RMS Voltage

The RMS or effective value of a sinusoidal waveform is the value of a steady DC voltage or current that produces the same heating (average power) in a resistive load as the AC waveform.

• The term RMS stands for Root-Mean-Squared. It is defined as the square root of the mean (average) of the square of the instantaneous values of a time-varying quantity. Symbols commonly used are V_RMS and I_RMS.
• An RMS value represents the equivalent DC magnitude in terms of power delivery and is often written as V_eff or I_eff when emphasising the effective (power-equivalent) value.
• The RMS concept applies to time-varying voltages and currents (sinusoidal or non-sinusoidal). For a steady DC quantity the RMS equals the DC value, so the concept reduces to the same magnitude for DC.
• For practical reference, the domestic mains voltage stated as 240 V (in the UK) is the RMS value of the sinusoidal supply. That means the AC waveform at the socket delivers the same average heating power as 240 V of steady DC.

RMS Voltage Equivalent

There are two common methods to determine the RMS value of a periodic waveform:

• Graphical method - useful for any waveform including non-sinusoidal shapes; it uses equally spaced instantaneous values (mid-ordinates) taken over a cycle.
• Analytical method - a mathematical procedure using calculus (integrals) to compute the mean of the squared instantaneous function for periodic waveforms; particularly simple for pure sinusoids.

RMS Voltage - Graphical Method

The graphical method approximates the RMS value by sampling the waveform at equally spaced instants (mid-ordinates), squaring each sample, averaging the squares, and then taking the square root of that average. This method works well for arbitrary or complex waveforms.

• Divide one period (or one half-cycle for symmetric waveforms) into n equal portions and record the instantaneous values at the mid-ordinates.
• Square each mid-ordinate value. This yields the "square" part of RMS.
• Take the arithmetic mean (average) of those squared values. This yields the "mean" part of RMS.
• Take the square root of that mean to obtain the RMS value - the "root" part.

For a discrete set of n mid-ordinate voltages v1, v2, ..., vn the RMS is:

Example: assume an alternating waveform has a peak voltage of 20 V and ten mid-ordinate values over one half cycle are taken as shown:

Using the graphical procedure, the RMS value computed from those mid-ordinates is:

The result of the graphical calculation for this example is 14.14 V.

RMS Voltage - Analytical Method (Sinusoidal)

For a pure sinusoid the RMS value can be derived by integration over one period. Let the instantaneous voltage be V(t) = V_m cos(ωt), where V_m is the peak (maximum) value and the period is T.

The RMS value definition for a continuous periodic function is the square root of the average of the square of the function over one period:

Evaluating the integral over one full period gives:

Carrying out the integral and simplifying (using ω = 2π/T) leads to the well-known result:

Thus for a sinusoid the RMS value is equal to the peak value divided by √2 (numerically ≈ 0.7071 × V_pk). The RMS value depends only on the amplitude of the sinusoid and is independent of frequency and phase.

Using the earlier example with V_pk = 20 V:

V_RMS = V_pk × 0.7071 = 20 × 0.7071 = 14.14 V

Both the graphical and analytical methods give the same numerical result for a sinusoidal waveform. The 0.7071 factor applies only to pure sinusoids. Non-sinusoidal waveforms require the graphical method or an equivalent integral method to compute RMS exactly.

Sinusoidal RMS Relationships

Common relations for a sinusoidal voltage are:

• V_RMS = V_pk / √2
• V_pk-to-pk = 2 V_pk
• Average (mean) value over a full cycle is zero for a pure sinusoid, so average value must be treated differently from RMS.

RMS Voltage - Summary

• To represent the magnitude of an alternating voltage or current the peak value and the RMS (effective) value are commonly used. The RMS (root-mean-square) value represents the equivalent DC magnitude in terms of heating (power) effect.
• The RMS value is not the same as the arithmetic average of instantaneous values; RMS involves squaring, averaging, and then taking a square root.
• The ratio V_RMS/V_max equals I_RMS/I_max for corresponding voltages and currents of the same waveform shape.
• Most standard handheld meters display RMS assuming the measured waveform is sinusoidal. For accurate measurement of non-sinusoidal waveforms a True-RMS multimeter is required.
• The heating power dissipated in a resistor R by a DC current I is I²R. For an alternating current the average power dissipated is I²_RMSR.
• When working with AC quantities they are normally quoted and used as RMS values unless stated otherwise. For example, an AC current of 10 A (RMS) produces the same heating as a DC current of 10 A and has a peak value of 10 × √2 ≈ 14.14 A.
• After finding the RMS value, other useful quantities such as the average value, peak-to-peak value, and instantaneous expressions can be calculated where required.