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# chapter 1: Magnetic Circuits and Magnetic Materials

Module by: NGUYEN Phuc. E-mail the author

Chapter 1: Magnetic Circuits and Magnetic Materials

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

• The objective of this course is to study the devices used in the interconversion of electric and mechanical energy, with emphasis placed on electromagnetic rotating machinery.
• The transformer, although not an electromechanical-energy-conversion device, is an important component of the overall energy-conversion process.
• Practically all transformers and electric machinery use ferro-magnetic material for shaping and directing the magnetic fields that acts as the medium for transferring and converting energy. Permanent-magnet materials are also widely used.
• The ability to analyze and describe systems containing magnetic materials is essential for designing and understanding electromechanical-energy-conversion devices.
• The techniques of magnetic-circuit analysis, which represent algebraic approximations to exact field-theory solutions, are widely used in the study of electromechanical-energy-conversion devices.

§1.1 Introduction to Magnetic Circuits

Assume the frequencies and sizes involved are such that the displacement-current term in Maxwell’s equations, which accounts for magnetic fields being produced in space by time-varying electric fields and is associated with electromagnetic radiations, can be neglected.

• H : magnetic field intensity, amperes/m, A/m, A-turn/m, A-t/m
• B : magnetic flux density, webers/m2, Wb/m2, tesla (T)
• 1 Wb = 108108 size 12{"10" rSup { size 8{8} } } {} lines (maxwells); 1 T = 104104 size 12{"10" rSup { size 8{4} } } {} gauss
• (1.1)From (1.1), we see that the source of H is the current density J .The line integral of the tangential component of the magnetic field intensity H around a closed contour C is equal to the total current passing through any surface S linking that contour.

c H . dl = s J . da c H . dl = s J . da size 12{ lInt rSub { size 8{c} } {H "." ital "dl"} = Int rSub { size 8{s} } {J "." ital "da"} } {}

• (1.2)Equation (1.2) states that the magnetic flux density B is conserved. No net flux enters or leaves a closed surface.There exists no monopole charge sources of magnetic fields.

s B . da = 0 s B . da = 0 size 12{ lInt rSub { size 8{s} } {B "." ital "da"} =0} {}

• A magnetic circuit consists of a structure composed for the most part of high-permeability magnetic material. The presence of high-permeability material tends to cause magnetic flux to be confined to the paths defined by the structure.

Figure 1.1Simple magnetic circuit.

• In Fig. 1.1, the source of the magnetic field in the core is the ampere-turn product N i , the magnetomotive force (mmf) F acting on the magnetic circuit.
• The magnetic flux φφ size 12{φ} {} (in weber, Wb) crossing a surface S is the surface integral of the normal component B :

φ=sB.daφ=sB.da size 12{φ= lInt rSub { size 8{s} } {B "." ital "da"} } {} (1.3)

• φcφc size 12{φ rSub { size 8{c} } } {} : flux in core, BcBc size 12{B rSub { size 8{c} } } {} : flux density in core

φc=BcAcφc=BcAc size 12{φ rSub { size 8{c} } =B rSub { size 8{c} } A rSub { size 8{c} } } {} (1.4)

• HcHc size 12{H rSub { size 8{c} } } {} : average magnitude H in the core. The direction of HcHc size 12{H rSub { size 8{c} } } {}can be found from the RHR.

F=Ni=HdlF=Ni=HclcF=Ni=HdlF=Ni=Hclcalignl { stack { size 12{F= ital "Ni"= lInt { ital "Hdl"} } {} # size 12{F= ital "Ni"=H rSub { size 8{c} } l rSub { size 8{c} } } {} } } {} (1.5)

• The relationship between the magnetic field intensity H and the magnetic flux density B:

{}B=μHB=μH size 12{B=μH} {} (1.6)

• Linear relationship?
• μ=μrμoμ=μrμo size 12{μ=μ rSub { size 8{r} } μ rSub { size 8{o} } } {} , μμ size 12{μ} {}: magnetic permeability, Wb/A-t-m = H/m
• μo=.107μo=.107 size 12{μ rSub { size 8{o} } =4π "." "10" rSup { size 8{ - 7} } } {}: the permeability of free space
• μrμr size 12{μ rSub { size 8{r} } } {}: relative permeability, typical values: 2000-80,000

A magnetic circuit with an air gap is shown in Fig.1.2. Air gaps are present for moving elements. The air gap length is sufficiently small. φφ size 12{φ} {} : the flux in the magnetic circuit.

