Lect08

(1)

EEE 498/598

Overview of Electrical

Engineering

Lecture 8: Magnetostatics:

Mutual And Self-inductance;

Magnetic Fields In Material

Media; Magnetostatic

Boundary Conditions;

Magnetic Forces And Torques

(2)

Lecture 8 Objectives

 To continue our study of To continue our study of

magnetostatics with mutual and magnetostatics with mutual and

self-inductance; magnetic fields self-inductance; magnetic fields

in material media; magnetostatic in material media; magnetostatic

boundary conditions; magnetic boundary conditions; magnetic

forces and torques. forces and torques.

(3)

Flux Linkage

 Consider two magnetically coupled circuitsConsider two magnetically coupled circuits

C₁ I₁

S₁ _S

2 C₂

(4)

Flux Linkage (Cont’d)

 The magnetic flux produced _{The magnetic flux produced}_I_I₁₁ linking the surface _{linking the surface}_S_S₂₂ is given by_{is given by}

 If the circuit _{If the circuit}_C_C₂₂ comprises _comprises_N_N₂₂ turns and the circuit _{turns and the circuit}_C_C₁₁ comprises _comprises_N_N₁₁ turns, then the total _{turns, then the total}

flux linkage is given by flux linkage is given by









2

2 1

12

S

s

d

B











(5)

Mutual Inductance

 The The mutual inductancemutual inductance between two circuits between two circuits

is the magnetic flux linkage to one circuit is the magnetic flux linkage to one circuit

per unit current in the other circuit: per unit current in the other circuit:

1

12 2

1 1

12 12

I

N

I

(6)

Neumann Formula for

Mutual Inductance



















2 2 2 1 1 2 1 2 1 1 2 1 1 12 2 1 1 12 12 C S

l

d

A

I

N

s

d

B

I

N

I

N

I

L

(7)

Neumann Formula for

Mutual Inductance

(Cont’d)





1 12 1 1 0 1

4

_C

R

l

d

I

A





 











1 2 2 12 2 1 2 1 0 2 1 1 2 1 12

4

_{C C}

C

R

l

d

l

d

N

l

d

A

I

N

L





(8)

Neumann Formula for

Mutual Inductance

(Cont’d)

 The Neumann formula for The Neumann formula for

mutual inductance tells us that mutual inductance tells us that

 LL₁₂₁₂ = L = L₂₁₂₁

 the mutual inductance depends the mutual inductance depends

only on the geometry of the

conductors and not on the current

(9)

Self Inductance

 Self inductance_{Self inductance} is a special case of mutual is a special case of mutual inductance.

inductance.

 The The self inductance_{self inductance} of a circuit is the ratio of the self of a circuit is the ratio of the self magnetic flux linkage to the current producing it:

magnetic flux linkage to the current producing it:

1

11 2

1 1

11 11

I

N

I

(10)

Self Inductance (Cont’d)

 For an isolated circuit, we call For an isolated circuit, we call the self inductance,

the self inductance, inductanceinductance,,

and evaluate it using and evaluate it using

I

N

I

L









(11)

Generation of Magnetic

Field

I

iron core

air gap with constant B field

N S

(12)

Equivalent of a Magnetic

Dipole

I

N

S

• Magnetic dipole can be viewed as a pair of magnetic charges by analogy with electric dipole.

(13)

Forces Exerted on a

Magnetic Dipole in a

Magnetic Field

N

(14)

Current Loops (Magnetic

Dipoles) in Atoms

 Electron orbiting nucleusElectron orbiting nucleus  Electron spinElectron spin

 Nuclear spinNuclear spin _negligible

 A complete understanding of these atomic mechanisms requires application of quantum mechanics.

(15)

Current Loops (Magnetic

Dipoles) in Atoms

(Cont’d)

 In the absence of an applied magnetic In the absence of an applied magnetic

field, the infinitesimal magnetic

dipoles

dipoles in most materials_{in most materials} are _are

randomly oriented, giving a net

macroscopic magnetization of zero.

macroscopic magnetization of zero.  When an external magnetic field is When an external magnetic field is

applied, the magnetic dipoles have a

tendency to align themselves with the

applied magnetic field.

