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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 45 Copyright © Orchard Publications Computation of Average Power from Fourier Series Then, (7.129) Next, (7.130) From (7.129) and (7.130), (7.131) b. The average power delivered by the voltage source is (7.132) or (7.133) Check: The average power absorbed by the capacitor is zero, and therefore, the average power absorbed by the resistor, must be equal to the average power delivered by the source. The average power absorbed by the resistor is v in1 t () 6 ω t cos = V 60 ° V = j ω 1 C ---------- j 10 3 10 3 3 × ------------------------------- j 3 == Z 1 1j 3 10 71.6 ° I C1 V Z 1 ° 10 71.6 ° 1 . 9 0 7 1 . 6 ° = i t 1.90 ω t7 1 . 6 ° + A cos = v in3 t 2 3 ω t cos 2 3 ω t1 8 0 ° + cos V 21 8 0 ° V = j ω 3 C j 31 0 3 × 10 3 3 × ----------------------------------------- j 1 Z 3 1 24 5 ° I C3 V Z 3 2 180 ° 5 ° ----------------------- 1 . 4 1 2 2 5 ° 1.41 225 135 ° = = i t 1.41 3 ω 3 5 ° A cos = i c t i c1 t i c3 t + 1.90 ω 1 . 6 ° + 1.41 3 ω 3 5 ° cos + cos P ave V 1RMS I θ 1 cos V 3RMS I θ 3 cos + = 6 2 ------ 1.90 2 --------- 71.6 ° cos 2 2 1.41 2 135 ° cos + = P 0.8 w = P 1 2 --I max 2 R 1 2 -- I 1max 2 I 3max 2 1 2 -- 1.90 2 1.41 2 0.8 w = =

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Chapter 7 Fourier Series 7 46 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 7.13 Evaluation of Fourier Coefficients Using Excel® The use of Fourier series is not restricted to electric circuit analysis. It is also applied in the anal- ysis of the behavior of physical systems subjected to periodic disturbances. Examples include cable stress analysis in suspension bridges, and mechanical vibrations. Quite often, it is necessary to construct the Fourier expansion of a function based on observed values instead of an analytic expression. Examples are meteorological or economic quantities whose period may be a day, a week, a month or even a year. In these situations, we need to eval- uate the integral(s) using numerical integration. The procedure presented here, will work for both the waveforms that have an analytical solution and those that do not. Even though we may already know the Fourier series from analytical methods, we can use this procedure to check our results. Consider the waveform of shown in Figure 7.47, were we have divided it into small pulses of width . Obviously, the more pulses we use, the better the approximation. If the time axis is in degrees, we can choose to be and it is convenient to start at the zero point of the waveform. Then, using a spreadsheet, such as Microsoft Excel, we can divide the period to in intervals, and enter these values in Column of the spreadsheet.
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