I first observed phase intermodulation in an audio amplifier when I was studying electronics at the University of Wales, around 1977, and an example from that time is shown near the end of this page. My finding then was that in even a simple amplifier the phase intermodulation effect can be very small. It is known that phase intermodulation is produced in significant quantities by all moving diaphragm type speakers because of the Doppler effect, so avoiding the effect completely is difficult.
There is a slightly different but related effect, which was described by Barrie Gilbert here. This is an effect which occurs even with a single sine-wave for which the closed-loop phase shift is a function not only of frequency but also of amplitude. (See footnote at end of page for a few comments about the cause of this effect.) My use of the term phase intermodulation (PIM) concerns the modulation of the phase of a low level high frequency component by a high level lower frequency component. I will here concentrate on this conventional PIM, but guess that the methods of avoiding these two effects will be more or less the same. (The footnote gives one example which I believe will eliminate the 'Gilbert effect' but not avoid conventional PIM.)
Two mechanisms which can be responsible for significant levels of PIM are illustrated in the following equivalent of a commonly used power amplifier circuit:
The differential input stage has mutual conductance gm. Assuming the second stage to have sufficient open-loop stage gain over the frequency range of interest, the amplifier open loop gain is given by gm/jwC. The closed loop gain is defined by the feedback resistors R1 and R2, and is fairly flat at low frequencies where the loop gain is high. The closed-loop gain and corresponding phase angle are shown in the next diagram:
The -3dB closed-loop frequency fc is proportional to gm/C.
Both gm and C may vary with signal voltage. Input stage gm variation is analysed in the 'Input Stage Distortion' article. For a single transistor second stage C includes the collector-base capacitance which has a highly non-linear voltage dependance. Any variation in gm or C resulting from an input signal will change the -3dB closed loop frequency, and so both amplitude and phase diagrams are moved backwards and forwards along the frequency axis. At any given frequency, particularly at high frequencies close to fc, the gain and phase shift will then vary with signal amplitude. A large low frequency signal will, for example, modulate both the phase and amplitude of a smaller high frequency signal, and so there will be both amplitude and phase intermodulation distortion.
The solution to this problem is not difficult, just improve the input stage linearity with local feedback, and either use a cascode second stage or add an emitter follower in front of the second stage so that the non-linear capacitance feeds back into a low impedance and has little effect.
One 'solution' sometimes suggested, which will help very little if at all, is to add a resistor in parallel with the compensation capacitor C, and some other equivalent methods intended to increase the open-loop -3dB frequency. The addition of any resistor in parallel with C will increase the input stage differential input at all frequencies, and therefore increases the nonlinearity in this stage, while also reducing distortion reduction from overall negative feedback. Even ignoring this increase in input stage nonlinearity, I did an approximate calculation here which showed that adding a resistor to increase open-loop -3dB frequency to 20kHz when using only 20dB feedback gave a reduction in phase modulation only by a factor of 1.1. As in the case of TID it is not the open-loop bandwidth we need to worry about, it is the feedback loop gain which we need to maximise at audio frequencies. If stability considerations permit this the reduction of C will be of benefit, and what this is doing is increasing the gain-bandwidth product, which is also what we need to increase to reduce TID.
The maths is not too difficult, but to avoid the maths entirely we only need to observe that the frequency and phase responses above are the closed-loop responses, and the effect of variations in fc will be greater the closer we get to fc, and will reduce the further below fc we go. To keep the effect low at 20kHz we need fc to be far greater than 20kHz. This can be achieved, for example, with a mosfet output stage and heavy overall feedback, which can give 60dB or more feedback loop gain at 20kHz and virtually flat amplitude and phase at that frequency with very low sensitivity to small changes in gm or C. The open-loop gain can be allowed to increase almost indefinitely with reducing frequency. As with TID the low feedback, wide open-loop -3dB bandwidth approach appears to be unhelpful. Actually, even the closed-loop -3dB frequency is not entirely relevant, because if we add a capacitor across the overall feedback resistor the -3dB closed loop frequency can be reduced without increasing PIM.
Another approach is to take the high frequency compensation capacitor back to the input as in the following example:
The following circuit is the class-B amplifier I used in 1977 for my distortion tests. The bridge-nulling technique was used to extract the distortion. Harmonic distortion at 2kHz was only 0.0006% at 7V rms output, but one reason it was so low is that a 22ohm resistive load was used. The purpose was to test the distortion extraction circuit, not the amplifier, which was just made as an example. It uses Darlington output transistors, and a two transistor driver stage.
