Magnetic Analysis 5 (3)

8/10/2019 Magnetic Analysis 5 (3)

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Nathan Ida, University of Akron, Akron, Ohio

3C H A P T E R

Magnetism1


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Coulombs LawThe nature of electromagnetism can besummarized by four vector quantities,their interaction, their relations with eachother and with matter. These four timedependent vector quantities are referred toas electromagnetic fields and include:electric field intensity E, electric fluxdensity D, magnetic field intensity Handmagnetic flux density B.

The study of electromagnetic fieldsbegins with the study of basic laws ofelectricity and magnetism and with theuse of some basic postulates. In particular,it is customary to start with Coulombs

law. This law states that the force betweentwo stationary charges is directly proportionalto the size of the charges and is inverselyproportional to the square of the distancebetween them.

Adding Gauss and Amperes lawsprovides a complete set of relationsdescribing all electrostatic, magnetostaticand induction phenomena, but not wavepropagation. To include wave propagationin electromagnetic field equations, thedisplacement current (continuityequation) is added to Amperes law. Doingso obtains Maxwells equations.

Alternatively, Maxwells equations mayserve as the basic postulates and, because

they form a complete set describing allelectromagnetic phenomena, the requiredrelations may be deduced. By choosingMaxwells equations as the starting point,an assumption of the equations accuracyis implicitly made. This is not moretroublesome than assuming thatCoulombs law applies or thatdisplacement currents exist. In either case,the proof of correctness is experimental.This is an important consideration: itspecifically states that Maxwells equationsand therefore the electromagnetic fieldrelations cannot be provenmathematically.

Use of Maxwells EquationsMaxwells equations do not take motioninto account and therefore do not includethe induction of currents due to motion.To do so, it is necessary to add the lorenzforce equation and the so-calledconstitutive relations. It may also be usefulto note that, by proper interpretation,relativistic effects can also be handled (thisapplication is found in the literature).2-6

The approach in this chapter is to start

with Maxwells equations and derive fromthem all the necessary relations. In doingso, the equations are accepted as the basicpostulates. In particular, at lowfrequencies, Maxwells equations areidentical to those of Coulomb, Faraday,Gauss and Ampere. The results derivedhere are general and, within theassumptions made in their derivation,apply to a wider range of applications.Only those electric and magneticphenomena most directly related tonondestructive testing and, in particular,to magnetic particle testing are consideredin detail here. Other electromagneticphenomena are mentioned briefly for

completeness of treatment.Maxwells equations are a set of

nonlinear, coupled, first order, timedependent partial differential equationswhose general solution is difficult toobtain. Some methods for the solution ofthe electromagnetic field equations arediscussed below.

Measurement UnitsOne major source of confusion whenapplying electromagnetic field theory hasbeen the units for measurement.Centimeter gram second (CGS) units, theelectromagnetic system of units (EMU)

and meter kilogram second ampere units(MKSA) are the most familiar, but othersystems such as the absolute magnetic, theabsolute electric and the so-callednormalized system have also been used.

More disturbing is the fact that mixedunits have also been used. For example,practitioners used to use MKSA units suchas the ampere for electrical quantities andEMU units such as the gauss for magneticquantities.

Only the International System (SI) ofUnits is used in this section. The benefitgained in consistency far outweighs anyinconvenience.

Another problem encountered inpractice is the confusion between magneticfield intensityH, sometimes called fieldstrength, and magnetic flux densityB. Theterm magnetic fieldis often used for Hor Bor both, depending on the situation. Toavoid such confusion, the quantity B isused consistently for the magnetic fluxdensity while H is the magnetic fieldintensity. Similarly, E is the electric fieldintensityand D is the electric flux density.

86 Magnetic Testing

PART 1. Fundamentals of Electromagnetism


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Maxwells equations are summarized inEq. 1 through Eq. 4 in differential formand in Eq. 5 through Eq. 8 in integralform. Equation 9 is the lorenz forceequation which describes the interactionof electric and magnetic fields withelectric charge:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

where d is differential length (meters), dsis differential area (square meters), E iselectric field intensity (Vm1), F is force(N),Iis current (A), J is current density(Am2), q is electric charge (coulomb) vischarge velocity (meter per second) and is the del operator.

Equations 1 and 5 are a statement ofFaradays law of induction. Equations 2and 6 are a modified form of Ampereslaw. The addition of the displacementcurrent term D(t)1 was Maxwells

contribution to the original laws ofelectricity. The displacement current,although often taken as an assumption, isnothing more than an expression that canbe derived from the continuity equation.Equations 4 and 8 are Gauss law formagnetic sources and they state thenonexistence of isolated magnetic chargesor poles. Equations 3 and 7 are Gauss lawfor electric charges.

At this point, the equations are neitherlinear nor nonlinear. This importantbehavior is determined through materialproperties and is not inherent in theequations. The material properties followthe constitutive relations:

(10)

(11)

(12)

If any of these relations is nonlinear,the field relations are nonlinear. Inparticular, the permeability is known tobe highly nonlinear for ferromagneticmaterials. In some cases, the conductivity

and permittivity may also be nonlinear.The electric conductivity , magnetic

permeability and the electricpermittivity are generally tensorquantities. Although for many practicalpurposes it can be assumed that they arescalar quantities, materials areencountered in practice that behavedifferently. These exceptions are definedthrough linearity, homogeneity andisotropy of the materials. Only for linear,isotropic, homogeneous materials are thematerial properties single-valued scalarquantities.

Linearity, Homogeneity andIsotropy

A medium is said to be homogeneous if itsproperties do not vary from point to pointwithin the material. A medium is linearina property when that property remainsconstant as the field changes. Thepermeability of most nonmagneticmaterials is considered to be independentof the field. These materials are usuallyconsidered to be linear in permeability. Amaterial is isotropicif its properties are

=

D E=

B H=

F E v B= + ( )q

B ds = 0s

D ds = Qs

H d D

ds = + I tSc

E d = ddtc

=B 0

=D

= +

H J

D

t

=

E B

t

87Magnetism

PART 2. Field Relations and Maxwells Equations


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independent of direction. This means thatthe permeability of an isotropic materialmust be the same in all three spatialdirections.

A good example of an anisotropicmaterial is a permanent magnet. Mostmaterials, including iron, are anisotropicon the crystal level. Because these crystalsare randomly oriented, it may often beassumed that a macroscopic material is

isotropic. This is not necessarily the casefor hard steels or for steels with large,preferred orientations.

