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Ballpark estimate is 110 amps for 120V and 55amps for 240V.

There's more to it than that though. Depending on the load you'll have to derate the circuit.

This is true. Depending on the devices, the breaker is sized on 125% of the continuous loads ( runs for 3 hours or more) plus the noncontinuous. (This is a rule of thumb for most circuits!!! Obviuosly this isn't always the case, for example, motor circuits)

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Q: What size circuit do you need for 20 devices with consumption of 640W per device?
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What size wire do you need to use to run 200 ft and have 40 amps of service?

To answer this question a voltage must be given. By using the term service I will draw a conclusion that it is a North American service of 120/240. A #6 copper conductor will limit the voltage drop to 3 percent or less when supplying 40 amps for 200 feet on a 240 volt system. If the circuit voltage is 120 volts, then a #2 copper conductor will limit the voltage drop to 3 percent or less when supplying 40 amps for 200 feet. <<<<>>>> Notice: The following calculations are wrong because a #10 copper conductor is only rated at 30 amps NOT 40 amps. To compute voltage drop (independent of voltage supply) 10 gauge wire is .9989 ohms/1000 ft to go 200 feet you need 400 ft of #10 conductor (out and back) just about 0.4 ohms I*R =E 40a*0.4Ohm =16v lost in heating the wire ( 640W ) Gauge Drop(400') #10 16v #6 6.3v #2 2.5v safety has to do with the insulation 3.4 Watts/ft will melt most plastics (#10) especially if its enclosed in conduit or worse in a wall that is insulated. bare conductors on ceramic insulators dont have a problem if this is for a resistive load the figures are correct if you are trying to run a single phase motor it may not start with #6 (starting current perhaps 7 times the running current) 45V drop if you are trying to start a 3 phase motor with #6 it may be ok only (19V drop) I do know a 7A 240V submersible well pump will not run with #10 wire (200' deep well 150' from the building) The wire coming out of the pump is #12 BTW (sales engineering) it is a GOOD idea to do some arithmetic before buying the wire.... the motor is marked with KVA class (start/run current)