23.12 Rlc series ac circuits  (Page 4/9)

Calculating the power factor and power

For the same RLC series circuit having a $\mathrm{40.0\; \Omega }$ resistor, a 3.00 mH inductor, a $\text{5.00 μF}$ capacitor, and a voltage source with a $V_{\text{rms}}$ of 120 V: (a) Calculate the power factor and phase angle for $f=\text{60}\text{.}0\text{Hz}$ . (b) What is the average power at 50.0 Hz? (c) Find the average power at the circuit’s resonant frequency.

Strategy and Solution for (a)

The power factor at 60.0 Hz is found from

We know $Z\text{= 531 Ω}$ from [link] , so that

This small value indicates the voltage and current are significantly out of phase. In fact, the phase angle is

Discussion for (a)

The phase angle is close to $\text{90º}$ , consistent with the fact that the capacitor dominates the circuit at this low frequency (a pure RC circuit has its voltage and current $\text{90º}$ out of phase).

Strategy and Solution for (b)

The average power at 60.0 Hz is

$I_{\text{rms}}$ was found to be 0.226 A in [link] . Entering the known values gives

Strategy and Solution for (c)

At the resonant frequency, we know $\text{cos}\varphi =1$ , and $I_{\text{rms}}$ was found to be 6.00 A in [link] . Thus,

$P_{\text{ave}}=(3\text{.}\text{00}\text{A})(\text{120}\text{V})(1)=\text{360}\text{W}$ at resonance (1.30 kHz)

Discussion

Both the current and the power factor are greater at resonance, producing significantly greater power than at higher and lower frequencies.