Friday, April 6, 2012

Why does the rotor need slots?

I wrote the other day about the importance of the air gap in an induction motor. It seems after all the rest of the world doesn't share my concern about this parameter. I looked up and down the internet and the unanimous opinion is that the air gap just has to be big enough to provide mechanical clearance for the rotor.

I found some good pictures of typical rotor stampings on this website, so we can see what we are talking about. Here is a good one:

You can see that the linear profile is about fifty percent copper and fifty percent iron. You want iron to carry the magnetic field, and you want lots of copper to provide a path for current to flow. (The fifty-fifty compromise is not that uncommon in these engineering situations, and not only because people are too lazy to work out the true optimum. But that's another story.)

But I'm still not convinced that this is entirely right. As I explained in my last article, if you want torque then the rotor current has to flow in a region where there is a strong magnetic field. What good is the magnetic field if it's inside the iron? It has to cut across the copper bars, and I can't for the life of me see why that should happen in this typical rotor configuration.

You have to understand something about what we call the magnetic permeability of iron. As compared to all ordinary materials, including in this case such things as air and copper, the relative permeability of magnetic iron is on the order of 1000:1. That means, in anthropomorphic terms, that a line of magnetic flux would just as rather pass through 1000 millimeters of iron rather than jump a gap of 1 millimeter through air or copper. If you'll forgive my shaky graphics, we can see what this means for this particular rotor in a typical magnetic field situation:

The field lines will do just about anything to stay inside the iron. So what good is all that copper if it's not in the magnetic field? I just don't get it.

Obviously people have been building motors that work for 100 years. So maybe I'm missing something here. I just can't see what's wrong with my analysis.

According to my theory, I wouldn't even have rotor slots. My rotor would be a solid iron cylinder, and my "squirrel cage" would be a smooth copper coating about 3 millimeters thick all around the suface of the rotor cylinder. That way the magnetic field lines would have to  cut through the copper to complete their circuit. Yes, it would mean the motor has lower inductance, which means it would need a higher magnetising current. But it does no good to design for low magnetising current if the tradeoff is that your magnetic field manages to avoid the rotor bars.

So that's my theory. You tell me what's wrong with it.


Dave Keenan said...

Your mistake is in thinking in terms of field "lines" "cutting through" a current carrying conductor. You can often get away with this popular model if you don't look too closely, but eventually it will cause grief. There are no lines. The equations don't say anything about lines. The equations are based on the field having a magnitude and direction at every point, and possibly changing with time.

So there are no lines to snap across the slot and it wouldn't matter if the flux density inside the copper was always zero (as it very nearly is).

And you have to consider, not individual rotor bars, but the current _loops_ consisting of a pair of bars joined by the end rings.

What matters is the rate of change in the total amount of flux passing through the loop, i.e. through the hole inside the loop. It makes no difference where in the hole that flux is.

-- Dave Keenan

Marty Green said...

Dave, I find it hard to agree with your claim that a motor can develop torque even if there is no flux passing through the copper. Is there no such thing as the (qv)x(B) force? If there is no B field where the charges are moving, where is the torque? It is true that you generate a voltage in the loop if the flux is changing anywhere inside the loop. To generate voltage you don't need flux in the wire. But torque is different.

Dave Keenan said...

My brane hertz when I try to think about this stuff. There's usually more than one way to look at it too.

I recommend The Feynman Lectures on Physics Vol II chapters 2, 16, 17, 18, 36.

You're right that at the most fundamental level there are no forces involved other than F = q(E+vxB). But don't forget the E. And don't forget curl(E) = -dB/dt. And the qE component of the force causes an increase in qv because of the low resistance of the rotor-bar/end-ring loops.

But try this one instead:
There is a small component of magnetic field due to the current circulating in the rotor bar loops. The mysterious (to me at least) quantum-mechanical effect we call ferromagnetism means that this small field causes the spins of the outer electrons of the iron atoms in the steel to align in such a way as to increase this field a thousand-fold. A spinning electron is equivalent to a tiny circulating current and their overall effect is equivalent to a sheet of current flowing, as if in a superconductor, around the surface of the steel. This "effective current" is 1000 times greater than the actual current in the rotor bars. It is the interaction between the stator field and this effective current that is responsible for 99.9% of the torque.

Marty Green said...

Let's agree there are no electrostatic forces and concentrate on the magnetics. I appreciate your point about the magnet-on-magnet interaction: I raise the issue myself in my very next blog post:

My problem with your analysis is that the circulating currents in the iron do not primarily originate with the rotor currents, but with the magnetising current of the stator. As in any transformer, the currents which flow by transformer action make very little difference to the magnetisation of the iron, which was set up by the magnetising current in the stator.

