KANA-EE-08/01/02-FEEDBACK0001
Analogue
Design For Teletubbies
By
Sir
Kevin Aylward B.Sc., Warden of the Kings Ale
Practical
Feedback Theory
Or
The
only guaranteed way to make a small fortune, is to start with a large one, and
lose some of it.
Abstract
This paper forms an introduction to feedback theory as it applies to
electronic amplifiers. The approach is that of engineering utility rather then
a mathematical precise expose, as it is not possible to fully understand
stability theory without an understanding of basic complex analysis. It should
be noted that this paper assumes some basic understanding of frequency and
phase response of amplifiers.
Overview
The fundamental purpose of feedback theory is to prevent an amplifier
oscillating, or to ensure an oscillator will, to wit:
"Amplifiers do, oscillators don’t", is the usual design
problem.
There are various arguments that are used to give conditions for
oscillations. Some of these are a bit dubious in the sense that they are not
usually sufficient or necessary. The general problem of stability in
quasi-linear systems was solved by Nyquist in the 1930's, or thereabouts. The
argument is rather technical, but is necessary because a loose analysis of the
issue results in a conclusion at odds with experiment.
A typical argument goes along the line of:
Suppose the gain is grater then unity, and that the net phase shift
around the loop is 0 degrees. If a small signal exists at the input of an
amplifier, it will become larger at the amplifiers output, and as its connected
to its input will amplifier again etc, until the amplifier limits and forces a
gain of exactly 1, such that this state will persist indefinitely. This is
sometimes called the Barkhausen Criteria,
and is false.
This argument is a bit daft really. By itself, it implies that the
amplifier will latch hard to one rail forever!
Even if this argument is extended in more detail it still leads to a
prediction that disagrees with experiment namely:
An amplifier can
have net positive feedback, with gain greater then one, and yet still be
stable.
The reason that attention is drawn to this statement is that it is not
unknown for individuals to have formal courses in stability theory, correctly
make predictions about transfer functions, yet are quite unaware of what the
physical implication of those transfer functions are. Indeed, even professors
discuss such "conditionally stable systems" in detail, quite
oblivious to the fact that it means that such systems are positive feedback
systems with gain greater than one.
Note:
A negative feedback signal is equivalent to a 180 phase lag. This means
that an additional 180 degrees of phase lag will generate a total of 360
degrees, or a net 0 degrees phase shift around the loop.
Stability Criteria
The understanding of stability criteria requires complex analysis, so
the results will be stated for reference but in practice they can be expressed
in simpler terms, which will be addressed below.
In general an amplifier will have a small
signal loop gain transfer
function given by:
That is, an amplifier will have a number of poles and zeros, or
alternatively a number of gain roll off and gain increases. B(s) is also of the
same form, and is typically the feedback network. Note that it is the gain
around the loop, not just the amplifier gain. The loop gain is thus the open
loop gain multiplied by the feedback divisor.
If there is a pole
in the right half plane of LG +1, it can be shown that the impulse response
of any transfer function described by LG+1, will result in an ever increasing
sinusoidal response. In practice this
increase will limit when the amplifier saturates at its supply rails. This
results in steady state oscillations. This implies that the amplifier is
analyzed such that it is reasonable linear for small signals, but can limit at
large signals in a way such that this limiting does not change the small signal
phase response at the limiting point.
Nyquist Theorem
The basic issue with the above transfer function is that it is usually
obtained as a ratio of polynomials, not in factored form from which the poles
and zeros can be determined. Nyquist's Theorem is an approach that determines whether or not there is a pole in the right
half plane by a graphical technique.
As far as general amplifier
design is concerned, Nyquist's Theorem, in the authors opinion, has no value whatsoever. It is
described here only for reference in a historical context. Modern and cheap
computer simulation programs have completely eliminated any requirement for
drawing "Nyquist Plots", even on the computer. There may be some
value in control systems analysis, but even in this case, there are fundamental
issues that still require stability to be addressed by other methods.
For those interested in an exact statement of Nyquist's Theorem, a
search on www.google.com
will turn up rather a lot.
A key issue in stability design is that a real system is nonlinear,
such that a small signal analysis that suggests stability is often incorrect.
Typically, when an amplifier goes into limiting, the phase response is much
worse than that indicated at the small signal bias conditions. Indeed, the only
practical way to determine stability accurately is to run transient
simulations, with the ac response being only used as a basic initial design
guide.
Stability and Bode Plots
There is some debate on exactly what a "Bode" plot is. To
wit, is it just a generic gain against frequency plot, or is it the
straight-line approximation to the exact gain against frequency. For the
purpose of stability analysis, the general technique is to discuss in terms of
the straight-line approximations.
