Stability factor
and available gain of amplifiers
Updated December
22, 2013
Click
here to go to our main page on amplifiers
Here's a page on unstable
amplifier examples.
Here's
a cool approximation for GMAX that you need
to know about
Back in 2006, we contacted with Microwave HallofFamer John
Rollett regarding his contribution to the art of microwaves
(the stability factor, "K"). Please note that Mr. Rollett's
last name rhymes with wallet; some engineers mistakenly try to give
it Frenchsounding pronunciation. Yet another fact that you won't
learn anywhere else but Microwaves101!
Below is John's perspective on
the stability factor that bears his name:
"In 1955 I joined a
team of scientists set up in London to make germanium transistors.
As the junior member, I was given the job (which no one else wanted)
of devising methods of assessing their capabilities. It emerged
that "f/T" was the preferred indicator of useful bandwidth,
and I designed more than one method of measuring this, and also
became interested in other transistor parameters, resulting in
the stability factor. In 1963, I joined what is now British Telecomms,
where a team was developing and manufacturing highly reliable
silicon transistors for the first transistorised transatlantic
cable amplifiers. All these transistors were measured in my highfrequency
gain (f/T) apparatus. At about the same time the stability factor
began to find application in microwave circuits, something I had
not envisaged! My interests then turned to active filters, and
eventually to speech processing (synthesis, recognition, compression
 GSM), far removed from microwaves. I retired in 1990.
It is interesting that a
concept which was developed largely in the abstract, and with
no specific applications in mind, should turn out to be useful
(as indeed I hope it is).
My opinion about the application
of the stability factor is not worth much, as it was developed
from theoretical principles, and I never used it on a real amplifier,
nor did I know of anyone who had, other than reading the literature.
I really have nothing useful
to say about the stability factor, as for me it was mainly an
exciting theoretical development, inspired by Sam Mason (who started
off a paper in the IEEE "In a hundred years the earth will
be covered with paper to a depth of eighty feet, and we shall
all be as bald as billiard balls." Great man!). It was a
pleasant surprise to see that it had useful applications. It also
provided a main topic of my PhD thesis, where I also demonstrated
that it could be measured directly, without being calculated from
twoport parameters."
Along with this fine page on
stability factor, we now have a download
spreadsheet for calculating stability factor and available gain
(and many other parameters) from vector Sparameters that you can
get from manufacturers' data sheets! It's called SParameter Utilities
101, here's a page that
explains how to use it.
Here is an index to this page and other content:
Stability 
what is it?
Passive
versus active devices
Conditional
stability versus unconditional stability
Rollett's
stability factor (Kfactor)
Maximum
gain
Available
gain
Stable gain
Other measures of stability
Mu1 and Mu2
B1
Examples
of an unstable amplifiers you could buy (separate page)
Stability
 what is it?
Before we get into this subject,
there are upwards of 1000 microwave eggheads out there who will
tell you "looking at Kfactor isn't enough to be 100% certain
that your amplifier won't oscillate, you need to examine every possible
loop for origin encirclements of right hand poles yadadadada..."
to which we say, "shut up". When you are picking out an
amplifier, always look at its Kfactor, ideally from zero
Hertz (DC) to the maximum frequency of oscillation Fmax of the device
that it is based on, at steps no larger than 1 GHz. If you are a
power amp designer, there is a lot more you should be looking at,
but we won't cover all of that here.
Stability, in referring to amplifiers,
refers to an amplifier's immunity to causing spurious oscillations.
The oscillations can be full power, largesignal problems, or more
subtle spectral problems that you might not notice unless you carefully
examine the output with a spectrum analyzer, one Hertz at a time!
Unwanted signals may be nowhere near your intended frequency but
will wreak system havoc all the same. In another extreme, instability outside your band may drop the gain of your amplifier by 20 dB inside the band, which should get your attention. These types of problems are ones that
you'd rather avoid.
Let's start with some definitions:
A passive
device contains nothing that could add energy to your signal.
The first law thermodynamics, conservation of energy, implies that
a passive device can't oscillate. An active device is one
in which an external energy source is somehow contributing to the
magnitude of one or more responses. Learn more about basic network
theory here.
Conditional
stability refers to a network that is stable when its input
and output "see" the intended characteristic impedance
Z_{0} (usually 50 ohms, sometimes 75 ohms), but if your
application presents a mismatch, there is a region of either source
or load impedances that will definitely cause it to oscillate (see
our page on the Smith chart for more
discussion of impedance). The phrase "potentially unstable"
refers to the same condition, the difference is the same as the
question of whether a glass is halffull or half empty.
Unconditional
stability refers to a network that can "see" any possible
impedance on the Smith chart from the center to the perimeter (up
to gamma=1.0) at any phase angle. Gamma < 1 means that the real
part of the impedance is positive. Note that any network can oscillate
if it sees a real impedance that is negative, so if your system
goes outside the normal Smith chart all bets on stability are off.
Stability has to be separately
evaluated at all frequencies where the amplifier could potentially
oscillate. This is generally up to the Fmax of the technology. For
power PHEMT parts, stability is not a problem past 50 GHz or so.
