05 Magnetostatics

Magnetostatics

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

UCF Surface (Sheet) Current

dwI S laJ dSJI

Volume current density for surface current

)(zSJJ

laSJ

SJ

J

Examplez

x

w

UCF Line Current

Volume current density for line current

zaJ )()( yxI Example

I

z

I

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

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

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

UCF Biot-Savarts Law (1)'dlI

0

'r r

RBd

20 '

4 RId RadlB

20 '4 RI RadlB

UCFBiot-Savarts Law (2)

'''' dVJdSI Jdldl

0

'r r

R Bd

20 '

4 RdVd RaJB

20 ')'(4 RdV RarJB

'dS

'dl

)J(r'


'''' dSdwJI SS Jdldl

0

'r r

R Bd

20 '

4 RdSd S RaJB

20 ')'(4 RdSS RarJB

'dw

'dl

)(r'J S


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

UCFMagnetic Gausss Law

0S

dSB

V S

BdS

Integral form

0)( V

dVB

0 B differential formFrom Biot-Savarts Law

Gausss Theorem

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.

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

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

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

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

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

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

UCF Perfect Magnetic Conductor (PMC)

0Han 0tH

0H

Media B

mSBan

(1)

(2)

na

PMC

(introduced)

UCF Energy Stored in Magnetostatic Field

HB 21

mw

V mm dVwW

Energy density

Total energy

Ferromagnetic Materials

T. Gonen, Electric machines with MatLab, 2nd Edition, p. 68, CRC press, 2012.

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

Hysteresis Loop (I)

Br remnant flux or residue fluxHc coercive flux or coercivity

Hysteresis Loop (II)

0CH

soft material

dVE

AIJ

AdR

RIV

resistance

EJ

dV

AI

Differential form:

I

A

+ V

E

dSince and , we have

or

where

integral form

Ohms Law

_

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.

_

Magnetic Circuit

length d

RHd

NIHd

NIL

F

dlH

R

R

NIA

d

area A

+_

Electric vs Magnetic Circuits

Ni

+_

_

Inductance

dsB

IN

IL

NB

Flux linkageI

total

NIR

total

NLR

2

NI total

+_

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

Example 1

Find magnetic flux density in two airgaps.

1g

1w 2w2g

r

N turns

I

l

1gR 2gR

1 2

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

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

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

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

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

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05 Magnetostatics