Concept explainers
For the circuit represented by Fig. 9.44, the two resistor values are R1 = 0.752 Ω and R2 = 1.268 Ω, respectively. (a) Obtain an expression for the energy stored in the capacitor, valid for all t > 0; (b) determine the settling time of the current labeled iA.
FIGURE 9.44
(a)
Find the expression for the energy stored in the capacitor, valid for all
Answer to Problem 20E
The expression for the energy stored in the capacitor, valid for all
Explanation of Solution
Given Data:
The value of the resistor
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,
The expression for the energy stored in the capacitor is as follows:
Here,
Calculation:
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 1 for
Refer to the redrawn Figure 1:
As parallel branches have same voltage so voltage across
The expression for the current flowing through
Here,
Substitute
The expression for the current flowing through the
Here,
Substitute
Substitute
As parallel branches have same voltage and the voltage across the short circuit branch has
The capacitor does not allow sudden change in the voltage and the inductor does not allow sudden change in current.
So,
Therefore, the voltage across the
The redrawn circuit diagram is given in Figure 2 at
l
Refer to the redrawn Figure 2:
As the voltage across the he
To find equivalent resistance across the capacitor, a
The redrawn circuit diagram is given in Figure 3.
Refer to the redrawn Figure 3:
Apply KVL in the circuit.
Here,
Substitute
Rearrange for
The expression for the equivalent resistance the circuit is as follows:
Here,
Substitute
Substitute
Substitute
Here, the exponential damping coefficient is greater than the resonating frequency,
So, the response of the parallel
Substitute
Substitute
Substitute
Substitute
The voltage across the capacitor at
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
Substitute
So, the energy stored in the capacitor, valid for all
Conclusion:
Thus, the expression for the energy stored in the capacitor, valid for all
(b)
Find the settling time of the current
Answer to Problem 20E
The settling time of current
Explanation of Solution
Calculation:
The redrawn circuit diagram is given in Figure 4.
Refer to the redrawn Figure 4:
The expression for the current flowing in the left hand mesh at
Here,
Substitute
Differentiate both side of the equation (19).
The maximum value is obtained when derivative is equated to zero.
Rearrange equation (20).
Take natural logarithm both sides.
Rearrange for
Substitute
So, the value of the maximum current flowing in the left hand mesh is
Settling time is the time at which the value of the current reaches
The expression for the current at
Here,
Substitute
The value of current flowing in the left hand mesh at
Substitute
Since the component
So, the new equation is:
Rearrange equation (22).
Take natural logarithm both sides.
Rearrange for
So, the settling time of the currentflowing in the left hand mesh
Conclusion:
Thus, the settling time of current
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