**Root-Mean-Square Quantities**

Flow of energy from emf to resistor in RLC circuit

Source of energy is the emf (ac generator).

Energy is disspated by resistor.

Average power dissipated?

The instantaneous power dissipated is

Using the definition of a time average of a periodic function over the period T

the average power dissipated is found to be

The time average of sine squared may be worked out directly by evaluating the integral
implied by the <sin^{2}> in the preceding formula.
Alternatively, you may argue that since sin^{2} + cos^{2} = 1 and each term
has the same shape, shifted by 90 degrees, that it must be true that
<sin^{2}> = <cos^{2}> = 1/2.
This is a result worth remembering!

rms quantities

In electrical and electronics contexts,
root-mean-square or rms quantities are conventionally used to indicate
the relative strength of ac signals rather than the magnitude of the phasor.
rms quantities simplifies the average power formula, that is,
<P> or P_{av} = *I*_{rms}^{2} R,
absorbing the factor of 1/2 in the formula for the average power stated earlier.

The rms value of any periodic time-varying function may be found by squaring the function, evaluating the mean or time-average of the squared function, and using the square root as the result. Note that for sinusoidal functions, the time average of the signal is simply zero, clearly not a useful representation of the function's magnuitude. The rms value of a sinusoidal function is 1/sqrt(2) = 0.707 times the magnitude of the function as shown below.

Multimeters used in ac mode report rms values. For example, the electrical power we commonly encounter at the wall socket is typically 110 to 115 V rms at 60 Hz. An rms emf of 110 V corresponds to a magnitude of 156 V or peak-to-peak value of 312 V.

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"http://www.physics.udel.edu/~watson/phys345/class/6-rms.html"

Last updated Sept. 17, 1998.

Copyright George Watson, Univ. of Delaware, 1998.