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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 21 Copyright © Orchard Publications Trigonometric Form of Fourier Series for Common Waveforms Figure 7.20. Full-wave rectifier circuit The input and output waveforms are shown in Figures 7.21 and 7.22 respectively. We will express as a trigonometric Fourier series, and we will assume that . Figure 7.21. Input sinusoid for the full rectifier circuit of Figure 7.20 Figure 7.22. Output waveform for full rectifier circuit of Figure 7.20 We choose the ordinate as shown in Figure 7.23. Figure 7.23. Full wave rectified waveform with even symmetry R + + v in t () v out t v t ω 1 = A A 0 π 2 π 3 π 4 π 0 2 4 6 8 10 12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 A π 2 π 3 π 4 π 0 2 4 6 8 1 2 π 2 π π π 0 A

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Chapter 7 Fourier Series 7 22 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications By inspection, the average is a non zero value. We choose the period of the input sinusoid so that the output will be expressed in terms of the fundamental frequency . We also choose the lim- its of integration as and , we observe that the waveform has even symmetry. Therefore, we expect only cosine terms to be present. The coefficients are found from where for this waveform, (7.49) and from tables of integrals, Since we express (7.49) as (7.50) To simplify the last expression in (7.50), we make use of the trigonometric identities and Then, (7.50) simplifies to (7.51) π + π a n a n 1 π -- ft () nt t d cos 0 2 π = a n 1 π At n t t d cos sin π π 2A π ------- tn t t d cos sin 0 π = = mx sin nx cos x d mn x cos 2n m ------------------------------ + x cos 2m n + ------------------------------ m 2 n 2 = xy cos y x cos x y cos cos xsiny sin + == a n π 1 2 n1 t cos --------------------------- + t cos + 0 π    = A π --- π cos ---------------------------- + π cos + ----------------------------- 1 ----------- 1 + = A π 1n ππ + cos + -------------------------------------- n 1 cos + = n + cos n cos cos n π sin π sin n π cos n cos n cos cos n π π sin + n π cos a n A π π cos + + ------------------------ π cos + A π 2 n π cos + + n π cos n 2 1 ------------------------------------------------------------------------------------- n π 1 + cos π n 2 1 ---------------------------------------- n 1 =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 23 Copyright © Orchard Publications Trigonometric Form of Fourier Series for Common Waveforms Now, we can evaluate all the coefficients, except , from (7.51). First, we will evaluate to obtain the value. By substitution of , we obtain Therefore, the value is (7.52) From (7.51) we observe that for all , other than , .

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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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