Category Archives: Jacob

Conclusions: A look at my model and a look forward to future work

My model is clearly a rough one, and is not entirely accurate to the performance of the average induction generator.  However, I believe that it captures the basic relationships between the variables involved in the operation of a generator, and that the work I have done has the potential to be useful, albeit after some refinement.

Over the course of my work, I have looked into several of models of standard wind turbines, and they have generally focused on the mechanics of the wind and its interaction with the blades, rather than the generator itself.  While the mechanics of this interaction is obviously important to consider when designing a wind turbine, I have been surprised by how little attention is paid to the design of the generator.  It might be said that this is because an induction generator is an induction generator, regardless of where it is, and we already know the most efficient designs.  However, some wind turbine manufacturers are switching to a new gearless design where in place of a gearbox to magnify the rotation rate of the rotor, there is a much larger rotor that can hold more magnets and thus generate more power.  Clearly there are still advances to be made in generator design for wind turbines, and modeling is a good way to explore these potentialities in a cost-effective manner.

If I were to continue my work, the first thing that I would do is construct a more complicated model that accounts for the electromagnetic torque exerted on the stator by the rotor (I believe this to be the largest flaw in my current model).  I would also attempt to look at not just the generator, but the various conversion processes that the generated current must go through before being fed into the grid, as this could inform my design.


A few observations

Having been stymied by the constraints that mathematica imposes on integrals of absolute value functions, I have forgone my attempt to create a smooth plot and moved on to obtaining discrete values and using them to create my plot.  Plotting the difference in rotation frequency between rotor and stator vs the the total induced emf produced by a single rotation, I have made a surprising discovery.  In my model, it does not seem to matter what I make the rotation rate: the emf induced remains constant.  The plot below gives values 1-10 for $\Delta \omega$ with corresponding values for $\epsilon_{induced}$.

Another oddity that comes to our notice is that our expression for $n$ plays exactly the same role in this equation as $\Delta \omega$.  That is to say, increasing the number of wire coils achieves the same objective as increasing the rotation rate of the rotor.

This is very interesting indeed, and quite handy as well.  A problem with some wind turbine designs (fixed-pitch, for example) is that they function suboptimally at lower rotation rates.  This causes a certain amount of inherent inefficiency in the generator (or rather adds to the inefficiency already present).  My model, while far from perfect, suggests that this might be ameliorated by increasing the number of wire loops in the stator.  As this inefficiency is largely a problem in cheaper turbine designs, it is my hope that this technique (being relatively inexpensive) may offer a cost-effective way to improve efficiency.


As in the last post, the mathematica code for the above can be found in


Preliminary Results

First, to set up my initial findings, I will show a comparison of a sine wave vs the Fourier Transform that I used to approximate a triangle wave.  Note how the transform is much closer to a linear function than the sine wave.

The link to the mathematica code for both of these can be found at the bottom of the page.

Now I will move on to the work that I have done with my triangle wave.  The first thing that I examined, as it seemed the most interesting, was the relative velocity $\omega_s – \omega_r$ which I simply called $d$ in my simulation.  This seemed the most interesting off the bat as it’s one of the only variables that appears both inside the sine term, and as a factor of the total expression’s amplitude.  Increasing the $d$ value on my plot of $B$ over a single cycle shrank the period dramatically but also increased the amplitude, yielding a much larger number of much skinnier spikes in $B$, as can be seen below.

It appears that the key is finding the balance between number and size of spikes (higher or lower relative rotation frequencies) that yields the highest overall total $B$, and thus the highest induced emf.  To this end, I tried to plot the integral of the absolute value of $B$ (the absolute value being introduced to account for all negative and positive values of $B$ as both contribute to the total emf) against an increasing $d$.  Unfortunately, Mathematica is unable to process this computation.  I have attempted to plug in points in between $d = 1$ and $d = 10$ to pinpoint a local max, or find that it continues to increase, but mathematica is being uncooperative with this computation as well.  Now that I know what exactly I’m looking for (in this vein at least), getting to it should not be too difficult.  For my next post, I will attempt to solve this problem using a 3-dimensional graph to include both $d$ and $t$ (the variable that cycles the sine waves) as variables.


Mathematica code for the all images:  continuous plots, data and discrete plot, sine and triangle wave.


Derivation of my Modeling Equation: version 1.5

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.

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


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



revised project plan

For my project I will be examining the effect of varying coil numbers, distances between stator and rotor, resistance of the wires involved, and relative frequencies of rotation of the stator and rotor on the induced current in the coils, and thus the power provided to the grid.

To derive the equation that I will be using for my model I have made several simplifying assumptions, including that the coils, the rotor, and the stator all have the same length, that coils and magnets are squares that pass straight by each other, and that the strength of the magnetic field passing through a given coil as the magnet moves by is sinusoidal.  The equation that I have derived is below.


Project Plan

My project will explore the effect of varying slip, rotation frequencies, and system size/configuration on the energy output of an induction generator.  If time and resources permit, I will also be testing how efficiency varies with slip, and how this effects my final results of which configuration is best.  This question (of bigger versus smaller systems with higher or lower rotation frequencies) is currently a topic of much debate in the field of wind turbine design, and thus is an interesting area to conduct my research.

The traditional design for an induction generator includes a gearbox with which to convert the relatively slow rotor rotations provided by the mechanical source (wind, steam, etc.) into rotations at a frequency that will generate power.  However, a newer model forgoes this for a much larger ring of permanent magnets, creating a stronger magnetic field that does not require the rapid rotation rate provided by the gearbox to generate a sufficient amount of power.  The basic equations that will form the backbone of my simulations are described below.

I will apply these equations to relate slip and the relative size, position, and number of loops of wire to the amount of current generated, and run simulations to find the most efficient configuration.



Induction generators: efficiency and applications

My work will focus on AC induction generators: exploring current designs and how to make them more efficient.  This offers a great deal of potential for modeling as there are many variables involved: rotor and stator size, slip, and materials to name a few.  This is a field that is currently quite popular due to the ongoing energy crisis, and the applications are ubiquitous.  To the extent that my work will explore specific applications of induction generators, I will be examining their use in harnessing wind power: specifically the unique restrictions that turbines and other types of casings impose.

Research questions will concern the effects of many different variables on overall efficiency, including magnet type, wire material, slip, and rotor/stator rotation frequencies.