Concept explainers
The current
Answer to Problem 5.68HP
The expression for the current flowing through the inductor is
Explanation of Solution
Calculation:
The given diagram is shown in Figure 1
The conversion from
The conversion from
The conversion from
The conversion from
The conversion from
For time
The required diagram is shown in Figure 2
From above, the expression for the initial voltage across the capacitor is given by,
Substitute
The expression for the voltage across the capacitor for time
Substitute
The expression for the current flowing through the inductor for time
Substitute
The inductor opposes sudden change in the current, thus the current
Substitute
Close the switch and redraw the circuit for time
The required diagram is shown in Figure 3
The expression for the voltage across the inductor is given by,
Apply KVL to the top node.
Substitute
The standard second order equation for the differential equation.
From above and from equation (1), the angular frequency is derived as,
Substitute
The expression for the damping coefficient is given by,
Substitute
The value of
The expression for the output response equation of the inductor current is given by,
The expression for the forced response of the system is given by,
Substitute
The expression for the natural response for the system is give by,
Substitute
The expression to calculate the damping frequency of the circuit is given by,
Substitute
Substitute
Substitute
Substitute
The differentiation of equation (5) with respect to
Substitute
Substitute
Substitute
Conclusion:
Therefore, the expression for the current flowing through the inductor is
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Chapter 5 Solutions
Principles and Applications of Electrical Engineering
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