RMS Voltage (Root Mean Square) – Definition, Formula and Significance in AC Circuits

Introduction

The abbreviation RMS stands for Root Mean Square. It is also known as the effective value or heating value
of an alternating current (AC) or voltage. RMS voltage is equal to the value of a DC voltage that would produce the same amount of heat in a resistor as the AC voltage when applied across the same resistance.

RMS voltage is not the same as average voltage. Depending on the shape of the waveform, the relationship between RMS value and peak value can vary. The RMS value is calculated as the square root of the mean of the squared instantaneous values.

Definition of RMS Value

The RMS value of an alternating waveform is defined as the square root of the average (mean) of the squared instantaneous values over one complete cycle.

Mathematically, it is expressed as:

V_RMS =
√[(1 / T) ∫ v²(t) dt]

where v(t) is the instantaneous voltage and T is the time period of one cycle.

RMS Value and Peak Value Relation

It takes a higher peak AC voltage to produce a given RMS value because AC voltage continuously fluctuates with time. For a sinusoidal waveform, the relationship between peak voltage and RMS voltage is:

V_RMS = 0.707 × V_PEAK

For example, to obtain 120 V RMS, the required peak AC voltage is:

V_PEAK = 120 / 0.707 ≈ 169 V

Therefore, a 169 V peak AC signal has the same heating effect as a 120 V DC source when applied to a resistor.

Measurement of RMS Voltage

Most multimeters, whether voltmeters or ammeters, display RMS values while assuming a pure sinusoidal waveform. These are known as average-responding RMS-calibrated meters. True RMS meters, on the other hand, can measure RMS values accurately for non-sinusoidal waveforms as well.

Conclusion

RMS voltage is one of the most important concepts in AC circuit analysis. It provides a true equivalent DC value that reflects the real power and heating effect of an AC signal. This makes the RMS value essential in the design, testing, and measurement of electrical systems.