Cost-Constrained Selection of Strand Wire and Number in a Litz-Wire Transformer Winding

C. R. Sullivan

Found in IEEE Industry Applications Society Annual Meeting, Oct. 1998, pp. 900–906.

°c 1998 IEEE. Personal use of this material is permitted. However, permission to reprint or republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

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e˜str—™t | hesign of litzEwire windings su˜ je™t to

en—lysis of ™ost is p erformed —t two levels in this p—E

™ost ™onstr—ints is —n—lyzedF en —pproxim—tion of norE

p erF pirstD — gener—l form for fun™tions des™ri˜ing

m—lized ™ost is ™om˜ined with —n—lysis of proximity

the ™ost of litz wire is hyp othesizedF „his le—ds to

ee™t losses to nd ™om˜in—tions of str—nd num˜er

gener—l —n—lyti™—l results des™ri˜ing the ˜ est ™hoi™e of

—nd di—meter th—t optim—lly tr—de o ™ost —nd lossF

litz wire for — given tr—nsformer windingD in terms of —

„he rel—tionship ˜ etween wire sizeD norm—lized ™ostD

™ost fun™tionF et the se™ond levelD results th—t —re less

—n norm—lized loss is shown to h—ve — gener—l form

th—t —pplies to — wide r—nge of designsF e pr—™ti™—l

gener—l ˜ut —re more expli™it —re o˜t—ined through

design pro ™edure is providedD —nd —pplied to —n exE

m—king the ™ost fun™tion expli™it with — p olynomi—l

—mple designD for whi™h it le—ds to less th—n h—lf the

™urve ttom—nuf—™turers9 pri™e quot—tionsF e deE

origin—l loss —t lower th—n the origin—l ™ostD or under

sign metho dologyD —ppli™—˜le to the gener—l ™—seD ˜ut

one fth the origin—l ™ost with the s—me loss —s the

eshed out in terms of the sp e™i™ ™ost fun™tionD is

origin—l designF

outlined —nd illustr—ted with — design ex—mpleF

s sF voss wodel

sF sntrodu™tion

ƒkin ee™t —nd proximity ee™t in litzEwire windE

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vitzEwire ™—n ˜ e used to redu™e the severe eddyE

ings m—y ˜ e divided into ˜undleElevel —nd str—ndElevel

™urrent losses th—t otherwise limit the p erform—n™e of

ee™tsD —s illustr—ted in pigF IF ‡ith prop erly ™hosen

highEfrequen™y m—gneti™ ™omp onentsF fut it is often

™onstru™tionD str—ndElevel proximity ee™t is the domE

—voided ˜y designers ˜ e™—use it ™—n ˜ e very exp ensiveF

in—nt ee™t th—t needs to ˜ e ™onsidered for ™ho osing

sn this p—p erD we develop — design metho dology ™onE

the num˜ er of str—nds ‘U“F

sidering ™ostF „his —ppro—™h en—˜les signi™—nt ™ost

‡e represent winding losses ˜y

redu™tion with no in™re—se in lossD or more gener—llyD

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en—˜les — designer to sele™t the minimum loss design

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du™ed ˜y ˜ etter th—n — f—™tor of ve with no in™re—se

resist—n™e whi™h —™™ounts for —ll winding lossesD given

in lossD ™omp—red to — design ˜—sed on — ™onvention—l

s — sinusoid—l ™urrent with rms —mplitude Fsntern—l

—™

rule of thum˜F

—nd extern—l str—ndElevel proximity ee™t loss ™—n ˜ e

—™™ounted for with the —pproxim—te expressionD

vosses in litzEwire tr—nsformer windings h—ve ˜ een

PPP PPT

™—l™ul—ted ˜y m—ny —uthors ‘IDPDQDRD SD T“D ˜ut relE

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—tively little work —ddresses the design pro˜lemX how

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to ™ho ose the num˜ er —nd di—meter of str—nds for —

p—rti™ul—r —ppli™—tionF sn ‘U“D the optim—l str—nding

giving minimum loss is ™—l™ul—tedF roweverD this ™—n

result in — very exp ensive solution with only slightly

lower loss th—n is p ossi˜le —t ™onsider—˜ly lower ™ostF

elthough ‘U“ —lso —ddresses the ™hoi™e of str—nding unE

der ™onstr—ints of minimum str—nd di—meter or m—xE

imum num˜ er of str—ndsD the re—l ™onstr—int is more

likely to ˜ e ™ost r—ther th—n one of these f—™torsF

I

litz wire ƒometimes the term is reserved for ™ondu™tors

™onstru™ted —™™ording to — ™—refully pres™ri˜ ed p—tternD —nd

str—nds simply twisted together —re ™—lled ˜un™hed wireF ‡e

pigF IF „yp es of eddyE™urrent ee™ts in litz wireF

litz wire will use the term for —ny insul—ted group ed str—ndsF

3 where is the r—di—n frequen™y of — sinusoid—l ™urE —round RH e‡q it h—s st—rted rising signi™—ntlyFRR

n x rentD is the num˜ er of str—ndsD is the num˜er e‡q is not—˜le —s the size —t whi™h the ™ost p er

d of turnsD is the di—meter of the ™opp er in e—™h unit length is — minimumF etRVe‡qD ™ost p er