Figure 1.2Magnetic circuit with air gap.

Bc=φAcBc=φAc size 12{B rSub { size 8{c} } = { {φ} over {A rSub { size 8{c} } } } } {} (1.7)

Bg=φAgBg=φAg size 12{B rSub { size 8{g} } = { {φ} over {A rSub { size 8{g} } } } } {} (1.8)

F=Hclc+HglgF=Hclc+Hglg size 12{F=H rSub { size 8{c} } l rSub { size 8{c} } +H rSub { size 8{g} } l rSub { size 8{g} } } {} (1.9)

F=Bcμlc+Bgμ0gF=Bcμlc+Bgμ0g size 12{F= { {B rSub { size 8{c} } } over {μ} } l rSub { size 8{c} } + { {B rSub { size 8{g} } } over {μ rSub { size 8{0} } } } g} {} (1.10)

F=φ(lcμAc+gμ0Ag)F=φ(lcμAc+gμ0Ag) size 12{F=φ $${ {l rSub { size 8{c} } } over {μA rSub { size 8{c} } } } + { {g} over {μ rSub { size 8{0} } A rSub { size 8{g} } } }$$ } {} (1.11)

• RcRc size 12{R rSub { size 8{c} } } {} , RgRg size 12{R rSub { size 8{g} } } {} : the reluctance of the core and the air gap, respectively,

Rc=lcμAcRc=lcμAc size 12{R rSub { size 8{c} } = { {l rSub { size 8{c} } } over {μA rSub { size 8{c} } } } } {} (1.12)

Rg=gμ0AgRg=gμ0Ag size 12{R rSub { size 8{g} } = { {g} over {μ rSub { size 8{0} } A rSub { size 8{g} } } } } {} (1.13)

F=φ(Rc+Rg)F=φ(Rc+Rg) size 12{F=φ $$R rSub { size 8{c} } +R rSub { size 8{g} }$$ } {} (1.14)

φ=FRc+Rgφ=FRc+Rg size 12{φ= { {F} over {R rSub { size 8{c} } +R rSub { size 8{g} } } } } {} (1.15)

φ=FlcμAc+gμ0Agφ=FlcμAc+gμ0Ag size 12{φ= { {F} over { { {l rSub { size 8{c} } } over {μA rSub { size 8{c} } } } + { {g} over {μ rSub { size 8{0} } A rSub { size 8{g} } } } } } } {} (1.16)

• In general, for any magnetic circuit of total reluctance RtotRtot size 12{R rSub { size 8{ ital "tot"} } } {}, the flux can be found as

φ=FRtotφ=FRtot size 12{φ= { {F} over {R rSub { size 8{ ital "tot"} } } } } {} (1.17)

The permeance P is the inverse of the reluctance

Ptot=1RtotPtot=1Rtot size 12{P rSub { size 8{ ital "tot"} } = { {1} over {R rSub { size 8{ ital "tot"} } } } } {} (1.18)

• Fig. 1.3: Analogy between electric and magnetic circuits:

Figure 1.3:Analogy between electric and magnetic circuits: (a) electric ckt, (b) magnetic ckt.

• Note that with high material permeability: Rc<<RgRc<<Rg size 12{R rSub { size 8{c} } "<<"R rSub { size 8{g} } } {} and thus Rtot<<RgRtot<<Rg size 12{R rSub { size 8{ ital "tot"} } "<<"R rSub { size 8{g} } } {}

φFRg=0Agg=Niμ0AggφFRg=0Agg=Niμ0Agg size 12{φ approx { {F} over {R rSub { size 8{g} } } } = { {Fμ rSub { size 8{0} } A rSub { size 8{g} } } over {g} } = ital "Ni" { {μ rSub { size 8{0} } A rSub { size 8{g} } } over {g} } } {} (1.19)

• Fig. 1.4: Fringing effect, effective AgAg size 12{A rSub { size 8{g} } } {} increased.

Figure 1.4 Air-gap fringing fields.

• In general, magnetic circuits can consist of multiple elements in series and parallel.

