To determine Derive the Fourier series of the given periodic voltage. Explanation Given data: Refer to the given figures for the given periodic voltage. Calculation: In studying the given periodic voltage function; consider that it can be visualized as the combination of two half-wave rectified sine waves. Hence, use the Fourier series for a half-wave rectified sine wave with a magnitude of V m . The periodic voltage v 1 ( t ) is drawn and it is shown in Figure 1. The periodic voltage v 2 ( t ) is drawn and it is shown in Figure 1. Consider that the expression for the Fourier coefficient a v . a v = 1 T ∫ 0 T / 2 v ( t ) d t Substitute V m sin ( 2 π T ) t for v ( t ) . a v = 1 T ∫ 0 T / 2 V m sin ( 2 π T ) t d t = V m T [ − cos ( 2 π T ) t ] 0 T / 2 = V m π Write the expression for the Fourier coefficient a k . a k = 2 T ∫ 0 T / 2 v ( t ) cos k ω 0 t d t Substitute V m sin ( 2 π T ) t for v ( t ) . a k = 2 T ∫ 0 T / 2 V m sin ( 2 π T ) t cos k ω 0 t d t = 2 V m T [ sin ( 2 π T ) t ( sin k ω 0 t k ω 0 ) − ( 2 π T ) cos ( 2 π T ) t ( − cos k ω 0 t k 2 ω 0 2 ) ] 0 T / 2 = V m π ( 1 + cos k π 1 − k 2 ) Consider that a k = 0 for k -odd and a k = 2 V m π ( 1 − k 2 ) for k -even. Write the expression for the Fourier coefficient b k . b k = 2 T ∫ 0 T / 2 v ( t ) sin k ω 0 t d t Substitute V m sin ( 2 π T ) t for v ( t ) . b k = 2 T ∫ 0 T / 2 V m sin ( 2 π T ) t sin k ω 0 t d t = 2 V m T [ sin ( 2 π T ) t ( − cos k ω 0 t k ω 0 ) − ( 2 π T ) cos ( 2 π T ) t ( − sin k ω 0 t k 2 ω 0 2 ) ] 0 T / 2 = 0 for k = 2 , 3 , 4 , ⋅ ⋅ ⋅ For k = 1 , consider that b 1 = V m 2 . Hence, the Fourier series can be written as, v ( t ) = a v + b 1 sin ω 0 t + ∑ n = 2 , 4 , 6 , ⋅ ⋅ ⋅ ∞ a n cos n ω 0 t + ∑ n = 2 , 4 , 6 , ⋅ ⋅ ⋅ ∞ b n sin n ω 0 t V Substitute V m π for a v , V m 2 for b 1 , 0 for b n , 2 V m π ( 1 − n 2 ) for a n . v ( t ) = V m π + V m 2 sin ω 0 t + ∑ n = 2 , 4 , 6 , ⋅ ⋅ ⋅ ∞ ( 2 V m π ( 1 − n 2 ) ) cos n ω 0 t + ∑ n = 2 , 4 , 6 , ⋅ ⋅ ⋅ ∞ ( 0 ) sin n ω 0 t V v ( t ) = V m π + V m 2 sin ω 0 t + 2 V m π ∑ n = 2 , 4 , 6 , ⋅ ⋅ ⋅ ∞ 1 ( 1 − n 2 ) cos n ω 0 t V (1) Substitute 100 V for V m and v 1 ( t ) for v ( t ) in equation (1) to find the Fourier series for the voltage v 1 ( t )

Electric Circuits, Student Value Edition Format: Unbound (saleable)

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ISBN: 9780134747170

Author: NILSSON, James W.^riedel, Susan

Publisher: Prentice Hall

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Electric Circuits, Student Value Edition Format: Unbound (saleable)

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 - Derive the Fourier series for the periodic voltage...Ch. 16 - Find the Fourier series expressions for the...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 - Derive the Fourier series for the periodic...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. 28P Ch. 16 - Prob. 30P Ch. 16 - Prob. 32P Ch. 16 - The periodic current shown in Fig. P16.33 is...Ch. 16 - The periodic voltage across a 10 Ω resistor is...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 - Use the first four nonzero terms in the Fourier...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 - A periodic function is represented by a Fourier...Ch. 16 - Prob. 53P Ch. 16 - Prob. 54P Ch. 16 - Prob. 55P Ch. 16 - Prob. 57P

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