Why is distortion harmonic?
- Thread starter dsavitsk
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dsavitsk said:
Good question. It has to do with the shape of the transfer characteristic of the devices. Suppose you have a pure squarelaw device and you input a sine wave with frequency f: sin(wt), where w=omega, or 2*pi*f. The square law means that at the output you get sin^2(wt). From mathematics we know that a sin^2(wt) = sin(2wt). There's your 2nd harmonic....
Amplifiers are never pure square law, but sometimes there are also 3rd order laws etc. All this leads to various mixtures of the fundamental and harmonics. But it can only be integer harmonics because of the mathematics involved.
Jan Didden
I guess the simple answer is that distortion products are mathematically related. With only one frequency to work with there's a finite number of relationships which are multiples. With real world music, the possible relationships become much more complex and is called intermodulation distortion. (Not to mention transient IM from feedback.)
When you put a single tone into an amplifier any "bad stuff" that happens to that tone happens at the same rate as the tone. IE 1000 times per second. Most distortions happen as the signal cross through the zero axis, or as they reverse direction, so it is easy to see how there are multiple opportunities for "stuff" to happen per cycle. These distortions are often called "harmonic distortion" for this reason.
Music is not just a single tone though. When you put two tones through an amplifier there will be some mixing of the two tones. If you put 100 Hz and 1000 Hz through a perfect amplifier you would get these two tones out. A real world amplifier would also give you some 900 Hz and some 1100 Hz from the intermingling of the two tones. These distortions are called "intermodulation distortion".
A real amplifier will have both types of distortion, so you will get 100 Hz, 1000 Hz (desired), 200 Hz, 2000 Hz (2nd harmonic) 900 Hz, 1100 Hz (IMD), 2100 Hz, 1900 Hz 1200 Hz 800 Hz (IMD of the 2nd harmonic) and so on.
Consider the IMD products of all of the harmonics and the IMD products from the residual power line hum and its harmonics (60 Hz, 120 Hz, 180 Hz etc.) and you can see how the picture (and the sound) gets cloudy real quick.
These are two common types of distortion that can be observed in steady state with an analyzer. There are other types of distortion that occur on transients and phase related distortions that are not easilly measured or observed. Music is composed of multiple tones and transients that change quickly. There are lots of opportunities for something to change, some of which aren't completely understood yet.
Music is not just a single tone though. When you put two tones through an amplifier there will be some mixing of the two tones. If you put 100 Hz and 1000 Hz through a perfect amplifier you would get these two tones out. A real world amplifier would also give you some 900 Hz and some 1100 Hz from the intermingling of the two tones. These distortions are called "intermodulation distortion".
A real amplifier will have both types of distortion, so you will get 100 Hz, 1000 Hz (desired), 200 Hz, 2000 Hz (2nd harmonic) 900 Hz, 1100 Hz (IMD), 2100 Hz, 1900 Hz 1200 Hz 800 Hz (IMD of the 2nd harmonic) and so on.
Consider the IMD products of all of the harmonics and the IMD products from the residual power line hum and its harmonics (60 Hz, 120 Hz, 180 Hz etc.) and you can see how the picture (and the sound) gets cloudy real quick.
These are two common types of distortion that can be observed in steady state with an analyzer. There are other types of distortion that occur on transients and phase related distortions that are not easilly measured or observed. Music is composed of multiple tones and transients that change quickly. There are lots of opportunities for something to change, some of which aren't completely understood yet.
The cheating answer is, "Because Fourier said so." More specifically, he showed that any repetitive waveform can be made up of a sinusoidal fundamental and a series of harmonics. Now, a distorted sine wave must be repetitive with each cycle (otherwise you'd get a different distortion depending on when you applied exactly the same signal). If you take the idea of applying a bust of sine wave at any time and getting the same answer each time a little further, you realise that any distortion must be higher in frequency (it can't be lower, because that would imply a result that changed depending on when you applied the signal). The distortion must be harmonically related to the input signal, otherwise it would beat with the input signal and produce a lower frequency, again causing a change dependent on when you applied your signal.l
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What a great question!