™

& str—ndD is the resistivity of the ™opp er ™ondu™torD unit length h—s in™re—sed dr—m—ti™—llyFpew m—nuE

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˜ is the ˜re—dth of the window —re— of the ™oreD —nd f—™turers will provide ™onstru™tions using ner str—nds

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k is — f—™tor —™™ounting for eld distri˜ution in multiE th—n thisD —nd though @RA is not ˜—sed on d—t— ˜ eyond

winding tr—nsformersD norm—lly equ—l to one ‘U“F por this p ointD it do es —ppropri—tely rise very r—pidlyF elE

w—veforms with — d™ ™omp onentD —nd for some nonE though @RA represents — smo oth fun™tionD wire ˜—sed

sinusoid—l w—veformsD it is p ossi˜le to derive — single on st—nd—rd sizes is ™he—p er th—n —r˜itr—ry ™hoi™esD

equiv—lent frequen™y th—t m—y ˜ e used in this —n—lysis —nd the —™tu—l ™ost fun™tion h—s signi™—nt ripples

‘U“F sn —n indu™torD the eld in the winding —re— deE ˜ e™—use of thisF sn p—rti™ul—rD evenEnum˜ ered sizes

p ends on the g—pping ™ongur—tionD —nd this —n—lysis —re gener—lly ™he—p er —nd more re—dily —v—il—˜le th—n

is not dire™tly —ppli™—˜le ‘V“F o ddEnum˜ ered sizesF „he extent of this v—ri—tions

highly sensitivetovolume|—t su™iently high volE

umesD there would ˜ e no p en—lty for using o ddD or

s s sF gost en—lysis

even ™ustom sizesF „husD su™hv—ri—tions —re omitted

from this —n—lysisY we —ssume the ™ost is des™ri˜ ed ˜y

ettempting to qu—ntify ™ost for —™—demi™ —n—lysis

the smo oth fun™tion shownF

is pro˜lem—ti™Y pri™es ™h—nge with volumeD m—nuf—™E

15

turerD timeD —nd negoti—tionF roweverD m—ny imp orE

t—nt results dep end only on the gener—l form of the

sn p—rti™ul—rD the gener—l solutions deE ™ost fun™tionF Cost per unit mass

Cost per unit length

rived in the —pp endix for optim—l ™ostGloss tr—deo

designs dep end only only the —ssumption th—t the ™ost 10

of — length of litz wire ™—n ˜ e —pproxim—tely des™ri˜ ed

˜y

P

g ost g g d dn– a@ C @ A A @QA

H m ™ ™

Relative Cost

g where is — ˜—se ™ost p er unit length —sso ™i—ted with

H 5

g d the ˜undling —nd serving op er—tionsD @ A is — ™ost

m ™

˜—sis fun™tion prop ortion—l to the —ddition—l ™ost p er

d n unit m—ss for — given str—nd di—meter D is the numE

™

– is the length of the wireF ƒin™e ˜ er of str—ndsD —nd 0

30 32 34 36 38 40 42 44 46 48

g d weh—ve not sp e™ied — form for @ AD the only loss ™

m Strand Diameter [AWG]

of gener—lity in —ssuming this form @QA is in the —sE

pigF PF xorm—lized ™ost p er unit m—ss —nd norm—lized ™ost

g d n sumption th—t dep ends only on D —nd not on F

m ™

p er unit lengthD —s mo deled˜y @RAF foth —re

ix—min—tion of pri™ing from litzEwire m—nuf—™tures

norm—lized su™h th—t the minimumv—lues —re oneD for

the purp ose of displ—y in this gr—phF

indi™—tes th—t this —ssumption is — v—lid —pproxim—E

tionF xote th—t for the purp ose of optimiz—tion with

g — xed winding lengthD we ™—n ignore D —nd ™onE

H

sider only the ™ost v—ri—tion whi™h is prop ortion—l to

s†F ghoosing xum˜er —nd hi—meter of

P

g d dn @ A F

m ™

™

ƒtr—nds

sn order to g—in intuition —˜ out the v—ri—tion of

™ostD —nd to provide sp e™i™ numeri™—l resultsD it is

„he design ™hoi™e of num˜ er —nd di—meter of

g d useful to nd —n —pproxim—te expression for @ AF

m ™

str—nds ™—n ˜ e ™on™eptu—lized —nd illustr—ted —s —

prom m—nuf—™turers9 pri™ingD we nd th—t the followE

twoEdimension—l sp—™eF sn the ™—se of — full ˜ o˜E

ing fun™tionD norm—lized to — v—lue of one for l—rgeE

˜inD the ™hoi™es in this sp—™e form — lineD —nd the

di—meter wireD is — go o d —pproxim—tion for — wide

tr—deo ˜ etween ™ost —nd loss ˜ e™omes — simple m—tE

n d r—nge of v—lues of —nd X

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ter of ev—lu—ting ˜ oth ™ost —nd loss —long this lineD

