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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
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.
Key Term Reference
 AC
 Appears in these related concepts: Driven Oscillations and Resonance, Resistors in AC Circuits, and Safety Precautions in the Household
 Radiation
 Appears in these related concepts: Radiation from Food, Time, and Biological Effects of Radiation
 amplitude
 Appears in these related concepts: Interference and Diffraction, Interference, and Period of a Mass on a Spring
 atom
 Appears in these related concepts: The Law of Multiple Proportions, John Dalton and Atomic Theory, and Overview of Atomic Structure
 capacitor
 Appears in these related concepts: Impedance, Combinations of Capacitors: Series and Parallel, and Introduction and Importance
 circuit
 Appears in these related concepts: Electric Potential Energy and Potential Difference, Microwaves, and Maxwell's Equations
 current
 Appears in these related concepts: Reporting LongTerm Liabilities, The Battery, and Magnetic Force Between Two Parallel Conductors
 damping
 Appears in these related concepts: Forced Vibrations and Resonance, Applications of SecondOrder Differential Equations, and Damped Harmonic Motion
 diagram
 Appears in these related concepts: Motion Diagrams, Bohr Orbits, and Muscles and Joints
 electric field
 Appears in these related concepts: Electric Flux, Properties of Electric Charges, and Potentials and Charged Conductors
 electromagnetic radiation
 Appears in these related concepts: Electromagnetic Spectrum, Gamma Decay, and Planck's Quantum Theory
 energy
 Appears in these related concepts: Surface Tension, Introduction to Work and Energy, and The Role of Energy and Metabolism
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 frequency
 Appears in these related concepts: Antennae, Period and Frequency, and Frequency of Sound Waves
 inductor
 Appears in these related concepts: Induced Charge, 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: Center of Mass and Translational Motion, TwoComponent Forces, and Moving Source
 period
 Appears in these related concepts: The Periodic Table, SinglePeriod Investment, and Annuities
 phase
 Appears in these related concepts: Borates: BoronOxygen Compounds, 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: Problem Solving With the Conservation of Energy, Defining Graviational Potential Energy, and Types of Energy
 power
 Appears in these related concepts: What is Power?, Sources of Power, and Weber's View of Stratification
 resistance
 Appears in these related concepts: Resistors in Parallel, Resisitors in Series, and Ecosystem Dynamics
 resistor
 Appears in these related concepts: Resistors and Capacitors in Series, The Loop Rule, and Current and Voltage Measurements in Circuits
 resonance
 Appears in these related concepts: Resonance, Bonding in Coordination Compounds: Valence Bond Theory, and Standing Waves and Resonance
 series
 Appears in these related concepts: Charging a Battery: EMFs in Series and Parallel, Combination Circuits, and Finding the General Term
 voltage
 Appears in these related concepts: The Millikan OilDrop Experiment, The Nernst Equation, and Electric Potential Due to a Point Charge
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Cite This Source
Source: Boundless. “Power.” Boundless Physics. Boundless, 01 Jul. 2015. Retrieved 01 Jul. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/inductionaccircuitsandelectricaltechnologies22/accircuits162/power5871645/