Chapter 12. Negative Feedback

It is difficult to convince people that this frequency is of no relevance whatsoever to the speed of amplifiers and that it does not affect the slew rate. Nonetheless, it is so, and any first-year electronics textbook will confirm this. High-gain op-amps with sub-1-Hz bandwidths and blindingly fast slewing are as common as the grass (if somewhat less expensive) and if that does not demonstrate the point beyond doubt then I really do not know what will.

Limited open-loop bandwidth prevents the feedback signal from immediately following the system input, so the utility of this delayed feedback is limited. No linear circuit can introduce a pure time delay; the output must begin to respond at once, even if it takes a long time to complete its response. In the typical amplifier the dominant-pole capacitor introduces a 90° phase shift between input pair and output at all but the lowest audio frequencies, but this is not a true time delay. The phrase delayed feedback is often used to describe this situation, and it is a wretchedly inaccurate term; if you really delay the feedback to a power amplifier (which can only be done by adding a time constant to the feedback network rather than the forward path) it will quickly turn into the proverbial power oscillator as sure as night follows day.

12.1. Amplifier Stability and Negative Feedback

In controlling amplifier distortion, there are two main weapons. The first is to make the linearity of the circuitry as good as possible before closing the feedback loop. This is unquestionably important, but it could be argued that it can only be taken so far before the complexity of the various amplifier stages involved becomes awkward. The second is to apply as much negative feedback (NFB) as possible while maintaining amplifier stability. It is well known that an amplifier with a single time constant is always stable, no matter how high the feedback factor. The linearization of the VAS by local Miller feedback is a good example. However, more complex circuitry, such as the generic three-stage power amplifier, has more than one time constant, and these extra poles will cause poor transient response or instability if a high feedback factor is maintained up to the higher frequencies where they start to take effect. It is therefore clear that if these higher poles can be eliminated or moved upward in frequency, more feedback can be applied and distortion will be less for the same stability margins. Before they can be altered—if indeed this is practical at all—they must be found and their impact assessed.

The dominant pole frequency of an amplifier is, in principle, easy to calculate; the mathematics are very simple. In practice, two of the most important factors, the effective beta of the VAS and the VAS collector impedance, are only known approximately, so the dominant pole frequency is a rather uncertain thing. Fortunately, this parameter in itself has no effect on amplifier stability. What matters is the amount of feedback at high frequencies.

Things are different with the higher poles. To begin with, where are they? They are caused by internal transistor capacitances and so on, so there is no physical component to show where the roll-off is. It is generally regarded as fact that the next poles occur in the output stage, which use power devices that are slow compared with small-signal transistors. Taking the Class-B design, the TO-92 MPSA06 devices have an Ft of 100 MHz, the MJE340 drivers about 15 MHz (for some reason this parameter is missing from the data sheet), and the MJ802 output devices an Ft of 2.0 MHz. Clearly the output stage is the prime suspect. The next question is at what frequencies these poles exist. There is no reason to suspect that each transistor can be modeled by one simple pole.

There is a huge body of knowledge devoted to the art of keeping feedback loops stable while optimizing their accuracy; this is called control theory, and any technical bookshop will yield some intimidatingly fat volumes called things like “control system design.” Inside, system stability is tackled by Laplace-domain analysis, eigenmatrix methods, and joys such as the Lyapunov stability criterion. I think that makes it clear that you need to be pretty good at mathematics to appreciate this kind of approach.

Even so, it is puzzling that there seems to have been so little application of control theory to audio amplifier design. The reason may be that so much control theory assumes that you know fairly accurately the characteristics of what you are trying to control, especially in terms of poles and zeros.

One approach to appreciating NFB and its stability problems is SPICE simulation. Some SPICE simulators have the ability to work in the Laplace or s-domain, but my own experiences with this have been deeply unhappy. Otherwise respectable simulator packages output complete rubbish in this mode. Quite what the issues are here I do not know, but it does seem that s-domain methods are best avoided. The approach suggested here instead models poles directly as poles, using RC networks to generate the time constants. This requires minimal mathematics and is far more robust. Almost any SPICE simulator, evaluation versions included, should be able to handle the simple circuit used here.