F=Hdl=kFk=kHklkF=Hdl=kFk=kHklk size 12{F= lInt { ital "Hdl"= Sum cSub { size 8{k} } {F rSub { size 8{k} } } } = Sum cSub { size 8{k} } {H rSub { size 8{k} } l rSub { size 8{k} } } } {} (1.20)

F=sJ.daF=sJ.da size 12{F= Int rSub { size 8{s} } {J "." ital "da"} } {} (1.21)

V=kRkikV=kRkik size 12{V= Sum cSub { size 8{k} } {R rSub { size 8{k} } i rSub { size 8{k} } } } {} (1.22)

nin=0nin=0 size 12{ Sum cSub { size 8{n} } {i rSub { size 8{n} } } =0} {} (1.23)

nφn=0nφn=0 size 12{ Sum cSub { size 8{n} } {φ rSub { size 8{n} } } =0} {} (1.24)

§1.2 Flux Linkage, Inductance, and Energy

cE.ds=ddtsB.dacE.ds=ddtsB.da size 12{ lInt rSub { size 8{c} } {E "." ital "ds"} = - { {d} over { ital "dt"} } Int rSub { size 8{s} } {B "." ital "da"} } {} (1.25)

• λλ size 12{λ} {}: the flux linkage of the winding, ϕϕ size 12{ϕ} {} : the instantaneous value of a time-varying flux,
• e : the induced voltage at the winding terminals

e=Ndt=dtλ=e=Ndt=dtλ=alignl { stack { size 12{e=N { {dϕ} over { ital "dt"} } = { {dλ} over { ital "dt"} } } {} # size 12{λ=Nϕ} {} } } {} (1.26)

• L : the inductance (with material of constant permeability), H = Wb-t/A

L=λiL=λi size 12{L= { {λ} over {i} } } {} (1.27)

L=N2RtotL=N2Rtot size 12{L= { {N rSup { size 8{2} } } over {R rSub { size 8{ ital "tot"} } } } } {} (1.28)

• The inductance of the winding in Fig. 1.2:

L=N2(g/μ0Ag)=N2μ0AggL=N2(g/μ0Ag)=N2μ0Agg size 12{L= { {N rSup { size 8{2} } } over { $$g/μ rSub { size 8{0} } A rSub { size 8{g} }$$ } } = { {N rSup { size 8{2} } μ rSub { size 8{0} } A rSub { size 8{g} } } over {g} } } {} (1.29)

• Magnetic circuit with more than one windings, Fig. 1.5:

Figure 1.5Magnetic circuit with two windings.

F=N1i1+N2i2F=N1i1+N2i2 size 12{F=N rSub { size 8{1} } i rSub { size 8{1} } +N rSub { size 8{2} } i rSub { size 8{2} } } {} (1.30)

φ=(N1i1+N2i2)μ0Acgφ=(N1i1+N2i2)μ0Acg size 12{φ= $$N rSub { size 8{1} } i rSub { size 8{1} } +N rSub { size 8{2} } i rSub { size 8{2} }$$ { {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} } } {} (1.31)

λ1=N1φ=N12(μ0Acg)i1+N1N2(μ0Acg)i2λ1=N1φ=N12(μ0Acg)i1+N1N2(μ0Acg)i2 size 12{λ rSub { size 8{1} } =N rSub { size 8{1} } φ=N rSub { size 8{1} } rSup { size 8{2} } $${ {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} }$$ i rSub { size 8{1} } +N rSub { size 8{1} } N rSub { size 8{2} } $${ {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} }$$ i rSub { size 8{2} } } {} (1.32)

λ1=L11i1+L12i2λ1=L11i1+L12i2 size 12{λ rSub { size 8{1} } =L rSub { size 8{"11"} } i rSub { size 8{1} } +L rSub { size 8{"12"} } i rSub { size 8{2} } } {} (1.33)

L11=N12μ0AcgL11=N12μ0Acg size 12{L rSub { size 8{"11"} } =N rSub { size 8{1} } rSup { size 8{2} } { {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} } } {} (1.34)

L12=N1N2μ0Acg=L21L12=N1N2μ0Acg=L21 size 12{L rSub { size 8{"12"} } =N rSub { size 8{1} } N rSub { size 8{2} } { {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} } =L rSub { size 8{"21"} } } {} (1.35)

λ2=N2φ=N1N2(μ0Acg)i1+N22(μ0Acg)i2λ2=N2φ=N1N2(μ0Acg)i1+N22(μ0Acg)i2 size 12{λ rSub { size 8{2} } =N rSub { size 8{2} } φ=N rSub { size 8{1} } N rSub { size 8{2} } $${ {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} }$$ i rSub { size 8{1} } +N rSub { size 8{2} } rSup { size 8{2} } $${ {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} }$$ i rSub { size 8{2} } } {} (1.36)

