## 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.

## Friday, March 30, 2012

### Is it pushing or pulling?

I'm starting to piece together the story of how the fields and currents interact in an induction motor. Last time I left off with this picture, showing the stator currents on top and the rotor currents on the bottom. The stator field is presumed to be moving from right to left oops! I meant left to right!:

This is how the field ought to look when the motor is starting. You can see that the maximum rotor current is flowing in a region where there is not much magnetic field. That's why even though there is a lot of current flowing, it's not very effective in creating torque. Another, more oblique way of putting it, is to say that because of the high slip frequency, the rotor current is mostly inductive.

As the motor comes up to speed, the slip frequency approaches zero, and the rotor current becomes more resistive than inductive. That means its phase relative to the stator shifts by as much as 90 degrees. This picture shows the rotor current near synchronous speed:

You can see that now the rotor current is flowing in a region where the magnetic field is strong. So it is effective in creating a lot of torque. Won't the maximum torque be created when the rotor current is exactly half way between, where the strongest field is located?

Actually, no. There are two things happening here at the same time. The phase is controlled by the slip frequency: as the slip frequency goes from zero to infinity, the reactance of the rotor goes from pure resistive to pure inductive, which means the phase shifts through 90 degrees. The most favorable torque occurs in pure resistive conditions, when the slip is zero, and the rotor current flows in the middle of the strongest magnetic field.

But that's only half the story. As the slip frequency decreases, the effective rotor voltage also decreases, going all the way to zero when the rotor reaches sychronous speed. So although the phase angle is most favorable, the current is now zero. The current increases with rotor voltage, which is directly proportional to slip frequency.

Where is the optiumum? It's one of those typical compromise situations where the maximum torque occurs at a phase angle of 45 degrees, or in other words, at that slip frequency where the resistive and inductive components of the rotor reactance are equal. That's when the motor develops its maximum torque. It's funny that from my pictures, it appeas that the currents are pushing on each other more than they are pulling. (Remember that when it comes to electric currents, opposites repel.)

But as Colombo used to say, there's just one thing still bothering me. I have a nice set of pictures, with the relative phases of rotor and stator current going through 90 degrees over the full range of operating conditions...this lines up with the idea of inductance vs. resistance. But am I quite sure I'm using the right pictures? Here's another picture that suggests a different mode of operations:

Here I'm showing the rotor currents lagging the stator instead of leading, and I've reversed the polarity so instead of repulsion I'm getting attraction between adjacent windings. (Like attracts like.) How do I know I haven't confused myself, and this isn't the true picture of how a motor works?

I find this a very hard question, but there is one way I can look at it that makes me fairly confident of what is the right answer. Oh, I know I can supposedly do those "right hand rule" tricks where I point my thumb along the direction of current flow and curl my fingers this way or that, but really now. If you're relying on that kind of stuff to tell you what's happening in a physical situation, you're obviously lost. You really need to argue from physical realities, and it's not that easy in this case. I think I have the right answer, but I'm going to leave it for next time.

## Saturday, March 24, 2012

### Welcome Back Russia

Wow. I thought it had been wiped from the face of the earth. No, not Israel, stupid: Russia. I haven't had a blog hit from Russia for at least two weeks. It's like they disappeared from the face of the earth. What happened, did the internet shut down over there?

Today they're back...just two hits, but enough to put the green color back in Russia on my audience chart. Welcome back, my Russian friends, and I hope you're here to stay. Maybe someone can explain your prolonged absence?

### 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.

## Tuesday, March 13, 2012

### The Basis States Revisited

This morning I tried to write out the basis states for two hydrogen atoms far apart. I made some mistakes, which I've left behind for the world to see; but now I'm going to get it right. Here are the four basis states, where A and B are two electrons and the graphs represent their location in two-dimensional phase space. (I don't need to work in six-dimensional phase space because the location of the electrons is sufficiently well specified by saying it's either at this atom or that atom.) The overall state is the superposition of up to four substates defined in terms of all sixteen possible combinations (including polarity) of {A-up}*{B-down} etc. Here are the energy eigenstates:

Essentially, each of the little colored circles in these diagrams represents a product state: so, for example, State III, with both electrons spin up, can be expanded in more conventional representation as  {B-up}*{A-up} - {A-up}*{B-up}, where the order of the product states is significant. It tells us which electron is at which atom. The singlet state I is the superposition of four product states: {A-up}*{B-dn} + {B-dn}*A{up} - {A-dn}*{B-up} - {B-up}*{A-dn}.   You can verify that for both these examples,  if you swap the letters A and B you get back the same state with a minus sign.

