Concept explainers
Sketch the line spectrum for the waveform shown in Figure 17.4c (limited to the five largest terms).
Answer to Problem 14E
The line spectrum for the waveform shown in Figure 17.4c is sketched as shown in Figure 1.
Explanation of Solution
Given data:
Refer to Figure 17.4c in the textbook.
Formula used:
Write the general expression for Fourier series expansion.
Write the general expression for Fourier series coefficient
Write the general expression for Fourier series coefficient
Write the general expression for Fourier series coefficient
Write the expression to calculate the fundamental angular frequency.
Here,
Calculation:
In the given Figure 17.4a, the time period is
The function
Substitute 2 for T in equation (5) to find
Applying equation (6) in equation (2) to find
Simplify the above equation as follows,
As the given function is an odd symmetry. Therefore,
Applying equation (6) in equation (4) to finding the Fourier coefficient
Equation (8) is simplified as,
Therefore, equation (8) becomes,
Consider the function,
Consider the following integration formula.
Compare the equations (10) and (11) to simplify the equation (10).
Using the equation (11), the equation (10) can be reduced as,
Simplify the above equation as follows,
Consider the function,
Compare the equations (12) and (11) to simplify the equation (12).
Using the equation (11), the equation (12) can be reduced as,
Simplify the above equation as follows,
Substituting the values of x and y in equation (9) as follows,
Converting the equation (1) which is in angular frequency into frequency.
Substitute the values of
For
For
The sketch for the line spectrum of
Conclusion:
Thus, the line spectrum for the waveform shown in Figure 17.4c is sketched as shown in Figure 1.
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Chapter 17 Solutions
Loose Leaf for Engineering Circuit Analysis Format: Loose-leaf
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