# Power percentage, square wave, Fourier series

1. Mar 31, 2013

### Jd303

1. The problem statement, all variables and given/known data

What is the percentage of power (out of the total power) contained up to the third harmonic (power in DC component, a1 , a-1 , a2 , a-2 , a3 , a-3 ) of the square waveform shown above? (the duty cycle = D = τ/T0= 0.5)

2. Relevant equations

3. The attempt at a solution

Hey all,
The following question refers to the attached diagram. I thought the question was simple enough due to the amplitude being 1, and the value of D being 0.5.

The simplified formula I calculated for the answer is 2/(k*pi) for odd harmonics (as even values of k result in a value of 0)

-However when adding 2/pi and 2/(3*pi) i get 0.8488 and hence 84.9%

However the multiple choice answers are 96.7, 95, 72.5 and 73.3%

I am not sure whether this is a simple mistake or a large misunderstanding of the theory.

Any help would be much appreciated!

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• ###### square_wave_002.jpg
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Last edited by a moderator: Apr 1, 2013
2. Apr 1, 2013

### Staff: Mentor

The waveform looks like it also has a DC component...?

3. Apr 1, 2013

### Jd303

Sorry only just starting out this topic, would the DC component be 0.5? How does this component chang emy calculations?

4. Apr 1, 2013

5. Apr 1, 2013

### Staff: Mentor

BTW, that article was the first hit on the list of my Google search for square wave harmonics amplitude.

6. Apr 1, 2013

### Jd303

Yes I have actually looked at that page, but it just isn't clicking, here is my understanding.
-The zero harmonic is the DC component and hence 0.5
-Even harmonics have a value of 0
-Odd harmonics have a value of 2/(pi*n)
-Total amplitude is 1
-So percentage power should be ((2/pi) + 2/(3*pi))/1 = 0.8488

7. Apr 1, 2013

### Staff: Mentor

That page is listing voltage component values. How is the power related to the voltage?

8. Apr 1, 2013

### Jd303

P = (V^2)/R
Hence power percentage would be 0.8488^2.!
Hopefully I have finally gotten that one right! Thanks for your persistence with me

9. Apr 1, 2013

### Staff: Mentor

10. Apr 7, 2013

### fbash

i think its
(2/pi)^2 + (2/3pi)^2
divided by
(2/pi)^2 + (2/3pi)^2 +(2/5pi)^2 + (2/7pi)^2

which gives you 95%