The big advantage to 3 phase is that motors can be smaller for the same horse power. Just as a single phase AC motor is smaller than a similar DC motor, so a 3 phase motor is smaller than a single phase. Airplanes use electric motors with even more phases where weight is even more important then size. In factories, it is common to have many high loads, and most of them use 3 phase motors and 240 or 480 volts. I once managed a small factory where almost everything was 480 3 phase or pneumatic. We had 3 phase motors ranging from 1/4 horse to 600 horsepower. I wouldn't even want to think about a 120 volt single phase 600 horse motor, or the conductors it would take to supply it. Most of the lighting was 277 volts pulling power from one phase of the 3 phase. The phases were 480 apart, but only 277 above ground. It is also easy to reverse a 3 phase motor, just switch any 2 leads. I even selected a vacuum cleaner that ran on compressed air because we weren't wired up to supply enough single phase 120 volts to run a large shop vac.
Perhaps the biggest advantage to single phase is that it is the way it has always been done. It works well with incandescent lights and other resistive loads too. It certainly is easy to understand why we stick to 120 volts in our houses. Enough people kill themselves with it, let alone 480.
Also, motors dont need a capacitor to start on 3 phase.
However, in my opinion, 3 phase is an overkill for the average hobbiest garage. Unless you are planning a major operation in a large shop, 3 phase wouldnt be worth it.
However if you will be running large machinery, then I recommend 3 phase.
If you are starting a restaurant anybit larger than a small coffe shop I recomend 3 phase, as most major food equipment is 3 phase.
You can draw single phase off of a three phase by just using one of the legs and a neutual.
You can set up an entire machine on three phase, then take one leg and a neutural and create a 120 volt single phase control circuit out of it, or take two hots and make a 240 volt control circuit.
480 volt makes for effecient use of electricity, but as stated with the 3 phase, its an overkill for many applicantions. 480 is seldom seen anywhere but the industrial world. You occastionally run into it in schools and universities.
3 phase is usually never available in single family and light multi-dwelling residential areas, or allowed in a residential dwelling even if it is on the street. All utility distribution is three phase, look at the top of utility poles on main roads and you will see three wires at the very top.
3 Phase is widely used for light commercial, such as stores, plazas, etc. 3 phase is widely used as the main feed for large residential building, such as large condos, apartment buildings, however only single phase is brought to the actual units. but having 3 phase brought to the building allows for the large motors such as elevators,boilers, etc.
To illustrate the typical service available from utilities:
Single phase 120/240 volt service at 100 amps can provide 24000 watts of power.
The largest single phase motor available is approximately 10 HP.
Typical 3 phase 120/208 volt service at 100 amps can provide 35984 watts of power.
3 Phase motors are available up to 3000 HP.
There is a 240 volt 3 phase service available, thou is rarely used, and is only for industrial applications, so I di not use that as an illustration.
Here are a couple of websites that explain three theory, the math is a bit more than you need, but the explaination is good:
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