# Root Mean Square Values

## The root mean square (RMS) voltage or current is the time-averaged voltage or current in an AC system.

#### 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

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

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

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

#### Figures

1. ##### 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 60-Hz 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.

2. ##### Average Power

AC power as a function of time. Since the voltage and current are in phase here, their product is non-negative and fluctuates between zero and I0V0. Average power is (1/2)I0V0.

3. ##### Waveforms

Sine, square, triangle, and sawtooth waveforms

## 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 continuous-time 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 {x1,x2,....,xn} , the RMS value is given by this formula:

<equation contenteditable="false">$x_{rms}=\sqrt{\frac{1}{n}(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})}$

The corresponding formula for a continuous function f(t) defined over the interval T1 ≤ t ≤ T2 is as follows:

$f_{rms}=\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[f(t)]^{2}dt}$

The RMS for a function over all time is below.

$f_{rms}=\lim_{T \to \infty }\sqrt{\frac{1}{T}\int_{0}^{T}[f(t)]^{2}dt}$

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 (Figure 1):

$V=V_{0}sin(2\pi ft)$

V is the voltage at time tV0 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:

$I=I_{0}sin(2\pi ft)$

Here, I is the current at time t, and I0=V0/R is the peak current. Now using the definition above, let's calculate the rms voltage and rms current. First, we have

$V_{rms}=\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[V_{0}sin(\omega t)]^{2}dt}$

Here, we have replaced 2πf with ω. Since V0 is a constant, we can factor it out of the square root, and use a trig identity to replace the squared sine function.

$V_{rms}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}\frac{1-cos(2\omega t)}{2}dt}$

Integrating the above, we have:

$V_{rms}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}[\frac{t}{2}-\frac{sin(2\omega t)}{4\omega }]_{T_{1}}^{T_{2}}}$

Since the interval is a whole number of complete cycles (per definition of RMS), the terms will cancel out, leaving:

$V_{rms}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}[\frac{t}{2}]_{T_{1}}^{T_{2}}}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}\frac{T_{2}-T_{1}}{2}}$

$=\frac{V_{0}}{\sqrt{2}}$

Similarly, you can find that the RMS current can be expressed fairly simply:

$I_{rms}=I_0/\sqrt{2}$

## 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:

$I_{rms}=\frac{V_{rms}}{R}$

The various expressions for AC power are below:

$P_{ave}=I_{rms}V_{rms}$

$P_{ave}=\frac{V_{rms}^{2}}{R}$

$P_{ave}=I_{rms}^{2}R$

We can see from the above equations that we can express the average power (Figure 2) as a function of the peak voltage and current (in the case of sinusoidally varying current and voltage):

$P_{ave}=I_{rms}V_{rms}=\frac{I_{0}}{\sqrt{2}}\frac{V_{0}}{\sqrt{2}}=\frac{1}{2}V_{0}I_{0}$

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 (Figure 3).

#### Key Term Glossary

AC
Alternating current.
##### Appears in these related concepts:
alternating current
(AC)—An electric current in which the direction of flow of the electrons reverses periodically having an average of zero, with positive and negative values (with a frequency of 50 Hz in Europe, 60 Hz in the US, 400 Hz for airport lighting, and some others); especially such a current produced by a rotating generator or alternator.
##### Appears in these related concepts:
application
the act of putting something into operation
##### Appears in these related concepts:
average
The arithmetic mean.
##### 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:
DC
Direct current; the unidirectional flow of electric charge.
##### Appears in these related concepts:
direct current
An electric current in which the electrons flow in one direction, but may vary with time.
##### Appears in these related concepts:
electric charge
A quantum number that determines the electromagnetic interactions of some subatomic particles; by convention, the electron has an electric charge of -1 and the proton +1, and quarks have fractional charge.
##### 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:
hertz
In the International System of Units, the derived unit of frequency; one (period or cycle of any periodic event) per second. Symbol: Hz
##### Appears in these related concepts:
Hertz
Measurement of sound frequency.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
magnitude
A number assigned to a vector indicating its length.
##### Appears in these related concepts:
ohm
in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω
##### 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:
power
A measure of the rate of doing work or transferring energy.
##### Appears in these related concepts:
quantity
A fundamental, generic term used when referring to the measurement (count, amount) of a scalar, vector, number of items or to some other way of denominating the value of a collection or group of items.
##### Appears in these related concepts:
resistance
The opposition to the passage of an electric current through that element.
##### Appears in these related concepts:
rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
##### Appears in these related concepts:
rms current
the root mean square of the current, Irms=I0/√2 , where I0 is the peak current, in an AC system
##### Appears in these related concepts:
rms voltage
the root mean square of the voltage, Vrms=V0/√2 , where V0 is the peak voltage, in an AC system
##### Appears in these related concepts:
root mean square
The square root of the arithmetic mean of the squares.
##### 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.
##### Appears in these related concepts:
waveform
The shape of a physical wave, such as sound or electromagnetic radiation.  The shape can be any function that repeats in space.