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Magnetic CircuitsOutline Ampere’s Law Revisited Review of Last Time: Magnetic Materials Magnetic Circuits Examples1
Electric Fields · dA o ESMagnetic Fields · dA 0BρdVVS QenclosedGAUSSGAUSSFARADAY d E · dl dtCAMPERE · dA B Sdλemf v dt2 · dl HC d · dA J · dA Edt SS
Ampere’s Law RevisitedIn the case of the magnetic field we can see that ‘our old’ Ampere’s law cannot be the whole story. Here is an example in which current does not givesrise to the magnetic field: 0?B BSideViewIIII B BConsider the case of charging up a capacitor C which is connected to very long wires.The charging current is I. From the symmetry it is easy to see that an application ofAmpere’s law will produce B fields which go in circles around the wire and whosemagnitude is B(r) μoI/(2πr). But there is no charge flow in the gap across the capacitorplates and according to Ampere’s law the B field in the plane parallel to the capacitorplates and going through the capacitor gap should be zero!This seems unphysical.3
Ampere’s Law Revisited (cont.)If instead we drew the Amperian surface as sketched below,we would have concluded that B in non-zero ! B BSideViewIIII BMaxwell resolved this problem by adding a term to the Ampere’s Law.In equivalence to Faraday’s Law,the changing electric field can generate the magnetic field: · d l HC d J · dA dtS4 · dA ESCOMPLETEAMPERE’S LAW
Faraday’s Law and Motional emfWhat is the emf over the resistor ?vΔtdΦmagemf dtLv BIn a short time Δt the bar moves adistance Δx v*Δt, and the fluxincreases by ΔФmag B (L v*Δt)ΔΦmag BLvemf ΔtThere is an increase in flux through the circuitas the bar of length L moves to the right(orthogonal to magnetic field H) at velocity, v.from Chabay and Sherwood, Ch 225
from Chabay and Sherwood, Ch 22Faraday’s Law for a CoilThe induced emf in a coil of N turns is equal toN times the rate of change of the magnetic flux on one loop of the coil.Moving a magnet towards a coil produces atime-varying magnetic field inside the coilRotating a bar of magnet (or the coil)produces a time-varying magnetic fieldinside the coilWill the current runCLOCKWISE or ANTICLOCKWISE ?6
Complex Magnetic SystemsDC Brushless Reluctance MotorStepper Motor · dl IenclosedHC · dA 0BInduction Motor f q v BSWe need better (more powerful) tools Magnetic Circuits: Reduce Maxwell to (scalar) circuit problemEnergy Method:Look at change in stored energy to calculate force7
Magnetic FluxMagnetic Flux DensityMagnetic Field IntensityΦ [Wb] (Webers)B [Wb/m2] T (Teslas)H [Amp-turn/m]due to macroscopiccurrents due to macroscopic& microscopic μo μr H μo H M μo H χm H BFaraday’s Lawemf emf N C · dl EdΦ magdtand 8Φmag · n̂dA B
Example: Magnetic Write HeadRing InductiveWrite HeadShieldGMRReadHeadRecording MediumHorizontalMagnetized BitsBit density is limited by how well the field can be localized in write head9
Review: Ferromagnetic MaterialsB rB C H sBB,J0 CHHInitialMagnetizationCurveH 0 r Bhysteresis s B0HSlope μiBehavior of an initially unmagnetizedCHr : coercive magnetic field strength material.Domain configuration during several stagesB S : remanence flux densityof magnetization.B : saturation flux density10
Thin Film Write HeadRecording CurrentMagnetic Head CoilMagnetic Head CoreRecording MagneticFieldHow do we apply Ampere’s Law to this geometry (low symmetry) ? · dl HC S11 J · dA
Electrical Circuit AnalogyCharge is conserved i Flux is ‘conserved’ · dA 0BSiiVФEQUIVALENTCIRCUITSElectricalMagneticφiV R12