whi™h ™—n ˜ e des™ri˜ ed ˜y using ™—l™ul—tions in ‘U“F

k k

I P

roweverD with ™ost ™onstr—intsD — full ˜ o˜˜in often

g d @ AaIC C @RA

m ™

T P

d d

™ ™

is not optim—lD —nd wemust ™ho ose — p ointintwoE

PT T

 d k X where is in metersD aII IH m D —nd dimension—l sp—™e r—ther th—n simply — p ointon—

I ™

W P

 k aP IH mF „his fun™tionD prop ortion—l to lineF

P

™ost p er unit m—ssD is shown in pigF PD —long with sn this se™tionD we explore this str—nd di—meE

P

g d d g the norm—lized ™ost p er unit lengthD @ A F terGnum˜ er sp—™e gr—phi™—llyD using the —pproxim—te

m ™ m

™

is —pproxim—tely ™onst—nt for l—rge di—metersD ˜ut ˜y ™urveEt ™ost fun™tion @RAF en —lge˜r—i™ deriv—tion 1000 1000

100 100

10 10 Number of strands (log scale) Number of strands (log scale)

1 1 30 35 40 45 50 55 60 30 35 40 45 50 55 60

Strand Gauge [AWG] Strand Gauge [AWG]

pigF QF iqu—lE™ost ™ontour linesF pigF RF iqu—lE™ost ™ontours shown with equ—lEloss ™ontoursF

hesigns with —n optim—l ™ostGloss tr—deo —re found

—t p oints where lines from these two sets —re t—ngentF

of equiv—lent ˜ut more gener—l resultsD indep endent

„he di—gon—l solid line ™urving up from the lower left

of the p—rti™ul—r ™ost fun™tion @RAD is provided in the

indi™—tes these p ointsF „he dotted line indi™—tes—

epp endixF

fullE˜ o˜˜in ™onstr—intF

‡e ™—n represe nt the tot—l ™ost @QA —s — set of ™onE

fun™tionD @RAF

tour lines in the sizeEofE —nd num˜ erEofEstr—nds sp—™e

sn the epp endixD the results shown in pigF S —nd T

@pigF QAF elong —ny given ™onst—ntE™ost ™urveD the

—re derived —n—lyti™—llyF „o plot the equiv—lentof

˜ est ™hoi™e is the p oint giving minimum lossF sn

pigF TD we ™—n use

pigF RD ™ontour lines for loss —re shown with the ™ost

™ontours from pigF QF „hese —re ˜—sed on —n ex—mE

I

p d ple design of — IREturn winding on —n ‚wS size ferrite @ AaIC @SA

rgv ™

P @A g d

m™

I

H

™oreD with I wrz ™urrent in the windingF „he ˜re—dth

@A g dd

™ ™

m

of the ˜ o˜˜in is RFWQ mmD —nd the ˜re—dth of the ™ore

g d with —ny given ™ost fun™tion @ AF

m ™

window TFQ mmF yn e—™h ™ost ™ontourD the t—ngent

elso from the epp endixD

p oint to the set of loss ™ontours is the minimum loss

p ointF „his set of p oints is —lso the set of minimum

p

g d I @ A

m ™

p ™ost p oints for —ny given loss ™onstr—intF „he set of

p d Y g a @ A I @TA

rgv ™ I

d



™

these p oints is —lso shown in pigF RF „he s—me set of

32

ts ™—n ˜ e plotted on —xes of ™ost —nd lossD from

p oin 10

whi™h — designer m—y™hose the —ppropri—te tr—deo

34 @pigF SAF

36

„he ™ostGloss tr—deo ™urvesD su™h —s in pigF SD

ve the s—me sh—p e reg—rdless of design p—r—metersF

h— 38

„husD norm—lized to the loss —nd ™ost for one referE

40

en™e str—nd di—meterD they —re identi™—l to the ™urve zed to th—t in pigF SD where ™ost —nd loss —re norm—li 42

Normalized Loss

‡q str—ndsF „his ™urve ™—n ˜ e used to ev—lE for RR e 44

1

the ™ostGloss tr—deos in —ny design —s long —s

u—te 46 t on this gr—ph

the ˜ o˜˜in is not fullF xote th—t — p oin 48

t the minimumEloss design for th—t

do es not represen 50

str—nd g—ugeY r—therD it represents the minimumEloss

0.1 1 10 100

en ™ostY the str—nd size used to —™hieve

design —t — giv Normalized Cost

this is indi™—tedF

pigF SF gost —nd lossD norm—lized to —n optim—l ™ostGloss

design using RR e‡q str—ndsF „his gr—ph —pplies to „he rem—ining inform—tion needed to re—lize — deE