(16)

Magnetized Materials

 A material is said to be A material is said to be magnetized magnetized

when induced magnetic dipoles

are present.

 The presence of the induced The presence of the induced

magnetic dipoles modifies the

magnetic field both inside and

outside of the magnetized

material.

(17)

Permanent Magnets

 Most materials lose their Most materials lose their

magnetization when the external magnetization when the external

magnetic field is removed. magnetic field is removed.  A material that remains A material that remains

magnetized in the absence of an magnetized in the absence of an

applied magnetic field is called a applied magnetic field is called a

permanent magnet

(18)

Magnetization Vector

 The The magnetizationmagnetization oror net magnetic net magnetic dipole moment per unit volume

dipole moment per unit volume is given _{is given} by

by

m

N

M



average magnetic dipole moment [Am2]

Number of

dipoles per unit volume [m-3]

(19)

Magnetic Materials

 The effect of an applied electric field on a The effect of an applied electric field on a

magnetic

magnetic material is to create a net magnetic material is to create a net magnetic dipole moment per unit volume

dipole moment per unit volume MM..

 The dipole moment distribution sets up The dipole moment distribution sets up

induced secondary fields:

ind app

B





Total field _{Field in free space} due to sources

Field due to

induced magnetic dipoles

(20)

Volume and Surface

Magnetization Currents

 A magnetized material may be A magnetized material may be

represented as an equivalent volume (

represented as an equivalent volume (JJm_m) )

and surface (

and surface (JJsm_sm) magnetization currents.) magnetization currents.

 These charge distributions are related to These charge distributions are related to

the magnetization vector by the magnetization vector by

n sm

m

a M

J

M J

ˆ  

  

(21)

Volume and Surface

Magnetization Currents

(Cont’d)

 Magnetization currents are equivalent Magnetization currents are equivalent

currents that account for the effect of the

magnetized material, and are analogous to

equivalent volume and surface polarization

charge densities in a polarized dielectric.

charge densities in a polarized dielectric.  If the magnetization vector is constant If the magnetization vector is constant

throughout a magnetized material, then

the volume magnetization current density

is zero, but the surface magnetization

current is nonzero.

(22)

Ampere’s Law in

Magnetic Media

 Ampere’s law in differential form Ampere’s law in differential form in free in free space

space::

 Ampere’s law in differential form Ampere’s law in differential form in a in a magnetized material

magnetized material::

J

B





₀







J

_m



B







(23)

Magnetic Field

Intensity

• define the magnetic field intensity as

M

B

H





0











J M B J M B M J J J B _m                       0 0 0 0 0 0      

(24)

General Forms of

Ampere’s Law

 The general form of Ampere’s law in differential form becomes_{The general form of Ampere’s law in differential form becomes}  _{The general form of Ampere’s law in integral form becomes}_{The general form of Ampere’s law in integral form becomes}

J

H







encl S

C

I

s

d

J

l

d

H











(25)

Permeability Concept

 For some materials, the For some materials, the net net

magnetic dipole moment per unit volume

is proportional to the

is proportional to the H_H field_field

H

M





_m

magnetic susceptibility (dimensionless)

• the units of both M and

(26)

Permeability Concept

(Cont’d)

 Assuming thatAssuming that

we havewe have

 The parameter The parameter  is the is the

permeability

permeability of the material._{of the material.}

H

M





_m



H

M







H

(27)

Permeability Concept

(Cont’d)

 The concepts of magnetization and magnetic The concepts of magnetization and magnetic

dipole moment distribution are introduced to

relate microscopic phenomena to the

macroscopic fields.

 The introduction of permeabilityThe introduction of permeability eliminates the eliminates the

need for us to explicitly consider microscopic

effects.

