|MadSci Network: Engineering|
The square root of three in the power equation is a result, not of any high scientific calculation, but because we take the easy way out in making measurements. It is easier to connect measuring instruments to the lines outside transformers, motors and generators than to make internal measurements. The power in watts is still equal to volts times amps, but we must use "phase" volts and "phase" amps to calculate power. Calculating these phase relationships requires a little bit of trigonometry. In a "Y" connected circuit, the voltage measured line to line is not the true "phase" voltage [phase to neutral voltage], but the combination of two voltages that are out of phase by 120 degrees. Assuming a balanced load, the current in any of the three phases is the same as that measured in each line because the line will be attached to one end of the phase so there cannot be any difference. But the line voltage is too high because it is composed of the sum of voltages from the two phases. We need the phase voltage to multiply times the phase current to get the phase power. There are two phases connected between each pair of lines in a "Y" circuit, and since the voltages are not in phase, they do not add together to make the line voltage twice the phase voltage. It turns out through some basic trigonometry, that the line voltage is equal to each of the two phase voltages times the sine of 120 degrees, and the sine of 120 degrees is "one-half" the square root of 3. Adding those two halves together gives the LINE voltage as the square root of 3 times the phase voltage. Or conversely the phase voltage is the line voltage DIVIDED by the square root of 3. So the power in any phase, assuming, again, a balanced load is the line voltage times the line current divided by the square root of 3. For the three phases, then, the total power is three times the power in any phase. 3 X the line voltage X the line current divided by the square root of 3. 3 divided by the square root of 3 simplifies to just the square root of 3. Multiplying that, as you noted, by the power factor converts volt-amps to watts assuming the power factor is other than one. One last item: In a delta connected circuit, the same problem exists except with the current measurement. Here, the line to line voltage measurement is the phase voltage, but the line current is composed of two currents that are out of phase by 120 degrees, and you guessed it, the "vector sum" of the two phase current components is the square root of three times the current in any phase. And, the phase current is the line current divided by the square root of 3, so the power equation works exactly the same for both the "Y" and the Delta connected circuits. I know, clear as mud, but you can thank Thomas Edison for this as he said there was no way to start a single phase motor. Today, there are five ways to do that. When the loads are unbalanced, as they usually are, things really get messy. Unbalanced loads produce circulating currents in delta connected transformers, so we try to avoid connecting that way. Unbalanced loads in "Y" circuits will cause a current flow in the neutral so it is usually as heavy a piece of wire as the phase conductors.
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