—ny design in whi™h the ˜ o˜˜in is not fullD given the

sign for —ny given p oint™hosen on pigF S ™—n ˜ e proE

™ost fun™tion @RAF €oints —re indexed with the e‡q

p vided in the form of — plot of v—lues for optim—l

r

str—nd size usedF xote th—t — p oint on this gr—ph

™ostGloss designs @pigF TAF vike pigF S @˜ut unlike

do es not represent the minimumEloss design for th—t

pigF RAD pigF T shows gener—l results th—t —pply to —ny

str—nd g—ugeY r—therD it represents the minimumEloss

tr—nsformer designD in the region where the ˜ o˜˜in

design —t — given ™ostY the str—nd size used to —™hieve

this is indi™—tedF is underlledF „he results dep end only on the ™ost

1.8

„efvi s

1.7

€—r—meters found for optim—l ™ostGloss designs using

—nd—rd str—nd sizesF

1.6 st

p str—nd g—uge rel—tive rel—tive for optim—l

1.5 r

@e‡qA ™ost loss ™ostGloss tr—deo

HFHQI WFR IFHRS

1.4 QP

QR HFHRW TFPP IFHTV

QT HFHUW RFIR IFIHR

1.3

QV HFIQI PFVH IFITI

RH HFPQR IFWH IFPRT

HFRS IFQS IFQUT 1.2 RP

Optimum ac resistance factor Fr

RR I I IFSQS

PFVQ HFUU IFTSS

1.1 RT

RV IHFS HFTI IFUIS

SH RT HFRV IFUQU 1 30 35 40 45 50

Strand diameter [AWG]

