Root-Mean-Square Value of Voltage and Current

Applying a sinusoidal voltage u(t) to a resistive load R causes the following current to flow through the load according to Ohm's law:

Because the voltage and current are time-dependent variables, so is the power produced in the resistor. It is defined by the following equation:

The diagrams below show the time characteristics of an AC voltage and current (upper diagram) along with the power (lower diagram).

The area enclosed between the power curve and time axis is a measure of the electrical energy converted by the resistor into heat. If a horizontal line is drawn parallel to the time axis at a height of p0/2, the areas above and below this line respectively (shaded in matching colours below) are equal in size. An average power p0/2 ascertained in this manner over several periods of oscillation would perform the same amount of work as the continuously changing instantaneous power p(t) does. This is illustrated by the diagram below.

The following animation demonstrates this relationship.

A DC voltage U that would be needed to develop the same power as the AC voltage in the resistor is determined as follows:

Resolving this equation in terms of the DC voltage U gives

This voltage U is termed the root mean square value of the alternating signal. Because it is a time-independent variable, it is designated in uppercase just like a direct voltage. Root mean square values of alternating current are specified in the same manner. In other words:

The rms values U and I of AC voltage and current in a resistor R develop the same power P as a DC current I and voltage U of equal magnitude.

The relationship between the rms and peak values of current and voltage in the case of the sinusoidal variables considered here is given by the following equations:

Accordingly, the rms value of a voltage or current is about 70% of the peak value.

Example: a mains voltage with an rms value U = 220 V has a peak value

Naturally, the rms values of non-sinusoidal periodic signals like triangular and rectangular forms can also be defined. In such cases, however, the mathematical relationship (i.e. conversion ratio) between the rms and peak (amplitude) values varies in accordance with the signal shape under consideration.

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