[SOLVED] Multiple pulse width ( 5 pulses ) inverter and lowest low order harmonics

[Easy peasy]

Hello All, we are looking at 5 pulse MPW for small inverters, 24VDC to 230Vac 50/60 Hz. ( unipolar for those tech savy in this area )

Is there a paper or person out there who has looked at the low order harmonics resulting from the 5 pulse approach and the spacing of the pulses and relative width at full power for lowest harmonic magnitude for the lower order harmonics? - particularly the 3rd, 5th, 7th, 11th, maybe the 15th

I guess I will be doing this from scratch from 1st principles to calc the harmonics resulting from the various arrangements possible - if someone has covered the ground before and zeroed in on the best relative spacing and relative widths I would be appreciative of some pointers in the right direction.

Most important is that the lowest harmonics have the lowest magnitude as, as you go up in freq the higher order are easier to filter.

Please no " guesses or 'helpful' suggestions " from those not au fait with fourier analysis or harmonic series mathematics or unipolar 5 pulse MPWM ( multiple pulse width modulation )

If some one has the ability to do this work - I would be happy to recompense for sound results

kind regards

- - - Updated - - -

additional info: the middle of the 5 pulses ( per half cycle ) is to be fixed, they are to vary in width to control Vout from no load ( skinny ) to full load ( maximum ideal width for min low order harmonics ) - ta.

 

[FvM]

Just for clarification, with 5 pulse unipolar, you are referring to a waveform like below?



It has 5 degrees of freedom, should allow to set fundamental magnitude + eliminate 4 harmonics (3, 5, 7, 9).

 

[FvM]

A similar modulation scheme with harmonic cancellation is referred to as Magic Sinewaves, for some reason using an even number of pulses per half cycle. https://tinaja.com/glib/msinprop.pdf

 

[Easy peasy]

Hello FvM, yes your sketch is correct, the magic sine waves page does not quite cover what I am seeking ( 5 pulse unipolar ) I note that LT Spice will give me an FFT of a waveform - so I will begin there before I jump into the heavy maths for an analytic solution, kind regards,

 

[FvM]

You can determine the optimal pulse timing for harmonic cancellation using fourier series and Excel solver. I'll send you an example tomorrow.

 

[Easy peasy]

Fourier analysis of 5 pulse MPWM ( multiple pulse width modulation ) for inverter

Hello All, thought I would re-post this here - as it essentially a mathematics problem.

Determine the placement and width of a 5 pulse multiple pulse width modulated (mpwm) system - unipolar on each half cycle
to get the least ( ideally zero ) lowest order harmonics for the 3rd, 5th, 7th, (9th) 11th, 15th

the lower the better for the lower orders as these are the hardest to filter ...




We are looking at 5 pulse MPW for small inverters, 24VDC to 230Vac 50/60 Hz. ( unipolar for those tech savy in this area )

Is there a paper or person out there who has looked at the low order harmonics resulting from the 5 pulse approach and the spacing of the pulses and relative width at full power for lowest harmonic magnitude for the lower order harmonics? - particularly the 3rd, 5th, 7th, 11th, maybe the 15th

I guess I will be doing this from scratch from 1st principles to calc the harmonics resulting from the various arrangements possible - if someone has covered the ground before and zeroed in on the best relative spacing and relative widths I would be appreciative of some pointers in the right direction.

Most important is that the lowest harmonics have the lowest magnitude as, as you go up in freq the higher order are easier to filter.


If some one has the ability to do this work - I would be happy to recompense for sound results




additional info: the middle of each of the 5 pulses ( per half cycle ) is to be fixed, they are to vary in width to control Vout from no load ( skinny ) to full load ( maximum ideal width for min low order harmonics ) - ta.

 

[KlausST]

Re: Fourier analysis of 5 pulse MPWM ( multiple pulse width modulation ) for inverter

Hi,

I see your idea:

There are positive signals causung some overtones...and there is the same signal negative ...and you want it to cancel all the frequencies.

For sure they include the same frequencies and for sure the same amplitudes....but there is more than this: Phase. Phase or each frequency.

You can only calculate FFT if you have a signal with fully specified timing.

