Harmonics and intermodulation products do not occur in ideally linear devices. Why do they occur in nonlinear devices? I am referring to amplifiers. It must be linked to the nonlinear characteristics, which are saturated at some point. But how does this lead to harmonics or intermodulation products? 

Also: An ideal amplifier doesn't have any phase distortion, does it?

I would also be glad about good literature hints!

Take a sine wave, \(x = \sin \omega t \).

Gin up a nonlinear function -- preferably something mild. \(y = x + ax^n\) with \(a\) small and \(n\) selectible.

Now, using only a pencil and paper, calculate \(y\), or at least the first few terms.

Do it again for \(x = \sin \omega_1 t + \sin \omega_2 t \).

thx, this really helps! 

I have tried both with a sine and a cos two-tone signal: 

x(t) = sin(wt) + a * sin(wt)^2

x(t) = sin(wt) + a/2 * (1 - cos(2 wt) ) 

x(t) = a/2 + sin(wt) - a/2 * sin(2 wt - pi/2) 

this is a little weird, because of the phase shift. I think it is still correct, but I don't see the harmonic component. Maybe you can help. 

With a cos signal, it is more obvious: 

x(t) = cos(wt) + a * cos(wt)^2

x(t) = cos(wt) + a/2 * (1 + cos(2wt) ) 

x(t) = a/2 + cos(wt) + a/2*cos(2wt) 

so apparently, there is a harmonic component at 2wt, with a scaled amplitude of a/2. There is also the constant term a/2 - but this is not a harmonic component, is it?

In your sine wave case you're getting a component out at \(2\omega t\).  That is the \(2^{nd}\) harmonic of your fundamental.

People can argue about whether the constant term is the "\(0^{th}\)" harmonic or not -- for radio the point is moot, but for power lines that are supposed to be purely AC, Bad Things can happen if there's significant DC current flowing.

the sine-wave at 2wt is also phase shifted by pi/2. Is this the phase distortion ? .. If I would draw the power spectrum of the output signal, would the term at 2wt be negative?

If you remove the \(\sin 2\omega t\) term from \(x(t)\), does the total power go up or down?  Does that answer your question about whether that term's power should be positive or negative in the power spectrum?

How do you define phase distortion?  Does the apparent phase of the \(2^{nd}\) harmonic have anything to do with any change in phase of the \(1^{st}\) harmonic?

Extra credit: Can a power spectrum have negative components?

ok, let me see: 

A power spectrum cannot have negative components. That means, with the sin(2wt) term, I am substracting a positive power. Leaving out this term means the power increases. 

The phase change of the second harmonic does not influence the phase of the first harmonic. But nonlinear amplitude and phase characteristics ( AM/AM and AM/PM ) are usually measured for a single-tone input. And the AM/PM gives the change in phase of this single-tone input. So when I model the nonlinearity with this mathematical formula, I wonder what represents the phase shift?

Get more basic yet.  What's the average value of \( \left( \cos \omega t \right)^2 \)?  What's the average value of \( \left( \cos \omega t + \sin 2\omega t \right)^2 \)?  Which is bigger?

Have you taken a course in signal processing?  You've been asking a lot of questions lately that really sum up to everything you need to know by taking a basic signal processing course (it was EE250 when I took it).

I'm not saying you shouldn't ask questions here -- but I am suggesting that given what you want to know, you may get more satisfaction out of taking the course (if you're a university student) or seeing if one of the "biggies" like MIT has an online version for free.

Ya, I thought so too. It is kind of urgent now, however. 

Can you still explain to me where I find the AM/PM in my mathematical expression? 

I don't know what you want me to see by thinking about the average value of (cos wt)^2, but I know it is 1/2. The average of (cos wt + sin 2wt)^2 however, is much more complicated to determine. I don't know what it is, but I expect it to be bigger. 

I'll go a bit out onto a limb and suggest perhaps a different framework for this.  So, I'll depart from the"why?" and come at it differently.  Perhaps more like "how?".

As long as the waveform coming out of the box is periodic - no matter the shape - then it can be represented by a Fourier Series.  We find this representation handy because it represents the waveform in terms of sinusoids that are harmonically related.  If they were not harmonically related then, except for some strange cases perhaps, the waveform would not necessarily be periodic.  But, the idea here is that the waveform *is* periodic.

So, start with an input sinusoid, observe the output waveform is *not* a sinusoid exactly but is periodic.  Then analyze the waveform with a Fourier Series and see the magnitude and phase of the harmonic components.  That's "how".

An ideal amplifier would not change the waveform.  Since it's ideal, you get to decide if it introduces a delay.  But a delay would not introduce a change in the waveform, only in its time reference.

I started to write something like this in my original answer and desisted because it was going to be Too Much Work.

It's a much better and more general-purpose answer than mine.  (Although, I do think that doing little "toy" test cases by hand helps to build intuition).

If an "ideal" amp is really "ideal" it doesn't add any distortion, amplitude or phase.  Otherwise it wouldn't be very "ideal", I think.  ;)

The easy example to see how harmonics get added is the extreme case where an amplifier in full clipping saturation turns a sine wave into a square wave.   The sine wave has a pure tone with a single harmonic element, the square wave has many harmonics because it's a square wave.   The non-linear behavior of the amp turned the sine wave into a square wave.

Anything in between is also fairly understood in terms of Fourier analysis, in that anything deviating from a sine wave needs more spectral content to create the resulting waveform.   So if a sine wave goes in and anything other than a sine wave comes out (i.e., it has distortion), then some new spectral components have to be present in the distorted waveform.

Distortion is the difference between the shape of the waveform that went into the amplifier and what came out.   If anything changes in the shape of the waveform, then it has been distorted.

The only way that an output signal is undistorted is if the amplifier characteristic in the region the signal occupies is linear.

Lucky_12,

Distorted signal means, you cannot recover the original signal or information.

Non-linearity causes distortion.  

Here is a webinar on Non-linearity and its impact on receiver.

https://ortenga.com/ortenga703/

Best Regards,

Shahram