Rollet's
stability factor (Kfactor)
John Rollett's 1962 IRE paper
titled Stability and PowerGain Invariants of Linear Twoports
forever links him to the stability factor, K, of an amplifier.
For this reason we've put him in our Microwave
Hall of Fame! From this one scalar dimensionless quantity you
get the most valuable measure of stability for a given frequency.
Although a lower case is sometimes
used, we'll follow Gonzalez and use upper
case K whenever we refer to stability factor on this site. Thanks,
Andy!
Whether you are designing an
amplifier, or merely picking out an amplifier from vendor data sheets
to use in a subsystem, you should always consider stability across
frequency. Rollett's stability factor is a one way to get an indication
of whether you'll have a problem or not.
Kfactor that is greater than
one tells you that your amplifier is unconditionally stable. If
K is less than 1, you may have a problem. Below is the equation
for Kfactor (in two parts):
Thanks to Ed we corrected the
top equation on April 17, 2007! The plus sign was previously shown
incorrectly as a minus sign.
Note that Kfactor is only defined for twoport networks. That's why you don't see "S31", "S33", etc. in the equation.
A good amplifier designer must pay attention to available gain (GMAX, see below) and Kfactor, being cognizant of the fact that high stability comes at the price of low available gain. The best amplifiers usually have Kfactor as low as 1.5 in the band.
Here's a page on unstable
amplifier examples which you should read before you ever buy
an RF amplifier!
Maximum
available gain versus maximum stable gain
The maximum
available gain (sometimes called MAG, sometimes called GMAX)
of a device is only defined where K is greater than one. Algebraically,
this is because the term under the squareroot becomes negative
for values of K less than 1. Another way to look at it is that maximum
available gain is infinite. Infinite gain means oscillator.
The maximum
stable gain (MSG) of a device is defined when maximum available
gain is undefined (K<1). It is merely the ratio of mag(S21)/mag(S12).
Under no circumstances should you try to tease more than this amount
of gain from a conditionally stable device. Better yet, you should
try to stabilize the device using resistive components, until K
is greater than 1, then you can optimize the matching networks for
all the gain you want.
Calculating
GMAX
Looking at available gain (GMAX)
is also helpful when you are looking for potential instabilities.
The equation below shows how GMAX is calculated from stability factor
K and the forward and reverse transmission coefficients:
Note that GMAX is only defined for twoport networks.
Available gain is undefined when
K is less than one. That's when the square root of (K^21) becomes
imaginary.
We found out the hard way that
the GMAX equation can give you a big problem when K gets really
big, which it often does. Our Sparameter Utilities spreadsheet
(available in the download area)
calculates GMAX, and whenever K became greater than about 20,000,000,
GMAX would become zero in the spreadsheet, which makes no physical
sense. It's because within the expression we are subtracting two
very large numbers that only differ by a small amount, and Excel
only maintains 15 digits of accuracy. So how does Agilent ADS deal
with this problem, short of doubling the precision of the number?
We found empirically that the
equation below gives the proper result with no worries about K being
large (but is inaccurate when K is not large, so watch out!). In
our free download Sparameter
Utilities spreadsheet we substitute this whenever K is greater
than 1,000,000:
If you are interested in proving
why this approximation is valid, we believe you'll need to perform
a Taylor series expansion of the original equation and then allow
K to go to infinity. We'll let some microwave professor assign that
math problem for extra credit, we just know that the equation works
correctly when K is greater than 1,000,000, having checked it against
some very expensive EDA software.
Update August 15 2008:
here's the math behind the approximation, thanks to Raj! There's
a mathematical approximation that comes into play, you can read
about it on Wikipedia
but here is the important part:
Applying this approximation to
the formula for GMAX when 1/K^2 is very small (K is very large),
we see that:
and hence:
Thanks, Raj!
Other measures of Stability
There are many other measures of stability, but Kfactor should remain the goto measure for designers.
Mu1 and Mu2
Mu1 measures the radius from the center of the Smith Chart (Z0) to the nearest unstable point in the output plane. Mu2 does the same thing for the input plane. Obviously you want both of them to be greater than one to move the unstable point off of the Smith Chart. The larger Mu1 and Mu2 are, the more margin your design has to variations in processing. Mu1 and Mu2 are only defined for twoport networks (just like Gmax and Kfactor).
The Mu factors tell you something that Kfactor does not. They tell you which side of your circuit is the likely culprit when it goes unstable. However, if K is much greater than one, the Mu factors will also be much greater than one.
We need to add the equations for Mu1 and Mu2 below... but they appear in our Sparameter Utilities spreadsheet that you can get for free in our download area.
B1
B1 is another stability factor similar to K. Be honest, does anyone ever use this?
B1 = 1+S112S222Δ2
where Δ is defined in the Kfactor calculation.
B1 should be greater than 0. This measurement like all of the others on this page) is only defined for twoport networks.
Examples
of unstable amplifiers you could buy
Moved to a
new page.
