S
Seed account
An amplifier that oscillates is not an amplifier, and most stability problems are found on the bench rather than in the simulator because people only checked K inside the operating band. Here is the workflow I use, with a worked example.
1. The two-port tests
With Delta = S11*S22 - S12*S21, the Rollett factor is
K = (1 - |S11|^2 - |S22|^2 + |Delta|^2) / (2 |S12 S21|)
The device is unconditionally stable (stable for any passive source and load) when K > 1 AND |Delta| < 1. Both conditions are needed.
A single-number alternative is the Edwards-Sinsky mu factor:
mu = (1 - |S11|^2) / (|S22 - Delta S11*| + |S12 S21|)
mu > 1 alone guarantees unconditional stability, and a larger mu means more margin, which K does not tell you directly.
2. Worked example
A small pHEMT at 2 GHz in its bias point: S11 = 0.8 at -60 deg, S21 = 5.0 at 110 deg, S12 = 0.05 at 40 deg, S22 = 0.5 at -40 deg.
|Delta| = 0.54, K = 0.80, mu = 0.77. Potentially unstable. Because K < 1 the maximum available gain is undefined, and the figure of merit is the maximum stable gain MSG = |S21|/|S12| = 20 dB.
3. Where exactly is it unstable? Stability circles
Load (output) circle: C_L = (S22 - Delta S11*)* / (|S22|^2 - |Delta|^2), r_L = |S12 S21| / ||S22|^2 - |Delta|^2|
Source (input) circle: C_S = (S11 - Delta S22*)* / (|S11|^2 - |Delta|^2), r_S = |S12 S21| / ||S11|^2 - |Delta|^2|
For the example: C_S = 1.63 at +72 deg with r_S = 0.72, and C_L = 5.34 at -81 deg with r_L = 6.11.
Since |S11| and |S22| are below 1, the center of the Smith chart (50 ohm) is a stable termination, so the side of each circle containing the center is the stable side. The unstable source region is the part of the source circle inside the chart: reflection magnitudes above about 0.92 around +72 deg, i.e. a high-Q inductive source. The unstable load region is a thin crescent near the chart edge. The trap: a narrowband input match built from a high-Q inductor can land right in that region at some frequency.
4. Check the whole band, not just the operating band
Transistor gain rises roughly 6 dB per octave as frequency drops, so K is usually worst well below the band (tens to hundreds of MHz) and sometimes near fT. Sweep the S-parameters from the lowest frequency your bias network passes up to the device fmax, including bias tees, decoupling and package parasitics. Out-of-band terminations are whatever your bias network happens to present, often close to a short or an open, i.e. right on the edge of the chart.
5. Stabilizing
Resistive loading is the reliable fix, at the cost of some gain and, on the input side, noise.
- Series resistor at the gate/base: in the example, 10 ohm raises K to 1.10, 15 ohm to 1.25 (|Delta| = 0.48), and MAG becomes 17.0 dB, i.e. we traded 3 dB of potential gain for unconditional stability at this frequency. On an LNA this resistor adds noise directly, so keep it small or move it behind a series inductor where possible.
- Shunt resistor at the drain/collector: interesting result for this device, 300 ohm alone only reaches K = 0.89, because the problem here is mostly on the input side. Do not assume one standard fix works for every device.
- Frequency-selective loading: put the resistor in series with a capacitor (or parallel with an inductor) so it only loads the circuit where K is poor, typically at low frequency, and leaves the in-band noise and gain almost untouched.
- Source/emitter inductance improves noise and input match together, but too much of it creates negative resistance at higher frequencies. Check K up to fmax after adding it.
- Bias feeds: a resistor in series with the gate bias choke (tens of ohm) kills low-frequency oscillation that the RF analysis never sees.
6. Remember what K does not cover
K and mu assume a linear two-port. Multistage amplifiers need each stage checked plus a loop check (e.g. a normalized determinant function or a probe-based loop-gain analysis), because feedback through shared supply lines and housing resonances creates paths the per-stage S-parameters cannot show. Large-signal parametric oscillation is another separate check.
