1. Fundamentals of Magnetics

Fundamentals ofMAGNETICS

Revised October 6, 2008

EEL 3211© 2008, Henry Zmuda

1. Fundamentals of Magnetics 1

Magnetic fields provide the fundamental mechanism by whichMagnetic fields provide the fundamental mechanism by which energy is converted from one from to another by means of:

•motors (electrical energy mechanical energy )

•generators (mechanical energy electrical energy)g ( gy gy)

•transformers (electrical energy electrical energy )



Four basic principles:

1. A current flowing in a wire produces a magnetic field in the vicinity of the wire.y

2. A wire in the presence of a time-varying magnetic field will induce a voltage across the wire (transformer action).

3 A current-carrying wire in the presence of a magnetic field3. A current-carrying wire in the presence of a magnetic field experiences a force exerted on the wire (motor action)

4. A a wire moving in the presence of a magnetic field has a l i d d i ( i )voltage induced upon it (generator action).

Each of these will be considered in detail.



The fundamental postulate governing the electrical behavior of i fi ld i i b A ’ lmagnetic fields is given by Ampere’s law:

enclosedC

H d I

: Vector Magnetic Field (amperes/meter)H

g ( p )

: Vector Closed Path of Integration (meters)N t C t l d bI C

: Net Current ecnlosed by enclosedI C



I

CC

H d I

I H

CC

Side view (along axis) H( g )



C

I H r

I

rH

Right Hand Rule Convention



C

I H r

rr

I 22C

IH d rH r I H rr



We assumed that field produced by the wire exists in air. The h f h i fi ld d d l d d hstrength of the magnetic field produced also depends on the

material in which the field exists. Certain materials, called ferromagnetic materials, have the property of confining the f g , p p y gfield inside of it. Ferromagnetic materials are characterized by their permeability, .

The for air (or free space):

7 Henrys74 10oHenrysmeter



The ratio of the permeability for a given material to that of free i k h l b lspace is known as the relative permeability,

r

o

For modern ferromagnetic materials,

1000 8000r

For an idealized case, r



Note the similarity:

From circuit theory, the magnetic energy stored by an inductor is

212mW L I2

I L

From electromagnetic theory, the magnetic energy stored by a magnetic field is

212mW H



The relationship that described the effect of the permeability ( h i l) h h i fi ld i i b h(the material) has on the magnetic field H is given by the relationship

B H

where B is the magnetic flux density.

Units:

2

Henry Ampere Ampere-Henrymeter meter meter

B H H

2

Weber Teslameter

H H



B is the magnetic flux density. If we integrate the density over b i h l fl an area we obtain the total magnetic flux

ˆ ˆB d B d B d d

A A AB da B nda B ndxdy

n̂

dB

A dx

dy



EXAMPLE

Mean path length c

Current i i

Cross sectional

N turns

Cross-sectionalarea AMagnetic

Core



EXAMPLE

enclosed ccC

NiH d I H Ni H

cC

NiB B H H

c

A

NiAB da

i

c

Ac

iiiiiiii

xx

xxxxxx ix



Circuit analogy for Magnetic circuits:

The simplifications and idealizations discussed imply that i l i i d l b d dsimple circuit models can be used to study magnetic circuits.

This idea is used extensively in the design of motors and y ggenerators to simplify what would otherwise be an extremely complicated electromagnetic problem.




10 turns




I Flux

+_V R +

_NiY c

Ae

Magnetomotive Reluctance

Flux

Force Reluctance

VIR

c c

NiA ANi

c c

Ye



e


Electrical Circuit Magnetic Circuit

Electromotive Force (emf)

VMagnetomotive Force (mmf)

NiYCurrent

IFlux

Y

eResistance ReluctanceVR

I I




Note the polarity – determined via a modified right-hand-rule

I

+_V+_V

I

+_

YII



Jargon – Just as the conductance is the reciprocal of the resistance,

1GR

the permeance, is the reciprocal of the reluctance.

R

cp p

1c ce

This circuit analogy is always an approximation to the real system.





mean path5 + 20 + 2.5 = 27.5

7.5 + 15 + 7.5 = 300



ANi

Ye

1c Y

e

left top right bottom

eY

e e e e eleft top right bottom



top bottome e

e

rightelefte





Magnetic cores often require “gaps”.

