The phases of the harmonics created by a distortion of a sine wave are always 0°, 90°, 180° and 270°. Why?

[Dan Bullard]

If you can capture it accurately, any distorted sine wave, when taken apart by an FFT will reveal that the phases of the harmonics will always be at 0°, 90°, 180° and 270° relative to the sine wave fundamental, without fail. This is not well known, since most people look only at the magnitude (amplitude) of the spectrum and not the harmonic phases, but if you do, you will "discover" this law. Even more interestingly, if you shift the wave over by 90° to make it a cosine wave, you will find that after doing the FFT, the harmonic phases are at exactly two values, 0° and 180° relative to the fundamental. In this animated GIF I apply 6 different distortions to a cosine wave and prove "Da er's Law" with a scatter plot showing the phases of all harmonics from the fundamental through the sampling frequency, aka Fs. This plot was made in Excel, it is not a hand drawing, this is real data taken from the functions mentioned.

Notice that no matter what the distortion is, whether it's peak clipping, asymmetrical or symmetrical, zero crossing clipping, either asymmetrical or symmetrical, a distortion in any portion of the transfer function, the phases of the harmonics in the scatter plot never vary from 0° or 180°. The same thing happens if I do this on a sine wave, but the phases move between four different values as mentioned.

If you can't capture the wave as a perfect sine or cosine wave, you are SOL because the phases of the harmonics will be all over the place, seemingly random numbers, although there is a way around this limitation. This is not true if you are talking about noise, this only applies to distortion caused by faults in the device's transfer function. Why is this law true?

[swansont]

11 hours ago, Dan Bullard said:

Da er's Law

The only hit I get on Google for "Da er's Law" is this thread

Can you provide a reference for it? Is it a typo?

[studiot]

14 hours ago, Dan Bullard said:

If you can capture it accurately, any distorted sine wave, when taken apart by an FFT will reveal that the phases of the harmonics will always be at 0°, 90°, 180° and 270° relative to the sine wave fundamental, without fail. This is not well known, since most people look only at the magnitude (amplitude) of the spectrum and not the harmonic phases, but if you do, you will "discover" this law. Even more interestingly, if you shift the wave over by 90° to make it a cosine wave, you will find that after doing the FFT, the harmonic phases are at exactly two values, 0° and 180° relative to the fundamental. In this animated GIF I apply 6 different distortions to a cosine wave and prove "Da er's Law" with a scatter plot showing the phases of all harmonics from the fundamental through the sampling frequency, aka Fs. This plot was made in Excel, it is not a hand drawing, this is real data taken from the functions mentioned.

Notice that no matter what the distortion is, whether it's peak clipping, asymmetrical or symmetrical, zero crossing clipping, either asymmetrical or symmetrical, a distortion in any portion of the transfer function, the phases of the harmonics in the scatter plot never vary from 0° or 180°. The same thing happens if I do this on a sine wave, but the phases move between four different values as mentioned.

If you can't capture the wave as a perfect sine or cosine wave, you are SOL because the phases of the harmonics will be all over the place, seemingly random numbers, although there is a way around this limitation. This is not true if you are talking about noise, this only applies to distortion caused by faults in the device's transfer function. Why is this law true?

 

So you are claiming that

 

[math]339.4\sin \left( {100\pi t} \right) + 67.9\sin \left( {300\pi t - \frac{{3\pi }}{4}} \right)[/math]

 

 

is not a sine wave distorted by 20% third harmonic at a phase angle of - 135^o ?

 

I think we need considerably more background about what you are trying to do and what sort of distortion you mean.

And by the way what is zero crossing clipping ?

How do you clip something that has zero value ?

