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Solved Problems-8

Problems-8
  1. Find the trigonometric Fourier series of the triangular waveform shown in the following figure.
Description: Description: 5174.png
  1. If this voltage is approximated by Description: Description: 5183.png, find the mean-square error.
  2. If this voltage waveform is applied to the network in the below figure, then find the current i(t) and draw the magnitude and phase spectra of i(t). Take ω0 = 1 radian/second for the waveform.
Description: Description: 925.png
Solution
  1. The wave is an odd function and is having half wave symmetry.
∴ an = 0 and a0 = 0
 
Now, Description: Description: 5190.pngDescription: Description: 5209.png
 
Description: Description: 5215.pngDescription: Description: 5221.png
 
∴ Description: Description: 5232.pngn is odd only.
 
Description: Description: 5238.png
 
Description: Description: 5245.png
 
Description: Description: 5255.png
 
Description: Description: 5261.png
Description: Description: 5267.png{∵ ωT = 2π}
 
∴ Description: Description: 5273.png, n = 1, 5, 9, …
 
= –Description: Description: 5279.pngn = 3, 7, 11, …
 
Hence,
Description: Description: 5285.png 
  1. The error is, Description: Description: 5294.png
The main square error is,
Description: Description: 5301.png
 
Since, the wave is having half–wave symmetry,
∴ Description: Description: 5312.png

Now, Description: Description: 5318.png; for 0 < t < T/4
∴ Description: Description: 5327.png 
  1. Here,
Description: Description: 5334.png
 
Description: Description: 5341.png
 
∴ Description: Description: 5347.png; for n = 1, 5, 9, …
 
Description: Description: 5353.png
 
and Description: Description: 5359.png; for n = 3, 7, 11, ...
 
∴ Description: Description: 5368.png
 
∴ Description: Description: 5375.png
 
∴ Description: Description: 5381.png
 
∴ Description: Description: 5390.png 

 





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