Modeling Litz-wire winding losses in high-frequency power inductorsPESC Record. 27th Annual IEEE Power Electronics Specialists Conference

(1)

Modeling Litz-Wire Winding Losses in

High-Frequency Power Inductors

Massimo Bartoli, Nicola Noferi, Albert0 Reatti, Member, IEEE

University of Florence, Department

of

Electronic Engineering,

Via S. Marta 3, 50139, Florence, ITALY

Fax +39-55-494569. E-mail: circuiti@speedynet.it

Marian K. Kazimierczuk, Senior Member, IEEE

Wright State University, Department

of

Electrical Engineering,

45435, Dayton,

OH.

Fax 513-873-5009. E-mail:

mkazim@valhalla.cs.wright

.edu

Abstract- T h e parasitic effects in stranded,

twisted, and Litz wire windings operating at high

frequencies are studied. T h e skin and proximity effects t h a t cause t h e winding parasitic resistance of an inductor t o increase with t h e operating frequency are considered. An expression for t h e ac resistance as a function of t h e operating frequency is given. T h e measured and calculated values of t h e inductor ac resistance and quality factor are plotted ver-

sus frequency and compared. T h e theoretical results

were in good agreement with those experimentally measured.

I. INTRODUCTION

T h e overall efficiency of power circuits such as

power converters and power amplifiers highly de- pends on t h e efficiency of inductors used in assem- bling t h e circuit. Therefore, t h e control of power losses in magnetic components is important for in- creasing t h e efficiency of t h e overall circuit. One of t h e main problems related t o t h e design of power

inductors is t h e calculation of t he winding ac resis-

tance

Rat.

Many efforts [1]-[6] have been made t o

derive expressions allowing for an accurate represen-

tation of frequency behavior of

Rat.

Stranded, twisted, and Litz wires are commonly

thought t o result in lower ac resistances than solid

wires. This is because t h e current through a solid

wire concentrates in t h e external part of the conduc-

tor a t high operating frequencies. If a multistrand

conductor is used, t h e overall cross-sectional area is

spred among several conductors with a small diame-

ter. For this reason, stranded, twisted, and Litz wire

results in a more uniform current distribution across

the wire section. Moreover, Litz wires ar e assembled

so that each single strand, in t h e longitudinal devel-

opment of th e wire, occupies all t h e positions in t he wire cross section. Therefore, not only t h e skin effect but also proximity effect is drastically reduced. However, t h e following limitations occur when using

Litz wires:

1) T h e utilization of t h e winding space inside a bob-

bin width is reduced with respect t o a solid

wire.

2) T h e dc resistance of a Litz wire is larger t h a n

th at of a solid wire with the same length and

equivalent cross-sectional area because each strand path is longer t h a n t h e average wire length.

Consequently, a Litz wire results in a lower a c re-

sistance than a solid wire only if t h e multistrand

wire windings are operated in a n appropriate fre-

quency range. This range can be predicted only if

a n expression for th e winding ac resistance of a Litz

wire as a function of frequency is available.

The purposes of this paper are (1) t o derive an

accurate analytical expression for Litz wire winding

ac resistance suitable for designing of low-loss high-

frequency power inductors, (2) t o compare t h e ac

(2)

ify t h e theoretical predictions with experimental results.

T h e significance of t h e paper is t h a t t h e derived expression for a c resistances allows for a n accurate prediction of t h e frequency behavior of a n inductor

Litzrwire windings. Both t h e expression for R,, a nd

t h e inductor model can be used t o design inductors for high frequency operation with reduced winding losses.

11. LITZ WIRE

AC

RESISTANCE

I n [l], t h e analytical expression for t h e ac resis-

tance of a solid wire winding has been derived under

t h e following assumptions:

T h e one-dimensional approach proposed in

[a]

a nd [3] has been followed.

A high magnetic permeability has been assumed

for t h e core material. Moreover, it was as-

sumed t h a t t h e winding layer wires fill t he en-

tire breadth of t h e bobbin. Under these as-

sumptions, t h e magnetic component behavior is close t o t h a t of a n ideally infinite long solenoid winding.

T h e curvature of t h e windings is neglected in calculating t h e magnetic field distribution across t h e winding layers.

T h e orthogonality principle [4] ,[5] has been ap-

plied.

T h e main advantage of this approach is t h a t it

allows for t h e calculation of t h e overall a c resistance of a solid round wire as t h e sum of a n a c resistance d u e t o t h e skin effect and a n ac resistance due to t h e proximity effect. T h e ac resistance of multiwire conductors, e.g., Litz wires, is d u e to four different parasitic effects:

1) skin effect in each single strand of the bundle.

