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Air gap in toroid

  1. Jun 20, 2010 #1
    1. The problem statement, all variables and given/known data
    (I have done parts i) through v) correctly. It is the remaining two I'm stuck on.

    A toroid of rectangular cross-section has a height of 10mm, an inner diameter of 68mm and an outer diameter of 82mm. It is made of iron and is wound with 118 closely-spaced turns. When a current of 2A flows, calculate;

    i)the mmf around the toroid
    ii)the magnetic field-strength within the toroid
    iii)Using B-H curve determine the flux density and flux in the ring,
    iv)the relative permeability at this point on the curve

    A 1mm air gap is then cut in the toroid. The current is still 2A. Calculate;
    v)the new flux density in the ring,
    vi)the mmf around the ring, and across the air gap.
    vii) It is then required to establish a flux density of 0.75T in the air gap. Calculate the current needed to do this.

    Answers to parts vi) and vii)
    vi) 53, 183 A-t
    vii) 8.0 A


    2. Relevant equations

    [tex]m.m.f.=NI=BS(R_{m}iron+R_{m}air)[/tex]

    [tex]R_{m}=\frac{l}{\mu_{0}\mu_{r}S}[/tex]

    where

    [tex]\mu_{r}=557[/tex],

    I have [tex]B=0.21T[/tex] (although the answer gives it as 0.23)

    [tex]R_{m}(iron)=4.787*10^6[/tex] and [tex]R_{m}(air)=11.365*10^6[/tex]

    [tex]S [/tex]= cross-sectional area = [tex]7*10^{-5}[/tex]


    3. The attempt at a solution

    I get the m.m.f. across the air gap correct if I use the value of B = 0.23T in answers. My answer was 0.21T and I found this was significant:

    [tex]R_{m}(air)=\frac{l}{\mu_{0}S}[/tex]

    [tex]m.m.f.=BS(R_{m}(air))[/tex]

    [tex]=0.23*7*10^-5*11.365*10^6=183.5At[/tex]

    I did as before with the iron, subtracting the 1mm from \piD for l (in metres)

    [tex]m.m.f.=BS(R_{m}(iron)[/tex] where

    [tex]R_{m}(iron)=\frac{l}{\mu_{0}\mu_{r}S}[/tex]

    but I get 70, which is far from the answer of 53.

    For the last part (vii) I don't know why it isn't simply

    [tex]I=\frac{BS}{N}R_{m}(air)[/tex]

    with this I get 5A rather than 8


    Your help is much appreciated.
     
    Last edited: Jun 20, 2010
  2. jcsd
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