While not really an answer.... I do remember looking at the power supply of a single rail amp on the o'scope one day. Funny thing was, when a 1K signal ran thru the amp, there was 2K on the power supply.
Had to scratch my head for a bit before it dawned on me.
Each half of the 1K signal was pulling on the PSU, sagging it a bit. Thus double the rate, or 2 KHz.
As so well explained in the posts above, there's a lot of stuff like that going on.
While not really an answer.... I do remember looking at the power supply of a single rail amp on the o'scope one day. Funny thing was, when a 1K signal ran thru the amp, there was 2K on the power supply.
Had to scratch my head for a bit before it dawned on me.
Each half of the 1K signal was pulling on the PSU, sagging it a bit. Thus double the rate, or 2 KHz.
As so well explained in the posts above, there's a lot of stuff like that going on.
tubelab.com said:
Any non-linearity in a circuit causes harmonics to be generated, regardless of whether the signal input to that circuit is a single frequency or multiple frequencies.
By definition, harmonics are multiples of the fundamental frequency. So a 2nd harmonic is 2000 Hz, a 3rd is 3000, etc.
So I don't think I understand what you are saying in the statement above...
Fourier found that any periodic function may be represented by multiple sine functions. However, is the signal is modulated by noise (that happens always because all electronics is non-linear and generate a noise) there are non-harmonic distortions generated, but their level is the less the better is linearity and the lower is the noise.
Wavebourn said:
I'm not sure if you were responding to my post above, but it seems like it.
To say that all electronics are non-linear and generate noise is two different statements and conditions. The non-linearity does not necessarily occur as a result of the noise, nor does the noise necessarily occur as a result of the non-linearity.
I am not a math major, by any means! But as I understand the math if I take a perfect sine wave at a single frequency (say 1KHz) and pass it through a non-linear but noiseless device (if there was such a thing), I will still get harmonic distortion due to nonlinearity. The perfect sine wave will no longer be perfect due to the device's nonlinearity, harmonics will be generated, and those harmonics by definition will be multiples of the 1KHz fundamental.
So I'm trying to figure out what Tubelab meant with his statement that all the bad stuff happens at 1K - the math says that can't be. I was seeking clarification about what he meant.
dsavitsk said:
First off, not all distortion is harmonic. If you were to modulate a SSB xmtr with a signal that had these components: 1.0KHz, 2.0KHz, 4.0KHz, 6.0KHz, you would create a radio wave where the carrier frequency was either added to those baseband frequencies (upper sideband) or subtracted from the carrier (lower sideband). If the xcvr's BFO was running at the wrong frequency, let's say it was off by 10Hz, you could get a baseband with these frequencies: 1010Hz, 2010Hz, 4010Hz, 6010Hz. Those are not harmonically related. The fact that SSB demod depends on a free running BFO that is neither frequency nor phase locked to the original carrier is what accounts for the unique sound of SSB. You also get the same distortion with a doubly balanced detector receiving AM since the DBM nulls out both the BFO and the RF carrier.
Other processes, such as aliasing during analog-to-digital conversion, intermodulation, and frequency division aren't harmonically related either.
As for why most distortion is harmonic, consider this: for musical instruments, the taught wires, or vibrating air columns, have one fundamental, and a series of stable arrangements of nodes and antinodes. These are all harmonically related. (The only exception here would be a drum head, which has a much more complex set of nodes and antinodes that can produce nonharmonic vibrations.) Electronic circuits, whether lumped or distributed, share the same property as physical resonant elements. Even if resonance doesn't play a part, every sort of nonlinear transfer function will generate harmonics, barring some other more complex phenomenon. Fourier analysis will demonstrate this.
The key question in thinking about this is whether the behaviour of the circuit is time dependent.
If it is independent of time, any distortion effects must be the same for each cycle. The output waveform will be the same for every cycle, and if we fourier analyse this we find that all the distortion will be harmonic.
Time dependent effects can easily result in the output varying from cycle to cycle, giving rise to non-harmonic components. A simple example is intermodulation with 100Hz (or 120Hz in the US) components from the power supply. This is often seen in power amps on the verge of clipping. The spectrum will show the usual high order (probably mostly odd) harmonics, and sidebands either side of the fundamental. More complex cases involve circuits which have some sort of nonlinear memory effect, which can give subharmonics. There are even circuits known that show a progression into chaotic behaviour as an appropriate variable like input amplitude is varied; an example of this is a resonator containing a nonlinear element, driven by an external sine wave.
