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When designing amplifiers, one goal is to get rid of any oscillations. The standard approach is to examine the so-called K-factor, or stability circles on the Smith chart. However, there are limitations to this approach. This paper examines those limitations and presents a new methodology to reduce oscillations by employing a complex sphere.

Historical glance

The origins of stability K-factor can be found in J. M. Rollett's article, “Stability and Power-Gain Invariants of Linear Twoports,” in IRE Transactions on Circuit Theory, March 19621, as it is often called by the author's name. Oddly enough, it contains merely transformation of an expression derived from previous sources [2,3,4]. Furthermore, the derived seemingly general form does not refer to S-parameters as commonly used nowadays. A relatively simple form for stability criterion was proposed, which appears probably most accepted so far:



Even though the expressions fail when applied to some complex amplifier chains, the approach is readily applicable to so-called class I networks5 on a single transistor chip. In the course of this article, I will also focus on this useful approach to describe stability of a single amplifying stage consistent (possibly) with S-parameters.

The two stability conditions above are not as convenient as a single one.

There was one such proposed in M. L. Edwards and J. H. Sinsky's article: “A New Criterion for Linear 2-Port Stability Using a Single Geometrically Derived Parameter,” in the IEEE Transactions on Microwave Theory and Techniques, December 19926:

However, it was not accepted nor perceived widely. There are opinions that K-factor alone could be sufficient. Just recently, a paper written by G. Lombardi and B. Neri titled “Criteria for the Evaluation of Unconditional Stability of Microwave Linear Two-Ports: A Critical Review and New Proof,” in the IEEE Transactions on Microwave Theory and Techniques, June 19997 appeared intended to help resolve the issue of sufficiency.

One thing I would like to point out about the article is that it reviews all the old conditions and concludes that the latest, single criterion (µ) can also be deduced.

Furthermore, as suggested in Edwards' article, it surpasses them with insightful meaning and makes possible to assess the degree of potential instability. Therefore, I propose that any stability condition sets based on K-factor should not be used.

Toward the single stability criterion

Now, it should be checked if the single criterion is also optimal. The declaration about the proposed µ criterion — that it ensures physical insight and even gives a measure of instability — can be disputed as will be shown further. All this suggests is the need to thoroughly verify the matter. I agree with Edwards and Sinsky that geometric illustration assist in derivation of stability criteria.

As one recalls, the expressions below are describing the input reflection coefficient ГIN of a two-port as function of its load reflection coefficient ГL and similarly at the output ГOUTS).

For a two-port described by S-parameters they are as follows:

It is very important to understand these functions, and their inverse ГLIN), ГSOUT) as mappings of circles into circles on complex plane. The unity circle in ГL plane encompassing every passive load reflection coefficient is transformed by (1a) to a circle on ГIN complex plane, similarly for (1b), as well for inverse functions. An intuitively understandable condition for a two-port unconditional stability requires:

as well as

From above the so-called stability circles can be derived, being the circles on ГL (or ГS) plane transformed through (1a) or (1b) into the unity circle in ГIN (or ГOUT) plane, respectively. The commonly known expressions for radii (r) and centers (c) of output and input stability circles are given below:

It is important to discern the three characteristic, non-intersecting locations of stability circle in reference to the unity Smith chart as shown in figure 1.

Only the two first cases relate to unconditional stability.

Now it is time to express the geometric meaning of the old K-factor. As it was proved, also in the Edwards article, K>1 is equivalent to non-intersecting of stability circles with the boundary of unity Smith chart (USC) as demonstrated in figure 1. It is evident that the K-factor alone is insufficient to guarantee stability and additional condition is needed. Also, this demonstrates that the K-factor is unable to quantify instability.

Some difficulty appears when studying near stability conditions. That is when the circles of figure 1a and 1b start to intersect with USC boundary. Instability regions appear of a different nature. In the first case, it is inside the circle, and in the second it is outside. There are known in mathematics, transformations between complex plane and complex sphere. It was important in the Edwards article to discern that presentation of stability circles on complex sphere eliminates the above difficulty. However, does it really show something concrete when the proposed µ factor reaches two, three or more? It looks like an indicator with marked critical point, but without limits. An indicator could be useful even not scaled with physical units, provided it has two limited sections showing the allowed and the prohibited regions. Here, it could be (for example) natural to set the marking point at zero, while instability region from -1 to 0, and, similarly, the stable region from 0 to +1. In such a case, the indicator level can be informative. Certainly, the µ factor defined as the minimum distance in the ГL plane between the origin of the USC and the unstable region, is far more insightful compared to old K-factor, yet it could be still considered as a non-optimum one.

