# Power

## Power delivered to an RLC series AC circuit is dissipated by the resistance in the circuit, and is given as $P_{avg} = I_{rms} V_{rms} cos\phi$. Here, $\phi$ is called the phase angle.

#### Key Points

• Phase angle ϕ is the phase difference between the source voltage V and the current I. See the phasor diagram in Figure 1.

• At the resonant frequency 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.

• Average power dissipated in an RLC circuit can be calculated by taking a time average of power, P(t) = I(t)V(t), over a period.

#### Terms

• Root mean square: a statistical measure of the magnitude of a varying quantity.

#### Figures

1. ##### 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.

2. ##### 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.

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

<equation contenteditable="false">$cos\phi = \frac{R}{Z}$, as diagramed in Figure 1.

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 Irms 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 Figure 2. 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.

#### Key Term Glossary

AC
Alternating current.
##### Appears in these related concepts:
amplitude
The maximum absolute value of some quantity that varies.
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
average
The arithmetic mean.
##### Appears in these related concepts:
capacitor
An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.
##### Appears in these related concepts:
circuit
A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T
##### Appears in these related concepts:
current
The time rate of flow of electric charge.
##### Appears in these related concepts:
damping
The reduction in the magnitude of oscillations by the dissipation of energy
##### Appears in these related concepts:
electric field
A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.
##### Appears in these related concepts:
radiation (quantized as photons) consisting of oscillating electric and magnetic fields oriented perpendicularly to each other, moving through space
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
frequency
The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.
##### Appears in these related concepts:
inductor
A passive device that introduces inductance into an electrical circuit.
##### Appears in these related concepts:
Inductor
A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.
##### Appears in these related concepts:
kinetic
Of or relating to motion
##### Appears in these related concepts:
motion
A change of position with respect to time.
##### Appears in these related concepts:
oscillation
the act of oscillating or the state of being oscillated
##### Appears in these related concepts:
period
The duration of one cycle in a repeating event.
##### Appears in these related concepts:
Period
The period is the duration of one cycle in a repeating event.
##### Appears in these related concepts:
phase
Any one point or portion in a recurring series of changes, as in the changes of motion of one of the particles constituting a wave or vibration; one portion of a series of such changes (in distinction from a contrasted portion) as the portion on one side of a position of equilibrium (in contrast with that on the opposite side).
##### Appears in these related concepts:
potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
##### Appears in these related concepts:
potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
##### Appears in these related concepts:
power
A measure of the rate of doing work or transferring energy.
##### Appears in this related concept:
Designates a portion of the electromagnetic spectrum having frequencies ranging from 300 GHz to 3 kHz, or equivalently, wavelengths from 1 millimeter to 100 kilometers.
##### Appears in these related concepts:
resistance
The opposition to the passage of an electric current through that element.
##### Appears in these related concepts:
resistor
An electric component that transmits current in direct proportion to the voltage across it.
##### Appears in these related concepts:
resonance
The increase in the amplitude of an oscillation of a system under the influence of a periodic force whose frequency is close to that of the system's natural frequency.
##### Appears in these related concepts:
rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
##### Appears in these related concepts:
series
A number of things that follow on one after the other or are connected one after the other.
##### Appears in these related concepts:
voltage
The amount of electrostatic potential between two points in space.
##### Appears in these related concepts:
wave
A moving disturbance in the energy level of a field.