Capacitor
Distortion

Applying a sine wave voltage and measuring the distortion in the current
through a capacitor can provide figures for harmonic distortion. At zero dc
bias non-polarised types with symmetrical construction should have mostly
third harmonic, 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.

Small levels of low order harmonic distortion should be of little concern,
but are there other problems beyond this? Dielectric absorption is often
mentioned as a cause of distortion, but the usual model of this effect is
entirely linear:

This 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. Different types of capacitor will have different equivalent
circuits, but the only effects of this in coupling capacitor applications are
small amplitude and phase variations.

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 a
coupling capacitor 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.

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 for the MJR7
amplifier is to use input filters with -1dB around 12Hz and 30kHz. 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.
My own MJR7 has a second order low-pass response, and the phase shift is
almost perfectly linear from 1kHz to 20kHz, equivalent to a constant time
delay of 3.3usec.

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 coupling capacitors in
other parts of an amplifier which handle much smaller signal currents, though
it may do no harm to choose types reputed to have lower distortion than
others, apart from possibly higher cost and also increased interference pickup
from physically larger sizes.

In my MJR6 and MJR7 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 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 470pF good types are polypropylene, polystyrene and NPO/COG ceramic.
The 2.2uF input coupling capacitor is more of a problem, polypropylene at
this value are both large and expensive. My tests revealed that a large
physical size for this component can lead to a large increase in interference
pickup, particularly if the signal source impedance is high.

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.

Transients

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.

The pulse test can be made more sensitive by subtracting the different
outputs with and without the DA effect added, and the result is shown
next.

The difference is more obvious now, but do real capacitors give the same
sort of result? It appears that they do. An article in 'The Audio amateur', A
Real-Time Signal Test for Capacitor Quality (1985) shows the difference
signal extracted for several pairs of capacitors, and apart from a short spike
at the pulse edge the results have a more or less similar shape to my
simulation result.
Unfortunately, just looking at this result we can no
longer see which of the two outputs is more accurate, and we could easily
assume the capacitor with DA is adding this difference signal compared to the
output of the 'ideal' capacitor, when it is the capacitor with DA which gives
a slightly more accurate output.

So is there a better approach? If we want to get some idea of how the
actual sounds differ it may be more instructive to look at the individual
error voltages, which are just the difference between the input voltage and
the output voltage. Ideally the input and output will be identical, so the
differences would be zero, but the two results are as shown
next:

The capacitor with DA has the error voltage shown in green, and this has a
lower magnitude peak value, but becomes slightly greater at around 40msec. It
is difficult to imagine why one of these error voltages would cause the
'smeared transients' and other audible effects supposedly caused by DA but the
other would not.

I could find no examples of testing real capacitors by extracting the
difference between input and output, so I tried a few tests myself, Capacitor
Distortion Part 2, using a 'musical' test signal. The results were not
particularly revealing, but given the simulation results on this page that is
not too surprising.

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 may be real causes of audible
differences, one I have observed is the increased pickup of interference by a
capacitor with large physical size, which is why I recommend small polyester
input capacitors for my amplifier designs rather than big polypropylene
types.

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 in some applications. The distortion levels are greatest when there is
significant signal voltage across the capacitor, so is less of a problem for
signal coupling where the idea is to avoid signal loss across the capacitor by
choosing a sufficiently high value.

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