For the RLC circuit shown in Fig. 16.88, find the complete response if v(0) = 100 V when the switch is closed.
Find the expression of voltage response
Answer to Problem 65P
The expression of voltage response
Explanation of Solution
Given data:
Refer to Figure 16.88 in the textbook.
The value of initial voltage across the capacitor
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 time
Apply Laplace transform for
Substitute
Substitute
Substitute
Using element transformation methods with initial conditions convert the Figure 2 into s-domain.
Apply Kirchhoff’s voltage law for the circuit shown in Figure 3.
Substitute
Simplify the above equation to find
Refer to Figure 3, the voltage
Substitute
Assume,
Substitute equation (5) in equation (4).
Take partial fraction for equation (5).
The equation (7) can also be written as follows:
Simplify the above equation as follows:
Simplify the above equation as follows:
Equate the co-efficient of constant term in equation (8) to find
Simplify the above equation to find
Equate the co-efficient of
Simplify the above equation to find
Substitute
Equate the co-efficient of
Substitute
Equate the co-efficient of
Substitute
Simplify the above equation to find
Equate the co-efficient of
Substitute
Substitute the equation (11) in equation (10).
Simplify the above equation as follows:
Simplify the above equation to find
Substitute equation (11) in equation (12).
Simplify the above equation as follows:
Substitute the equation (13) in above equation to find
Simplify the above equation to find
Substitute
Substitute
Substitute
Substitute
Substitute
Simplify the above equation as follows:
Take inverse Laplace transform for above equation to find
Conclusion:
Thus, the expression of voltage response
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Chapter 16 Solutions
Fundamentals of Electric Circuits
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