(a) Explanation Given data: The given function is v ( t ) = { 0 , − 2 < t < 0 5 , 0 < t < 6 0 , 2 < t < 4 − 5 , 4 < t < 6 Calculation: Draw the required waveform as per the given function. The required waveform is shown in Figure 1. The timer period T can be defined from the above waveform as, T = 8 The value of ω 0 is given by the value, ω 0 = 2 π T (1) The Fourier series of v ( t ) is, v ( t ) = a 0 + ∑ n = 1 ∞ [ a n cos n ω 0 t + b n sin n ω 0 t ] (2) The value of a 0 is given by, a 0 = 1 T ∫ T g ( t ) d t (3) The value of a n is given as, a n = 2 T ∫ T g ( t ) cos ( n ω 0 t ) d t (4) The value of b n , b n = 2 T ∫ T g ( t ) sin ( n ω 0 t ) d t (5) Substitute 8 for T in the equation (1) ω 0 = 2 π 8 = π 4 rad / s Substitute π 4 for ω 0 in equation (2). v ( t ) = a 0 + ∑ n = 1 ∞ [ a n cos n π 4 t + b n sin n π 4 t ] (6) Substitute 8 for T in equation (3). a 0 = 1 8 [ ∫ − 2 0 [ 0 ] d t + ∫ 0 2 [ 5 ] d t + ∫ 2 4 [ 0 ] d t + ∫ 4 6 [ − 5 ] d t ] = 1 8 [ 0 + 5 ∫ 0 2 d t + 0 − ∫ 4 6 d t ] = 5 8 [ ∫ 0 2 d t − ∫ 4 6 d t ] = 5 8 [ [ t ] 0 2 − [ t ] 4 6 ] Solve further as, a 0 = 5 8 [ [ 2 − 0 ] − [ 6 − 4 ] ] a 0 = 0 Substitute π 4 for ω 0 and 8 for T in equation (4) a n = 2 8 [ ∫ − 2 0 ( 0 ) cos ( n π t 4 ) d t + ∫ 0 2 ( 5 ) cos ( n π t 4 ) d t + ∫ 2 4 ( 0 ) cos ( n π t 4 ) d t + ∫ 4 6 ( − 5 ) cos ( n π t 4 ) d t ] = 1 4 [ 5 ∫ 0 2 cos ( n π t 4 ) d t − 5 ∫ 4 6 cos ( n π t 4 ) d t ] = 5 4 [ ∫ 0 2 cos ( n π t 4 ) d t − ∫ 4 6 cos ( n π t 4 ) d t ] = 5 4 [ sin ( n π t 4 ) n π 4 ] 0 2 − [ sin ( n π t 4 ) n π 4 ] 4 6 Solve further as, a n = 5 4 ( [ sin ( 2 n π 4 ) − sin ( 0 ) n π 4 ] − [ sin ( 6 n π 4 ) − sin ( 4 n π 4 ) n π 4 ] ) = 5 4 × 4 n π ( [ sin ( n π 2 ) − 0 ] − [ sin ( 3 n π 2 ) − sin ( n π ) ] ) = 5 n π [ sin ( n π 2 ) − sin ( 3 n π 2 ) ] For the even values of n , sin ( n π 2 ) and sin ( 3 n π 2 ) are zero (b) To determine The Fourier series for the given voltage waveform.

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Author: Hayt, William H. (william Hart), Jr, Kemmerly, Jack E. (jack Ellsworth), Durbin, Steven M.

Publisher: Mcgraw-hill Education,

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Fourier Transform

In mathematics, the Fourier transformation is a mathematical transformation that rotates responsibilities by using region or time into tasks depending on the local or temporal frequency, such as the rendering of a musical track in step with existing volu…

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Chapter 17.3, Problem 5P

(a)

To determine

The Fourier series for the given voltage waveform.

(b)

To determine

The Fourier series for the given voltage waveform.

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Chapter 17.2, Problem 4P

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