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Capacitor Run Single Phase Induction Motor have very high starting currents, possibly in the order of 35-40 Amps upon startup for a 1 hp device, whereas the nominal current usually lies in the order of 4-5 Amps. How does one choose the appropriate wire gauge for the windings?

If we decide that our wire is to handle 40 amps, we will be using 8 AWG, which is probably a bit too thick for windings of a device of small size. If we use the nominal current the size could go down to 16-18 AWG.

I assume that in practice it will be a compromise between the two, but how do we know our compromised mid-sized wire is capable of handling current above its max rating? I cannot find specs on wires which give a max continuous, and max peak (and given duration). Where can I find wire with such specs?

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You can't find this in a wire's datasheet since it almost entirely depends on the application. For example, any wire can take as much current as you want as long as the pulse is short enough.

You need to calculate this yourself: Choose a wire diameter that can handle your continuous current (there are tables for that), then figure out what happens during the turn-on current spike. To do this, you have to calculate the total energy dissipated in the wire (you can estimate this as E=I²Rt, where I is the peak current, R is the resistance of the wire and t the duration of the pulse). Then you divide the energy by the thermal capacity of your wire and you get the temperature delta of the wire after the pulse (deltaT=E/(Sm), where S is the specific heat of copper and m is the total mass of copper in your wire). Based on the heating of the wire, you can then determine whether the choosen wire diameter is okay or whether you should make the wire bigger.

Given that the current spike is generated by a capacitor, you can also estimate the energy to be dissipated in the wire as being equal to the energy that has to be pushed into the capacitor to charge it up.

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Induction start currents are usually 5x to 8x rated but only for a short duration relative to the thermal time constant of the wire, so can use the same as the rated load wire current.

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Most motors that are rated in horsepower rather than watts are in areas of the world where the US National Electrical Code or similar codes apply. That would generally mean sizing the wires for 125% of the rated current marked on the nameplate or for the current listed in the applicable NEC table, whichever is greater. In any case, the branch circuit conductors would not be less than 14 AWG. Motors are generally not rated for starting and stopping so frequently that a larger wire size would be necessary.

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