# Homework Help: AC waveform question help

1. Apr 20, 2012

### oxon88

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

An A.C. voltage, V comprises of a fundamental voltage of 100V rms at a frequency of 120Hz, a 3rd harmonic which is 20% of the fundamental, a 5th harmonic which is 10% of the fundamental and at a phase angle of 1.2 radians lagging.

i) Write down an expression for the voltage waveform.

ii) Sketch the waveforms of the harmonic compnents.

iii) Determin the voltage at 20ms.

iv) Given an ideal V = 100V rms, what is the percentage error at 20ms

3. The attempt at a solution

part i)

V = Vrms * sqrt2 = 100* 1.414 = 141.4V at 120Hz

3rd harmonic = 20% of 141.4 = 28.28V at 360Hz

5th harmonic = 10% of 141.4 = 14.14V at 600Hz

v = (141.4sin(240∏t)) + (28.3sin(720∏t)) + (14.1sin(1200∏t-1.2))

does this first part look correct?

2. Apr 20, 2012

### Staff: Mentor

Yes, it looks okay.

3. Apr 20, 2012

### oxon88

Ok thanks. So for the sketch I just need to plot the 3rd harmonic and 5th harmonic. What time period would be appropriate?

4. Apr 20, 2012

### Staff: Mentor

A couple of cycles (of the fundamental) should do. But at least up to 20ms. (Note: the fundamental can also be referred to as the first harmonic.)

5. Apr 20, 2012

### Staff: Mentor

The fundamental should be plotted too (it's the "first harmonic").

Choose a time period that will display at least one full cycle of the fundamental.

6. Apr 20, 2012

### oxon88

should it look something like this?

7. Apr 20, 2012

### Staff: Mentor

You're sketch for fifth harmonic should show x5 the frequency!

8. Apr 20, 2012

### mahnoorbloch

i think the ans is right...

9. Apr 24, 2012

### oxon88

Ok so i have changed the time scale and got this....

does this look as it should?

10. Apr 24, 2012

### Staff: Mentor

That looks more like it.

11. Apr 24, 2012

### oxon88

ok great.

iii) Determin the voltage at 20ms.

v = (141.4sin(240∏*0.02)) + (28.3sin(720∏*0.02)) + (14.1sin(1200∏*0.02-1.2))

v = 83.11 + 26.91 - 13.14 = 96.88 v

does this look correct?

12. Apr 24, 2012

### Staff: Mentor

It does.

Might be worth going back and checking to make sure that the problem did not specify a phase angle for the 3rd harmonic.

13. Apr 25, 2012

### oxon88

Many thanks. I have checked the original question and there is no mention of a phase angle for the 3rd harmonic.

14. Apr 25, 2012

### oxon88

ok so last part

iv) Given an ideal V = 100V rms, what is the percentage error at 20ms

100v - 96.845v = 3.155v

so would it be 3.155% ?

15. Apr 25, 2012

### Staff: Mentor

I'm sure you can't mix RMS and instantaneous values to get anything meaningful.

16. Apr 25, 2012

### oxon88

ok so would i need to work out the RMS at 20ms?

96.845 / (√2) = 68.5v

then work this out as a % of the 100v rms?

100v - 68.5v = 31.5v = 31.5% error?

am i making any sense?

17. Apr 25, 2012

### Staff: Mentor

The only calculation that would make sense to me would be based on:

ideal instantaneous value at that time - actual instantaneous value

18. Apr 26, 2012

### oxon88

Actual instantaneous value at 20ms v = (141.4sin(240∏*0.02)) + (28.3sin(720∏*0.02)) + (14.1sin(1200∏*0.02-1.2))

v = 83.11 + 26.91 - 13.14 = 96.88 v

Ideal instantaneous value at 20ms = 141.42Sin(240∏*0.02) = 83.11v

error = [(83.11v - 96.845v)/100]*100 = -13.735%

19. Apr 26, 2012

### oxon88

does the answer above look ok?

i have ploted a graph to show actual Vs. Ideal

20. Apr 26, 2012

### Staff: Mentor

I would not have divided by 100 when determining fractional error. Some other data value would seem more appropriate.

21. Apr 26, 2012

### oxon88

ok but the question asks for a % error?

22. Apr 26, 2012

### Staff: Mentor

On second thought, it would be better to swap them:

actual instantaneous value - ideal instantaneous value at that time

So if the actual value were a few volts too high, the error would be +ve.

23. Apr 26, 2012

### Staff: Mentor

I was referring to where you divided by 100.

24. Apr 26, 2012

### oxon88

ah right ok. so....

error = [(96.845v - 83.11v)/83.11]*100 = 16.5%

25. Apr 26, 2012

### Staff: Mentor

How did you decide that 96.845 would be the appropriate denominator here?

EDIT Yes, 83.1V does seem the better choice.

Last edited by a moderator: Apr 19, 2017