14 hours ago, Dan Bullard said:

zero crossing clipping

Edited November 4, 2021 by studiot
to correct formula

[Dan Bullard]

No I am not claiming that339.4sin(100πt)+67.9sin(300πt−3π4)339.4sin(100πt)+67.9sin(300πt−3π4)

Distortion in an amplifier (or DAC, ADC, etc) are caused by a failure of the transfer function to accurately pass on the input wave on to the output. If a distortion happens at any angle, it will automatically be replicated later, so for example, if a distortion appears at, say, 66°, it will happen again (90-66) + 90, or 114°, absolutely guaranteed. That will cause all harmonics created to occur with a phase offset relative to 90°. If the distortion happens at, say, 200°, a perfect sine wave will hit it again at (270-200)+270 or 340°. It helps if you understand how electronics work to start with. Every amplifier, DAC or ADC (which I have written code to test millions of) will treat a sine wave as a two way street, what goes up, must come down, and what goes down, must come up. As I say in one of my videos, if a distortion doesn't hit you by the time you get to 270° in a sine wave, it won't hit one, period. If you jump on a trampoline and don't crash into the roof on the way up you will not hit the roof on the way down, absolutely guaranteed. 

Nothing happens only once in a sine wave, except for two places, 90° and 270°. Anything that happens only once will not cause "measurable harmonics," that is, yes, there will be harmonics, but they will disappear instantly, before your instrumentation gets a chance to see them.

 

It's Daver's Law, somehow a typo snuck in. And you won't find it anywhere other than LinkedIn or my website. Here is a reference:

https://www.linkedin.com/pulse/gets-even-better-dan-bullard/

Allow me to address the Zero Crossing issue. Back in the old days there arose a Push-Pull amplifier topology that was called a Class B amplifier. Class B amplifiers worked great, but not as good as they first thought because for a small portion of the transfer function, both transistors would be off as the bases were grounded. So for 1.2V to 1.4V both transistors would be off. Now, this doesn't sound like too much of a problem because the voltage of the sine wave is near zero at that point, so who will miss that small portion of the zero crossing, right? Nobody can even see it, so who cares?

Well, not so fast. Assume the transistors are will matched (P and N having similar barrier voltages). This will result in a perfectly symmetrical distortion right at the zero crossing, and because they are symmetrical, there will be no Even harmonics, only Odd harmonics. And the odd harmonics do not start out right away thanks to the other topic that I mentioned that has been tyrannically closed. The phase angle determines the harmonic signature and it's very different at the zero crossing (180°) than it is near the positive or negative peak. It will look like this

There is a problem there, because nobody (NOBODY) looks at harmonics for THD above the 5th harmonic, or the 7th, or in rarer cases, the 9th. Remember, these are odd harmonics, so count by two. The fundamental comes first, then the 3rd, then the 5th, then the 9th. The 9th is only at -65dB or so, so nobody cares, right? WRONG!!! Maxim Integrated Products got bit by this one so now they cover the entire audio band (DC-20KHz) when testing their THD values, but most companies could care less (Maxim is better than most). If the distortion is asymmetrical, like in this case where I distort just one side of the transfer function.

 

Now you get a completely different spectrum. Notice how the even harmonics (yes, the red ones) start off very high and then dive down low, but the odd (yes, the blue ones) start off down in the noise floor and rise up to a peak at the point where the even harmonics are now at their first minimum.

This is why the phase angle where the distortion occurs is so important, nearer the peaks the odd and even harmonics follow a similar trajectory,  but in a zero crossing (distortion can happen anywhere, even now that Class B amplifiers are out of favor) the odd and even harmonics appear out of phase and that can cause some serious problems. Counting some random number of harmonics to get a THD number is a fools errand, because you have no idea where the distortion may have happened, so you can't even begin to guess what the harmonics are going to do. If you can get very unlucky you may get something like this.