λ2=L21i1+L22i2λ2=L21i1+L22i2 size 12{λ rSub { size 8{2} } =L rSub { size 8{"21"} } i rSub { size 8{1} } +L rSub { size 8{"22"} } i rSub { size 8{2} } } {} (1.37)

L22=N22μ0AcgL22=N22μ0Acg size 12{L rSub { size 8{"22"} } =N rSub { size 8{2} } rSup { size 8{2} } { {μ rSub { size 8{0} } A rSub { size 8{c} } } over {g} } } {} (1.38)

• Induced voltage, power (W = J/s), and stored energy:

e=ddt(Li)e=ddt(Li) size 12{e= { {d} over { ital "dt"} } $$ital "Li"$$ } {} (1.39)

e=Ldidt+idLdte=Ldidt+idLdt size 12{e=L { { ital "di"} over { ital "dt"} } +i { { ital "dL"} over { ital "dt"} } } {} (1.40)

p=ie=idtp=ie=idt size 12{p= ital "ie"=i { {dλ} over { ital "dt"} } } {} (1.41)

ΔW=t1t2pdt=λ1λ2idλΔW=t1t2pdt=λ1λ2idλ size 12{ΔW= Int rSub { size 8{t rSub { size 6{1} } } } rSup {t rSub { size 6{2} } } { ital "pdt"} size 12{ {}= Int rSub {λ rSub { size 6{1} } } rSup {λ rSub { size 6{2} } } { ital "id"λ} }} {} (1.42)

ΔW=λ1λ2idλ=λ1λ2λL=12L(λ22λ11)ΔW=λ1λ2idλ=λ1λ2λL=12L(λ22λ11) size 12{ΔW= Int rSub { size 8{λ rSub { size 6{1} } } } rSup {λ rSub { size 6{2} } } { ital "id"λ} size 12{ {}= Int rSub {λ rSub { size 6{1} } } rSup {λ rSub { size 6{2} } } { { {λ} over {L} } } } size 12{dλ= { {1} over {2L} } $$λ rSub {2} rSup {2} } size 12{ - λ rSub {1} rSup {1} } size 12{$$ }} {} (1.43)

W=12Lλ2=L2i2W=12Lλ2=L2i2 size 12{W= { {1} over {2L} } λ rSup { size 8{2} } = { {L} over {2} } i rSup { size 8{2} } } {} (1.44)

§1.3 Properties of Magnetic Materials

• The importance of magnetic materials is twofold:
• Magnetic materials are used to obtain large magnetic flux densities with relatively low levels of magnetizing force.
• Magnetic materials can be used to constrain and direct magnetic fields in well defined paths.
• Ferromagnetic materials, typically composed of iron and alloys of iron with cobalt, tungsten, nickel, aluminum, and other metals, are by far the most common magnetic materials.
• They are found to be composed of a large number of domains.
• When unmagnetized, the domain magnetic moments are randomly oriented.
• When an external magnetizing force is applied, the domain magnetic moments tend to align with the applied magnetic field until all the magnetic moments are aligned with the applied field, and the material is said to be fully saturated.
• When the applied field is reduced to zero, the magnetic dipole moments will no longer be totally random in their orientation and will retain a net magnetization component along the applied field direction.
• The relationship between B and H for a ferromagnetic material is both nonlinear and multivalued.
• In general, the characteristics of the material cannot be described analytically but are commonly presented in graphical form.
• The most common used curve is the B  H curve.
• Dc or normal magnetization curve:
• Hysteresis loop (Note the remanance):

Figure 1.6B-H loops for M-5 grain-oriented electrical steel 0.012 in thick.

Only the top halves of the loops are shown here. (Armco Inc.)

Figure 1.7 Dc magnetization curve for M-5 grain-oriented electrical steel 0.012 in thick.

(Armco Inc.)

Figure 1.8Hysteresis loop.

