Question

In a series RCL circuit the dissipated power drops by a factor of 8.0 when the frequency of the generator is changed from the resonant frequency to a nonresonant frequency. The peak voltage is held constant while this change is made. Determine the power factor of the circuit at the nonresonant frequency. Note: The ac current and voltage are rms values and power is an average value unless indicated otherwise.

Answer 1

`cos\phi = (1)/(2\sqrt2)`

Step-by-step explanation:

Average power is defined as

`P = I_(rms)V_(rms) cos\phi`

so here we know that

`I_(rms) = (V_(rms))/(z)`

`P = (V_(rms)^2)/(z) cos\phi`

here we know that

`cos\phi = (R)/(z)`

so we will have

`P_(resonance) = 8 P_(nonresonance)`

`(V_(rms)^2)/(z_1) * (R)/(z_1) = 8 (V_(rms)^2)/(z_2) * (R)/(z_2)`

so we have

`z_1^2 = (z_2^2)/(8)`

`z_1 = (z_2)/(2\sqrt2)`

at resonance power factor is given as

`(R)/(z_1) = 1`

so when it is at non resonance condition then we have

`cos\phi = (R)/(z_2)`

`cos\phi = (z_1)/(2\sqrt2 z_1)`

`cos\phi = (1)/(2\sqrt2)`