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Seminar 08 Magnetostatics-PM

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S IMON DUBI TS KY
H IGHER
SCHOOL OF
H IGH V OLTAGE E NERGY
Autumn 2021
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
1
Week 8: Static Magnetic Field
1. What is the source of magnetic field?
2.Which potential is used?
3.The governing equation in 2D symmetry.
4.Boundary conditions: Dirichlet, Neumann.
5.Calculated field quantities: locals and integrals
6.Example: applying external magnetic field
7. Example: DC magnetic shielding.
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
2
Source of magnetic field
1. Current
2. Permanent Magnet
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
3
Starting from Maxwell’s equations
Integral Form
Gauss’ theorem Magnetic flux leaving any closed
surface S is zero
for magnetic
B  ds = 0
field
S

Ampere’s Law
Differential Form
Conclusion
Magnetic field has no
sources nor sinks
No magnetic charge exists.
It allows as introducing of
vector magnetic potential A.
divB = 0
Magnetic field is created by
H-field circulation around a closed
Magnetic field is not
currents.
loop L is equal to free current
conservative in areas
through the surface S enclosed by L. where the current exists. Magnetic scalar potential Ψ
only exists in current-free areas
 H  dl = I
rotH = J
S
Material
Equation
B = H = 0 (H + M )
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
The state of material in
magnetic field is determined
by both external current and
material magnetization
4
Which potential we use?
Electric Field
B
rotE = −
=0
t
Magnetic Field in current-free region
Electrostatic field is
conservative
Scalar potential φ exists: E=-grad φ
rotH = J +
D
t
Magnetic field is conservative
in current-free area
Scalar magnetic potential ψ only exists
in current-free areas: H=-grad ψ
Magnetic Field everywhere
divB = 0
Magnetic field is solenoidal
(divergent-free)
Vector magnetic potential A always exists:
B=rotA
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
5
What happens with vector potential in 2D ?
Only out-of-plane component
Magnetic Flux
A=izAz
J=izJz
Stokes’
theorem
Φ =  B  ds 


S

   =  rotA  ds =  A  dl
S
L
B = rotA 
B=ixBx+ iyBy
H=ixHx+ iyHy
A 

 Bx = y 
B = rotA  

 B = − A 
 y
x 
Note!
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
6
Governing Equation of static magnetic field
rot(
1

rotA) = J + rotHc
(
 = f rotA
)
 H cy H cx 
  1 A    1 A 
 = − J + 

+ 
−
x   y x  y   x y 
y 
 x
Poisson equation
(coordinate independent form
both 3D and 2D)
2D in Cartesian coordinates (x, y)
(plane-parallel symmetry)
  1  (rA)    1  (rA) 
 H cr H cz  2D IN cylindrical coordinates (r, z)

 + 
 = − J + 
−

r  r z r  z  rr z 
r  (axial symmetry)
 z
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Magnetic boundary conditions
A = f ( x, y )
Plane
rA = f ( x, y )
Axisymmetric
Dirichlet Condition:
the known magnetic potential A (or rA)
(may depend on coordinates)
1 A
Ht =
=  ( x, y )
 n
Neumann condition:
the known tangential H-field,
equal to surface current density
A = const  Bn = 0
Zero normal flux (superconductor):
A special type of Dirichlet condition
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Magnetic vs. electric boundary conditions
E
U lines
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
field lines
B
9
Magnetic vs. electric boundary conditions
Electric field
Dirichlet
Condition:
Neumann
condition
(zero):
Magnetic field
Electric field lines are normal
to a Dirichlet surface
Magnetic field lines are tangential
to a Dirichlet surface
Electric field lines are tangential
to a Neumann surface
Magnetic field lines are normal
to a Neumann surface
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
10
Calculation results
1. Locals
Magnetic flux density (B-Field):
B = rotA
Magnetic field strength (H-Field): H = 1 B − H c

Magnetic energy density:
Linear material
Nonlinear material
BH
w=
2
1B
w =  H ( B  ) dB 
20
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
11
Calculation results
2. Integrals
Mechanical force:
F=
1
2
 ( H ( B  n ) + B ( H  n) − n ( H  B ))ds
S
Mechanical torque:
1
T =  ((r  H )(B  n ) + (r  B )(H  n) − (r  n )(H  B ))ds
2S
Magneto-motive force (MMF):
MMF =  H  dl
L
Flux Linkage:
1
 =  A  ds
SL
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Example: Lab 5 – DC Magnetic Shielding
Steel spherical shield
in uniform magnetic field B0
B0=12.6 mT
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Sizing and material properties
Sizing:
External diameter 54 mm
Internal diameter 44 mm
Outer diameter
500 mm
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
B, T
0.030
0.206
0.325
0.445
0.535
0.600
0.650
0.685
0.720
0.745
0.770
0.800
0.820
0.840
0.860
H, A/m
50
150
200
250
300
350
400
450
500
550
600
650
700
750
800
14
Z-uniform field boundary condition
1  (rA) 