§1.4 AC Excitation

• In ac power systems, the waveforms of voltage and flux closely approximate sinusoidal functions of time. We are to study the excitation characteristics and losses associated with magnetic materials under steady-state ac operating conditions.
• Assume a sinusoidal variation of the core flux ϕ(t)ϕ(t) size 12{ϕ $$t$$ } {}:

ϕ(t)=φmaxsinωt=AcBmaxsinωtϕ(t)=φmaxsinωt=AcBmaxsinωt size 12{ϕ $$t$$ =φ rSub { size 8{"max"} } "sin"ωt=A rSub { size 8{c} } B rSub { size 8{"max"} } "sin"ωt} {} (1.45)

Where φmax=φmax= size 12{φ rSub { size 8{"max"} } ={}} {}amplitude of core flux  in webers

Bmax=Bmax= size 12{B rSub { size 8{"max"} } ={}} {} amplitude of flux density BcBc size 12{B rSub { size 8{c} } } {} in teslas

ω=ω= size 12{ω={}} {}angular frequency =2πf=2πf size 12{ {}=2πf} {}

f = frequency in Hz

• The voltage induced in the N-turn winding is

e(t)=ωNφmaxcos(ωt)=Emaxcosωte(t)=ωNφmaxcos(ωt)=Emaxcosωt size 12{e $$t$$ =ωNφ rSub { size 8{"max"} } "cos" $$ωt$$ =E rSub { size 8{"max"} } "cos"ωt} {} (1.46)

Emax=ωNφmax=fNAcBmaxEmax=ωNφmax=fNAcBmax size 12{E rSub { size 8{"max"} } =ωNφ rSub { size 8{"max"} } =2π ital "fNA" rSub { size 8{c} } B rSub { size 8{"max"} } } {} (1.47)

• The Root-Mean-Squared (rms) value:

Frms=1T0Tf2(t)dtFrms=1T0Tf2(t)dt size 12{F rSub { size 8{ ital "rms"} } = sqrt { { {1} over {T} } Int rSub { size 8{0} } rSup { size 8{T} } {f rSup { size 8{2} } $$t$$ ital "dt"} } } {} (1.48)

Erms=2fNAcBmax=2πfNAcBmaxErms=2fNAcBmax=2πfNAcBmax size 12{E rSub { size 8{ ital "rms"} } = { {2π} over { sqrt {2} } } ital "fNA" rSub { size 8{c} } B rSub { size 8{"max"} } = sqrt {2} π ital "fNA" rSub { size 8{c} } B rSub { size 8{"max"} } } {} (1.49)

Note that the rums value of a sinusoidal wave is 12 times its peak value.

• Excitation phenomena, Fig. 1.9:
• ϕϕ size 12{ϕ} {} vs iϕiϕ size 12{i rSub { size 8{ϕ} } } {} size 12{ dlrarrow } {}BcBc size 12{B rSub { size 8{c} } } {}vs HcHc size 12{H rSub { size 8{c} } } {}, iϕiϕ size 12{i rSub { size 8{ϕ} } } {} : exciting current.
• Note that ϕ=BcAcϕ=BcAc size 12{ϕ=B rSub { size 8{c} } A rSub { size 8{c} } } {} and that iϕ=Hclc/Niϕ=Hclc/N size 12{i rSub { size 8{ϕ} } =H rSub { size 8{c} } l rSub { size 8{c} } /N} {}.

Figure 1.9Excitation phenomena. (a) Voltage, flux, and exciting current;

(b) corresponding hysteresis loop.

Iϕ,rms=IcHc,rmsNIϕ,rms=IcHc,rmsN size 12{I rSub { size 8{ϕ, ital "rms"} } = { {I rSub { size 8{c} } H rSub { size 8{c, ital "rms"} } } over {N} } } {} (1.50)

ErmsIϕ,rms=2πfNAcBmaxIcHrmsNErmsIϕ,rms=2πfNAcBmaxIcHrmsN size 12{E rSub { size 8{ ital "rms"} } I rSub { size 8{ϕ, ital "rms"} } = sqrt {2} π ital "fNA" rSub { size 8{c} } B rSub { size 8{"max"} } { {I rSub { size 8{c} } H rSub { size 8{ ital "rms"} } } over {N} } } {} (1.51)

=2πfNBmaxHrms(Aclc)=2πfNBmaxHrms(Aclc) size 12{ {}= sqrt {2} π ital "fNB" rSub { size 8{"max"} } H rSub { size 8{ ital "rms"} } $$A rSub { size 8{c} } l rSub { size 8{c} }$$ } {} (1.52)

