05 magnetostatics
DESCRIPTION
150280611-TRANSCRIPT
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Magnetostatics
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UCF Volume Current
LSVQ VV
mobility called is
tQI
vSIJ V
vSt
LStQI VV
dSJI
Definition:
Volume current density
tyconductivi VEEJ V
vJ VEv
V
S
v
L
J
or
1 resistivity
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UCF Surface (Sheet) Current
dwI S laJ dSJI
Volume current density for surface current
)(zSJJ
laSJ
SJ
J
Examplez
x
w
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UCF Line Current
Volume current density for line current
zaJ )()( yxI Example
I
z
I
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UCFConservation of Charge
Any current leaving a closed surface S implies a decrease of charge within that closed surface
V
VS
dVdtddSJ
dtdQI
tV
J
V
VV
dVdtddVJ
V S
JdS
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UCFSteady State Current
0 J
C
G
QI
DJ
EDEJ
0t
For steady state current
V EJ 0)( VIf is uniform, 02 V Laplaces equation
A steady state current problem is analogues to an electrostatic problem
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UCF Example for Steady State Current
zaEJ dV2
zaE dVV 2
Find resistance (conductance)
zd
VzV 2)(
d
1Then
)()(
2
2top
22top
dS
VI
RG
SdVSJI
dV
dVJ
zz aa
0
V2
zaS
Recall for parallel plate capacitor
dSC
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UCF Biot-Savarts Law (1)'dlI
0
'r r
RBd
20 '
4 RId RadlB
20 '4 RI RadlB
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UCFBiot-Savarts Law (2)
'''' dVJdSI Jdldl
0
'r r
R Bd
20 '
4 RdVd RaJB
20 ')'(4 RdV RarJB
'dS
'dl
)J(r'
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UCFBiot-Savarts Law (3)
'''' dSdwJI SS Jdldl
0
'r r
R Bd
20 '
4 RdSd S RaJB
20 ')'(4 RdSS RarJB
'dw
'dl
)(r'J S
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UCFBiot-Savarts Law (4)
Example 1
Example 2
Example 3
Find magnetic flux density on xoy plane (z=0) from a line current I on z axis from -a to a.
z
xy
0
-a
a
Find magnetic flux density from infinite line current I on z axis.
Find magnetic flux density on z axis from loop current in xoy plane.
z
x
y0
a
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UCFMagnetic Gausss Law
0S
dSB
V S
BdS
Integral form
0)( V
dVB
0 B differential formFrom Biot-Savarts Law
Gausss Theorem
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UCFAmperes Circuital Law
LdL
enclosedIL
dLH
HI
Integral form
LS
dLHdSH )(From Stokess Theorem
SS
dSJdSH )(
JB 0
Differential form
From Biot-Savarts Law
Define HB 0JH
Can be extended to general magnetic media.
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UCFConstitutive Relationship
rHHB r 0 permeability
relative permeability
In isotropic media
material ticferromagne :1
1 materialsmost For material icparamagnet :1,1
vacuum:1material cdiamagneti :1,1
r
r
r
r
r
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UCFScalar Potential for Magnetostatics
02 mV
0 mVB
0 HIn isotropic media
mVH
Or
mV HB
0mV Then
If the permeability distribution is uniform
Laplaces Equation
Can only define for source free region
Define
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UCFVector Potential for Magnetostatics
JA 20 A
From math, we have
and
0 Bwe can define
0)( TSince
AB where A is called vector potential.In simple media, from JH
JA )1(
HB
If is uniform, JA )(1AAA 2 )()(From math
Define gauge
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UCF Solution of Vector Potential
( )4
V dVVR
r
Since the solution of Poissons Equation VV 2
JA 2For equationIn Cartesian coordinate system
zyx
zyx
aaaJaaaA
zyx
zyx
JJJAAA
2222
the solution for equation (1) is
is
To be proved in the following slide.
