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How does 277 and 440 3 phase work and how is it different from 120 and 220 single phase?

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First, it's 277/480 and 120/240. But you're trying to relate what I call in the trade 'high voltage' 3 phase and 'low voltage' single phase.
Let's make it easier, with 'low voltages' only: you want to know the difference between 120/208 (3 phase) and 120/240 (split single phase).
First, look at your power source. Commercial services are often 3 phase, meaning you are using all 3 phases of generated AC in the building service. (look up AC generation for more information).
Residential services are really only using 1 of those 3 originally generated phases. Let's say a subdivision is made up of 30 houses, there will be approximately 10 houses on each phase from the power company to keep the load balanced.
On the ground somewhere (if underground service) or on a pole (overhead service) there is a transformer that converts single phase to split single phase. Some people incorrectly call this two phase power. Keep in mind you only have one phase from the power company's generator.
Split single phase enters a residence as two 120 volt conductors, and a neutral conductor. This is done by the pole or ground transformer's center tapped secondary winding. (Google this for more info) This center tap is actually where the neutral conductor is connected, and the two ends of the winding are where the 120v conductors come from.
A measurement from hot wire to hot wire in a residence produces about 240 volts, the sum of the two 120 volt legs. In fact, this is how your large electrical appliances operate. Your receptacles are connected to either 120v leg and a neutral.
Commercial services are actually simpler to explain. The power company produces 3 phases of power at very high voltages (to aid transmission over long distances), and those voltages are periodically stepped down lower the closer they get to a building's service entrance. As with the example earlier, there will be a ground mounted or pole mounted transformer that does a final step down before the wires enter the building.
When they do enter the building, they are the same 3 phases generated miles away at the power company. The difference is they aren't stepped down as low as residential services. Commercial services come in at 277 volts phase to ground, or 480 volts from phase to phase. Inside the building, there are more transformers that produce the 120 volt 'household' voltages for receptacles by stepping down even further.
Notice that in the residential example, the voltages are 120v phase to ground and 240 volts phase to phase. (Exactly double)
In the commercial example, the voltage is 277v phase to ground and 480v phase to phase, or 120v and 208v (depending on if you are talking high voltage or low). The multiplier here is not 2.0, it is 1.73, due to a different type of transformer used. I won't explain this in detail though. If you want to understand this, you need to do your homework.
Long winded, but if this question was easy to explain and understand, everyone could be an electrician overnight!

If you don't know what you are doing three phase will kill you dead in half a second, so don't mess with it, call a professional.

The 220V takes a phase from the 440V 3 phase, to neutral to give you single phase.
Say you had Red Yellow Blue, then take the Red phase (live) and a neutral and you will have half the 440V
To understand 277 3 phase and 440 3 phase you need a knowledge of how AC electricity actually works.

The two live power lines of single phase power in a house come from a single pole transformer. They are 180 degrees out of phase, due to the grounded center tap in the transformer winding.
The three phases are actually produced at the generator. They are sent down a set of three live wires and one neutral wire. Those are the wires you see on high tension power lines. There is always a multiple of three wires on big insulators, plus one ground wire. Three transformers are needed to step down the power for distribution, and three pole transformers feed the industry the three phases.
The three phases are 120 degrees apart from each other. This makes the sum of the currents in the three hot wires equal to zero. The currents flowing in one direction exactly equal the currents flowing in other direction in other hot wires. If the three phase system is properly loaded, no current flows through the neutral.
Three phases are the perfect source for a heavy motor. The three phases naturally provide a rotating magnetic field. A single phase motor needs a capacitor or shading coils to make the magnetic field rotate. And to reverse a three phase motor, a switch trades two of the phase wires.

The 120 degree difference in timing between the phases is why the voltages of two phases do not add to twice the voltage. Due to the phase difference, two 120 volt phases produce 208 volts, and two 277 volt phases produce 480 volts.
The 440 or 480 confusion exists because there is also 440/760 three phase power available.

ANSWER:You need to think of the relationship between the voltages in terms of vectors. Start at a point in the center of your paper and draw 3 lines originating from that point 120 degrees apart (Like a clock with one hand at the 12, another at the 4 and a third at the 7 - Approx). Each line should be the same length and end with an arrow pointing out. Now draw three dotted lines connecting the three arrow heads together. These dotted lines represent the potential (voltage) between each hot leg of the 3 phase system... So pick one (480V, 240, 208, [400 Europe]) and label your dotted lines with this voltage. The dot in the middle of your drawing = ground and we are interested what the voltage to ground is going to be.

Time for some trig! We know the angle between our 3 arrow lines is 120 degrees and we know the line opposite that angle = (Your voltage) mine is 208V. We are looking for the voltage of the adjacent line to my 120 degree angle, so we use the TAN function to give us the ratio between these two sides TAN of 120 degrees = -1.732. Since we don't care about the vector location, I can eliminate the "-" sign and call it 1.732. Since TAN = Opposite / Adjacent I can write the equation: 1.732 = 208 / Adjacent then 1.732(adjacent) = 208 then adjacent = 208 / 1.732 or adjacent = 120.09V. As you can see, 208 3phase was chosen for the availability of our common 120v single phase.
If you do the same calculation on 480V, you end up with 277V to ground. This is why you see so many light ballasts rated for 208/277V. If you run the calculation at 400V you get 240V to ground - the common European single phase voltage.
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Thanks for the feedback!

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