> how do you work out the square root of zero?
Use a calculator. Mine says zero.
Never trust a calculator. Check: 0*0 equals zero. The 4-bit idiot may be right.
> surely that neutral conductor is now underrated?
3 is a big number. Look at a "2-phase" (actually split single-phase) supply. When current flows one way in loop A it flows the other way in loop B. If loads A and B are equal, the neutral has NO current. If A and B don't balance, the *difference* (not sum) flows in "neutral". If B has been evicted and only A is pulling power, the A-line and "neutral" currents are equal.
In = Ia-Ib or In = Ib-Ia depending on magnitude.
The neutral current is never(*) more than either line current. Nominal max load: A-line and B-line flow 100 Amps, neutral flows zero (why we call it neutral): Worst-case A-line and "neutral" flow 100 Amps, B-line flows zero Amps. Either way we have two conductors flowing 100A and one loafing.
3-phase theory is trickier but the result is the same. The neutral current is never(*) more than any of the line currents.
Look in to high order harmonics.
Often "it may be assumed" to be much less. A factory may be a lot of 3-phase motors. These will always draw equally from each phase, and the neutral current will sum to zero. Ah, but you may have a 1-Ph tea-pot. So maybe 600A for motors and 10A for tea-pot. The neutral is more than decoration but can be much smaller than the lines. Nevertheless, most codes require it be 50%-80% of line capacity.
No argument, with the proviso for harmonics.
So even if this were a factory converted to flats, it is un-likely the tenant loads would be both large and SO un-balanced as to burn the "neutral". (If loads are not-large, which in residences is true 95% of the time, then even a large percent unbalance is a small current and not a problem.)
But it happens on a regular basis.
A typical residence may pull 0A-5A for lights, 0A-5A for TV etc, and 25A for holiday cooking. If A B and C are busy in the kitchen, the loads near-balance and the neutral current is small. If A is cooking but B and C went out, that's 25+A in A-line but maybe 2A in B and C. The "neutral" has strong current but B and C lines are cool, the cable won't overheat. While the "neutral" will run warm, the negligible heat in B and C conductors means it is probably safe; anyway cooking loads may run 2 hours but not a full 8 hours steady.
The idea of correctly sizing cable is that they don't run "warm"
(*) There is a special case when MOST loads are DC power supplies. Capacitor-input rectifiers draw large spikes which violate the "sine current" assumptions of classic theory. The rectifier spikes do NOT cancel in the neutral, and can exceed current in any line. When LARGE computer systems grew, some neutrals burned off. The first fix is an oversize neutral. Even so the cost and spikes are troubling. Recently most large DC supplies are "PFC" to draw a sine-like current waveshape which is less nasty, and does mostly cancel to the neutral.
