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where (14) has been used. Considering a linear network with an applied voltage which is periodic, we would expect that the resulting current would contain the same harmonic terms as the voltage, but with harmonic amplitudes of di erent relative magnitude, since the impedance varies with n!. It is possible that some harmonics would not appear in the current; for example, in a pure LC parallel circuit, one of the harmonic frequencies might coincide with the resonant frequency, making the impedance at that frequency in nite. In general, we may write X X v V0 Vn sin n!t n and i I0 In sin n!t n 17 with corresponding e ective values of q 2 2 2 Vrms V0 1 V1 1 V2 2 2 q 2 2 2 I0 1 I1 1 I2 2 2
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The average power P follows from integration of the instantaneous power, which is given by the product of v and i: h ih i X X p vi V0 Vn sin n!t n I0 In sin n!t n 19 Since v and i both have period T, their product must have an integral number of its periods in T. (Recall that for a single sine wave of applied voltage, the product vi has a period half that of the voltage wave.) The average may therefore be calculated over one period of the voltage wave: ih i X X 1 Th P V0 Vn sin n!t n I0 In sin n!t n dt 20 T 0 Examination of the possible terms in the product of the two in nite series shows them to be of the following types: the product of two constants, the product of a constant and a sine function, the product of two sine functions of di erent frequencies, and sine functions squared. After integration, the product of the two constants is still V0 I0 and the sine functions squared with the limits applied appear as Vn In =2 cos n n ; all other products upon integration over the period T are zero. Then the average power is P V0 I0 1 V1 I1 cos 1 1 V2 I2 cos 2 1 V3 I3 cos 3 2 2 2 21
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where n n n is the angle on the equivalent impedance of the network at the angular frequency n!, and Vn and In are the maximum values of the respective sine functions. In the special case of a single-frequency sinusoidal voltage, V0 V2 V3 0, and (21) reduces to the familiar P 1 V1 I1 cos 1 Veff Ieff cos  2 Compare Section 10.2. Also, for a dc voltage, V1 V2 V3 0, and (21) becomes P V0 I0 VI Thus, (21) is quite general. Note that on the right-hand side there is no term that involves voltage and current of di erent frequencies. In regard to power, then, each harmonic acts independently, and P P0 P1 P2 17.8 APPLICATIONS IN CIRCUIT ANALYSIS
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It has already been suggested above that we could apply the terms of a voltage series to a linear network and obtain the corresponding harmonic terms of the current series. This result is obtained by superposition. Thus we consider each term of the Fourier series representing the voltage as a single source, as shown in Fig. 17.10. Now the equivalent impedance of the network at each harmonic frequency n! is used to compute the current at that harmonic. The sum of these individual responses is the total response i, in series form, to the applied voltage.
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EXAMPLE 17.5 A series RL circuit in which R 5
and L 20 mH (Fig. 17-11) has an applied voltage v 100 50 sin !t 25 sin 3!t (V), with ! 500 rad/s. Find the current and the average power. Compute the equivalent impedance of the circuit at each frequency found in the voltage function. Then obtain the respective currents. At ! 0, Z0 R 5
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