Furthermore, if I did believe that the rotor currents created microscopic currents in the iron which were a thousand times greater than the rotor currents, I would have a real problem with the magnitude of the forces. We can make a pretty good engineering estimate of motor torque by assuming a field of one volt-sec per meter squared (typical for magnetic iron) and a current density in the copper of 1000 amps per mcm. Pushing numbers, I get a reasonable motor horsepower for typical dimensions. If you want me to use a value for current that is 1000 times higher, it throws everything way out of whack.

Dave Keenan said...

I agree there are no electrostatic forces. But that's not the same as saying there is no E field. The E field I referred to is not the diverging kind that you get from unbalanced charges, but the circulating kind that you get from changing flux. You already referred to it indirectly when you agreed there was a voltage induced in the conducting loop due to a change in the flux passing thru the loop. But whether this E field is relevant to our discussion, is another question.

I agree that the effective circulating currents do not primarily originate with the rotor currents. That's why I referred to it as a "component". But the component due to the rotor current is not aligned with the component due to the stator current. If it was, there would be no torque. You can take the component due to the rotor current and resolve it into two further components at right angles to each other, one being aligned with the field due to the stator current, and one at right angles to the field due to the stator current. It is only the component of rotor field that's at right angles to the stator field that could have anything to do with produces the torque. And so it would only be the effective circulating current corresponding to that component that you might need to consider in calculating the torque.

1 weber/metre^2 = 1 tesla sounds right for the magnetic flux density in the steel, but around 2000 amps/mm^2 for the current density in the copper? That sounds way too high to me.

Does this help.$file/abb_technical%20guide%20no.7_revc.pdf
It says, "The maximum torque of an induction motor is proportional to the square of the magnetic flux".

You could make a toy motor with sloppy slots to find out if the bulk of the torque is on the copper or the steel.

Marty Green said...

I think I was correct in my use of the term mcm for wire size: 1000 mcm is one million circular mills, or an equivalent of one-inch diameter circular wire.

As to your point about which component of the magnetic field is effective, I urge you to consider the basic theory of a transformer: the magnetic field in the core is established by the magnetising current in the primary, and has almost nothing to do with the working currents flowing in either the primary or the secondary. Whatever additional fields result from the working current in the primary are essentially cancelled out by the corresponding secondary currents.

I find it very hard to accept that I cannot analyze the torque in a motor by looking at the (qv)xB forces in the rotor bars.

Dave Keenan said...

Your definition of MCM is correct, and I used the correct conversion 1 MCM ~= 0.5 mm^2, but your current density is too high by about a factor of 1000. A realistic current density is more like 1 A/MCM or 2 A/mm^2. See This could explain why you get roughly the right answer and why a further factor of 1000 throws everything way out of whack.

I totally agree with what you say about transformers. But unlike a transformer, an induction motor producing torque has an angular displacement between primary and secondary fields, so the fields due to working currents don't completely cancel. If they are 90 degrees apart, they don't cancel at all and you get maximum torque.

The ABB paper saying that maximum torque is proportional to flux squared, implies to me that the permeability of the steel enters into the equation in a squared manner too, and I can interpret this as entering once for the field due to stator current and once for the field at right angles to that due to rotor current (which we can then interpret as a multiplied rotor current).

Marty Green said...

Okay, I couldn't believe I got it wrong, but after reading it over five times, I see I wrote "1000 amps per mcm" when I meant 1000 amps per THOUSAND mcm, or 1000 amps per square inch. Whatever I wrote, the fact is I used the correct density in my calculation, more or less the same density that you use. So I agree, I quoted the wrong current density, but my calculation was still correct after all.

That means if I use your hypothetical current density in the iron, I would get a torque a thousand times too big. The other thing about using those hypothetical circulating currents is that the iron saturates. It's not can't just add the fields due to all the sources. In fact, the iron was already near saturation from the effect of the magnetising current. You can't get more magnetisation in the iron by adding more external currents, because the microscopic loops are already fully aligned.

And as far as I know, the curves for motor torque show no strange behavior associated with the iron being saturated at high currents. Ultra-high rotor currents, in fact, give ultra-high torques strictly proportional to the currents. If the iron was invovled you should see dramatic saturation effects.

Anonymous said...

The current generated in iron part of rotor can't be utilized for torque as it is formed in loops and has zig-zag path and is very low due to high resistivity of iron.
The current passing through copper bars, generated by whatsoever amount of flux, is high (due to low resistivity of copper) and unidirectional. So that current carrying conductor can experience proper force and torque.

Anonymous said...

the current in iron part of rotor is called eddy current.