™onrming the ™—t—log re™ommend—tionF roweverD

p pigF TF egEresist—n™e f—™torD D for optim—l ™ostGloss

r

this is only ™orre™t for —n isol—ted litz ˜undleD —nd

tr—deo designs —s — fun™tion of str—nd di—meterF

it do es not t—keinto —™™ount the extern—l proximE

„hese d—t— —re v—lid for —ny geometry or frequen™yD

ity ee™t th—t domin—tes —™ resist—n™e in — typi™—l

given the ™ost fun™tion mo deled˜y @RAF

tr—nsformerF sing @PA to —™™ur—tely predi™t the —™

resist—n™e of this ˜undleD we o˜t—in —n —™ resist—n™e

g g where is the ™ost with the ™onst—nt term su˜E

I H

p X f—™tor of a W PF „his le—ds to QFI ‡ of loss in

r

tr—™tedD —nd



e—™h windingD —nd — tot—l temp er—ture rise of VU g

p d @ A

in™luding ˜ oth windings —nd the ™ore lossD ˜—sed on

rgv ™

p

d @UA tot—l loss

™



—n empiri™—l therm—l resist—n™e of U gG‡ ‘IH“F

p d @ A I

rgv ™

„he ™—l™ul—tion used here @PA is not v—lid for str—nds

d iqu—tions @TA —nd @UA ™—n ˜ e used with —s —

mu™h l—rger th—n — skin depthF elthough str—nds will

™

p—r—meter to gener—te plots su™h —s pigF S for —ny

very r—rely ˜ e this l—rge in — go o d litz designD the

g d ™ost fun™tion @ AF

design ™—l™ul—ted —˜ ove is f—r enough from — go o d

m ™

design th—t wemust ™he™kF „he skin depth in ™opp er

—t ISH krz is —˜ out HFIU mm|the di—meter of QQ or QR

†F hesign ix—mple

e‡q wire|—nd so @PA is v—lid in the r—nge of interestF

xote th—t even for this p o orly ™hosen designD the —™

sn this se™tionD we illustr—te the use of the —˜ ove

resist—n™e is lower th—n it would ˜ e for —ny singleE

results with — design ex—mpleY — gener—l metho d will

str—nd designY the optimum singleEstr—nd design in

˜ e outlined in the following se™tionF

this ™—se is — singleEl—yer winding th—t would h—ve

„he design ex—mple is — QHEturn to QHEturn tr—nsE

—lmost triple the —™ resist—n™e of the rst designF

former on — igEUH ferrite ™ore with — ISH krzD V

‡enow —pply the results o˜t—ined in ƒe™tion s†

e rms sine w—ve ™urrent in ˜ oth windingsF ‡hile

to this tr—nsformerF pirstD we —ssume RR e‡q wireD

for present purp oses it is not ne™ess—r y to know the

—nd ™—l™ul—te the num˜ er of str—nds to o˜t—in the ™orE

volt—geD we ™—nD for the s—ke of ™on™reten ess D —ssume

resp onding —™ resist—n™e f—™tor shown in pigF T @—lso

— QHH † squ—reEw—vevolt—ge @THH † pEpAD —s would

p X shown in „—˜le sAF ‡e nd a I SQS with IIQI

o ™™ur in — p—r—llelElo—ded reson—nt ™onverterF „his

r

str—nds of RR e‡qF elthough this h—s higher d™ reE

would le—d to — ux —mplitude of TH m„D — ™ore loss

sist—n™e th—n the rst design @IIHH of 5RHAD its over—ll

—round IFR ‡ in — typi™—l p ower ferrite m—teri—lD —nd

—™ resist—n™e is SW7 lowerD —nd furthermoreD the preE

—power output of PITH ‡F „he ˜re—dth of the ™ore

di™ted rel—tive ™ost is PS7 lowerF

˜ X windowis aRR TmmY the ˜ o˜˜in —llows — winding

™

˜ X —re— of aRISmm˜y PR mm highY e—™h of the two

„—˜le s s ™olle™ts d—t— on these —nd further deE

w

windings m—y then t—ke up — height ofIPmmF signsF „he ™ost —nd loss gures —re shown norm—lized

e st—nd—rd design pro ™edure might ˜ e to st—rt with to ˜ oth the origin—l design ˜—sed on m—nuf—™turers9

— m—nuf—™turer9s ™—t—logD whi™h re™ommends RH e‡q d—t—D —nd to this new optim—l ™ostGlost design usE

str—nd litz wire for the IHH to PHH krz r—ngeF pitE ing RR e‡q wireF ‡ith this l—tter norm—liz—tionD

ting QH turns in the —llotted window —re—D we nd the ™ostGloss p ossi˜ilities —re m—pp ed out ˜y pigF SF

the l—rgest p ermissi˜le st—nd—rd ˜undle of RH e‡q yne ™—n now sele™tD on this plotD the desired ™ostGloss

str—nds h—s IIHH str—ndsF en —n—lysis of intern—l tr—deoF por ex—mpleD one ™ould ™hose to keep the loss

proximity ee™t losses ‘W“D —s outlined in wire m—nE ™onst—nt —t the level in the origin—l designD or ™ould

uf—™turers9 —ppli™—tion notes predi™ts — mild —™ resisE optimize for minimum tot—l ™ost in™luding the ™ost of

t—n™e f—™tor of IFIW for this ™onstru™tionD seemingly the energy dissip—ted over the life of the equipmentD

„efvi s s

ƒtr—nding yptions for ix—mple hesign

hesign xum˜er ƒtr—nd vossD p er voss €redi™ted gost e™tu—l gost

of ƒtr—nds q—uge ‡inding xorm—lized toX xorm—lized toX xormF to yrigF hesF

@w—ttsA origF desF RR e‡q desF origF desF RR e‡q desF wfrF e wfrF f

f—sed on IIHH RH SFSS I PFRQ I IFQS I I

™—t—log rule

of thum˜

yptimum IIQI RR PFPV HFRI I HFUR I

™ostGloss

with 5RR

glosest IHSH RR PFQR HFRP IFHPS HFTW HFWQ HFUS HFWV

™—t—log size

winF ™ost with IHH QV SFQP HFWT PFQQ HFIPW HFIU HFIIW HFIUI

origin—l loss

winF loss —t PPHDHHH TQ HFTS HFIIU HFPVS PTVDHHHB QTIDHHHB

—ny ™ost

@theoreti™—lA

en exp ensive SPHH RV IFQW HFPS HFTI UFU IHFQ

˜ut pl—usi˜le

lowEloss design

ƒingleEl—yer I IT ISFI PFUP TFTP H ISB HPB `X `X

singleEstr—nd

B sndi™—tes v—lues th—t —re extr—p olo—tions th—t —re not exp e™ted to ˜ e —™™ur—teF

—nd other ™osts th—t indire™tly result from lower eE ogy to redu™e ™ostD lossD or ˜ othF sn p—rti™ul—rD the

™ien™y —nd higher he—t pro du™tionF IHSH str—nd RR e‡q design —™hieves — SV7 loss reE

du™tion —t less th—n the origin—l ™ostD —nd the IHH

„he designs in „—˜le s s in™lude IHH str—nds of QV

str—nd QV e‡q design —™hieves under oneEfth the

e‡q wireD for —˜ out the s—me loss —s the origin—l

origin—l ™ost —t the s—me lossF

design —t IQ7 of the ™ostD —nd IHSH str—nds of RR

e‡qD — st—nd—rd ™—t—log ™onstru™tion ™lose to the

™—l™ul—ted ™hoi™e of IIQI str—nds for this sizeD —nd

†sF hesign €ro™edure

providing simil—r ™ost —nd loss redu™tionsF ‡ith this

designD the temp er—ture rise would ˜ e redu™ed from

eow™h—rt for — re™ommended design pro ™edure is

 

