MadSci Network: Engineering

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

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 

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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