tailieunhanh - Electric Circuits, 9th Edition P65

Electric Circuits, 9th Edition P65. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 616 Fourier Series Example Finding the Fourier Series of an Odd Function with Symmetry Find the Fourier series representation for the current waveform shown in Fig. . In the interval 0 s t T 4 the expression for z t is i t l n AAA T Figure 0 T 2 t 3 T 2 2t 5 T 2 3 t I i The periodic waveform for Example . Thus g r 44 bk t sinket dt 1 Jo 1 32 m I sin ka lt T2 k2 t cos ka t k i Solution We begin by looking for degrees of symmetry in the waveform. We find that the function is odd and in addition has half-wave and quarter-wave symmetry. Because the function is odd all the a coefficients are zero that is av 0 and ak 0 for all k. Because the function has half-wave symmetry bk 0 for even values of k. Because the function has quarter-wave symmetry the expression for bk for odd values of k is 81 m . kir 7 sin ir2k2 2 k is odd . The Fourier series representation of z r is 1 UTT . z t y sm sm zict Ot 1 .n 2 SI 1 y-1 sin cdqî - sin 3w 7T 9 8 fT 4 bk I i t sin dt. T Jo 1 e 1 sm 5w0i - sin 7w t assessment problem Objective 1 Be able to calculate the trigonometric form of the Fourier coefficients for a periodic waveform Derive the Fourier series for the periodic voltage shown. Answer 12VW æ sin Z7î 3 . ------7 --------2------Sin TT . n NOTE Also try Chapter Problems and . An Alternative Trigonometric Form of the Fourier Series 617 An Alternative Trigonometric Form of the Fourier Series In circuit applications of the Fourier series we combine the cosine and sine terms in the series into a single term for convenience. Doing so allows the representation of each harmonic of v t or z i as a single phasor quantity. The cosine and sine terms may be merged in either a cosine expression or a sine expression. Because we chose the cosine format in the phasor method of analysis see Chapter 9 we choose the cosine expression here for the alternative form of the series. Thus we write the Fourier series in Eq. as 0 av JXcosQw - 0