Study of commonly encounter waveform in the power electronics system - EX 2 #3

Example-2

Consider shown waveform for Fourier series analysis. Here shown waveform is the output waveform of the semi-controlled converters and thyristor firing is the same for the positive and negative half cycle.

$i(t)$ is not even or odd function. so $a_n$ and $b_n$ is not equal zero. But $a_0$ is zero because of the net are of the waveform in one period of the cycle is zero.

Fourier series of $i(t)$  is as follows,

For simplification of Fourier analysis of this type of problems, we can convert this signal to the odd function as follows :

Here, the signal is shifted by half of the switching angle. Shifted signal x(t) is an odd function of time and because of that only $b_n$ calculation is required for Fourier series analyses.

Fourier series of x(t) is as follows :

Fourier series of i(t) can be calculated by shifting signal x(t).

  RMS current (i) :

Example-3

Fourier series analysis delayed signal i(t) :

  

• x(t) can be obtained by shifting signal i(t) and transformation relations are shown in the figure.

Fourier series of  x(t)  (Example-1) (https://eefacts.blogspot.com/2019/06/study-of-commonly-encounter-waveform-in.html)

 

And Fourier series of  i(t) is as follows