Random Noise

Being, educationally, from a Physics background I missed learning a lot of the sort of things any good electronics engineer would be expected to know about. Being also somewhat lazy I don't feel any enthusiasm for going back and filling in the missing knowledge, unless I find some good reason why I should want to. Wandering around the audio sites on the internet I encountered the name 'Middlebrook' with reference to feedback analysis. I don't know if this is something I should want to know about. I started by looking at an article 'The General Feedback Theorem: A Final Solution for Feedback Systems' This at least starts with a statement of a problem, that 'the analysis methods you learned in college simply don't work'. I don't remember ever learning any analysis methods, but if there are colleges teaching methods which don't work I hope someone has told them.

What I think is true is that there are often several different ways to approach a problem and different people understand things in different ways. One example; my analysis of the RIAA feedback network with 4 time-constants was a page or two of 'simple' algebra, and yet one of my references was to an analysis using 'two-port network parameters' and 'synthesized using a Foster expansion', which again I don't know much about. The final results were identical, so they are just two different ways to solve the same problem.
There are other examples, my explanation of the Quad 'current dumping' method on my MJR9 amplifier page ignores the usual bridge balance approach.
In Physics having two different but equivalent approaches can happen even at a fundamental theory level, and has been given a name: 'duality'.

Another example, a different way to think about feedback, as a way to generate an inverse of a nonlinear transfer function:
Generally, for a nonlinear transfer function there exists an inverse function which will reverse its effect. (There are exceptions, for example there is irreversible loss of information during clipping, slew rate limiting, and class-B dead-zones.) I remember seeing an example with one common-emitter stage driving another similar stage, and it was found that adjusting the relative gains and operating currents the second stage distortion partly cancels that of the input stage.
Apart from the exceptions, if we have two identical nonlinear amplifiers we can put one of them in the feedback loop of an op-amp and generate an inverse function to 'predistort' the signal so that passing it through the second amplifier will return us close to the original undistorted signal. We don't need to go to such lengths, any amplifier with overall negative feedback already generates this sort of predistorted signal at the input of the output stage to at least partly cancel output stage distortion. That is just another way to think about negative feedback, that it generates the inverse function of the output stage nonlinearity at the input of the output stage. To create a perfect inverse would need infinite feedback loop gain, so in practice we never reach zero output distortion.
If the output stage is the primary source of nonlinearity then with heavy feedback applied the relative levels of harmonics of the output signal are almost identical to those of the 'predistorted' signal at the input of the output stage. That is why increasing feedback changes the distortion spectrum, it changes from the spectrum of the original nonlinear function towards the spectrum of the inverse function.
Update: I found where I first saw the example with two emitter-followers partly cancelling their distortions, it was in Wireless World Nov 1972, page 521-522, a 'Letter to the Editor' from G.W.Short, 'High-load transistor voltage amplifiers'. The partial cancellation occurs because the first stage inverts the signal, so the curvature of the transfer function acts in the opposite direction for the second stage. The second stage plus inversion is not a very accurate inverse of the first, but near enough to significantly reduce overall distortion.

Back on 17-Jan-2020 I wrote about my article on elementary EM theory, calling it an advanced idiot's guide, 'that being my own level'. I was reminded of that when reading a piece by a well known mathematician, John Baez, (he is related to the singer Joan Baez, they are cousins). Although I occasionally read his website I find much of it beyond my comprehension. But then, reading his article in Nautilus about algebraic geometry he mentions the classic introductory book by Robin Hartshorne, and says even the first chapter is seriously difficult, 'like trying to catch up with centuries of geniuses running as fast as they could.' The point is that it is easy to be impressed by other people's level of understanding, but less easy to comprehend just how far the scale extends upwards. The Hartshornes and Grothendiecks of the world are so far ahead of most of us, that it's difficult to visualise just how far. Myself, I'm happy to remain just an 'advanced idiot'.

There is very little on this website about subjective effects, personally I was already happy with my sound system more than 30 years ago, and apart from an occasional backward step I remained happy. What I did write on this subject was about testing methods, and I suggested ways to avoid some common errors. Carrying out accurate and meaningful listening tests is difficult, and a useful first step, often omitted, is to determine whether it is possible to play the same music twice without changing anything. That is more difficult than it appears, if we have not intentionally changed anything how could there be any significant change? A useful exercise is to try to think of 10 different effects which could change the acoustic output between two auditions of the 'same' music. I previously mentioned one obvious example, the increase in speaker voice coil temperature after the first play will increase its resistance and reduce the sound level of the second play unless we wait long enough for the temperature to return to its original level. Another example is that if we measure the bass resonance frequency of a speaker before and after playing some music the values will possibly differ, I measured one full range driver changing from 150Hz to 115Hz after a few hours use. Only part of the change is a temperature effect, if we wait an hour for it to cool down the resonance frequency will increase again, but not back to the original value. Are there another 8 effects? I got up to 4 before losing interest, but there must be a few more.

