Measuring Capacitor Distortion.
Applying a sine wave current and measuring the distortion in the voltage across a capacitor can provide figures for harmonic distortion. At zero dc bias non-polarised types with symmetrical construction have mostly third harmonic, as expected from their symmetry, and higher order harmonics are generally found to be much smaller. Adding a dc bias increases the 2nd harmonic, as expected for any device with a primarily cubic non-linearity. Some capacitor types, e.g. high-k ceramics, are found to have unusually high levels of distortion and should be avoided in some audio applications. Most other types have very low distortion levels compared to other parts of the audio chain. Even electrolytics can have low distortion provided we take care to avoid reverse bias and keep the signal voltage across the capacitor small, which can be ensured by using a sufficiently large value, and keeping the current low, which is easy in some applications, but not in the case of a speaker coupling capacitor. D.Self found that a 'standard' 6800uF driving 40watts into 8ohms added no more than 0.0025% distortion down to 20Hz. Polarised electrolytics are asymmetric, and have a relatively high second harmonic at zero bias, but more surprising is the observation published by C.Bateman that the second harmonic falls as the dc bias is increased up to a certain point, though the minimum value is at an unpredictable and usually low voltage, so this is not generally very useful.
Small levels of low order harmonic distortion should be of little concern, but more worrying are published tests using bridge techniques, and transient type signals, which can show all sorts of apparently serious effects. To give one example, if a reference standard capacitor C2 (teflon seems to be a favourite) is compared to the tested capacitor C1 using a bridge to compare the outputs, and a square or pulse wave test signal applied, then accurate nulling adjusting VR1 is impossible, and what may appear to be a distortion residual remains. With ideal capacitors and resistors exact balance should be possible giving zero output from the bridge. It is known that all capacitors have resistive losses, but even including an adjustable resistor VR2 to try to compensate for this will not completely eliminate the bridge output:
What must be taken into acount in this sort of test is that the transient signal includes frequency components over a wide bandwidth, and the equivalent series resistance of any capacitor is a function of frequency, and so inevitably the single resistor adjustment will be of limited effectiveness. Dielectric absorption is often mentioned as a cause of distortion, but the usual model of this effect is entirely linear:
A widely used equivalent circuit for dielectric absorption in a typical capacitor includes a number of resistors and capacitors. ('Circuits, Systems & Standards' by Bob Pease, reprinted in Electronics World Oct.1992 p832-835). The series resistors typically range from 200k to 1000M in a 1uF mylar capacitor, and the parallel capacitors from 0.0006uF to 0.006uF. At any single frequency this is equivalent to a single capacitor and series resistor, but this equivalent series resistance will be different at different frequencies. Using just a single series resistance to compensate when using a rectangular wave test signal in a bridge circuit some frequency components will be cancelled better than others, and the resulting output will be nothing like the test signal, but almost the entire effect may be caused by small unbalanced amplitude and phase variations.
I have now found and reread an article I read some years ago which partly inspired this page, a pdf file here from 1985. There is little in the article I really disagree with, apart from some typical audiophile excesses, such as passing a rectangular pulse through a 100kHz low-pass filter to 'more realistically' simulate an audio signal. (The slew rate will be at least 10 times greater than any audio signal I have ever measured). It appears to be implied that the differing decay characteristics between different capacitor types after the pulses end is relevant to coupling capacitor applications, but I can see nothing in the article explaining or justifying this in any detail. For any capacitor type the output signal will continue after the input pulse has ended, and it is not clear why one such decay is to be preferred to another, unless one is lower than the other (see footnote 2.), which does not appear to be the claim, merely that they are different, and the difference also continues after the input pulse has ended. Capacitors with lower dielectric absorption appear to be assumed to be in some way better based on this difference. The point is made that different capacitors have different frequency dependance of capacitance and resistance, and their phase shifts will therefore be different functions of frequency, which is about the only result we would expect from a linear effect. Unfortunately this test method produces results which combine both nonlinear distortion and linear DA effects. In a given application such as an input coupling capacitor, the linear effect can be easily measured separately just by measuring the gain and phase shift as a function of frequency, as in the following example:
The phase differences between capacitors may be thought to be a problem, and in some applications careful selection is needed, but in the case of an input coupling capacitor for example, the low frequency phase shift from even an ideal capacitor is in principle undesirable. A typical non-ideal capacitor will merely give a slightly different undesirable phase shift. Both are wrong, but if we compare them this can lead to a surprising conclusion.
