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[solved] Crest Factor or PAPR of a full-wave rectified sine with Villard voltage doublers
Move Thread LAN_403
PhilGood
Tue Jul 02 2013, 08:57AM Print
PhilGood Registered Member #3806 Joined: Sat Apr 02 2011, 09:20PM
Location: France
Posts: 259
 		Hi all,

I am writting a small javascript app to help me in my VTTC calculations. I need to know the peak power/average power ratio (PAPR) for different waveforms. I don't need complex formulas but constants that would make calculations easy.

From what I found in Crest Factor Wiki article, I got these values with PAPR=C^2 (C as Crest Factor)

DC: 1 (ok)
Halfwave rectified sine: 4 (ok)
Fullwave rectified sine: 2 (ok)

and from RMS value of a sine wave with DC offset (see below for details)

Halfwave rectified sine with voltage doubler (shifted sine): 2.667 (ok)

But I have a doubt about this last one: (Edit2: found it, see below)

Fullwave rectified sine with voltage doubler (shifted rectified sine): 1.442 (ok)


My researches history:

For the halfwave rectified sine with voltage doubler, the voltage doubler is a classic Villard circuit:

Voltag10

I first thought the waveform of such a voltage doubler would be something weird like this:

Weird 10

I quickly built such a voltage doubler and scoped it, in fact the waveform is just a shifted sine:

Shifte10

Edit: Ok I think I've been a bit lazy, after more researches I found this article: RMS value of a sine wave with DC offset,

VRMS = sqrt(Vdc^2 + (Vac^2)/2)
With Vdc as DC offset and Vac as AC component peak

In this case V = Vdc = Vac (in the above pic, V=two divisions)

so we can write VRMS = (v^2 + (V^2)/2) = sqrt(6)/2 V = sqrt(6)/4 2V

PAPR = (1/sqrt(6)/4)^2 = 8/3 = 2.667

I have a last doubt about the fullwave rectified sine with voltage doubler, the circuit consists in two transformers + Villard circuit 180° out of phase as described by Steve:
Steve Conner wrote ...

The solution is to use a voltage doubler. Then you can ground both the core and the negative output. Indeed that is why MOTs have one end of the secondary connected to the core, because they are designed for use with a doubler.

You get twice the output voltage, which you may not want, but you can fix that by reducing the input voltage to 120V.

You also have halfwave rectification, but you can fix that by using two MOT/doubler arrangements 180 degrees out of phase. (and maybe connect the primaries in series for 240V operation?)
I think the output waveform should be (image edited from above one):

Double10

Just gotta find the correct Crest Factor or PAPR for this waveform...

Edit 2:

I finally found out with the help of my brother who is better at Maths than I am

For this waveform, we have:
DC component value = V
Vpeak = V*2
one cycle is made of two similar parts from 0 to T/2 and from T/2 to T

VRMS = sqrt(1/T * integral from 0 to T of v(t)^2 dt) = sqrt(2/T * integral from 0 to T/2 of v(t)^2 dt)

between 0 and T/2 we have v(t) = V sin(2pi t/T) + V

so we have VRMS = sqrt(2/T * integral from 0 to T/2 of (V sin(2pi t/T) +V)^2 dt)

Xcas is an awesome tool for solving such equation:
VRMS = simplify(evalf(simplify(sqrt(2/ T*int((V*sin(2*pi*t/T)+V)^2,t,0,T/2))))) = 1.6653 V = 0.83265 V*2

Crest factor is 1/0,83265 = 1.2
PAPR = 1.2^2 = 1.44

I found another interesting article that may help for complex waveforms:
DETERMINING THE EFFECTIVE OR RMS VOLTAGE OF VARIOUS WAVEFORMS WITHOUT CALCULUS

sorry for the monologue and thanks for reading :)
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