Static FieldsBy setting to zero all time derivatives inMaxwells equations, the equations for thestatic electric and magnetic fields areobtained. The four equations become:

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

Equations 13 and 15 are the governingequations for electrostatic fields (Faradays

and Gauss laws). Equations 14 and 16 areAmperes and Gauss (magnetic) laws formagnetostatic applications. Note thatEqs. 13 and 15 do not contain themagnetic field whereas Eqs. 14 and 16 donot depend on the electric field. Thus, theequations for electrostatics andmagnetostatics are completely decoupledand electrical quantities can be calculatedwithout resorting to the magnetic fieldand vice versa.

As with the general system ofequations, the lorenz equation has tosupplement these equations. Because ofthe decoupling of the two sets, the forceequation (Eq. 9) should be used in itselectrostatic form for electrostatic fields:

(21)

For magnetostatic fields, only themagnetic force exists:

(22)

The electric current or, moreconveniently, the electric current density Jis the source of the magnetic fieldintensity (Eq. 14).

Equation 18 is particularly usefulbecause it allows the calculation of themagnetic field intensity and thedetermination of its direction. If theintegration is taken such that the normal

to the surface enclosed by the contour Cis in the direction of the current, then thefield is described by the right hand rule: ifthe current is in the direction of thethumb, the field is in the direction of thefingers (Fig. 1). The total current IinEq. 18 is the current enclosed within thecontour.

Magnetic Vector PotentialEquation 14 is a cross product and resultsin a vector quantity J. Because the crossproduct of a vector is also a vector, the

magnetic flux density B may be written asthe curl (cross product) of anothervector A:

(23)

Thus, Amperes law (Eq. 14) can bewritten as:

B A=

F v B= q

F E= q

B ds = 0s

D d s = Qs

H d = Ic

E d = 0c

=B 0

=D

=H J

=E 0

88 Magnetic Testing

FIGURE 1. Right hand rule: if the thumb ofthe right hand is in the direction of thecurrent, the fingers show the direction ofthe magnetic field.

Magnetic flux lines

R

I


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(24)

The vector A is called a magnetic vectorpotential. For an isotropic linear medium,the following vector identity can be usedto simplify this expression:

(25)

A vector is only defined when both itsdivergence and curl are specified. Thedivergence may be specified in differentways. The simplest is to set it equal tozero in Eq. 25. Amperes law thenbecomes:

(26)

which is the vector poisson equation.Equation 23 defines the magnetic

vector potential. This definition of themagnetic vector potential A allows the

use of a simpler poisson equation insteadof the original field equations.

Biot-Savart LawThe purpose of field relations is to solvefield problems. Any of the relationsobtained previously may be used for thispurpose. In particular, Eq. 14 can be usedfor general field problems while Eq. 18 isuseful for solution of highly symmetricproblems. In problems where no suchsymmetry can be found but which aresimple enough not to require the solution

of the general Eq. 26, another method canbe used. For these types of problems, themagnetic vector potential is used in stillanother form. Considering the current ina straight wire in Fig. 2, the following canbe written at pointP:

(27)

In Eq. 27, the definition of the magneticvector potential in Eq. 23 has been used.In order to find the total magnetic vectorpotential, this differential is integratedover the closed contour formed by thecurrent to obtain:

(28)

In order to find the field intensity orthe flux density, it is necessary to find thecurl of this expression by performing theoperations in Eq. 23.

If the magnetic vector potential inEq. 28 is substituted back into itsdefinition, the following expression isobtained for the flux density:

(29)

where^

R denotes a unit vector in thedirection of R in Fig. 2.

This expression, known as theBiot-Savart law, allows the calculation ofthe flux density directly. It is particularlyuseful for situations where the currentpaths are clear and the required contourintegration is in the direction of thecurrents.

B d R

R=

I

c4 2

A d

R=

I

c4

d I

A d

R=

4

= 2A J

= ( )

A A A2

( ) =1

A J

89Magnetism

FIGURE2. Straight current carrying wire andrelation of current and field at point P.

I

R

P

d


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Steady State AlternatingCurrent FieldsLow frequency alternating current fieldsare unique in that a simpler form ofMaxwells equations can be used. Thedisplacement current term (Dt1) inEq. 2 or Eq. 6 depends on frequency and isvery small for low frequencies. In fact,this term may be neglected withinconducting materials and for allfrequencies below about 10E + 13 Hz. Ifthis assumption is introduced and theterm neglected, thepre-Maxwell set ofequations is obtained.

By introducing a phasor notation forall vectors, the time dependency is notexplicitly used and the transformedsystem is both simpler in presentationand in solution. A general vector can beexpressed as a phasor by the followingdefinition:

(30)

wherej is current density (Am2), tis time(seconds) and is angular frequency

(radians per second).Maxwells equations (in differential

form) can now be written as:

(31)

(32)

(33)

(34)

In this form, all time derivatives werewritten as:

(35)

This version of Maxwells equations isoften called the quasistatic form and isvery convenient for many alternatingcurrent calculations at low frequencies,including alternating current leakagefields. The following equation is obtainedby a method similar to that used to obtainEq. 26:

(36)

This equation, often called the curl-curlequation is a diffusion equation and is thebasis of many analytical and numericalmethods. By using the vector relation inEq. 25, Eq. 36 can be written as:

(37)

Here, the divergence of the magneticvector potential may be chosen in anyconvenient, physically sound method. Bychoosing a zero divergence and rewritingthe laplacian 2

~A in its differential form, a

partial differential equation can be writtenfor two-dimensional (Eq. 38),axisymmetric (Eq. 39) and

three-dimensional geometries (Eq. 40).Equations 37 through 40 assume linearpermeability:

(38)

(39)

(40) 1 2

2

2

2

2

2

A A A A

x y zJ js+

+

= +

1 12

2

2

2 2

A

r r

A

r

A

z

A

rJs

+

+

= + jj A

1 2

2

2

2

A

x

A

yJ

j A

z zs

z

+

=

+

( ) = +2 A A J A j

( ) = +1

A J Aj

A At

j=

=B 0

=D

= + H J Dj

= E Bj

A x y z t x y z ej t, , , , ,( ) = ( )

real A

90 Magnetic Testing

PART 3. Electromagnetic Fields and Boundaries


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Skin Depth

In a linear isotropic material, aftersubstitution of the constitutive relationsin Eqs. 10 through 12 and using thevector identity in Eq. 25, Eqs. 1 and 2become:

(51)

(52)

Thus, E and B satisfy identical waveequations with a damping (dissipative)term proportional to the conductivity andmagnetic permeability of the material. Ina good conductor such as most metals,the second order derivative may beneglected for low frequencies since it isdue to the displacement current in

Maxwells second equation. For example,Eq. 52 becomes:

(53)

This is a simple diffusion equation. If analternating magnetic field intensityH0exp(jt) orE0(jt) is applied parallel tothe surface of the conductor, the electricfield intensity E or the magnetic fieldintensity H is attenuated exponentiallywith distance below the surface. Theattenuation is exp (x1) where x is the

distance below the surface and is theskin depth given by:

(54)

This is an important factor to considerfor alternating current magnetic particletesting because, even at 60 Hz, the skindepth can be quite small. For example, fora typical ferromagnetic material with aconductivity = 5 106, a relativepermeability r= 100 and a frequency

f=60 Hz, the skin depth = 3 mm. This isprobably an overestimate because a linearmaterial was assumed in the derivation.

For this reason, alternating currentmagnetic particle methods, such as theso-called swinging fieldmethods, generallydetect only discontinuities which areopen to the surface. These methods aremore sensitive to surface breakingdiscontinuities because the applied field isrelatively more intense at the surface.

Electromagnetic BoundaryConditionsElectromagnetic fields behave differentlyin different materials. The constitutiverelations in Eqs. 10 through 12 are astatement of this behavior. Whendifferent materials are present, the fieldsacross the boundaries between these

materials must undergo some changes toconform to both materials. In such cases,the field may experience a discontinuityat the boundary. In order to find thenecessary conditions that apply atmaterial boundaries, assume two differentmaterials as in Fig. 3 and apply Maxwellsequations at the boundary. Forconvenience, the integral form is used. Bydoing so, the following four conditionsare obtained:

(55)

(56)

(57)

(58)

The boundary conditions are the samefor the magnetostatic and time varyingfields. These conditions are summarized asfollows:

1 The tangential component of theelectric field intensityE and thenormal component of the magneticflux densityB are continuous acrossthe boundary.

2. The normal component of the electricflux densityD and the tangentialcomponent of the magnetic fieldintensityHare discontinuous acrossthe boundary. The discontinuitydepends on the existence of surfacecharges and currents. For situationswhere no such charges or currentsexist, either component may becontinuous, depending on thematerials and the fields involved.

The four conditions presented inEqs. 55 through 58 can be used in orderto describe the fields in different materialsand across their boundaries.

The four conditions are not entirelyindependent and should be specified withcare. For example, in time varying fields,specification of the tangential componentofE (Eq. 55) is equivalent to thespecification of the normal component ofB (Eq. 58). Similarly, specification of the

B Bn n1 2=

n ( ) =D Dn n s1 2

n J ( ) =H H s1 2

E E1 2 =

= 2

=2 0H H

t

=22

2 0H

H H

t t

=22

2

0E E E

t t

92 Magnetic Testing


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tangential component ofHis equivalentto that of the normal component ofD.Only two of the four may be specifiedindependently (the tangential componentofE and the tangential component ofHor any other acceptable combination).Overspecification of boundary conditionsmay result in contradiction of conditionsand may therefore be in error.

The boundary conditions in Eqs. 55

through 58 were obtained by usingMaxwells equations directly. In order torender these relations more useful, it isconvenient to introduce the constitutiverelations in these conditions and find theinterface conditions for some specialclasses of common materials. Two suchgroups of materials often found inpractice are:

1. boundary conditions between twolossless media (a lossless medium isone that has zero conductivity witharbitrary permittivity andpermeability; two perfectly insulatingmaterials are considered here); and

2. boundary conditions between alossless material and a goodconductor.

At the boundary between two goodinsulators, no current densities and freecharges are normally present. Thus, allfour components in Eqs. 55 through 58are continuous. These then can berewritten using the constitutive relationsin Eqs. 10 and 12 as:

(59)

(60)

(61)

(62)

At the interface between a goodconductor and an insulator, both surfacecurrent densities and free charges may

exist. The electric field is zero inside aperfect conductor and both the tangentialcomponent of the electric field intensityand the normal component of the electricflux density must be zero inside theconductor. The boundary conditions(Eqs. 55 to 58) then become:

(63)

(64)

(65)

(66)

Note that while Eqs. 63 through 66 are

correct for the static field, for the timevarying field, both B and Hmust also bezero inside a perfect conductor. Thus,Eqs. 63 through 66 must be modified forthe time varying case to:

(67)

WhenH2 = 0, then:

(68)

WhenD2n = 0, then:

(69)

(70)

Note that the boundary conditions inEqs. 67 through 70 only apply for perfectconductors. This rarely arises except forsimplified problems and forsuperconductors. In the case of asuperconductor, these boundaryconditions are also correct for the static

field.Proper application of the field

equations and imposition of the correct

B Bn n1 2 0= =

n =D n s1

n J =H s1

E E1 2 0 = =

B Bn n1 2=

n =D n s1

n J ( ) =H H s1 2

E1 0 =

1 1 2 2H Hn n=

1 1 2 2E En n=

B

B1

2

1

2

=

D

D1

2

1

2

=

93Magnetism

Figure 3. Boundary conditions betweentwo materials.

A1

A1

A1n

A2

A2

A2n

n1

2

2

2

1

1

1

n2

Legend

A = magnetic vector potentialn = normal component of vector = permittivity = permeability = conductivity


10/24


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Material Properties andConstitutive RelationsMagnetic properties are important becauseof their effect on the behavior of materialsunder an external field (under activeexcitation) or when the external field isremoved (residual magnetism). Themagnetic properties are often discussedusing the magnetic permeability ofmaterials. This important quantity isdefined through the constitutive relationin Eq. 10.

Permeability governs two importantfeatures of the magnetic field and

therefore affects any application that usesthe magnetic field. For ferromagneticmaterials below saturation, flux densityBis generally the quantity of interest andhas higher values for high values of thepermeability for a given source fieldintensityH. Secondly, the permeabilityalso defines whether the field equation islinear or nonlinear.

