what is root mean square voltage? and why is there a root mean square voltage and a peak voltage? for instance, 120v ac has an RMS voltage of 120v and has a peak voltage of 170 volts
In: Physics
So root mean square is just an average but we get rid of the minuses before we take the average.
So AC has positive and negative components that are equal to each other, if we actually average this at all points we get 0 volts, clearly 0 volts of electricity is not going to power everything so something must be wrong.
The trick is that electricity does work in both directions, whether current is flowing forward or backwards it still will light up a lightbulb or heat your water.
So for all useful measures of say energy use or something, we care about the total amount voltage with both positive and negative voltages contributing positively. To get rid of these negatives, we square the voltage first (since negative times negative = positive), take the average, then square root it to undo the square.
AC power isn’t continuous; it’s a sine wave, which goes back and forth across zero dozens of times per second (60 hz and 50 hz are common, which is 60 and 50 cycles per second). So the voltage goes from 0 to 120, back to 0, to -120 (which for things designed for AC power is equivalent to 120), to 0, to 120, etc. very very fast.
Root mean square is the method to figure the average voltage (you can also do it for average power) of that cycle. For a sine wave with a peak of 170v, the average is 120v.
You can’t just take the average over time of an alternating current, because it keeps changing direction, so over time it will average to zero, and that doesn’t tell you how much voltage there is.
What you can do is square the voltage, so it all becomes positive, then take the average, then square root it again so it’s in units of voltage. This quantity will be representative of the amount of voltage you have, and it’s related to both the peak voltage and the waveform.
If the voltage was constant DC, then the rms would just be the average. Because it alternates, the times it goes through zero will bring the average down, and the rms will come out less than the peak voltage.
For sine waves, which are the easiest waveform to generate and the one you get out of your wall, it’s a relatively simple calculation to show that the rms is the peak divided by the square root of 2. So that’s where those specific numbers come from.
It is the equivalent voltage of alternating current that delivers the same amount of energy as direct current at that voltage.
Alternating currents in the power grid are sine waves that form from plus to minus peak voltage. in the shape of a sine wave. DC and AC look like https://www.matsusada.com/column/uploads/dc_vs_ac_img_dcac.png
If you look at how a load like a simple resistive heater is heated. For DC it is simple Power = voltage * current.
But with AC the voltage and current constantly change and are sometimes zero. If you just take the peek value and multiply them you get a to high value. What you can do is take the voltage and current at each moment in time and multiply them. If the times steps are identical you can calculate the average power as the sum of all of them divided by the number of steps. This is really quite impractical.
If you look at the maths and how electricity works you will notice that we can drive what DC voltage provides the same power. Why it just is the RMS (root mean square) is not that relevant to what matters if we have found a way to get this voltage?
For sine waves RMS voltage is Peek voltage / square root of 2. The square root of two is approximately 1.41. for 120V the result is 120*1.41 = 169.2V. It is rounded to 170V. Grid voltages have an allowed range, it is +-5% in the US so 114 V to 126 V, so a tiny 0.8V rounding do not matter.
So the RMS voltage and current is the useful values for AC current because it you can use it as a DC voltage and current would be used, at least for resistive loads.
It is quite similar to how if you drive in a city 5 km with lots of other traffic and, so the speed varies. The top speed is not relevant if you look at travel time, it is the average speed that matters. The average speed is like the RMS voltage and the top speed is like the peak voltage.
The top speed can be relevant too like if a cop gives you a speeding ticket because it was to high. For the same reason, peek voltage is relevant for some design parameters of devices like distance to separate the high and low voltage side of a device. But as a usage of the power grid just RMS is enough.
When you have AC in a circuit, the electrons are being pushed forward for a fraction of a second, and then backwards for a fraction of a second. Between each of those movements, there’s a moment that the force is slowing down and reversing direction, and it’s not accomplishing much.
If AC power is spending some of its time at 170 volts, and some of its time at 0 volts during the transitions, we need a way to figure out how the power output compares to a steady voltage in one direction. Because power is proportional to voltage ***squared***, we look at the voltage squared over time, add it all up, and take the square root of it to find the equivalent voltage.
Looking at it very roughly, 120^2 + 120^2 (constant voltage at time #1 and time #2) is very close to 170^2 + 0^2 (varying voltage between a max at time #1 and nothing at time #2). So the total power of a sine wave with a max of 170 voltage is close to a flat line at 120 voltage.
AC is wibbly-wobbly. It makes a nice up-down curve.
It goes from minus some voltage up to plus some voltage.
So at points it’s far higher than 120V and that’s the peak voltage.
But because it’s up and down all the time, sometimes there’s no voltage at all. When it’s all averaged out, those wibbly wobblies would “flatten” to be at the RMS voltage.
Imagine a series of hills and valleys and then you level them out to one flat surface. The height of the levelled-out land is the RMS voltage. The height of the top of the hill before you start is the peak voltage.
In effect the voltage will vary between +170V and -170V, but the amount of actual power if you average it out over time is about 120V.
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