The 15pF high frequency compensation capacitor is connected to the input stage, an idea I adapted from an earlier design by John Linsley Hood (Hi Fi News Nov.1972-Feb.1973). I found an even earlier use of a similar method in '30 watt High Fidelity Amplifier' by Arthur R. Bailey, (Wireless World, May 1968), which takes a compensation capacitor to the input transistor emitter in a non-inverting design but has similar advantages. The article also mentions the problem of non-linear driver stage transistor capacitance, and the importance of selecting transistor type to minimise this. This high frequency compensation method helps to reduce slew rate and TID problems. The open loop gain is now primarily determined by a fixed passive resistor and capacitor, almost entirely avoiding the PIM mechanism described above. (In practice this may not be totally effective because it is sometimes necessary to add components such as the 3n3 plus 10R in the diagram above to stabilise the feedback loop.) Phase effects still exist in the amplifier tested, including those caused by the various non-linear transistor capacitances, as shown in the following test results.
The test signal is a sum of 2kHz and 20kHz sine waves, each 5V peak amplitude, shown in the top traces. The distortion is shown in the lower traces, and clearly there is some breakthrough of undistorted signal here. What can still be clearly seen is that the 20kHz signal is cancelled well only over part of the 2kHz wave in the first diagram. Adjusting amplitude compensation in the bridge circuit could make no improvement to the uncancelled signal, but adjusting the phase gave the second diagram, where the cancellation is improved over one part of the 2kHz wave but worsens over the part where it was originally good. This shows that the phase of the 20kHz signal is being modulated by the 2kHz signal. The relative amplification of the lower traces compared to the upper traces is around 80dB, and the calculated peak to peak amplitude of phase modulation is about 0.006 degrees. That such a low level can be detected shows that this is a very sensitive test method. For comparison, a 5kHz signal from a speaker drive unit in the presence of a lower frequency causing just a 1mm cone excursion will have phase modulated by 6 degrees, i.e. a thousand times higher than the effect of this simple amplifier.
To summarise, both TID and phase intermodulation in the audio frequency range can be reduced by an increase in gain-bandwidth product, but in addition to this local feedback to linearise the input stage and steps to reduce the effects of non-linear capacitances can be helpful.
The effect described by Barrie Gilbert in the link given above includes a mathematical analysis, which produces a formula for phase shift which includes both frequency and amplitude squared. His analysis may give the right answer but I find it difficult to get any real understanding from it of what is actually causing the effect. My impression is that it involves nothing more than changes in average input stage gm with signal amplitude, leading to the sort of phase shifts I described earlier. If this is correct then I think I am right to guess that this effect will be smaller than the intermodulation effect I have been writing about because the average gain varies less than the instantaneous gain.
If my understanding is correct there is an alternative way to avoid the 'Gilbert effect' which is to use a square-law device as the input stage, e.g. a single undegenerated fet. For a sine wave applied to a square-law device there is a second harmonic and a dc term in its output, both of which are proportional to signal amplitude squared, but the fundamental term has the same gain at all signal levels. With the more common differential stage using either fets or bjts, there will be a cubic term, (and other higher order terms), and then there is an additional output term at the fundamental frequency proportional to input amplitude cubed. The gain of the stage therefore varies with signal amplitude as shown here. I read somewhere (maybe on DiyAudio) that the 'Gilbert effect' is associated with third harmonic distortion, so maybe I am right in thinking a square-law device will avoid it. (Sept.2007... I have now found the the relevant thread on DiyAudio, and one contributor does suggest that a square-law device at the input will avoid the Gilbert effect. The reason I am not entirely sure about this is that the closed-loop distortion resulting from a square-law input device includes a third harmonic term, even though this is absent in the open-loop distortion. Any effect from this however seems likely to be very small.) The square-law device unfortunately only avoids (or reduces) the phase shift effect for a single sine-wave, but with two frequencies there will still be phase intermodulation. To reduce both effects an input stage having only small variation of gm with changing output current level is needed, and this was compared for different input stages in my 'Input Stage Distortion' article. The best of those compared was the bjt complementary feedback pair, which has primarily second harmonic distortion, which as suggested may also be some advantage.
The conclusions in the Gilbert article about requiring very high maximum slew rates or avoiding high feedback seem to apply under a very limited range of conditions if at all. To reduce both PIM and TID a highly linear input stage is more generally useful, and with very high overall feedback loop gain the input stage signal will be very small, then input stage distortion will be very low. The 'JLH method' of high frequency compensation mentioned above is an alternative approach.
To simplify the maths the analysis used by Barrie Gilbert works out the distorted input needed to produce an undistorted output, the idea being that this will be similar to the output distortion for an undistorted input, and I was glad to see he checked the results using a simulation. The same method was used many years ago by Peter Baxandall, and I remember not being at all convinced then about its validity. An obvious example is if we apply a sinewave to a square-law device, when the output will have only second harmonic distortion. Working out the input signal needed to produce an undistorted output however we then find an infinite series of harmonics are required. Not quite the same thing.