Klaus

 

[BradtheRad]

Re: Fourier analysis of 5 pulse MPWM ( multiple pulse width modulation ) for inverter

I assume the filter is an inductor or transformer? The tendency is that voltage falls immediately after a pulse ends. I think the aim is to add a few brief pulses, to slow the drop.

My simulation makes each pulse adjustable as to length and when it occurs in the cycle. Diodes steer them to a joint output. The load gets the positive portion of a sine-like waveform. The inductor value was selected so as to slow the upward transition of the long pulse so it resembles the rise of a sine wave.



It may work just as well to make pulse #2 the long pulse. It leaves more remaining pulses which can be more finely spaced.

 

[FvM]

Re: Fourier analysis of 5 pulse MPWM ( multiple pulse width modulation ) for inverter

As mentioned before, the fundamental and harmonic magnitudes can be best calculated and optimized using Fourier series.



You can superimpose 5 instances of the elementary double pulse with alternating sign to model the unipolar 5-pulse pattern.

See the below appended Excel table. Excel solver must be installed to perform the optimization.

 

[Easy peasy]

Re: Fourier analysis of 5 pulse MPWM ( multiple pulse width modulation ) for inverter

as it is symmetric about pi/2 ( 90 deg ) this helps in the maths

 

[FvM]

Re: Fourier analysis of 5 pulse MPWM ( multiple pulse width modulation ) for inverter

I get the below shown edge timing for cancellation of 3/5/7/9th harmonic. Time scale is rad. Magnitudes without the 4a/pi factor.

I have set an arbitrary minimal pulse width of 0.05 rad, the yellow background marks limited pulse/gap width, respectively incomplete cancellation.

 

[Easy peasy]

great work FvM, unfortunately it is a bit hard to see what you have presented, the middle pulse should be the major one - as per your sketch, the next outer pulses should be about half the width of the middle one and the outer pulses should be about half again in width - this will get near to low order cancellation ( <30dB ).

I have been reconstructing a "filtered" "sine wave" output by putting the 1st, 3rd, 5th, 7th, 9th, 11th, 13, 15, 17, 19, 21 back together after seeing what they are from the MPWM, this gives me good optics on the progress, and the effect of each harmonic. LT spice seems to be good for this - generating the harmonic amplitudes from the MPWM, and for adding the harmonic sine waves in various harmonics and phases ....

 

[FvM]

The emphasis of the presented results is on the feasibility of harmonic cancellation over a certain magnitude range. I'm assuming that this solution, if achievable, is also unique.

Because the magnitude of higher harmonics isn't represented in the error function, the solution isn't necessarily optimal according to more general criteria, e.g. meeting power quality standards with or without low-pass filter.

Alternatively you can include higher harmonics in the error function and optionally apply weighting. Lowest four harmonics won't be exactly cancelled, but total T.H.D. or some weighted error function can be minimized.

 

[Easy peasy]

Hello FvM, thank you for your kind remarks, can you please explain what r1, f1 is in the above? and the ratio or exact width of the 3 pulses in the above also ( middle, outer, next outer ) - from inspection and from common sense 5 pulses ( in total ) will be able to greatly reduce the 3rd, 5th, 7th, and one other ( 9th or 11th - whichever gives the lowest overall THD ) - however none of that is easily viewable in your presentation.

kind regards,

 

[FvM]

Sorry for lack of clarity. r1,f1 etc. are the edge times (rising 1, falling 1,...) in rad. To get the full picture, you should review the original Excel table from post #9. You can also try alternative optimizations.

I add the achieved harmonic magnitudes in the below table.

 

[Easy peasy]

that is impressive - are the rise & fall times in radians?

 

[BradtheRad]

By definition the output which closely resembles a sinewave has the least harmonic content. Or from another viewpoint, the less filtering needed, the better.

Here is my simulation which puts the first pulse as the widest. The first upward-going transistion is slowed by the inline inductor. The remaining pulses are timed to allow a gradual drop in amplitude. (And not the typical sharp drop). The result is sine-like. A sinewave serves as reference.



A switching arrangement unfolds the output polarity, so that the load receives bipolar AC.

 

[FvM]

are the rise & fall times in radians?

Yes, exactly.