leakage impedance of transformer windings

Leakage Impedance of Leakage Impedance of Transformer WindingsTransformer Windings

Dept. Of Information Engineering – DEIDept. Of Information Engineering – DEI

University of PadovaUniversity of Padova

Prof. Giorgio SpiazziProf. Giorgio Spiazzi

Leakage Impedance of Leakage Impedance of Transformer WindingsTransformer Windings

• High Frequency induced effects on transformer High Frequency induced effects on transformer windingswindings

• Qualitative analysis of transformer winding Qualitative analysis of transformer winding leakage impedanceleakage impedance

• Quantitative analysis of transformer winding Quantitative analysis of transformer winding leakage impedanceleakage impedance

• Dowell curvesDowell curves• ExamplesExamples

Outline: Outline:

Ref.:Ref.:

P.L. Dowell, “Effects of Eddy Currents in Transformer Windings,” Proc. of IEE, Vol.13, No.8, August 1966, pp.1387-1394.

Simple Transformer Winding Simple Transformer Winding ArrangementArrangement

Secondary Secondary windingwinding

Primary Primary windingwinding

Isolation gapIsolation gap

Magneto-motive Force in the Magneto-motive Force in the Core WindowCore Window

m.m.f.m.m.f.(dc)(dc)

00

NIdH.f.m.mm

Magneto-motive Force in the Magneto-motive Force in the Core WindowCore Window



Isolation gapIsolation gap


00

Leakage FluxLeakage Flux

The leakage flux in the core window causes The leakage flux in the core window causes eddy currents in the windingseddy currents in the windings

Leakage fluxLeakage flux

Power loss in the winding resistancePower loss in the winding resistance

Leakage fluxLeakage flux

Magnetic energy stored in the core window crossed Magnetic energy stored in the core window crossed by the leakage fluxby the leakage flux

Magnetic energy stored in the transformer leakage Magnetic energy stored in the transformer leakage inductanceinductance

Leakage FluxLeakage Flux

Skin EffectSkin Effect

High frequency currents in the conductor High frequency currents in the conductor generate a variable magnetic field that induces generate a variable magnetic field that induces

voltages and, consequently, currents. The voltages and, consequently, currents. The latter are directed in such a way to reinforce the latter are directed in such a way to reinforce the current flowing close to the conductor surfacecurrent flowing close to the conductor surface

DDPENPEN

JJREALREAL JJEQUIVALENTEQUIVALENT

DDWW

Current linesCurrent lines

DDPENPEN = skin depth = skin depth

Proximity EffectProximity Effect

• The current in a close path distributes The current in a close path distributes itself in such a way so as to minimize the itself in such a way so as to minimize the energy drawn from the source. energy drawn from the source.

PCBPCB

WW

++ ++ ++ ++ ++ ++ ++ ++ ++ ++

•• •• •• •• •• •• •• •• •• ••

Example: faced PCB tracksExample: faced PCB tracks

HF Current Distribution in HF Current Distribution in WindingsWindings

Inductor: single winding Inductor: single winding

........................

++++++

++++++

++++++

++++++

gg

FF FF

HF

Hg

g

F

H

The current concentrates The current concentrates on winding inner surfaceon winding inner surface


Transformer: single layerTransformer: single layerPrimary: 4 turns - 3A / Secondary: 1 turn - 12APrimary: 4 turns - 3A / Secondary: 1 turn - 12A

..

........

..............................

++++++

++++++

++++++

++++++

FF FF

++

..

++++++++++++++++++++++

....

..

The magnetic field is almost zero outside the The magnetic field is almost zero outside the two windings but is high between themtwo windings but is high between them

LF Current Distribution in LF Current Distribution in WindingsWindings

Transformer Transformer with multiple with multiple

layerslayers

PP11 PP22 SS33 SS22 SS11

Current Current homogeneously homogeneously

distributed inside distributed inside conductors conductors

W [J]W [J]

F=NIF=NI

Energy Energy densitydensity

[J/m][J/m]

LF Current Distribution in LF Current Distribution in WindingsWindings

Multiple winding transformer Multiple winding transformer with interleaved with interleaved

primary/secondary windingsprimary/secondary windings

PPaa PPbbSSbbSSaa

F=NIF=NI

Energy Energy densitydensity

[J/m][J/m] W [J]W [J]