X the origin—l VU g to RP S g with no in™re—se in ™ostF

shown in pigF UF „his pro ™edure will provide designs

por ™omp—risonD the minimum loss design ™—l™uE

with the minimum loss for —ny given ™ost @—nd the

l—ted using the metho ds of ‘U“ is —lso in™luded|

lowest ™ost for th—t lossAD m—king use of the d—t— preE

for this tr—nsformer th—t metho d indi™—tes PPHDHHH

sented in previous plotsD —nd ™olle™ted for st—nd—rd

str—nds of TQ e‡qwould pro du™e the minimum lossF

str—nd sizes in „—˜le sF „he pro ™edure ™—n ˜ e impleE

st is not ™le—r th—t su™h — litz wire ™ould in pr—™ti™e

mented on — ™omputerY howeverD it ™—nnot ˜ e ™omE

˜ e pro du™ed —t —ny ™ostD mu™h less th—t the estim—te

pletely —utom—tedD —s it requires the user to m—ke deE

pro du™ed ˜y @RA is ™orre™tF roweverD it would —lE

™isions reg—rding the ™ostGloss tr—deoF sn —dditionD

low redu™ing the loss to —˜ out one qu—rter the loss

™onsulting — m—nuf—™turer to o˜t—in —™tu—l ™urrent

o˜t—ined with IHSH str—nds of RR e‡q wireF e RV

pri™e quotes is v—lu—˜leD —nd in ™—ses with — full ˜ o˜E

e‡q design is in™luded to illustr—te — more pr—™ti™—l

˜inD it m—y ˜ e ne™ess —ry to exp eriment—lly me—sure

highE™ostD low loss ™onstru™tionF

p—™king f—™torF

efter p erforming this design workD we o˜t—ined „he ™hoi™e of ™onstru™tion under the ™onstr—intof

pri™ing from m—nuf—™turers for some of the wire ™onE —v—il—˜le wire sizes is explored further in pigF VD whi™h

stru™tions listedF „hese norm—lized pri™es —re —lso in™ludes the ide—l ™ostGloss tr—deo ™urve of pigF SD

shown on „—˜le s sF elthough they do not ex—™tly ˜ut —lso h—s ™urves for e—™h wire sizeF st is —pp—rE

follow the pri™es predi™ted ˜y our mo delD they folE ent th—t the ex—™t wire size is mu™h less imp ort—nt

low the exp e™ted trendsF xote th—t th—t in @QAD we for sm—ller g—uge num˜ ers @˜ elowRHe‡qA|simil—r

dropp ed the ™onst—nt p ortion of the ™ostY determinE ™ost —nd loss p erform—n™e is —v—il—˜le with ne—r˜y

ing this ™onst—nt for the quoted pri™es would improve sizesF roweverD with ner wireD there is more in™enE

the —™™ur—™y our predi™tionsF fut reg—rdlessD weh—ve tive to ™onsider —n o dd str—nd sizeF „he —™tu—l ™ost of

™onrmed the usefulness of the mo del —nd metho dolE the wire with —n o dd str—nd size m—y dep end on the

ƒee next p—ge for gure

pigF UF hesign pro ™edure th—t —llows the user to ™ho ose ™ostGloss tr—deo —nd gu—r—ntees minimum loss for the sele™ted ™ost