This will be seen by most users that measure THD the traditional way as a perfect wave, because nothing happens until the 10th harmonic, well after most programs and instruments have stopped looking at harmonics to sum up into a THD number (except at Maxim). But it is a problem because the harmonic signature continues to rise up out of the noise floor, and while some can't hear the difference, some can, and you want all your customers happy, not just a few. Plus, if this is supposed to used in an RF function, you are probably counting on those harmonics to reproduce your wave accurately, and one customer actually tried to get away with something like this for transmitting data. Sure, I got a free trip to Florida and $100 an hour consulting for two days before I convinced then that you can't transmit harmonics on radio when you are confined to a very small bandwidth, the harmonics will never make it out to the antenna.

 

 

 

Edited November 4, 2021 by Dan Bullard

[studiot]

Thank you for your answers.

I'm sorry to report that your answer to my first question was lost somewhere in the translation. (about the sine wave plus 20% third harmonic at a phase of -135^o.)

Can you please complete it because it is a straightforward counterexample to your earlier claim that

On 11/3/2021 at 10:57 PM, Dan Bullard said:

will reveal that the phases of the harmonics will always be at 0°, 90°, 180° and 270° relative to the sine wave fundamental, without fail.

This discrepancy needs explaining.

 

I wondered if your zero crossing distortion was what is commonly called crossover distortion by another name and that would appear to be the case.

I don't think the renaming is a good idea since 'zero crossing' is a recognised technique to reduce/theoretically eliminate intense RF harmonic activity in power control circuitry and not in anyway connected to harmonic distortion in circuits.

I would further like to observe that there are other (non harmonic) forms of distortion which can be of greater importance than any harmonic distortion in a well designed electronic/electric system.

 

[Dan Bullard]

Crossover distortion is zero crossing distortion. The idea that using Class B amplifiers to "reduce/theoretically eliminate intense RF harmonic activity in power control circuitry and not in anyway connected to harmonic distortion in circuits" is bunk. The harmonics are there, just lower in amplitude than you can see and claiming something like this just means that you don't have the resolution to see it. I've used 18 bit, 20 bit and 24 bit digitizers (Credence, LTX, Schlumberger, Applicos, etc) and I have seen harmonics creep in at levels down below the -100dB level.  Just because you can't see it doesn't mean it's not happening, and in an RF application, as I mentioned you cannot transmit harmonics. I used to work on 40KW HF transmitters in the outback in Western Australia, and the final stage was a Class C air cooled tube amplifier (triode) that had monster RF inductors and tunable vacuum capacitors which served to eliminate harmonics created in the output tube before it went out to the antenna. And just down the road from me was (at the time) the tallest man-made structure in the Southern hemisphere, the antenna array for a 2MW VLF transmitter used to send nuclear launch codes to submerged US subs in event of a nuclear war. The inductors there were the size of a small car, and you could hold up a 40W florescent light bulb and watch it light without any wires connected to it while the transmitter was operating, it was amazing. Again, the purpose of the inductors was to use along with giant capacitors to form a resonant tank circuit in the 10KHz range. The inductor was built on wooden frames and was tunable! All in an effort to remove harmonics caused by the ten water cooled Class C tubes used as power amps. If you try to transmit harmonics your signal will step on a lot of other applications and I think the FCC has a rule about that. That is what I had to tell my customer in Florida, after getting a nice holiday in the middle of an Oregon snowstorm. It cost me a $100 an hour job but continuing to spend their VC's $1 million on their "new technology" was pointless and I knew it, and now, so do they.

