RMS and Average Values of a Sine Wave

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Alternating current flows periodically first in one direction and then in the opposite direction. One direction is called a positive alternation and the other direction is called a negative alternation. A complete positive and negative alternation is called one cycle. The number of complete cycles that occur each second is the frequency and is designated in hertz, abbreviated Hz. Therefore, if one complete cycle occurs per second, the frequency is 1 Hz; if 5 cycles are completed per second, the frequency is 5 Hz, and so on.

An ac wave can have many wave shapes; for example, it can be a sine wave, square wave, sawtooth wave, etc. AC meters are calibrated based on sine waves. When an ac meter is used to measure non-sinusoidal waveforms, only an approximate indication of values is obtained. Sometimes the indicator could be so far off that the reading is meaningless. Therefore, other measuring instruments such as oscilloscopes should be used instead of ac meters to measure non-sinusoidal waveforms.

RMS and Average Values of a Sine Wave

The root-mean-square (RMS) value of a sine wave is very important in the study of meters. The basic electrical units, that is, the ampere and the volt, are based on de. Therefore, a method had to be derived to relate AC to DC. The maximum, or peak, value of a sine wave could not be used because a sine wave remains at its peak value for only a very short time during an alternation. Thus, a sine wave with a peak current of 1 ampere is not equal to a dc current of 1 ampere from an energy standpoint since the dc current always remains at 1 ampere.

A relationship based on the heating effects of ac and dc was derived. It was found that a current equal to 0.707 of the peak ac wave produced the same heat, or lost the same power, as an equal DC current for a given resistance. For example, a sine wave with a peak value of 3 amperes has a heating effect of 0.707 x 3 or 2.121 amperes of de.

The value of 0.707 can be derived in the following manner: The heating effect of current is based on the basic power formula; that is, P = 12 R, where P is the power dissipated as heat. From the formula, you can see that the heat varies as the square of the current.

When a sine wave reaches its peak value, the heat dissipated becomes maximum. Lesser heat values are dissipated for all values of current below the peak value. To find the heat dissipated during an entire sine wave cycle, each instantaneous value of current is first squared and then added. Then the mean (or average) of this sum is found. After this, the square root of the mean is found, and the answer is called the root-mean-square (RMS) value of the sine wave. Often the RMS value of a sine wave is called the effective value because 0.707 of the peak value of a sine wave has the same effect as an equal amount of de.

Another sine wave characteristic that is important in the study of meters is the average value of the sine wave. The average value is obtained during one alternation and is equal to 0.637 of the peak value of the sine wave.