The energy eigenstates turn out to be completely specified by their spin state. Since there are two electrons, there are four spin states: the three triplet states (shown in purple) and the singlet state (shown in green). The parity of the spin state governs the parity of the spatial wave function. Since the overall parity must be odd, the odd parity (singlet state) must have even parity for the position state, and the even parity triplet states must have odd parity position states. You should be able to convince yourself that the parity of the position states is  found by checking for reflection about the y=x axes, with blue and red representing opposite polarites of the wave function.

If it is true that all these four states provide a true basis for all possible ground state configurations of two hydrogen atoms, then we are faced with some very clear restrictions. Primarily, since these are all states of definite parity, and odd parity at that, the parity of any composite states must preserve the odd parity of the basis states of which it is composed. The most immediate consequence is that we cannot say "electron A is at this atom with spin up, and electron B is at that atom with spin down." That would be represented by a single blue dot in the upper-right subquadrant of one of our charts, and that is clearly impossible to construct from the basis states.

On the other hand, we can readily arrange to have an electron at A with spin up, and one at B with spin down...as long as we are careful not to define which electron is where. We do this simply by adding together states I and IV. Red dots cancel blue dots and the result is:

But there's a catch. Remember that state IV, the triplet state, has a slightly higher energy than state I, the singlet state. So the complex phases of those two states precess at slightly different rates. After a time, the two states initially in phase will be 180 degrees out of phase, and instead of adding the two basis states we will have to subtract them:

The spins have reversed themselves, and they will keep on reversing themselves as long as the system is in a superposition of the I and IV states.

I think that's enough for now. When we come back, I think I'm going to try to show how to get the spin to point sideways. Because we can't really believe that the spin direction on one atom somehow restricts the spin direction on the other atom. We have to be able to construct arbitrary combinations of spin using our limited set of four basis states. It should be interesting.

### The Basis States for Two Hydrogen Atoms

I'm going to digress for a moment from the discussion on electric motors to return to the question of the double-potential well, which has surfaced on a couple of discussion groups in the last month. A year ago I described the wave function for the two-electron well...not two wells with two electrons, but one well with two electrons. And then I generalized this to the case of two hydrogen atoms far apart, which is basically the equivalent problem to the double-potential well.

(EDIT: It turns out I'm going to make so many mistakes in what's I'm about to do, that I should probably delete the whole post and start again from scratch. And yet I'm not going to do that. I think it's more realistic to show how the thought process works including the false steps. If you want to skip ahead to the cleaned up version, just click on the next post.)

At that time, I described only the lowest-energy state, which happens to be the spin-singlet state. There are actually three more spin-triplet states that need to be considered. For the two-electron well, these states are significantly higher in energy, but for the case of two hydrogen atoms far apart (the double well) they are so close in energy as to be virtually degenerate. Last year when I sketched them I used a couple of different graphic styles, and this year I've come up with yet another way of drawing the pictures. I hope this one is self-explanatory: (EDIT: This picture is still correct!)

The graphs represent the spatial distribution of the wave function, with blue and red indicating positive and negative polarity. The total state is the sum of the four states shown. (Two of them are empty in this case.) Each little circle represents a "product state", where A is here and B is there. The ground state of two hydrogen atoms must therefore, according to my representation, be composed of the sum of four product states. You ought to verify that if you exchange A and B, you get back the same state except with a sign reversal. That's the overriding condition that must be fulfilled by any "legal" wave function.

There are three more states needed to complete the picture, and they are the triplet states. The first two are nice and easy. (EDIT: Here's where I start screwing up.) First, both electrons with spin up:

Then, both electrons with spin down:
(EDIT: You can see these cases must be wrong because when you reverse A and B, you get back the same state. I'll fix it by reversing the polarity on one of the blue dots.)

And finally, the state that confuses people: the triplet state with spin zero. What makes this state physically different from the singlet state? It's a funny question. Here is what the state looks like:

If I've drawn these right, and I think I have, (EDIT: I know I haven't! Still working on fixing them...) then anything that two hydrogen atoms can do has to be representable as the superposition of two or more of these systems. For example, if you want to say that this atom has spin up and the other one has spin down, you have to figure out how to generate that outcome as a superposition of these four basis states.

I don't have time to do that right now but I'll come back to it later.

15 MINUTES LATER: Okay, I said the triplet state was confusing, and it was. So confusing that I got it wrong too. Here is the REAL triplet state:

When I come back, once I've got all the basis states fixed up, we'll see what we can do with them. The first problem is to show how we can have one atom with spin up and the other with spin down. That will be easy. What gets harder is to show how we can put the basis states together to form any arbitrary combination of spin states.

## Monday, March 12, 2012

### Is it attraction or repulsion?

I really opened a can of worms when I decided to talk about induction motors. I left off yesterday with the unanswered question: is the rotor pushed by the stator from behind, via magnetic repulsion, or is it dragged along by the force of attraction? The more I think about it, the more disturbing it gets.