Keeping the capacitor fixed at 1uF and adding another 1uF in series with 100k to simulate a rather extreme example of the DA effect we actually find that the amplitude errors are little different, but there is a clear reduction in phase error at low frequencies when the DA is added (shown below in green). It would be misleading to suggest that DA is therefore a good thing. Increasing the coupling capacitor to 10uF will give a far greater reduction of amplitude and phase errors. (Actually, it is debatable whether the 1uF with added DA described here should be regarded as a 1uF or a 2uF capacitor, it depends on how we choose to define or measure capacitance.) Just comparing the two phase graphs the DA effect reduces the phase advance, easily interpreted as a relative phase delay if we mistakenly take the ideal capacitor to be our reference standard, which could lead to the conclusion that DA adds time delay when in this application it actually adds less phase advance. (If anyone is still interested in results using a pulse test signal with a dc component I have added a simple simulation, see footnote 2.)
A more appropriate reference standard would be the direct coupled response, shown in blue, with no amplitude or phase errors. Direct coupling has its own problems, and generally the effects of phase shifts from input filters must be balanced against the increased noise, switch-on thumps from signal sources, danger of interference, offset voltages etc. My own choice is to use input filters with -1dB around 15Hz and 25kHz. Without the rest of the audio chain, particularly the speakers, having flat response down to dc there is little to be gained by using direct coupled amplifiers, and extending the -3dB frequency to something like a tenth that of the speakers is enough to keep the increase in phase errors relatively insignificant.
The high frequency response needs far less extension beyond the nominal 20kHz upper limit, most of the phase error from a typical low-pass filter is equivalent to a constant time delay, which will be inaudible and can be ignored. An article by Dr Leach, The Differential Time-Delay Distortion and Phase-Shift Distortion as Measures of Phase Linearity, examined this and concluded that for a less than 5deg phase nonlinearity up to 20kHz a first-order low-pass filter needs to be -3dB at 35kHz or more, while a second-order Bessel filter can be -3dB as low as 25kHz for the same error, though here the gain error may be considered more important. He suggested that higher order Bessel filters add even less error.
I have never personally heard any difference between different capacitors, which may be either a hearing deficiency on my part, or just a lack of imagination. There are well documented listening tests with relevance to 'capacitor sound'. One example is the famous Quad amplifier test in which the conventional class-B type 303 using an electrolytic output capacitor was compared with the direct coupled feedforward 'current dumping' 405 and the Quad II transformer coupled valve (tube) amplifier. ('Valves versus transistors' by James Moir, Wireless World July 1978 p.55-58.) Using top quality master-tape recordings and two different experienced listening panels no statistically significant differences could be found, either for the group averages or for any individual member. Care had been taken to accurately match gains and avoid clipping to eliminate these common causes of audible difference. The test was originally intended as a challenge to those audio reviewers who claimed to hear clear differences between what were known to be good amplifiers. Peter Walker said that Quad would stake their reputation on the outcome, predicting that no differences could be heard, even though these are radically different designs. In earlier tests ('Dynamic testing of audio amplifiers', Hi-Fi News, Nov.1970, p1655), the distortion of the 303 including output capacitor was extracted while using a music test signal, and the distortion alone without the masking effect of the music was found to be inaudible, and had to be increased many times before becoming audible. If output coupling electrolytic capacitor distortion can have so little effect it seems unlikely that we need to worry too much about capacitors in other parts of an amplifier which handle much smaller signals, though of course it does no harm (other than financially in some cases) to choose types reputed to have lower distortion than others.
The addition of something like 0.0025% low order harmonic distortion resulting from using an output electrolytic would not be expected to be audible, so the results from the Quad tests are not really a big surprise. The failure to observe 'obvious audible effects' when properly conducted blind testing is used is similarly nothing unusual.
