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Root Mean Square Values
The root mean square (RMS) voltage or current is the timeaveraged voltage or current in an AC system.
Learning Objectives

Relate the average power in an alternating current circuit with the root mean square voltage and current

Relate the root mean square voltage and current with the peak voltage and current

Compare the alternating current and voltage with the direct current and voltage
Key Points
 Recall that unlike DC current and voltage, which are constant, AC current and voltage vary over time. This is called alternating current because the direction alternates.
 The root mean square (abbreviated RMS or rms) is a statistical measure of the magnitude of a varying quantity. We use the root mean square to express the average current or voltage in an AC system.
 The RMS current and voltage (for sinusoidal systems) are the peak current and voltage over the square root of two.
 The average power in an AC circuit is the product of the RMS current and RMS voltage.
Terms

rms voltage
the root mean square of the voltage, Vrms=V0/√2 , where V0 is the peak voltage, in an AC system

rms current
the root mean square of the current, Irms=I0/√2 , where I0 is the peak current, in an AC system

root mean square
The square root of the arithmetic mean of the squares.
Full Text
Root Mean Square Values and Alternating Current
Recall that in the case of alternating current (AC) the flow of electric charge periodically reverses direction. Unlike direct current (DC), where the currents and voltages are constant, AC currents and voltages vary over time. Recall that most residential and commercial power sources use AC. It is often the case that we wish to know the time averaged current, or voltage. Given the current or voltage as a function of time, we can take the root mean square over time to report the average quantities.
Definition
The root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when the function alternates between positive and negative values, e.g., sinusoids.The RMS value of a set of values (or a continuoustime function such as a sinusoid) is the square root of the arithmetic mean of the squares of the original values (or the square of the function). In the case of a set of n values {x_{1},x_{2},....,x_{n}} , the RMS value is given by this formula:
The corresponding formula for a continuous function f(t) defined over the interval T_{1} ≤ t ≤ T_{2} is as follows:
The RMS for a function over all time is below.
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples.
Application to Voltage and Current
Consider the case of sinusoidally varying voltage :
Sinusoidal Voltage and Current
(a) DC voltage and current are constant in time, once the current is established. (b) A graph of voltage and current versus time for 60Hz AC power. The voltage and current are sinusoidal and are in phase for a simple resistance circuit. The frequencies and peak voltages of AC sources differ greatly.
V is the voltage at time t, V_{0} is the peak voltage, and f is the frequency in hertz. For this simple resistance circuit, I=V/R, and so the AC current is as follows:
Here, I is the current at time t, and I_{0}=V_{0}/R is the peak current. Now using the definition above, let's calculate the rms voltage and rms current. First, we have
Here, we have replaced 2πf with ω. Since V_{0} is a constant, we can factor it out of the square root, and use a trig identity to replace the squared sine function.
Integrating the above, we have:
Since the interval is a whole number of complete cycles (per definition of RMS), the terms will cancel out, leaving:
Similarly, you can find that the RMS current can be expressed fairly simply:
Updated Circuit Equations for AC
Many of the equations we derived for DC current apply equally to AC. If we are concerned with the time averaged result and the relevant variables are expressed as their rms values. For example, Ohm's Law for AC is written as follows:
The various expressions for AC power are below:
We can see from the above equations that we can express the average power as a function of the peak voltage and current (in the case of sinusoidally varying current and voltage):
Average Power
AC power as a function of time. Since the voltage and current are in phase here, their product is nonnegative and fluctuates between zero and I0V0. Average power is (1/2)I0V0.
The RMS values are also useful if the voltage varies by some waveform other than sinusoids, such as with a square, triangular or sawtooth waves .
Waveforms
Sine, square, triangle, and sawtooth waveforms
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Key Term Reference
 AC
 Appears in these related concepts: Power, Resonance in RLC Circuits, and Safety Precautions in the Household
 DC
 Appears in these related concepts: Resistors and Capacitors in Series, Impedance, and Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams
 Hertz
 Appears in these related concepts: Antennae, Frequency of Sound Waves, and Characteristics of Sound
 Law
 Appears in these related concepts: Newton and His Laws, Mechanical Work and Electrical Energy, and Models, Theories, and Laws
 Ohm's law
 Appears in these related concepts: Energy Usage, Current and Voltage Measurements in Circuits, and RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram
 alternating current
 Appears in these related concepts: Education and the Professions, The Second Industrial Revolution, and Energy in a Simple Harmonic Oscillator
 application
 Appears in these related concepts: What is Potential Energy?, Work Done by a Variable Force, and Physics and Other Fields
 circuit
 Appears in these related concepts: Forced Vibrations and Resonance, Photon Energies of the EM Spectrum, and Combinations of Capacitors: Series and Parallel
 current
 Appears in these related concepts: Reporting LongTerm Liabilities, The Battery, and Magnetic Force Between Two Parallel Conductors
 direct current
 Appears in these related concepts: Resistors in AC Circuits, Different Types of Currents, and The Inventions of the Telephone and Electricity
 electric charge
 Appears in these related concepts: The Junction Rule, Overview of Electric Current, and Static Electricity, Charge, and the Conservation of Charge
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 frequency
 Appears in these related concepts: Energy and Momentum, General Case, and Sound
 magnitude
 Appears in these related concepts: Adding and Subtracting Vectors Using Components, Multiplying Vectors by a Scalar, and Components of a Vector
 ohm
 Appears in these related concepts: Poiseuille's Equation and Viscosity, Phase Angle and Power Factor, and Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams
 period
 Appears in these related concepts: SinglePeriod Investment, Annuities, and Ending Punctuation
 power
 Appears in these related concepts: Weber's View of Stratification, Authority, and Power
 resistance
 Appears in these related concepts: Resistors in Parallel, Resisitors in Series, and Ecosystem Dynamics
 rms
 Appears in these related concepts: Temperature, Speed Distribution of Molecules, and Nuclear Size and Density
 series
 Appears in these related concepts: Charging a Battery: EMFs in Series and Parallel, Combination Circuits, and Finding the General Term
 sinusoidal
 Appears in these related concepts: Driven Oscillations and Resonance, Position, Velocity, and Acceleration as a Function of Time, and Sinusoidal Nature of Simple Harmonic Motion
 voltage
 Appears in these related concepts: The Nernst Equation, Electric Potential Due to a Point Charge, and Principles of Electricity
 wave
 Appears in these related concepts: Particle in a Box, Waves, and Properties of Waves and Light
 waveform
 Appears in these related concepts: Period and Frequency and QuantumMechanical View of Atoms
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Source: Boundless. “Root Mean Square Values.” Boundless Physics. Boundless, 12 Aug. 2015. Retrieved 01 Dec. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/electriccurrentandresistance19/alternatingcurrents148/rootmeansquarevalues52611282/