### Re: Why is the Square root of Three a factor in 3 phase power calculations?

Date: Fri Jan 19 11:12:01 2001
Posted By: Donald Howard, Staff, Nuclear Engineering, Retired
Area of science: Engineering
ID: 978034639.Eg
Message:
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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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