In my own recent amplifier designs the speaker coupling capacitor was included in the overall feedback loop, primarily to improve the low frequency damping factor, but this will also minimise distortion, and the low measured distortion figures for the feedforward bjt output amplifier included the effect of this and all the other capacitors. Only the input filtering capacitors are expected to add significant distortion if badly chosen, these being outside the feedback loop. Looking at published distortion measurements by Bateman and others there seems general agreement that at 1nF good types are polystyrene and NPO/COG ceramic. The 2.2uF input coupling capacitor is more of a problem, one of the best is polypropylene, but these are both large and expensive. More convenient in both respects are polyphenylene sulphide. (Available in the UK from Farnell.)
My measured distortion was with a polyester input capacitor. I used a very small 2.2uF 100V polyester, Epcos type B32560 from Farnell. Tests published by C.Bateman included Epcos polyester types, and found third harmonic at -90db at 1kHz at 4V capacitor voltage. The distortion is relative to the voltage across the capacitor, which in my amplifier will be more than 400 times lower at 1kHz, and third harmonic can be expected to be proportional to the square of the signal level, so capacitor distortion relative to total input voltage at 1kHz could be something like -234dB. At 20Hz the figure will be higher, about -142dB. Worrying about this sort of distortion level in capacitors seems pointless when in most audio amplifiers far higher distortion is produced by the nonlinear semiconductor junction capacitances.
The 1978 Quad amplifier tests mentioned above used the Yamaha NS1000 speakers, which puzzled me at the time. These were fairly high sensitivity, and apparently not a difficult load, so why did the participating reviewers from the Hi-Fi press insist on this choice of speaker? I recently saw a review of the Yamaha in the April 1998 issue of Hi-Fi World which mentioned an earlier review, around 1977, in Practical Hi-Fi, which claimed excellent results using a Quad II amplifier, while other reviewers using solid-state amplifiers thought it sounded "fatiguing and fizzy". The amplifiers were claimed to be responsible, but of course these differences were not found in the Quad test. More recently there have been claims that the NS1000 is actually a very difficult load, and so amplifiers capable of very high current into low impedance are needed. The Quad 405 has been criticised for being unable to drive difficult loads of this type, so again Quad's tests were remarkable for failing to reveal these claimed problems. Of course Quad were careful to operate the amplifiers below their clipping level, and turning up the volume too far will make anything sound terrible.
I did a simple simulation to see how the DA effect changes pulse response. The pulse is a single 1V positive pulse starting from zero and returning to zero after 5 msec. This pulse is applied to a capacitor which drives a 1k load. The green trace is the output across the resistor for an ideal 8uF capacitor and the red trace is for the same capacitor with a series 1uF plus 10k connected in parallel to simulate a DA effect, similar to the circuit used earlier to investigate phase shifts. For clarity the plots start 1msec before the start of the input pulse.
It can be seen that the two results are very similar, both have significant negative output voltage for some time after the input pulse has ended at 6msec. The capacitor plus DA has slightly less of this continuing output and so if we wanted to minimise this 'error' we again find that DA could actually be a benefit. Again however a far greater improvement can be achieved just by increasing the capacitor value. Real capacitors are more complex, but extracting and amplifying the small difference between two capacitors seems unhelpful as a method of choosing audio frequency coupling capacitors, where linear effects such as DA generally do no harm. Using a higher value coupling capacitor can be expected to reduce errors, while choosing one with low DA is not necessarily of any benefit, and may even increase errors slightly.
There is some evidence that capacitor types with high dielectric absorption also tend to have relatively high distortion, possibly because both effects can be worse with materials having polar molecules. The distortion is usually nothing much worse than low levels of third harmonic, and is only at a serious level in a few types, such as high-k ceramics, which should certainly be avoided. The distortion levels are greatest when there is significant signal voltage across the capacitor, so is less of a problem in coupling applications where the idea is to avoid signal loss across the capacitor by choosing a sufficiently high value.