Figure 12.1 shows the basic model, with SPICE node numbers. The scheme is to idealize the situation enough to highlight the basic issues and exclude distractions such as nonlinearities or clipping. The forward gain is simply the transconductance of the input stage multiplied by the transadmittance of the VAS integrator. An important point is that with correct parameter values, the current from the input stage is realistic, as are all the voltages.

Figure 12.1. Block diagram of system for SPICE stability testing.

The input differential amplifier is represented by G. This is a standard SPICE element—the VCIS, or voltage-controlled current source. It is inherently differential, as the output current from Node 4 is the scaled difference between the voltages at Nodes 3 and 7. The scaling factor of 0.009 sets the input stage transconductance (g^m) to 9 mA/V, a typical figure for a bipolar input with some local feedback. Stability in an amplifier depends on the amount of NFB available at 20 kHz. This is set at the design stage by choosing the input g^m and C^dom, which are the only two factors affecting the open-loop gain. In simulation it would be equally valid to change g^m instead; however, in real life it is easier to alter C^dom as the only other parameter this affects is slew rate. Changing input stage transconductance is likely to mean altering the standing current and the amount of local feedback, which will in turn impact input stage linearity.

The VAS with its dominant pole is modeled by the integrator E^vas, which is given a high but finite open-loop gain, so there really is a dominant pole P¹ created when the gain demanded becomes equal to that available. With C^dom=100 pF, this is below 1 Hz. With infinite (or as near-infinite as SPICE allows) open-loop gain, the stage would be a perfect integrator. As explained elsewhere, the amount of open-loop gain available in real versions of this stage is not a well-controlled quantity, and P¹ is liable to wander about in the 1- to 100-Hz region; fortunately, this has no effect at all on HF stability. C^dom is the Miller capacitor that defines the transadmittance, and since the input stage has a realistic transconductance, C^dom can be set to 100 pF, its usual real-life value. Even with this simple model we have a nested feedback loop. This apparent complication here has little effect, as long as the open-loop gain of the VAS is kept high.

The output stage is modeled as a unity-gain buffer, to which we add extra poles modeled by R¹, C¹ and R², C². E^out1 is a unity-gain buffer internal to the output stage model, added so that the second pole does not load the first. The second buffer E^out2 is not strictly necessary as no real loads are being driven, but it is convenient if extra complications are introduced later. Both are shown here as a part of the output stage but the first pole could equally well be due to input stage limitations instead; the order in which the poles are connected makes no difference to the final output. Strictly speaking, it would be more accurate to give the output stage a gain of 0.95, but this is so small a factor that it can be ignored.

The component values here are of course completely unrealistic and chosen purely to make the math simple. It is easy to appreciate that 1 Ω and 1 μF make up a 1-μs time constant. This is a pole at 159 kHz. Remember that the voltages in the latter half of the circuit are realistic, but the currents most certainly are not.

The feedback network is represented simply by scaling the output as it is fed back to the input stage. The closed-loop gain is set to 23 times, which is representative of most power amplifiers.

Note that this is strictly a linear model, so that the slew-rate limiting associated with Miller compensation is not modeled here. It would be done by placing limits on the amount of current that can flow in and out of the input stage.

Figure 12.2 shows the response to a 1-V step input, with the dominant pole the only time element in the circuit. (The other poles are disabled by making C¹, C² 0.00001 pF, because this is quicker than changing the actual circuit.) The output is an exponential rise to an asymptote of 23 V, which is exactly what elementary theory predicts. The exponential shape comes from the way that the error signal that drives the integrator becomes less as the output approaches the desired level. The error, in the shape of the output current from G, is the smaller signal shown; it has been multiplied by 1000 to get mA onto the same scale as volts. The speed of response is inversely proportional to the size of C^dom and is shown here for values of 50 and 220 pF as well as the standard 100 pF. This simulation technique works well in the frequency domain, as well as the time domain. Simply tell SPICE to run an AC simulation instead of a TRANS (transient) simulation. The frequency response in Figure 12.3 exploits this to show how the closed-loop gain in an NFB amplifier depends on the open-loop gain available. Once more elementary feedback theory is brought to life. The value of C^dom controls the bandwidth, and it can be seen that the values used in the simulation do not give a very extended response compared with a 20-kHz audio bandwidth.

