Page 653 - The Mechatronics Handbook

0066_Frame_C20.fm Page 123 Wednesday, January 9, 2002 1:47 PM

It should be emphasized that we use the 3-2-1 orthogonal transformation matrix for the z-y-x Euler
rotation sequence, and θ x , θ y , θ z are the rotation Euler angle about the x, y, and z axes.
The field B and gradients of B produced by the microcoils fixed in the inertial frame and expressed
assuming that the electromagnetic fields can be described by the second-order Taylor series. Expanding
B about the origin of the x, y, z system as a Taylor series, we have [18]

B e = B + ( r ∇)B + 1 ) B
-- r ∇⋅(
⋅
2
2
or
2
B ei = B i + --------r + 1 T ∂ B i
∂B i
--r ----------r
∂r 2 ∂r 2
where

∂B i ∂B i ∂B i
∂------ ∂------ ∂------
∂x
∂x
∂x
--------- --------- ---------
∂x ∂y ∂z
2 ∂B ∂B ∂B
-------- = ∂B i ∂B i ∂B i and ---------- = ∂------ i ∂------ i ∂------ i
∂ B i
∂B i
∂y
∂y
∂y
∂r -------- -------- -------- ∂r 2 --------- --------- ---------
∂y
∂x
∂z
∂x ∂y ∂z
∂B i ∂B i ∂B i
∂------ ∂------ ∂------
∂z
∂z
∂z
--------- --------- ---------
∂x ∂y ∂z
We denote
∂B i
∂------
∂j
B ij = ∂B i and B ij()k = ----------
--------
∂j ∂k
Then,
2 B ix()x B ix()y B ix()z
-------- = [ B ix B iy B iz ] and ---------- =
∂B i
∂ B i
∂r ∂r 2 B iy()x B iy()y B iy()z
B iz()x B iz()y B iz()z
Hence, the first-order gradients are given as
∂B i
∂------
∂j
B eij = B ij + ----------r = B ij + [B ij()x B ij()y B ij()z ] r
∂r
The expanded field is expressed in the permanent-magnet coordinates as
2
⋅
B e = B + ( r ∇)B + 1 ) B
-- r ∇⋅(
2
where B = T r B and ∇ = T r ∇ .
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