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Input and Output Stability Circles: Reading the Regions

By John · Updated · 5 min read

Each stability circle marks the terminations where |Γ_in| = 1 or |Γ_out| = 1. With |S11| < 1 and |S22| < 1 the origin is stable, so the side of the circle that holds Γ = 0 is the safe side.

Engineer’s note: I treat the out-of-band load as the dangerous one. Outside the band the matching network is nowhere near 50 Ω, so overlay its impedance trace on the stability circles across the full sweep. And check |S11| before the origin test, because above 1 it flips the answer.

For practitionersNew to this? Plain-language version

For practitioners

A two-port with gain and some reverse transmission (S12 ≠ 0) can show a negative resistance at one port when the other port sees the wrong termination. Stability circles show you exactly which terminations cause it. They are drawn on the Smith chart, so you can lay them over gain circles, noise circles or a matching network trace and see at once whether a planned source or load is safe.

What a stability circle marks

Start from the input and output reflection coefficients of a loaded two-port:

Γin=S11+S12S21ΓL1−S22ΓL\Gamma_{\text{in}} = S_{11} + \frac{S_{12} S_{21} \Gamma_L}{1 - S_{22} \Gamma_L}
Γout=S22+S12S21ΓS1−S11ΓS\Gamma_{\text{out}} = S_{22} + \frac{S_{12} S_{21} \Gamma_S}{1 - S_{11} \Gamma_S}

When |Γ_in| > 1, the input port reflects more power than it receives. That means a negative real part of input impedance, and with a suitable source the circuit can oscillate. The boundary case is |Γ_in| = 1. If you take every Γ_L that gives |Γ_in| = 1, the points form a circle in the Γ_L plane. This is the output stability circle. It is called "output" because it constrains the load, even though its edge is defined by a condition at the input. The input stability circle is the same idea in the other direction: the set of Γ_S that gives |Γ_out| = 1.

The circle edge is only the boundary. To use the circle, you must also know which side of it is stable.

The circle equations

Both circles use the determinant of the S-matrix:

Δ=S11S22−S12S21\Delta = S_{11} S_{22} - S_{12} S_{21}

Output stability circle, Γ_L plane:

CL=(S22−ΔS11∗)∗∣S22∣2−∣Δ∣2,rL=∣S12S21∣∣ ∣S22∣2−∣Δ∣2 ∣C_L = \frac{\left(S_{22} - \Delta S_{11}^{*}\right)^{*}}{|S_{22}|^2 - |\Delta|^2}, \quad r_L = \frac{|S_{12} S_{21}|}{\left|\,|S_{22}|^2 - |\Delta|^2\,\right|}

Input stability circle, Γ_S plane:

CS=(S11−ΔS22∗)∗∣S11∣2−∣Δ∣2,rS=∣S12S21∣∣ ∣S11∣2−∣Δ∣2 ∣C_S = \frac{\left(S_{11} - \Delta S_{22}^{*}\right)^{*}}{|S_{11}|^2 - |\Delta|^2}, \quad r_S = \frac{|S_{12} S_{21}|}{\left|\,|S_{11}|^2 - |\Delta|^2\,\right|}

The two sets have the same shape with S11 and S22 swapped. C_L and C_S are complex numbers, so they give both the distance and the angle of each center from the chart origin. The radius is always positive because of the outer magnitude bars in the denominator. When |S22| is close to |Δ|, the denominator is small and the output circle becomes very large. In the limit it turns into a straight line through the chart.

Which side is stable: the origin test

The quickest way to read a circle is to test one known point. The best point to use is the chart center.

If Γ_L = 0, the formula above gives Γ_in = S11. So if |S11| < 1, a matched load is stable, and the region of the Γ_L plane that contains the origin is the stable region. In practice:

  • If the origin lies outside the output circle (|C_L| > r_L), the outside of the circle is stable and the inside is unstable.
  • If the origin lies inside the output circle (|C_L| < r_L), the inside is stable and everything outside is unstable.

The source plane uses the same argument with Γ_S = 0, which gives Γ_out = S22. This test needs |S22| < 1.

If |S11| > 1, the device is already unstable with a matched load, so the origin falls on the unstable side and the reading flips. Always check the reflection magnitude before you apply the origin test.

