Return of the Induction Motor

It's been almost two weeks since I left off my discussion of the induction motor. An argument came up over the basis states of two hydrogen atoms, which derailed me for a while, and then I was off on personal stuff for about a week. Today I'm back on induction motors.

There are so many confusing things about induction motors, it's hard to know where to begin. I keep coming back to what I consider my fundamental principle: the torque is at its greatest when the rotor impedance is consists of equal parts inductance and resistance. This cryptic declaration turns out to have tremendously useful consequences. Let's try and see how it works.

I've drawn here the air gap of an induction motor, unrolled so it's linear. On top I have the stator poles, and on the bottom the rotor. I show a very simplified form of winding, with a very minimal number of slots. In fact, an actual motor has an intricate weave whereby the three phases cascade one after another, and the magnetic field is a sinusoidal wave that circles the air gap at synchronous frequency (30 or 60 cycles depending on the winding.) It's hard to show that in a still picture. We almost have to imagine that the stator winding (on top of the diagram) is moving from left to right, and the rotor winding is pulled along behind it:

Well, there it is. And the funny thing is, the rotor seems to be just ahead of the stator. That can't be right, can it? The rotor goes slower than the stator, so it ought to be lagging behind...not so?  Here's where it starts getting confusing: it's about keeping track of all the phase lags. There is the time lag and there is the spatial lag, and it's hard to keep track of which is which.

I've gone over this again and again, and I think this picture has to be right. I'm trying to show the field configurations when starting the motor, so the rotor is basically stationary. When we last left off this discussion, we drew a stationary transformer and talked about the relative polarity of the primary and secondary currents.  First we consider the no-load case, where the rotor windings are open: then we get pure inductive current flowing in the stator, creating the magnetisation. Then, when we close the rotor circuits, the rotor acts as an additional inductive load, so it draws additional current in phase with the pre-existing magnetisation current. This additional current creates no new magnetic fields because all the additional stator current is balanced out by an equal and opposite rotor current. So the phase of the rotor current is opposite that of the magnetising current in the stator. That's how I've drawn it.

Except I haven't drawn it quite opposite, because if I did, there would be no starting torque at all. I've drawn the rotor current displaced just a little to the right, so that it falls in a region with non-zero magnetic field. Look at the blue arrows: they are strongest midway between the magnetizing current. (Even though I've drawn a simplified winding diagram, which would give flat magnetic poles, I've drawn the field arrows more realistically, as they would be in a motor with sinusoidally distributed windings.) If the rotor currents were directly opposite the stator currents, there would be no torque. Hence, the displacement in my sketch.

But why have I displaced the rotor so that it seems to be ahead  of the stator? Shouldn't it be the other way around, getting dragged along behind? Yes, that's what you'd think, but there is a problem: like currents attract, and opposites repel. If I drew the displacement the other way, wouldn't the rotor winding be pushing back against the stator winding, giving you the wrong motor rotation? No, the displacement has to be the way I've shown.

Does it make sense that I've drawn a very small displacement? Yes it does. At startup there are very large currents flowing, but they do not generate a commensurately large motor torque. The reason is that the stator currents flow in a region of rather weak magnetic field. It's all about the relative phases. We will see as the motor comes up to speed, those relative phases beome more favorable...in other words, we will find the rotor currents to be flowing in a region where the magnetic field is stronger.

But how do we get from here to there? Let's leave that for another day.