Reduced leakage Reduced leakage inductance inductance


F=NIF=NI

PP22PP11SS33 SS22 SS11

Multiple winding transformer

Conductor thickness >> DConductor thickness >> DPENPEN

Magnetic field only between layers Magnetic field only between layers


F=NIF=NI

SS33 SS22 SS11

++.. .... ++++++

++

++

+1+1-1-1+2+2-2-2+3+3I=I=

Conductor thickness >> DConductor thickness >> DPENPEN

Secondary winding: 1ASecondary winding: 1A

Magnetic field only between layers : Magnetic field only between layers : different different currents induced on layer surfacescurrents induced on layer surfaces

Passive LayersPassive Layers

F=NIF=NI

SS33 SS22 SS11

++.. .... ++++++

++

++

+1+1-1-1+2+2-2-2+3+3I=I=

++++

++ ......-3-3+3+3

PP

Winding carrying zero current in a given Winding carrying zero current in a given instant instant

((e.g. one primary winding in a push-pull e.g. one primary winding in a push-pull transformer, one secondary winding in transformer transformer, one secondary winding in transformer

with center tapped secondary, EMI shieldwith center tapped secondary, EMI shield))

High losses! High losses!

Leakage ImpedanceLeakage Impedance

The leakage flux crossing a winding layer The leakage flux crossing a winding layer determines both its ac leakage resistance and determines both its ac leakage resistance and

inductanceinductance

When considering the leakage impedance due When considering the leakage impedance due to a particular layer, it is necessary to consider to a particular layer, it is necessary to consider the other layers of the windings insofar as they the other layers of the windings insofar as they

affect the flux in the layer being consideredaffect the flux in the layer being considered


From the behavior of the From the behavior of the m.m.f. in a core window we m.m.f. in a core window we can say :can say :

The leakage flux distribution The leakage flux distribution across any layer depends only across any layer depends only on the current in that layer and on the current in that layer and the total current between the the total current between the layer and an adjacent position layer and an adjacent position of zero m.m.f.of zero m.m.f.

P1 P2 S3 S2 S1


00

Winding portionsWinding portions


For leakage impedance For leakage impedance calculation purposes we can calculation purposes we can consider the whole winding consider the whole winding subdivided in parts containing subdivided in parts containing each a position of zero m.m.f.. each a position of zero m.m.f.. Such parts will be termed Such parts will be termed “winding portions”.“winding portions”.


00


An intersection gap can be An intersection gap can be considered to be part of either considered to be part of either of the adjacent portions; thus, of the adjacent portions; thus, the leakage impedance due to the leakage impedance due to these gaps will be referred to these gaps will be referred to the primary if they are the primary if they are associated to a primary associated to a primary winding portion or to the winding portion or to the secondary if the gaps are secondary if the gaps are associated to a secondary associated to a secondary winding portion.winding portion.


GapGap

00

Winding portionsWinding portions

Interleaved WindingsInterleaved Windings

A winding portion can contain a A winding portion can contain a half layer (if the corresponding half layer (if the corresponding winding section is composed by winding section is composed by an odd number of layers) an odd number of layers)

Pa PbSbSa


00

Two cases: Two cases:

1)1) The winding portion contains The winding portion contains mm full layers full layers

2)2) The winding portion contains The winding portion contains mm full layers full layers ++ a a halfhalf layer layer

Frequency-Independent Components Frequency-Independent Components of Leakage Impedanceof Leakage Impedance

• Increasing the frequency Increasing the frequency will affect the current will affect the current distribution across each distribution across each conductor, but the total conductor, but the total net current will remain net current will remain unaltered. Consequently, unaltered. Consequently, the magnetic field H and the magnetic field H and its associated energy in its associated energy in the intersection gaps will the intersection gaps will be independent of the be independent of the frequency. frequency.

aa

uu gg

bb

00

Winding portionWinding portion


Frequency-Independent Components Frequency-Independent Components of Leakage Impedanceof Leakage Impedance

• Hp:Hp:• Square section conductor having the Square section conductor having the

same section of circular onessame section of circular ones

• Average turn length Average turn length TT

• Uniform magnetic field in the core windowUniform magnetic field in the core window

