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  • 7/29/2019 Inductor Losses

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    Po w er El ec t ron i c s Te c hn ol ogy Ap r i l 2 0 05 w w w.p o w e re l e c t ron i c s . c o m14

    switch-m ode p ower supply incur s loss

    in many areas of its circuitry, includ-

    ing the MO SFETs, inpu t and o utp ut ca-

    pacitors, quiescent controller current

    and indu ctor. The p ower dissipated in

    the inductor arises from two separate sources: the losses

    associated with the ind uctor core and t hose associated with

    the inductor windings. Though determining these losses

    with p recision can r equire comp lex measurem ents, an easier

    alternative exists. Inductor losses may be estimated using

    readily available data from core and indu ctor supp liers along

    with the relevant power supply application parameters.

    Inductor BasicsAn inductor consists of wire wound around a core of

    ferrite material that includes an air gap. A subset within

    the broad inductor category, power inductors operate as

    energy-storage devices. They store energy in a magnetic

    field during the power supplys switching-cycle on time

    and deliver th at energy to th e load du ring th e off time. To

    un derstand p ower loss in indu ctors, you m ust first un der-

    stand t he basic par ameters associated with ind uctor s. These

    include m agnetom otive force F(t), m agnetic-field strength

    Inductors dissipate power in the core and in thewindings. Although exact calculations of theselosses can be complex and difficult, they can bereadily estimated using data sheet parametersavailable from m agnetic comp onent supp liers.

    AH(t), magnetic flux (t), magnetic-field density B(t),

    permeability , and reluctance .

    To avoid the complicated physics of electromagnetic

    fields, we offer only a brief treatment of these parameters.

    The magnetic field strength generated by an inductor is

    measured in amperes multiplied by turns per meter. The

    magnetic field is created when current flows in the turns

    of wire that wrap around the magnetic core. For switch-

    mode power inductors, we can approximate the magnetic

    field by assum ing it is com pletely contain ed within the core.

    Magnetic-field density, measured in teslas, is equal to

    the m agnetic-field strength, H (t), m ultiplied by th e mag-

    netic-core permeability, :

    Magnetic flux, which is measured in webers, equals the

    magnetic-field density, B(t), multiplied by the cross-sec-

    tional area of the core, AC:

    Permeability, measured in henrys/m, expresses the ca-

    pability of a specific m aterial to allow the flow of m agnetic

    flux m ore easily. Thu s, higher perm eability enables a m ate-

    By Travis Eichhorn ,Application s Engi neer,Maxim Int egrated Products, Sunn yvale, Calif.

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    Po w er El ec t ron i c s Te c hn ol ogy Ap r i l 2 0 05 w w w.p o w e re l e c t ron i c s . c o m16

    r ia l to pass more

    magnet ic f lux .

    Permeabil i ty is a

    product:

    in which 0 isthe p ermeability of

    free space (0

    = 4

    10-7 H/m) and R

    is the materials

    relative permeabil-

    ity (a dimensionless

    quan tity). For exam ple, R

    for iron is approximately 5000

    an d R

    for airthe other extremeis 1. The core of a

    power indu ctor contains an air gap and ferrite material, so

    its effective is som ewhere between th at of ferrite and air.

    Magnetom otive force, F(t), is approximated in o ur case

    as the magnetic-field strength, H(t), multiplied by theeffective length of t he core, l

    E:

    F t = H t lE( ) ( )

    where the un its for F(t) are amp eres m ultiplied by turn s.

    Effective length is the length of the path followed by the

    m agnetic flux aroun d th e core. In a m agnetic circuit, F(t)

    can be regarded as the generator of m agnetic flux (Fig. 1).

    Finally, reluctance, which is measured in amperes multi-

    plied by turn s/weber, is the resistance of a material to m ag-

    netic fields. Reluctance is also the ratio of magnetomotive

    force, F(t), to magnetic flux, (t), and therefore depends

    on the physical construction of the core. Substitution ofthe above equations for F(t) an d (t) yields the following

    equation for reluctance:

    Indu ctors operate accordin g to th e laws of Am pere and

    Faraday. Amperes Law relates current in the windings

    or turns of wireto the magnetic field in the core of the

    indu ctor. As an app roximation, one assumes the m agnetic

    field in t he indu ctors core is uniform th rou ghout the core

    length ( lE

    ). That assum ption lets us write Amperes Law as:

    H t l = n i tE( ) ( )

    where n is the number of wire turns around the in-

    ductor core and i(t) is the inductor current.

