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chapter 1: Magnetic Circuits and Magnetic Materials

Module by: NGUYEN Phuc

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
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=4π.10−7μo=4π.10−7 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 2
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 3
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} } } {}
φ≈FRg=Fμ0Agg=Niμ0Aggφ≈FRg=Fμ0Agg=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 4
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=0∑nin=0 size 12{ Sum cSub { size 8{n} } {i rSub { size 8{n} } } =0} {} (1.23)
∑nφn=0∑nφn=0 size 12{ Sum cSub { size 8{n} } {φ rSub { size 8{n} } } =0} {} (1.24)
§1.2 Flux Linkage, Inductance, and Energy
  • Faraday’s Law:
∮cE.ds=−ddt∫sB.da∮cE.ds=−ddt∫sB.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=Ndϕdt=dλdtλ=Nϕe=Ndϕdt=dλdtλ=Nϕalignl { 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 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=idλdtp=ie=idλdt 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=∫λ1λ2λLdλ=12L(λ22−λ11)ΔW=∫λ1λ2idλ=∫λ1λ2λLdλ=12L(λ22−λ11) 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 6
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 7
Figure 1.7 Dc magnetization curve for M-5 grain-oriented electrical steel 0.012 in thick.
(Armco Inc.)
Figure 8
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=2πfNAcBmaxEmax=ωNφmax=2πfNAcBmax 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=1T∫0Tf2(t)dtFrms=1T∫0Tf2(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)