Concept explainers
Obtain the Fourier series for the periodic waveform in Fig. 17.57.
Find the Fourier series for the periodic waveform in Figure 17.57.
Answer to Problem 19P
The Fourier series
Explanation of Solution
Given data:
Refer to Figure 17.57 in the textbook.
Formula used:
Write the expression to calculate the fundamental angular frequency.
Here,
Write the general expression to calculate trigonometric Fourier series of
Here,
Write the expression to calculate the dc component of the function
Write the expression to calculate Fourier coefficients.
Calculation:
The given waveform is drawn as Figure 1.
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
Simplify the above equation to find
Substitute
Simplify the above equation to find
Assume the following to reduce the equation (6).
Substitute the equations (7) and (8) in equation (6) to find
Consider the following integration formula.
Compare the equations (7) and (10) to simplify the equation (7).
Using the equation (10), the equation (7) can be reduced as,
Simplify the above equation to find
Compare the equations (8) and (10) to simplify the equation (8).
Using the equation (10), the equation (8) 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 (11).
Substitute the equations (12) and (13) in equation (11) to find
Compare the equations (12) and (10) to simplify the equation (12).
Using the equation (10), the equation (12) can be reduced as,
Simplify the above equation to find
Compare the equations (13) and (10) to simplify the equation (13).
Using the equation (10), the equation (13) can be reduced as,
Simplify the above equation to find
Substitute
Substitute
Conclusion:
Thus, the Fourier series
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Chapter 17 Solutions
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