SAY IT WITH ME: ROOT-MEAN-SQUARE

If you measure a DC voltage, and want to get some idea of how “big” it is over time, it’s pretty easy: just take a number of measurements and take gennemsnittet. If you’re interested in the average power over the same timeframe, it’s likely to be pretty close (though not identical) to the same answer you’d get if you calculated the power using the average voltage instead of calculating instantaneous power and averaging. DC voltages don’t move around that much.

Try the same trick with an AC voltage, and you get zero, or something nearby. Hvorfor? With an AC waveform, the positive voltage excursions cancel out the negative ones. You’d get the same result if the flip were switched off. Clearly, a simple average isn’t capturing what we think of as “size” in an AC waveform; we need a new concept of “size”. enter root-mean-square (RMS) voltage.

To calculate the RMS voltage, you take a number of voltage readings, square them, add them all together, and then divide by the number of entries in the average before taking the square root: . The rationale behind this strange averaging procedure is that the resulting number can be used in calculating average power for AC waveforms through simple multiplication as you would for DC voltages. If that answer isn’t entirely satisfying to you, read on. hopefully we’ll help it make a little more sense.

Nødvendighed

When it comes to averages, the ideas of “big” and “little” for AC and DC voltages are fundamentally different. DC waveforms are roughly constant, and what matters is the distance from zero. AC waveforms are always wiggling around a center point, and this is often ground. If the waveform is symmetric, and you take enough samples, it’s going to average out to zero.

Average Power close to Power at average Voltage

Average Voltage = 0, average Power Not Zero

One way to measure the size of AC voltages is to take the maximum and minimum over time: the peak-to-peak voltage. another possibility would be to take the absolute value of each voltage and average them together. That works too. A third choice is to square all of the individual voltage measurements before adding them up. This has the same effect as taking the absolute value — all of the individual terms are positive now and don’t cancel out — and has the additional side-effect of making the big values bigger and the small values smaller. Which do we choose?

Physics

Using the squared voltages in the average gets the physics right. If you’re interested in the power that you can get out of the AC signal, it’s the squares of the voltage that are relevant anyway. Let’s pretend you’re driving a resistive load for now — maybe you’re heating your apartment or using an electric stove — and do a tiny bit of algebra.

Remember that power is equal to the current flowing through our imaginary device times the voltage being dropped across it: P = IV. and who could forget Ohm’s Law? V = IR or I = V / R. put them together, and P = V² / R. The power in the system, at any given instant, is proportional to the voltage squared. The average power over time is thus proportional to the average of the squared voltages. Sounding familiar? since the average of squared instantaneous voltages is in units of volts squared, taking the square root at the end (“root of the mean of the squares”) brings it on home.

The same logic holds for RMS current measurements as well. Substituting Ohm’s law the other way, you get P = I² R and power is proportional to current squared. average current in a balanced AC waveform is zero, but RMS-averaged current, squared, is proportional to power.

By [AlanM1], Public DomainAgain, the big takeaway is that RMS voltage is the measure of average AC voltage or current that lets you pretend it was a DC average to get the average power. By doing the squaring inside the average, you avoid voltages of opposite signs cancelling, and by taking the square root at the end, it gets the units right.

If you have an AC voltage that’s riding on top of a DC component, the RMS value still delivers. in that case, the squared DC component adds up n times before dividing by n again, and you get something like this: , where v is just the pure AC voltage.

Tommelfingerregler

One place you’ll see RMS voltages is in mains power. Indeed, the 120 V in the us (or 230 V in the EU) coming out of your walls right now is an RMS figure. For sine waves, like what you get from the electrical company, the peak voltage is a factor of sqrt(2) higher than the RMS voltage. The peak voltage in the states is something like 120 V * sqrt(2) = 170 V, and the peak-to-peak is 340 V. That’s 650 V peak-to-peak in Europe; yikes!

This also means that if you’re lacking an RMS meter and need a quick-and-dirty estimate of something that’s sine-wave-like, you can take the amplitude and divide by 1.414, or take the peak-to-peak and divide by twice that.

Another waveform you mJeg er opmærksom på, at den PWM’ed Square Wave, som vi ofte bruger til at køre motorer fra mikrocontrollere. Det er klart, at hvis du skifter mellem nulvolt og tolv volt, leverer den kun strøm til motoren, når den er på tolv volt. Tilsvarende vil du ikke blive overrasket over at høre, at rms spænding af en PWM Waveform er den firkantede rod af pligtcyklusen gange spændingen.

Wikipedia har du dækket af trekantbølger og andre sjove bølgeformer.

Rms overalt

Det viser sig, at du ofte er bekymret for kvadratiske mængder. Kinetisk energi er proportional med hastighed kvadreret, for eksempel så RMS hastighed anvendes til beregning af temperatur fra den gennemsnitlige hastighed af molekyler i en gas. Hvis du har en måleprocedure, der kan være i gennemsnit, men du er bekymret for spredningen af ​​resultaterne, kan du måske lide at minimere RMS-fejl. Statistikens koncept for standardafvigelse er ens, med den gennemsnitlige værdi subtraheret på forhånd.

Du beregner endda hypotenuse af en trekant efter samme procedure, bare uden at dividere med n. (Ok, det er en strækning, men firkantede rødder af summer af firkanter er overalt!) Jeg vil forlade det til de matematiske filosoffer blandt jer at duke det ud i kommentarerne om, hvorfor L2-normen virker så ofte. For de elektriske hackere derude er det nok at huske Ohms lov begrundelse: Når du er interesseret i magt, er du interesseret i firkanter.