EM 2 Magnetism_f

8/11/2019 EM 2 Magnetism_f

1/31

Magnetism

Dr SMR Jones

Electromagnetism

EM 2


2/31

2

Permanent magnets

Permanent magnets are made from hardferromagnetic materials which retaintheir magnetisation. (more on this later)

Magnets always have two poles, North (N) and South (S).

Like poles repel, whilst opposite poles attract.

The north pole is that which is attracted towards the Earths North Pole.

The magnetic field may be visualised, e.g. dipping a magnet in iron-filings

Poles occur in pairs: there is no magnetic equivalent of an isolated electric

charge. Magnetic field lines form continuous closed loops

around a generating current (may originate in an

electron orbit or spin or current in a wire).

The magnetic equivalent of Gausss Law states thatthe total outward component of magnetic flux

flowing through any closed surface is zero. Because field lines form closed loops, all flux

entering the surface of a region, e.g. enclosing the

south pole of a magnet, must also leave it.


3/31

3

First Law of Electro-magnetism (Oersted, 1820)

Electric currents produce a magnetic field that circulates the current

Readily verified by using compass to plot thefield around a long straight wire (carrying a

fairly substantial current)

For an infinite (long) straight wire the magneticfield strength, H is proportional to current iand

inversely proportional to the radial distance r

from the wire. There is no component of field parallel to the

wire.

The direction of circulation of the magneticfield around the wire may be determined by

Maxwells rule:

make a fist with your righthand align your thumb with the current i

Field points along fingers towards tips


4/31

4

In a manner directly comparable to the electrostatic case, we define a magneticflux densityBand a magnetic field strengthHrelated by

B= H For most dielectrics and metals is constant and in most cases = 0 Forferromagneticmaterials howeverBversusHis non-linear.

Units for H are A/m, (compare to V/m for electrostatic lines of force).

0= 4107[H/m] is thepermeabilityof free space.

Electromagnetic Field

Field lines form a continuous loop aroundthe wire.

Circumference at radius ris 2r

So if we integrate the field around the loop,

we get the encircled current. Direction of field indicated by unit vector

in the -direction(polar-coords aligned with the wire)

[A/m]2

r

i

H

FieldH- [A/m]

Units: H - Henries


5/31

5

Biot-Savart (-Ampere-Laplace) Law

If the field due to an infinite straight wire is given by H= i/2r, whatis the contribution dH of each short segment dof wire ?

Biot-Savart (and also both Ampere and Leplace) showed that

Using the geometry shown, it is straightforward to show that thisformulation for dHgives the required result for an infinitely long

straight wire.

22 4

)sin(

4

r

di

r

rdiHd

ofdirectionin2

)]1(1[4

cos4

,0wire,infinitefor

sin1

sin1

,sin

cos,

sin,

cos

cos4

sin4

sin1

4

2

1

2

1

2

1

2

1

21

22

2

r

i

r

i

r

iH

dr

dR

r

d

dRrR

dRd

r

id

r

id

R

iH


6/31

6

Magnetic field on the axis of a solenoid

The Biot-Savart-Ampere-Laplace formula allows us to calculate the field or fluxdensity (B= 0H) for any configuration of electrical conductors - even (inprinciple) wires arranged like a plate of spaghetti !

E.g. consider a solenoid (coil) length L with circular cross-section radius r

It can be shown that the magnetic field at any point on the axis either inside oroutside of the solenoid (see graph) is given by

Nis the number of turns in the coil,xdistance from the centre of the coil

Note that field is directly proportional to iand N/L (turns per unit length)

/cos

)(/)(coscoscos

222

22

1

2121

xrx

xLrxL

L

iNH


7/31

7

Magnetic field at the centre of a circular coil Each element dof the circular coil, radius rmakes a

contribution dHto the total fieldHat the centre (C).

Since is the angle between the short segment dand theradius r, =/2 at all points on the coil, so sin= 1

The total field is obtained by summing all the contributions,i.e. by integrating around the coil.

ForNturns we must integrateNtimes round the loop.

In practice, all turns do not have the same radius, but anaverage value of r gives a good approximation.

The flux density

24)sin(

rdiHd

]T[200

r

iNHB

r

iNr

r

iNd

r

iNH

coil 22

44 22

For air = 0 (the magnetic permeability of free space)is a very goodapproximation. 0 = 4107 [H/m]

Units: T Tesla

or webers / m2


8/31

8

Lorentz Force

Due to Hendrik Lorentz (1882) and Oliver Heaviside (1889).

