Molly Ayala
The root mean square (RMS) is a mathematical quantity used in many math fields to compare both alternating and direct currents or voltages. It is the square root of the time average of the voltage squared. The RMS value of an AC voltage or current is the equivalent value of a DC voltage or current that produces the same heating effect in a resistive load. Electrical engineers use the RMS to calculate the power dissipated by an electrical resistance. The calculation is easy when there is a constant current through the resistance. However, if the current is a time-varying function, the formula must be adjusted to account for the varying instantaneous power.
| Characteristics | Values |
|---|---|
| Definition | Root mean square (RMS) is a mathematical quantity used to compare both alternating and direct currents (or voltage) |
| Use | Useful in calculating average power in AC circuits |
| Formula | The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values |
| Formula for continuous function | fRMS = sqrt { [1 / (T2 - T1)] * integralT1^T2 [f(t)]^2 dt} |
| RMS of alternating electric current | Equals the value of constant direct current that would dissipate the same power in a resistive load |
| RMS of AC+DC | sqrt {VDC2 + Vrms2} |
| RMS of AC only | RMS(signal) = stdev(signal) if the mean signal is 0 |
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What You'll Learn
Calculating the AC only RMS of a signal
The root mean square (RMS) is a mathematical quantity used in various fields, including electrical engineering, to compare both alternating and direct currents or voltages. It is particularly useful for calculating the average power in AC circuits. The RMS value of a set of values is the square root of the set's mean square. In other words, it is the square root of the mean (average) value of the squared function of the instantaneous values.
The RMS value of an AC waveform is the amount of AC power that produces the same heating effect as DC power. This is calculated by taking the square root of the time average of the voltage squared. For example, the RMS voltage of a 60-hertz, 120-volt alternating current is 170 volts. This is calculated as 120/0.707, or 170 volts, which is the RMS voltage.
The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. This can be done through the analytical method, which is a mathematical procedure for finding the RMS value of any periodic voltage or current using calculus. The positive half of the waveform is divided into any number of equal portions or mid-ordinates, and the height of each mid-ordinate is equal to the instantaneous value of the waveform at that time along the x-axis. Each mid-ordinate value is then squared and added to the next.
In the common case of alternating current when I(t) is a sinusoidal current, the RMS value is easy to calculate from the continuous case equation. If Ip is defined as the peak current, then:
> I_RMS = sqrt[(1 / (T2 - T1)) * integral from T1 to T2 of [Ip * sin(ωt)]^2 dt]
Where t is time and ω is the angular frequency.
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RMS value of a set of values
The root mean square (RMS) is a mathematical quantity used in many fields, including electrical engineering. It is particularly useful in carrying out power calculations, and is used to compare both alternating and direct currents (or voltage).
The RMS value of a set of values is the square root of the arithmetic mean of the squares of the values. In other words, it is the square root of the set's mean square. The formula for the RMS value of a set of values is:
> ! [x_{\text{RMS}}={\sqrt {{\frac {1}{n}}\left({x_{1}}^{2}+{x_{2}}^{2}+\cdots +{x_{n}}^{2}\right)}}](https://katex.org/cgi-bin/server.cgi?symbolstyle=unicode&format=png&dpi=150&latex=x_%7B%5Ctext%7BRMS%7D%7D%3D%7B%5Csqrt%20%7B%7B%5Cfrac%20%7B1%7D%7Bn%7D%7D%5Cleft%28%7Bx_%7B1%7D%5E%7B2%7D%2Bx_%7B2%7D%5E%7B2%7D%2B%5Ccdots%20%2B%7Bx_%7Bn%7D%7D%5E%7B2%7D%5Cright%29%7D)
The RMS value of a set of values can be calculated by first squaring the values, then taking the average of the squared values, and finally taking the square root of the average.
In the context of electrical engineering, the RMS value of voltage is used to characterise the source of electromotive force. It is also used to determine the equivalent DC value of an AC waveform.
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RMS of an alternating electric current
The root mean square (RMS) is a mathematical quantity used in many fields, including electrical engineering. It is particularly useful in calculating average power in AC circuits. The RMS of an alternating electric current is equal to the value of constant direct current that would dissipate the same power in a resistive load.
