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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

Compare the alternating current and voltage with the direct current and voltage

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

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

root mean square
The square root of the arithmetic mean of the squares.

rms current
the root mean square of the current, Irms=I0/√2 , where I0 is the peak current, in an AC system
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 :
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):
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 .
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Key Term Reference
 AC
 Appears in this related concepts: Power, Resonance in RLC Circuits, and Impedance
 DC
 Appears in this related concepts: Resistors and Capacitors in Series, Phasors, and Discrete Fourier Transform Examples
 Hertz
 Appears in this related concepts: Time, Frequency of Sound Waves, and Characteristics of Sound
 Law
 Appears in this related concepts: TwoComponent Forces, Damped Harmonic Motion, and Models, Theories, and Laws
 Ohm's law
 Appears in this related concepts: Energy Usage, Current and Voltage Measurements in Circuits, and Ohm's Law
 alternating current
 Appears in this related concepts: Education and the Professions, Safety Precautions in the Household, and The Second Industrial Revolution
 application
 Appears in this related concepts: Physics and Other Fields, The First Law, and XRay Imaging and CT Scans
 circuit
 Appears in this related concepts: Combinations of Capacitors: Series and Parallel, Microwaves, and Maxwell's Equations
 current
 Appears in this related concepts: Reporting LongTerm Liabilities, Magnetic Force Between Two Parallel Conductors, and The Junction Rule
 direct current
 Appears in this related concepts: Resistors in AC Circuits, Different Types of Currents, and The Inventions of the Telephone and Electricity
 electric charge
 Appears in this related concepts: Gauss's Law, Mass Spectrometer, and Overview of Electric Current
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 frequency
 Appears in this related concepts: Antennae, General Rules for Assigning Electrons to Atomic Orbitals, and Sound
 magnitude
 Appears in this related concepts: Roundoff Error, Multiplying Vectors by a Scalar, and Components of a Vector
 ohm
 Appears in this related concepts: Poiseuille's Equation and Viscosity, Phase Angle and Power Factor, and Null Measurements
 period
 Appears in this related concepts: The Periodic Table, Number of Periods, and Atomic Size
 power
 Appears in this related concepts: What is Power?, Sources of Power, and Power
 resistance
 Appears in this related concepts: Resistors in Parallel, Resisitors in Series, and Ecosystem Dynamics
 rms
 Appears in this related concepts: Temperature, Speed Distribution of Molecules, and Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams
 series
 Appears in this related concepts: Taylor Polynomials, Charging a Battery: EMFs in Series and Parallel, and Finding the General Term
 sinusoidal
 Appears in this 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 this related concepts: Conductors, The Millikan OilDrop Experiment, and Principles of Electricity
 wave
 Appears in this related concepts: Waves, Properties of Waves and Light, and Other Forms of Energy
 waveform
 Appears in this related concepts: Period and Frequency and QuantumMechanical View of Atoms
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Source: Boundless. “Root Mean Square Values.” Boundless Physics. Boundless, 06 Jan. 2015. Retrieved 20 May. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/electriccurrentandresistance19/alternatingcurrents148/rootmeansquarevalues52611282/