 Knowing the permeabilityKnowing the permeability of a magnetic material of a magnetic material

tells us all we need to know from the point of

view of macroscopic electromagnetics.

(28)

Relative Permeability

 The The relative permeabilityrelative permeability of a of a

magnetic material is the ratio of magnetic material is the ratio of

the permeability of the magnetic the permeability of the magnetic

material to the permeability of material to the permeability of

free space free space

0



(29)

Diamagnetic Materials

 In the absence of applied magnetic field, In the absence of applied magnetic field,

each atom has net zero magnetic dipole

moment.

 In the presence of an applied magnetic In the presence of an applied magnetic

field, the angular velocities of the

electronic orbits are changed.

 These induced magnetic dipole moments These induced magnetic dipole moments

align themselves

align themselves oppositeopposite to the applied to the applied field.

field.

(30)

Diamagnetic Materials

(Cont’d)

 Usually, diamagnetism is a very Usually, diamagnetism is a very

miniscule effect in natural

materials - that is

materials - that is r_r  1. 1.

 Diamagnetism can be a big effect in Diamagnetism can be a big effect in

superconductors

superconductors and in _{and in}artificial materialsartificial materials._.

 Diamagnetic materials are repelled Diamagnetic materials are repelled

from either pole of a magnet.

(31)

Paramagnetic Materials

 In the absence of applied magnetic field, In the absence of applied magnetic field, each atom has net non-zero (but weak)

each atom has net non-zero (but weak)

magnetic dipole moment. These magnetic

dipoles moments are randomly oriented

so that the net macroscopic magnetization

is zero.

 In the presence of an applied magnetic In the presence of an applied magnetic field, the magnetic dipoles align

field, the magnetic dipoles align

themselves with the applied field so that

themselves with the applied field so that 

(32)

Paramagnetic Materials

(Cont’d)

 Usually, paramagnetism is a very Usually, paramagnetism is a very miniscule effect in natural

miniscule effect in natural materials - that is

materials - that is r_r  1. 1.

 Paramagnetic materials are Paramagnetic materials are

(weakly) attracted to either pole (weakly) attracted to either pole

of a magnet. of a magnet.

(33)

Ferromagnetic

Materials

 Ferromagnetic materials include iron, nickel Ferromagnetic materials include iron, nickel

and cobalt and compounds containing these

elements.

each atom has very strong magnetic dipole

moments due to uncompensated electron

spins.

 Regions of many atoms with aligned dipole Regions of many atoms with aligned dipole

moments called

moments called domainsdomains form. form.

 In the absence of applied magnetic field, the In the absence of applied magnetic field, the

domains

domains are randomly oriented so that the net are randomly oriented so that the net macroscopic magnetization is zero.

(34)

Ferromagnetic Materials

(Cont’d)

 In the presence of an applied In the presence of an applied

magnetic field, the domains align

themselves with the applied field.

 The effect is a very strong one with The effect is a very strong one with



m_m >> 0 and >> 0 and rr >> 1. >> 1.

 Ferromagnetic materials are Ferromagnetic materials are

strongly attracted to either pole of

a magnet.

(35)

Ferromagnetic Materials

(Cont’d)

 In ferromagnetic materials:In ferromagnetic materials:

 the permeability is much larger the permeability is much larger

than the permeability of free space

than the permeability of free space  the permeability is very non-linearthe permeability is very non-linear  the permeability depends on the the permeability depends on the

previous history of the material

(36)

Ferromagnetic

Materials (Cont’d)

 In ferromagnetic materials, the relationship In ferromagnetic materials, the relationship BB

=

= H_H can be illustrated by means of a can be illustrated by means of a magnetization curve

magnetization curve (also called (also called hysteresis loophysteresis loop).).

B

H

coercivity remanence

(37)

Ferromagnetic Materials

(Cont’d)

 Remanence (retentivity)Remanence (retentivity) is the value is the value of

of BB when when HH is zero. is zero.

