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In Reply to:
RE: Phase Distortion, dv/dt, and Slew Rate posted by **Lee of Omaha**
on December 17, 2015 at 09:55:23:

You are correct that some sound waves can line up in phase at certain times, but aside from the harmonics resulting from a single voice or instrument, most won't. As Geoff mentioned, you need some luck. I will try to show that via math at the bottom of this post.

I think what we really care about is crest factor, the ratio of the highest peak level of the music over the RMS (average) level of the music. I've heard that the crest factor of the human voice is around 12 dB, whereas the crest factor for percussion could be over 24 dB. So I would not expect choral music to be the most stressing. Back in the days before 24-bit digital recording, it was typical to use a compressor in the recording chain, at least for percussion. These days, the compression is usually applied after the fact in software. Either way, the end product usually has some compression in it. So the recordings we play back through our systems have crest factors of no more than about 16-18 dB at the high end, in the case of a relatively uncompressed recording from before the loudness wars.

Given the crest factor, the average level you would like to listen at, loudspeaker sensitivity and impedance, how far away you sit from the speakers, and an approximate room gain, you can calculate how much peak amplifier power you need and what the peak voltage will be. Suppose it's 200W and 40V.

Then you pick a maximum frequency. For CD, 22.05 KHz. The formula from my previous post gives a maximum slew rate requirement of 2 * pi * 40 * 22050 = 5.54V/usec. This calculation is conservative because the power spectra of music falls off at high frequency so you would never get a 20KHz component at 0dBFS.

But suppose you want to be ultra conservative. Let's say the amplifier has a wide 100KHz bandwidth and its voltage rails are at 50V. In order to make sure there is no possibility whatsoever of slew rate limiting regardless of what input signal you feed it, you would need a maximum slew rate of 2 * pi * 50 * 100000 = 31.4V/usec. For modern solid state amplifiers where a maximum slew rate specification is available, a typical value is around 40V/usec.

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Here is the math bit I promised:

Start with the formula for a sine wave:

V(t) = A * sin(2*pi*f*(t-t0))

V is volts, A is the amplitude, and 2*pi*f*(t-t0) is the phase. The phase has a time varying term 2*pi*f*t and a constant term 2*pi*f*t0 where t0 is the time offset and f is the frequency of the sine wave.

If you have two sine waves:

V1(t) = A1 * sin(2*pi*f1*(t-t1))

V2(t) = A2 * sin(2*pi*f2*(t-t2))

The sum V(t) = V1(t) + V2(t)

The derivative dV/dt = dV1/dt + dV2/dt

dV1/dt = A1 * cos(2*pi*f1*(t-t1))

dV2/dt = A2 * cos(2*pi*f2*(t-t2))

Now we're concerned with the maximum slew rate, i.e. the maximum absolute value of dV/dt. You can see that IF the phase of the two sine waves lines up, the maximum will be A1 + A2. So the important question is under what conditions will the phase line up.

The maximum of value of cos(x) is 1 and it occurs at values of x=0, x=2*pi, x=4*pi, ... The minimum of value is -1 and it occurs at values of x=pi, x=3*pi, x=5*pi, ...

So the maximum value of dV1/dt occurs at t = t1 + n * 1/f1 where n=0,1,2,3...,

and the maximum value of dV2/dt occurs at t = t2 + m * 1/f2 where m=0,1,2,3...

Likewise, the minimum value of dV1/dt occurs at t = t1 + pi + n * 1/f1 where n=0,1,2,3...,

and the minimum value of dV2/dt occurs at t = t2 + pi + m * 1/f2 where m=0,1,2,3...

In order for a maximum of dV1/dt to line up in time with a maximum of dV2/dt, or alternatively for two minima to line up, there has to be a pair of integers m,n where t1 + n * 1/f1 = t2 + m * 1/f2. Rearrange this to be n = m * f1/f2 + f1*(t2-t1). Since n and m have to be integers, the sum of m * f1/f2 + constant has to result in an integer value.

In the lucky case of t1=t2, there is at least one solution: m=n=0. That is, if t1=t2 then the two maxima are guaranteed to line up at least once at time t=t1=t2 but not necessarily before or after. In the similarly lucky case where f1*(t2-t1) happens to fall on an integer value other than 0, then there are certain combinations of f1 and f2 which will also produce a solution at one time but not necessarily before or after. These lucky occurrences aren't very interesting solutions because they only produce the maximum of A1+A2 at one time value. I'm more interested in knowing under what conditions this can recur, and the answer is time/phase aligned harmonics.

If t1=t2 and f1 is an integer multiple of f2 or vice versa, then you will get a periodic series of solutions. This would occur if f1 is a harmonic of f2 or vice versa. Suppose you have the case of t1=t2 and f1 = 3*f2. Then there will be solutions at t=0, t=3/f1, t=6/f1, and so on.

Any time you are adding harmonics, you will get a pattern repeating at the frequency of the fundamental. But in order for the pattern to include maximum values of A1+A2, the harmonics have to align in time/phase. That itself is a lucky occurence given that the phase response of a microphone is generally not a constant vs. frequency.

Also, this is an analysis of just two sine waves. It can be extended to account for additional frequency components, but as you increase the number of components, the number of solutions which can produce a pathological maximum of A1+A2+A3+... become significantly more rare. Which explains why we don't see these in music.

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Follow Ups

RE: Phase Distortion, dv/dt, and Slew Rate-Dave_K13:57:41 12/18/15(3)

- RE: Phase Distortion, dv/dt, and Slew Rate -
Lee of Omaha14:13:39 12/19/15(2)

- RE: Phase Distortion, dv/dt, and Slew Rate -
Tony Lauck19:06:35 12/21/15(0)

- RE: Phase Distortion, dv/dt, and Slew Rate -
Dave_K07:29:05 12/21/15(0)