Mistakes were made throughout this process. Some were as simple as a dropped variable and others were intensely more complicated. In the end I was still able to produce what I set out to do: An interactive animation that allows you to see how a single coil in a stator of magnetic induction based wind turbine creates current.
Much of the work I did was simply writing the code to get the interactive animation to function properly. Below is a screenshot, but the important code is linked to at the end of this post.
Just to recap some of the derivation I did in earlier posts, I will present a short breakdown here. I started with the basic equation for B, but in order to have the magnetic field change over time based on my magnet array, I had to use a triangle wave. In my previous post I inserted values for the period, but in my Mathematica code, I made it dependent on the velocity so that all the graphs would match up.
Triangle Wave Function:
Magnetic Field Equation
Once I had my B(t) I quickly found the flux, but to get to the voltage I had to find the derivative of my triangle wave based B(t) function. I made a square wave approximation to solve for the derivative. In my last post I again plugged in constants in order to show the look of the graph, but in my code I made it dependent on all of the variables.
Using this square wave approximation I was able to find the voltage in the coil. Once I had a formula for the voltage, the induced current was simple to find.
At this point I noticed a grave mistake with my approach to this problem. By approximating the magnetic field with a triangle wave, I made it so the voltage simply flipped back and forth from negative to positive with no gradation. In hindsight it would have been better to use a version of a sin wave as an approximation. That would present its own issues, as the magnetic field would have a fairly linear change. With more time I would have attempted to find a better approximation for the magnetic field that was easier to use.
It should also be noted that my last post had a few typos in it that could be very confusing. I wrote the equation:
but it should read:
The mu was confused for most of the post, but I was able to correct the issues I had with it for all of the formulas in this post and the code.