    Faradays Law relates the voltage applied acro ss the in -

    du ctor to the m agnetic flux contained within th e core:

    where (t) is the m agnetic flux and n is the num ber

    of wire turns around the core. The functional diagram of

    Fig. 1 shows a power inductor an d its equivalent magneticcircuit. As shown , the air gap p laces a high-r eluctance ele-

    ment (AIR

    ) in series with a low-reluctan ce ferrite m aterial

    (Fe

    ), thereby locating the bu lk of the magnetom otive force,

    ni(t), at a desired location that o f the air gap. The indu c-

    tor value is calculated as:

    Because ferrite materials have high permeability, they

    offer an easy path for m agnetic flux (low reluctan ce). That

    characteristic helps contain th e flux within the ind uctors

    core, which in turn enables the construction of inductors

    with high values and small size. This advantage is evident

    in th e inductan ce equation above, in which a core m aterial

    with high value allows for a smaller cross-sectional ar ea.

    Inductor OperationThe power inductor in a buck or boost converter oper-

    ates as follows. Turning on the primary switch applies a

    source voltage VIN

    across the inductor, causing th e current

    to increase as:

    di t

    dt=

    V

    L

    IN( )

    This changing current, di(t)/d t, induces a changing

    magnetic field in the core material according to Amperes

    Law:

    dH t

    dt

    =n

    l

    di t

    dtE

    ( )

    ( )

    In turn, magnetic flux through the inductors core in-

    creases as:

    and that increase can be rewritten in terms of mag-

    netic-field density:

    dB t

    dt=

    n

    A

    di t

    dt

    ( )

    ( )

    The primary switch opens during the off time and re-

    moves VIN

    , causing the magnetic field to decrease. In re-

    INDUCTOR POWER LOSS

    Fig. 1. This m agn etic circuit (a) is represented by th e equi valent circuit m od el (b).

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    INDUCTOR POWER LOSS

    sponse, a decreasing d/dt in th e in ductor s core induces

    (accord ing to Faradays Law) a voltage -n d/dt across the

    inductor.

    A graph of B(t) as a fun ction of H (t) for

    a sinusoidal inpu t voltage prod uces the hys-

    teresis loop shown in bold lines on Fig. 2.

    B(t) is measured as H(t) is increased. The

    response of B(t) versus H(t) is non linear and

    exhibits hysteresis, hence the name hyster-

    esis loop. Hysteresis is one of t he core-m ate-rial characteristics that causes power loss in

    the inductor core.

    Power Loss in the Inductor CoreEnergy loss due to the changing m agnetic

    energy in the core during a switching cycle

    equals the difference between magnetic en-

    ergy put in to the core dur ing the on time and

    the m agnetic energy extracted from the core

    during the off time. Total energy (ET) into

    the indu ctor over one switching period is:

    Using Amperes Law:

    i t = H tl

    n

    E( ) ( )

    and Faradays Law:

    v t = n Ad B t

    d t( )

    ( )

    the equation for ET can be rewritten as:

    E = A l H dBT E

    Thu s, the total energy put in to th e core over on e switch-

    ing period is the area of the shad ed region within t he B-H

    loop ofFig. 2 multiplied by the volume of the core. The

    m agnetic field decreases as indu ctor curren t r amp s down,

    tracing a different path (following the direction of th e ar-

    rows in Fig. 2) for magnetic flux density. Most of the en-

    ergy goes to the load, but the difference between stored

    energy and delivered energy equals the energy lost. En-

    ergy lost in th e core is the area traced ou t by the B-H loop

    multiplied by the cores volume, and the power lost is this

    energy (ET) m ultiplied by the switching frequency.

    Hysteresis loss varies as a function ofBn, where (for

    m ost ferrites) n lies in the r ange 2.5 to 3. This expression

    applies on the conditions that the core is not driven into

    saturation, and the switching frequency lies in th e intended

    operating r ange. The shaded ar ea in Fig. 2, which occupies

    the first qu adran t of the B-H loop, represents the op erat-

    ing region for positive flux-density excursions, because

    typical buck and boost converters operate with positive

    inductor currents.

    The second type of core loss is due to eddy currents,

    which are indu ced in the core m aterial by a time-varying

    flux d/d t. According to Lenzs Law, a chan gin g flux in -

    Fig. 2. A plot of m agnetic field density B(t) versus mag netic

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