Power formula for 3-phase motor

[Ahmedzica]

What is the power formula for 3-phase Y connected motor. I've read that

P = sqrt(3) Vt*Ia*Cos(theta)
and P = sqrt(3) Vt*E*sin(sigma)/Xs

Where sigma is the the angel between Vt and E
Is that 2 formulas hold true for both delta and star connected 3 phase motor? and if so, Where the sqrt(3) came from?

 

[psparky]

If you are referring to the amps actually coming thru the lines (what would be read by an ampmeter) you would use the formula P=V*I*1.73. This is the total power needed from power company to make the motor run. This is what is referred to as VA.

The actual power from the shaft is going to be somewhat less. Like you say above P=V*I*1.73*PF (power factor)...or you could say Cos(theta) instead of PF..same thing. This number is expressed in watts or horsepower (HP)

So to deliver 10 HP (10*745=7450 watts) to the shaft of a motor, it may take 14 HP (14*745=10,430 VA) worth of electricity to make this happen. Some say the extra power is "used" in the magnetic field of motor. You can alter PF with capacitors, but that is another story.

Where does 1.73 come from? Take the vector form and set 3 vectors 120 degrees apart from each other. If want line to line, you must subtract phase B from phase A for example...voltage is always the DIFFERENCE between to points. So knowing this you must flip the vector B a complete 180 degrees (the opposite or DIFFERENCE). This will now give you 60 degrees between the two vectors, which you now add. Using simple trigonometry or geometry, you will see the resultant vector (line to line voltage) is now 1.73, or the square root of three multipled by your line to neutral voltage. (original length of vectors). Also, the resultant vector just shifted 30 degrees.

To answer the question of different formulas for star or delta motors...the power formula (VA) will be the same if the current is measured outside of the motor, P=V*I*1.73

If you measure the current inside the motor in the delta to be exact, that is the only time you will need to alter the power formula.

 

[psparky]

Couple interesting links:

Page 267 of this link shows what I'm talking about above. The entire article will easily answer your questions.
http://www.ahmetkucuker.com/wp-content/uploads/2011/05/Three-Phase_Circuits.pdf

This link shows the three current vectors in rotation with their resultant vector:
http://www.ece.umn.edu/users/riaz/animations/spacevectors.html

 

[Ahmedzica]

Thanks psparky!

 

[alphy]

Yes, the two formulas you have mentioned are valid for both delta and star connected 3 phase motors. The sqrt(3) term in the formulas is a constant factor that takes into account the relationship between line voltage and phase voltage in a 3 phase system. In a 3 phase Y connected system, the line voltage (Vt) is equal to the phase voltage (Vp), while in a delta connected system, the line voltage is sqrt(3) times the phase voltage. This is why the sqrt(3) term appears in the power formula for both types of connections. It is important to note that these formulas assume a balanced 3 phase system, where all three phases have equal voltage and current.