## Power in *RLC* Series AC Circuits

If current varies with frequency in an *RLC* circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in this figure, voltage and current are out of phase in an *RLC* circuit. There is a **phase angle** \(\varphi \) between the source voltage \(V\) and the current \(I\), which can be found from

\(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =\cfrac{R}{Z}\text{.}\)

For example, at the resonant frequency or in a purely resistive circuit *\(Z=R\)*, so that \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =1\). This implies that \(\varphi =0º\) and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because \({I}_{\text{rms}}\) is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the *average power* is

\({P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi ,\)

Thus \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi \) is called the **power factor**, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =1\).

### Example: 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 F\(\text{5.00 μF}\) 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

\(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =\cfrac{R}{Z}\text{.}\)

We know \(Z\text{= 531 Ω}\) from this example, so that

\(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =\cfrac{\text{40}\text{.}0\phantom{\rule{0.25em}{0ex}}\Omega }{5\text{31}\phantom{\rule{0.25em}{0ex}}\Omega }=0\text{.}\text{0753 at 60.0 Hz.}\)

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

at 60.0 Hz.\(\varphi ={\text{cos}}^{-1}\phantom{\rule{0.25em}{0ex}}0\text{.}\text{0753}=\text{85.7º at 60.0 Hz.}\)

**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

\({P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}\text{cos}\phantom{\rule{0.25em}{0ex}}\mathrm{\varphi .}\)

\({I}_{\text{rms}}\) was found to be 0.226 A in this example. Entering the known values gives

\({P}_{\text{ave}}=(0\text{.}\text{226}\phantom{\rule{0.25em}{0ex}}\text{A})(\text{120}\phantom{\rule{0.25em}{0ex}}\text{V})(0\text{.}\text{0753})=2\text{.}\text{04}\phantom{\rule{0.25em}{0ex}}\text{W at 60.0 Hz.}\)

**Strategy and Solution for (c)**

At the resonant frequency, we know \(\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =1\), and \({I}_{\text{rms}}\) was found to be 6.00 A in this example. Thus,

\({P}_{\text{ave}}=(3\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{A})(\text{120}\phantom{\rule{0.25em}{0ex}}\text{V})(1)=\text{360}\phantom{\rule{0.25em}{0ex}}\text{W}\) 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.

Power delivered to an *RLC* series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next tutorial on electromagnetic radiation, but it can also be suppressed as is the case in this tutorial.

The circuit is analogous to the wheel of a car driven over a corrugated road as shown in this figure. The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels’ motion is a maximum if the bumps in the road are hit at the resonant frequency.

A pure *LC* circuit with negligible resistance oscillates at \({f}_{0}\), the same resonant frequency as an *RLC* circuit. It can serve as a frequency standard or clock circuit—for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time. This figure shows the analogy between an *LC* circuit and a mass on a spring.

### PhET Explorations: Circuit Construction Kit (AC+DC), Virtual Lab

Build circuits with capacitors, inductors, resistors and AC or DC voltage sources, and inspect them using lab instruments such as voltmeters and ammeters.