2) skin effect on t h e overall bundle t h a t causes t h e

absence of current in t h e internal strands;

3 ) proximity effect between t h e single strands;

4) proximity effect between t h e bundles of t he dif-

ferent turns.

T h e proximity an d skin effects can be assumed to

be independent from each other, t h a t is, t h e orthog-

onality principle can be applied t o multiwire wind-

ings along with t h e other assumptions used for t h e

solid wire windings. T h e twisted wires have t h e advantage of lower proximity lmses t h a n t h e stranded ones. Actually, th e internal strands a r e wounded togheter, so t h a t each single s t r an d has a n helicoy- dal movement around t h e center of t h e bundle, with

a constant distance from t h e center itself. Since z

is t h e longitudinal axis along t h e direction of t h e

bundle an d X is t h e twisting pitch, t h e istantaneus

phase of t h e helix with respect to t h e center of t h e bundle J ( 2 ) is given by

(1)

z

19

( 2 ) = 271.

-.

x

In t he expression for t h e current density inside t h e twisted wire, a multipling factor s i n J ( z ) is con-

tained. Averaging this effect over one wavelength A,

one obtains: (2) 1

1

s i n 2 8

( z )

dz =

-

x

2 ' 0

Twisting allows a 50% reduction of t h e proximity

losses. However, there ar e no evident advantages

of reducing t h e skin effect. Improvements in this

direction can be achieved by t h e use of Litz wire.

Litz wire is a bundle of s tr an d conductors, where

t h e strands a r e twisted with others so t h a t all t h e strands alternatively occupy all t h e positions of t h e

wire cross section. W ith respect to a t h e twisted

wires, in th e Litz wires t h e s tr an d distance t o t h e center of t h e bundle is not fixed. This allows t h e

current t o flow also in t h e internal strands even at

higher frequencies a n d a more unifor current distri-

bution is achieved. In this way, a n averaging of t h e

overall skin effect is obtained a n d also at high fre-

quencies, t h e current density d u e t o t h e mere skin effect is almost t h e same in all t h e strands. Previous considerations allow us t o deduce a n analitycal ex-

pression for t h e ac resistance of a Litz wire winding

in the following way.

Taking th e one-dimensional approach

[a],

[3], a n d

using t h e orthogonality principle, [4], [SI, a n analyt-

ical expression for th e ac resistance of a coil winding

has been derived in [l] . For a solid round wire wind-

ing composed of Nl layers, t h e procedure given in [l]

yields:

4

(Nf

- 1)

(3)

where

ber2 yber'y - beiyybei'y ber2y

+

bei 2y d

y = -

S J Z

(4) 7 = porosity factor = and (6) 4P R d c = TLNlT = - N ~ T T d2

is t h e dc winding resistance,

S

=

d

m

is the

skin depth, p r is t h e relative magnetic permeabil-

ity of t h e conductor material ( p r = 1 for a copper

conductor), po = 4n10-l' H

/

mm, d is the solid

round conductor diameter in mm,

t

is the distance

between t h e centers of two adjacent conductors in mm, N is t h e number of t u r n s of t he winding, IT

(mm) is t h e average length of one turn of the wind-

ing, and p is t he conductor resistivity a t a given

operating temperature ( p = 1 7 . 2 4 ~ 1 0 - 6 Rmm for

a copper conductor a t a n operating temperature of 20 "C). Wire d a t a sheets usually give values of rL a t 20

"C

and 100 'C; the latter is typically 1.33 times t h e former.

This approach can be applied also t o Litz wire

and it allows for calculating the skin effect losses

separately from t h e proximity ones. In Fig. 1, the

cross section of a Litz wire conductor subjected t o

a n external magnetic field He is shown.

Fig. 1. Cross section of a Litz wire.

T h e analitycal expression for t h e ac resistance of

t h e Litz wire winding has been derived under the

following assumptions:

1) T h e one-dimensional approach proposed in [2]

and [3] has been followed

T h e expression derived for t h e ac resistance is in t h e rectangular coordinates and is deduced ap-

proximating a winding layer t o a foil conductor

with the magnetic field assumed t o b e parallel

t o th e conductor surface.

Assumptions 1) and 2) are guaranteed if the core material has a high magnetic permeability, and t h e winding layer wires fill t h e entire breadth of t h e bobbin. Under these assumptions, t he magnetic component behavior is close t o th at of an ideally infinite long solenoid windings. T h e curvature of t he windings is neglected

when calculating t h e magnetic field distribution across th e layers.

