Explanation Given data: Refer to Figure 16.11 in the textbook. Formula used: Write a general expression to calculate the impedance of a resistor in s -domain. Z R = R (1) Here, R is the value of resistor. Write a general expression to calculate the impedance of a capacitor in s -domain. Z C = 1 s C (2) Here, C is the value of capacitor. Calculation: The given circuit is redrawn as shown in Figure 1. For time t < 0 , the switch is at position ‘b’ and the capacitor acts as open circuit at steady state condition. Now the Figure 1 is redrawn as shown in Figure 2. Refer to Figure 2, the resistor and capacitor is connected in parallel. For the parallel connection, the voltage is same. Therefore, the voltage across the capacitor is equal to the source voltage V o . v C ( 0 − ) = V o The capacitor is always continuous so that, v ( 0 ) = v C ( 0 − ) = v C ( 0 + ) = V o For time t > 0 , the switch is at positon A and the Figure 1 becomes as shown in Figure 3. Apply Laplace transform for i ( t ) to find I ( s ) . I ( s ) = I o s { ∵ L ( u ( t ) ) = 1 s } Using element transformation methods with initial conditions convert the Figure 3 into s -domain using equation (1) and (2). Apply nodal analysis at the node V ( s ) for Figure 4. − I o s + V ( s ) R + V ( s ) ( 1 s C ) − C v ( 0 ) = 0 − I o s + V ( s ) R + s C V ( s ) − C V o = 0 { ∵ v ( 0 ) = V o } I o s + C V o = V ( s ) R + s C V ( s ) I o s + C V o = V ( s ) [ 1 R + s C ] Rearrange the above equation to find V ( s ) . V ( s ) = I o s + C V o [ 1 R + s C ] = I o s ( 1 R + s C ) + C V o ( 1 R + s C ) = I o s C ( s + 1 R C ) + C V o C ( s + 1 R C ) V ( s ) = ( I o C ) s ( s + 1 R C ) + V o ( s + 1 R C ) (3) Assume, V 1 ( s ) = ( I o C ) s ( s + 1 R C ) (4) Substitute the equation (4) in equation (3) to find V ( s ) . V ( s ) = V 1 ( s ) + V o ( s + 1 R C ) (5) Take partial fraction for equation (4). V 1 ( s ) = ( I o C ) s ( s + 1 R C ) = A s + B s + 1 R C (6) The equation (6) also written as follows: V 1 ( s ) = ( I o C ) s ( s + 1 R C ) = A ( s + 1 R C ) + B s s ( s + 1 R C ) Simplify the above equation as follows: I o C = A ( s + 1 R C ) + B s (7) Substitute 0 for s in equation (7) to find A

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16.2, Problem 3PP

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Find the expression of voltage v(t) for t>0.

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