Γ_L = 0C_Lr_L|Γ_L| = 1|Γ_in| = 1unstable Γ_Lstable Γ_L(origin side)Output stability circleC_L = (S22 − Δ S11*)*/ (|S22|² − |Δ|²)r_L = |S12 S21|/ | |S22|² − |Δ|² |Δ = S11 S22 − S12 S21|S11| < 1, so at Γ_L = 0Γ_in = S11: origin stable
Output stability circle in the Γ_L plane partly overlapping the Smith chart, with the origin side stable and the overlap unstable. Concept drawing.

The figure shows the common case. The output circle sits off to one side and partly overlaps the unit circle. The origin lies outside the circle, and since |S11| < 1 it is stable, so the region outside the circle is stable. The part of the circle that lies inside the unit circle is the set of passive loads that produce |Γ_in| > 1.

Worked example

Take an illustrative set of S-parameters at one frequency. They are all real so the arithmetic stays visible:

S11 = 0.6, S22 = 0.5, S21 = 4, S12 = 0.1

Determinant: Δ = (0.6)(0.5) − (0.1)(4) = 0.3 − 0.4 = −0.1, so |Δ|² = 0.01. Also |S12 S21| = 0.4.

Output circle. Denominator: |S22|² − |Δ|² = 0.25 − 0.01 = 0.24. Numerator: S22 − Δ·conj(S11) = 0.5 + 0.06 = 0.56. So C_L = 0.56 / 0.24 ≈ 2.33 and r_L = 0.4 / 0.24 ≈ 1.67. On the real axis the circle runs from about 0.67 to 4.0. Its near edge, Γ_L = 2/3, is inside the unit circle, so the circle cuts into the chart. To check the edge, put Γ_L = 2/3 into the Γ_in formula: 0.6 + 0.4·(2/3) / (1 − 1/3) = 0.6 + 0.4 = 1.0. That is the expected boundary value.

The origin is outside the circle (2.33 > 1.67) and |S11| = 0.6 < 1, so the outside is stable. The unstable sliver is the region near Γ_L = +1, which corresponds to high-impedance loads. In a 50 Ω system, Γ_L = 2/3 corresponds to Z_L = 50·(5/3)/(1/3) = 250 Ω. So purely resistive loads above 250 Ω are unstable. At Γ_L = 1 (an open), Γ_in = 0.6 + 0.4/0.5 = 1.4.

Input circle. Denominator: 0.36 − 0.01 = 0.35. Numerator: 0.6 + 0.05 = 0.65. So C_S = 0.65 / 0.35 ≈ 1.86 and r_S = 0.4 / 0.35 ≈ 1.14. The near edge is at Γ_S = 5/7 ≈ 0.71, which corresponds to 300 Ω in a 50 Ω system. The origin is outside the circle and |S22| < 1, so the outside is stable. Resistive sources above 300 Ω are unstable.

Both circles cut into the unit circle, so the device is conditionally stable at this frequency. This agrees with the Rollett factor: K = (1 − 0.36 − 0.25 + 0.01) / (2·0.4) = 0.5, which is below 1.

Now reduce the feedback to S12 = 0.02 and keep everything else the same. Then Δ = 0.22, C_L ≈ 1.83 with r_L ≈ 0.40, and C_S ≈ 1.57 with r_S ≈ 0.26. The near edges sit at about 1.43 and 1.32. Both are outside the unit circle, so every passive termination is stable.

The three geometric cases

With |S11| < 1 and |S22| < 1, each circle falls into one of three cases:

  1. Entirely outside the unit circle (|C| − r > 1). The origin side contains the whole Smith chart. The circle sets no limit on passive terminations.
  2. Enclosing the unit circle (r − |C| > 1). The origin is inside the circle, so the interior is stable. The interior contains the whole chart. This case is also unconditionally stable.
  3. Cutting the unit circle. Some passive terminations fall on the unstable side. The device is conditionally stable.

The device is unconditionally stable at a frequency only if both circles fall into case 1 or case 2.

A conditionally stable device can still be used, but the source and load must stay clear of the unstable regions at every frequency where the device has gain, not just in the design band. Out-of-band terminations are where unplanned oscillations usually come from, because the matching network seldom presents 50 Ω there.