These assumptions are t h e same as those utilized

in [l] t o deduce expression (3):

If only the skin effect is considered and a current

I m / n s , where ns is the number of strands, is as-

sumed t o flow through each strand, t h e power losses of t he m-th layer of th e winding are given by

where Rs s t r a n d is given by [I]

R.9 s t r a n d m

where

y,

= d,/S& and d, = 2 r, is t h e single strand conductor diameter in mm, and R d c s t r a n d is t h e internal strand dc resistance.

The skin effect losses may b e approximated by t h e sum of the effects in t h e n, Litz wire internal

strands. Moreover, thanks t o th e Litz wire geometry

t h e overall skin effect in t h e bundle can b e nglected.

As a consequence, power losses in t h e m-th layer of

t h e winding can be calculated as

Lets now evaluate th e contributions of t h e two proximity effect components (internal and external) t o t he power losses. I t has been demonstrated in [5]

(4)

t h a t t h e overall proximity losses can b e calculated

as t h e sum of t h e contributions due to the internally

generated magnetic field ( H i ) and t o t h e external

one ( H e ) :

Pp m = Pp m i n t

+

p p m e x t . ₍₉₎

Under t h e assumption t h a t all t h e strands in the bundle behave in t h e same way with respect t o t h e

external field, Pp ezt is given by [l]:

where

77, = external porosity factor =

58

(12)

t 0

where

to

is t h e distance between the centers of two

adjacent Litz wire conductors.

A

good approximation of t h e overall internal prox-

imity losses is given by [l], [ 5 ] :

(1.3) 72,

'$ KZ

strand I:n

8 ~2 rg

Pp m i n t =

where

q2 = internal porosity factor =

"8

(14)

t S

ro

is t h e Litz wire radius excluding the insulating

layer a n d

t ,

is t h e distance between the centers of two adjacent strands, taking into account the effective turn-to-turn, layer-to-layer, and strand-to- strand distances and t h e non-uniformity of t h e magnetic field.

I n [l], t h e importance of t h e introduction in (3) of

t h e porosity factor, q 2 , has been underlined. Actu-

ally, it takes into account t h e effective turn-to-turn

a n d layer-to-layer distances and the absence of uni-

formity of t h e magnetic field. For these same rea-

sons, coefficients 771 a n d 772 have been introduced

also in (10) and (13), respectively. T h e introduction

of two different factors is necessary in t h e analysis of Litz wires because of t h e geometrically different operating influences of t h e external ( H e ) and internal ( H i ) magnetic field on t h e single strand of t h e bundle. The two proximity components of t h e ac

resistance, for t h e m-th layer of a Litz wire winding,

are then given by

7 s 2

%

m ext = R d c strand m- ber2"ysber'ys - beizysbei'ys ber2ys

+

bei 2 ~ s

[

-2Tns771 and

7 S

R p m i n t = R d c strand m- 2

p berzy, ber'y, - bei2ys bei'y, 21r ber2ys

+

bei 2^/s

-2Tr72

-

where t h e packing factor of a Litz wire is defined as

and do = 2 ro is t h e Litz wire diameter excluding

t h e insulating layer.

Fkom (7), (15) a n d (16) and applying t h e orthog-

onality principle as done in [l] t o obtain (3), t h e overall ac resistance for t h e m-th layer is given by

Rw-m = R d c - m 7 r ( -

I

bery, bei'y, - beiy, ber'y,

2 ns bert2ys

+

beit2yS

berzySber'ys

-

beizysbei'ys ber2y,

+

bei 2ys

As a consequence, if a n Nl-layer Litz wire winding

is considered, where all layers have t h e same number of turns and then t h e same R d c s t r a n d m , its overall

a c resistance can be calculated as follow:

-2T

[

4 (Nf - 1) i l l

berzy,ber'y,

-

bei2ySbei'ys ber2ys

+

bei2ys

(5)

where

is t h e Litz wire dc resistance, N is the number of

t ur n s of t h e winding, IT (mm) is t h e average length of one t u r n of t h e winding.

T h e expression for t h e ac resistance given in (19)

provides a n accurate prediction of t h e Litz wire ac

resistance over a wide frequency range. Moreover, it

also allows for choosing t h e Litz wire with t he lowest

resistance for any given application. T h e theoretical results a r e in good agreement with t h e experimental

tests. T h e results of this paper can be effectively

used in t h e design of high quality factor, low-loss, high-frequency power inductors.