I'm going to start by going right back to the basic operation of a transformer. I've drawn a picture of a transformer with two windings, primary and secondary. I'm showing the iron core to be cut in two places, so it has air gaps just like a motor. The primary side winding represents the stator winding, and the secondary winding on the floating part of the core represents the rotor:
I've shown a switch in the rotor winding, so we can imagine what happens if we don't allow current to flow. Let's apply 120 volt AC power to the primary. We know that inductive current flows through the coil; by "inductive", we mean that the current is 90 degrees out of phase with the voltage, and that it is lagging rather than leading. This current creates a magnetic flux that goes all the way around the iron core, jumping across the air gap in two places. The magnetic flux is oscillating sinusoidally, and in doing so it generates a voltage. In fact, it generates the 120 volts needed to balance the applied voltage, and it generates it everwhere throughout the iron core. In an ordinary motor or transformer, it doesn't take an awful lot of magnetising current to generate this back-voltage. And make no mistake, it's a real voltage: we can measure it if we like with a voltmeter across the switch on the secondary winding. (Actually, it's 40 volts compounded over three loops as I've shown in the picture. If I only drew two loops on the secondary side, you would measure just 80 volts.)

What can we say about the magnetisation of the iron? Since the flux is going the same direction everywhere, the poles line up across the air gaps: in other words, if there is a North pole during the positive half of the AC cycle, then it faces a South pole across the air gap, and vice versa. The poles are always opposite, so they attract each other.

What about when we close the switch? Now current flows in the secondary. How much current? You might think it's limited by inductance, just as it was on the primary side: but it's not that way at all! There is virtually nothing limiting the current flow on the secondary side. The current flowing in the secondary does not create a new magnetic field in the iron core, because it is accompanied by an equal-and-opposite current in the primary side. Remember it was the magnetic flux that limited the primary current: but there is basically no new magnetic flux associated with the secondary current, because it is cancelled out by the effect of the new primary current.

Here's where it gets messed up: the primary and secondary currents are flowing in opposite directions, and opposite currents...repel! Yes, I know that opposites "attract", but that's just for charges. (And sexes.) Opposite currents repel, and like currents attract. Yes, current is gay.

Before I closed the switch, I looked at the magnetisation of the iron and I had North facing South. It lookes like attraction. But now, when I let currents flow in the secondary winding, I get opposite currents facing each other. Opposites repel. So what's it going to be...attraction or repulsion.

We're just getting started.

## Sunday, March 11, 2012

### How does a motor work?

I always thought I knew a lot about motors. Last week I realized there was a very basic question that I couldn’t answer. Does a motor work by magnetic attraction or by repulsion? It’s a funny question.

I know we all built little motors in grade six. There were two magnets; the idea was that you switch the polarity on the rotating magnet every time the pole passes under the stationary magnetic. So it’s always attracting. Actually, if you think about it, it’s repelling just as much as it’s attracting. But that’s not the case I want to talk about.

Ninety percent of all motors in the world are AC induction motors. Maybe 98%. I’ve always thought of the induction motor as one of the most ingenious inventions of all time: because even after you’ve seen it work, it’s really hard to explain exactly what’s happening. (By the way, I think Tesla gets credit for the invention; I don’t know if it was an evolutionary thing that came about naturally, or if he just pulled it out of his ass one day. But that’s another story.)

There is a small complication with understanding induction motors, and that is the issue of rotation of the field, which is problematic on small household motors running on single-phase power. The problem goes away when we consider three-phase motors, which are the workhorse of heavy industry. With three-phase power, you easily get a rotating magnetic field set up in the stationary part of the motor (the “stator”). My purpose today is not to explain how that works; I’m just going to take note of the fact that the magnetic field rotates in sync with the AC power frequency. The rotor gets dragged along by this rotating field: the question is, how exactly does this happen?

There are of course no wires connected to the rotor. Current flows in the rotor, but it flows via inductive pickup. The rotating magnetic field in the stator induces currents in the rotor: the interaction between those two systems of currents creates a torque, and the motor turns. Simple? I don’t think so. If you think it’s simple, I’m going to aks a question that goes to the very core of how the motor works, and I guarantee you won’t be able to answer it. Bear with me...

I read something many years ago that I thought gave me a tremendous insight into how motors work. I don’t remember where I read it but this is what it said: At the speed when a motor develops its maximum torque, the resistance of the rotor is equal to its inductive reactance. That’s it: that’s the key to understanding induction motors.