@—nd the lowest ™ost for th—t lossAF

‚eferen™es qu—ntity pur™h—sedD —nd so it is not p ossi˜le here to

determine when it is e™onomi™—lly —dv—nt—geousF fut

‘I“ tF eF perreir—D ’smproved —n—lyti™—l mo deling of ™ondu™E

pigF V highlights where it is worth ™onsideringF

siii „r—ns—™tions tive losses in m—gneti™ ™omp onentsF4D

on €ower ile™troni™s DvolF WD ppF IPU{QID t—nF IWWRF

†s sF gon™lusion

‘P“ eF ‡F vot —nd pF gF veeD ’e high frequen™y mo del

in gonferen™e for litz wire for swit™hEmo de m—gneti™s4D

‚e™ord of the „wentyEiighth seƒ ennu—l weeting DvolF PD

gom˜ined —n—lysis of loss —nd ™ost of litzEwire

ppF IITW{USD y™tF IWWQF

windings ™—n le—d to su˜st—nti—l improvements in

‘Q“ w—ssimo f—rtoliD xi™ol— xoferiD el˜ erto ‚e—ttiD —nd w—rE

™ostD lossD or ˜ othF „he —n—lysis le—ds to gener—l exE

i—n uF u—zimier™zukD ’wo deling litzEwire winding losses

pressions des™ri˜ing the rel—tionship ˜ etween ™ost —nd

in PUth ennu—l in highEfrequen™ypower indu™tors4D

loss in optim—l designsD in terms of — ™ost fun™tionF sn

siii €ower ile™troni™s ƒpe™i—lists gonferen™e DvolF PD

—dditionD this ™ost fun™tion ™—n ˜ e —pproxim—ted ˜y—

ppF ITWH{ITWTD tune IWWTF

p olynomi—lD le—ding to numeri™—l d—t— th—t f—™ilit—tes

ƒoft perritesD €roperties —nd eppli™—tions ‘R“ iF gF ƒnellingD D

— simple design pro ™ess th—t le—ds to minimum loss

futterworthsD se™ond editionD IWVVF

designs —t —ny given ™ostD or minimum ™ost designs

‘S“ €F xF wurg—troydD ’g—l™ul—tion of proximity losses in

for —ny given lossF

sii €ro™eed ingsD €—rt multistr—nded ™ondu™tor ˜un™hes4D

e DvolF QTD ppF IIS{IPHD IWVWF

‘T“ fF fF eustinD ’„he ee™tive resist—n™e of indu™t—n™e ™oils

„he ‡ireless ingineer —t r—dio frequen™y4D DvolF IID ppF

IP{ITD t—nF IWQRD ƒumm—ryof work ˜y ƒF futterworthF

‘U“ gh—rles ‚F ƒulliv—nD ’yptim—l ™hoi™e for num˜er of

in PVth enE str—nds in — litzEwire tr—nsformer windingF4D

nu—l siii €ower ile™troni™s ƒpe™i—lists gonferen™e D ppF

PV{QSD IWWUF

‘V“ gh—rles ‚F ƒulliv—nD ’yptimiz—tion of sh—p es for roundE

in €roE wire highEfrequen™y g—pp edEindu™tor windings4D

™eedings of the IWWV siii sndustry eppli™—tions ƒo™iety

ennu—l weeting D IWWVF

‚—dio ingineer9s r—nd˜ook prederi™k immons „erm—nD D

10 ‘W“

32

w™qr—wErillD IWRQF

ƒteef eF wulderD ’eppli™—tion note on the design of lowE

34 ‘IH“

prole highEfrequen™y tr—nsformers4Dw—y IWWHD €hillips ts v—˜ or—tory ‚ep ortF 36 gomp onen

38 eppendix 40

Normalized Loss

heriv—tion of optim—l ™ostEloss ™urve 42 eF 1

44

 t terms —s Dwe ™—n express @PA 46 vumping ™onst—n

48 —s

PT

p n d X aIC @VA

50 r ™ 0.1 1 10 100

Normalized Cost g n et — given ™ostD Dwe wish to nd the ™hoi™e of

I

pigF VF gost —nd lossD norm—lized to —n optim—l ™ostGloss

d —nd th—t gives minimum tot—l lossF „ot—l loss is

™

design using RR e‡q str—ndsF „he ide—l rel—tionship

p prop ortion—l to tot—l resist—n™e f—™tor D

rt

shown —s the ˜ ottom ™urve —ssumes —ny str—nd

di—meteris —v—il—˜leF gurves for individu—l even wire

—™ resist—n™e of litzEwire winding

p p p Y a a

sizes —re —lso plotted to show the p en—lty for using —

rt d™ r

d™ resist—n™e of singleEstr—nd winding

st—nd—rd wire sizeF por l—rge di—meter wireD the ™urve

@WA

—re ™lose to one —notherD indi™—ting th—t the ex—™t

p where is the r—tio of d™ resist—n™e of the litz wire

d™

™hoi™e of di—meter is unimp ort—ntFroweverD for ne

to the d™ resist—n™e of — single str—nd windingD using

wireD the ™hoi™e of — st—nd—rd even size m—yent—il —

signi™—nt p en—ltyF wire with the s—me di—meter —s the litzEwire ˜undleF Ασσυε ΑΓ 44 στρανδσ, ανδ χαλχυλατε νυβερ οφ στρανδσ φορ

Φρ = 1.535 (2)

Χαλχυλατε τηε νυβερ οφ στρανδσ το φιλλ Φιτσ ιν Νο τηε βοββιν ωιτη τηε χυρρεντ στρανδ γαυγε ωινδοω? υνδερ χονσιδερατιον, ανδ χαλχυλατε τηε οπτιαλ νυβερ οφ στρανδσ (ωιτη ανψ γαυγε ανδ νο χοστ χονστραιντ) υσινγ τηε Ψεσ αναλψσισ ιν [7]. Τεντατιϖελψ ωηιχηεϖερ οφ τηεσε ηασ λαργερ στρανδσ, ανδ εϖαλυατε τηε Οβταιν α πριχε θυοτε φορ τηισ ωιρε (ορ φορ α στανδαρδ λοσσ ανδ χοστ. Αδϕυστ τηε νυβερ οφ προδυχτ ωιτη α σιιλαρ νυβερ οφ στρανδσ) ανδ στρανδσ το τραδε οφφ χοστ ανδ λοσσ, χαλχυλατε λοσσεσ. Τηεσε τωο δατα προϖιδε τηε ασσυινγ α φυλλ βοββιν. νοραλιζατιον σχαλε φορ τηε νοραλιζεδ χοστ ανδ λοσσ σηοων ιν Φιγ. 5 ανδ Ταβλε Ι Γιϖεν τηισ σχαλε, σελεχτ α δεσιραβλε λοσσ/χοστ τραδεοφφ.

