Explanation Given data: Refer to Figure 17.49 in the textbook. Formula used: Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (1) Here, T is the period of the function. Write the general expression to calculate trigonometric Fourier series of z ( t ) . z ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos n ω 0 t + b n sin n ω 0 t ) (2) Here, a 0 is the dc component of z ( t ) , a n and b n are the Fourier coefficients, n is an integer, and ω 0 is the angular frequency. Write the expression to calculate the dc component of the function z ( t ) . a 0 = 1 T ∫ 0 T z ( t ) d t (3) Write the expression to calculate Fourier coefficients. a n = 2 T ∫ 0 T z ( t ) cos n ω 0 t d t (4) b n = 2 T ∫ 0 T z ( t ) sin n ω 0 t d t (5) Calculation: The given waveform is drawn as Figure 1. Refer to the Figure 1. The Fourier series function of the waveform is defined as, z ( t ) = { 7 , 0 < t < π − 14 , π < t < 2 π The time period of the function in Figure 1 is, T = 2 π Substitute 2 π for T in equation (1) to find ω 0 . ω 0 = 2 π 2 π = 1 Substitute 2 π for T in equation (3) to find a 0 . a 0 = 1 2 π ∫ 0 2 π z ( t ) d t = 1 2 π ( ∫ 0 π z ( t ) d t + 1 2 π ∫ π 2 π z ( t ) d t ) = 1 2 π ( ∫ 0 π 7 d t + 1 2 π ∫ π 2 π ( − 14 ) d t ) = 1 2 π ( ∫ 0 π 7 d t − 1 2 π ∫ π 2 π 14 d t ) Simplify the above equation to find a 0 . a 0 = 1 2 π ( 7 [ t ] 0 π − 14 [ t ] π 2 π ) = 1 2 π ( 7 [ π − 0 ] − 14 [ 2 π − π ] ) = 1 2 π ( 7 π − 28 π + 14 π ) = − 3.5 Substitute 2 π for T and 1 for ω 0 in equation (4) to find a n . a n = 2 2 π ∫ 0 2 π z ( t ) cos n ( 1 ) t d t = 1 π ∫ 0 2 π z ( t ) cos n t d t = 1 π ( ∫ 0 π z ( t ) cos n t d t + ∫ π 2 π z ( t ) cos n t d t ) = 1 π ( ∫ 0 π 7 cos n t d t + ∫ π 2 π ( − 14 ) cos n t d t ) Simplify the above equation to find a n . a n = 1 π ( ∫ 0 π 7 cos n t d t − ∫ π 2 π 14 cos n t d t ) = 1 π ( 7 [ sin n t n ] 0 π − 14 [ sin n t n ] π 2 π ) = 1 π ( 7 [ sin n π n − sin n ( 0 ) n ] − 14 [ sin n ( 2 π ) n − sin n π n ] ) = 1 π ( 7 [ sin n π n − sin 0 ° n ] − 14 [ sin 2 n π n − sin n π n ] ) Simplify the above equation to find a n

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Question

Chapter 17, Problem 5P

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Find the Fourier series expansion for the given waveform z(t).

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Chapter 17, Problem 4P

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