Electrical Circuit AnalogyElectromotive force (charge push) v Magneto-motive force (flux push) · d lE · d l IenclosedHCi iiVФEQUIVALENTCIRCUITSVElectricalMagneticiφ R13
Electrical Circuit AnalogyMaterial properties and geometry determine flow – push relationshipφiOHM’s LAW RV μ o μr H B DCJ σE BRecovering macroscopic variables: I V E · dA σ AllρlV I I IRσAA σJ · dA14 HN i Φ
Reluctance of Magnetic BarMagnetic “OHM’s LAW”N i Φ lAl μAφ15
Flux Density in a Toroidal CoreCore centerlineHμN iB 2πRR2RN turncoilμN i 2πRB lBllB Φmmf N i μμA(of an N-turn coil)immf Φ 16
Electrical Circuit AnalogyElectricalMagneticVoltage vCurrent iResistance RConductivity 1/ρCurrent Density JElectric Field EMagnetomotive Force N iMagnetic Flux φReluctance Permeability μMagnetic Flux Density BMagnetic Field Intensity H17
Toroid with Air GapMagneticfluxElectriccurrentA cross-section areaWhy is the flux confined mainly to the core ?Can the reluctance ever be infinite (magnetic insulator) ?Why does the flux not leak out further in the gap ?18
Fields from a Toroid · dl HC Magneticflux J · dASElectriccurrent IenclosedA cross-section area NiH 2πR M μo HB NiB μ2πRμAΦ BA N i2πR19
Scaling Magnetic FluxMagneticfluxN i Φ &l μA2πR μAElectriccurrentA cross-section areaμAΦ BA N i2πRSame answer as Ampere’s Law (slide 9)20
Magnetic Circuit for ‘Write Head’Core Thickness 3cm2cml μA8cm 2cm0.5cm corei gapφN 500 A cross-section areaΦ 21 -
Parallel Magnetic Circuits10cm1cmGap ai10cm0.5cmGap bA cross-section area22
A Magnetic Circuit with Reluctances in Series and Parallel“Shell Type” TransformerMagnetic Circuit 1 v1N1 turns l2-l1 v2-N2 turns φa 3N1 i1N2 i2Depth Al l1 l2l1 1 μA23φb 2 φc-φcl2 2 μA
Faraday Law and Magnetic CircuitsФi1 sinusoidal- v1-i2N1N2PrimaryLoadSecondaryLaminated Iron CoreFlux linkage v2-λ NΦStep 1: Estimate voltage v1 due to time-varying flux Step 2: Estimate voltage v2 due to time-varying flux 24A cross-section areadλv dtv2 v1
Complex Magnetic SystemsDC BrushlessReluctance MotorStepper MotorInduction MotorPowerful tools Magnetic Circuits: Reduce Maxwell to (scalar) circuit problemMakes it easy to calculate B, H, λEnergy Method:Look at change in stored energy to calculate force25
Stored Energy in InductorsIn the absence of mechanical displacement WS Pelec dt ivdt dλi dt i (λ) dλFor a linear inductor:λi (λ) LStored energy WS 26λ0λ λ2dλ L2L
Relating Stored Energy to ForceLets use chain rule WS (Φ, r) WS dΦ WS dr dt Φ dt r dtThis looks familiar drdWS i · v frdtdtdidr iL frdtdtComparing similar terms suggests WSfr r27
Energy Balancei·velectricalheatdWSdtdWS (λ, r) WS dλ WS dr dt λ dt r dtFor magnetostatic system, dλ 0 no electrical power flow dWSdr frdtdt28dr frdtmechanicalneglect heat
Linear Machines: Solenoid ActuatorxlCoil attached to coneIf we can find the stored energy, we can immediately compute the force lets take all the things we know to put this together 1 λ2WS (λ, r) 2 L WS λfr r29
KEY TAKEAWAYS · d l HCOMPLETE AMPERE’S LAWC d J · dA dtSElectricalMagneticVoltage vCurrent iResistance RConductivity 1/ρCurrent Density JElectric Field EMagnetomotive Force N iMagnetic Flux φReluctance Permeability μMagnetic Flux Density BMagnetic Field Intensity H30 · dA ESRELUCTANCEl μA
MIT OpenCourseWarehttp://ocw.mit.edu6.007 Electromagnetic Energy: From Motors to LasersSpring 2011For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
not be the whole story. Here is an example in which current does not gives rise to the magnetic field: Consider the case of charging up a capacitor C which is connected to very long wires. The charging c
