes the sound bigger, you get a loud squeal. Nobody made that sound. The loop did.
An amplifier has the same risk. A little of the output leaks back to the input through S12, and the gain is S21. When the returning part is big enough, the amplifier squeals electrically. That is oscillation.
Where the analogy breaks down: a speaker and a microphone sit in one fixed room. An amplifier's loop also depends on what you connect to it (the source and the load), and on the frequency. The same amplifier can be calm in one setup and unstable in another. K tells you whether any setup can cause trouble.
The same example in simple steps
Use device A: S11 = 0.5, S22 = 0.5, S12 = 0.1, S21 = 2.
- Work out delta: 0.5 * 0.5 = 0.25. Then 0.1 * 2 = 0.20. Subtract: 0.25 - 0.20 = 0.05.
- Work out the top of K: 1 - 0.25 - 0.25 + 0.0025 = 0.5025.
- Work out the bottom of K: 2 * 0.1 * 2 = 0.4.
- Divide: 0.5025 / 0.4 = 1.256.
- Check the rules. K is above 1, and delta (0.05) is below 1. The amplifier is stable.
Now change S12 to 0.15 and S21 to 4. Delta becomes -0.35, and K becomes 0.519. K is below 1, so the amplifier may be unstable for some source and load.
Mistakes beginners make
- Checking only K. You need both rules: K above 1 and the size of delta below 1.
- Checking only the working band. The amplifier can oscillate at a frequency you never planned to use, as long as it still has gain there.
- Reading "K below 1" as "it oscillates". It means it can, with the wrong source or load. It is a warning, not a verdict.
- Forgetting that delta is a complex number. Use its size (magnitude), not just its real part.
Remember
- Stable at a frequency means K is above 1 and the size of delta is below 1.
- A small S12 (little feedback from output to input) makes a high K easier to reach.
- Check every frequency where the transistor has gain, not only the band you use.
Related pages on RF View
Frequently asked questions
What K factor value means an amplifier is stable?
K must be greater than 1, and |delta| must also be less than 1. Both conditions together mean the amplifier is unconditionally stable at that frequency, for any source and load.
What is the delta term in stability analysis?
Delta is S11 * S22 - S12 * S21. Its magnitude must be below 1 for unconditional stability, and it also enters the numerator of the K factor.
Does K less than 1 mean the amplifier will oscillate?
No. It means the amplifier is potentially unstable: some source and load impedances can cause oscillation, while others may not. You can then check which terminations are safe or add stabilizing loss.
Why check stability outside the working band?
An amplifier can oscillate at any frequency where the transistor has gain. If you check only the working band, an out-of-band oscillation goes unnoticed.