Concept explainers
Determine the Fourier series representation of the function in Fig. 17.63.
Figure 17.63
Find the Fourier series representation of the function given in Figure 17.63.
Answer to Problem 25P
The Fourier series representation
Explanation of Solution
Given data:
Refer to Figure 17.63 in the textbook.
Formula used:
Write the expression to calculate the fundamental angular frequency.
Here,
Write the general expression to calculate the Fourier series of
Here,
Calculation:
The given waveform is drawn as Figure 1.
Refer to Figure 1, each half cycle is the mirror image of the next half cycle. Therefore, the waveform is called as half-wave symmetry and it is expressed as,
Write the expressions for Fourier coefficients for the half-wave odd symmetric function.
Refer to the Figure 1. The Fourier series function of the waveform is defined as,
The time period of the function in Figure 1 is,
Substitute
Substitute
Simplify the above equation to find
Assume the following to reduce the equation (5).
Substitute the equation (6) in equation (5) to find
Consider the following integration formula.
Compare the equations (6) and (8) to simplify the equation (6).
Using the equation (8), the equation (6) can be reduced as,
Simplify the above equation to find
Substitute
Substitute
Simplify the above equation to find
Assume the following to reduce the equation (9).
Substitute the equation (10) in equation (9) to find
Compare the equations (8) and (10) to simplify the equation (10).
Using the equation (8), the equation (10) can be reduced as,
Simplify the above equation to find
Substitute
Substitute
Conclusion:
Thus, the Fourier series representation
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Chapter 17 Solutions
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