inductor losses

7/29/2019 Inductor Losses

1/7

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.


2/7


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).


3/7


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 field strength H(t) reveals the

m ajor an d m inor h ysteresis loops associated w ith an in ductor core.

CIRCLE 213 on Reader Service Card or freeproductinfo.net/pet

From 1 Wattup to 300 Watts

Transformers

Inductors

Filters

EMI/RFI CMCs

For UL/CSA recognized magneticcomponents...for SMPS applica-

tions...designed for use

with leading semiconductors...for

application notes and reference cir-

cuits...call Premier, the

"Innovators in Magnetics Technology".

www.premiermag.com/switcher

20381 BarentsSea CircleLake Forest, CA 92630 Tel. (949) 452-0511

Minor hysteresis loop


4/7


INDUCTOR POWER LOSS

Fig. 3.AC core loss for a pa rt icular ferrit e mat erial is plo tt ed as afunctio n o f flux density at different frequencies. (Data courtesy of

Span g an d Co.)

duces a current that itself induces a flux in opp osition to

the initial flux. This eddy current flows in the conductive

core material and p rodu ces an I2R, or V2/R, p ower loss.

That effect also can be seen via Faradays Law. If youimagine the core as a lumped resistive element with resis-

tance RC, then the voltage v

I(t) in du ced across R

Caccord-

ing to Faradays Law is:

where AC

is the cross-sectional area of the core. The

power loss in th e core due to edd y cur rents is

P =v t

RE

I

2

C

( )

This power loss is prop ortion al to the square of th e rate

of change of flux in the core. Since the rate of change of

flux is directly proportional to the applied voltage, the

power loss due to eddy curren ts increases as the square of

the applied inductor voltage and directly with its pulse

width. Thus:

P

V

R

t

TE

L

2

C

APPLIED

P

where VL

is the voltage applied to the inductor, tAPPLIED

is the on or off time, and TP

is the switching period. Be-

cause the core material has high resistance, losses due to

eddy current s in the core are usually mu ch less than t hose

du e to h ysteresis. The dat a given for core losses usually in-

clud es the effects of both hysteresis and core eddy cur rent s.

Core-loss measurements are difficult because they re-

quire complicated setups for measuring flux density and

because they involve the estimation of hysteresis-loop

areas. Many inductor manufacturers do not supply this

data, but curves are available from ferrite manufacturers

to h elp you approximate t he core loss in an indu ctor. Such

curves ind icate power loss in W /kg or W /cm 3 as a fun ction

of p eak-to-peak flux d ensity, B(t), an d frequency (f).

The m agnetics division o f Spang an d Co. in Pittsbur gh

supplies ferrite m aterial for inductor m anu facturers. From

the website www.mag-inc.com, you can o btain material data

sheets that include curves for core loss versus flux density

at various frequencies. If you know the particular ferrite

m aterial and th e volum e of the indu ctors core, these curves

enable you to m ake a good estimate of core loss.

Such cur ves for a given ferrit e material (Fig. 3) are takenwith a sinu soidal applied voltage using bipolar flux swings.

When estimating the core loss for dc-dc converters that

operate with u nipolar flux swings and rectangular applied

voltages, which con sist of h igher-frequency harm onics, you

can approximate th e loss using the fun dam ental frequency

and o ne-h alf the peak-to-p eak flux density:

The core volum e can usually be estimated with a rou gh

measurement.A few inductor m anu facturers do offer core-loss graph s

or equation s that enable mor e accurate estim ations of core

power loss. For example, Pulse Engineering in San Diego

provides inductor core-loss equations in some of its in-

ductor data sheets (see www.pulseeng.com). See SMT

power inductors P1172/P1173 for examples. These data

sheets include an equation using constants (K-factors) th at

enable the calculation of core loss as a fun ction o f frequen cy

and p eak-to-peak ripple in the indu ctor current.

