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FOURIER METHOD OF WAVEFORM ANALYSIS
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Find the e ective voltage, e ective current, and average power supplied to a passive network if the applied voltage is v 200 100 cos 500t 308 75 cos 1500t 608 (V) and the resulting current is i 3:53 cos 500t 758 3:55 cos 1500t 78:458 (A). Ans: 218.5 V, 3.54 A, 250.8 W A voltage v 50 25 sin 500t 10 sin 1500t 5 sin 2500t (V) is applied to the terminals of a passive network and the resulting current is i 5 2:23 sin 500t 26:68 0:556 sin 1500t 56:38 0:186 sin 2500t 68:28 Find the e ective voltage, e ective current, and the average power. Ans: A
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53.6 V, 5.25 A, 276.5 W
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A three-element series circuit, with R 5
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, L 5 mH, and C 50 mF, has an applied voltage v 150 sin 1000t 100 sin 2000t 75 sin 3000t (V). Find the e ective current and the average power for the circuit. Sketch the line spectrum of the voltage and the current, and note the e ect of series resonance. Ans: 16.58 A, 1374 W A two-element series circuit, with R 10
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and L 20 mH, has current i 5 sin 100t 3 sin 300t 2 sin 500t Find the e ective applied voltage and the average power. Ans: A 48 V, 190 W
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A pure inductance, L 10 mH, has the triangular current wave shown in Fig. 17-59, where ! 500 rad/s. Obtain the exponential Fourier series for the voltage across the inductance. Compare the answer with the result of Problem 17.8. Ans: vL 200 j 1 e j3!t je j!t je j!t j 1 e j!t 3 3 2 V
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A pure inductance, L 10 mH, has an applied voltage with the waveform shown in Fig. 17-60, where ! 200 rad/s. Obtain the current series in trigonometric form and identify the current waveform. Ans: i 20 1 1 sin !t 1 sin 3!t 25 sin 5!t 49 sin 7!t A ; triangular 9 
Figure 17-61 shows a full-wave-recti ed sine wave representing the voltage applied to the terminals of an LC series circuit. Use the trigonometric Fourier series to nd the voltages across the inductor and the capacitor.
2 Ans: vL
FOURIER METHOD OF WAVEFORM ANALYSIS 3
[CHAP. 17
7 4Vm 6 4!L 6  2!L  cos 2!t   cos 4!t 7 5 1 1  4 3 2!L 15 4!L 2!C 4!C 2 4Vm 61 6  42
vC
7 1 1   cos 2!t   cos 4!t 7 5 1 1 3 2!C 2!L 15 4!C 4!L 2!C 4!C
Fig. 17-61
A three-element circuit consists of R 5
in series with a parallel combination of L and C. At Find the total current if the applied voltage is given by ! 500 rad/s, XL 2
, XC 8
. v 50 20 sin 500t 10 sin 1000t (V). Ans: i 10 3:53 sin 500t 28:18 (A)
APPENDIX A
Complex Number System
A1 p A complex number z is a number of the form x jy, where x and y are real numbers and j 1. We write x Re z, the real part of z; y Im z, the imaginary part of z. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. COMPLEX NUMBERS
COMPLEX PLANE
A pair of orthogonal axes, with the horizontal axis displaying Re z and the vertical axis j Im z, determine a complex plane in which each complex number is a unique point. Refer to Fig. A-1, on which six complex numbers are shown. Equivalently, each complex number is represented by a unique vector from the origin of the complex plane, as illustrated for the complex number z6 in Fig. A-1.
Fig. A-1
Fig. A-2
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COMPLEX NUMBER SYSTEM
[APP. A
VECTOR OPERATOR j
In addition to the de nition of j given in Section A1, it may be viewed as an operator which rotates any complex number (vector) A 908 in the counterclockwise direction. The case where A is a pure real number, x, is illustrated in Fig. A-2. The rotation sends A into jx, on the positive imaginary axis. Continuing, j 2 advances A 1808; j 3 , 2708; and j 4 , 3608. Also shown in Fig. A-2 is a complex number B in the rst quadrant, at angle . Note that jB is in the second quadrant, at angle  908.
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