Explanation Given data: Refer to Figure 16.63 in the textbook. The current source is, i 1 ( t ) = 5 u ( t ) A (1) The initial voltage v ( 0 ) = 0 V and initial current i ( 0 ) = 1.25 A . Formula used: Write a general expression to calculate the impedance of a resistor in s -domain. Z R = R (2) Here, R is the value of resistance of the resistor. Write a general expression to calculate the impedance of an inductor in s -domain. Z L = s L (3) Here, L is the value of inductance of the inductor. Write a general expression to calculate the impedance of a capacitor in s -domain. Z C = 1 s C (4) Here, C is the value of capacitance of the capacitor. Calculation: Take Laplace transform of equation (1). I 1 ( s ) = 5 s { ∵ ℒ [ u ( t ) ] = 1 s } Using equation (2), (3), and (4), the given circuit is redrawn as shown in Figure 1 in s -domain with initial conditions of capacitor voltage v ( 0 ) = 0 V and inductor current i ( 0 ) = 1.25 A . Since, the initial inductor current through the inductor is 1.25 A. That is, i ( 0 ) = 1.25 A in Figure 1 which is shown in s -domain as 1.25 s . Apply Kirchhoff’s current law in Figure 1. V 2 + V ( 2 s ) + V s = 5 s − 1.25 s ( 1 2 + s 2 + 1 s ) V = 3.75 s ( s + s 2 + 2 2 s ) V = 3.75 s ( s 2 + s + 2 ) V = 7.5 Rearrange the equation as follows, V = 7.5 s 2 + s + 2 (5) From equation (5), the roots of s 2 + s + 2 is calculated as follows. s = − 1 ± 1 2 − 4 ( 1 ) ( 2 ) 2 ( 1 ) { ∵ x = − b ± b 2 − 4 a c 2 a } = − 1 ± − 7 2 = − 0.5 ± j 1.3229 Therefore, the roots of s 2 + s + 2 is ( s + 0.5 + j 1.3229 ) ( s + 0.5 − j 1.3229 ) . Substitute ( s + 0.5 + j 1.3229 ) ( s + 0.5 − j 1.3229 ) for s 2 + s + 2 in equation (5). V = 7.5 ( s + 0.5 + j 1.3229 ) ( s + 0.5 − j 1.3229 ) (6) Expand V using partial fraction method. V = 7.5 ( s + 0.5 + j 1.3229 ) ( s + 0.5 − j 1.3229 ) = A s + 0.5 + j 1.3229 + B s + 0.5 − j 1.3229 (7) Here, A and B are the constants. Find the constants by using residue method. Constant A : A = ( s + 0.5 + j 1.3229 ) V | s + 0.5 + j 1.3229 = 0 (8) Substitute equation (6) in equation (8) to find the constant A . A = ( s + 0.5 + j 1.3229 ) × 7.5 ( s + 0.5 + j 1.3229 ) ( s + 0.5 − j 1.3229 ) | s + 0.5 + j 1.3229 = 0 = 7.5 s + 0.5 − j 1.3229 | s = − 0.5 − j 1.3229 = 7.5 − 0.5 − j 1.3229 + 0.5 − j 1.3229 = 2.835 j Convert into polar form, A = 2.835 ∠ 90 ° Constant B : B = ( s + 0.5 − j 1.3229 ) V | s + 0.5 − j 1.3229 = 0 (9) Substitute equation (6) in equation (9) to find the constant B . B = ( s + 0.5 − j 1.3229 ) × 7.5 ( s + 0.5 + j 1.3229 ) ( s + 0.5 − j 1.3229 ) | s + 0.5 − j 1.3229 = 0 = 7.5 s + 0.5 + j 1.3229 | s = − 0.5 + j 1.3229 = 7.5 − 0.5 + j 1.3229 + 0.5 + j 1.3229 = − 2.835 j Convert into polar form, B = 2.835 ∠ − 90 ° Substitute 2.835 ∠ 90 ° for A and 2.835 ∠ − 90 ° for B in equation (7) to find V . V = 2.835 ∠ 90 ° s + 0.5 + j 1.3229 + 2.835 ∠ − 90 ° s + 0.5 − j 1.3229 (10) Take inverse Laplace transform for the equation (10). v ( t ) = [ ( 2.835 ∠ 90 ° ) e − ( 0.5 + j 1.3229 ) t u ( t ) + ( 2.835 ∠ − 90 ° ) e − ( 0.5 − j 1.3229 ) t u ( t ) ] { ∵ ℒ [ e − a t u ( t ) ] = 1 s + a } = [ ( 2.835 ∠ 90 ° ) e − 0.5 t e − j 1.3229 t + ( 2.835 ∠ − 90 ° ) e − 0.5 t e j 1.3229 t ] u ( t ) = [ ( 2.835 e j 90 ° ) e − 0.5 t e − j 1.3229 t + ( 2.835 e − j 90 ° ) e − 0.5 t e j 1.3229 t ] u ( t ) { ∵ ∠ θ = e j θ } = 2.835 e − 0.5 t [ e j 90 ° e − j 1.3229 t + e − j 90 ° e j 1.3229 t ] u ( t ) Reduce the equation as follows, v ( t ) = 2.835 e − 0.5 t [ e − j ( 1.3229 t − 90 ° ) + e j ( 1.3229 t − 90 ° ) ] u ( t ) = [ 5

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ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16, Problem 40P

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Find the voltage v(t) and the current i(t) for t>0 in the circuit of Figure 16.63.

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Chapter 16, Problem 39P

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Find the voltage v ( t ) and the current i ( t ) for t > 0 in the circuit of Figure 16.63.

Solution Summary: The author explains the formula used to calculate the impedance of a resistor in s -domain.

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Find the voltage v(t) and the current i(t) for t>0 in the circuit of Figure 16.63.

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Chapter 16 Solutions

Find the voltage v ( t ) and the current i ( t ) for t &gt; 0 in the circuit of Figure 16.63.

Solution Summary: The author explains the formula used to calculate the impedance of a resistor in s -domain.

Videos

Find the voltage v(t) and the current i(t) for t>0 in the circuit of Figure 16.63.

Want to see the full answer?

Chapter 16 Solutions

Find the voltage v ( t ) and the current i ( t ) for t > 0 in the circuit of Figure 16.63.