Concept explainers
With regard to the circuit presented in Fig. 9.42, (a) obtain an expression for v(t) which is valid for all t > 0; (b) calculate the maximum inductor current and identify the time at which it occurs; (c) determine the settling time.
FIGURE 9.42
(a)
Find the equation for voltage
Answer to Problem 16E
The equation of voltage is
Explanation of Solution
Formula used:
The expression for the exponential damping coefficient is as follows:
Here,
The expression for the resonating frequency is as follows:
Here,
The expression for the two solutions of the characteristic equation of a parallel
Here,
Here,
Calculation:
The redrawn circuit is shown in Figure 1 as follows:
Refer to the Figure 1,
At
The current across inductor is zero because inductor is not energized.
The expression for voltage
Here,
The redrawn circuit at
Refer to the Figure 2:
The expression for the voltage across capacitor at
Substitute
The expression for the natural response of voltage for capacitor in the parallel
Substitute
Rearrange for
At
Therefore,
So,
The redrawn circuit is shown in Figure 3 as follows:
Refer to the Figure 3:
The current across resistor
The expression for KCL at node
Here,
Substitute
The expression for current flowing through capacitor for
Here,
At
The circuit diagram is redrawn as shown in Figure 4 for
Refer to the redrawn Figure 4:
Substitute
Substitute
As value of exponential frequency
Substitute
Substitute
Differentiate equation (6) both the sides with respect to time
Substitute
Substitute
Substitute
Rearrange for
Rearrange for
Substitute
Substitute
Therefore, voltage across inductor
Conclusion:
Thus, the voltage
(b)
Find the maximum inductor current and the time of occurrence.
Answer to Problem 16E
The maximum inductor current
Explanation of Solution
Calculation:
Refer to the Figure 4:
The expression for inductor current is:
Substitute
For the maximum positive value of inductor current, take the derivative of its equation and then, equate it to zero to find the time at which maximum value of current occurs.
Differentiate equation (14) both the sides with respect to time
Equate the equation to zero.
Rearrange the equation.
Take log both the sides.
Rearrange for
Substitute
This is the maximum inductor current.
Conclusion:
Thus, the maximum inductor current is
(c)
Find the settling time.
Answer to Problem 16E
The settling time of the voltage
Explanation of Solution
Calculation:
The settling time is the time at which inductor current reaches to
Substitute
The settling time is the time at which the inductor current is decreased to
Equation (17) is solved by scientific calculator which can determine the value of time
Conclusion:
Thus, the settling time of the inductor current
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Chapter 9 Solutions
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