High Temperature Superconducting Magnetic Levitation

224 Ë 7 Numerical simulations of HTS Maglev

adjacent PM elements in the PMG, into account. According to the Biot-Savart law, the
elemental flux density dBcan be described as,

dB=

휇 0

4 휋

Idl×l×r

r^2

, (7.21)

where휇 0 is the magnetic permeability of vacuum,ris a coordinate vector from the
element of length dlto an observation pointP, andIis the magnitude of the current.
Therefore, the three components of the dBin thex,y, andzdirections produced
by the element currentIdyof the element area ABCD in Fig. 7.2 can be expressed as

dBx 1 =

휇 0 M 0 dy耠

4 휋

0

X

−tPM

(y耠−y)dz耠

((x−(l+g))^2 +(y−y耠)^2 +(z−z耠)^2 )^3 /^2

, (7.22)

dBy 1 =휇^0 M^0 dy

耠

4 휋

0

X

−tPM

(x−(l+g))dz耠

((x−(l+g))^2 +(y−y耠)^2 +(z−z耠)^2 )^3 /^2

, (7.23)

dBz 1 = 0 , (7.24)

whereM 0 is the magnetization of the PM,gis the half of the gap between the adjacent
PMs. dBxi, dByi, and dBzi(i=1, 2, 3) can be obtained from the element areas CDEF,
EFGH, and GHAB with the same method. We also introduce the following expressions
to simplify the description as done in [43]

K=

휇 0 M 0

4 휋

, 휓i(휙 1 ,휙 2 ,휙 3 )=

휙i

(휙 12 +휙 22 +휙^23 )^3 /^2

(i= 1 , 2 , 3 ), (7.25)

Γ(훾 1 ,훾 2 ,훾 3 )=ln

y훾 12 +(훾 2 −y耠)^2 +훾 32 −훾 3

y훾 12 +(훾 2 −y耠)^2 +훾 32 +훾 3

, (7.26)

휑(휙 1 ,휙 2 ,휙 3 )=

. 6

> 6

F

arctan 휙^1

휙 3

휙 2 −y耠

휙^21 +(휙 2 −y耠)^2 +휙 32

¡ 휙 3 =/ 0 ,

0 , 휙 3 = 0.

(7.27)

Therefore, we have

dBx=

4

H

i= 1

dBxi=Kdy耠

0

X

−tPM

[휓 2 (x−(L+g),y耠−y,z−z耠)

+휓 2 (x−g,y耠−y,z−z耠)]dz耠, (7.28)