N

Ae

“ideal” gapgg

A

e

Sgap

o A e



Fringing fields in a gap. Note that the cross-sectional area of the gap is greater than that of the metal.g p g

N “real” gapgapg

Ae

eff

eff

o A

A A

g

S, 1.05ffTypically A A, 1.05ffeTypically A A



i

Model:core

c

Ae

+_NiY

gape

core A

gapg p

o effAe





Mean path0.37 m 0.37 m

0.37 m0.37 m

0.37 m 0.37 m



+_

NiY



1 / 3e 3 / 3e

e

5e1 / 6e 3 / 6e

left right

+_

2e4e

Ni/ 3e/ 6e / 3e / 6e

fgap

ggap

1 / 3e1 / 6e3 / 3e 3 / 6e



1e 4e5e

+_

2e e2 3e



1e 4e 5e

+_

2e eNi

2 3e

TOT 5 1 2 3 4 e e e e e e







Example – A Simple Synchronous Machine – Assume that h d h i fi i bili i d h ithe rotor and stator have infinite permeability. Find the air-gap

flux and the flux density Bg. Use I = 10 A, N = 1000 turns, gap length g = 1 cm, and Ag = 2000 cm2.g p g g , g



Example – A Simple Synchronous Machine

gap0, ,2 2

gap o gapcgap

o gap gap gap

NI AA A

Y

e ee

gape

o gap gap gap

gap

erotore

e+_

statore statore

Y

gape



Example – A Simple Synchronous Machine

2 2o gap

gap

NI A

Ye

7

2 2

1000 10 4 10 0.2 0 13

gapgap gap

Webers

e

0.132 0.01

Webers

0.13 0.650 2

gapgapB Tesla

A

0.2gapA



Real Magnetic Materials



Recall from Slide 14:

NiH Y

c c

H

and

andNiBA A AHA Y

cc



Saturation or Magnetization Curve

Saturated Region

A Y

Linear

Y

NiY

LinearRegion

NiY

The permeability is constant only for air (o).



Recall from Slide 14:

NiH Y

c c

H

and

an

BA H

dNi AA A Y

c c




B Saturated Region

Slope = o

Linear

The slope of this curve definesthe permeability of a given material

Slope =

H

LinearRegion

HSince generators/motors rely on the flux to produce voltage/torque, we wish to produce as much flux as possible. In practice we operate near the knee of the curve



but not in the saturated region.


B Saturated Region

Linear

The slope of this curve definesthe permeability of a given material

H

LinearRegion

HSince generators/motors rely on the flux to produce voltage/torque, we wish to produce as much flux as possible. In practice we operate near the knee of the curve



but not in the saturated region. Why?

.B vs HH

.B vs Ho H

SaturatedRegion

LinearRegion



The advantage of using a ferromagnetic material for cores in electric machines and transformers is that one gets many times more flux formachines and transformers is that one gets many times more flux for a given magnetomotive force (mmf) with iron than with air. However, if the resulting flux has to be (approximately) proportional

h li d f h h b d i h dto the applied mmf, then the core must be operated in the unsaturated region of the magnetization curve.



Hysteresis Effects in Magnetic Materials

i2

t1

2

, B2

, HY

2

1 , HY




i2

t1 3

, B2

, HY

2

3

, HY1



Hysteresis Effects in Magnetic Materials ii

2

t1 3

4

, B2

, HY3

11

4



4


i2

6

t1 3

45

4, B

26

, HY

3

1

5

1

4




Residual Flux

Coercive mmf

Y

Coercive mmf

A Typical Hysteresis Loop – Present flux in the core is determined not only by the applied mmf but also on the previous history of the fluxin the core.



What causes hysteresis effects?MODELMODEL

The permanent magnetic moment of an atom comes about from:

Electron orbitElectron spin (primary contributor in solids)Nuclear dipole momentNuclear dipole moment

Ferromagnetic materials (Fe, Ni, Co, Cd., & alloys)These tend to be high conductivity metals (like iron and steel) and also

it lquite lossy.