 CoercivityCoercivity is the value of is the value of HH when when BB is zero.

is zero.

 The The hysteresishysteresis phenomenon can be phenomenon can be used to distinguish between two used to distinguish between two

states. states.

(38)

Antiferromagnetic

Materials

 Antiferromagnetic materials include Antiferromagnetic materials include

chromium and manganese.

 In antiferromagnetic materials, the In antiferromagnetic materials, the

magnetic moments of individual atoms

are strong, but adjacent atoms align in

opposite directions.

 The macroscopic magnetization of the The macroscopic magnetization of the

material is negligible even in the

presence of an applied field.

(39)

Ferrimagnetic Materials

 Ferrimagnetic materials include Ferrimagnetic materials include

oxides of iron, nickel, or cobalt.

 The magnetic moments of adjacent The magnetic moments of adjacent

atoms are aligned opposite to each

other, but there is incomplete

cancellation of the moments because

they are not equal.

 Thus, there is a net magnetic moment Thus, there is a net magnetic moment

within a domain.

(40)

Ferrimagnetic Materials

(Cont’d)

the

the domainsdomains are randomly oriented so that _{are randomly oriented so that}

the net macroscopic magnetization is zero.

the net macroscopic magnetization is zero.  In the presence of an applied magnetic In the presence of an applied magnetic

field, the domains align themselves with

the applied field.

 The magnetic effects are weaker than in The magnetic effects are weaker than in

ferromagnetic materials, but are still

substantial.

(41)

Ferrites

 Ferrites are the most useful Ferrites are the most useful

ferrimagnetic materials.

 Ferrites are ceramic material containing Ferrites are ceramic material containing

compounds of iron.

 Ferrites are non-conducting magnetic Ferrites are non-conducting magnetic

media so eddy current and ohmic losses

are less than for ferromagnetic materials.

 Ferrites are often used as transformer Ferrites are often used as transformer

cores at radio frequencies (RF).

(42)

Fundamental Laws of

Magnetostatics in

Integral Form

0 









S S C

s

d

B

s

d

J

l

d

H

B





Gauss’s law for magnetic field

Ampere’s law

(43)

Fundamental Laws of

Magnetostatics in

Differential Form

0

 



 



B

J H

H

B





Ampere’s law

Gauss’s law for magnetic field

(44)

Fundamental Laws of

Magnetostatics

 The integral forms of the fundamental laws The integral forms of the fundamental laws

are more general because they apply over

regions of space. The differential forms are

only valid at a point.

 From the integral forms of the fundamental From the integral forms of the fundamental

laws both the differential equations

governing the field within a medium and

the boundary conditions at the interface

between two media can be derived.

(45)

Boundary Conditions

1



2



n

a

ˆ

 Within a Within a

homogeneous homogeneous

medium, there are no medium, there are no

abrupt changes in abrupt changes in HH

or

or BB. However, at the _{. However, at the} interface between two interface between two

different media different media

(having two different (having two different

values of

values of , it is _{, it is}

obvious that one or obvious that one or both of these must both of these must

change abruptly. change abruptly.

(46)

Boundary Conditions

(Cont’d)

 The normal component of a The normal component of a solenoidalsolenoidal

vector field is continuous across a

material interface:

 The tangential component of a The tangential component of a

conservative

conservative vector field is continuous _{vector field is continuous}

across a material interface:

across a material interface: n

n

B

₁



₂

0 ,

2

1t  H t J s 

(47)

Boundary Conditions

(Cont’d)

 The tangential component of The tangential component of HH is is

continuous across a material

interface, unless a surface current

exists at the interface.

 When a surface current exists at When a surface current exists at

the interface, the BC becomes





_s

n H H J

(48)

Boundary Conditions

(Cont’d)

 In a perfect conductor, both the In a perfect conductor, both the electric and magnetic fields must

electric and magnetic fields must

vanish in its interior. Thus,

s n

n

J

H

a

B







ˆ

0

• a surface current must exist

• the magnetic field just outside the perfect

conductor must be tangential to it.

