{"id":1400,"date":"2012-04-15T18:38:59","date_gmt":"2012-04-15T22:38:59","guid":{"rendered":"http:\/\/blogs.vassar.edu\/magnes\/?p=1400"},"modified":"2013-07-11T10:28:07","modified_gmt":"2013-07-11T14:28:07","slug":"derivation-of-my-modeling-equation-version-1-5","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2012\/04\/15\/derivation-of-my-modeling-equation-version-1-5\/","title":{"rendered":"Derivation of my Modeling Equation: version 1.5"},"content":{"rendered":"

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. \u00a0The 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. \u00a0The majority of the information here comes from common sense equations (d = rt, for example). \u00a0The 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. \u00a0Now, on to the derivation.<\/p>\n

\"\"<\/a><\/p>\n

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 $w=\\frac{2 \\pi b}{n}$. \u00a0The height of the wire coil is irrelevant as it is directly proportional to the number of coils, which we will not be dealing with.<\/p>\n

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

  <\/span>   <\/span>\"\[\Phi = \int \vec{B} \cdot d\vec{a} = BA_{loop} = Blw = Bl\frac{2 \pi b}{n}.\]\"<\/p>\n

Simple enough. \u00a0Now we only have to find $B$. \u00a0As we know that magnetic field strength varies as the inverse cube of the distance from a magnet, we can first say that<\/p>\n

  <\/span>   <\/span>\"\[B = \frac{1}{r^3} B_0 = \frac{B_0}{(b-a)^3}\]\"<\/p>\n

,<\/p>\n

$B_0$ being the magnetic field strength of the magnet. \u00a0We also know that flux changes with time periodically, and linearly. \u00a0A sine wave seeming inappropriate even as an approximation in this case, we use the Fourier Transform of a triangle wave to approximate. \u00a0This gives us<\/p>\n

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

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). \u00a0To make this a bit simpler to think about, we imagine the period $T$.<\/p>\n

<\/p>\n

  <\/span>   <\/span>\"\[T = \frac{\text{angle passed through}}{\text{velocity with which magnets pass through angle}} = \frac{2 \cdot (2 \pi / n)}{\omega_s - \omega_r} = \frac{4 \pi}{n(\omega_s - \omega_r)}\]\"<\/p>\n

where $\\omega_s$ and $\\omega_r$ are the rotation frequencies of the stator and the rotor, respectively. \u00a0From here, it follows that<\/p>\n

  <\/span>   <\/span>\"\[f = \frac{1}{T} = \frac{n(\omega_s - \omega_r)}{4 \pi}\]\"<\/p>\n

..<\/p>\n

With this in mind, we proceed to the next step of our approximation, leaving $f$ in for the sake of simplicity. \u00a0We 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.<\/p>\n

  <\/span>   <\/span>\"\[\Phi = \frac{16B_0b}{(b-a)^3n \pi} [\sin(ft) -\frac{1}{9}\sin(3ft) + \frac{1}{25}\sin(5ft)]\]\"<\/p>\n

.<\/p>\n

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.<\/p>\n

  <\/span>   <\/span>\"\[\varepsilon = -\frac{4B_0bn(\omega_s - \omega_r)}{(b-a)^3\pi ^2} [\cos(ft) - \frac{1}{3}\cos(3ft) + - \frac{1}{5}\cos(5ft)]\]\"<\/p>\n

with <\/p>\n

  <\/span>   <\/span>\"\[f = \frac{n(\omega_s - \omega_r)}{4 \pi}\]\"<\/p>\n

.<\/p>\n","protected":false},"excerpt":{"rendered":"

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. \u00a0The 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 […]<\/p>\n","protected":false},"author":1599,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,29899,29905],"tags":[],"class_list":["post-1400","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-jacob","category-spring-2012"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1400","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/users\/1599"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=1400"}],"version-history":[{"count":9,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1400\/revisions"}],"predecessor-version":[{"id":2375,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1400\/revisions\/2375"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=1400"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=1400"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=1400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}