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ttokoro
ttokoro21-Topaz I
21-Topaz I
May 27, 2025 1:25 AM UTC
Solved

Make equivarent sine waveform that has Em = 1, Ea = pi/2 and E = 1/sqrt(2) .

Make equivarent sine waveform that has maximum value = 1, average value = pi/2 and RMS = 1/sqrt(2).

Below waveform has RMS=1/sqrt(2) but Ea=2/3 not pi/2.

image.png

image.png, image.png

 

Hard probrem. If you can make it, find the Fourier series of the waveform. Calculate the Ea and RMS values for 10, 100 and 1000 harmonics include. Symbolic evaluation by using below programming may causing a trouble.  

image.png

 

ttokoro
ttokoroAnswer
21-Topaz I
Jun 10, 2025 2:51 AM UTC

Step 1. Solve a and b for the function f(x) that has same RMS and average values of sine wave. 

 

image.png 

Step 2. Only use the function of 1/4 T to calculate the Fourier series for odd function with same f(t)=-f(t-T/2) and symmetory function. It only has odd B Fourier series. 

image.png

Step 3. Calculate the odd B Fourier series symbolically.

image.png

Step 4. Find the (2*n-1)th B Fourier series function symbolically.

image.png

image.png

Step 5. Make waveform by numerical engine with N=1000, that is 1999th harmonics included.

image.png

image.png

 

Step 6. Plot the waveform that has 1st to (2*N-1)th spectum signals what you want . 

 

image.pngimage.png

t.t.
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3 replies

ppal_25568718-Opal
18-Opal
May 30, 2025 1:56 AM UTC

Is this sampled data  and you are creating phasors at the fundamental? What is being asked here?

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ttokoro
ttokoro21-Topaz IAuthor
21-Topaz I
May 31, 2025 12:03 AM UTC

My question is how to make equivqrent sine wave. 

image.pngimage.png

 

The answer of same RMS waveform is solved by below.

image.png

 

Therefore, one of the same average and RMS values waveform is also solved as below.

image.pngimage.pngimage.png

 

These are good questions for electrical engineering students.

How to calculate the Fourier series by using Mathcad is also good question.

image.png

But this attempt is fail when increase the number of Fourier series Nt.

So this is good charenging practice for Mathcad users to calculate all 1 to 1999 harmonics and make the waveform below.

image.png

 

 

 

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t.t.
ttokoro
ttokoro21-Topaz IAuthorAnswer
21-Topaz I
Jun 10, 2025 2:51 AM UTC

Step 1. Solve a and b for the function f(x) that has same RMS and average values of sine wave. 

 

image.png 

Step 2. Only use the function of 1/4 T to calculate the Fourier series for odd function with same f(t)=-f(t-T/2) and symmetory function. It only has odd B Fourier series. 

image.png

Step 3. Calculate the odd B Fourier series symbolically.

image.png

Step 4. Find the (2*n-1)th B Fourier series function symbolically.

image.png

image.png

Step 5. Make waveform by numerical engine with N=1000, that is 1999th harmonics included.

image.png

image.png

 

Step 6. Plot the waveform that has 1st to (2*N-1)th spectum signals what you want . 

 

image.pngimage.png

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t.t.
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