[sdiy] Harmonic effect of rectification

[Tom Wiltshire]

Never mind, I've worked it out.

I needed a trigonometric identity. sin(a)*sin(b) = (cos(a-b) - cos(a+b)) / 2

That identity explains where the phase shift comes from - you put sines in, but get cosines out. People don't usually mention that when they talk about what happens with a ring mod. I suppose mostly it doesn't matter.

Thanks,
Tom

On 20 Apr 2012, at 19:18, Tom Wiltshire wrote:

> Do the distortion products all have the same phase as the original square or sine signal?
> (This being rectification, we know that the two signals going into our rectifier-ringmod are the same phase)
> 
> I'm trying to graph it, and whilst I can do the ring modulation ( sin(x) times sin(x)+sin(3x)/3+sin(5x)/5+etc ) and get a rectified sine wave, I can't get the same result adding together the harmonic products.
> 
> If I rectify a sinewave, I should get two harmonic series out, one generated by a+b, and the other by a-b.
> 
> a+b is a squarewave sequence, where all the harmonics have been shifted up by the fundamental frequency. So if the harmonics of the ordinary square are 2n+1 (1,3,5,7,9,11, etc) then these harmonics are 2n+2 (2,4,6,8,10,12,etc).
> Similarly, a-b is shifted down, so 2n+1 becomes 2n (0,2,4,6,8,etc).
> 
> But adding together these two series doesn't give me the right result.
> 
> The harmonic amounts are unchanged, so these are still 1/2n+1 (1, 1/3, 1/5, 1/7, etc). I figured that 1x1/3 is still a 1/3rd. Or is this a mistake? Do the harmonic amounts not simply multiply like the waveforms? Instead they follow Bessel functions like FM? RM/AM can certainly sound like it.
> 
> Thanks
> Tom
> 
> 
> On 20 Apr 2012, at 13:03, Richie Burnett wrote:
> 
>> There are two common types of rectification.  Half-wave and full-wave
>> rectification.  Both introduce DC and even order distortion components.
>> 
>> You can think of both types of rectification either as a wave-shaping function
>> or alternatively as a modulation process.
>> 
>> Thinking of rectification as an amplitude modulation process might not seem so
>> intuitive, but it might give you a better insight into what is going on in the
>> frequency domain.
>> 
>> For example if you put a sinewave through a half-wave rectifier the positive
>> half-cycle is multiplied by +1 and the negative half-cycle is multiplied by 0. 
>> This is equivalent to multiplying a sinewave with a squarewave that exists
>> between 0 and +1.  This is amplitude modulation (or ring-modulation where the
>> square-wave input has a DC bias.)
>> 
>> For the example of a full-wave rectifier the positive half-cycle is multiplied
>> by +1, and the negative half-cycle is multiplied by -1 in order to flip it about
>> the x-axis and make it become positive.  This is equivalent to multiplying a
>> sinewave by a squarewave that exists between -1 and +1.  It's the same ring-mod
>> process as above but this time there's no DC bias applied to the squarewave.
>> 
>> Thinking of sinewave rectification as amplitude modulation by a squarewave might
>> help you to figure out where all those harmonics come from.
>> 
>> I hope this helps,
>> 
>> -Richie,
>> 
>>> On 20 Apr 2012, at 11:56, Neil Johnson wrote:
>>> 
>>>> Hi Tom,
>>>> 
>>>> (to list this time!)
>>>> 
>>>>> What's the harmonic effect of rectification? Has this been studied anywhere?
>>>> 
>>>> http://www.rfcafe.com/references/electrical/periodic-series.htm
>>>> 
>>>> Just apply the Fourier transform and some undergrad maths.
>>>> 
>>>> Neil
>>> 
>>> I can apply the fourier transform to the output for a particular case and find
>>> out what it did to X or Y wave, but that doesn't help me know what will happen
>>> when I feed in Z wave. Given that sometimes it just changes the overall
>>> frequency and leaves the harmonic structure alone, sometimes leaves the
>>> frequency alone and changes the harmonic structure, and sometimes changes both,
>>> it doesn't seem straightforward to predict what will happen.
>>> 
>>> But hang on a minute...I think I've got it...Rectification is an example of a
>>> waveshaping function, and the required theory is all worked out for waveshaping.
>>> Ok, sorry to have bothered you.
>>> 
>>> Tom
>>> 
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>> 
> 
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