Figure 12.2. SPICE results in the time domain. As C^dom increases, the response V(7) becomes slower and the error (g1) declines more slowly. The input is the step-function V(3) at the bottom.

Figure 12.3. SPICE simulation in the frequency domain. As the compensation capacitor is increased, the closed-loop bandwidth decreases proportionally.

In Figure 12.4, one extra pole P² at 1.59 MHz (a time constant of only 100 ns) is added to the output stage, and C^dom stepped through 50, 100, and 200 pF as before. 100 pF shows a slight overshoot that was not there before; with 50 pF there is a serious overshoot that does not bode well for the frequency response. Actually, it's not that bad; Figure 12.5 returns to the frequency-response domain to show that an apparently vicious overshoot is actually associated with a very mild peaking in the frequency domain.

Figure 12.4. Adding a second pole P² causes overshoot with smaller values C^dom, but cannot bring about sustained oscillation.

Figure 12.5. The frequency responses that go with the transient plots of Figure 12.4. The response peaking for C^dom=50 pF is very small compared with the transient overshoot.

From here on C^dom is left set to 100 pF, which is its real value in most cases. In Figure 12.6, P² is stepped instead, increasing from 100 ns to 5 μs, and while the response gets slower and shows more overshoot, the system does not become unstable. The reason is simple: sustained oscillation (as opposed to transient ringing) in a feedback loop requires positive feedback, which means that a total phase shift of 180° must have accumulated in the forward path and reversed the phase of the feedback connection. With only two poles in a system the phase shift cannot reach 180°. The VAS integrator gives a dependable 90° phase shift above P¹, being an integrator, but P² is instead a simple lag and can only give a 90° phase lag at infinite frequency. So, even this very simple model gives some insight. Real amplifiers do oscillate if C^dom is too small, so we know that the frequency response of the output stage cannot be modeled meaningfully with one simple lag.

Figure 12.6. Manipulating the P² frequency can make ringing more prolonged but it is still not possible to provoke sustained oscillation.

A certain president of the United States is alleged to have said: “Two wrongs don't make a right—so let's see if three will do it.” Adding in a third pole, P³, in the shape of another simple lag gives the possibility of sustained oscillation.

Stepping the value of P² from 0.1 to 5 μs with P³=500 ns shows sustained oscillation starting to occur at P²=0.45 μs. For values such as P²=0.2 μs, the system is stable and shows only damped oscillation. Figure 12.7 shows over 50 μs what happens when the amplifier is made very unstable (there are degrees of this) by setting P²=5 μs and P³=500 ns. It still takes time for the oscillation to develop, but exponentially diverging oscillation like this is a sure sign of disaster. Even in the short time examined here the amplitude has exceeded a rather theoretical half a kilovolt. In reality, oscillation cannot increase indefinitely, if only because the supply rail voltages would limit the amplitude. In practice, slew rate limiting is probably the major controlling factor in the amplitude of high-frequency oscillation.

Figure 12.7. Adding a third pole makes possible true instability with an exponentially increasing amplitude of oscillation. Note the unrealistic voltage scale on this plot.

We have now modeled a system that will show instability. But does it do it right? Sadly, no. The oscillation is about 200 kHz, which is a rather lower frequency than is usually seen when an amplifier misbehaves. This low frequency stems from the low P² frequency we have to use to provoke oscillation; apart from anything else this seems out of line with the known Ft of power transistors. Practical amplifiers are likely to take off at around 500 kHz to 1 MHz when C^dom is reduced, which seems to suggest that a phase shift is accumulating quickly at this sort of frequency. One possible explanation is that there are a large number of poles close together at a relatively high frequency.

A fourth pole can be simply added to Figure 12.1 by inserting another RC–buffer combination into the system. With P²=0.5 μs and P³=P4=0.2 μs, instability occurs at 345 kHz, which is a step toward a realistic frequency of oscillation. This is case B in Table 12.1.