Pa=ErmsIϕ,rmsmass=2πfρcBmaxHrmsPa=ErmsIϕ,rmsmass=2πfρcBmaxHrms size 12{P rSub { size 8{a} } = { {E rSub { size 8{ ital "rms"} } I rSub { size 8{ϕ, ital "rms"} } } over { ital "mass"} } = { { sqrt {2} πf} over {ρ rSub { size 8{c} } } } B rSub { size 8{"max"} } H rSub { size 8{ ital "rms"} } } {} (1.53)

PaPa size 12{P rSub { size 8{a} } } {}: the exciting rms voltamperes per unit mass,

• The rms exciting voltampere can be seen to be a property of the material alone. It depends only on BmaxBmax size 12{B rSub { size 8{"max"} } } {}because HrmsHrms size 12{H rSub { size 8{ ital "rms"} } } {} is a unique function of BmaxBmax size 12{B rSub { size 8{"max"} } } {} .

Figure 1.10Exciting rms voltamperes per kilogram at 60 Hz for

M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)

• The exciting current supplies the mmf required to produce the core flux and the power input associated with the energy in the magnetic field in the core.
• Part of this energy is dissipated as losses and results in heating of the core.
• The rest appears as reactive power associated with energy storage in the magnetic field.

This reactive power is not dissipated in the core; it is cyclically supplied and absorbed by the excitation source.

• Two loss mechanisms are associated with time-varying fluxes in magnetic materials.
• The first is ohmic I2RI2R size 12{I rSup { size 8{2} } R} {} heating, associated with induced currents in the core material.
• Eddy currents circulate and oppose changes in flux density in the material.
• To reduce the effects, magnetic structures are usually built of thin sheets of laminations of the magnetic material.
• Eddy-current loss f2,Bmax2f2,Bmax2 size 12{ prop f rSup { size 8{2} } ,B rSub { size 8{"max"} } rSup { size 8{2} } } {}.
• The second loss mechanic is due to the hysteretic nature of magnetic material.
• The energy input W to the core as the material undergoes a single cycle

W=iϕ=(HclcN)(AcNdBc)=AclcHcdBcW=iϕ=(HclcN)(AcNdBc)=AclcHcdBc size 12{W= lInt {i rSub { size 8{ϕ} } } dλ= lInt { $${ {H rSub { size 8{c} } l rSub { size 8{c} } } over {N} }$$ } $$A rSub { size 8{c} } ital "NdB" rSub { size 8{c} }$$ =A rSub { size 8{c} } l rSub { size 8{c} } lInt {H rSub { size 8{c} } } ital "dB" rSub { size 8{c} } } {} (1.54)

• For a given flux level, the corresponding hysteresis losses are proportional to the area of the hysteresis loop and to the total volume of material.
• Hysteresis power loss ff size 12{ prop f} {}.
• Information on core loss is typically presented in graphical form. It is plotted in terms of watts per unit weight as a function of flux density; often a family of curves for different frequencies are given. See Fig.1.12.

Figure 1.11Hysteresis loop; hysteresis loss is proportional to the loop area (shaded).

Figure 1.12Core loss at 60 Hz in watts per kilogram for

M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc).

§1.5 Permanent Magnets

• Certain magnetic materials, commonly known as permanent-magnet materials, are characterized by large values of remanent magnetization and coercivity. These materials produce significant magnetic flux even in magnetic circuits with air gaps.
• The second quadrant of a hysteresis loop (the magnetization curve) is usually employed for analyzing a permanent-magnet material.
• BrBr size 12{B rSub { size 8{r} } } {} : residual flux density or remanent magnetization,
• HcHc size 12{H rSub { size 8{c} } } {}: coercivity, (1) a measure of the magnitude of the mmf required to demagnetize the material, and (2) a measure of the capability of the material to produce flux in a magnetic circuit which includes an air gap.
• Large value (> 1 kA/m): hard magnetic material, o.w.: soft magnetic material
• Fig. 1.13(a): Alnico 5, Br1.22TBr1.22T size 12{B rSub { size 8{r} } simeq 1 "." "22"T} {}, Hc49 kA/mHc49 kA/m size 12{H rSub { size 8{c} } simeq - "49"" kA/m"} {}
• Fig. 1.13(b): M-5 steel, Br1.4T ,Hc6 kA/mBr1.4T ,Hc6 kA/m size 12{B rSub { size 8{r} } simeq 1 "." 4"T "," "H rSub { size 8{c} } simeq - 6" kA/m"} {}
• Both Alnico 5 and M-5 electrical steel would be useful in producing flux in unexcited magnetic circuits since they both have large values of remanent magnetization.
• The significant of remanent magnetization is that it can produce magnetic flux in a magnetic circuit in the absence of external excitation (such as winding currents).