Rule I use: mu > 1.05 from the lowest bias-network frequency up to fmax, with the actual layout parasitics included, before any board is ordered.
1. The two-port tests
With Delta = S11*S22 - S12*S21, the Rollett factor is
K = (1 - |S11|^2 - |S22|^2 + |Delta|^2) / (2 |S12 S21|)
The device is unconditionally stable (stable for any passive source and load) when K > 1 AND |Delta| < 1. Both conditions are needed.
A single-number alternative is the Edwards-Sinsky mu factor:
mu = (1 - |S11|^2) / (|S22 - Delta S11*| + |S12 S21|)
mu > 1 alone guarantees unconditional stability, and a larger mu means more margin, which K does not tell you directly.
2. Worked example
A small pHEMT at 2 GHz in its bias point: S11 = 0.8 at -60 deg, S21 = 5.0 at 110 deg, S12 = 0.05 at 40 deg, S22 = 0.5 at -40 deg.
|Delta| = 0.54, K = 0.80, mu = 0.77. Potentially unstable. Because K < 1 the maximum available gain is undefined, and the figure of merit is the maximum stable gain MSG = |S21|/|S12| = 20 dB.
3. Where exactly is it unstable? Stability circles
Load (output) circle: C_L = (S22 - Delta S11*)* / (|S22|^2 - |Delta|^2), r_L = |S12 S21| / ||S22|^2 - |Delta|^2|
Source (input) circle: C_S = (S11 - Delta S22*)* / (|S11|^2 - |Delta|^2), r_S = |S12 S21| / ||S11|^2 - |Delta|^2|
For the example: C_S = 1.63 at +72 deg with r_S = 0.72, and C_L = 5.34 at -81 deg with r_L = 6.11.
Since |S11| and |S22| are below 1, the center of the Smith chart (50 ohm) is a stable termination, so the side of each circle containing the center is the stable side. The unstable source region is the part of the source circle inside the chart: reflection magnitudes above about 0.92 around +72 deg, i.e. a high-Q inductive source. The unstable load region is a thin crescent near the chart edge. The trap: a narrowband input match built from a high-Q inductor can land right in that region at some frequency.
4. Check the whole band, not just the operating band
Transistor gain rises roughly 6 dB per octave as frequency drops, so K is usually worst well below the band (tens to hundreds of MHz) and sometimes near fT. Sweep the S-parameters from the lowest frequency your bias network passes up to the device fmax, including bias tees, decoupling and package parasitics. Out-of-band terminations are whatever your bias network happens to present, often close to a short or an open, i.e. right on the edge of the chart.
5. Stabilizing
Resistive loading is the reliable fix, at the cost of some gain and, on the input side, noise.
- Series resistor at the gate/base: in the example, 10 ohm raises K to 1.10, 15 ohm to 1.25 (|Delta| = 0.48), and MAG becomes 17.0 dB, i.e. we traded 3 dB of potential gain for unconditional stability at this frequency. On an LNA this resistor adds noise directly, so keep it small or move it behind a series inductor where possible.
- Shunt resistor at the drain/collector: interesting result for this device, 300 ohm alone only reaches K = 0.89, because the problem here is mostly on the input side. Do not assume one standard fix works for every device.
- Frequency-selective loading: put the resistor in series with a capacitor (or parallel with an inductor) so it only loads the circuit where K is poor, typically at low frequency, and leaves the in-band noise and gain almost untouched.
- Source/emitter inductance improves noise and input match together, but too much of it creates negative resistance at higher frequencies. Check K up to fmax after adding it.
- Bias feeds: a resistor in series with the gate bias choke (tens of ohm) kills low-frequency oscillation that the RF analysis never sees.
6. Remember what K does not cover
K and mu assume a linear two-port. Multistage amplifiers need each stage checked plus a loop check (e.g. a normalized determinant function or a probe-based loop-gain analysis), because feedback through shared supply lines and housing resonances creates paths the per-stage S-parameters cannot show. Large-signal parametric oscillation is another separate check.
Rule I use: mu > 1.05 from the lowest bias-network frequency up to fmax, with the actual layout parasitics included, before any board is ordered.