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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 29 Copyright © Orchard Publications Circuit Analysis with Trigonometric Fourier Series Figure 7.28. Input waveform for the circuit of Figure 7.27 Solution: In Subsection 7.4.1, Page 7 11, we found that the waveform of Figure 7.28 can be represented by the trigonometric Fourier series as (7.79) Since this series is the sum of sinusoids, we will use phasor analysis to obtain the solution. The equivalent phasor circuit is shown in Figure 7.29. Figure 7.29. Phasor circuit for Example 7.2 We let represent the number of terms in the Fourier series. For this example, we are only inter- ested in the first three odd terms, that is, . By the voltage division expression, (7.80) With reference to (7.79) the phasors of the first 3 odd terms of (7.80) are (7.81) (7.82) 0 π 2 π T ω t A A v in t () ft 4A π ------- ω t 1 3 -- 3 ω t 1 5 5 ω t sin ++ sin + sin   = + 1 C V 1 j ω ----- V C R n n 1 3 and 5 ,, = V C n 1j n ω 11j n ω + ------------------------------V n 1 n + ------------- V n == π t sin π t9 0 ° cos = V in1 π 90 ° = π 1 3 3t sin π 1 3 90 ° cos = V in3 π 1 3 90 ° =

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Chapter 7 Fourier Series 7 30 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (7.83) By substitution of (7.81) through (7.83) into (7.80), we obtain the phasor and time domain volt- ages indicated in (7.84) through (7.86) below. (7.84) (7.85) (7.86) Thus, the capacitor voltage in the time domain is (7.87) Assuming that in the circuit of Figure 7.27 the capacitor is initially discharged, we expect that capacitor voltage will consist of alternating rising and decaying exponentials. Let us plot relation (7.87) using the MATLAB script below assuming that . t=0:pi/64:4*pi; Vc=(4./pi).*((sqrt(2)./2).*cos(t 135.*pi./180)+. .. (sqrt(10)./30).*cos(3.*t 161.6.*pi./180)+(sqrt(26)./130).*cos(5.*t 168.7.*pi./180)); plot(t,Vc) Figure 7.30. Waveform for relation (7.87) 4A π ------- 1 5 -- 5t sin π 1 5 90 ° () cos = V in5 π 1 5 90 ° = V C1 1 1j + ---------- π 90 ° 1 24 5 ° -------------------- π 90 ° == π 2 2 ------ 135 ° π 2 2 t1 3 5 ° cos = V C3 1 3 + ------------- π 1 3 90 ° 1 10 71.6 ° --------------------------- π 1 3 90 ° π 10 30 --------- 161.6 ° π 10 30 3t 161.6 ° cos = V C5 1 5 + π 1 5 90 ° 1 26 78.7 ° π 1 5 90 ° π 26 130 168.7 ° π 26 130 168.7 ° cos = v C t π 2 2 3 5 ° cos 10 30 161.6 ° cos 26 130 168.7 ° cos +++ = A1 =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 31 Copyright © Orchard Publications The Exponential Form of the Fourier Series The waveform of Figure 7.30 is a rudimentary presentation of the capacitor voltage for the circuit of Figure 7.27. However, it will improve if we add a sufficient number of harmonics in (7.87).

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