Consider the circuit depicted in Fig. 9.40. (a) Obtain an expression for iL(t) valid for all t > 0. (b) Obtain an expression for iR(t) valid for all t > 0. (c) Determine the settling time for both iL and iR.
■ FIGURE 9.40
(a)
Obtain an expression for
Answer to Problem 13E
The current across inductor
Explanation of Solution
Formula used:
The expression for the exponential damping coefficient in parallel
Here,
The expression for the resonating frequency in parallel
Here,
The expression for the two solutions of the characteristic equation of a parallel
Here,
The expression for the natural response of the parallel
Here,
Calculation:
The redrawn circuit is shown in Figure 1 as follows:
Refer to the Figure 1:
At
Here,
The redrawn circuit at
Refer to the Figure 2:
Substitute
The expression for voltage
Here,
Substitute
Substitute
Rearrange for
At
The voltage across inductor is same as voltage across capacitor due to parallel circuit and thus, the expression for voltage across inductor is:
The redrawn circuit is shown in Figure 3 as follows:
Refer to the Figure 3:
Substitute
Substitute
Differentiate equation (5) both the sides with respect to time
The expression for the voltage across inductor at time
At
Substitute
Rearrange for
Substitute
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
Substitute
Substitute
Solve for
Rearrange for
Substitute
Rearrange for
Substitute
Conclusion:
Thus, the current across inductor
(b)
Find the equation for current across resistor for
Answer to Problem 13E
The equation of current
Explanation of Solution
Calculation:
Refer to the Figure 3:
The expression for current across resistor at
At
Therefore,
At
Substitute
At
Substitute
Conclusion:
Thus, the equation of current
(c)
Find the settling time for both
Answer to Problem 13E
The settling time for
Explanation of Solution
Calculation:
The settling time is the time at which current reaches to
Since the inductor current is exponential in nature and time cannot be taken as negative, therefore, inductor current takes its maximum value at
Substitute
The maximum value of current is:
The expression for current at settling time
Substitute
The settling time is the time at which the current is decreased to
Equation (24) is solved by scientific calculator which can determine the value of time
Take log both the side in equation (25).
Rearrange for
Substitute
The maximum value of current is:
The expression for current at settling time
Substitute
The settling time is the time at which the current is decreased to
The equation can be approximated for
Take log both the sides of equation (28).
Rearrange for
Conclusion:
Thus, the settling time for
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Chapter 9 Solutions
Loose Leaf for Engineering Circuit Analysis Format: Loose-leaf
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