The permeability of free space is0 = 4 10

7Hm1. Other materials mayhave larger or smaller permeabilities.Table 1 lists the relative permeabilities ofsome important materials.

The magnetic properties of materials

are defined through the interaction ofexternal magnetic fields and movingcharges in the atoms of the material(static charges are not influenced by themagnetic field since no magnetic forcesare produced in Lorenz law). Atomic scalemagnetic fields are produced inside thematerial through orbiting electrons. Theseorbiting electrons produce an equivalentcurrent loop that has a magnetic moment:

(77)

where a2 is the area of the loop,Iis the

equivalent current (Fig. 4a) and^

z is a unitvector normal to the plane of currentflow.

Many such atomic scale loops ormagnetic moments exist and the materialvolume contains a certain magneticmoment density. If Nmagnetic momentsper unit volume are present, and if thesemoments are aligned in the samedirection, a total magnetization isgenerated. The magnetization M is thengiven by:

(78)

The magnetic flux density of the materialis then given by:

(79)

The terms Hin, m and Mare vectors.This implies that a net magnetic field orflux density can only exist if these vectorsare aligned in such a way that a total netvector Mexists. If the independentvectors m are randomly oriented, as isoften the case, the net magnetization iszero.

MaterialsFor the purposes of this chapter, threetypes of magnetic materials are important:diamagnetic,paramagneticandferromagnetic.

B Min =

H M min = = N

m z= I a 2

95Magnetism

PART 4. Effect of Materials on Electromagnetic

Fields

TABLE 1. Relative permeabilities of magnetic materials.Values given for ferromagnetic materials representapproximate maximum relative permeabilities.

RelativeMagnetic Material Permeability

Diamagnetic materials

Gold 0.999964

Silver 0.99998

Copper 0.999991

Lead 0.999983

Water 0.999991

Mercury 0.999968

Bismuth 0.99983

Paramagnetic materials

Vacuum (nonmagnetic) 1.0

Air 1.00000036

Aluminum 1.000021

Ferromagnetic materials

Cobalt (99 percent annealed) 250

Nickel (99 percent annealed) 600

Iron (99.8 percent annealed) 6000

Iron (99.95 percent annealed in hydrogen) 2.0 105

Nickel alloy

Annealed nickel alloy (controlled cooling)a 1.0 106

Steel (0.9 percent carbon) 100

Iron (98.5 percent, cold rolled) 2000

a. By weight 79 percent nickel; 5 percent molybdenum; iron.


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Diamagnetic Materials

In these materials, the internal magneticfield due to electrons is zero under normalconditions. In an external magnetic field,an imbalance occurs and a net internalfield opposing the external field isproduced. Thus, M in Eq. 78 is negativewith respect to the applied field. Themagnetization is proportional to the

external field through a quantity calledthe magnetic susceptibilityof thematerialxm:

(80)

In terms of the applied flux density, thisbecomes:

(81)

The magnetic permeability of anymaterial can be written as:

(82)

In diamagnetic materials, the magneticsusceptibility is very small and negative.Its magnitude is usually on the order of105. The net effect is that the relativepermeabilities exhibited by diamagneticmaterials are slightly smaller than 1.0.This group of materials includes manyfamiliar metals including pure copper andlead.

Under special conditions such astemperatures less than 150 C, somematerials may become superconducting.An ideal superconductor has a magnetic

susceptibility of 1 and a permeability of0. A superconductor expels magnetic flux(the meissner effect) from its interior.

Paramagnetic MaterialsThis group of materials exhibits propertiessimilar to diamagnetics except that themagnetic susceptibility is positive. In thepresence of an applied magnetic field

intensity, the atomic magnetic dipolemoments can align to form a netmagnetic dipole density. The effect is stillrelatively small, producing observedrelative permeabilities slightly larger than1.0.

The permeability of paramagneticmaterials remains constant over a largerange of applied magnetic field intensities.Examples of materials in this group areair, aluminum and some stainless steels.

Ferromagnetic Materials

Ferromagnetic materials vary fromdiamagnetic and paramagnetic materialsin two critical ways: (1) their susceptibilityis very large and (2) there is a pronouncedvariation in the internal structure of theirmagnetic moments. In these materials,many atomic moments are aligned in acertain direction within a very smallregion called a magnetic domain.Neighboring domains have a similarstructure, with the net magnetic domainin one direction. In the demagnetizedstate, the magnetic domains tend to bealigned randomly, exhibiting a netinternal field that is either very small orzero.

This domain model is depicted in

Fig. 5. When an external magnetic field isapplied, those domains that have a netfield aligned in the direction of theapplied field grow in size while the otherdomains shrink. The internal field and theexternal fieldHare aligned in the samedirection producing a larger total flux

= +( )0 1 xm

B H= +( )0 1 xm ex

M H= xm ex

96 Magnetic Testing

Figure 4. Representation of materialproperties: (a) field due to current loop;(b) current loops created by spinningelectrons.

(a)

(b)

Legend

B= magnetic flux densityI= current

I

I

B

Figure 5. Magnetic domains inferromagnetic material: domain 8 is alignedwith field and will grow as magnetic fieldintensity H is increased; domain 3 is alignedagainst field and will shrink as H isincreased.

I

2

H

3

4

5

6

7

8


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densityB. The above argument is relatedto the hysteresis curve of a ferromagneticmaterial and explains why any such curvehas a saturation region: beyond a certainfield, all the magnetic domains arealigned with the field and an increase inthe magnetic field intensity cannotincrease the net magnetization. Materialstypical of this group are iron, steels, nickeland some stainless steels. Table 1

summarizes some of the more importantferromagnetic materials and theirpermeabilities. Table 2 lists conductivitiesof various materials and Table 3 is a listingof dielectric constants.

As is evident from any hysteresis curve,the permeability of ferromagneticmaterials is not constant but varies withthe field. This is exhibited through theslope of the initial magnetization curve towhich the permeability is tangent. Thus,most ferromagnetic materials are highlynonlinear materials.

97Magnetism

TABLE 2. Electrical conductivities of somematerials.