111. COMPARISON OF LITZ AND SOLID ROUND

WIRE RESISTANCES

Two different windings with a nearly equal dc re-

sistance a re considered as follows:

1) solid round wire AWG#28, with d = 0.32 mm,

transversal conducting area A = 0.081 mm2,

an d dezt = 0.37 mm;

2) Litz wire 2 0 ~ 0 . 0 8 , with a number of strands

n, = 20, conducting diameter d, =

0.08 nim, a total transversal conducting area

A = 0.1005 mm2, a n d dezt = 0.554 mm.

Both wires have been wound on t h e same E 2 5

Micrometals core with a number of turns N = 114

in order t o provide t h e same inductance. T h e solid

round wire results in a 3-layer winding and Litz in

more t h a n a 4-layer winding.

In Fig. 2, two winding ac resistances calculated

from (19) a r e plotted as functions of t h e inductor

operating frequency. These plots show t h a t t he a c

resistance of a Litz wire winding is lower tha n t hat

of a solid round wire in a frequency range from 20

kHz t o 800 kHz. Above this frequency, the Litz wire

resistance becomes much higher t h a n t h e solid wire

resistance. This demonstrates t h a t Litz wire pro-

vides lower losses only in a limited frequency range.

Since at low frequencies t h e d c resistances of t he two wires a r e nearly t h e same, t h e solid round wire

is preferred because of its better utilization of t he

winding width.

I n Fig. 3, t h e resitanceas of a solid round wire,

and Litz wire windings with a different number of strands ar e compared. T h e wire overall cross-

sectional areas were chosen to result in t h e s ame d c

resitance. T h e plots of Fig. 3 lead t o t h e following conlcusions:

1) T h e Litz and solid wire conductors have t h e same

resistance u p to f = 1 0 ~ 1 5 kHz, depending

o n t h e number of strands.

2) T h e ac resistances of Litz wires a r e lower t h a n

t h e solid wire resistances in a frequency range

depending on t h e number of strands as follows

for a 10 s tr an d Litz wire;

for a 20 strand Litz wire;

for a 60 strand Litz wire.

15

+

800 kHz

15 kHz

c

1 MHz

15 kHz

+

1.7 MHz

3) For frequencies higher t h a n those shown above

t h e Litz wire ac resistances a r e higher th an t he solid wire resistances.

Therefore, if t h e number of strands of t h e Litz

wire is increased, its a c resistance is lower a n d is lower th an t h a t of a solid round wire over a wider

frequency range. T h e limitation of a high strand

number Litz wire is t h a t it has a large winding dis-

placement. 104 I O 5

r

103

I

Rac (0) 102 10' lO-'l , , , ,

I

Frequency (Hz)

Fig. 2. Resistances R,, of a solid an d Litz wire

versus frequency.

IV. EXPERIMENTAL RESULTS

To verify th e accuracy of expression (19), t h e

(6)

shown in Fig. 4 was used. 4(a) , L is

t h e nominal inductance, R,, is t h e winding resis-

tance, R, is t h e magnetic core resistance, and C is

t h e inductor self-capacitance. Both resistances are

frequency dependent and, if t h e core resistance R,

is much lower t h a n t h e winding ac resistance R,,,

t h e inductor model simplifies to the form depicted

in Fig. 4(b). Most L C R meters measure a n equiv-

alent series reactance X , and an equivalent series

resistance R, of two-terminal devices as shown in

Fig. 4(c). T h e impedance of t h e equivalent circuit of Fig. 4(c) is In Fig. R,, i-

j w L

(1 - w2LC -

z,

= = R, + j X , . (1 - w2LCI2

+

w2C2R;, (21)

For frequencies f much lower than the first self-

resonant frequency

f,.

,

t h e equivalent series reac-

tance X , has a n inductive character and can be ex-

press as X , =

wL,.

Hence, t h e equivalent series

inductance is L, =

X,/w

as shown in Fig. 4(d).

T h e inductor quality factor at a given frequency is

)X,)

R, R,, jwL (I

-

w2LC -

*)

1 & = - =

. (22) 20 10 5 RLitz 0.5 0.1 i o 2 i o 10 I O l o 6 l o 7

Frequency

(Hz)

Fig. 3. Resistances R,, of 10 strands Litz wire, 20

strands Litz wire, and 60 strands Litz wire

normalized with respect t o a solid round wire

resistance.

To verify t h e accuracy of expression (19), tests

were performed using a n HF'4192A "LF Impedance

Analyzer" equipped with a n HP16047A test fixture

to achieve a higher accuracy in minimizing resid-

ual parameters and contact resistances. The inductor with negligible core losses was built using low-

permeability core materials. Tests have been per-

formed on a n inductor assembled with 114 turns in

4-layers of Lit2 2 0 ~ 0 . 0 8 wire wound on a Micromet-

als E25 material mix #52 with p,=75 core with a

gapped central leg of 9 mm a n d a n effective magnetic

permeability pe= 6. An inductance of L = 168 pH

and a resonance frequency f r = 2.620 MHz were

measured.