Many years ago I was given a problem to write a computer program to illustrate the operation of a damped mechanical oscillator. That usually involves writing out the relevant differential equation and working out the solutions. I was never very good at that sort of thing, but then I remembered that there are numerical methods for solving differential equations which should work fine for a computer simulation. (I did a course in 'numerical maths' at Durham University back in the days of mechanical adding machines). That involves starting at some time T and putting in initial values for displacement and velocity, then use the equation to work out their values at T+dt, substitute back in the equation, then solve at T+2dt and so on in a loop. The result was exactly what was needed, and demonstrated graphically the difference between under-damped, over-damped and critically damped. This is relevant to the understanding of transient response of feedback amplifiers, it may be easier for the mathematically adept to solve the equations, but an alternative is to put in some starting conditions then go round and round the feedback loop updating the voltages at each time step. We would expect the time interval should be reduced towards zero for greatest accuracy, but there is a time delay round the feedback loop anyway, possibly less than one nsec, so maybe no point going beyond that.
I believe this is more or less how transient analysis works in Spice, but using the 'Backward Euler' method.

I never entirely resolved the question of common-emitter output impedance, but I look out for any useful information. There is a discussion on diyAudio starting at post 10084 about impedance of current sources which looks relevant. There are equations there, which are either wrong or don't apply to zero external emitter resistor, giving R_OUT not affected by RS with R1 = 0, but more interesting a Spice simulation giving a plausible result. My own Spice simulation failed to demonstrate any effect of RS, so maybe some Spice models work better than others, or maybe Spice makes the same 'error' as the equation, and only works well for the current source version. Still puzzled. The simple model I ended up with is really just a variation of the hybrid-pi model, which at least matches the measured results, but it looks like another of those 'wrong but useful' models.

Sept-3-2020: Still doing house repairs, fixing garage roof and so on. Also needed to upgrade the computer, I had Windows XP, but too many problems with not being supported, even BBC iPlayer gave dire warnings. I was given a Windows 7 dvd, which the previous owner had replaced by Windows 10, so I'm still outdated, but the big problem is lack of drivers for my soundcard and TV card. I found an old 'M-Audio Audiophile 2496' card which has Win7 drivers available, so installed that, but there was nothing to be done with the Leadtek TV card, until I found a suggestion to use Windows Media Center, which can work with a TV card to watch live TV. It actually works better than the original Leadtek Winfast PVR. So far all my favourite programs still work ok, even MGI PhotoSuite (Ver.8.06), an ancient Windows 95 program, which I used for all the circuit diagrams on the website. I extracted all the component symbols from various published diagrams. My favourite simulator AIM-Spice also has a Win7 version, so all is well. I also installed a hardware monitor program: CPUID HWMonitor (version 1.41.0). That revealed the processor was running at up to 60 deg, so added a better cooler, and now it's a healthier 30 deg.

Back to amplifier design: There are a few common problems and errors, and I have tried to cover some of these in the theoretical pages. One important error concerns 'the unity gain frequency' of a feedback loop, the problem being that the unity gain frequency is not fixed, it can vary over a wide range close to clipping or slew-rate limiting, and also depends on the load. There may even be more than one unity gain frequency if the load has dips in impedance at various frequencies. Designing for stability is more difficult than just avoiding too much phase shift at one easily specified unity gain frequency.

Another problem is with 'symmetry' and there is a piece in my 'archive' section about some of the problems and limitations. At best symmetry may cancel some even harmonic distortion, but even in my MJR7 with little or no circuit symmetry the second harmonic at 1kHz is under 0.0001%. Even that low level is almost entirely a result of differences between the p-channel and n-channel power mosfets. The seemingly symmetric 'complementary' output pair are actually the greatest source of distortion.

I have seen a few designs in the past few years with input stages similar to my MC Phono Preamp, with npn and pnp input transistors with separate feedback networks. My circuit immediately connects the collectors of these two transistors, but there are power amplifier designs where they drive separate following stages. I only once saw any mention of accurately matching the feedback networks, and see no evidence that anyone has worked out what happens if there is significant mismatch.
Update: ok, having said there is a problem with unmatched feedback networks I should include an opinion about what bad effects are possible. I haven't calculated or simulated, so I claim it only as an 'opinion', but I am fairly sure the operating currents of the following stages will be modulated by the signal, and if this reaches the output stage with nothing to limit the modulation then there could be severe cross-conduction on one half-cycle. In other words high dissipation with risk of failure. With just standard 1% tolerance components and a high feedback loop gain it is just a matter of luck whether such an amplifier works fine or there is enough mismatch to self-destruct. Using a Vbe multiplier to bias the output stage may help prevent excessive cross-conduction, but the driver stage may still be at risk.