Explanation Given data: Refer to Figure 17.74(a) and 17.74(b) in the textbook. The inductor L is 1 H . Formula used: Write the general expression for Fourier series expansion. f ( t ) = a 0 + ∑ n = 1 ∞ a n cos n ω 0 t + ∑ n = 1 ∞ b n sin n ω 0 t (1) Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (2) Here, T is the period of the function. Calculation: Refer to Figure 17.73(a) in the textbook. The source voltage v s ( t ) over the period is, f ( t ) = { 7 , 0 < t < 1 0 , 1 < t < 2 (3) The time period of Figure 17.17(a) is, T = 2 Substitute 2 for T in equation (2) to find the angular frequency ω 0 . ω 0 = 2 π 2 = π Finding the Fourier coefficient a 0 . a 0 = 1 T ∫ 0 T v s ( t ) d t (4) Applying equation (3) in equation (4) as follows, a 0 = 1 2 [ ∫ 0 1 7 d t + ∫ 1 2 0 d t ] { ∵ T = 2 } = 1 2 [ 7 [ t ] 0 1 + 0 ] = 1 2 [ 7 [ 1 − 0 ] ] = 7 2 Finding the Fourier coefficient a n . a n = 2 T ∫ 0 T v s ( t ) cos n ω 0 t d t (5) Applying equation (3) in equation (5) as follows, a n = 2 2 [ ∫ 0 1 ( 7 ) cos n π t d t + ∫ 1 2 ( 0 ) cos n π t d t ] { ∵ T = 2 , ω 0 = π } = 7 [ sin n π t n π ] 0 1 + 0 = 7 n π [ sin n π ( 1 ) − 0 ] = 7 n π [ sin n π ] The above equation becomes, a n = 0 { ∵ sin n π = 0 } Finding the Fourier coefficient b n . b n = 2 T ∫ 0 T v s ( t ) sin n ω 0 t d t (6) Applying equation (3) in equation (6) as follows, b n = 2 2 [ ∫ 0 1 ( 7 ) sin n π t d t + ∫ 1 2 ( 0 ) sin n π t d t ] { ∵ T = 2 , ω 0 = π } = 7 [ − cos n π t n π ] 0 1 + [ 0 ] = − 7 n π [ cos n π − cos n π ( 0 ) ] = − 7 n π [ cos n π − cos ( 0 ) ] Simplify the equation as follows, b n = − 7 n π [ cos n π − 1 ] { ∵ cos ( 0 ) = 1 } = 7 n π [ 1 − cos n π ] = 7 n π [ 1 − cos n ω ] { ∵ ω = π } = 7 n π [ 1 − ( − 1 ) n ] { ∵ cos n ω = ( − 1 ) n } The above equation becomes, b n = { 14 n π , n = odd 0, n = even (7) Substituting the Fourier coefficients in equation (1) as follows, v s ( t ) = 7 2 + ∑ n = 1 ∞ ( 0 ) cos n ω 0 t + 14 π ∑ k = 1 ∞ 1 n sin n π t , n = 2 k + 1 { ω 0 = π } v s ( t ) = 3

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 17, Problem 38P

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Find the voltage across the inductor vo(t) of the circuit shown in Figure 17.74(b).

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Chapter 17, Problem 37P

Chapter 17, Problem 39P

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Chapter 17 Solutions

Fundamentals of Electric Circuits

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