On a macroscopic scale, a piece of ferromagnetic material is made up of magnetic domains where each domain is in effect a smallof magnetic domains, where each domain is in effect a small permanent magnet. These domains are randomly oriented.



What causes hysteresis effects? With no externally applied mmf or magneticapplied mmf or magnetic field, individual magnetic domains are randomly li d d h flaligned and the net flux

is zero.

Wi h llWith an externally applied mmf or magnetic field, individual magnetic , gdomains align and a net flux occurs.

When no further alignment is possible,

iEEL 3211© 2008, Henry Zmuda


saturation occurs.

First Major Effectj

Faraday’s Law – Induced voltage due to a time-changing magnetic fieldtime-changing magnetic field.



Faraday’s Law – Induced voltage due to a time-changing magnetic field. g

i t

t




i t

t




A look ahead: Transformer Action

Pi t Si t

+ Pv t Sv t

PN SN P S

PN SN



Faraday’s Law: If flux passes through a turn of a coilFaraday s Law: If flux passes through a turn of a coil of wire, a voltage will be induced in the turn of wire that is directly proportional to the rate of change of the flux.

appliedinduced

de

d

applied t

AppliedFlux:

einduced is the induced voltage

induced dt+

-inducede

is the flux passing through the loop

li dd t 0appliedd t

dt



Lenz’s Law (explains the minus sign in Faraday’s Law)

First, a slight variant of the right-hand-rule:

II I



Lenz’s Law: For a short-circuited loop, the applied change of flux produces a current that would cause a flux opposing the

flux produces a current that would cause a flux opposing the original flux change. This a statement of the conservation of energy. Applied

applied t inducedB t

Flux:

i

0appliedd t

dt

induced

i dtInduced

Flux: induced tThe induced current is such as to OPPOSE the CHANGE inapplied field.



Lenz’s Law: The polarity of the induced voltage at the terminals of the coil is such that the current produced would cause a flux in a pdirection opposing the original flux.

Applied

i d dB t

ppFlux: applied t

+

_inducede

i

inducedB t

iInduced

Flux:

0appliedd t

dt

induced tAn induced electromotive force generates a current that induces a counter magnetic field that opposes the magnetic field generating the current



the current.

For N turns all with the same flux,

i t Applied Flux

0d t

dt

+

inducedde Ndt

dt

_

Induced (opposing) flux

N t h thi i l i l GENERATOREEL 3211© 2008, Henry Zmuda


Note how this is also a simple GENERATOR.

Flux Linkage:

d de N inducede N

dt dt

Flux Linkage: N



In practice the polarity of the induced voltage isIn practice, the polarity of the induced voltage is determined from physical considerations. As a result, the minus sign in Faraday’s law is often omitted. The

(Ch i f i d ) i h i itext (Chapman is from industry) omits the minus sign for the remainder of the book. This could be confusing.

inducedde Ndt

Text omits the minus sig .n

dt



Example – Consider a coil of wire (10 turns) wrapped around an ironcore. If a flux has a peak value of 0.1 Weber and varies sinusoidally at ap yfrequency of 60 Hz. What is the value and polarity of the inducedvoltage (flux is increasing in the direction shown).

10 turns



Solution –

0.1sin 120t t

10 0.1 120 cos 120inducedde N tdt

120 cos 120dt

t



Solution – (Note how this one-half of a transformer – the secondary)

ii +

inducede10 turns_

120 cos 120inducede t



Example – A Primitive AC Generator

B Fi ld Cross-Sectional Area A = 0.4 m2B – Field(1 Tesla)

10-Turn Coil

Axis of RotationS d 3600

Find the induced voltageat the terminals.



Speed = 3600 rpm


A = 0.4 m2

B = 1 T

Prc

ojectedos

AB da B A

ProjectedArea

t

3600 rpmAngular Speed:

t

2 1360060 sec

radians revolutions radians minutesecond minute revolution onds

120 377 60radians Hertzsecond




B Fi ldB – Field(1 Tesla)

A

A

cosB A B A




B = 1 T

A = 0.4 m2 cosAB da B A

t

cost BA t

cos

sininduced

t BA tde t N NBA tdt

1508sin377dt

t volts

What is the polarity of the induced emf?EEL 3211© 2008, Henry Zmuda


What is the polarity of the induced emf?

