Vrms

Vrms, or root mean square voltage, is a measure of the effective magnitude of an alternating voltage. For a periodic waveform v(t) with period T, Vrms is defined as the square root of the average of the squared voltage over one period: Vrms = sqrt( (1/T) ∫_0^T [v(t)]^2 dt ). This value represents the equivalent direct current (DC) voltage that would deliver the same average power to a resistor.

In circuits, Vrms is used together with Irms to calculate power. The real power is P = Vrms × Irms × power factor. In a purely resistive circuit, P = Vrms^2 / R. Thus Vrms provides a practical measure of voltage when comparing electrical heating or energy delivery, independent of waveform shape.

For a sinusoidal voltage v(t) = Vm sin(ωt), the RMS value is Vm/√2, where Vm is the peak amplitude. This relationship shows why RMS values are typically smaller than peak values and are the standard for specifying AC voltages such as mains electricity.

RMS values depend on waveform shape. While the sine wave has a simple closed-form Vrms, non-sinusoidal waveforms require the general definition or true RMS measurements to determine their effective voltage. Modern meters labeled as true RMS can accurately capture Vrms for complex or distorted signals.

In everyday use, Vrms is the conventional way to express AC voltages, such as household mains, enabling direct comparison to DC-equivalent power delivery.