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First, let's discuss multi-wire branch circuits. This is where we have multiple hot conductors paired with a single neutral. This installation can take the form of non-metallic cable (with a black, red, white and ground conductor) or even a standard MC cable/pipe and wire method. Under normal circumstances this is a perfectly acceptable practice, provided (among other things) you have connected the hot conductors to TWO SEPARATE phases. One hot in phase A and the other hot in phase B for example. This way you don't overload the neutral with return current from the same phase. Another thing the NEC requires us to do is make sure the breaker handles on multi-wire branch circuits have a common handle. This can be accomplished via a two-pole (or three-pole, if a three-phase, multi-wire branch circuit) breaker OR another approved breaker handle clip. By doing this, if you turn off a breaker, you should have no return current from any of the other circuits in the multi-wire branch onto the shared (or common) neutral conductor.
To make it a little easier to understand, imagine that while the hot black conductor pushes current through the wire/load, the white conductor pulls it through. And if the red/white assembly did the same thing, then the current flowing on the white conductor of a multi-wire branch circuit would cancel it out and we would have no problem. This would essentially be the same as a 2 pole 240V circuit with two hot spots. Now, this would only be true if the exact same load was on each circuit. Let's say 10 amps. What would end up happening is that the 12 ohm resistance (120 V divided by 10 A 12 ohms. Ohm's basic law) would be doubled on double the voltage (12 Ohms 12 Ohms on a circuit of 240 V now). The current is still able to make the full loop from the source, through the load(s) and back to the source, creating a viable electrical circuit, and everything works as it should.
Let's go back to the statement that the load is exactly the same size on each circuit and unpack it for a minute. We discussed what would happen if the loads were balanced. But what happens if the loads are unbalanced? Say 10a on one circuit and 15a on the other. Well, the neutral conductor (grounded conductor) would bring that 5a imbalance back to the source. This doesn't necessarily pose a problem other than the fact that there is 5A of current flowing through something other than the hot conductor. If we were to look at it from two single branch circuits (with the same amount of current on each), instead of one multi-wire branch circuit, it would look pretty much the same. 10a going out on the black wire, through the load and back onto the white conductor, and the same with the red/white assembly. But at the point where these neutral conductors came together (at the neutral bus bar), the current flow would become zero because they would cancel each other out. So overall the neutral only carries the unbalanced amount of current!
A big problem that can arise is when we don't use hots of different phases. Instead of each carrying 10A of current working in tandem (one pushing the other pulling for example), they are now both trying to push that 10A of current since they are both traveling in the same direction. So even if the breaker may be working properly (10 amps of current on it), the neutral, at the time they came together, would have 20 amps of current on it since there was nothing to carry the other 10 amps!
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