When a fifth output stage pole is grafted on, so that P³=P⁴=P⁵=0.2 μs, the system just oscillates at 500 kHz with P² set to 0.01 μs. This takes us close to a realistic frequency of oscillation. Rearranging the order of poles so that P²=P³=P⁴=0.2 μs, while P⁵=0.01 μs, is tidier and the stability results are of course the same; this is a linear system so the order does not matter. This is case C in Table 12.1.

Table 12.1. Instability Onset: P2 is Increased Until Sustained Oscillation Occurs

Case	Cdom	P2	P3	P4	P5	P6
A	100 p	0.45	0.5	–	–		200 kHz
B	100 p	0.5	0.2	0.2	–		345 kHz
C	100 p	0.2	0.2	0.2	0.01		500 kHz
D	100 p	0.3	0.2	0.1	0.05		400 kHz
E	100 p	0.4	0.2	0.1	0.01		370 kHz
F	100 p	0.2	0.2	0.1	0.05	0.02	475 kHz

Having P², P³, and P⁴ all at the same frequency does not seem very plausible in physical terms, so case D shows what happens when the five poles are staggered in frequency. P² needs to be increased to 0.3 μs to start the oscillation, which is now at 400 kHz. Case E is another version with five poles, showing that if P⁵ is reduced, P² needs to be doubled to 04 μs for instability to begin.

In the final case F, a sixth pole is added to see if this permitted sustained oscillation is above 500 kHz. This seems not to be the case; the highest frequency that could be obtained after a lot of pole twiddling was 475 kHz. This makes it clear that this model is of limited accuracy (as indeed are all models—it is a matter of degree) at high frequencies and that further refinement is required to gain further insight.

12.2. Maximizing Negative Feedback

Having freed ourselves from fear of feedback, and appreciating the dangers of using only a little of it, the next step is to see how much can be used. It is my view that the amount of NFB applied should be maximized at all audio frequencies to maximize linearity, and the only limit is the requirement for reliable HF stability. In fact, global or Nyquist oscillation is not normally a difficult design problem in power amplifiers; the HF feedback factor can be calculated simply and accurately, and set to whatever figure is considered safe. (Local oscillations and parasitics are beyond the reach of design calculations and simulations and cause much more trouble in practice.)

In classical control theory, the stability of a servomechanism is specified by its phase margin, the amount of extra phase shift that would be required to induce sustained oscillation, and its gain margin, the amount by which the open-loop gain would need to be increased for the same result. These concepts are not very useful in amplifier work, where many of the significant time constants are known only vaguely. However, it is worth remembering that the phase margin will never be better than 90° because of the phase lag caused by the VAS Miller capacitor; fortunately, this is more than adequate.

In practice, the designer must use his judgment and experience to determine an NFB factor that will give reliable stability in production. My own experience leads me to believe that when the conventional three-stage architecture is used, 30 dB of global feedback at 20 kHz is safe, providing an output inductor is used to prevent capacitive loads from eroding the stability margins. I would say that 40 dB was distinctly risky, and I would not care to pin it down any more closely than that.

The 30-dB figure assumes simple dominant-pole compensation with a 6-dB/octave roll-off for the open-loop gain. The phase and gain margins are determined by the angle at which this slope cuts the horizontal unity loop-gain line. (I am deliberately terse here; almost all textbooks give a very full treatment of this stability criterion.) An intersection of 12 dB/octave is definitely unstable. Working within this, there are two basic ways in which to maximize the NFB factor.

1. While a 12-dB/octave gain slope is unstable, intermediate slopes greater than 6 dB/octave can be made to work. The maximum usable slope is normally considered to be 10 dB/octave, which gives a phase margin of 30°.This may be acceptable in some cases, but I think it cuts it a little fine. The steeper fall in gain means that more NFB is applied at lower frequencies and so less distortion is produced. Electronic circuitry only provides slopes in multiples of 6 dB/octave, so 10 dB/octave requires multiple overlapping time constants to approximate a straight line at an intermediate slope. This gets complicated, and this method of maximizing NFB is not popular.
2. The gain slope varies with frequency so that maximum open-loop gain and hence NFB factor is sustained as long as possible as frequency increases; the gain then drops quickly, at 12 dB/octave or more, but flattens out to 6 dB/octave before it reaches the critical unity loop-gain intersection. In this case the stability margins should be relatively unchanged compared with the conventional situation.

