Explanation Given data: The given function is, x ( t ) = cos t + ∫ 0 t e λ − t x ( λ ) d λ (1) Formula used: Write the general expression to find the convolution integral. y ( t ) = f 1 ( t ) ∗ f 2 ( t ) = ∫ 0 t f 1 ( λ ) f 2 ( t − λ ) d λ (2) Write the general expression to find the Laplace transform function. L [ cos ω t ] = s s 2 + ω 2 (3) L [ sin ω t ] = ω s 2 + ω 2 (4) L [ e − a t ] = 1 s + a (5) Here, s is the complex variable. Write the general expression for the Laplace transform. F ( s ) = L [ f ( t ) ] (6) Calculation: Assume the following to reduce the equation (1). y ( t ) = ∫ 0 t e λ − t x ( λ ) d λ y ( t ) = ∫ 0 t e − ( t − λ ) x ( λ ) d λ (7) Substitute the equation (7) in equation (1) to find x ( t ) . x ( t ) = cos t + y ( t ) (8) On comparing the equation (2) and equation (7). y ( t ) = x ( t ) ∗ e − t Substitute x ( t ) ∗ e − t for y ( t ) in equation (8) to find x ( t ) . x ( t ) = cos t + x ( t ) ∗ e − t (9) Apply the Laplace transform function of equation (3), (4), (5) (6) in equation (9). X ( s ) = s s 2 + 1 2 + X ( s ) ( 1 s + 1 ) X ( s ) − X ( s ) ( 1 s + 1 ) = s s 2 + 1 X ( s ) ( 1 − 1 s + 1 ) = s s 2 + 1 Simplify the equation as follows, X ( s ) ( s + 1 − 1 s + 1 ) = s s 2 + 1 X ( s ) ( s s + 1 ) = s s 2 + 1 X ( s ) = ( s s 2 + 1 ) ( s + 1 s ) X ( s ) = s + 1 s 2 + 1 (10) Case a: Reduce the equation as follows, X ( s ) = s s 2 + 1 + 1 s 2 + 1 Taking inverse Laplace transform for the above equation as follows. L − 1 [ X ( s ) ] = L − 1 [ s s 2 + 1 ] + L − 1 [ 1 s 2 + 1 ] x ( t ) = L − 1 [ s s 2 + 1 2 ] + L − 1 [ 1 s 2 + 1 2 ] x ( t ) = cos ( t ) + sin ( t ) Case b: Expand X ( s ) in equation (10) using partial fraction method. X ( s ) = s + 1 s 2 + 1 = s + 1 ( s + j ) ( s − j ) X ( s ) = A s + j + B s − j (11) Here, A and B are the constants. Find the constants by using residue method. Constant A : A = ( s + j ) X ( s ) | s + j = 0 (12) Substitute equation (10) in equation (12) to find the constant A . A = ( s + j ) × s + 1 s 2 + 1 | s + j = 0 = ( s + j ) × s + 1 ( s + j ) ( s − j ) | s = − j = s + 1 s − j | s = − j = − j + 1 − j − j Reduce the equation as follows, A = 1 − j − 2 j = 0

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ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Transfer function

A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.

Convolution Integral

Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is …

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Chapter 15, Problem 53P

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Find the value of x(t) using Laplace transform.

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Chapter 15, Problem 52P

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Chapter 15 Solutions

Fundamentals of Electric Circuits

Ch. 15.2 - Prob. 1PP Ch. 15.2 - Prob. 2PP Ch. 15.3 - Prob. 3PP Ch. 15.3 - Prob. 4PP Ch. 15.3 - Prob. 5PP Ch. 15.3 - Prob. 6PP Ch. 15.3 - Obtain the initial and the final values of...Ch. 15.4 - Prob. 8PP Ch. 15.4 - Find f(t) if F(s)=48(s+2)(s+1)(s+3)(s+4)Ch. 15.4 - Obtain g(t) if G(s)=s3+2s+6s(s+1)2(s+3)

Ch. 15.4 - Find g(t) given that G(s)=20(s+1)(s2+4s+13)Ch. 15.5 - Graphically convolve the two functions in Fig....Ch. 15.5 - Given g(t) and f(t) in Fig. 15.20, graphically...Ch. 15.5 - Use convolution to find vo(t) in the circuit of...Ch. 15.6 - Prob. 15PP Ch. 15.6 - Prob. 16PP Ch. 15 - Prob. 1RQ Ch. 15 - Prob. 2RQ Ch. 15 - Prob. 3RQ Ch. 15 - Prob. 4RQ Ch. 15 - Prob. 5RQ Ch. 15 - If F(s) = 1/(s + 2), then f(t) is (a) e2t u(t) (b)...Ch. 15 - Prob. 7RQ Ch. 15 - Prob. 8RQ Ch. 15 - Prob. 9RQ Ch. 15 - Prob. 10RQ Ch. 15 - Prob. 1P Ch. 15 - Prob. 2P Ch. 15 - Prob. 3P Ch. 15 - Prob. 4P Ch. 15 - Prob. 5P Ch. 15 - Prob. 6P Ch. 15 - Prob. 7P Ch. 15 - Prob. 8P Ch. 15 - Prob. 9P Ch. 15 - Prob. 10P Ch. 15 - Find F(s) if: (a) ft=6etcosh2t (b) ft=3te2tsinh4t...Ch. 15 - If g(t) = 4e 2t cos 4t, find G(s).Ch. 15 - Prob. 13P Ch. 15 - Prob. 14P Ch. 15 - Prob. 15P Ch. 15 - Prob. 16P Ch. 15 - Prob. 17P Ch. 15 - Prob. 18P Ch. 15 - Prob. 19P Ch. 15 - Prob. 20P Ch. 15 - Prob. 21P Ch. 15 - Prob. 22P Ch. 15 - Prob. 23P Ch. 15 - Design a problem to help other students better...Ch. 15 - Let F(s)=18(s+1)(s+2)(s+3) (a) Use the initial and...Ch. 15 - Determine the initial and final values of f(t), if...Ch. 15 - Prob. 27P Ch. 15 - Prob. 28P Ch. 15 - Prob. 29P Ch. 15 - Prob. 30P Ch. 15 - Find f(t) for each F(s): (a) 10ss+1s+2s+3 (b)...Ch. 15 - Prob. 32P Ch. 15 - Prob. 33P Ch. 15 - Prob. 34P Ch. 15 - Obtain f(t) for the following transforms: (a)...Ch. 15 - Prob. 36P Ch. 15 - Prob. 37P Ch. 15 - Prob. 38P Ch. 15 - Determine f(t) if: (a)...Ch. 15 - Show that...Ch. 15 - Prob. 41P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 43P Ch. 15 - Prob. 44P Ch. 15 - Given h(t) = 4e2tu(t) and x(t) = (t) 2e 2tu(t),...Ch. 15 - Given the following functions...Ch. 15 - A system has the transfer function...Ch. 15 - Find f(t) using convolution given that: (a)...Ch. 15 - Prob. 49P Ch. 15 - Prob. 50P Ch. 15 - Given that v(0) = 5 and dv(0)/dt = 10, solve...Ch. 15 - Prob. 52P Ch. 15 - Prob. 53P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 55P Ch. 15 - Solve for v(t) in the integrodifferential equation...Ch. 15 - Prob. 57P Ch. 15 - Given that dvdt+2v+50tv()d=4u(t) with v(0) = 1,...Ch. 15 - Solve the integrodifferential equation...Ch. 15 - Prob. 60P

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