NTSC, HDTV, and virtually all forms of modulation from QAM to AM find clever ways to reduce harmonics. QAM is one of the best, but still, those harmonics do exist and the output amplifier has some form of filtering to eliminate them. I doubt that there are any forms of "non-harmonic distortion" as the only way that a non-sinewave can exist is with the presence of harmonics. I suspect that you are trying to minimize my discoveries by claiming that other things in electronics are more interesting. Well, maybe. I've worked on Intel and Motorola uPs, Analog Devices MEMS chips, F16 HUD chips, Trident nuclear missile chips, chips for China's Ministry of Aerospace (I could tell you more but then I would have to kill you), a Maxim HDTV tuner-on-a-chip where I created a production test program that had a test time under 10 seconds, including calibration of the chip's tuner map, Karaoke chips (my newest boom-box uses a chip I was working on in 1999!), the list goes on forever. It was all interesting, and myself and other people knew about the principles of those devices, but when it came to harmonics the best those other people could do was guess. Well, we don't have to guess anymore, I know how harmonics work and I am trying to spread that knowledge far and wide. To quote Dr. Gad Saad from his latest book, "Many professors forget that their professional responsibility is not only to generate new knowledge (as I did) but also to seek to maximally disseminate it."  That is what I am doing here. If this site prefers to ignore new discoveries in science, so be it. Kick me off the site like so many others have. I'll quickly follow that with a YouTube video on my channel informing my 4000 subscribers that this site prefers to bury its head in the sand in the face of new discoveries like I did with Stack Overflow.

Edited November 5, 2021 by Dan Bullard

[studiot]

 

22 minutes ago, Dan Bullard said:

I suspect that you are trying to minimize my discoveries by claiming that other things in electronics are more interesting.

 

I said no such thing.

But I did ask you a simple question of Mathematics, which in your 25 lines of self congratulation you have failed to address.

I hope you will stoop from your exalted position to to answer it soon as I fear this thread will onterwise beclosed as contributing nothing to Modern and Theoretical Physics.

You have made a mathematical claim, Sir.

Back it up with Mathematics.

[Phi for All]

14 hours ago, Dan Bullard said:

If this site prefers to ignore new discoveries in science, so be it. Kick me off the site like so many others have. 

!

Moderator Note

We would prefer it if you'd focus on answering the science, and leave the trumping out of it. This might account for your past discussion forum failures, so let's approach this differently this time, shall we? If you're banned, it will be because you broke the rules, not because you challenged mainstream concepts.

 

[Dan Bullard]

I am asking for you "science experts" to answer why this is true. I have a gut feeling, and proof that it's true, as did Mendel with his laws of inheritance but I don't know the math behind why. But one thing you did for me for sure, you once again proved that nobody here understands the first thing about waves and harmonics. That is telling, and probably means there is no reason to hang out here.

YouTube video removed by moderator

Edited November 8, 2021 by Phi for All
no more videos as support

[Phi for All]

!

Moderator Note

You need to present your case here. We have no way of knowing if you're simply trying to drive traffic to your YT channel, which is against the rules here. You need to support your assertions with mainstream science. Gut feelings aside, this is science discussion, so you need MORE RIGOR if you're going to persuade anyone here you might be onto something. We're rooting for you because it would be exciting if you're right, but you need to tick all the boxes and support your idea with evidence.

 

[studiot]

1 hour ago, Dan Bullard said:

But one thing you did for me for sure, you once again proved that nobody here understands the first thing about waves and harmonics.

Being rude about fellow members whom you can have no possible knowledge of doesn't help your case either.

Answer my question please, without trying to put words into my mouth.

For your information it had nothing to do with class B or any other amplifiers.

It was actually a national certificate electrical power engineering question.

Again FYI, just as I already said, harmonics play a very important role in the national power grid because they can play havoc with the grid system if not managed properly.

Edited November 8, 2021 by studiot

[John Cuthber]

On 11/3/2021 at 10:57 PM, Dan Bullard said:

If you can capture it accurately, any distorted sine wave, when taken apart by an FFT will reveal that the phases of the harmonics will always be at 0°, 90°, 180° and 270° relative to the sine wave fundamental, without fail.

Not really.
This circuit generates a distorted sine wave in which the phase of the distortion is arbitrary WRT the sine wave.
https://en.wikipedia.org/wiki/Phase-fired_controller

Obviously, you can resolve any arbitrary phase into a sin and cosine component.
In that case, you only need two phases, as long as you don't mind one of the amplitudes being negative.