A new approach

The objects of matter here are stability circles, and they require proper “round” domain to represent effectively and non-ambiguously. In reality, it was the traditional presentation of stability circles on complex plane (that is on Smith chart with its infinite exterior), which caused problems. The Smith chart, created by simulator software on a monitor screen, is easily visible. However, the stability include loci far beyond the chart. Due to this difficulty, the matter could be simpler with the presentation of stability circles on the complex sphere instead of infinite complex plane. Stereographic presentation is accomplished in many kinds of software today. Let us imagine the creation of complex sphere from the plane. The ball is first put tangentially under the center of the chart so that the point of ideal matching places as the north pole, the unity circle appears as the equator while after wrapping all the infinity gathers in the southern pole. The Smith chart goes this way into the northern hemisphere while all the exterior goes into the southern hemisphere. Now the two stable situations of figures 1a and 1b can be shown as in figures 2a and 2b.

Looking at these stereographic pictures shows how nicely the planar ambiguities vanish. What was strangely difficult on complex plane now indicates vary similar situations: circles encircle the south pole or not.

The designer's goal to maintain stability reduces simply to keep the stability circles well on the southern hemisphere, that is, below the equator. The required stability condition becomes intuitively evident; simply like the distance of a stability circle to the chart equator.

The complex sphere considered so far is not very convenient for manipulation. Let us compress it, from north pole to the south, to get the flat, two-sided disk.

Let the acquired entity be referred to as unity disk chart (UDC). The word “disk” underlines the essential feature: the something has two sides, lower and higher or bottom and upper or negative and positive. We do not need to see both at the same time, but perceive the other all the time. The advantage of UDC is in keeping only one graphical arrangement in form of the well-acquainted Smith chart. One can visualize every complex point, especially their sets like stability circles inside the circumference. We are interested in analyzing the stability circles here and they are still maintained with this visualization but can be placed under or on the disk. Let their drawings on the monitor screen be distinguished in such a way that the bottom ones are green (allowable) dashed circles and the upper ones are red (prohibited) continuous line circles. Now checking absolute stability at a frequency point reduces to assuring the stability circle to lay well inside disk and be green. And the possibility of unstable operation appears like the circle coming out beyond the bottom side and folding around the edge of the disk onto the upper side, where its folded part appears in red. Both such cases are shown in figure 3.

The stability factor, referred to as S, can be intuitively derived now as the distance between the edge of the UDC and a stability circle as depicted in figure 3a. Note that the UDC circular edge is simply unity magnitude circle of reflection coefficient domain. Thus, checking S for stability means to have S>0 and its stability range is from 0 to 1. Similarly, the instability range is from -1 to 0. Therefore, such a factor fulfills the requirement considered earlier for an effective indicator.

Now let us express the S-factor in analytical form. First, the UDC should be described analytically. Its upper side states the usual Smith chart, encompassing the reflection coefficient loci of magnitude <1. Its bottom side is created through transformation of all complex points having modules m>1, according to

General transformation of the whole complex plane into UDC is accomplished thru inversion of modules for points outside the unity circle, as well as through ascribing to these new points/objects an additional side parameter of value -1, enabling them to differentiate (in color on screen for example) from the upper side objects. Those two functions are needed to define REV for reversion and SIG for signing. Function REV translates the complex plane into two sides of UDC while function SIG takes value +1 for upper side and -1 for lower side.