Force on a particle carrying charge qmoving with velocity vin an electric fieldEand a magnetic fieldBis

Force on element dcarrying current i through magnetic flux of density B

Magnetic component of Lorentz force is

If B and are perpendicular, F = B i (principle of electric motor)

Bv

Bvn

vBqnEqBvEqF

tofromturnedifmovewouldscrewadirectionin the

andbothlar toperpendicurunit vectoaiswhere

sin

sinsinforcetotal,lengthofirestraight wFor

sinsin

andbutsin

iBndBinFdF

BdinBdt

ddqnFd

dt

dv

dt

dqivBdqnFd


9/31


10/31

10

Interesting magnetic facts !

Earths magnetic field produced by movement of liquid magma in its core.

Compasses indicate magnetic north, aiding navigation. Varies a few degrees fromtrue North. The magnetic North pole (in Canadian Arctic) is currently moving

approximately northwest at 40 km per year.

Year Latitude ( N) Longitude ( W)

2004 82.3 113.4

2005 82.7 114.4

The Earths magnetic field protects us from the solar wind, a lethal stream of

charged particles emitted by the sun.The fast moving particles are deflected by

the Lorentz-Heaviside force.

Humans appear unable to sense Earths magnetic field without a compass,however many animals such as sharks, dolphins, pigeons, bees and magnetotactic

bacteria can detect magnetic fields for accurate navigation.

Magnetic North

drifting at around

30/year in the UK.


11/31

11

Principle of electricity generation

When the bar magnet moves towards the coil,the flux linking the coil changes.

The induced emf produces a current as shown,which produces a North pole nearest the magnet

The resulting repulsion opposes the motion.

When the bar magnet is stationary, the fluxlinkage does not change with time.

No emf is produced and no current flows. When the magnet moves away from the coil anemf is produced so that the current in the coil

produces a south pole nearest the magnet.

The resulting attractive force opposes themotion.

(Maxwells rule gives current direction).


12/31

12

Generators and motors

Worldwide, electrical power is generated by rotating large conducting coilsthrough a magnetic field.

To rotate the generator requires energy to overcome the magnetic forces thatoppose the rotation.

That the electrical forces oppose the mechanical rotation is an example of the lawof conservation of energy.

The electrical energy cannot be created from nothing, it is simply converted frommechanical into electrical form.

This energy may be supplied by turbines driven by coal, gas, nuclear energy,water power, wind, tides.

Conversely, if a coil situated in a magnetic field is supplied by an electricalcurrent, mechanical torque is produced so as to rotate the coil.

This is the basis for an electrical motor.

Generators and motors exemplify electro-mechanical energy conversion.


13/31

13

Electric Motor

The basic principle is shown (right)

A coil of area A= 2r is placed between thepole pieces of a permanent magnet so that it

can rotate about its axis.

A D.C. voltage connects through the coil viathe split rings so that current flows.

A force (recall F =B i) acts on the coil

perpendicular to both current and field,causing the coil to rotate.

Rotation of the coil in the field causes a change in the flux cutting the coil.

This causes a back-emf in a direction opposing the applied voltage.

The two opposing voltages act to control the current flowing in the coil.

When =0, F has no rotational component about the axis, so the split rings swapthe direction of current when =0 and . (Angular momentum carries the coilover the join).


14/31

14

Motor properties

Detailed analysis of the D.C. Motor is beyond the scope of this module.

Basically: Voltage or flux controls speed, flux and current controls torque.

If the load is such as to prevent the motor turning, there is no back-emf to limitthe current.

This is likely to cause coil burn out.

An efficient motor design has several coils set at uniformly spaced angles.

Several coils (armatures) means multiple segments around the split ring.

Coils operate in the region where they produce maximum torque (/2)

The permanent magnets are often replaced by electro-magnets.

To improve efficiency, the coils are wound on (usually laminated) ferromagneticcores to help concentrate the flux in the region of the armature coil.

The construction of generators is very similar to motors.


15/31

15

Flemings left-hand rule

Flemings left-hand rule for motors (remember motors are driven on the left-handside of the road !)

Thu Mb points in the direction of Motion Forefinger points in the direction of the Field

M Iddle finger points in the direction of the current (I)

The force is maximum when the current is perpendicular to the field

( F = Bi)


16/31

16

Ferromagnetic cores

Current-carrying coils with ferromagnetic cores are an important source ofmagnetic fields.

The field strength and flux densities close to the core are very much larger thancan be obtained with a coil alone.