Figure 1.13 (a) Second quadrant of hysteresis loop for Alnico 5; (b) second quadrant of hysteresis loop for M-5 electrical steel; (c) hysteresis loop for M-5 electrical steel expanded for small B. (Armco Inc.)

• Maximum Energy Product: a useful measure of the capability of permanent-magnet material.
• The product of B and H has the dimension of energy density ( J/m3J/m3 size 12{J/m rSup { size 8{3} } } {})
• Choosing a material with the largest available maximum energy product can result in the smallest required magnet volume.
• (1.55) can be obtained:

Bg=AmAgBmBg=AmAgBm size 12{B rSub { size 8{g} } = { {A rSub { size 8{m} } } over {A rSub { size 8{g} } } } B rSub { size 8{m} } } {} (1.55)

HmlmHgg=1HmlmHgg=1 size 12{ { {H rSub { size 8{m} } l rSub { size 8{m} } } over {H rSub { size 8{g} } g} } = - 1} {} (1.56)

Bg2=μ0(lmAmgAg)(HmBm)=μ0(VolmagVolairgap)(HmBm)Bg2=μ0(lmAmgAg)(HmBm)=μ0(VolmagVolairgap)(HmBm) size 12{B rSub { size 8{g} } rSup { size 8{2} } =μ rSub { size 8{0} } $${ {l rSub { size 8{m} } A rSub { size 8{m} } } over { ital "gA" rSub { size 8{g} } } }$$ $$- H rSub { size 8{m} } B rSub { size 8{m} }$$ =μ rSub { size 8{0} } $${ { ital "Vol" rSub { size 8{ ital "mag"} } } over { ital "Vol" rSub { size 8{ ital "airgap"} } } }$$ $$- H rSub { size 8{m} } B rSub { size 8{m} }$$ } {} (1.57)

Volmag=VolairgapBg2μ0(HmBm)Volmag=VolairgapBg2μ0(HmBm) size 12{ ital "Vol" rSub { size 8{ ital "mag"} } = { { ital "Vol" rSub { size 8{ ital "airgap"} } B rSub { size 8{g} } rSup { size 8{2} } } over {μ rSub { size 8{0} } $$- H rSub { size 8{m} } B rSub { size 8{m} }$$ } } } {} (1.58)

• Equation (1.58) indicates that to achieve a desired flux density in the air gap the required volume of the magnet can be minimized by operating the magnet at the point of maximum energy product.
• A curve of constant B-H product is a hyperbola.
• In Fig.1.13, the maximum energy product for Alnico 5 is 40 kJ/m3kJ/m3 size 12{ ital "kJ"/m rSup { size 8{3} } } {}, occurring at the point B =1.0 T and H=40 kA/m .

Figure 1.14Magnetization curves for common permanent-magnet materials.

Figure 1.14 shows the magnetization characteristics for a few common permanent magnet materials. Alnico 5 is a widely used alloy of iron, nickel, aluminum, and cobalt, originally discovered in 1931. It has a relatively large residual flux density. Alnico 8 has a lower residual flux density and a higher coercivity than Alnico 5. Hence, it isless subject to demagnetization than Alnico 5. Disadvantages of the Alnico materialsare their relatively low coercivity and their mechanical brittleness.

Ceramic permanent magnet materials (also known as ferrite magnets) are made from iron-oxide and barium- or strontium-carbonate powders and have lower residual flux densities than Alnico materials but significantly higher coercivities. As a result, they are much less prone to demagnetization. One such material, Ceramic 7, is shown in Fig.1.14, where its magnetization characteristic is almost a straight line. Ceramic magnets have good mechanical characteristics and are inexpensive to manufacture; as a result, they are the widely used in many permanent magnet applications.

Samarium-cobalt represents a significant advance in permanent magnet technology which began in the 1960s with the discovery of rare earth permanent magnet materials. From Fig.1.14 it can be seen to have a high residual flux density such as is found with the Alnico materials, while at the same time having a much higher coercivity and maximum energy product. The newest of the rare earth magnetic materials is the neodymium-iron-boron material. It features even larger residual flux density, coercivity, and maximum energy product than does samarium-cobalt.

Figure 1.15 Magnetic circuit including both a permanent magnet and an excitation winding.

