# inductor losses

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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.

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 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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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 m18

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