Fourier scan702, Study notes of Electrical Circuit Analysis

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6.3 FOURIER SERIES The exponential Fourier series for a periodic signal was developed in Section 5.4 as a linear combination of orthonormal functions having the property of being a least-square-error approximation. Also shown was the fact that when the signal in question has well-defined average power, its Fourier series may be deemed a valid signal representation for most engineering purposes. That premise is adopted here, where we pul the (theory to work, Specifically, we shall use the Fourier series to express periodic signals as sums of phasors, from which their line spectra drop out immediately. Then, in Section 6.5, the phasor sum is employed to carry out periodic steady-state analysis. Exponential Fourier series Let v(t) be a periodic power signal with fundamental period 7). It can be expanded as a linear combination of phasors via the exponential Fourier series LN = DT cen Q note where (is the fundamental angular frequency 2a =a 2 To (2) iL also happens in this case that the corner frequency numerically equals the half-power bandwidth and, at that point, [Mf(B) ‘a, = ~20 logo (1/V2) = 3.01 ~ 3 db, which explains why 8 commonly is referred to as the 3-db bandwidth, 6.3 FOURIER SERIES | 187