Consider the magnetic circuit of Fig.1.15. This includes a section of hard magnetic material in a core of highly permeable soft magnetic material as well as an N-turn excitation winding. With reference to Fig.1.16, we assume that the hard magnetic material is initially unmagnetized (corresponding to point a of the figure) and consider what happens as current is applied to the excitation winding. Because the core is assumed to be of infinite permeability, the horizontal axis of Fig.1.16 can be considered to be both a measure of the applied current i=Hlm/Ni=Hlm/N size 12{i= ital "Hl" rSub { size 8{m} } /N} {} as well as a measure of H in the magnetic material.

Figure 1.16 Portion of a B-H characteristic showing a minor loop and a recoil line.

As the current i is increased to its maximum value, the B-H trajectory rises from point a in Fig.1.16 toward its maximum value at point b. To fully magnetize the material, we assume that the current has been increased to a value/max sufficiently large that the material has been driven well into saturation at point b. When the current is then decreased to zero, the B-H characteristic will begin to form a hysteresis loop, arriving at point c at zero current. At point c, notice that H in the material is zero but B is at its remanent value Br.

As the current then goes negative, the B-H characteristic continues to trace out a hysteresis loop. In Fig. 1.16, this is seen as the trajectory between points c and d. If the current is then maintained at the value i(d)i(d) size 12{ - i rSup { size 8{ $$d$$ } } } {}, the operating point of the magnet will be that of point d. Note that, this same operating point would be reached if the material were to start at point c and, with the excitation held at zero, an air gap of length g=lm(Ag/Am)(μoH(d)/B(d))g=lm(Ag/Am)(μoH(d)/B(d)) size 12{g=l rSub { size 8{m} } $$A rSub { size 8{g} } /A rSub { size 8{m} }$$ $$- μ rSub { size 8{o} } H rSup { size 8{ \( d$$ } } /B rSup { size 8{ $$d$$ } } \) } {} were then inserted in the core. Should the current then be made more negative, the trajectory would continue tracing out the hysteresis loop toward point e. However, if instead the current is returned to zero, the trajectory does not in general retrace the hysteresis loop toward point c. Rather it begins to trace out a minor hysteresis loop, reaching point f when the current reaches zero. If the current is then varied between zero and i(d)i(d) size 12{ - i rSup { size 8{ $$d$$ } } } {}, the B-H characteristic will trace out the minor loop as shown.

As can be seen from Fig. 1.16, the B-H trajectory between points d and f can be represented by a straight line, known as the recoil line. The slope of this line is called the recoil permeability μRμR size 12{μ rSub { size 8{R} } } {}. We see that once this material has been demagnetized to point d, the effective remanent magnetization of the magnetic material is that of point f which is less than the remanent magnetization BrBr size 12{B rSub { size 8{r} } } {} which would be expected based on the hysteresis loop. Note that should the demagnetization be increased past point d, for example, to point e of Fig.1.16, a new minor loop will be created, with a new recoil line and recoil permeability.

The demagnetization effects of negative excitation which have just been discussed are equivalent to those of an air gap in the magnetic circuit. For example, clearly the magnetic circuit of Fig.1.15 could be used as a system to magnetize hard magnetic materials. The process would simply require that a large excitation be applied to the winding and then reduced to zero, leaving the material at a remanent magnetization BrBr size 12{B rSub { size 8{r} } } {} (point c in Fig.1.16).

Following this magnetization process, if the material were removed from the core, this would be equivalent to opening a large air gap in the magnetic circuit, demagnetizing the material. At this point, the magnet has been effectively weakened, since if it were again inserted in the magnetic core, it would follow a recoil line and return to a remanent magnetization somewhat less than BrBr size 12{B rSub { size 8{r} } } {}. As a result, hard magnetic materials, such as the Alnico materials of Fig.1.14, often do not operate stably in situations with varying mmf and geometry, and there is often the risk that improper operation can significantly demagnetize them. A significant advantage of materials such as Ceramic 7, samarium-cobalt and neodymium-iron-boron is that, because of their "straight-line" characteristic in the second quadrant (with slope close to μoμo size 12{μ rSub { size 8{o} } } {}), their recoil lines closely match their magnetization characteristic. As a result, demagnetization effects are significantly reduced in these materials and often can be ignored.

At the expense of a reduction in value of the remanent magnetization, hard magnetic materials can be stabilized to operate over a specified region.

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