Root Mean Square




Now, when measuring the signal's mean power with a resistor & thermometer we don't always know anything about the signal pattern — i.e. we don't always know if it's d.c., a sinewave, or more complicated. All we know are a power and a resistance. A given measured average power, P, could be produced by any signal pattern whose average squared voltage was . The three signals (d.c., sinewave, & complicated) shown in figure 4.1 can now all be said to have the same ‘size’ if they produce the same amount of power — i.e. when

equation


Hence this technique of measuring a mean power gives us a way to compare the sizes of signals without having to worry about the details of their variations.

fig2.gif - 19Kb


From equation 7 we can now define a voltage, , calculated from the measured power

equation


i.e.

equation


This quantity, , can be used to represent the size of the signal's voltage no matter what the signal pattern is. The rms stands for root mean square and indicates that it is a value which depends upon the square root of the time average (mean) of the voltage squared. Root mean square voltages (and other quantities, like currents) are a convenient way to represent the signal size since we don't have to bother with the signal's shape. As a result rms voltages and current values are widely used to indicate the sizes of signals. In effect, when we say a given signal has an rms voltage level, , we're actually saying that it carries the same power as a steady d.c. voltage of volts. For a sinewave we can expect that . Since we've already found that we can use an oscilloscope to determine a sinewave's rms voltage by noting the peak-to-peak voltage an using

equation


Note, however, that equation 10 is only correct for sinewaves. Waves like squarewaves or trianglewaves have different ratios. Some waveforms — e.g. the complicated one shown in figure 4.1c on the previous page — don't even have a unique peak-to-peak value!




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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.