Peak, Average and RMS


Maybe the following explanation will clear up some of the confusion over the terms peak, average and RMS voltage and current, peak versus average power, why RMS voltage and current are not the same as average voltage and current, and why average power is not calculated by multiplying average voltage times average current.

The root-mean-square (RMS) value of an alternating voltage or current is defined as the equivalent DC voltage or current that would deliver the same amount of energy to a resistor as the AC does over a complete cycle.

The formulae given in the textbooks for alternating currents assume a sine wave.  The sine waveform as seen on an oscilloscope can be thought of as the projection of a circular rotation, viewed edge-on as it moves steadily across the field of vision, to a two-dimensional plane.  Mathematically this is called a “rotating vector”.  This is where all the trigonometric calculations involving angles, degrees and π (pi) come from.  

Another term for R.M.S. is the quadratic mean. That is, the mean, or average, of a varying quantity taken to the second power.  Why the second power (square)?  Simple Ohm's law.  The power dissipated in a resistor is equal to the square of the voltage across the resistor, divided by the resistance (V˛/R), or the square of the current through the resistor multiplied by the resistance (I˛R).

For to-day’s digitally-minded in particular, this might be easier to visualise if we consider a pulsating DC voltage varying between some fixed value and zero, instead of a continually varying sinusoidal AC voltage for which we have to maintain the mental image of the rotating vector simultaneously to pondering all the peaks, averages, squares and square roots.  

Imagine that we have a battery that puts out exactly 2 volts, a 1Ω resistor and a SPST switch, as shown in the attached diagram. Imagine someone with superhuman reflexes opening and closing the switch precisely every millisecond, so that we have a series of square pulses, each exactly one millisecond in length, separated by a space of one millisecond.  Or think of the switch as an electronic keyer with the dit paddle held down and the speed set to 500 dits per second. See the attached drawing.

When the switch is closed, the voltage across the resistor is 2 volts. That would be the peak voltage.  But since the switch is open 50% of the time, the average voltage across the resistor is only 1 volt.  Connect an electromechanical analogue DC voltmeter across the resistor and it will read exactly 1 volt.  

But how hot does the resistor get?  Look at the power dissipated in the resistor.  While the switch is closed, we have 2 volts across 1 ohm.  As stated above, power (P) = V˛/R.  So we are dissipating 4 watts in the 1Ω resistor while the switch is closed.  This is the peak power. But since the switch is open 50% of the time, the average power dissipated in the resistor is 2 watts.

So, what is the RMS voltage across the resistor?   The equivalent steady DC voltage that would deliver the same quantity of energy per second (average power) to the resistor as does our pulsating DC, can be calculated, since we know the average power dissipated in the resistor.  What steady DC voltage would get our resistor just exactly as hot as our pulsating 2 volts?  Rearranging the equation P = V˛/R gives us V˛ = PR. We have shown that the average power is 2 watts and our resistor is 1ohm.  So V˛ = 2 watts X 1Ω, or 2 volts˛.  The RMS voltage, that is, equivalent steady DC voltage that would have the same heating effect on our resistor is the square root of V˛:  Vrms = √2, or 1.414 volts.  

So we see that our peak voltage is 2 volts, our average voltage is 1 volt and our RMS voltage is 1.414 volts.

Our peak power is 4 watts, and our average power is 2 watts.  Note that the term "RMS power" is not used; there is no such thing.

Where does the term “RMS” (root means square) come from?  In the circuit described above we tried to clarify matters by using a 1Ω resistor, so that power (P) is simply equal to the voltage squared. RMS is the square root of the mean (average value) of the square of the varying quantity, as it is taken through one complete cycle.

I just ran across this little titbit while doing a search, regarding rms and average voltmeters and ammeters.

The only true rms-reading meter capable allowing accurate measurement of average output power, until recently, was the thermocouple rf ammeter, but with the advent of microprocessor controlled instruments, for example the Bird APM-16, a true rms reading voltmeter is now possible. The scale of this meter may be calibrated to directly read watts in the case of a specifically defined non-reactive load, 50 ohms for example.

RF wattmeters like the Bird 43, and the Hammy Hambone wattmeters cannot accurately measure average power except for a steady sine wave, since these are really average-reading voltmeters with a scale calibrated in watts, based on the assumption of a specifically defined non-reactive load.  This type of meter will accurately read the average output power only from a frequency-modulated or unmodulated carrier. Any other envelope waveform requires a true rms-reading instrument, since average (or mean) power is a product of rms voltage times rms current, NOT average voltage times average current, as already explained in the original post above.

Root mean square (RMS) amplitude is used especially in electrical engineering: the RMS is defined as the square root of the mean over time of the square of the vertical distance of the graph from the rest state.

For complex waveforms, especially non-repeating signals like noise, the RMS amplitude is usually used because it is both unambiguous and has physical significance. For example, the average power transmitted by an acoustic or electromagnetic wave or by an electrical signal is proportional to the square of the RMS amplitude (and not, in general, to the square of the peak amplitude).

For alternating current electrical power, the universal practice is to specify RMS values of a sinusoidal waveform... Some common meter types used in electrical engineering are calibrated for RMS amplitude, but actually operate on a DC input. Both digital voltmeters and moving coil meters are in this category. Such meters require the AC input to be first rectified. They are not true RMS meters, but rather, are reading proportional to either rectified average or peak amplitude. The RMS calibration is only correct for a sine wave input since the ratio between peak, average and RMS values is dependent on waveform. Until recently, true RMS meters were mostly used only in radio frequency measurements. These instruments based their measurement on detecting the heating effect in a load resistor with a thermistor. The advent of microprocessor controlled meters capable of calculating RMS by sampling the waveform has made true RMS measurement commonplace.

Here is another good link for a discussion of peak, average and rms values of a.c. waveforms.

Why there is no such thing as 'RMS watts' or 'RMS power' and never has been:


[0] Message Index

AMfone - Dedicated to Amplitude Modulation on the Amateur Radio Bands