## Saturday, March 31, 2012

### In Which the Induction Motor is Explained

When I left off yesterday, I said there were two possible explanations for how an induction motor worked: by pushing or by pulling. I've been leaning towards the "pushing" option, but how can I be sure it's not the other way around? I carefully sketched out the magnetic field and current distributions around the air gap, and these were the two alternative models. First, the repulsion theory:

We understand the stator field (on top) to be sweeping from left to right. Here I show the rotor currents as being opposite in polartiy to the stator currents, meaning that there is repulsion. It is rather ineffective in generating torque because of the unfavorable angle...as the motor speeds up, the rotor currents shift to the right, with the most favorable angle occuring when the rotor and stator are perfectly staggered, in other words offset by 90 degrees. That however corresponds to the no-slip condition: although the angles are favorable, driving voltage diminishes to zero at that point, so there is no rotor current. The maximum torque occurs at the compromise condition of 45 degree displacement.

That was Theory A. Theory B was exactly the same except it worked by attraction instead of repulsion. Here is a sketch of the rotor and stator currents under the Attraction Theory:

Here you see we have similar currents, so the stator field pulls the rotor along behind it. Like Theory A, Theory B has maximum current in locked rotor conditions, with the currents almost lined up with each other so the effective torque is rather weak. As the motor speeds up, the relative phases shift back through 90 degrees, just like Theory A, except this time moving to the left instead of the right. The question is: which theory is correct?

I find this to be a very difficult and confusing question, yet I am able to answer it with 99 percent certainty by considering the locked-rotor situation. Here we have essentially a fixed transformer: the stator is the primary side, and the rotor is the secondary side. The secondary load is a short circuit, and the current is limited only by the inductance of the rotor...that is, the portion of the inductance which is not linked to the primary.

From transformer theory, we know that this inductive load on the secondary must appear as an inductive load on the primary. Therefore, the TOTAL primary current consists of the  magnetizing current plus the secondary load current, both of these in phase with each other because both loads are purely inductive.

(Actually, in the sketches here, what I am showing as the stator current is really only the magnetizing portion of the current. The drawing gets much too confusing if I try to include the load current in addition to the magnetizing current. Remember, that the magnetic field (the blue arrows) is due ONLY to the magnetizing current. The additional current which flows under load does not create any new fields, because it only neutralizes the field due to the currents flowing in the secondary.)

It is by carefully considering the nature of the primary side current that we can definitively choose between Theory A and Theory B. Remember that in a transformer, the load current in the secondary causes an equal and opposite current to flow in the primary. It must be opposite because otherwise the two currents would reinforce to create large, new magnetic fields in the transformer core: and a transformer cannot work this way. The iron is already fully saturated by the magnetizing field which is set up in no-load conditions; the primary current which flows in response to the secondary current only serves to neutralize the new fields which those secondary currents would otherwise create.

Remember further that the polarity of the primary load current is the same as the polarity of the magnetising current: then it follows that the polarity of the secondary current must be opposite, as shown in Theory A.

But it goes far beyond that. Imagine if the currents were parallel, as in Theory B. Then to cancel out the new fields generated by the rotor current, we would need to draw a primary current opposite in polarity to the magnetising current. That means under the right load conditions, the stator current could decrease almost to zero! Actually, that can happen in a synchronous motor if you over-excite the rotor, making it act as a capacitor...but it definitely doesn't happen in induction motors. Therefore, we must rule out Theory B and accept Theory A.

So the rotor currents flow ahead of the stator currents: the phase difference advances from zero to 90 degrees as the slip frequency goes from zero to infinity; and the maximum torque occurs at that particular slip frequency when the rotor currents are 45 degrees ahead of the stator currents.

God knows how anyone figured this out in his head without ever having seen a motor work, but somehow it happened. History credits Tesla with the invention, but history does not record, so far as I know, how the invention evolved within his mind.

I have not yet commented on the important role of the air gap in motor design. That is a topic for a future blog post.