PHYS345 Electricity and Electronics

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.

Resistor: Voltage, current, and power

Average power dissipated?
The instantaneous power dissipated is

[instantaneous power]

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

[time average]

the average power dissipated is found to be

[power average]

The time average of sine squared may be worked out directly by evaluating the integral implied by the <sin2> in the preceding formula. Alternatively, you may argue that since sin2 + cos2 = 1 and each term has the same shape, shifted by 90 degrees, that it must be true that <sin2> = <cos2> = 1/2. This is a result worth remembering!

[sine squared average]

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 Pav = Irms2 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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Last updated Sept. 17, 1998.
Copyright George Watson, Univ. of Delaware, 1998.