IB Math SL Topic (A mathematical investigation of the Prius)

Here's one idea, where I'd actually be interested in seeing your paper when it's done, because I've been interested myself, and probably won't really find the time to sit down and get to the bottom of it myself, and there's a spot where the sort of slacker non-paywalled information sources get a little thin (but you probably don't face the paywalls through your school's library).

There are two big motors up front. They are essentially "permanent magnet synchronous motors" (PMSM), where the rotor is built with permanent magnets, and there is a stator of electrical windings around it. The windings are divided into three circuits or phases, and they are wound in coils around the stator forming several different poles per phase.

The car can send electric current through the phases, making magnetic fields, and by control of how much current through which phase windings, it controls the direction of the fields. By constantly changing that direction so it is ahead of the position of the rotor, it pulls the rotor magnets along. Poof! it's a motor.

There are good easy-to-find references on how that math works. Here's about a ten-minute video that covers the basics, and it has a companion video that picks up where it left off, and shows exactly how you can use six on/off transistor switches (per motor) to do the job, which is what the Prius does.

There's a longer (about an hour) video from TI that's a bit more in depth.

One of the key ideas is the math gets messy while you are thinking about these three currents in the three phase windings and having them all constantly changing (three phase-offset sine waves in the ideal picture) to produce magnetic fields with directions constantly changing to stay ahead of the rotor while it turns. You can cut through the messiness by using coordinate transformations to a (q,d) reference plane that goes around with the rotor, and poof, the messy AC you were dealing with changes to a couple vectors that you just control to a fixed relationship to each other in this coordinate space.

That's all covered pretty well in easy-to-find videos like the ones above.

The car can also use the motors for braking. It is easy to get an intuition of how that works, but there are fewer ready-made videos out there to just go watch for the details. This is the part of the story that exists more in the form of papers, and a lot of those might be behind paywalls. One of them that is easily accessible is here.

There are two different ways you can use a motor to slow down a car. In one, you can let the motor be a generator. The rotor is still spinning forward from the car's momentum, but you change the voltage/current phasing in the stator so that magnetic field is pulling on the rotor the other way. A resulting voltage is generated and you get electrical power out of the motor. You can use it to charge a battery. If you draw the mechanical flow of power, it is from the wheels to the motor, and when you draw the electrical power (which, in AC, depends on the voltage and the current and their relative phasing), it is out from the motor (generator). This should bring a smile to your inner physics sense of energy conservation. So that's your regenerative braking.

There's another way you can slow the car down with the motor: you can shift to reverse and press the go pedal. (The Prius won't let you unless you're below about 7 mph, but you can try it below that speed. It doesn't hurt the car, and it does just smoothly slow you down from 7, cross zero and then accelerate you in reverse.)

In this case (considering the first part where you're still moving forward, before your speed crosses 0), mechanically, everything looks like the regenerative case. The rotation is forward, the mechanical power flow is the wheels to the motor, and whatever is happening inside the motor is pulling on the rotor in the opposite direction. But in this case, you're not doing it by getting electrical power out of the motor. You're doing it by putting power in. (Or are you? Your inner energy-conservation bean counter might be saying "hmm..." about now.)

Instead of "regenerative braking", this is a case of "energy consumed braking".



That figure's from the paper linked above, page 7. It's using the (q,d) coordinate space that's familiar from the Clarke and Park transforms in the earlier videos. The videos tended to show only the current vector i, but I think this figure is also showing EMF (voltage) vectors E.

What's going on in the two different braking cases? What goes on as you transition between the two braking cases? And what do you say to your bean counter?

Maybe you'd even end up making a video that would cover the braking cases the way those good existing ones already cover the motor side.

It might be obvious that this isn't the kind of question where I already know the whole story and just pose it for you to work out. I don't know the whole story, and you might end up finding I didn't even get the questions quite right. So if you pick the topic, I'd be interested in where you end up.

It looks clear that which quadrant the U vector is in makes a difference. (There's one more quadrant not shown. I wonder what would be happening if U were there?)

 

H᠎i, @Ueki,

Any updates? What topic did you end up choosing?

 

Wow, I wish this was a few yrs ago when all my marbles still rolled around properly.
I have a much simpler one, but it might be too late to tackle it.
The difference in actual usable energy (watt hrs) between the battery chemistries, NiMh, LiFeP04 and LTO.
My angle is finding out just how many Wh each chemistry can deliver using the same base Ah capacity. Each chemistry drops its voltage under load, so the loads presented by the Prius in EV mode would probably be the easiest to use as base line for that part.
We all understand energy can not be destroyed, only changed from one form to another, so that calculation as where the energy went would also be important to balance the equations ....

T1 Terry