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Stability circles and the gain versus noise trade-off

A small signal amplifier design is three questions asked in order. Will it oscillate? How much gain can I have? How quiet can it be at that gain? The Smith chart answers all three as overlapping circles.

Question one: will it oscillate

An amplifier oscillates when it can present a negative resistance to something, which happens when the source or load impedance falls in the wrong region. Rollett's criterion tells you whether such a region exists at all.

Take the device from Hewlett-Packard Application Note 970 at 6 GHz, as Gonzalez presents it:

From those four numbers:

K is 1.5 and |Delta| is 0.3, so this device is unconditionally stable at 6 GHz: there is no passive source or load impedance anywhere that can make it oscillate. That is a comfortable place to start and it is not the usual case.

When K is below 1 the device is only conditionally stable, and the stability circles matter. QuickSmith draws them on both planes: the circle divides the chart into a region where the device is stable and a region where it is not, and you have to keep your source and load impedances out of the bad region. The catch is that which side is which depends on |S11| and |S22|, so read the labels rather than assuming the inside is bad.

Enter Parameters → S-Parameters takes the four values, and the stability figures appear with them. If you have an .s2p file from a network analyser you can import that instead and skip the typing.

Question two: how much gain

For an unconditionally stable device the ceiling is the maximum available gain:

GMSG is the maximum stable gain, the figure you would get if you could neutralise the feedback entirely. GMAX is what you can actually have with a simultaneous conjugate match at both ports, and it is about 4 dB lower because S12 is not zero. Real devices leak backwards.

Below the ceiling, every gain figure is a circle on the source plane rather than a point. Ask for 10.55 dB and you get a locus of source reflection coefficients that all give you 10.55 dB. That degree of freedom is what makes the next question answerable.

Question three: how quiet

Noise figure is also a family of circles on the source plane, centred on the optimum source reflection coefficient from the data sheet. The minimum noise figure sits at a point, and every value above it is a circle around it.

Here is the design tension in one sentence: the source impedance for maximum gain and the source impedance for minimum noise are not the same impedance, and they are usually not close.

So you do not pick a point, you pick an intersection. Draw the 10.55 dB gain circle. Draw a noise circle. Shrink the noise figure until the two circles just touch. Where they touch is the source reflection coefficient that gives you exactly the gain you asked for at the lowest noise figure that gain allows.

In QuickSmith: Circles → Available Gain Circles for 10.55 dB, then Noise Circles, and decrement the noise figure until the circles intersect. Then drag GammaS to the intersection and read off the source impedance, with the corresponding load impedance appearing on the other plane.

For this device that lands at ZS = 13.98 + j7.57 and ZL = 17.92 + j35.64.

Then it is an ordinary matching problem, twice

You now have two impedances and a chart, which is the rest of this site. Transfer → Source Impedance sends ZS across to the schematic so you can design a network for it, and Transfer → Load Impedance does the same for ZL.

One thing to be careful about: for maximum power transfer the network has to present the conjugate of what you worked out. It is an easy sign to lose.

Where this comes from

The worked version is Example 7, following HP Application Note 970 (February 1978) as presented by Gonzalez in Microwave Transistor Amplifiers (Prentice-Hall, 1984) and by Dale Henkes in RF Design, December 1993.

K = 1.504 and |Delta| = 0.301 are stated in the note, and the regression suite asserts them rather than pinning whatever the program happens to produce. If the stability maths ever drifts, the tests fail.