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Power
Power delivered to an RLC series AC circuit is dissipated by the resistance in the circuit, and is given as
Learning Objectives

Determine power delivered to an RLC series AC circuit from the current and the voltage

Identify conditions when the source voltage and the current are in phase
Key Points
 Phase angle ϕ is the phase difference between the source voltage V and the current I. See the phasor diagram in.
 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.
Term

rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
Full Text
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
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
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
(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.
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Key Term Reference
 AC
 Appears in these related concepts: Driven Oscillations and Resonance, Resistors in AC Circuits, and Impedance
 Radiation
 Appears in these related concepts: Scattering of Light by the Atmosphere, Antennae, and Time
 amplitude
 Appears in these related concepts: Interference and Diffraction, Period of a Mass on a Spring, and Sound
 atom
 Appears in these related concepts: The Law of Multiple Proportions, Stable Isotopes, and John Dalton and Atomic Theory
 capacitor
 Appears in these related concepts: Combinations of Capacitors: Series and Parallel, Introduction and Importance, and ParallelPlate Capacitor
 circuit
 Appears in these related concepts: Forced Vibrations and Resonance, Energy Usage, and Photon Energies of the EM Spectrum
 current
 Appears in these related concepts: Reporting LongTerm Liabilities, The Battery, and Magnetic Force Between Two Parallel Conductors
 damping
 Appears in these related concepts: Applications of SecondOrder Differential Equations, Damped Harmonic Motion, and Back EMF, Eddy Currents, and Magnetic Damping
 diagram
 Appears in these related concepts: Motion Diagrams, The Third Law, and XRay Diffraction
 electric field
 Appears in these related concepts: Relation Between Electric Potential and Field, Potentials and Charged Conductors, and Maxwell's Predictions and Hertz' Confirmation
 electromagnetic radiation
 Appears in these related concepts: Other Forms of Energy, Gamma Decay, and Planck's Quantum Theory
 energy
 Appears in these related concepts: Energy Transportation, Surface Tension, and Introduction to Work and Energy
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 frequency
 Appears in these related concepts: Beats, Frequency of Sound Waves, and General Case
 inductor
 Appears in these related concepts: Energy in a Magnetic Field, RL Circuits, and Inductance
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Sculpture
 motion
 Appears in these related concepts: Motion with Constant Acceleration, Newton and His Laws, and Mechanical Work and Electrical Energy
 period
 Appears in these related concepts: The Periodic Table, Number of Periods, and Atomic Size
 phase
 Appears in these related concepts: Wavelength, Freqency in Relation to Speed, The Phase of Orbitals, and The Production of Electromagnetic Waves
 phasor
 Appears in these related concepts: Resonance in RLC Circuits, Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams, and Phasors
 potential
 Appears in these related concepts: Maslow's Hierarchy of Needs, Conservative and Nonconservative Forces, and Linear Expansion
 potential energy
 Appears in these related concepts: The Chain Rule, Energy Conservation, and Electric Potential Energy and Potential Difference
 power
 Appears in these related concepts: What is Power?, Sources of Power, and Power
 resistance
 Appears in these related concepts: Resistors in Parallel, Resisitors in Series, and Ecosystem Dynamics
 resistor
 Appears in these related concepts: Safety Precautions in the Household, Resistors and Capacitors in Series, and The Loop Rule
 resonance
 Appears in these related concepts: Resonance, RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram, and Standing Waves and Resonance
 series
 Appears in these related concepts: Taylor Polynomials, Charging a Battery: EMFs in Series and Parallel, and Finding the General Term
 voltage
 Appears in these related concepts: Conductors, The Millikan OilDrop Experiment, and The Nernst Equation
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Cite This Source
Source: Boundless. “Power.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 13 Oct. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/inductionaccircuitsandelectricaltechnologies22/accircuits162/power5871645/