# Solved Problems-8

Problems-8

- Find the trigonometric Fourier series of the triangular waveform shown in the following figure.

- If this voltage is approximated by , find the mean-square error.
- 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*Ï‰*_{0}= 1 radian/second for the waveform.

Solution

- The wave is an odd function and is having half wave symmetry.

âˆ´
Now, ;
;
âˆ´ ;

{âˆµ
âˆ´
= â€“,
Hence,

Since, the wave is having halfâ€“wave symmetry,

âˆ´

Now, ; for 0 <

âˆ´
âˆ´ ; for
and ; for
âˆ´
âˆ´
âˆ´
âˆ´

*a*= 0 and_{n}*a*_{0}= 0*n*is odd only.{âˆµ

*Ï‰T*= 2*Ï€*}*,**n*= 1, 5, 9, â€¦*n*= 3, 7, 11, â€¦- The error is,

âˆ´

Now, ; for 0 <

*t*<*T*/4âˆ´

- Here,

*n*= 1, 5, 9, â€¦*n*= 3, 7, 11, ...