Concept explainers
(a) Assuming the passive sign convention, obtain an expression for the voltage across the 1 Ω resistor in the circuit of Fig. 9.41 which is valid for all t > 0. (b) Determine the settling time of the resistor voltage.
■ FIGURE 9.41
(a)
Obtain an expression for voltage across
Answer to Problem 15E
The expression for voltage across
Explanation of Solution
Formula used:
The expression for the exponential damping coefficient or the neper frequency 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,
The expression for the natural response of the parallel
Here,
Calculation:
The redrawn circuit diagram is given in Figure 1 for
Refer to the redrawn Figure 1:
Substitute
Substitute
As value of neper frequency
Substitute
Substitute
The unit-step forcing function as a function of time which is zero for all values of its argument less than zero and which is unity for all positive values of its argument.
Here,
So, at
The capacitor and the inductor are connected in the circuit for long time.
So, the capacitor behaves as open circuit and the inductor behaves as short circuit.
The redrawn circuit diagram is given in Figure 2 for
Refer to the redrawn Figure 2:
The expression for the current flowing in the
Here,
Substitute
The expression for the voltage across the
Here,
Substitute
The switch closes at
The capacitor does not allow sudden change in the voltage and the capacitor does not allow sudden change in the current.
So,
As parallel branches have same voltage across them, so, voltage across resistor
Therefore,
The redrawn circuit diagram is given in Figure 3 for
Refer to the redrawn Figure 3:
The expression for the current flowing in the resistor
Here,
Substitute
Apply KCL at node
Here,
Substitute
Rearrange for
Substitute
Substitute
The voltage across the resistor
Substitute
Rearrange for
The expression for the current flowing through the
Substitute
Rearrange for
Substitute
The current flowing through the
Substitute
Rearrange for
Substitute
Rearrange for
Substitute
Substitute
Conclusion:
Thus, expression for voltage across
(b)
Find settling time of the resistor voltage.
Answer to Problem 15E
The settling time is
Explanation of Solution
Calculation:
Function for voltage across resistor
The maximum value of voltage
Substitute
So, the maximum value of voltage
Settling time is the time at which the value of the voltage
The expression for the voltage
Here,
Substitute
Substitute
Since the component
So, the new equation is:
Rearrange equation (20).
Take natural logarithm both sides.
Conclusion:
Thus, the settling time is
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