On the other hand, Coil t ronics, headquartered in

Boynton Beach, Fla., presents core loss for many of its in-

ductors in graph form (see FLAT-PAC 3 series power in-ductors for example at www.coiltronics.com). Fig. 4 shows


5/7

w w w. p o w e re l e c t ro n i c s . c o m Pow er El e c t ron i c s Tec hn ol o gy Ap r i l 20 0521

the curve for core power loss versus flux

density and frequency from a Coiltronics

Flat-Pac 3 data sheet.

Power Loss in Inductor WindingsThe preceding discussion presented

losses in the inductor core, but losses alsooccur in th e inductor windings. Power loss

in the windings at dc is due to the wind-

ings dc resistance and the RMS current

through the inductor (IRM S

2z

RDC

). Resis-

tance (R) is defined as:

where is the resistivity of the winding material. This

material is usually copper, for which =1.724 z10 -8(1+

.0042 z(TC-20C) )m ). Physically sm aller ind ucto rs typi-

cally use smaller wire, and thus exhibit a higher dc resis-tance du e to the sm aller cross-sectional area of th e wire.

Larger-value ind uctors have mor e turn s of wire, and there-

fore also have higher resistance due to the longer length.

Windin g losses at dc are d ue to the dc resistance (RDC

)

of the windings and are given in the inductor data sheet.

With in creasing frequency, the wind ing resistance increases

due t o a p henom enon called skin effect, caused by a chang-

ing i(t) within the conductor. The changing current in-

duces a changing flux (d/dt) perpen dicu lar to th e cu r-

rent that indu ced it.

Accord ing to Lenzs Law, the chan ging flux induces eddycurrents that induce a flux themselves, in opposition to

the initial changing flux. These eddy cur rents are of a p o-

larity opposite that of th e initial current . The indu ced flux

is strongest at the conductors center and weakest at the

surface, causing the cur rent density at th e center to decline

from its dc value with increasing frequency. As a result,

current gets pushed to the surface of the conductor, pro-

INDUCTOR POWER LOSS

Fig. 4. AC core loss for a pa rticu lar ind ucto r is plo tt ed versus flux density at different

frequencies. (Dat a cou rtesy of Coilt ron ics.)


Kooler Inductors

P.O. Box 11422 Pittsburgh, PA 15238-0422Phone 412.696.1333 Fax 412.696.0333

1-800-245-3984email: [email protected]

www.mag-inc.com

Inductors made from Magnetics Kool M E cores

run cooler than those made with gapped ferritecores. Eddy currents, caused by the fringing fluxacross the discrete air gaps of a gapped ferrite, canlead to excessive heat due to heavy copper losses.The distributed air gaps inherent in Kool M canprovide a much cooler inductor.

Kool M E cores are available in many industrystandard sizes. Magnetics now offers cores in14 sizes (from 12 mm to 80 mm) and fourpermeabilities (26, 40, 60, and 90).New sizes are being added. Standardbobbins are also available.

If you are using gapped ferrite E cores for inductorapplications, see what Kool M E cores can do foryou. You may even be able to reduce core size inaddition to having a cooler unit. Productionquantities are now in stock. For more information,contact Magnetics.

New Sizes Available!New Sizes Available!


6/7


ducing a lower current density at the center and a higher

cur rent den sity at the surface. Resistance increases because

the res i s t iv i ty of copper remains cons tan t and the

conductors effective current carrying area decreases.

The wind ings ac resistance is foun d by determ ining the

depth, known as penetration depth, to which current ex-

ists in the cond uctor at a particular frequency. Current d en-sity at that p oint falls to 1/e times the current density at the

surface, or at dc. This depth (DPEN

) can be calculated as:

where is the resistivity of the conductor (usually cop-

per) and is the conductors permeability ( = 0

z

R,

where R

= 1 for copp er). This calculation is accurate when

the conductor is a flat surface or when the radius of the

conductor is much larger th an th e penetration depth. Note

that ac resistance (RAC) acts as a power loss only to the accurrent , which for buck and boost converters is the indu c-

tor-curr ent ripple. DC curr ent in the indu ctor only creates

power loss in RDC

.

