Concept explainers
Find the current through the inductor
Answer to Problem 39P
The current through the inductor
Explanation of Solution
Given data:
Refer to Figure 17.75(a) and 17.75(b) in the textbook.
The inductor L is
The capacitor
Formula used:
Write the general expression for Fourier series expansion.
Write the expression to calculate the fundamental angular frequency.
Here,
Calculation:
Refer to Figure 17.73(a) in the textbook.
The source voltage
The time period of Figure 17.75(a) is,
Substitute 2 for T in equation (2) to find the angular frequency
Finding the Fourier coefficient
Applying equation (3) in equation (4) as follows,
The above equation becomes,
Finding the Fourier coefficient
Applying equation (3) in equation (5) as follows,
Finding the Fourier coefficient
Applying equation (3) in equation (6) as follows,
Simplify the equation as follows,
The above equation becomes,
Substituting the Fourier coefficients in equation (1) as follows,
Refer to Figure 17.75(b) in the textbook.
For the DC component, the current
The inductor acts as a short circuit for DC.
For the kth harmonic,
Consider Figure 17.75(b) for AC analysis.
The impedance of the inductor is calculated as follows,
Substitute
The impedance of the capacitor is calculated as follows,
Substitute
The modified circuit is shown in Figure 1.
In Figure 1, the impedance is,
Simplify the above equation as follows,
The impedance
Substitute
The current
Substitute
The current
Substitute
Simplify the equation as follows,
The magnitude and phase angle of the current
In the time domain,
Where,
The phase angle
And,
The current
Conclusion:
Thus, the current through the inductor
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Chapter 17 Solutions
Fundamentals of Electric Circuits
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