One Leg or Two?

Dear Swami,

 If a load draws 10 amps at 208 volts, is that 10 amps per leg or 10 amps total?? I recently worked on a show where I calculated 10 amps total and I was shocked to find I was short of amps. Please clarify.

 Signed, Shorty

Dear Shorty,

What you have done is a classic miscalculation when you are using 3-phase power. When you connect a 208V load across two legs, you are completing a circuit in which the current is passing through both legs, therefore that would be a total of 20 amps. But something happens when you connect a second 208V load that is not using the same two legs of your 3-phase system. It will also draw 10 amps through both legs, and since one leg will be shared (there is no other option), then the current passing through the shared leg is the combination of the 10 amps being drawn by one of the loads and 10 amps being drawn by the other. The trick is that the currents are 120 degrees out of phase with each other, meaning that the alternating current waveforms start at different times, and there is some phase cancellation. So the result is that, rather than combing to make 20 amps in the common leg, they combine for a total of 17.32 amps in that leg.

Why 17.32? Because that's the square root of 3 times 10 amps, and in 3-phase systems, that number—the square root of 3 or 1.732—often factors in.

Now, if you connect a third load across two different phases you now have a balanced 3-phase load in which all three legs have a combination of 10 amps plus 10 amps 120 degrees out of phase with respect to one another, which combines for a total of 17.32 amps in each of the three legs. That makes a total of about 52 amps (17.32 x 3 = 51.96 amps).

You could have calculated that using the 3-phase power formula, which is:

P = V x I x PF x 1.732

And in this case, 

(2080 x 3) = 208 x I x PF x 1.732

Assuming PF (or power factor) = 1

6240 = 208 x I x 1 x 1.732
6240 = 208 x I x 1.732
6240 = 360.3 x I
I = 6240 / 360.3 = 17.32 per leg
or Itotal = 52 amps

Now that you know the 3-phase power formula, I'll let you in on a little secret. You can use the single-phase power formula and get the same answer as long as you use the correct voltages. When you use the 3-phase power formula, to get the right answer you have to use the phase-to-phase voltage, which, in North America is 208V, but when you use the single-phase power formula you have to use the phase-to-neutral voltage, which is 120V in North America.

To demonstrate, the single-phase power formula is:

P = V x I x PF

And in this case,

6240 = 120 x I x 1
I = 6240 / 120 = 52 amps

Signed,
Swami Candela of the Third Millennium

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