ConductivityMaterial (Sm1)

Silver 6.1 107

Copper (pure) 5.8 107

Gold 4.1 107

Aluminum 3.5 107

Tungsten 1.8 107

Brass 1.1 107

Iron (pure) 1.0 107

Soft steel 0.8 107

Carbon steel (1 percent carbon) 0.5 107

Nickel chromium stainless steela 1.4 106

Nickel chromium alloyb 0.9 106

Mercury 1.0 106

Graphite 1.0 105

Carbon 3.0 104

Sea water 4.0

Germanium 2.3

Silicon 3.9 104

Phenolic resin, cured 1.0 109

Glass 1.0 1012

Rubber 1.0 1013

Mica 1.0 1015

Quartz 1.0 1017

a. 18 percent nickel, 8 percent chromium.

b. 80 percent nickel, 20 percent chromium.

TABLE 3. Dielectric constants (relativepermittivities) for some materials.

RelativeMaterial Permittivity

Vacuum 1

Air 1.0006

Rubber 3

Paper 3

Phenolic resin, cured 5

Quartz 5

Glass 6

Mica 6

Water 81

Barium titanate 1200

Barium strontium titanate 10000


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Magnetic CircuitsThe two equations that define the staticmagnetic field are Eq. 14 and Eq. 16.These are written below in differential andintegral forms in terms ofB:

(83)

(84)

(85)

(86)

These are Amperes and Gauss laws forthe static field. They can also be viewed asdefining a vector quantity B through itscurl and divergence.

The line integral of the magnetic fieldintensity around a closed path is definedas a magnetomotive force.

(87)

The units of the magnetomotive forceare customarily expressed as ampere turnsalthough the correct unit is the ampere.The modification fromIto NIsimplystates that, if the total current inside theclosed contour is divided into Nwires,then the number of turns may be used forconvenience.

Circuit Theory

A magnetomotive force Vm = NIcauses amagnetic flux to exist within the closed

contour mentioned in Eq. 87. If for anyreason this flux is contained within amaterial, it may be assumed that a fluxflows within the material. This conceptallows flux to be treated much the sameway as current and therefore circuittheory concepts may be used for thesolution of some specific field problems.

To develop this concept, it isconvenient to use a toroid (Fig. 6). Thegap is assumed to be small and the fluxdensities inside the toroid and the gap are

assumed to be the same. This assumption

in effect neglects any fringing effects inthe gap. If the field intensity is denoted inthe gap asHg and in the toroid asH, thenthe fields can be calculated in terms of theflux densityB in the toroid and thepermeabilities of the gap and of the toroid(0 and ) as:

(88)

and:

(89)

By substituting these in Eq. 87, themagnetic flux density is found to berelated to gap lengths g and the length(2r g) of toroid material, where ris themean radius of the toroid:

(90) B NI

r

=( ) +

0

0 2 g g

H B

g =

0

H B

=

V NI

c

m = =

H d

B ds = 0s

B d = Ic

=B 0

=B J

98 Magnetic Testing

PART 5. Magnetic Circuits and Hysteresis

Figure 6. Toroid with air gap used to define

magnetic circuit concept.

A

A

Ig

I

r

o

A

A

s

LegendA = reference pointIg = gap distancer= radius of toroids= cross sectional area of toroid = permeability = flux


15/24

If it is assumed that the magnetic fluxdensity is uniform within a material (it isoften uniform inside a toroid but rarely inother shapes), the flux can be calculated:

(91)

The total flux through the toroid or thegap is therefore:

(92)

Written in terms of the magnetomotiveforce Vm, the equation for the flux can bewritten as:

(93)

where:

(94)

and:

(95)

The forms of Eqs. 94 and 95 areanalogous to that of the direct currentresistance (R = a1) and are thereforecalled magnetic resistances or reluctances.The reluctance of the gap isRg andR is

the reluctance of the material in thetoroid. The units for reluctance are 1 perhenry (1H1). Similarly, if magnetomotiveforce is considered analogous to voltageand flux analogous to current, Eq. 93 isanalogous to Ohms law.

For any closed magnetic path, theequation can be written as:

(96)

Similarly, by using the divergence ofthe magnetic flux density B = 0, the lawfor a junction is:

(97)

For simplicity, an analogous magneticcircuit can be defined as in Fig. 7. Becauseof its simplicity, this approach has foundconsiderable use in many areas, especiallyin devices with closed paths (transformersand machines). The approach is quitelimited in scope because of theapproximations used to derive theconcept. First, the fringing effects cannotbe neglected for large air gaps. Second,there are always some leakage fields thatcannot be taken into account. Finally, thepermeability has been assumed to be

constant. In most cases of practicalimportance, the permeability of a materialis field dependent (Eq. 98).

HysteresisThe constitutive relation between themagnetic field intensity and the magneticflux density is shown in Eq. 10. Thebehavior of the field within differentmaterials has been described above.However, these do not describe allphenomena that exist within materials.

Inspecting Eq. 10 shows that by

increasing the magnetic field intensity H,the flux density B increases by a factor of. However, for ferromagnetic materials,Eq. 10 must be written as a nonlinearequation:

(98)

An alternative way to look at thisphenomenon is to inspect the domainbehavior of a ferromagnetic material.Initially, the domains are randomlyoriented. As the applied field increases,domains begin to grow by displacing

other domains and eventually occupyingmost of the material volume. Any furtherincrease of the field has little effect on thedomains and therefore has little effect onthe flux density in the material; thus thepermeability depends strongly on theapplied field.

Magnetization Curves

A plot of the relation in Eq. 98 describingthe flux density as a function of the fieldintensity is a useful way to look at

B H= ( ) H

ii

= 0

N I Rii

i j j

j

=

Rs

gg=

0

Rs

=

=+

V

R Rm

g

=

+

NI

r

s s

2

0

g g

= Bs

99Magnetism

Figure 7. Equivalent magnetic circuitrepresentation.

Legend

Rg = gap resistanceRt = toroid resistanceVm = magnetomotive force

= flux

Vm

+

Rt

Rg


16/24

magnetic materials. For linear materials(materials for which the permeability isconstant at any field value), this curve is astraight line whose slope is equal to thepermeability. Ferromagnetic materialsbehave differently. The curves in Fig. 8describe the behavior of iron. Initially, theapplied field intensity is zero and so is theflux density.