The efficiency of transformers, motors, relays, loudspeakers, actuators,electromagnets are greatly enhanced by the use of ferromagnetic cores.

Correct design using magnetic circuits requires an understanding of thesematerials.


17/31

17

Magnetic materials

Magnetisation of a material is associated with atomic current loops, primarily dueorbital motions of electrons and electron spin.

The behaviour of the material depends on the interaction of its crystallinestructure to an applied magnetic field.

Materials are classified as

diamagnetic,

paramagnetic and

ferromagnetic.

Diamagnetic materials have no permanent magnetic dipole moment and theinduced field opposes any applied field. r1

Paramagnetic materials have only weak magnetic dipoles. The induced fieldaligns with the applied field. r1

Ferromagnetic materials have strong magnetic dipole moments whose alignmentdepends on previous magnetic conditions (hysteresis). r>> 1


18/31

18

Ferromagnetic materials

Common ferromagnetic materials are iron, nickel and cobalt.

The crystalline structure of ferromagnetic materials is characterised bymagnetised domains of the order of 1010 m3within which magnetic dipoles arealigned.

In the absence of an external magnetising field the domain fields have randomlydistributed directions, resulting in zero net magnetisation.

An external applied field produces partially alignment of domains.

Thus the degree of magnetisation will depend on the previous magnetisationhistory of the material.

When a ferromagnetic material is subject to an external applied field H, themagnetic flux density inside the material is given by

whereMis the net magnetisation of the material, (i.e. magnetic moment per unit

volume in fieldH) and 0is the permeability of free space.

]T[0 HMB


19/31

19

Hysteresis loops

If the applied field is now progressively reduced, the flux density reduces, butfollowing the upper trajectory, until at H=0, flux density is still B rsince the material

retains some magnetisation.

A reverse field of BHcis needed to reduce B to zero, defining the coercivity.

Increasing negative applied H saturates the material in the opposite direction.

Cycling H back, B follows the lower trajectory.

The non-linear relationship of BandH is described through its B-H

magnetisation curve. The sample is initially unmagnetised

with H=0, B=0.

As the applied field H is increased,the material becomes progressively

more magnetised so the flux density

B increases (non-linearly) until thematerial is fully magnetised

(saturated).


20/31

20

Minor hysteresis, parasitic loops and energy loss,

The saturation loop provides a repeatablebasis for characterisation of the material.

If not cycled into saturation, the B-Hcurve follows a minor hysteresis loop(tips follow initial magnetisation curve).

The energy dissipated per unit volume incycling the material once round the loopis given by the area enclosed by the loop.

Depending on the initial magnetisationand field variation, endless trajectoriesare possible, e.g. parasitic loops causeenergy loss

Demagnetise material by cycling roundwhilst progressively reducing maximumapplied field (slowly withdraw from a.celectromagnet).


21/31

21

Soft and hard magnetic materials

Hard magnetic materials strongly retain magnetisation (high BHc) They are thus used to make permanent magnets. The narrow loops of soft magnetic materials dissipate less energy.

Soft materials with high saturation magnetisations, characterised by high provide high flux densities and low loss required in transformers and otherapplications.


22/31

22

M-H loops

The M-H plot is an alternative way ofrepresenting the hysteresis

Recall

Removing the free-space (0H)contribution to B and dividing by 0we can plot M versus H.

The M-H plot of magnetisation of thematerial versus applied field separates

the intrinsiccontribution, i.e. that of

the material alone.

This allows us to clearly identify the

magnetisation at saturation MS, anddefine the intrinsic coercivity iHcof

the material.

]T[0 HMB


23/31


24/31


25/31

25

Mutual Inductance

If the flux generated by one circuit linkswith a second circuit, then there is a

magnetic interaction. If the two circuits 1,2 carry currentsI1

andI2then flux 21is that generated bycircuit 1 that links with circuit 2.

12is the flux generated in circuit 2 that links with 1.

11

22

link only with circuit 1 and 2 respectively.

If the number of turns in circuit 1 and 2 are N1and N2respectively.

We define theMutual Inductance M12as the ratio of flux linkage in circuit 1related to currentI2, i.e.

It can be shown that for fields in linear media (non ferromagnetic) that

2112 MM

[H]2

12112

I

NM


26/31

26

Transformers

Two coils whose fluxes link form a transformer.

To ensure maximum magnetic coupling, they are

usually wound on the same ferromagnetic core.

The supply-side coil is called the PRIMARY.