You find RAC

by calculatin g the effective condu cting area

of the copper wire at a given frequency. For conductors

that h ave radii larger then the skin depth at the given op er-

ating frequency, the effective conducting area is the sur-

face area of a condu cting ring with th ickness equal to th e

skin depth. Because resistivity remains constant, the ratio

of RAC

to RDC

is simply the ratio of the two areas:

Furthermore, RAC

/RDC

m ultiplied by RDC

is the effective

resistance at a given frequency for a straight wire in free

space.

Eddy currents in the indu ctor windings are also induced

by other nearby condu ctors, a phenom enon known as the

proxim ity effect. For ind ucto rs with m any overlappin g wire

tur ns and adjacent wires, the increased eddy curr ents cause

a resistance considerably higher than that from the skin

effect alone. The proximity effect becomes complicated,

however, due to the various configurations an d d istances

with which cond uctors can b e placed relative to each oth er.

Because such calculations are beyond the scope of this ar-ticle, the reader should refer to the references provided.

A simple circuit illustrates losses in the inductor (Fig.

4). RC

repr esents the core losses, and RAC

and RDC

represent

the ac- and dc-depen dent win ding losses. RC

is determ ined

by core loss calculations or estimates, while RDC

is the dc

winding resistance and RAC

is the ac resistance due to skin

effect, proximity effect or both. An example of this loss

model can be developed using the MAX5073 switching

power supply. We operate the MAX5073 as a buck con-

verter with VIN

= 12 V, VOU T

= 5 V, fSW

=1 MH z, and IOU T

= 2

A. A 4.7-H inductor (FP3-4R7 from Coiltronics) p roduces

an indu ctor current ripple (I(t)) of 621 mA.A graph of core loss versus flux density and frequency

is shown in Fig. 4. Peak-to-peak flux density (B) is what

matters. It traces out a sm all hysteresis loop within the larger

hysteresis loop (see the inner loop in Fig. 2). You can find

B using the equation given in th e inductor data sheet:

where K is a constan t given in t he dat a sheet (K = 105 in

our case), and L is the ind uctance in m icrohenries. In th is

example:

As an alternative, you can estimate B(t) using the in-

ductor volt-second prod uct divided by the num ber of turns

and the core area within the turns:

Goin g to the FP3 dat a sheet, core loss at 613 gauss and

fSW

= 1 MH z is approximately 470 mW. RC

in Fig. 5is the

equivalent parallel resistance that accounts for power loss

in th e indu ctor core. That resistance is calculated from the

RMS voltage across the inductor and the core power loss:

V = V D = 12 V 0.417 =7.75 VRMS IN RMS

INDUCTOR POWER LOSS



7/7


INDUCTOR POWER LOSS

RC

is then 60.1 V2/0.470 W= 128 , where VIN D is

the RMS value of a rectangular wave with duty cycle D

and am plitude VIN .RDC

from t he data sheet is 40 m, assumin g a zero tem -

peratu re rise for the indu ctor, which would ot herwise in-

crease the value of RDC

. The penetration depth for a 1-MH z

switching frequency, using only the fundamental of the

triangular current ripple at TA

= +20C, is 0.065 mm. A

rough m easurem ent o f the cond uctors radius gives 0.165

m m , which results in an RAC

value of:

This resistance only dissipates power due to the RMS

ac current. The RMS value of indu ctor curr ent r ipple is:

Thus, the total estimated losses are:

PETech

References1. Erickson, Robert W., and D ragan M aksimovic. Fundamen-

tals of Power Electron ics, 200 1. C hap ter s 13 an d 14, pp. 491-

562.

2. Kassakian,Joh n G., M ar tin F. Sch lech t , an d George C.

Verghese. Principles of Power Electronics, 19 91 . Chap ter

20, pp. 565-601.

3. D ix on , L loyd H . Magnetics Design for Switching Power

Supplies. Sections 1- 5.

Fig. 5. An equivalent loss m odel for a p ower indu ctor includes terms

representing the a c- and dc-dependent win ding losses (RAC

and RDC

)and th e core losses (R

C).

&UJI3EMICONDUCTOR)NC(IGHLANDER7AY

#ARROLLTON4853!

0HONE

&AX

6ISITOURWEBSITEAT

WWWFUJISEMICOMFORMOREDETAILS


inductor losses

Documents