As the field is increased, the flux

density also increases but, unlike linearmaterials, the curve is not linear. At somefield valueH1, the curve starts bendingand the slope of the curve is reducedsignificantly. Any increase beyond thefieldH3 increases the flux density but notat the same rate as at lower points on thecurve. In fact, the slope in this section ofthe curve approaches unity, meaning thatthe relative permeability approaches 1.This region is called saturation and isdependent on the material tested. Thewhole curve described in Fig. 8a is called amagnetization curve. Since it starts withzero applied field it may also be called an

initial magnetization curve.

Hysteresis Curves

Reducing the applied field moves thecurve to the left, rather than retracing theinitial magnetization curve (Fig. 8b). Theflux density is reduced up to the pointBr,where the applied field is zero. Thisresidual flux is called remanence orretentivityand is typical of allferromagnetic materials. Applying a

reverse magnetic field further reduces theflux density to the pointHc, where anapplied field intensity exists without anassociated flux density. The field intensityat this point is called the coercivityor thecoercive force of the material. Furtherincrease in the negative field intensitytraces the magnetization curve throughpointP2 where a saturation point hasagain been reached, except that in thiscase the field intensity and the fluxdensity are negative.

If the applied field is decreased to zero,a point symmetric toBr is reached.Similarly, by increasing the applied fieldintensity to a value equal (but positive) to

Hc, the flux density is again zero. Furtherincrease in the field intensity brings itback to the pointP1. Repeating theprocess described above results in aretracing of the outer curve but not thatof the initial magnetization curve. Thisunique magnetization curve is called thehysteresis curve and is typical of allferromagnetic materials (hysteresis curvesof different materials, including theircoercive forces and remanence, aremarkedly different).

The slope of this curve at any point isthe magnetic permeability. The slope isrelatively high in the lower portions of

the initial magnetization curve and isgradually reduced to unity. At this point,the material has reached magneticsaturation. A curve describing the slope ofthe initial magnetization curve of Fig. 8ais shown in Fig. 9. Figure 9 shows that forthis material (iron), the initial relativepermeability is low, increases graduallyand then, as the field approachessaturation, decreases and approaches 1.

The hysteresis curve in Fig. 8b has fourdistinct sections described by the fourquadrants of the coordinate system.Particularly important are the first andsecond. The curve in the first quadrant is

created by an applied field or source andis therefore called a magnetization curve. Inparticular, the initial magnetization curvecan only be described by starting with anunmagnetized sample of the material andthen increasing the field within thematerial. This section of the curve isreferred to as the active part of the curve.All direct current applications of magneticparticle testing that depend on activemagnetization are governed by thissection of the curve.

100 Magnetic Testing

Figure 8. Hysteresis curve: (a) initialmagnetization curve; (b) hysteresis curve.

(a)

(b)

B

H1 H3

H

B

Hc

Br

P2

P1

Hc

Br

H1

H

Legend

B= magnetic flux densityH= magnetic field intensityP= saturation point


17/24

The second quadrant (with the limitsat Br and Hc) is called the demagnetizationcurve. It is important for two reasons.First, any magnetic material, after beingmagnetized, relaxes to the point Br ormore commonly to a point in the secondquadrant. Secondly, this is the quadrantin which permanent magnets operate. Thecoercivity and remanence offerromagnetic materials are very different

from each other and define to a largeextent the classification of magneticmaterials. The coercivity and remanenceof some important materials are shown inTable4.

The area under the hysteresis curverepresents energy. This is understood byreferring to the poynting theorem. Indevices such as transformers, this is adetrimental property because the energy is

dissipated, primarily in the magnetic coreof the device. In other cases, includingpermanent magnets or switchingmagnetic devices, this property is useful.

Magnetization

In order to magnetize a sample, it isnecessary to apply a magnetic field to thesample. The form in which this field isapplied may vary depending on practicalconsiderations but the same basic effectmust be obtained: the field in the samplemust be increased to a required value.

In general, if a sample is initiallydemagnetized, the field is graduallyincreased through the initialmagnetization curve to a required point.If a residual method is being used, thefield is reduced to zero and the materialrelaxes to a point in the second quadrantof the hysteresis loop. For previouslymagnetized samples, it is usually better todemagnetize the sample first and then tomagnetize it to the required point.

Demagnetization

The hysteresis curve indicates that whenthe source of a field is reduced to zero,there is a remanent flux density in thematerial. This remanent or residual field issometimes used for testing purposes butin many cases it is desirable todemagnetize a test object before acontrolled field is applied or todemagnetize it after a test.

Demagnetization cannot be achievedsimply by creating a field opposing theoriginal source field. The demagnetizationprocess is complicated by shape effects

that usually cause different operating

101Magnetism

Figure 9. Initial permeability curve for iron.

H1 H3

H

Permeability(relativescale)

Magnetic field intensity (relative scale)

TABLE 4. Coercivity and remanence of some important materials. Values for Hc and Br areapproximate and strongly depend on thermoelectrical history.

Coercive Remanent SaturationForce Flux Density Flux DensityHc Br Bs

(Am1) (Wbm2) (Wbm2)

Soft magnetic materials

Annealed nickel alloy (controlled cooling)a 0.2 104 0.8

Nickel zinc ferrite 16 0 0.34

Silicon iron (4 percent silicon) 20 0.5 1.95

Iron (pure annealed) 100 1.2 2.16Steel (0.9 percent carbon, hot rolled) 4000 1.0 2.0

Hard magnetic materials

Carbon steel (0.9 percent carbon) 4000 1

Aluminum nickel alloy, type 5 44 000 1.2

Aluminum nickel alloy, type 8 126 000 1.04

Samarium cobalt 560 000 0.84

a. 79 percent nickel; 5 percent molybdenum; iron.


18/24

points to exist in different sections of thematerial (see the curve in Fig. 8).

Effective demagnetization of materialscan be achieved by heating the materialbeyond the curie temperature and thencooling it in a zero field environment.Under most circumstances, this method isimpractical because of the metallurgicaleffects associated with it.

A practical demagnetization approach

is to cycle the material through thehysteresis curve while gradually reducingthe magnetic field intensity to zero. Theeffect is shown in Fig. 10. If started with ahigh enough field intensity and reducedslowly, this procedure results in a properlydemagnetized sample. In practice,demagnetization is performed by applyingan alternating current field and graduallyreducing its amplitude to zero. Completedemagnetization is usually a very timeconsuming process. In practical situations,it is usually limited to reducing the fluxdensity to some acceptable level.

