the maths behind one phase of 415V 3 phase power being 240V and not whatever 415/3 is
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A magnet has a north pole and a south pole. Imagine there is a line that goes through the magnet in the north-south direction.
When you move a wire near the magnet things happen.
If you move the wire such that it crosses that line, electric current is generated in the wire.
If you move the wire perpendicular to this line, no current is generated through the wire.
If you move it in a different direction, it will generate a weaker current, depending on how similar the direction is to perpendicular movement.
A generator is a wire moving around a magnet. Imaging the magnet north is to the right and the magnet south is to the left, and the wire is moving clockwise.
When the wire is above the magnet it is moving right, in parallel to the magnet, so no current is generated. when the wire is to the right of the magnet is it moving down, in a straight angle to the magnet line, then the current generated in the wire is very strong. When the wire is below the magnet, it is moving left, in parallel with the magnet again, and no current is generated. When the wire is to the left of the magnet it is moving up, in a straight angle to the magnet line, and therefore the current is at it’s strongest again, this time in the other direction it was when the wire was on the right, because it moves in the opposite direction.
If you draw a graph of the current over time, you would get what we call a sine wave. This is the reason that AC power is in the shape of a sine wave.
Now you know what Alternating Current (AC) power is
A 3 phase generator is 3 wires on a circle, 120 degrees from each other, thus dividing the circle into 3, rotating around a magnet. Therefore if you draw their graphs on top of one another, each wave would be a third of the way behind the previous one. This one wave being ahead or behind another wave is what we call a phase. So since the wires on the circle are 120 degrees from each other, we say that their phases are 120 degrees apart.
Now you know what 3 phase Alternating Current (AC) power is.
Now lets understand the 240V and 415V numbers come from.
What is 240V AC power?
AC power numbers are the answer to this question:
If you connected an AC power source to a heating element, and connected the same heating element to a Direct Current (DC) power source, what Direct Current (DC) voltage would cause the heating element to produce the same amount of heat?
So the mains power of 240V is actually a sine wave that goes all the way up to 340V. The way you calculate this is by dividing the peak voltage (the amplitude) by the square root of 2, which is 1.4142.
This is called the power output of the power supply. It is also called “RMS Voltage” – it stands for “root mean square” (RMS voltage is what you want to google if you want to know more)
Now – the calculation of the 415V figure is a bit more complicated, and you can derive it from the 240V figure with trigonometry like other people did here in the comments – but essentially, the power output of the combined 3 sine waves with a peak of 340V when their phases are 120 degrees apart is equivalent to the power output of 415V of Direct Current (DC)
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