The other coil is called the SECONDARY.

A sinusoidally alternating voltage V1connected to the primary produces analternating current (a.c.) I1in coil 1 and alternating flux in the core.

By Faradays Law, the e.m.f. produced across coils 1 and 2 are

If ohmic losses in the coil are negligible, V1= E1etc so V1/V2= N1/ N2

If flux leakage is negligible the flux through both coils is the same, so

2

1

2

12211 thereforeand,

N

N

E

E

dt

dNE

dt

dNE

1

2

2

12211 ,

N

N

I

IININ


27/31

27

Ideal Transformer

i.e. the load impedance is transformed in the ratio of turns squared.

[W], 22111

2

2

1

2

1

2

1 IVIVN

N

I

I

N

N

V

V Combining

For an ideal transformer, power(W = VI) is conserved.

The transformer is represented bythe equivalent circuit shown

L1,2 are the self-inductances.

M is the mutual inductance.

LLIN

L

L

L

LIN

ZNNZ

LLZ

LL

LjZ

ZLj

LjZ

ZLjLLkZ

kLLkMN

N

L

L

2

2

1

2

1

1

1

L21

2

1

1

1

2

121

22

1

1

21

2

2

2

1

2

1

IV

Z,If

I

V

coupling)(perfect

1korf

1

I

V

t.coefficiencouplingtheiswhere,and


28/31

28

Eddy currents and magnetic flux penetration

When an alternatingmagnetic flux penetrates a ferromagnetic material, theFaraday-Lenz law tells us to expect an induced emf in such a direction as to

produce currents that oppose the change in flux.

The resulting currents are known as eddy currents and are responsible for heatgeneration and power loss in the material.

The opposing magnetic field produced by the eddy currents causes a progressivereduction in the flux density as we move into the material, i.e. it limits the

penetrationof the magnetic flux into the material. This reduces the amount bywhich a ferromagnetic core increases the flux

To counteract these effects, cores are usually made as a sandwich of thinlaminates of ferromagnetic material with insulating coatings.

Lamination reduces eddy currents, thus improving penetration, reducing lossesand increasing flux densities.

An optimum thickness can be calculated for the laminates, for 3.2% Si-Fe theoptimum thickness is about 0.3 mm.


29/31

29

Magnetic circuits

We define Magneto-motive force with units of Ampere-turns

We also define Reluctance with units of Ampere-turns per Weber

The magnetic equivalent of Ohms Law, with flux is then

Low reluctance of ferromagnetic cores concentrate MMF across the gap[A.t]RF

[A.t]NIF

S1 = S3= 3cm2

S2 = 6 cm2

l1 = l3 = 20 cml2= 10 cm

lg= 0.2 mm

Require

Bg= 1.3 Wb/m2

[A.t / Wb]S

R


30/31

30

Example: Magnetic circuits Magnetic equivalents of Kirchhoffs current and voltage laws apply

i.e. Magneto-motive forces (MMF) around any closed loop sum to zero

Total flux into any node is zero (taking account of direction)

MMF around left and right -hand loopsNI= H11+ H22 = H22+ H33+ Hgg

g= 3, 2= 1+ 3Bg= g/ Sg effective= 1.3 Wb/m2 (required)

Sg effective=(3+g)2 = 3.07 cm2

Hg= Bg/0= 1.3/4107 = 1.03106A/m

B3

= 3

/ S3

= g

/ S3

= Bg

Sg effective

/ S3

= 1. 3 3.07 104/ 3104 = 1.333 T

H3= 475 A/m (from curve)

][A.t

HB

SNI

RF


31/31

31

Example (continued)Knowing Hg and H3we obtain

H11= H33+ Hgg= 475 0.2 + 1.031060.2 103

H1= 302.4 / 0.2 = 1512 A/m

B1= 1.56 T (from curve)

1= B1S1= 1.56 3 104 = 4.68 104 Wb

2= 1+ 3 = 4.68 104+ 1.333 3 104 = 8.68 104 Wb

B2= 2/ S2= 8.68 104/ 6 104 = 1.447 TH2= 750 T (from curve)

NI= H11+ H22= 302.4 + 750 0.1 = 377.4 [A-t]

This could be provided, e.g., by 3774 turns carrying 0.1 A

Note the important difference that unlike their electrical counterparts [S/m] andJ[A/m2], permeability varies with flux densityBin the ferromagnetic

components of the circuit.

EM 2 Magnetism_f

Documents

Transcript of EM 2 Magnetism_f