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Fourier Method of Waveform Analysis
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17.1 INTRODUCTION In the circuits examined previously, the response was obtained for excitations having constant, sinusoidal, or exponential form. In such cases a single expression described the forcing function for all time; for instance, v constant or v V sin !t, as shown in Fig. 17-1(a) and (b).
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Certain periodic waveforms, of which the sawtooth in Fig. 17-1(c) is an example, can be only locally de ned by single functions. Thus, the sawtooth is expressed by f t V=T t in the interval 0 < t < T and by f t V=T t T in the interval T < t < 2T. While such piecemeal expressions describe the waveform satisfactorily, they do not permit the determination of the circuit response. Now, if a periodic function can be expressed as the sum of a nite or in nite number of sinusoidal functions, the responses of linear networks to nonsinusoidal excitations can be determined by applying the superposition theorem. The Fourier method provides the means for solving this type of problem. In this chapter we develop tools and conditions for such expansions. Periodic waveforms may be expressed in the form of Fourier series. Nonperiodic waveforms may be expressed by their Fourier transforms. However, a piece of a nonperiodic waveform speci ed over a nite time period may also be expressed by a Fourier series valid within that time period. Because of this, the Fourier series analysis is the main concern of this chapter. 420
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FOURIER METHOD OF WAVEFORM ANALYSIS
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TRIGONOMETRIC FOURIER SERIES
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Any periodic waveform that is, one for which f t f t T can be expressed by a Fourier series provided that (1) (2) (3) If it is discontinuous, there are only a nite number of discontinuities in the period T; It has a nite average value over the period T; It has a nite number of positive and negative maxima in the period T.
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When these Dirichlet conditions are satis ed, the Fourier series exists and can be written in trigonometric form: f t 1 a0 a1 cos !t a2 cos 2t a3 cos 3!t 2 b1 sin !t b2 sin 2!t b3 sin 3!t
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The Fourier coe cients, a s and b s, are determined for a given waveform by the evaluation integrals. We obtain the cosine coe cient evaluation integral by multiplying both sides of (1) by cos n!t and integrating over a full period. The period of the fundamental, 2=!, is the period of the series since each term in the series has a frequency which is an integral multiple of the fundamental frequency. 2=! f t cos n!t dt
2=!
2=!
0 2=!
1 a cos n!t dt 2 0
2=! a1 cos !t cos n!t dt
an cos2 n!t dt
2=! b1 sin !t cos n!t dt
b2 sin 2!t cos n! dt
The de nite integrals on the right side of (2) are all zero except that involving cos2 n!t, which has the value =! an . Then an !  2=! f t cos n!t dt
T f t cos
2nt dt T
Multiplying (1) by sin n!t and integrating as above results in the sine coe cient evaluation integral. ! bn  2=!
2 f t sin n!t dt T
T f t sin
2nt dt T
An alternate form of the evaluation integrals with the variable 2 radians is an bn 1  1  2 F cos n d
0 2
!t and the corresponding period
5 6
F sin n d
where F f =! . The integrations can be carried out from T=2 to T=2,  to , or over any other full period that might simplify the calculation. The constant a0 is obtained from (3) or (5) with n 0; however, since 1 a0 is the average value of the function, it can frequently be determined by 2 inspection of the waveform. The series with coe cients obtained from the above evaluation integrals converges uniformly to the function at all points of continuity and converges to the mean value at points of discontinuity.