Explanation Given data: Refer to the given figures in the textbook for the voltage functions v 1 ( t ) , v 2 ( t ) , and v 2 ( t + T / 8 ) . Calculation: From the given figure, v 2 ( t + T / 8 ) is even. Hence, b k = 0 for all k . Calculate the value of Fourier co-efficient a v . a v = ( V m / 2 ) ( T / 4 ) T = V m 8 Write the expression for the Fourier co-efficient a k . a k = 4 T ∫ 0 T / 8 v ( t ) cos k ω 0 t d t Substitute V m 2 for v ( t ) . a k = 4 T ∫ 0 T / 8 V m 2 cos k ω 0 t d t = 2 V m T [ sin k ω 0 t k ω 0 ] 0 T / 8 = 2 V m T ( sin k ω 0 ( T / 8 ) k ω 0 ) = 2 V m ( 2 π ω 0 ) ( sin k ω 0 ( 2 π ω 0 ) 8 k ω 0 ) { ∵ T = 2 π ω 0 } Re-arrange the Equation. a k = V m k π sin k π 4 = V m n π sin n π 4 { ∵ k = n } Write the expression for the Fourier series of v 2 ( t + T / 8 ) . v 2 ( t + T / 8 ) = a v + ∑ n = 1 ∞ a k cos n ω 0 t Substitute V m 8 for a v and V m n π sin n π 4 for a k . v 2 ( t + T / 8 ) = V m 8 + ∑ n = 1 ∞ ( V m n π sin n π 4 ) cos n ω 0 t v 2 ( t + T / 8 ) = V m 8 + V m π ∑ n = 1 ∞ 1 n sin n π 4 cos n ω 0 t (1) From Equation (1), the expression for v 2 ( t ) is, v 2 ( t ) = V m 8 + V m π ∑ n = 1 ∞ 1 n sin n π 4 cos n ω 0 ( t − T / 8 ) Re-arrange the Equation. v 2 ( t ) = V m 8 + V m π ∑ n = 1 ∞ 1 n sin n π 4 cos n ω 0 ( t − 2 π 8 ω 0 ) { ∵ T = 2 π ω 0 } = V m 8 + V m π ∑ n = 1 ∞ 1 n sin n π 4 cos ( n ω 0 t − n π 4 ) = V m 8 + V m π ∑ n = 1 ∞ 1 n sin n π 4 [ cos n ω 0 t cos n π 4 + sin n ω 0 t sin n π 4 ] v 2 ( t ) = V m 8 + V m π ∑ n = 1 ∞ ( 1 n sin n π 4 cos n π 4 ) cos n ω 0 t + ( 1 n sin 2 n π 4 ) sin n ω 0 t (2) Consider the expression for the voltage function v ( t )

10th Edition

ISBN: 9780133760033

Author: James W. Nilsson, Susan Riedel

Publisher: PEARSON

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Fourier Transform

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Question

Chapter 16, Problem 18P

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Derive the expression for the Fourier series of v2(t) and verify the Fourier co-efficients.

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Chapter 16 Solutions

Electric Circuits (10th Edition)

Ch. 16.2 - Objective 1–Be able to calculate the trigonometric...Ch. 16.2 - Prob. 2AP Ch. 16.3 - Derive the Fourier series for the periodic voltage...Ch. 16.4 - Compute A1 – A5 and θ1 – θ5 for the periodic...Ch. 16.5 - The periodic triangular-wave voltage seen on the...Ch. 16.5 - The periodic square-wave shown on the top is...Ch. 16.6 - a. 16.7 The periodic voltage function in...Ch. 16.8 - Derive the expression for the Fourier coefficients...Ch. 16.8 - Calculate the rms value of the periodic current in...Ch. 16.9 - Prob. 10AP

Ch. 16 - Prob. 1P Ch. 16 - Find the Fourier series expressions for the...Ch. 16 - Prob. 3P Ch. 16 - Prob. 4P Ch. 16 - Prob. 5P Ch. 16 - Prob. 6P Ch. 16 - Prob. 7P Ch. 16 - Prob. 8P Ch. 16 - Prob. 9P Ch. 16 - Prob. 10P Ch. 16 - Prob. 11P Ch. 16 - Prob. 13P Ch. 16 - Prob. 14P Ch. 16 - Prob. 15P Ch. 16 - Prob. 16P Ch. 16 - Prob. 17P Ch. 16 - Prob. 18P Ch. 16 - Prob. 20P Ch. 16 - Prob. 21P Ch. 16 - Derive the Fourier series for the periodic...Ch. 16 - Prob. 23P Ch. 16 - Prob. 24P Ch. 16 - Prob. 25P Ch. 16 - Show that for large values of C Eq. 16.24 can be...Ch. 16 - Prob. 29P Ch. 16 - Prob. 30P Ch. 16 - Prob. 32P Ch. 16 - Prob. 33P Ch. 16 - Prob. 34P Ch. 16 - The triangular-wave voltage source, shown in Fig....Ch. 16 - Prob. 36P Ch. 16 - Find the rms value of the voltage shown in Fig....Ch. 16 - Prob. 38P Ch. 16 - Estimate the rms value of the periodic...Ch. 16 - Estimate the rms value of the full-wave rectified...Ch. 16 - Prob. 41P Ch. 16 - Prob. 42P Ch. 16 - Prob. 43P Ch. 16 - Prob. 44P Ch. 16 - Prob. 45P Ch. 16 - Prob. 46P Ch. 16 - Prob. 48P Ch. 16 - Make an amplitude and phase plot, based on Eq....Ch. 16 - Prob. 50P Ch. 16 - Prob. 51P Ch. 16 - Prob. 52P Ch. 16 - Prob. 53P Ch. 16 - Prob. 54P Ch. 16 - Prob. 55P Ch. 16 - Prob. 57P

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