tailieunhanh - Electric Circuits, 9th Edition P70

Electric Circuits, 9th Edition P70. 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. | 666 The Fourier Transform W1U - F a 2da . 77 Ja x Note that expressing the integration in the frequency domain as i F w 2 dio 2 77 xj instead of 1 I F aj 2 da 17 Jo allows Eq. to be written in the form . z CcH Win z- lf l2 rf o F P . ft J 2. IT Figure shows the graphic interpretation of Eq. . Examples illustrate calculations involving Parseval s theorem. Example Applying Parseval s Theorem The current in a 40 il resistor is i 20e 2 n t A. What percentage of the total energy dissipated in the resistor can be associated with the frequency band 0 co 2V3 rad s and W40n 40 f 400 --------z da 77 Jo 4 it 16 000 1 ------ tan T TT 2 2 Solution The total energy dissipated in the 40 il resistor is 8000 tt ------ - 4000 J. 77 2 The energy associated with the frequency band 0 w 2V3 rad s is 40 f- 400 W4on - T dco 77 1 J_ -4 cc 4000 J. -4 o We can check this total energy calculation with Parseval s theorem F co 20 2 ja Therefore 16 000 77 8000 77 _ 8000 7T 3 J 3 Hence the percentage of the total energy associated with this range of frequencies is F co 20 V4 co2 8000 3 V - X 100 . 4000 Parseval s Theorem 667 Example Applying Parseval s Theorem to an Ideal Bandpass Filter The input voltage to an ideal bandpass filter is v t 120e 24 u t V. The filter passes all frequencies that lie between 24 and 48 rad s without attenuation and completely rejects all frequencies outside this passband. a Sketch V w 2 for the filter input voltage. b Sketch V co 2 for the filter output voltage. c What percentage of the total 1 il energy content of the signal at the input of the filter is available at the output Solution a The Fourier transform of the filter input voltage is The total 1 il energy available at the output of the filter is 1 f48 14 400 600 _ co 48 -----------r aco ----tan 1 - . 24 576 co2 24 24 600. . - tan tan 4 600 77 T Z84 J. The percentage of the input energy available at the output is V x 100 . .