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Concept
Moyenne quadratique
Résumé
In mathematics and its applications, the root mean square of a set of numbers (abbreviated as RMS, or rms and denoted in formulas as either or ) is defined as the square root of the mean square (the arithmetic mean of the squares) of the set.
The RMS is also known as the quadratic mean (denoted ) and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted ) can be defined in terms of an integral of the squares of the instantaneous values during a cycle.
For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load.
In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.
The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In physics, the RMS current value can also be defined as the "value of the direct current that dissipates the same power in a resistor."
In the case of a set of n values , the RMS is
The corresponding formula for a continuous function (or waveform) f(t) defined over the interval is
and the RMS for a function over all time is
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous.
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