Explanation Formula used: Consider the general expression for the Initial-Value theorem as follows: lim t → 0 + f ( t ) = lim s → ∞ s F ( s ) Consider the general expression for the Final-Value theorem as follows: lim t → ∞ f ( t ) = lim s → 0 s F ( s ) Calculation: Case 1: Given data: F ( s ) = 320 s 2 ( s + 8 ) (1) Calculation: Write the expression for F ( s ) using the Initial-value theorem as follows: lim t → 0 + f ( t ) = lim s → ∞ [ 320 s 2 ( s + 8 ) ] ( s ) = lim s → ∞ [ 320 s ( s + 8 ) ] = lim s → ∞ [ 320 ( s 2 + 8 s ) ] Divide the numerator and denominator of the Right hand side by highest power of s , in the denominator ( s 2 ) and find the value for the limit 1 s → 0 as follows: lim t → 0 + f ( t ) = lim 1 s → 0 [ 320 ( 1 s 2 ) 1 + 8 ( 1 s ) ] f ( 0 + ) = [ 320 ( 0 ) 1 + 8 ( 0 ) ] f ( 0 + ) = 0 Write the expression for F ( s ) using the Final-Value theorem as follows: lim t → ∞ f ( t ) = lim s → 0 [ 320 s 2 ( s + 8 ) ] ( s ) Write the poles of F ( s ) before applying the final-value theorem as follows: The second order poles of F ( s ) are at 0 and one pole is at − 8 respectively from Equation (1). Since, two poles are located at the origin of the complex plane; Final Value Theorem is not applicable to this function. The initial value of f ( t ) is 0 and the final value theorem is not applicable to this function. Case 2: Given data: F ( s ) = 80 ( s + 3 ) s ( s + 2 ) 2 (2) Calculation: Write the expression for F ( s ) using the Initial-value theorem as follows: lim t → 0 + f ( t ) = lim s → ∞ [ ( 80 s + 240 ) s ( s + 2 ) 2 ] ( s ) = lim s → ∞ [ ( 80 s + 240 ) ( s + 2 ) 2 ] = lim s → ∞ [ ( 80 s + 240 ) s 2 + 4 s + 4 ] Divide the numerator and denominator of the Right hand side by highest power of s , in the denominator ( s 2 ) and find the value for the limit 1 s → 0 as follows: lim t → 0 + f ( t ) = lim 1 s → 0 [ 8 s + 24 s 2 + 4 s + 4 ] f ( 0 + ) = [ 80 ( 1 s ) + 240 ( 1 s 2 ) 1 + 4 ( 1 s ) + 4 ( 1 s 2 ) ] = [ 80 ( 0 ) + 240 ( 0 ) 1 + 4 ( 0 ) + 4 ( 0 ) ] = 0 Write the expression for F ( s ) using the Final-Value theorem as follows: lim t → ∞ f ( t ) = lim s → 0 ( 80 ( s + 3 ) s ( s + 2 ) 2 ) ( s ) = lim s → 0 ( 80 ( s + 3 ) ( s + 2 ) 2 ) Write the poles of F ( s ) before applying the final-value theorem as follows: The poles of F ( s ) are at 0 and two poles are at − 2 respectively from Equation (2). Since, the poles are located on the left hand of the complex plane, use the Final-value theorem as follows: lim t → ∞ f ( t ) = ( 80 ( 0 + 3 ) ( 0 + 2 ) 2 ) = 240 4 f ( ∞ ) = 60 The initial value of f ( t ) is 0 and the final value of f ( t ) is 60. Case 3: Given data: F ( s ) = 60 ( s + 5 ) ( s + 1 ) 2 ( s 2 + 6 s + 25 )

10th Edition

ISBN: 9781292060545

Author: James W. Nilsson, Susan Riedel

Publisher: Pearson Education Limited

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Chapter 12, Problem 52P

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Implement the initial and the final-value theorems to the transform pair in Problem 12.42.

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Chapter 12, Problem 51P

Chapter 12, Problem 53P

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Chapter 12 Solutions

Electric Circuits, Global Edition

Ch. 12.4 - Prob. 1AP Ch. 12.5 - Prob. 2AP Ch. 12.7 - Prob. 3AP Ch. 12.7 - Prob. 4AP Ch. 12.7 - Prob. 5AP Ch. 12.7 - Prob. 6AP Ch. 12.7 - Prob. 7AP Ch. 12.7 - Prob. 8AP Ch. 12.7 - Prob. 9AP Ch. 12.8 - Use the initial- and final-value theorems to find...

Ch. 12 - Prob. 1P Ch. 12 - Prob. 2P Ch. 12 - Use step functions to write the expression for...Ch. 12 - Prob. 4P Ch. 12 - Explain why the following function generates an...Ch. 12 - Find the area under the function shown in Fig....Ch. 12 - Prob. 7P Ch. 12 - Prob. 8P Ch. 12 - Prob. 9P Ch. 12 - Prob. 10P Ch. 12 - Prob. 11P Ch. 12 - Prob. 12P Ch. 12 - Prob. 13P Ch. 12 - Prob. 14P Ch. 12 - Prob. 15P Ch. 12 - Prob. 16P Ch. 12 - Prob. 17P Ch. 12 - Prob. 18P Ch. 12 - Prob. 19P Ch. 12 - Prob. 20P Ch. 12 - Prob. 21P Ch. 12 - Prob. 22P Ch. 12 - Prob. 23P Ch. 12 - Prob. 24P Ch. 12 - Prob. 25P Ch. 12 - Prob. 26P Ch. 12 - Prob. 27P Ch. 12 - The switch in the circuit in Fig. P12.28LO has...Ch. 12 - Prob. 29P Ch. 12 - Prob. 30P Ch. 12 - Prob. 31P Ch. 12 - Prob. 33P Ch. 12 - Prob. 39P Ch. 12 - Prob. 40P Ch. 12 - Prob. 41P Ch. 12 - Prob. 42P Ch. 12 - Prob. 43P Ch. 12 - Prob. 44P Ch. 12 - Prob. 45P Ch. 12 - Use the initial-value theorem to find the initial...Ch. 12 - Prob. 47P Ch. 12 - Prob. 48P Ch. 12 - Prob. 49P Ch. 12 - Prob. 50P Ch. 12 - Prob. 51P Ch. 12 - Prob. 52P Ch. 12 - Prob. 53P Ch. 12 - Prob. 54P Ch. 12 - Prob. 55P Ch. 12 - Prob. 56P

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