For describing the S-factor, expression for radii and centers according to expression (2) will be used, the module of centers will be taken as:

The distance of a stability circle on a complex plane can be marked as:

Now the S-factor can be described for two cases, as in Figure 3a and 3b, like:

To obtain a single expression, the transformation functions are needed to employ:

Such defined stability factor can be used as a single parameter in contrast to the traditional K-factor, which required auxiliary conditions. More-over, it has meaningful geometrical derivations, as well as it can serve as a useful indicator of stability potential, varying from -1 to +1 with 0-crossing as the critical point. Any stability factors or conditions postulated in the last few decades attempted to express directly with S-parameters. Nevertheless, the expressions were too complicated to calculate quickly on a calculator. Furthermore, to analyze such condition throughout any frequency region, a computer simulator is required. Therefore, I have abandoned any attempt to express the factor in a relatively simple S-parameter expression. The final, simple form gives readable meaning for an engineer while implementation in any simulator software is straightforward with a set of expressions and conditional functions REV and SIG easily defined.

An example

A comparison of the traditional stability approach and the proposed one will be shown with an RF transistor BFP420 analyzed on Genesys simulator software from Eagleware Corp. (www.eagleware.com). Figure 4 shows K-factor and the supplementary B1 parameter across wide frequency range.

The particular, additional stability condition implemented in the software suite is:

The proposed stability S-factor was computed in Genesys software using postprocessing script as below:

using linear1.sch1
function conj(X)
return 2*re(X)-X
function rev(X)
return iff(X>1,1/X,X)
function sig(X)
return iff(X>1,-1,1)

The script refers to the output of the device, while the similar one can be written for the input. The results of S-factor computation are shown in figure 5.

As the pictures show, both K and S factors have similar appearance, typical for RF transistors, indicating potential instability below 1.9 GHz. Yet, when embedding the chip with additional circuitry, some differences arise. Inserting an inductor in the emitter branch one makes possible instability at higher frequencies. With K-factor, it is not directly evident, as it goes close to 1; only the additional B1 factor — going below 0 — confirms the state. Quite simpler with S factor, the alone characteristic dropping below zero, until -1, clearly envisions the potential of instability.

On the other hand, one can examine the growth of stability making resistive coupling at the transistor output, for example. The K-plus-B1 approach would result in both growing to more and more positive values, with no direct relation of stability level. When analyzing the S parameter, the result comes closer and closer to +1, which corresponds to complete isolation of an active device with resistive matching in the port.

With evident meaning and known, fixed limits, the S factor states the useful stability indicator.


Strangely enough, in such general theoretical subject-like stability conditions, and after long research time, new papers have appeared in recent years. Not many years ago a single stability factor was proposed, although such a need seems to be evident. It is even more strange that this hardly gains interest. My intention was to enforce the way, giving definition of the insightful and useful single stability factor. Furthermore, its derivation can be intuitively obtained through clear geometrical manipulation. With the same approach, the way was shown to envision every stability conditions with the stability circles drawn at the unity disk chart appearing like the well-known Smith chart.


  1. J. M. Rollett, “Stability and Power-Gain Invariants of Linear Twoports,” I.R.E. Transactions on Circuit Theory, March 1962.

  2. F. B. Llewellyn, “Some Fundamental Properties of Transmission Systems”, Proceedings of the I.R.E., March 1952.

  3. A. P. Stern, “Stability and Power Gain of Tuned Transistor Amplifiers,” Proceedings of the I.R.E., March 1957.

  4. E. F. Bolinder, “Survey of Some Properties of Linear Networks,” I.R.E. Transactions on Circuit Theory, September 1957.

  5. D. Woods, “Reappraisal of the Unconditional Stability Criteria for Active 2-Port Networks in Terms of S parameters,” IEEE Transactions on Circuits and Systems, February 1976.

  6. M. L. Edwards and J. H. Sinsky, “A New Criterion for Linear 2-Port Stability Using a Single Geometrically Derived Parameter,” IEEE Transactions on Microwave Theory and Techniques, December 1992.

  7. G. Lombardi, B. Neri, “Criteria for the Evaluation of Unconditional Stability of Microwave Linear Two-Ports: A Critical Review and New Proof,” IEEE Transactions on Microwave Theory and Techniques, June 1999.

About the Author

Stan Alechno is a microwave and RF design engineer at WZR RAWAR, a Warsaw, Poland, based scientific-industrial group of professional electronics. He received his M.S.E.E. degree from the Warsaw University of Technology, and previously worked for the Microwave Instruments Establishment of Polish Academy of Sciences, and LAMINA, the Electron Components Establishment. His main area of his work is microwave frequency synthesis. He can be reached at alechno@onet.pl.