Harmonics

[Basra123]

Voltage Crest factor is defined aa the Vpeak/Vrms. if the voltage contains 10% 3rd harmonics that is phase shifted by 30 deg. then how to calculate the Vpeak and Vrms. Please not the phase shift on the harmonics,i.e., it is not in phase with the fundamental.

V(T)= 100* sqrt 2 * Sin(w.t)+ 10 * sqrt 2* Sin(3w.t + 30); where w is fundamental angular freq.

 

[steve66]

Re: Harmonics

That is a calculus problem of finding the maximum point (the peak). Basically, you have to differentiate the equation you have, and find the point where the differential equation is zero. That tells you where the maximum is. Then you find the value of the peak.

If this isn't for a calculus class, use a graphing calculator to find your peak.

Steve

 

[charlie b]

Re: Harmonics

Originally posted by steve66: Basically, you have to differentiate the equation you have, and find the point where the differential equation is zero. That tells you where the maximum is.

Oh come on Steve. We have a golden opportunity to do some hard core dazzling with our brilliance on this one. No need to oversimplify here.

 

[Basra123]

Re: Harmonics

Charlie,

This is just a sinosoidal function and thus the total RMS is calculate as follow:

RMS = [(100/sqrt 2)^2+ (10/sqrt2)^2]^1/2
= 71.0693

as to the peak, your approach is right since we know a sine wave will have maxima, minima and inflection point. However, if you draw the pahsor diagram wiht the first term as a reference you will see that 2nd term is leading the first by 30 degrees and thus only its cosine component will contribute to the total peak. What are your thoughts on this? it looks simple so I am not so sure of it.

 

[steve66]

Re: Harmonics

I am going to cheat and use my graphing calculator. I get a peak of about 133.2 at about 107.7 deg.

If I use Basra123's formula for RMS (I can't verify if it's right or not, but I think it might be since the sine waves are different frequencies, or for Charlie B, they are orthogonal.) I get 100.5.

Thus, the crest factor is 133/100.5 or 1.32.

If you want something more formal, I'm sure Charlie or Rattus will help The problem I see is that when you take the deriverative, you get the sum of two cosines. Trying to solve that for zero may be just as hard as trying to find the max. for your first equation.

Steve

 

[Basra123]

Re: Harmonics

Steve,

when calculating Rms you need to divide the peak values given by sqrt of 2 first.

 

[Basra123]

Re: Harmonics

Charlie,
Correction to the this phasor diagram explaination below. Actually using the phasor diagram one can use the law of cosines and find the total peak. so both the normal and horizontal components will contribute to the peak. it turns out to be very close since the normal component is very small and the total peak is 108.77 instead of the inital value of 108.66
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However, if you draw the pahsor diagram wiht the first term as a reference you will see that 2nd term is leading the first by 30 degrees and thus only its cosine component will contribute to the total peak. What are your thoughts on this? it looks simple so I am not so sure of it.

 

[steve66]

Re: Harmonics

Originally posted by Basra123:
Steve,

when calculating Rms you need to divide the peak values given by sqrt of 2 first.

I meant to mention that I was using the equation in your first post where you multiply by the sqrt (2). Thus, I am using 100V RMS fundamental, and a 10V RMS harmonic.

You can either multiply by sqrt(2) as in your first post, or divide by sqrt(2) as in your second post, but you can't do both.

Steve

 

[steve66]

Re: Harmonics

If you are trying to draw these on a phasor diagram, don't forget one wave is 3x the frequency of the fundamental. Thus, by the time the fundamental peaks, the harmonic is negative. As a result, the peak of the combined wave is actually less than the peak of the fundamental.

Steve

 

[Basra123]

Re: Harmonics

Steve,

In your earlier post you used the graphing calc and came up with a peak of 133 but now you are saying the total paek of the fundamental and 3rd harmonic is less than the fundamental which is 100??? can you explain this conflict?

 

[steve66]

Re: Harmonics

Originally posted by Basra123:
Steve,

In your earlier post you used the graphing calc and came up with a peak of 133 but now you are saying the total paek of the fundamental and 3rd harmonic is less than the fundamental which is 100??? can you explain this conflict?

Yes, I used 100V RMS for the fundamental, and 10V RMS for the harmonic. Thus, the peak of the fundamental is 100*sqrt(2) = 141.4 volts.

I was supprised that the peak of the combined wave was less, and at first I thought I had made a mistake. But I double checked the math, and after graphing all three waves, it makes sense.

It makes me wonder what use the "Crest factor" is if the crest factor can be less with a harmonic than it is with a pure sine wave?

Steve