Arguable, stability in 99.9% of cases, in general amplifier design, can
be summed up as follows:
A - If the slope of the loop gain is -20db/dec when
the loop gain finally falls to 0db, the system will be stable, and unstable if
the slope is greater then -20db.
Prior to simulation tools, checking for this condition was a reasonable
method. With the advent of simulation tools, the phase is checked directly,
which results in the following:
B - If the phase of the loop gain is less than
180 degrees when the loop gain finally falls to 0db, the system will be stable,
and unstable if the phase is greater then 180 degrees.
This is the probably the most useful concept with which to tackle the
stability problem. The general procedure is to set up a circuit simulation that
plots the gain and phase of the loop gain. The response is then checked to
ensure the above condition is satisfied. Note the use of the word
"final", it is the phase shift at the final 0db-gain point that
determines stability. It is immaterial if the phase goes to 180 degrees prior
to the 0db point so long as it gets to less then 180 degrees before the final
unity gain point.
There is one provider to the above. If the phase dose go to 180 degrees
then recovers, this is usually a system known as conditionally stable. The
issue here is that if the loop gain falls due to component tolerances, power
supply ramp ups etc, it is possible for the gain to become 0db at a frequency
such that the phase has not recovered, and hence the amplifier could become
unstable.
For reference, sometimes A is equivalently expressed as:
If the intersection of open
loop gain and closed loop gain is at a rate of -20db/dec, the system will be
stable, and unstable if greater then -20db
Because the loop gain is the difference between open loop gain and
closed loop gain when expressed in db's. In this procedure, both the open loop
gain and closed loop gain are plotted together, and their intersection
examined.
So, a gain roll off of 40db/dec will be unstable, noting that minus
means a decrease in gain, and "greater" then implying a larger negative roll off, as usually
understood, but possible a bit misleading. One should use a bit of logic in the
following as to whether there is an implied negative to the gain response
rates.
Statement (A) and (B) implies that if a higher order roll off of say
80db/dec, is then "compensated" such that when the gain goes through
the 0db point it is only at a 20db slope, then the system will be stable. This
is despite the fact that in the 80db/dec region the phase may approach n*90
degrees, which would have violated the naive interpretation of a stability
requirement.
Technical point
Technically, a negative feedback system with only a pure second order
roll off can not be unstable, as a second order system can only approach 180
degrees phase shift at infinity. However, it is understood that any practical amplifier will always have at
least few degrees of excess phase shift that will always result in the system
oscillating.
Basic Compensation
The key concept in stability compensation is that each 20db/dec roll
off creates a phase shift that tends to generate additional 90 degrees lags.
The idea is to somehow generate loop gain boosts that increases the phase from
excessive negative values at the 0db loop gain point.
General Concepts
The loop gain frequency response will have terms that result in:
1 Gain roll offs (poles) of n.20db/dec or phase shifts tending to -90
degrees, often called a lag.
2 Gain increases (zeros) of m.20db/dec, or phase shifts tending to +90
degrees often called a lead.
2 Flat gain regions.
It is also possible to get lag phase shifts with flat gain response.
These are called "all pass" networks, and generally sound a death
knell in a feedback amp.
For example, consider a (ladder) set of series resistors (A), with
their junctions connected to a series resistor and capacitor to ground (B).,
with an input and output at the resistor string ends. If the resistors in
series with the capacitors are much smaller then the (A) resistors then each of
the (A) resistors and its capacitors will generate a 20db/dec roll off. At a
sufficiently high frequency the (B) resistors and capacitors will generate a
20db/dec increase, such that at the highest frequencies the response will be
attenuated but flat.
The only guaranteed way to
make a small fortune, is to start with a large one, and lose some of it.
For general amplifiers, a basic design principle is that there is no
extra gain available at high frequencies.
This means that if you want to turn a 40db/dec slope into a 20db/dec
slope, you have to through away some gain at lower frequencies and recover it
back again at high frequencies. This can be done in a number of ways
1 Place a series RC across the input of the amplifier.
2 Place a series RC across an internal gain setting node of the amplifier.
These techniques all reduce the loop gain prematurely because of the
capacitor, but at high frequencies the series resistor limits the shunting
effect of the capacitor, thereby removing the phase shift introduced by the
capacitor. A key idea here, is that
reducing the gain earlier in frequency means that additional poles in the
amplifier will (hopefully) only occur well after the 0db point, hence will not
effect the amplifiers stability.
Another technique, but basically the same principle in disguise, is to:
3 Bypass a feedback resistor with a capacitor.
This increases the loop gain because the feedback resistors had already
attenuated the gain.
(...Work in Progress...)
Mixed Mode Simulation from
AnaSoft Ltd.
© Kevin Aylward, 2002, all
rights reserved.
This paper may be reproduced
so long as no charge is made,
and that the paper is
reproduced in full, with full credit to it author given.
KANA-EE-08/01/02-FEEDBACK0001