An induction motor is a kind of rotating transformer. The stator is the primary winding, and the rotor is the secondary winding. The alternating current in the primary winding (the stator) sets up an alternating magnetic field. The alternating magnetic field sets up an alternating voltage in the secondary winding (the rotor). The alternating volatage drives the rotor current. The catch is, that the rotor is rotating, which means that the frequency of the AC voltage in the rotor is different from the frequency in the stator. If the rotor is stationary, it sees the same frequency as the primary, or 60 Hz. But if it winds up to sychronous speed, there is zero frequency in the second winding. With no frequency, there is no induced voltage, and no current. If there is no current, there is no torque, and the motor doesn’t turn. That’s why an induction motor can never run at synchronous speed.

In practice, an induction motor almost always runs just a little slower than the line frequency, so that from the point of view of the rotor, there is always an alternating voltage to drive the current, even if that voltage only reverses once or twice a second. The smaller this driving frequency, the less voltage is available. That’s because it is the field which is constant, and the amount of voltage is proportional to the frequency of the alternating field. So as the motor approaches synchronous speed, the voltage in the rotor approaches zero.

Doesn’t that mean that the current also approaches zero; that the maximum flows when you have maximum voltage, which occurs at maximum frequency, which from the rotor’s point of view is when the rotor is stationary? Here’s where it gets interesting.

The amount of current is not the only thing that determines the torque. You also have to consider the relative displacement between the rotor currents and the stator currents. There is no torque unless there is some relative displacement. There is force, but the force is all in the radial direction, so it has no effect. Ideally, the magnetic fields of the rotor and stator should be at ninety degrees to each other, just like in the little toy DC motor you wound in Grade Six. But just what is it that allows you to control the relative angle of the fields in an induction motor?

In Electrical Engineering we learn abous something called leading and lagging current. In AC power, when the voltage is a sine wave, the current also flows in a sine wave. For a simple resistive device like a toaster, the current flows in sync with the voltage. The two sine waves track each other. But for other types of devices, this doesn’t always hold. For devices with magnetic windings, the current tends to lag behind the voltage, to a maximum of ninety degrees. For other types of devices, known as capacitors, the current can even flow ahead of the voltage; it’s called a leading power factor and we don’t need to worry about that for now.

The nature of a motor winding is that is is partially resistive, like a toaster, and partially inductive. If the rotor winding were made of superconducting copper, it would be pure inductance, but that’s not how is is. There is always some resistance, and we will see that the resistance becomes significant.

It is the nature of inductance that as the driving frequency increases, the inductive resistance also increases; wheras the pure, “toaster-style” resistance doesn’t change. The combination of ordinary and inductive resistance is called the impedance, and it depends on the driving frequency. In the rotor of an induction motor, it turns out that in starting conditions, when the rotor sees the full 60-Hz of the line frequency, the rotor impedance is mostly inductive; so the current lags the voltage by almost 90 degrees. On the other hand, as the motor comes up to speed, the inductance impedance gets lower and lower until finally, at some point, it is less than the ordinary resistance. Even though the copper may be very good copper, it still has resistance, and in the absence of inductance that is the only thing to limit the current. That’s why you don’t need a lot of voltage to drive the rotor currents near synchronous speed: although the available voltage has become very low, there is at the same time very little resistance.

The important thing to note is that as the rotor goes from zero 100% of synchronous speed, the impedance changes from inductive to resistive; and therefore, the relative phase of the two magnetic fields changes by ninety degrees. To be clear, when we talk about relative phase we usually mean the time lag between the driving voltage and the driven current; but in the case of motors, this time lag translates directly into a physical angle of displacement between the two fields.

And that’s where the optimization comes in. As you go from zero to 100%, the rotor voltage is decreasing. So the current is going down. But at the same time, the phase angle is advancing through ninety degrees, as the impedance of the rotor changes from a pure inductance to a pure resistance. It turns out that at the same time the current is become “less favorable” to the development of torque, the phase angle is becoming…more favorable. Somewhere there is an optimum.

And I already told you where it occurs. It’s something I read in a book and it always stuck in my mind: the maximum torque of a motor occurs at that speed where the inductive impedance of the rotor becomes equal to its resistive impedance. At that speed, the torque is a maxium and the physical angle between the stator field and the rotor field is 45 degrees.

This is a tremendously useful insight, and we will have more to say about it. But there is one more nagging question that is still not answered. What is the polarity of the relative phase? If we identify the north pole of the stator field, then where is the rotor field? Because there are two possibilities. We might have a south pole in the rotor, trailing behind the rotor by 45 degrees, and getting dragged along because north attracts south. Or we might just as well have a north pole, 45 degrees ahead of the stator field, getting pushed along because north repels north!

Which one is it? It can’t be both; it has to be one or the other. Does the induction motor work by attraction or repulsion?