Concept explainers
Find the Fourier series of the given voltage source.
Answer to Problem 12P
The Fourier series
Explanation of Solution
Given data:
The voltage source of the periodic waveform is,
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:
Given voltage source function is,
The time period of the given voltage source function is,
Substitute
Substitute
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
Assume the following to reduce the equation (11).
Substitute the equation (12) 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
Substitute
Substitute
Substitute
Simplify the above equation to find
Assume the following to reduce the equation (14).
Substitute the equations (15) and (16) in equation (14) to find
Compare the equations (15) and (10) to simplify the equation (15).
Using the equation (10), the equation (15) can be reduced as,
Simplify the above equation to find
Compare the equations (16) and (10) to simplify the equation (16).
Using the equation (10), the equation (16) can be reduced as,
Simplify the above equation to find
Assume the following to reduce the equation (18).
Substitute the equation (19) in equation (18) to find
Compare the equations (19) and (10) to simplify the equation (19).
Using the equation (10), the equation (19) can be reduced as,
Simplify the above equation to find
Substitute
Substitute
Substitute
Conclusion:
Thus, the Fourier series
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Chapter 17 Solutions
Fundamentals of Electric Circuits
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