So the answer to the OP question is that distortion is not always purely harmonic, if the circuit has time dependent behaviour.
If it is independent of time, any distortion effects must be the same for each cycle. The output waveform will be the same for every cycle, and if we fourier analyse this we find that all the distortion will be harmonic.
Time dependent effects can easily result in the output varying from cycle to cycle, giving rise to non-harmonic components. A simple example is intermodulation with 100Hz (or 120Hz in the US) components from the power supply. This is often seen in power amps on the verge of clipping. The spectrum will show the usual high order (probably mostly odd) harmonics, and sidebands either side of the fundamental. More complex cases involve circuits which have some sort of nonlinear memory effect, which can give subharmonics. There are even circuits known that show a progression into chaotic behaviour as an appropriate variable like input amplitude is varied; an example of this is a resonator containing a nonlinear element, driven by an external sine wave.
So the answer to the OP question is that distortion is not always purely harmonic, if the circuit has time dependent behaviour.
What I was trying to say is that a distorted 1KHz sine wave will create "stuff" at 1KHz and its integer multiples. I was thinking faster than I was typing and a few thoughts did not get translated into words.
Fourier states that any repettitive signal can be represented by a series of summed sine waves. In theory a 1KHz tone distorted through an amplifier can only create 1KHz and its integer multiples. This represents harmonic distortion. Here the "bad stuff" happens at a 1KHz (and its multiples) rate. There are distortions that do not happen at the fundamental (or its harmonics) rate.
You will not get 1.3KHz out of a distorted 1KHz sine wave unless some other signal is present. It is possible for 300 Hz and 1KHz to be applied to the same amplifier simultaneously to create 1.3 KHz through intermodulation distortion. (the SSB analogy presented above). It is possible for 100 Hz and 1KHz to enter an amplifier and have 1.3 KHz come out if harmonic distortion and intermodulation distortion are both present. It is possible under some conditions for the second signal to be created inside the amplifier.
I have seen cases where 1KHz goes into an amplifier and some unrelated tones come out. The usual cause here is oscillation (and IMD). When an amplifier is oscillating at an ultrasonic rate the oscillation is rarely a prefect sine wave. It may have harmonics and noise associated with it that gets mixed in with the intended signal creating a broad spectrum of low level spurious signals.
I have seen an amplifier overdriven by a 1KHz tone produce frequencies below 1KHz. Fourier says this can't happen, but it does. The usual culprit is blocking distortion. If the time constant is small enough (guitar amps) it is possible to drive the output stage into cutoff so that only every second (or third, fourth, etc.) peak of the sine wave makes it through to the speaker. There is a secondary signal being created by the output stage turning on and off at a rate slower than the input signal. With a radically unbalanced push pull amp operating in this manner some really unusual "stuff" can be created. Blocking distortion and power supply sag together can create some unusual sounding distortion, often at a subharmonic of the input signal.
Jim McShane said:
I was responding to the original question. If common practice of measurement of distortions is to measure harmonic distortions it does not mean that only they exist. Harmonic distortions are easy to measure in order to compare amplifiers. But such measurements have to be weighted against human perceptions that has no standard defined yet.
Hi, guys......Sorry, but some of you are making a terrible mess out of something that's relatively simple from a math POV, although the math may seem hairy for some. I'll also state clearly that I have no intentions of being condecending here- I've been teaching radio theory at polytecnic level for several years, and seen quite alot of students struggling with this.....
Harmonic and intermodulation are the two sides of the same coin - harmonic distortion is a single frequency item - injecting F1 results if F1+n x F1, where n is an integer.
Intermodulatioin is usually described with two frequencies - F1 and F2, resulting in intermodulation products ( n x F1 +- m x F2).
Ex. above with 100Hz and 1000Hz gives 900,1100,1900,2100,1200, 800, 1800, and so on, all depending on the nonlinearites in the transfer function.
