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 den gjennomsnittlige. 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.
Nødvendighet
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.
Tommelfingerregel
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 might Care om er PWM’ed Square Wave som vi ofte bruker til å kjøre motorer fra mikrokontroller. Klart, hvis du skifter mellom null volt og tolv volt, leverer det bare strøm til motoren når den er på tolv volt. Tilsvarende vil du ikke bli overrasket over å høre at RMS-spenningen til en PWM-bølgeform er kvadratroten av pliktsyklusen ganger på spenningen.
Wikipedia har du dekket for trekantbølger og andre morsomme bølgeformer.
Rms overalt
Det viser seg at du ofte er opptatt av kvadratiske mengder. Kinetisk energi er proporsjonal med hastighetsspørsmål, for eksempel, slik at RMS-hastigheten brukes til å beregne temperaturen fra gjennomsnittshastigheten til molekyler i en gass. Hvis du har en måleprosedyre som kan være rett i gjennomsnitt, men du er bekymret for spredningen av resultatene, kan du også like å minimere RMS-feilen. Statistikerens konsept om standardavvik er lik, med gjennomsnittsverdien trukket av på forhånd.
Du beregner selv hypotenuse av en trekant med samme prosedyre, bare uten å dele med n. (OK, det er en strekk, men kvadratrøtter av summer av firkanter er overalt!) Jeg skal legge det til matematiske filosofer blant dere for duke det ut i kommentarene til hvorfor L2-normen vises så ofte. For de elektriske hackere der ute, er det nok å huske Ohms lovgradering: Når du er interessert i makt, er du interessert i firkanter.
