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 ) = 135 s ( s + 3 ) 3 (1) Calculation: Write the expression for F ( s ) using the Initial-value theorem as follows: lim t → 0 + f ( t ) = lim s → ∞ [ 135 s ( s + 3 ) 3 ] ( s ) = lim s → ∞ [ 135 ( s + 3 ) 3 ] = lim s → ∞ [ 135 s 3 + 9 s 2 + 27 s + 27 ] Divide the numerator and denominator of the Right hand side by highest power of s , in the denominator ( s 3 ) and find the value for the limit 1 s → 0 as follows: lim t → 0 + f ( t ) = lim 1 s → 0 [ 135 ( 1 s 3 ) 1 + 9 ( 1 s ) + 27 ( 1 s 2 ) + 27 ( 1 s 3 ) ] f ( 0 + ) = [ 135 ( 0 ) 1 + 9 ( 0 ) + 27 ( 0 ) + 27 ( 0 ) ] f ( 0 + ) = 0 Write the expression for F ( s ) using the Final-Value theorem as follows: lim t → ∞ f ( t ) = lim s → 0 [ 135 s ( s + 3 ) 3 ] ( s ) = lim s → 0 [ 135 ( s + 3 ) 3 ] Write the poles of F ( s ) before applying the final-value theorem as follows: The poles of F ( s ) are at 0 and three poles are at − 3 respectively from Equation (1). Since, the poles are located on the left hand of the complex plane, use the Final-value theorem as follows: lim t → ∞ f ( t ) = [ 135 ( 0 + 3 ) 3 ] = 135 27 f ( ∞ ) = 5 The initial value of f ( t ) is 0 and the final value of f ( t ) is 5. Case 2: Given data: F ( s ) = 10 ( s + 2 ) 2 ( s 2 + 2 s + 2 ) 2 . Calculation: Find the roots of the quadratic term in the denominator as follows: s 2 + 2 s + 2 = − 2 ± 2 2 − 4 ( 1 ) ( 2 ) 2 ( 1 ) = − 2 ± − 4 2 = − 2 ± j 2 2 = − 1 ± j 1 Rewrite the given function F ( s ) as follows: F ( s ) = 10 ( s + 2 ) 2 ( s + 1 − j 1 ) 2 ( s + 1 + j 1 ) 2 (2) Simplify F ( s ) as follows: F ( s ) = 10 s 2

ELECTRIC CIRCUITS-W/MASTERINGENGINEERING

11th Edition

ISBN: 9780134894300

Author: NILSSON

Publisher: PEARSON

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Chapter 12, Problem 55P

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Implement the initial and the final-value theorems to the transform pair in Problem 12.43.

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Chapter 12, Problem 54P

Chapter 12, Problem 56P

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ELECTRIC CIRCUITS-W/MASTERINGENGINEERING

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. 12.7 - Prob. 10AP

Ch. 12.7 - Prob. 11AP Ch. 12.7 - Prob. 12AP Ch. 12.8 - Be able to find and plot the poles and zeros for a...Ch. 12.9 - Use the initial- and final-value theorems to find...Ch. 12 - Prob. 1P Ch. 12 - Use step functions to write the expression for...Ch. 12 - Prob. 3P 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 - Evaluate the following integrals: . . Ch. 12 - Prob. 9P Ch. 12 - Prob. 10P Ch. 12 - Prob. 11P 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. 40P Ch. 12 - Prob. 41P Ch. 12 - Prob. 42P Ch. 12 - Prob. 43P Ch. 12 - Prob. 44P Ch. 12 - Prob. 45P Ch. 12 - Find the poles and zeros for the s-domain...Ch. 12 - Prob. 47P Ch. 12 - Use the initial-value theorem to find the initial...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 Ch. 12 - Prob. 57P Ch. 12 - Prob. 58P

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