(1)
( )4
dVR
J rA
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UCF Solution of Poissons Equation
RVdrV
VdrdQ
V
V
4)(
)(
r
O
r )(rV
Rvolume charge density
Vd
This is the solution of Poissons Equation
VV 2
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UCF Magnetic Boundary Conditions
current surfacefor
current or volume current nofor 0
21
LJLHLH
S
tt
dSJdLH2
0h
0Volume
0dSB 20for 0)(
)(
S12n
S12n
JHHaJHHa
1 LMedia 1Media 2
H1
H2
H1t
H2t
0h1
S
Media 1
Media 2
B1
B2
B1n
B2n
021 SBSB nn
0)(or 0
12
12
BBannn BB
(1)
(2)
na
na
JS
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UCF Perfect Magnetic Conductor (PMC)
0Han 0tH
0H
Media B
mSBan
(1)
(2)
na
PMC
(introduced)
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UCF Energy Stored in Magnetostatic Field
HB 21
mw
V mm dVwW
Energy density
Total energy
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Ferromagnetic Materials
T. Gonen, Electric machines with MatLab, 2nd Edition, p. 68, CRC press, 2012.
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Magnetic Domains
T. Gonen, Electric machines with MatLab, 2nd Edition, p. 69, CRC press, 2012.
(a) Magnetic domains oriented randomly;(b) Magnetic domains becoming magnetized;(c) Magnetic domains fully magnetized (lined up) by the magnetic field H
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Hysteresis Loop (I)
Br remnant flux or residue fluxHc coercive flux or coercivity
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Hysteresis Loop (II)
0CH
soft material
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dVE
AIJ
AdR
RIV
resistance
EJ
dV
AI
Differential form:
I
A
+ V
E
dSince and , we have
or
where
integral form
Ohms Law
_
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Magnetic Ohms Law
dH F
AB
AdR
RF
reluctance
HB
dAF
Differential form:
A
+
H
dSince and , we have
or
where
integral form
F
dSB
F is called magnetomotive force (MMF), whose unit is A.
_
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Magnetic Circuit
length d
RHd
NIHd
NIL
F
dlH
R
R
NIA
d
area A
+_
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Electric vs Magnetic Circuits
Ni
+_
_
-
Inductance
dsB
IN
IL
NB
Flux linkageI
total
NIR
total
NLR
2
NI total
+_
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Mutual Inductance
ipIN
ipII
L
pi
i
pi
iii
,0
,0
2
R
nnnnnn
ninjij
iiiii
nn
nn
ILILIL
ILILILIL
ILILILILILIL
2211
11
22221212
12121111
Self inductance
I1
I2
In
n
jpII
L pj
iij ,0Mutual inductance
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Example 1
Find magnetic flux density in two airgaps.
1g
1w 2w2g
r
N turns
I
l
1gR 2gR
1 2
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Example 2
cm 10 cm 200 cm 1A 10 1000 wlgINFind flux and airgap flux density Bg.
w
wlA
Ag
g0R
g
NIR2
ABg
gR
gR
gR2
l
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Example 3
5% cross-section increase for fringing in airgap
Find: (a) total reluctance of the flux path;(b) current required to produce B = 0.5 T in the air gap; (c) inductance of the coil.
r=2000
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Example 4
(1) How much is current required to produce 0.016Wb of flux in the core?(2) What is cores relative permeability at that current level?(3) What is its reluctance and inductance at this level?
M5 Steel at DC
N = 400, A = 150 cm2lc = 55 cm
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Example 5
2221212
2121111
ILILILIL
r
N1 turns
g
N2 turns 0g
g
gAR
(1) Let I2 = 0
r
N1 turns
I1
N2 turns
Find self and mutual inductances.
(2) Let I1 = 0
+_
11INgR
r
N1 turns
I2
N2 turns
+_
22 INgR
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Example 6
2221212
2121111
ILILILIL
1g
2g
r
N1 turns
I1N2 turns
I2
10
11
gg A
gR
Find self and mutual inductances.
20
22
gg A
gR
(1) Let I2 = 0
1gR2gR
(2) Let I1 = 0