Υσε Φιγ. 6 το δετερινε τηε ϖαλυε οφ Φρ φορ τηε στρανδ γαυγε σελεχτεδ, ανδ (2) το δετερινε τηε Νο Φρ βελοω νυβερ οφ στρανδσ φορ τηισ ϖαλυε οφ Φρ.. Το τραδε οφφ χοστ ανδ λοσσ ιν φινερ ινχρεεντσ τηαν στανδαρδ ωιρε ϖαλυε ιν σιζεσ, ψου αψ ϖαρψ τηε νυβερ οφ στρανδσ. Φιγ. 6?

Ψεσ Φιτσ ιν Νο ωινδοω?

Ψεσ

Οβταιν α φιναλ πριχε θυοτε, ανδ χηεχκ τηατ τηε στρανδ διαετερ ισ νοτ ορε τηαν τωο σκιν δεπτησ ατ τηε ηιγηεστ σιγνιφιχαντ ηαρονιχ φρεθυενχψ ιν τηε χυρρεντ ωαϖεφορ. Ιν τηε ηιγηλψ υνλικελψ χασε τηατ ιτ ισ, τηε λοσσ πρεδιχτιον ισ χονσερϖατιϖε ανδ τηε δεσιγν ισ νοτ οπτιυ. Ηοωεϖερ, ιτ ωιλλ περφορ βεττερ τηαν πρεδιχτεδ.

f—sed on this denitionD „husD using @ITAD

P

p

p d @ A

d

rgv ™

P

™ss

PT

p

p d d @IVA a

p n d a@IC A @IHA

rt g v ™

rt

™ss

™

P

dn

p d @ A I

rgv ™ ™

d where is the di—meter of the l—rgest singleEstr—nd

™ss sf @IVA —nd @ITA —re norm—lized —s in pigF SD the

wire th—t would tF „his ™onst—ntm—y ˜ e dropp ed

 ™onst—nts sp e™i™ to — p—rti™ul—r design pro˜lemD

for the purp oses of optimiz—tionY wework with

d d —nd D drop outF „husD with —s — p—r—meterD

™ss ™

@IVA —nd @ITA ™—n ˜ e used to plot — ™urve of norm—lized

I

R

™ost —nd loss for —ny given ™ost fun™tionD —s shown in

p  nd X C @IIA

rt

™

P

dn

™

pigF S for @RAF

„o minimize tot—l lossD holding ™ost ™onst—ntD we

n ™—n elimin—te from @IIA ˜y using @QAD to o˜t—in

P

g d d g @ A

I m ™

™

p  Y C @IPA

rt

@ A g g d

I m ™

where is the sp e™ied ™ostD with the ™onst—nt term g

I

g in @QA su˜tr—™tedF ƒetting the deriv—tive of this

H

d expression with resp e™t to equ—l to zeroD we o˜t—in

™

H P P

g d g g @ A

™

I I m

H P

d d g d Y P @ Aa H @IQA

™ ™

m ™

P

@ A @ A g d g d

™ m ™

m

H

g d g d where @ A is the deriv—tiveof @ A with resp e™t

™ m ™

m

d to F

™

g d qiven — represent—tion of @ A —nd — ™ost sp e™E

m ™

g i™—tion D @IQA ™ould ˜ e numeri™—lly solved for —

I

d optimum v—lue of FroweverD it is p ossi˜le to derive

™

sever—l more gener—l results th—t provide —ddition—l

insight —nd le—d to pigsF T —nd S in ƒe™tion s†F ƒolvE

 ing @IQA for D su˜stituting th—t result into @IHAD —nd

n —g—in elimin—ting using @QA le—ds to

I

p d @ Aa IC @IRA

rgv ™

P @A g d

m™

I

H

@A g dd

™™

m

„his expression des™ri˜ es the rel—tionship ˜ etween

p wire size —nd the optim—l ™ostGloss v—lue of D deE

r

p d noted @ AD —s shown in pigF TF „he gener—lity

rgv ™

of the result is indi™—ted ˜y the indep enden™e of @IRA

 from the design det—ils lump ed in the ™onst—nt F

„he gener—lity of the rel—tionship shown in pigF S

™—n ˜ e seen —s followsF prom @IQAD

g d @ A

m ™

q

g a @ISA

I

P @A g d

m™

P

d @I A

H

™

@A dg d

™ ™

m

or

p

g d I @ A

m ™

p

g p d X a @ A I @ITA

I rgv ™

d



™

@A g d

m ™ I

sing the rel—tionship a Dwe ™—n write

P

dn g

I

™

g d @ A

m ™

P

p p d d X a @ A @IUA

rt g v rgv ™

™ss

g I