In plain words · for students and beginners

Stability circles in plain words

An everyday picture

Picture a map with a fenced-off swamp on it. The fence is the stability circle. You do not know straight away which side of the fence is the swamp. But you do know that your own house, at the center of the map, is on dry ground. So the dry side of the fence is the side your house is on.

On the Smith chart, the "house" is the center point, Γ = 0. Γ (gamma) is the reflection coefficient: how much of a wave bounces back from a termination. The center of the chart means a perfect match, with nothing reflected. A termination is whatever is connected to the amplifier's input (the source) or output (the load).

Where the picture breaks down: the house is on dry ground only if |S11| < 1, which means the amplifier on its own, with a matched load, reflects less than it receives. If |S11| > 1, the house is in the swamp, and the rule flips.

The three shapes you will see

stableΓ = 0|Γ| = 1stability circleunconditionally stablecircle outside chartΓ = 0stableunstableconditionally stablecircle cuts chartstableΓ = 0stability circleunconditionally stablecircle encloses chart
The three ways a stability circle can sit relative to the Smith chart when |S11| is less than 1, with the stable side always being the one that contains the center point Γ = 0. Concept drawing.
  • Left: the circle sits completely outside the chart. No real termination can touch it. The amplifier is unconditionally stable, meaning it will not oscillate with any passive termination.
  • Middle: the circle bites into the chart. The overlap is unstable. The amplifier is conditionally stable: it is safe only if you avoid that area.
  • Right: the circle is so big that it swallows the whole chart. The center is inside it, so the inside is safe, and the whole chart is inside. The amplifier is unconditionally stable again.

The worked example, step by step

  1. Write down the S-parameters: S11 = 0.6, S22 = 0.5, S21 = 4, S12 = 0.1.
  2. Work out Δ = 0.6 × 0.5 − 0.1 × 4 = −0.1.
  3. Output circle: center about 2.33, radius about 1.67. It reaches in to 0.67, which is inside the chart (the chart edge is at 1).
  4. Is the center of the chart inside this circle? Its distance from the circle's center is 2.33, which is more than the radius, so no. Since |S11| = 0.6 is less than 1, the center is safe. So the outside of the circle is safe.
  5. The unsafe bit is near the right edge of the chart. For a 50 Ω system, that means resistive loads above 250 Ω.
  6. Do the same for the input circle. Sources above 300 Ω are unsafe.
  7. Both circles bite into the chart, so this amplifier is conditionally stable.

Mistakes beginners make

  • Assuming the inside of the circle is always the bad side. It depends on where the origin lies. Test it every time.
  • Mixing up the planes. The output circle goes on the load chart (Γ_L) and the input circle goes on the source chart (Γ_S).
  • Checking only one frequency. S-parameters change with frequency, and so do the circles.
  • Forgetting the |S11| < 1 condition before using the center point as the safe reference.

Remember

  • The circle edge is where the other port's reflection reaches exactly 1.
  • The side that holds the chart center is the safe side, as long as |S11| < 1 (load plane) or |S22| < 1 (source plane).
  • If a circle cuts into the chart, some real terminations can make the amplifier oscillate.

Frequently asked questions

How do I know which side of a stability circle is stable?

Test the origin. If |S11| < 1, then Γ_L = 0 gives Γ_in = S11, which has a magnitude below 1, so the side of the output circle that holds the origin is stable. The input circle works the same way if |S22| < 1.

What does the edge of a stability circle represent?

On the output stability circle, drawn in the Γ_L plane, every point gives |Γ_in| = 1. On the input stability circle, drawn in the Γ_S plane, every point gives |Γ_out| = 1. The edge is the boundary between terminations that make the device unstable and those that do not.

When are stability circles showing unconditional stability?

When |S11| < 1 and |S22| < 1, and each circle either lies entirely outside the unit circle or fully encloses it. If either circle cuts into the unit circle, the device is only conditionally stable at that frequency.

Do I need to check stability circles at more than one frequency?

Yes. The circles are computed from S-parameters at a single frequency, so a device that is unconditionally stable at one frequency can be conditionally stable at another. Repeat the check across the whole range where the device has gain.

References and further reading

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