7.4 Calculation of the magnetic field of PMG Ë 223Fig. 7.1:Cross-sectional view of the surface current model
for a vertically magnetized PM of width 2wand thicknessd
andbinfinite along thex-direction (normal to the paper) of
a Cartesian coordinate systemx,y,z.Through the integral operation upon Eq. (7.18) fromz耠=0 andz耠= −d, one can formally
arrive at the analytic equation to calculate the magnetic vector potentialAxat point
(y,z) generated by PM in Fig. 7.1,
Ax=휇 0 M 0
4 휋
(z−z耠)ln(y+w)(^2) +(z−z耠) 2
(y−w)^2 +(z−z耠)^2
- 2 (y+w)arctanz−z
耠
y+w
− 2 (y−w)arctanz−z
耠
y−w
¡
0−d. (7.19)
Once the expression of the magnetic vector potential is obtained, the two components
of the magnetic flux density,ByandBz, can be deduced to be
By=휇 0 M 0
4 휋
ln[(y+w)(^2) +z (^2) ][(y−w) (^2) +(z+d) (^2) ]
[(y+w)^2 +(z+d)^2 ][(y−w)^2 +z^2 ]
,
Bz=휇 0 M 0
2 휋
arctan z
y−w+arctanz+d
y+w−arctan z
y+w−arctanz+d
y−w,
(7.20)
according to the relations thatBy=휕Ax/휕zandBz= −휕Ax/휕y.
The contribution of the PM with other magnetization directions and locations in
a certain PMG can be estimated by the geometrical operations of translation and/or
rotation.
7.4.2Three-dimensional case [42]
The 3D model is realistic and can take into account the non-uniformity of the magnetic
field along the forward direction, which may be caused by the existent gap between the