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CHAP. 17]



Fig. 17-9


EFFECTIVE VALUES AND POWER The effective or rms value of the function



f ðtÞ ¼ 12 a0 þ a1 cos !t þ a2 cos 2!t þ    þ b1 sin !t þ b2 sin 2!t þ    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð12 a0 Þ2 þ 12 a21 þ 12 a22 þ    þ 12 b21 þ 12 b22 þ    ¼ c20 þ 12 c21 þ 12 c22 þ 12 c33 þ   


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 different 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 infinite. In general, we may write X X v ¼ V0 þ Vn sin ðn!t þ n Þ and i ¼ I0 þ In sin ðn!t þ n Þ ð17Þ with corresponding effective values of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vrms ¼ V02 þ 12 V12 þ 12 V22 þ   


Irms ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I02 þ 12 I12 þ 12 I22 þ   


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 ¼ vi ¼ V0 þ Vn sin ðn!t þ n Þ I0 þ In sin ðn!t þ n Þ ð19Þ 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: ð ih i X X 1 Th P¼ V0 þ Vn sin ðn!t þ n Þ I0 þ In sin ðn!t þ n Þ dt ð20Þ T 0 Examination of the possible terms in the product of the two infinite 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 different 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 ðVn In =2Þ cos ðn  n Þ; all other products upon integration over the period T are zero. Then the average power is P ¼ V0 I0 þ 12 V1 I1 cos 1 þ 12 V2 I2 cos 2 þ 12 V3 I3 cos 3 þ