 BZ = r r 
B = rotA = 

(
)
 B = − 1  rA 
 R
r z 
Choose as: rA
R
Bz0=12.6 mT
Air
= 0.5r2BZ0
Check it by differentiating:
1

 r 2 BZ 0 
1 2
=B
BZ =
Z0
r
r
Axis of
rotation
Steel
Z
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
15
Z-uniform field boundary condition
Dirichlet boundary condition: rA = 0.5r2BZ0
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Shielding Factor
Field outside B0=12.6 mT
Air gap across the field
Field inside shield Bi=0.807 mT
The shielding factor:
k = 12.6/0.807 = 15.6
Air gap along the field
Field inside shield Bi=0.158 mT
The shielding factor:
k = 12.6/0.158 = 79.7
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
17
Field created by a permanent magnet
Permanent Magnet
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
18
Material Magnetic Properties
Magnetic Permeability
Examples
Free Space: μr = 1
Paramagnetic:
μr > 1
Diamagnetic:
μr < 1
Tungsten, Cesium, Aluminum, Magnesium, Sodium
Soft magnetic:
B = f(H),
Hc ≈ 10…103 A/m
Hard magnetic:
B = f(H),
Hc ≈ 104 …106 A/m
Bismuth, Mercury, Silver, Lead, Graphite, Copper,
Water
Supermalloy (16Fe:79Ni:5Mo)
Permalloy (Fe:4Ni)
Elecrtic steel (96Fe:4Si)
Ferrite (Zn, Fe, Ni, O)
Disk drive recording media (Cr, Co, Pt)
Neodimium magnets (Nd, Fe, B)
Samaruim Cobalt magnets (SmCo5)
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
19
Typical hysteresis loop
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
20
Hard and Soft Magnetic Materials
Hard
Soft
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
21
Magnetic properties of Hard Magnetics
Definitions by IEC
Magnetic
remanence
Br
Coercive
(magnetic)
field strength
Hc
Remanent magnetic flux density in a
substance, when it departs from magnetic
saturation by monotonic reduction to zero of
the applied magnetic field strength
Magnetic field strength to be applied to a
magnetic substance to bring the magnetic
flux density to zero.
(any intersection with H-axis of the
magnetization curve (for flux density,
polarization or magnetization)
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
22
Material Magnetic Properties in QuickField
Non magnetized state (HC=0) :
B=f(H)
Magnetized up to maximum (HC=147.3 kA/m)
B = f(H+M)
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
23
Material Magnetic Properties in QuickField
The modern magnets,
both Neodymium
and Samarium-Cobalt,
are almost linear:
The constant permeability
μr = 1.05 = const
The magnitude and
direction of initial
magnetization
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Examples of Magnetization Direction
Polar
Cartesian
M
α=0
Horizontal
M
α=45°
α=0
Radial
α=90°
Tangential
Sloped
2.38125
M
α=90°
α=180°
Vertical
M
Horizontal
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Example: Pulling Force of Disc Magnet
Manufacturer’s Data:
δ
Neodymium Disc Magnet
N40 3/16"x1/32"
Gauss Rating: 12800 gauss
Pulling Force: 0.49 lbs
Price: 0.06 $
t
Ød
Geometry
d
4.7625 mm (R = 2.38125 mm)
t
0.7937 mm
δ
h/20 =0.04 mm
D
100 mm
Magnet
Exercise:
Find pulling force with
δ=0.04 mm
δ=0.08 mm
δ=0.12 mm
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
μr
1.05
Br
12800 Gs = 1.28 T
Hc
Br / (μ0μr) = 970 kA/m
F
0.49 lbs = 1.8 N
26
One Side Magnet: The Halbach Array
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Example 2: Refrigerator Magnet
(Triangle Halbach Array)
Given:
D = 1 mm,
H = 1 mm,
B = 20 mm,
Magnet material - N35,
Steel fridge door thickness 1 mm,
relative magnetic permeability 1000.
Coercive HC=100 000 A/m
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Refrigerator Magnet (Halbach Array)
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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Comparing pulled force
Manufacturer Data
QuickField Calculation
Film size = 2cm * 0.5cm = 1cm2
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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How to magnetize the fridge magnet?
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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How the magnetizer looks like?
SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM
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