Read Back Signals in Magnetic Recording - Research Group Fidler

Numerical Methods
j =−σgrad Φ.
(4.42)
With the calculated electric potential (4.41) we are able to determine the current distribution
for each tetrahedron ijkl. Inserting in (4.42) gives
( grad grad grad grad )
j =−σ Φ ⋅ ϕ +Φ ⋅ ϕ +Φ ⋅ ϕ +Φ ⋅ ϕ . (4.43)
ijkl ijkl i i j j k k l l
4.4.4 Magnetic Field of Current
When the current distribution is known for each element, the next task is to determine the
additional magnetic field due to the current. The current is uniform for each tetrahedron. So
the Biot-Savart Law (2.46) can be written as
or
1
Hijkl () r = jijkl
×
4π
∫
Vijkl
r−r′ r−r′ 3
dV ′
(4.44)
1 1
Hijkl () r = j ′ ijkl × ∇ dV ′
4 ∫
. (4.45)
π r−r′ Vijkl
Integration delivers (see Appendix A)
1 1
Hijkl () r = jijkl × d ′
4 �∫
A . (4.46)
π r−r′ ∂Vijkl
So the field of a tetrahedron with uniform current density can be calculated by integrating
over its surface triangles:
1 ⎛ 1 1 1 1 ⎞
Hijkl () r = jijkl × ⎜ d ′ + d ′ + d ′ + d ′ ⎟
4 ⎜ ∫ A
− ′ ∫ A
− ′ ∫ A
− ′ ∫ A . (4.47)
π
− ′ ⎟
⎝
r r r r r r r r
Δijk Δjlk Δilk Δijl
⎠
Fortunately the integrals in (4.47) can be solved analytically for triangle surfaces, resulting in
quite complicated formulas [27] [28]. The total field due to the current can be obtained by
summation over all tetrahedral elements ijkl. Now we are able to determine the magnetic field
at all nodes r i of the magnetic model. Generally the effective field can be written as
multiplication of the interaction matrix and the current density vector.
∑
Hr ( ) = A ⋅j
. (4.48)
i ij j
jl
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