RMS is used to compare both alternating and direct currents or voltages. The RMS value of AC current is the direct current that, when passed through a resistor for a given period, would produce the same heat as the alternating current when passed through the same resistor for the same time. This is useful for electrical engineers in calculating the "AC only" RMS of a signal.
The RMS value of alternating current is given by direct current flowing through a resistance. The RMS value of a sine current wave can be determined by the area covered in half-cycle. This is applicable to all waves, including sinusoidal, non-sinousoidal, symmetrical, and asymmetrical waves. The RMS value of AC is greater than the average value.
To calculate the RMS of an alternating electric current, you can sample the current at tiny intervals of time. Square each value, add up the squares (which are all positive), and divide by the number of samples to find the average square or mean square. Finally, take the square root of that value, and you have the RMS average value.
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RMS of waveform combinations
The root mean square (RMS) is a mathematical way to find the DC equivalent voltage of an AC sinusoidal waveform. It is a common calculation used to define the voltage or current of an AC waveform. The term is only applied to sinusoidal voltages, currents, or other complex waveforms and not in DC applications where the voltage is constant over time.
RMS is also used in completing power calculations. The RMS value of an alternating voltage or current waveform can be calculated by first determining the average value (VAVG) of an alternating voltage. The RMS value of a signal is calculated as the square root of the average of the squared values of the signal. The RMS value of a sinusoidal voltage (V(t)) can be calculated by integrating through with limits taken from 0 to 360 degrees or 'T', the period. This gives:
> Vm = Vmax * cos(ωt)
Dividing through further as ω = 2π/T, the complex equation above eventually reduces to:
> VRMS = Vm * 0.7071
The RMS value of a signal can also be computed using frequency domain components.
The RMS value is always greater than or equal to the average, as the RMS includes the squared deviation (error) as well. The RMS value is also the same as the standard deviation when the signal has a zero-mean.
A special case of the RMS of waveform combinations is:
> RMS(AC+DC) = sqrt(VDC^2 + RMS(AC)^2)
Where VDC refers to the direct current (or average) component of the signal.
In the case of a time-varying voltage, V(t), with an RMS value of VRMS, the average power can be found using the same method as for any periodic waveform.
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RMS voltage and current
The root mean square (RMS) is a mathematical quantity used in many math fields, including electrical engineering. It is used to compare both alternating and direct currents (or voltage). The RMS value of an alternating current (AC) is the direct current (DC) that, when passed through a resistor for a given period, produces the same heat as the AC passed through the same resistor for the same time.
RMS voltage, also referred to as the effective value, depends on the magnitude of the waveform and is not a function of either the waveform's frequency or its phase angle. The RMS value of a sinusoidal source of electromotive force (Vrms) is used to characterise the source. The RMS voltage of a sinusoidal waveform is the same heating effect as an equivalent DC power. The RMS value is the square root of the mean (average) value of the squared function of the instantaneous values.
The RMS value of AC voltage is useful in AC circuits (which are linear as opposed to rectifier circuits) as it is used to find the time-average power delivered. The rms values of current and voltage multiplied together give the actual power. This is a vital fraction when trying to do quantitative power and energy experiments such as specific thermal capacity.
To calculate the RMS value, you need the average value of sin^2 as time goes on. The graph of sinωt and the graph of cosωt look the same, except for a shift of origin. Because they are the same pattern, sin^2ωt and cos^2ωt have the same average as time goes on. But sin^2ωt + cos^2ωt = 1. Therefore the average values of either of them must be 1/2. Therefore, the RMS value of I0sinωt must be I0√ 2. The RMS value is 0.707 times the peak value, and the peak value is 1.41 times the value the voltmeter shows.
For example, the voltage across a lamp was 6 volts (V). The energy transferred each second, measured in watts (W), was 36 watts. With direct current, an AC supply would need to be set to 12 V rms to achieve the same 36 W power output. The rms voltage of an AC signal is always less than peak voltage because the peak voltage only occurs twice in the cycle (one positive, one negative), and the rest of the cycle is less.
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Frequently asked questions
The root mean square (RMS) of a set of values is the square root of the set's mean square.
The RMS value of an AC voltage is the equivalent value of a DC voltage that produces the same heating effect in a resistive load. The average voltage or current is irrelevant but the average power is key.
The RMS value of a waveform is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal.