Minor Hysteresis LoopsIt often happens while a sample is atsome operating point on the hysteresiscurve (either on the initial magnetizationcurve or on the outer loop) that arelatively small change in magnetizationoccurs. An example of this is a large directcurrent corresponding to a point on thehysteresis curve and a small alternatingcurrent superimposed on it.

Alternatively, if the magnetizingcurrent is suddenly decreased and thenincreased again, the same effect is created.

This situation results in a change in thehysteresis curve as shown in Fig. 11. Thus,a small oval curve similar to the hysteresiscurve is described at the initial point.These loops are called minor hysteresis

loops to distinguish them from the normal(or major) hysteresis loop. Becausepermeability is defined as the ratio of |B|and |H|, the permeability of a minor loopmay be defined as BH1:

(99)

Also called an incremental permeability,this quantity depends on the location ofthe minor curve on the hysteresis loopand decreases as the normalmagnetization increases. The slope ofminor loops is always smaller than that ofthe major loop at a given point. Thus, theincremental permeability is lower thanthe normal permeability at any point onthe hysteresis curve. As the materialapproaches saturation, the relativeincremental permeability approachesunity.

Hysteresis Curve As Classifier

When applying electromagnetic fields, itis necessary to distinguish betweenapplications, specialties and frequencyranges. For example, electromagneticnondestructive testing is classified as adiscipline separate from paleomagnetism(terrestrial magnetism), even thoughexactly the same principles are involvedand, often, the same methods are used.Moreover, within each discipline differentapplications are distinguished.

In nondestructive testing, active leakagefield, residual leakage field, eddy currentandother electromagnetic phenomena areused. This distinction helps focus the

treatment of different problems. Often,the distinction parallels that of thevarious areas of electromagnetic fields:active leakage fields are associated withmagnetostatics; residual leakage fields

inc =

B

H


Figure 10. Demagnetization offerromagnetic materials.

B

H

Legend

B = magnetic flux densityH = magnetic field intensity

Figure 11. Major and minor hysteresisloops.

B

H

B H

Legend



19/24

with source-free magnetostatics; and eddycurrents with steady state alternatingcurrent fields.

It is far more practical to distinguishbetween the various applications based onthe point of operation on the hysteresiscurve. This offers a visual description aswell as some physical insight into theapplication.

Active leakage field methods are those

that employ the initial magnetizationcurve (Fig. 12a). The point on the initialmagnetization curve is obtained byincreasing the current that increasesintensityHfrom zero to somepredetermined value. It is possible to

apply a field to an initially magnetizedsample but this is usually not donebecause of the difficulty in determiningthe exact operating point.

Residual leakage fields are obtainedwhen an active excitation is removed andthe operating points of the material areallowed to relax into the second quadrant(Fig. 12b). Similarly, alternating currentleakage methods may be defined as those

that employ a normal hysteresis curve.The operating point is on the major loop(Fig. 12c).

Eddy current methods requirealternating current excitation but this isusually very low. In terms of the hysteresiscurve, it may be said that the operatingpoint is at the origin although smallhysteresis loops are described around theorigin as in Fig. 12c.

Energy Lost in Hysteresis Cycle

The energy stored in the magnetic field isgiven as a volume integral of an energydensity w:

(100)

After integrating over the hysteresiscurve or over any part of it, the areaunder the curve may be written as:

(101)

The units of this integral are those of avolume energy density and, underlinearity assumptions (dB = dH), theenergy density becomes w= 0.5 H2.

If this is then integrated over thevolume of material in which the magneticfield exists, the total work done byexternal sources can be written as:

(102)

The fact that energy is transformed in

the process becomes apparent byconsidering that work needs to beperformed in order to change themagnetic field in the volume of amaterial. The expression in Eq. 102 is thework done for a complete cycle over thehysteresis loop. If the field changes at acertain frequency, the energy per cycle inEq. 102 must be multiplied by thefrequency to obtain:

W HdB dv

B

v

=

0

w HdB

B

= 0

W wdv

v

=

103Magnetism

Figure 12. Classification of testingmethods: (a) active leakage fields (directcurrent); (b) residual leakage fields;(c) alternating current operation.

Legend


H

B

H

B

H

B

(a)

(b)

(c)


20/24

(103)

This equation is exact but of limiteduse because it requires integration overthe hysteresis loop. Being a complexfunction and in many cases only knownexperimentally, the hysteresis loop isdifficult to integrate. For practicalpurposes, an approximate expression interms of the maximum induced fluxdensityBmax is often used. The dissipatedpower is:

(104)

This expression is credited to CharlesSteinmetz and assumes that the constant is known. It ranges from 0.001 forsilicon steels to about 0.03 for hard steels.It is an experimental value and theequation is only correct for relatively largesaturation fields (above 0.1 T, or 1 kG).For low saturation flux densities, the

equation cannot be used.

Eddy Current Losses

In addition to hysteresis losses, thechange in flux density inside conductingmaterials generates inducedelectromagnetic forces in those materials.The existence of this electromagneticforce, and the relatively large conductivityof most metals, results in a relatively largecurrent flowing inside the material in apath that is a mirror image of the sourcegenerating the field. It is difficult tocalculate the eddy current generated forany particular situation but the relative

quantity involved is easy to obtain.For any conductor, the electric field

due to the induced electromagnetic forceis directly proportional to the magneticfield as:

(105)

The dissipated energy due to heatinglosses (I2R) is related to the square of theelectric field. In terms of Eq. 105 and themagnetic field, this becomes:

(106)

This relation clearly indicates that thelosses due to eddy currents can be very

large, especially for large flux densitiesand higher frequencies. Eddy currentlosses may be reduced: (1) by reducing theconductivity of the materials involved(ferrites); (2) by special alloying toproduce very narrow hysteresis curves(silicon steels); and (3) by breaking theeddy current paths (laminated cores).

The total losses in magnetic materialsdue to hysteresis and eddy current lossescan be summarized in terms of the actualfield as:

(107)

where Nis the number of turns.In terms of the saturation flux density

Bmax, total losses may be written as:

(108)

The constants k, k1 and k2 depend ongeometry as well as on materialproperties.

