Power

If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. However, the average power is not simply current times voltage, as is the case in purely resistive circuits. As seen in previous Atoms, voltage and current are out of phase in an RLC circuit. There is a phase angle ϕ between the source voltage V and the current I, given as

$cos\phi = \frac{R}{Z}$, as diagramed in .

Phasor Diagram for an RLC Series Circuit

Phasor diagram for an RLC series circuit. \phi is the phase angle, equal to the phase difference between the voltage and current.

For example, at the resonant frequency $(\nu_0 = \frac{1}{2\pi \sqrt{LC}})$ or in a purely resistive circuit, Z=R, so that cosϕ=1. This implies that ϕ=0º and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance, because voltage and current are out of phase and I_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

$P_{avg} = I_{rms} V_{rms} cos\phi$

(an equation derived by taking a time average of power, P(t) = I(t)V(t), over a period. I(t) and V(t) are current and voltage at time t). Thus cosϕ 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, cosϕ=1.

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 energy out of the circuit. Rather, they transfer energy back and forth to one another, with the resistor dissipating the exact amount that the voltage source gives the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor (such as radio waves).

The circuit is analogous to the wheel of a car driven over a corrugated road, as seen in . 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.

Forced Damped Motion of a Wheel on a Car Spring

The forced but damped motion of the wheel on the car spring is analogous to an RLC series AC circuit. The shock absorber damps the motion and dissipates energy, analogous to the resistance in an RLC circuit. The mass and spring determine the resonant frequency.