Magnetic Circuits

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Transcript of Magnetic Circuits

  • Magnetic Circuits

    Outline Amperes Law Revisited Review of Last Time: Magnetic Materials Magnetic Circuits Examples

    1

  • Electric Fields Magnetic Fields

    dA =

    oE dV B

    S V

    dA = 0

    S

    = Qenclosed

    GAUSS GAUSS

    FARADAY AMPERE

    dE dl = B dA H dl

    C dt

    (S

    )

    C

    =

    dJ dA+

    E dA

    dtd S S

    emf = v =dt

    2

  • Amperes Law Revisited

    In the case of the magnetic field we can see that our old Amperes law can 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 current is I. From the symmetry it is easy to see that an application of Amperes law will produce B fields which go in circles around the wire and whose magnitude is B(r) = oI/(2r). But there is no charge flow in the gap across the capacitor plates and according to Amperes law the B field in the plane parallel to the capacitor plates and going through the capacitor gap should be zero! This seems unphysical.

    Side View

    B

    BB

    B = 0??

    I

    II I

    3

  • Amperes Law Revisited (cont.)

    If instead we drew the Amperian surface as sketched below, we would have concluded that B in non-zero !

    BB Side View

    I

    I I I

    BMaxwell resolved this problem by adding a term to the Amperes Law. In equivalence to Faradays Law, the changing electric field can generate the magnetic field:

    d l

    C

    COMPLETE H d =

    AMPERES LAW S

    J dA+dt

    S

    E d A4

  • Faradays Law and Motional emf What is the emf over the resistor ?

    demf = mag

    In a short time t the bar moves a distance x = v*t, and the flux increases by mag = B (L v*t)

    There is an increase in flux through the circuit as the bar of length L moves to the right

    (orthogonal to magnetic field H) at velocity, v.

    from Chabay and Sherwood, Ch 22

    dt

    emf =magt

    = BLv

    L

    vt

    vB

    5

  • Faradays Law for a Coil

    from Chabay and Sherwood, Ch 22

    Will the current run CLOCKWISE or ANTICLOCKWISE ?

    Rotating a bar of magnet (or the coil) produces a time-varying magnetic field

    inside the coil

    Moving a magnet towards a coil produces a time-varying magnetic field inside the coil

    6

    The induced emf in a coil of N turns is equal to

    N times the rate of change of the magnetic flux on one loop of the coil.

  • Complex Magnetic Systems

    DC Brushless Stepper Motor Reluctance Motor Induction Motor

    We need better (more powerful) tools

    Magnetic Circuits: Reduce Maxwell to (scalar) circuit problem

    Energy Method: Look at change in stored energy to calculate force

    H

    C

    dl = Ienclosed

    BS

    dA = 0 f = q(v B

    )

    7

  • Magnetic Flux [Wb] (Webers) Magnetic Flux Density B [Wb/m2] = T (Tesla

    Magnetic Field Intensity H [Amp-turn/m]

    due to macroscopic & microscopic

    s)

    due to macroscopic currents

    Faradays Law

    ( H

    )B = o +M

    )= o

    (H + mH = orH

    demf = mag

    dt

    emf =

    ENC dl and mag =

    B ndA

    8

  • Example: Magnetic Write Head

    Bit density is limited by how well the field can be localized in write head

    Horizontal Magnetized Bits

    Ring Inductive Write Head

    Shield

    GMR Read Head

    Recording Medium

    9

  • Review: Ferromagnetic Materials

    hysteresis

    0

    B

    B,J

    H

    C

    r H : coercive magnetic field strength S B : remanence flux density

    B : saturation flux density

    H 0

    Initial Magnetization

    Curve

    Slope = i Behavior of an initially unmagnetized

    material. Domain configuration during several stages of magnetization.

    BsBr

    HC HC

    Br Bs

    H > 0

    10

  • Thin Film Write Head

    How do we apply Amperes Law to this geometry (low symmetry) ?

    Recording Current

    Magnetic Head Coil

    Magnetic Head Core

    Recording Magnetic Field

    H

    C

    dl =

    JS

    dA

    11

  • Electrical Circuit Analogy

    Charge is conserved Flux is conserved

    Electrical

    EQUIVALENT

    CIRCUITS

    Magnetic

    i

    ii

    + V

    S

    B d A = 0

    V

    i

    R

    12

  • Electrical Circuit Analogy

    Electromotive force (charge push)= Magneto-motive force (flux push)=

    Electrical

    Magnetic

    EQUIVALENT CIRCUITS

    V

    i

    i

    C

    H dl = Ienclosedv =

    E dl

    + V

    ii

    R

    13

  • Electrical Circuit Analogy

    Material properties and geometry determine flow push relationship

    B = OHMs LAW oJ = E rHDCB

    Recovering macroscopic variables:

    I =

    VJ d =

    A E dA = H

    lA

    l lV = I = I = IR

    A A Ni =

    V

    i

    R

    14

  • Reluctance of Magnetic Bar

    Magnetic OHMs LAW

    A

    l Ni =

    l =A

    15

  • Flux Density in a Toroidal Core

    N turn coil

    (of an N-turn coil) i

    Core centerline

    H

    NiR B =2R

    2R

    Ni = 2RB = lB

    lBmmf = Ni =

    =

    l

    A

    mmf =

    16

  • Electrical Circuit Analogy

    Electrical Magnetic

    Voltage v Magnetomotive Force = NiCurrent i Magnetic Flux Resistance R Reluctance

    1/Conductivity Permeability Current Density J Magnetic Flux Density BElectric Field E Magnetic Field Intensity H

    17

  • Toroid with Air Gap

    Why 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 ?

    A = cross-section area

    Electric current

    Magnetic flux

    18

  • Fields from a Toroid

    A = cross-section area

    Electric current

    Magnetic flux

    H

    C

    dl =

    JS

    dA

    = Ienclosed

    NiH =

    B = o

    ( H +M

    )Ni

    B = 2R 2R

    A = BA = Ni

    2R

    19

  • Scaling Magnetic Flux

    &

    Same answer as Amperes Law (slide 9)

    A = cross-section area

    Electric current

    Ni = Magnetic flux

    l = 2R=A

    A

    A = BA = Ni

    2R

    20

  • i N=500

    2cm 8cm

    2cm

    0.5cm

    Core Thickness = 3cm

    Magnetic Circuit for Write Head

    A = cross-section area

    + -

    core gap

    l =A

    21

  • Parallel Magnetic Circuits

    A = cross-section area

    10cm 10cm

    0.5cm 1cm i

    Gap a Gap b

    22

  • A Magnetic Circuit with Reluctances in Series and Parallel

    Shell Type Transformer

    + + N1 turns vv 2 N2 turns 1 - -

    Depth A l2 l1 l=l1+l2

    N1i1

    N2i2

    a b

    c

    c

    Magnetic Circuit

    1

    3 + 2

    - +

    -

    l 11 =A

    2 = l2A

    23

  • Faraday Law and Magnetic Circuits

    Flux linkage

    Step 2: Estimate voltage v2 due to time-varying flux

    + + sinusoidal v1 v2

    - -

    A = cross-section area

    Step 1: Estimate voltage v1 due to time-varying flux

    N1 N2

    + -

    i1 i2

    Load

    Primary Secondary

    Laminated Iron Core

    d = N v =

    dt

    v2v1

    =

    24

  • Complex Magnetic Systems

    DC Brushless Stepper Motor Reluctance Motor Induction Motor

    Powerful tools

    Magnetic Circuits: Reduce Maxwell to (scalar) circuit problem

    Makes it easy to calculate B, H,

    Energy Method: Look at change in stored energy to calculate force

    25

  • Stored Energy in Inductors

    In the absence of mechanical displacement

    For a linear inductor:

    Stored energy

    dWS =

    Pelecdt =

    ivdt =

    i =dt

    i () d

    i () =

    L

    WS =

    2d =

    0 L 2L

    26

  • Relating Stored Energy to Force

    Lets use chain rule

    This looks familiar

    Comparing similar terms suggests Wfr = S

    r

    WS (, r) WS d WS dr= +

    dt dt r dt

    dWS dr= i

    dt v fr

    dtdi dr

    = iLdt

    frdt

    27

  • Energy Balance

    l dWSdt

    heat

    electrica mechanical

    For magnetostatic system, d=0 no electrical power flow

    dWS dr=

    dtfr

    dt

    i v drfrdt

    dWS (, r) WS d WS dr= + neglect heat

    dt dt r dt

    28

  • Linear Machines: Solenoid Actuator

    Coil attached to cone

    If we can find the stored energy, we can immediately compute the force

    lets take all the things we know to put this together

    W S 1 2

    r| WS (, r) =

    2 Lfr =

    x

    l

    29

  • KEY TAKEAWAYS COMPLETE AMPERES LAW

    RELUCTANCE

    Electrical Magnetic

    Voltage Magnetomotive Force Current Magnetic Flux Resistance Reluctance Conductivity Permeability Current Density Magnetic Flux Density Electric Field Magnetic Field Intensity

    R

    i

    1/

    J

    E

    v = Ni

    B

    H

    C

    H dl =S

    J d A+ ddt

    S

    E d A

    = lA

    30

  • MIT OpenCourseWare http://ocw.mit.edu

    6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011

    For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.