In the circuit of Fig. 16.74, find i(t) for t > 0.
Find the expression of current
Answer to Problem 51P
The expression of current
Explanation of Solution
Given data:
Refer to Figure 16.74 in the textbook.
Formula used:
Write a general expression to calculate the impedance of a resistor in s-domain.
Here,
Write a general expression to calculate the impedance of an inductor in s-domain.
Here,
Write a general expression to calculate the impedance of a capacitor in s-domain.
Here,
Calculation:
The given circuit is redrawn as shown in Figure 1.
For a DC circuit, at steady state condition when time
Now, the Figure 1 is reduced as shown in Figure 2.
Refer to Figure 2, the switch is open and there is no source connected to the circuit. Therefore, the voltage across the capacitor and current through inductor are zero.
The current through inductor and voltage across capacitor is always continuous so that,
For time
Substitute
Substitute
Substitute
Substitute
Refer to Figure 3, the circuit for
Apply Laplace transform for above to find
Convert the Figure 3 into s-domain.
Apply nodal analysis at node
Simplify the above equation as follows:
Simplify the above equation as follows:
Simplify the above equation to find
From the above equation, the characteristic equation is
Write a general expression to calculate the roots of quadratic equation
Comparing equation (5) with the equation
Substitute
Simplify the above equation to find
Substitute the roots of characteristic equation in equation (4) to find
Take partial fraction for above equation.
The equation (7) can also be written as follows:
Simplify the above equation as follows:
Substitute
Simplify the above equation as follows:
Simplify the above equation to find
Substitute
Simplify the above equation to find
Substitute
Simplify the above equation to find
Substitute
Refer to Figure 3, the current through resistor
Substitute
Take inverse Laplace transform for above equation to find
Simplify the above equation to find
Conclusion:
Thus, the expression of current
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Chapter 16 Solutions
Fundamentals of Electric Circuits
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