Find the expression of current
Answer to Problem 26P
The expression of current
Explanation of Solution
Given data:
Refer to Figure 16.49 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 the switch is in position A at time
Now, the Figure 1 is reduced as shown in Figure 2.
Refer to Figure 2, the short circuited inductor is connected in parallel with resistors
Now, the Figure 2 is reduced as shown in Figure 3.
Refer to Figure 3, the current flow through the inductor is same as the value of current source
Refer to Figure 3, there is no capacitor placed in a circuit. Therefore, the voltage across the capacitor is zero.
The current through inductor and voltage across capacitor is always continuous so that,
For time
Substitute
Substitute
Substitute
Using element transformation methods with initial conditions convert the Figure 4 into s-domain.
Apply Kirchhoff’s current law for the circuit shown in Figure 5.
Substitute
Simplify the above equation to find
From the equation (4), the characteristic equation is
Write a general expression to calculate the roots of quadratic equation
Comparing the 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 to find
Substitute
Simplify the above equation to find
Substitute
Refer to Figure 5, the current
Substitute
Assume,
Substitute equation (10) and (11) in equation (9).
Take partial fraction for equation (10).
The equation (13) can also be written as follows:
Simplify the above equation as follows:
Substitute
Simplify the above equation to find
Substitute
Simplify the above equation to find
Substitute
Take partial fraction for equation (11).
The equation (13) can also be written as follows:
Simplify the above equation as follows:
Substitute
Simplify the above equation to find
Substitute
Simplify the above equation to find
Substitute
Substitute
Apply 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
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