(49)

Overview of Magnetic

Forces and Torques

 The experimental basis of magnetostatics The experimental basis of magnetostatics

is the fact that current carrying wires is the fact that current carrying wires

exert forces on one another as described exert forces on one another as described

by Ampere’s law of force. by Ampere’s law of force.

 A number of devices are based on the A number of devices are based on the

forces and torques produced by static forces and torques produced by static

magnetic fields including DC electric magnetic fields including DC electric

motors and electrical instruments such motors and electrical instruments such

as voltmeters and ammeters. as voltmeters and ammeters.

(50)

Magnetic Forces on

Moving Charges

 The force on a charged particle The force on a charged particle moving with velocity

moving with velocity vv in a in a magnetostatic field

magnetostatic field

characteristic by magnetic flux characteristic by magnetic flux

density

density BB is given by_{is given by}

B

v

q

(51)

Lorentz Force Equation

 The force on a charged particle The force on a charged particle moving with velocity

moving with velocity vv in a in a

region where there exists both a region where there exists both a

magnetostatic field

magnetostatic field BB and an _{and an} electrostatic field

electrostatic field EE is given by_{is given by}



E

v

B



q

(52)

Lorentz Force Equation

(Cont’d)

 The Lorentz force equation can be The Lorentz force equation can be

used to obtain the equations of

motion for charged particles in

various devices including cathode ray

tubes (CRTs), microwave klystrons

and magnetrons, and cyclotrons.

 The Lorentz force equation also The Lorentz force equation also

explains the

explains the Hall effectHall effect in conductors in conductors and semiconductors.

(53)

Magnetic Force on

Current-Carrying

Conductors

 When a current carrying wire is placed in When a current carrying wire is placed in a region permeated by a magnetic field, it

a region permeated by a magnetic field, it

experiences a net magnetic force given by

B

l

Id

F

C

(54)

Torque on a Current

Carrying Loop

 Consider a small Consider a small

rectangular

current carrying

loop in a region

permeated by a

magnetic field.

x y

I

B Fm1

F_m2 L W

(55)

Torque on a Current

Carrying Loop (Cont’d)

 Assuming a uniform magnetic field, the force on the upper wire is_{Assuming a uniform magnetic field, the force on the upper wire is}

 The force on the lower wire isThe force on the lower wire is

ILB

a

F

_m₁





ˆ

_z

ILB

a

(56)

Torque on a Current

Carrying Loop (Cont’d)

 The forces acting on the loop The forces acting on the loop have a tendency to cause the have a tendency to cause the

loop to rotate about the x-axis. loop to rotate about the x-axis.

 The quantitative measure of the The quantitative measure of the tendency of a force to cause or tendency of a force to cause or

change rotational motion is change rotational motion is

torque

(57)

F

r

T





Torque on a Current

Carrying Loop (Cont’d)

 The The torquetorque acting on a body with acting on a body with respect to a reference axis is

respect to a reference axis is given by

given by

distance vector from the reference axis

(58)

B

ILW

a

ILWB

a

F

W

a

F

W

a

T

z x m y m y

















ˆ

2 ˆ

₁ ₂

magnetic dipole moment of loop

Torque on a Current

Carrying Loop (Cont’d)

(59)

B

m

T





Torque on a Current

Carrying Loop (Cont’d)

 The torque acting on the loop The torque acting on the loop tries to align the magnetic

tries to align the magnetic

dipole moment of the loop with dipole moment of the loop with

the B field the B field

holds in general regardless of

(60)

Energy Stored in

Magnetic Field

 The magnetic energy stored in a The magnetic energy stored in a region permeated by a magnetic region permeated by a magnetic

field is given by field is given by

dv

H

dv

H

B

W

V V

m











2

1

2

(61)

Energy Stored in an

Inductor

 The magnetic energy stored in The magnetic energy stored in an inductor is given by

an inductor is given by

2