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Mutual-Inductance of Rogowski Coil (Toroid of Circular Cross-section)

  1. Aug 28, 2014 #1
    1. The problem statement, all variables and given/known data
    Calculate Mutual-inductance of a circular cross section toroid.
    circular cross-section radius :a
    toroid mean radius :R
    Previous attempt for self inductance :https://www.physicsforums.com/showthread.php?t=537149

    2. Relevant equations
    B=μNI/(2pi(R+y)) (Cartesian coordinates)
    flux=∫∫B.dA
    dA=dxdy, x from 0 to √(a^2 -y^2), y from (-a to a) (using symmetry, only integrate over half of x)

    3. The attempt at a solution
    Discarding the constants for 'unclutteredness' we have
    ∫∫(1/(R+y))dx dy
    integrating with respect to x the integral becomes
    ∫(√(a^2 -y^2))/(R+y) dy
    using a trig substitution of y=asin(θ); the boundaries (y=a and -a) become (pi/2 and -pi/2)
    the integral then simplifies to
    ∫(acos(θ)^2)/(R+asin(θ)) dθ
    which is equal to ln(R+asin(θ))
    solving with the boundaries, it give ln((R+a)/(R-a)) which is essentially ln(B/A) with B the most outer radius, A the most inner radius of the toroid.

    from, the self inductance can be equated as L=(flux)*N/I
    The solution seems strange, but it may be my own perspective that is blurred because I have been struggling with this problem for about two weeks.

    any comment would be appreciated
     
    Last edited: Aug 28, 2014
  2. jcsd
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