Concept explainers
Design a problem to better understand the circuit analysis using Laplace transform using Figure 16.36.
Explanation of Solution
Problem design:
Find the value of voltage across resistor
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.
Apply Laplace transform to the voltage source
Substitute
Substitute
Substitute
Substitute
Convert the Figure 1 into s-domain.
Apply Kirchhoff’s current law at node
Simplify the above equation as follows,
Simplify the above equation to find
From equation (4), the characteristic equation is written as follows,
Write an expression to calculate the roots of characteristic equation
Here,
Compare equation (5) with quadratic equation
Substitute
Now, the equation (4) is written as follows,
Take partial fraction for equation (7).
The equation (8) 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 as follows,
Simplify the above equation to find
Substitute
Simplify the above equation as follows,
Simplify the above equation to find
Substitute
Take inverse Laplace transform for equation (10) to find
Simplify the above equation to find
Conclusion:
Thus, the problem to better understand the circuit analysis using Laplace transform is designed.
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Chapter 16 Solutions
Fundamentals of Electric Circuits
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