12.3. Maximizing Linearity Before Feedback

Make your amplifier as linear as possible before applying NFB has long been a cliché. It blithely ignores the difficulty of running a typical solid-state amplifier without any feedback to determine its basic linearity.

Virtually no dependable advice on how to perform this desirable linearization has been published. The two factors are the basic linearity of the forward path and the amount of NFB applied to further straighten it out. The latter cannot be increased beyond certain limits or else high-frequency stability is put in peril, whereas there seems no reason why open-loop linearity could not be improved without limit, leading us to what in some senses must be the ultimate goal—a distortionless amplifier. This book therefore takes as one of its main aims the understanding and improvement of open-loop linearity.

Further Reading

Attwood, November 1983 Attwood B., Design parameters important for the optimisation of PWM (Class-D) amplifiers JAES 31 November 1983842-

Baxandall, Dec. 1978 Baxandall P., Audio power amplifier design: Part 5 Wireless World Dec. 197853- (This superb series of articles had six parts and ran on roughly alternate months, starting in January 1978.)

Blomley, February 1971 Blomley P., A new approach to Class-B Wireless World February 197157-

Buitendijk, 1991 Buitendijk P., A 40W Integrated car radio audio amplifier IEEE Conf on Consumer Electronics 1991174- Session THAM 12.4 (Class-H)

Cherry, May 1982 Cherry E., Nested differentiating feedback loops in simple audio power amplifiers JAES 30 5 May 1982295-

Evenson, November 1988 Evenson R., Audio amplifiers with tailored output impedances November 1988 AES convention (Los Angeles) Preprint for

Feldman, August 1976 Feldman L., Class-G high-efficiency hi-fi amplifier Radio-Electronics August 197647-

Gilbert, 1990 Gilbert, B. (1990). Current mode circuits from a translinear viewpoint Ch 2, Analogue IC design: The current-mode approach ed Toumazou. Lidgey & Haigh, IEE.

Goldberg and Sandler, February 1991 Goldberg, Sandler, Noise shaping and pulse-width modulation for all-digital audio power amplifier JAES 39 February 1991449-

Hancock, September 1991 Hancock J., A Class-D amplifier using MOSFETS with reduced minority carrier lifetime JAES 39 September 1991650-

Lin, September 1956 Lin H.C., Transistor audio amplifier Electronics September 1956173-

Linsley-Hood, April 1969 Linsley-HoodSimple Class-A amplifier Wireless World April 1969148-

Mills and Hawksford, March 1989 Mills, Hawksford, Transconductance power amplifier systems for current-driven loudspeakers JAES 37 March 1989809-

Olsson, December 1994 Olsson B., Better audio from non-complements? Electronics World December 1994988-

Peters, June 1975 Peters A., Class E RF amplifiers IEEE J. Solid-State Circuits June 1975168-

Raab, May 1986 Raab F., Average efficiency of Class-G power amplifiers IEEE Transactions on Consumer Electronics CE-22 May 1986145-

Sampei, August 1978 Sampei, ...Highest efficiency & super quality audio amplifier using MOS-power FETs in Class-G IEEE Transactions on Consumer Electronics CE-24 August 1978300-

Sandman, September 1982 Sandman, A Class S: A novel approach to amplifier Distortion Wireless World September 198238-

Sinclair et al., 1993 Sinclair, Audio and hi-fi handbook, Newnes, p. 541, 1993.

Stochino, October 1994 Stochino G., Audio design leaps forward? Electronics World October 1994818-

Sweeney and Mantz, June 1988 Sweeney, Mantz, An informal history of amplifiers Audio June 198846-

Tanaka, January/February 1981 Tanaka S.A., New biasing circuit for Class-B operation JAES January/February 198127-

Thus et al., 1992 Thus, Compact bipolar Class AB output stage, IEEE Journal of Solid-State Circuits, p. 1718, December 1992.

Walker, December 1975 Walker P.J., Current dumping audio amplifier Wireless World December 1975560-