Fig. 4. Equivalent circuit of inductors.

(a) Lumped parameter equivalent circuit.

(b) Simplified equivalent circuit for

R,

<<

Rac.

( c ) Series equivalent circuit. (d) Equivalent circuit

assumed by many LCR meters.

10 10 102

Rs

10 O 1 I I lo-'

'

i __1 l o 2 l o 3 lo4 l o 5 l o 6 l o 7

Frequency

(Hz)

Fig. 5 . Equivalent series resistance

R,

of t h e

experimentally tested inductor. -- calculated;

(7)

100

80

e

6o 40 20 0

I

lo3

lo4 l o 5 l o 6 l o 7

Frequency

(Hz)

Fig. 6. Quality factor Q of t h e experimentally

tested inductor. -- calculated; o-o- measured.

Fig. 6 shows t h e plots of t h e theoretical and measured equivalent series resistances of the inductor

modelled as in [6]-[8]. Note t h a t R, is approximately

one hundred times Rdc at 1 MHz and R, increases at

higher frequencies. T h e plots of the inductor qual-

ity factor Q is shown in Fig. 6. Theoretical and

experimental results were in good agreement.

V. CONLCUSIONS

An expression for t h e a c resistance of inductor

winding wound by using Litz wire has been derived.

T h e accuracy of this expression has been verified

by comparing theoretical results with experimental

results derived from testing a n iron-powder core in-

ductor. A relatively large air gap was introduced

to minimize t h e core losses. T h e theoretical results were in agreement with measured results.

T h e expression for t h e Litz wire winding a c resistance was used to compare t h e frequency behavior of Litz wire a n d solid wire windings. This comparison has shown t h a t t h e resistances of t h e windings remains nearly t h e same u p to about 15 kHz. For frequencies higher t h a n 12-18 kHz, t h e ac resistance of Litz wire winding is much lower than t h a t of solid

wire windings. At frequencies higher than some hun-

dreds of kilohertz, t h e solid wire results in a lower

ac resistance t h a n t h e Litz wire.

T h e expression for t h e a c resistance for Lztz wire winding shown in this work, along with the expression for t h e solid wire winding shown in other papers

by t h e same authors, can be used to design power inductor with reduced power losses. Therefore, this paper represents a n useful tool for designing high- efficiency power converter.

T h e extension of t h e presented results to induc-

tor winding operated at non sinusoidal current such

as in inductors used in P W M DC-DC converters is

highly recommended.

REFERENCES

[I] M. Bartoli, N. Noferi, A. Reatti, and M. K . Kaz- imierczuk, “Modelling winding losses in high-

frequency power inductors,” World Scientific

Journal o f Circuits, Systems and Computers,

Special Issue on Power Electronics, P a r t 11,

vol. 5 , NO. 4, pp. 607-626, Dec. 1996.

[a]

E . Bennet and S. C. Larson, “ERective resistailcc

of alternating currents of multilayer windings,”

Transactions of American Institute of Electrical Engineering, vol. 59, pp. 1010-1017, 1940.

[3] M. P. Perry, “Multiple layer series connected

winding design for minimum losses,” IEEE

Trans. Power Application Systems, vol. PAS-98, pp. 116-123, Jan./Feb. 1979.

[4] J. A. Ferreira, Electromagnetic Modeling of

Power Electronic Converters, Kluwer Academic Publishers: Boston-Dordrecht-London, 1989. [5] J. A. Ferreira, “Improved analytical modeling

of conductive losses in magnetic components,”

IEEE Trans. Power Electronics, vol. 9, No. 1, pp. 127-131, Jan. 1994.

[6] M. Bartoli, A. Reatti, and M. K . Kazimier-

czuk, “Modeling iron-powder inductors at high

frequencies

,”

Proc. of 1994 IEEE-IAS Annual

Meeting, Denver, CO, Oct. 2-7, 1994, vol. 2, pp. 1225-1232.

[7] M. Bartoli, A. Reatti, a n d M. I<. Kazimierczuk, “Predicting the high-frequency ferrite-core in-

ductor performance,” Proc. of Electrical Manu-

facturing & Coil Windings Association Meeting,

Rosemont, Chicago, IL, USA, Sept. 27-29, 1994,

pp. 409-413.

[8] M. Bartoli, A. Reatti, a n d M. K. Kazimierczuk,

“High-frequency models of ferrite cores induc-

tors,” Proc.

of

IECON’94, Bologna, Italy, Sept.