## Minggu, 11 Maret 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?