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Mutual inductance using bessel function

  1. Mar 19, 2009 #1
    Hi, I am en electrical engineering grad student and I have to solve an equation to calculate the mutual inductance between an antenna and a micro-inductor. I think it is a Bessel equations but I don't know how to solve.

    M(a,b,d)=(1.45x10^-8)*integral(J1(1.36x)*J1(0.735x)*exp(-x-13.6))dx
    the integration is from 0 to infinity


    Please help!
     
  2. jcsd
  3. Mar 21, 2009 #2
    Hello salla2,

    I have rewritten your problem as:

    [tex]I=d e^{-c} \cdot \int_{0}^{\infty}e^{-x}J_1(ax)J_1(bx)dx[/tex]

    Leaving out the factor before the integral it comes down to:

    [tex]I=\int_{0}^{\infty}e^{-x}J_1(ax)J_1(bx)dx[/tex]

    This is not easy. I tried several ways.

    *) The online mathematica integrator: NOK
    *) xmaxima: NOK
    *) The following integral book:
    "Integraltafel, zweiter teil bestimmte integrale" by grobner - hofreiter
    gives on page 203 the following formula:

    [tex]\int_{0}^{\infty}e^{-ax}J_{v}(bx)J_{v}(cx)dx=\frac{1}{\pi\sqrt{bc}}Q_{v-\frac{1}{2}}
    \left( \frac{a^2+b^2+c^2}{2bc}\right)[/tex]
    [tex]Q_{v-\frac{1}{2}}(z)=\frac{\sqrt{2\pi}\Gamma\left(v+\frac{1}{2}\right)}
    {2^{v+1}\Gamma\left(v+1\right)} \cdot z^{-v-\frac{1}{2}}
    F\left(\frac{2v+1}{4},\frac{2v+3}{4},v+1;\frac{1}{z^2}\right)[/tex]
    Which is a "Legendresche Function" F is the hypergeometric series.

    *) The following book:
    "Table of Laplace Transforms" by Roberts and Kaufman
    gives on page 57, the following function for the Laplace transform of [itex]J_{v}(at)J_{v}(bt)[/itex]
    which can be used considering the definition of the Laplace transform, s is then afterwards set to 1.

    [tex]\frac{1}{\pi\sqrt{ab}}Q_{v-\frac{1}{2}}
    \left( \frac{s^2+a^2+b^2}{2ab}\right)[/tex]
    The definition of [itex]Q_{v-\frac{1}{2}}[/itex] was given as:
    [tex]Q_v^{\mu}(x)=\frac{e^{\mu \pi i}\pi^{1/2}\Gamma\left(\mu+v+1\right)(x^2-1)^{\mu/2}}
    {2^{v+1}\Gamma\left(v+\frac{3}{2}\right)x^{\mu +v+1}} \cdot {\left.}_2F_1{\right.}
    \left[\frac{\mu+v+1}{2},\frac{\mu+v+2}{2};v+\frac{3}{2};\frac{1}{x^2}\right][/tex]
    In which [itex]\mu=0[/itex]

    I can't help you any further on this, hope it helps a bit. It is a difficult question...

    coomast
     
  4. Mar 24, 2009 #3
    I didn't understand a single word but thanks for your help anyway!!
     
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