Amps at short circuit end ring

[Gyo]

Hi. For a caged induction motor, what is the current at one end of the short circuit end ring i.e. is it the summation of all of the rotor bar currents?

 

[jraef]

Not sure what you are asking here, or why. All rotor bar current must pass through the end rings, the ring completes each circuit. The current flowing IN the ring itself will vary depending on where you measure it and how much each set of bars are being energized by the stator fields at any given moment of time within the section of the ring you are measuring, but that is constantly changing as the rotor rotates within the stator fields. If you are asking if it is additive, then the answer is yes, all rotor bar sets that are being induced by the stator fields will be adding to the total current flowing through any point of measurement on the ring you may theorize on (meaning I don't see how it could actually be measured). So if you were to manufacture a AC induction motor rotor collection ring, that current value would be the basis of your design consideration.

So now, why do you want to know this?

"Will work for (the memory of) salami"

 

[electricpete]

For an endring the current is function of time and angle.
I(t) = I0*cos(w*t-p*theta) where theta takes on Nr discrete values, Nr is number of rotor bars
But the theta dependence doesn’t affect the heating (it simply means the currents in different portions of the ring are out of phase with each other). It has the same heating as if the current I0 were flowing around the ring

Induction Machine Handbook equation (15.37) gives:
Iendring = Ibar / [2 * sin(pi*p1/Nr)] where I think p1 is pole pairs

There is no derivation I could put my hands on but I've seen it before somewhere. Draw a current vector for each bar. They are are the Nr spokes of a wheel. We could also draw a current vector for each ring segment. It also resembles Nr spokes of a wheel (but different magnitude). How to compare these two wheels to develop a relation between bar current and ring current? By KCL, each ring segment vector differs from the adjacent ring segment vector by a vector equal to the current of the bar between them. So a graphical procedure would be to start with the spokes of a wheel representing the bar vectors. Then translate those vectors (without rotating them or resizing them) into the ring of a new wheel. Now the radius of that new wheel represents the length of a ring segment current vector. We have a diagram, just needs a little trigonometry to come up with the formula. It is reminiscent of the vector diagram used for deriving distribution factor of a stator winding.


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(2B)+(2B)' ?

 

[electricpete]

Correction

There is no derivation I could put my hands on but I've seen it before somewhere. Draw a current vector for each bar. They are are the Nr spokes of a wheel. We could also draw a current vector for each ring segment. It also resembles Nr spokes of a wheel (but different magnitude). How to compare these two wheels to develop a relation between bar current and ring current? By KCL, each ring segment vector differs from the adjacent ring segment vector by a vector equal to the current of the bar between them. So a graphical procedure would be to start with the spokes of a wheel representing the bar vectors. Then translate those vectors (without rotating them or resizing them) into the ring of a new wheel. Now the radius of that new wheel represents the length of a ring segment current vector. We have a diagram, just needs a little trigonometry to come up with the formula. It is reminiscent of the vector diagram used for deriving distribution factor of a stator winding.

should've been:

There is no derivation I could put my hands on but I've seen it before somewhere. Draw a current vector for each bar. They are are the Nr spokes of a wheel. We could also draw a current vector for each ring segment. It also resembles Nr spokes of a wheel (but different magnitude). How to compare these two wheels to develop a relation between bar current and ring current? By KCL, each ring segment vector differs from the adjacent ring segment vector by a vector equal to the current of the bar between them. So a graphical procedure would be to start with the spokes of a wheel representing the bar vectors. Then translate those vectors (without rotating them or resizing them) into the ring rim of a new wheel. Now the radius of that new wheel represents the length of a ring segment current vector. We have a diagram, just needs a little trigonometry to come up with the formula. It is reminiscent of the vector diagram used for deriving distribution factor of a stator winding.

=====================================
(2B)+(2B)' ?

 

[Gyo]

Thanks for the clear explanation.
I was trying to find a quick way to determine the cross section area of an end ring without doing any thermal analysis or any detailed calculations. If there are 26 rotor bars with each bar of x cross section area carrying y current, I wasn't sure if I could size the end ring to have at least the same current density as one rotor bar.

 

[electricpete]

Just to follow up on your latest post (sorry if this is obvious)

Symbols:
Ar = area of ring.
Ab = area of bar (unknown... to solve for)
Ir = current in the ring
Ib = current in a bar
p1 = number of pole pairs (twice the number of poles)
Nr = number of rotor slots

ASSUME the current is uniformly distributed **

"Question": What area Ab is required to give the same current density in ring as the bar?
Ir / Ar = Ib / Ab

solve for Ar:
Ar = Ab * Ir / Ib

substitute Ir = Ib / [2 * sin(pi*p1/Nr)]
Ar = Ab * Ib / [2 * sin(pi*p1/Nr)] / Ib

cancel Ib
Ar = Ab / [2 * sin(pi*p1/Nr)] ("answer")

You can plug Nr = 26 and p1 = 1 (for 2-pole motor) or p2 = 2 (for four pole) to get an answer...
** But the assumption about uniform current distribution is probably not good considering the large size of the bars and rings.

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(2B)+(2B)' ?

 

[electricpete]

Here is the vector diagram / trigonometry to derive the equation relating bar and ring currents.

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(2B)+(2B)' ?