Derivation of my Modeling Equation: version 1.5 | Modeling and Experimental Tools with Prof. Magnes

Below is my updated equation that I will use to model the emf produced by my induction generator, as well as the derivation that led me to it. The objective of my work and the variables dealt with remain the same as in my last post, this is more meant to expose the inner workings of how I came up with the equations that I did. The majority of the information here comes from common sense equations (d = rt, for example). The only more complicated equations thatI am using are Griffiths 7.13: relating emf to change in flux, and the expression for the Fourier Transform of a triangle wave, taken from Wolfram Mathworld. Now, on to the derivation.

We look at a cylindrical rotor of radius $a$ contained within a cylindrical stator of radius $b$, with $n$ coils of wire of length $l$ and width $w=\frac{2 \pi b}{n}$. The height of the wire coil is irrelevant as it is directly proportional to the number of coils, which we will not be dealing with.

We begin by looking at the flux through a single loop, assuming $\vec{B}$ is parallel to the normal vector $d\vec{a}$.

$\Phi = \int \vec{B} \cdot d\vec{a} = BA_{loop} = Blw = Bl\frac{2 \pi b}{n}.$

Simple enough. Now we only have to find $B$. As we know that magnetic field strength varies as the inverse cube of the distance from a magnet, we can first say that

$B = \frac{1}{r^3} B_0 = \frac{B_0}{(b-a)^3}$

$B_0$ being the magnetic field strength of the magnet. We also know that flux changes with time periodically, and linearly. A sine wave seeming inappropriate even as an approximation in this case, we use the Fourier Transform of a triangle wave to approximate. This gives us

$B = \frac{B_0}{(b-a)^3} [\frac{8}{\pi ^2} \Sigma_{k = 1, 3, 5}^{\infty} \frac{(-1)^{(k-1)/2}}{k^2} \sin(fkt)]$

where $f$ is the frequency with which a coil undergoes a full cycle between North and South magnetic fields (ie, the time that it takes for a North and South magnet to pass by the coil). To make this a bit simpler to think about, we imagine the period $T$.

$T = \frac{\text{angle passed through}}{\text{velocity with which magnets pass through angle}} = \frac{2 \cdot (2 \pi / n)}{\omega_s - \omega_r} = \frac{4 \pi}{n(\omega_s - \omega_r)}$

where $\omega_s$ and $\omega_r$ are the rotation frequencies of the stator and the rotor, respectively. From here, it follows that

$f = \frac{1}{T} = \frac{n(\omega_s - \omega_r)}{4 \pi}$

With this in mind, we proceed to the next step of our approximation, leaving $f$ in for the sake of simplicity. We now approximate the Fourier Series with the first three terms, and multiply by our previously established area to give us magnetic flux through a loop.

$\Phi = \frac{16B_0b}{(b-a)^3n \pi} [\sin(ft) -\frac{1}{9}\sin(3ft) + \frac{1}{25}\sin(5ft)]$

Finally, we take the time derivative of this expression and multiply by a negative n to give us the emf induced in every coil in the stator as the rotor spins.

$\varepsilon = -\frac{4B_0bn(\omega_s - \omega_r)}{(b-a)^3\pi ^2} [\cos(ft) - \frac{1}{3}\cos(3ft) + - \frac{1}{5}\cos(5ft)]$

with

$f = \frac{n(\omega_s - \omega_r)}{4 \pi}$