P k f B k f B

d = +11 6

22

2

max. max

P Nf kB f

d = +2 2

E f B2 2 2

E B

B d

dtf

P f Bd = max.1 6

P Wfd =



21/24

Energy in ElectromagneticFieldIn order to examine the energy in amagnetic field it is convenient to lookfirst at the general time dependentexpression for energy. This expressionincludes stored magnetic energy, storedelectric energy and dissipated energy. Thefollowing vector identity is used:

(109)

Into this expression, a substitution ismade: the expression for the curl of E andH from Maxwells first and secondequations:

(110)

Assuming the energy flows in a volumevbounded by an area s, it is then possibleto integrate this expression over thevolume v. Before this, transform the leftside from a volume integral to an areaintegral using a divergence theorem.

(111)

The left side of the expressionrepresents the total flow of energythrough the area bounding the volume.The expression E H is a power densitywith units of Wm2. The power density iscalled apoynting vector:

(112)

The advantage of such an expression isthat it also indicates the direction of theenergy flow, information that isimportant for wave propagationcalculations.

The first term on the right side ofEq. 111 represents the time rate ofincrease in the potential or stored energyin the system. It has two components: the

stored magnetic energy and the storedelectric energy. These energy densitiesreduce to simpler expressions for thestatic electric and magnetic fields:

(113)

(114)

The second term on the right side of

Eq. 111 is the power dissipated and thepower due to sources that may exist in thevolume v. If no such sources exist, thisterm represents ohmic losses.

The poynting theorem describes all of asystems energy relations: electrostatic,magnetostatic or time dependent. Becausethe cross product between the electricfield and the magnetic field is taken, thesetwo quantities must be related, otherwisethe results have no meaning.

The expression in Eq. 111 is aninstantaneous quantity. For practicalpurposes, a time averaged quantity ismore useful. This can be done byaveraging over a time T(usually a cycle of

the alternating current field):

(115)

Force in Magnetic FieldThe force in the magnetic field isgoverned by the lorenz force equationgiven in Eq. 9. For the purposes here, theelectric force (Coulombs law) Fe = qE is

not important and was removed from theequation. This in effect assumes that thecharge q only experiences a magneticforce:

(116)

Here, v is the velocity of the charge.While forces on charges are important inthemselves, the force on current carrying

F v B= ( )q

PT

P t dt

T

av = ( )10

w H

m = 2

2

w E

e =

2

2

P E H=

E H s H B E D

E J

( ) =

+

s v

v

t

dv dv

2 2

( ) = E H H B

E D

E J

t t

( ) = E H H E E H

105Magnetism

PART 6. Characteristics of Electromagnetic Fields


22/24

conductors is more important inconjunction with magnetic fields. If it isassumed that an element of conductor d,with a cross sectional area s carries Ncharge particles per unit volume movingwith an average velocity v, then themagnetic force that this conductorexperiences is:

(117)

Since Nqsvis the total current in theconductor, the magnetic force becomes:

(118)

To obtain the force due the completeconductor, integration is taken over thelength of the conductor:

(119)

Another important consideration isthat of the force exerted on a currentcarrying conductor due to the field of asecond conductor. This is treated byassuming that the fieldB is due to oneconductor. If there are two conductors,the force on conductor 1 due to the fieldof conductor 2 is:

(120)

Similarly, the force on conductor 2 dueto conductor 1 is:

(121)

In these equations, the integration isassumed to be over the entire (closed)path of the currents. This is not merely aconvenience but is required to ensure thatthe forcesF12 andF21 are equal and

opposite in direction. In other words,integration over part of the contoursviolates Newtons third law (action andreaction forces).

Forces in the magnetic field may alsobe expressed in terms of the energy storedin the magnetic field. A systemsmechanical work is done at the expenseof its potential energy, so that:

(122)

where is angular frequency (radians persecond).

The force due to this reduction in thestored energy is therefore:

(123)

This expression for the force isparticularly convenient when the actual

current distributions are not known or aretoo complicated to permit calculation ofthe flux densities of each currentseparately.

Because the magnetic field energy maybe expressed in terms of inductances(Eq. 131), the force in the magnetic fieldmay also be expressed in terms ofinductances. Thus, for example, the forcebetween two conductors carrying currentsI1 andI2, having inductancesL1,L2 andmutual inductanceL12 can be written as:

(124)

The stored energy is calculated as:

(125)

Torque in Magnetic FieldThe torque on a current carrying systemmay be calculated by using the definitionof torque: the product of force and themoment arm length. For simplicity,

consider the square loop in Fig. 13. If thisdefinition is used, the torque is equal to:

(126)

where s is the area of the loop and s is aunit vector normal to the plane of thecurrent loop. The product of current andarea is defined as a magnetic moment m.

(127)

The magnetic moment has a direction

normal to the area s and in the directiondescribed by the right hand rule. Thus,torque is a vector quantity and can bewritten as:

(128) T m B=

m s= I

T s= sinIs B

W L I L I I L I = + +1

2

1

21 12

12 1 2 2 22

F = ( )I I L1 2 12

F =

F d d = ( )

F I IR

12 2 1

2 1 12

1224

12

= ( )

d d R

CC

F I IR

21 1 21 2 21

2124

21

= ( )

d d R

CC

F d B= I C

d IF d B=

d Nqs v F d B=



23/24

InductanceInductance is a property of thearrangement of conductors in a system. Itis a measure of the flux linked within thecircuit when excited and a measure of themagnetic energy stored in the system ofconductors. Flux linkage is defined as theflux that links the whole system of

conductors, multiplied by the number ofconductors or turns in the system. For asimple solenoid, this is defined as:

(129)

In effect, this includes only the flux thatpasses through the center of the solenoid.The integration is over the cross sectionalarea of the solenoid. A more complicateddefinition can be used, one that includesflux linkages that do not link all theconductors, but this has little practical use

because of the difficulties in calculation.Ifthe system under consideration is linear,the field is directly proportional to thecurrent and the inductance in henries (H)of a system (a coil) may be defined as theratio of the flux linkage and the current:

(130)

The inductance in Eq. 130 can becalculated, provided that the flux linkagescan be obtained. In many practicalsituations, it is more convenient to use anenergy relation.

(131) W LI=

2

2

L

N

Is=

B ds

= = N Ns

B d s

107Magnetism

Figure 13. (a) Rectangular loop;(b) direction of forces and torque onrotating loop.

Legend

B = magnetic flux densityF = mechanical forcem = magnetic movement (Eq. 127)I = currents = area within loop

I

s

F

mB

F

(a) (b)


24/24

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Magnetic Analysis 5 (3)