Explanation Given data: Refer to Figure 16.35 in the textbook. Formula used: Write a general expression to calculate the impedance of a resistor in s- domain. Z R ( s ) = R (1) Here, R is the value of resistance. Write a general expression to calculate the impedance of a inductor in s- domain. Z L ( s ) = s L (2) Here, L is the value of inductance. Write a general expression to calculate the impedance of a capacitor in s- domain. Z C ( s ) = 1 s C (3) Here, C is the value of capacitance. Write a general expression to calculate the Laplace transform of the step input. ℒ ( u ( t ) ) = 1 s Given data: The given circuit is redrawn as shown in Figure 1. Apply Laplace transform to the voltage source v . V ( s ) = 8 ( 1 s ) { ∵ ℒ ( u ( t ) ) = 1 s } = 8 s Substitute 1 for R in equation (1) to find Z R . Z R ( s ) = 1 Substitute 1 for L in equation (2) to find Z L . Z L ( s ) = s ( 1 ) = s Substitute 1 for C in equation (3) to find Z C . Z C ( s ) = 1 s ( 1 ) = 1 s Convert the Figure 1 into s- domain. Refer to Figure 2, the impedance of resistor; inductor and capacitor are connected in series. Write an expression to calculate the equivalent impedance of series connected resistor, inductor and capacitor. Z ( s ) = Z R ( s ) + Z L ( s ) + Z C ( s ) (4) Substitute 1 for Z R ( s ) , s for Z L ( s ) , and 1 s for Z C ( s ) in equation (4) to find Z ( s ) . Z ( s ) = 1 + s + 1 s = s 2 + s + 1 s The reduced diagram Figure 2 is shown in Figure 3. Write an expression to calculate the current in s- domain. I ( s ) = V ( s ) Z ( s ) (5) Substitute 8 s for V ( s ) , and s 2 + s + 1 s for Z ( s ) in equation (5) to find I ( s ) . I ( s ) = ( 8 s ) ( s 2 + s + 1 s ) I ( s ) = 8 s 2 + s + 1 (6) From equation (6), the characteristic equation is written as follows, s 2 + s + 1 = 0 (7) Write a general expression to calculate the roots of characteristic equation in the form of a s 2 + b s + c = 0 . s 1 , 2 = − b ± b 2 − 4 a c 2 a (8) Here, a is the coefficient of second order term, b is the coefficient of first order term, and c is the coefficient of constant term. Comparing the equation (7) with quadratic equation a s 2 + b s + c = 0 , a = 1 b = 1 c = 1 Substitute 1 for a , 1 for b , and 1 for c in equation (8) to find s 1 , 2

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16, Problem 12P

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Find the expression of current i(t) in the circuit of Figure 16.35.

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Find the expression of current i ( t ) in the circuit of Figure 16.35.

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Find the expression of current i(t) in the circuit of Figure 16.35.

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