The optimum transfer function is Aout=k x sin(wt), where k is your amplification factor -graphically displayed as a straight line where the rise angle coefficient is the Amp factor.
The problems arise when the curve deviates from the straight line, introducing higher order factors - sin^2, sin^3 and so on.
These are the factors responsible for the higher order distortion products. If only sin^2 were present, there would only be 1 order products 2 x F1 and F1 +- 2 x F2.
The reference to SSB modulation, in essence AM and all its derivates, ( SSB being one of them), is a situation where you strive to achieve a perfectly quadratic transfer function based on sin^2. In the frequency mixer circuits used for this purpose, one actually uses purposely developed diodes wiith a well controlled dforward current to optimize the 2nd order behaviour - nothing but well controlled IMD.
Mingling these processes with clipping, noise, oscillation etc. is trying to describe the interaction of highly different processes - truly a daunting task from a theoreticacl POV.
Harmonic and intermodulation are the two sides of the same coin - harmonic distortion is a single frequency item - injecting F1 results if F1+n x F1, where n is an integer.
Intermodulatioin is usually described with two frequencies - F1 and F2, resulting in intermodulation products ( n x F1 +- m x F2).
Ex. above with 100Hz and 1000Hz gives 900,1100,1900,2100,1200, 800, 1800, and so on, all depending on the nonlinearites in the transfer function.
The optimum transfer function is Aout=k x sin(wt), where k is your amplification factor -graphically displayed as a straight line where the rise angle coefficient is the Amp factor.
The problems arise when the curve deviates from the straight line, introducing higher order factors - sin^2, sin^3 and so on.
These are the factors responsible for the higher order distortion products. If only sin^2 were present, there would only be 1 order products 2 x F1 and F1 +- 2 x F2.
The reference to SSB modulation, in essence AM and all its derivates, ( SSB being one of them), is a situation where you strive to achieve a perfectly quadratic transfer function based on sin^2. In the frequency mixer circuits used for this purpose, one actually uses purposely developed diodes wiith a well controlled dforward current to optimize the 2nd order behaviour - nothing but well controlled IMD.
Mingling these processes with clipping, noise, oscillation etc. is trying to describe the interaction of highly different processes - truly a daunting task from a theoreticacl POV.
Wavebourn said:
It is absolutely a fact that no others exist. If they would exist, you would see them on the spectrum, simple as that.
You are correct that we don't know exactly how we (our ear/brain system) reacts to distortion. But that's another unrelated kettle of fish, of course.
Jan Didden
As to the above...
Noise is stocastic in nature - i.e non predictable vs. time. That's the reasons we use pink noise as a test signal, as pink noise has a constant energy content pr. octave ( or parts of an octave), but it's frequency is impossible to predict in absolute time. Thus room resonances and reflections are not excited ( mostly..).
As such, mixing a fixed frq. AND noise is an interesting thought - giving the possible production of all sorts of IM products - i.e. .all frequencies, but this requires the noise to have a significant amplitude - which we don't want, do we?
Somewhat like listening to music frrom a weak shortwave station -- all hiss and whistles - and maybe some music..........
EDIT : An overdriven, worn down guitar amp may be an interesting case for its function - creating strange guitar sounds ( and I'm actally currently planning one for my son) , but this has little relevance in HiFi - and I wouldn't even dream of trying to describe one in math terms
Noise is stocastic in nature - i.e non predictable vs. time. That's the reasons we use pink noise as a test signal, as pink noise has a constant energy content pr. octave ( or parts of an octave), but it's frequency is impossible to predict in absolute time. Thus room resonances and reflections are not excited ( mostly..).
As such, mixing a fixed frq. AND noise is an interesting thought - giving the possible production of all sorts of IM products - i.e. .all frequencies, but this requires the noise to have a significant amplitude - which we don't want, do we?
Somewhat like listening to music frrom a weak shortwave station -- all hiss and whistles - and maybe some music..........
EDIT : An overdriven, worn down guitar amp may be an interesting case for its function - creating strange guitar sounds ( and I'm actally currently planning one for my son) , but this has little relevance in HiFi - and I wouldn't even dream of trying to describe one in math terms