Polarity of induced emf:

d B Ade t N N

inducede t N Ndt dt

:

?

Two questions

is B A

?disdt



Polarity of induced emf: A mathematical approach

A

Convention for positive loop orientation



Polarity of induced emf: induced

d B Ae t N

dt

90 0B A

d B Ad

90 0B A

d B Ad

0 increasing

d B Addt dt

0 decreasingd B Ad

dt dt

A

A

B

B

+ _



_ +


In this example, is constant but is changing with time.B

B A

A

A

B

B



B cos cos


cosB A B A

d0B A

d

cos sind t tdt

sin

sin

d B A tdt

B A t

sin

induced

B A t

d B Ae t N

sin

induced dtN B A




iiLoopOrientation

_ +

Orientation

+ sininducede t N B A

inducede t




Approach used in text:Current direction opposesoriginal change in flux.

cos

sin

B A

dN N B A t

i

g gsin

decreasing

N N B A tdt

i

_ +

B

sininducede t N B A +

inducede t



Second Major Effectj

Lorentz Force Law: A magnetic field induces a force on a current-carrying wireinduces a force on a current-carrying wire in the presence of the field.



Lorentz Force Law – Force exerted on charges

EF qE

of little interest here

MF q v B



BB

B

C C

B

CA

A

Right-Hand Rule for determining the direction of A B C



Right-Hand Rule for determining the direction of F q v B





Lorentz Force Law – a more useful form for our application

is the velocity of the charge MF F q v B v q

s t e ve oc ty o t e c a geM q v v q

qq B B

q B Bt t

i B

i B



Lorentz Force Law

F i B

:F force

::

F forcei current

:

:

oriented conductor path length

B magnetic flux density

:B magnetic flux density



Lorentz Force Law BF i

x x x x x x x x x x x x x xx wire

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

xB into page

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x F

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x x x x x x x x x x x x x xx



Lorentz Force Law

F i B

F

B

sinF F i B i

Where is the angle between vectors and B



Lorentz Force Law

Recall Faraday’s Law: induceddedt

andB A

For constant B (with some hand-waving)

d dA d x dxB B B B vdt dt dt dt

What is the polarity?



Lorentz Force Law

d B B vdt

What is the polarity?



Lorentz Force Law

Recall: q v BFF qE Eq q

q q

F B

E v B

For constant B,

inducede E d v B induced



Lorentz Force Law

B inducede v B



Lorentz Force Law – An intuitive approach

wire x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x wire

+++++

F q v B

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

B into page v

B

+

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

xB into page inducede v B

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x_

____ F q v B

x x x x x x x x x x x x x xx

The direction for is such that is positive. (upwards in B v



this example)

Example – A Primitive Generator Revisit our previous example.

i Coil End LengthendF

B

iL

gCoil Side Length

i i

ix

endF

sin 90F Ni B

Bx

FEEL 3211© 2008, Henry Zmuda


sin 90endF Ni B endF

Example – A Primitive Generator

i Coil End Length iL

gCoil Side Length

F Bi i

sideFsideF

i

sideF NiLB

B




Axial view:

BdF NiLB

d

F

sideF NiLB

sideF xt



Torque

r

F

F

sin

T r F

T r F Newton Meters

sinT r F Newton Meters




B F NiLB d

F

sideF NiLB

sideF x

t

2 sin2

sin

T F

NBiL

sinsin ,

NBiLNBiA A L loop area



Observations:

•As the coil turns, the forces on the ends will vary with position, but will be equal and opposite, thus creating no net force on the coil as a whole, nor any torque about the axis of rotation.

•The forces on the sides of the coil will also be equal and oppositeThe forces on the sides of the coil will also be equal and opposite in direction, and will not vary with position, but will exert a torque about the axis of rotation.

•Note that the torque resists motion in the given direction, which generates a current. In other words, mechanical energy is being converted to electrical energy as the coil is turned.




Mechanical Power: sinmP T i NBA t

Electrical Power: sininducede t i i N B A

Mechanical Power = Electrical Power

M ch more on this laterMuch more on this later...



Documents

1. Fundamentals of Magnetics