22

a4

D

2

Da

DD aa

Leakage Inductance of Isolation Leakage Inductance of Isolation Gap g Gap g

aa

gg

bb

00


Integer number Integer number mm of layers of layers

b

IN

b

ImNH 1p1

g

2

1gg

2

g0gIL

2

1VH

2

1W

NN = Number of turns per layer= Number of turns per layer

NNp p = Number of turns of the = Number of turns of the

whole winding portionwhole winding portion

gb

NL T

2p0

g

Tg bgV

Leakage Inductance of Isolation Leakage Inductance of Isolation Gap gGap g

aa

gg

bb

00


Integer number Integer number mm of layers of layers ++ halfhalf layer layer

b

IN

2

1m

b

INH 1p1

g

2

1gg

2

g0gIL

2

1VH

2

1W

NNp p = Number of turns of the = Number of turns of the

whole winding portionwhole winding portion

gb

NL T

2p0

g

Tg bgV

Leakage Inductance of Gaps u Leakage Inductance of Gaps u Between LayersBetween Layers

aa

uu

bb

00



b

INpH 1

up

ppthth Gap Gap

ppthth Gap Gap

Tup buV

21upup

2up0up IL

2

1VH

2

1W

2T2

0up p

b

uNL


aa

uu

bb

00


Overall inductance:Overall inductance:

1m

1p

2T2

0U p

b

uNL

1m1m26

mp

1m

1p

2

m2

11

b3

UNL T

2p0

U

u1mU (Total gap width)(Total gap width)


aa

bb

00



uu

2

1p

b

INH 1

up

ppthth Gap Gap

p = 1p = 1mm

Tup buV

21upup

2up0up IL

2

1VH

2

1W

2T

20

up 2

1p

b

uNL


aa

bb

00


uu

m

1p

2T

20

U 2

1p

b

uNL

Overall inductance:Overall inductance:

1m21m212

m

2

1p

m

1p

2

1m2

1m2

b3

UNL T

2p0

U

muU (Total gap width)(Total gap width)

DC Winding InductanceDC Winding Inductance

Since the flux in the intersection gaps have Since the flux in the intersection gaps have already been taken into account, the energy already been taken into account, the energy associated to the flux crossing the conductor associated to the flux crossing the conductor layers of a given winding portion can be done layers of a given winding portion can be done considering the layers close to each other considering the layers close to each other without gaps. without gaps.

Being the current density constant at dc, the Being the current density constant at dc, the current and the associated magnetic field current and the associated magnetic field vary vary linearlylinearly with position x as indicated in the figure with position x as indicated in the figure (x=0 corresponds to the position of zero m.m.f.).(x=0 corresponds to the position of zero m.m.f.).


a

h

b

HH

00 mhmh xx

dxdx

b

ImNH 1

max

xbh

INx

mh

HxH 1max

dvxH2

1dw 2

0

210w

21

32T

0

mh

0

T2

0

W

0

IL2

1

b3

hImN

2

1

bdxxH2

1dwW


b3

hmNL

32T

00w


Integer number Integer number mm of layers of layers ++ halfhalf layer layera

h

b

HH

00 xx

dxdx

2

1m

b

INH 1

max

(m+ )h(m+ )h1122

Same procedure, only Same procedure, only substitute m with m+1/2substitute m with m+1/2

32T

00w 2

1m

b3

hNL

DC Winding ResistanceDC Winding Resistance

a

h

b


bh

mN

ah

mNR T

2T

0w

ab

NForm factor. It is equal to 1 Form factor. It is equal to 1 when the turns of the same when the turns of the same layer are close to each layer are close to each other.other.

DC Winding ResistanceDC Winding Resistance

The resistance of the The resistance of the winding portion is half of winding portion is half of the resistance of the the resistance of the winding section (made up winding section (made up by 2m+1 layers) having that by 2m+1 layers) having that portion as half sectionportion as half section

a

h

b


2

1m

bh

N

ah

1m2N

2

1R T

2T

0w

AC Winding Leakage ImpedanceAC Winding Leakage Impedance

• Only the flux crossing the winding layers is Only the flux crossing the winding layers is considered. considered.

• The current density inside each layer is calculated.The current density inside each layer is calculated.• The voltage developed across each layer is calculated The voltage developed across each layer is calculated

as the sum of a resistive component plus an induced as the sum of a resistive component plus an induced voltage due to the linked flux.voltage due to the linked flux.

• The total voltage across the winding portion is The total voltage across the winding portion is calculated summing the voltage across each layer.calculated summing the voltage across each layer.

• The leakage impedance of the winding portion is The leakage impedance of the winding portion is calculated whose real and imaginary parts represent calculated whose real and imaginary parts represent the leakage resistance and inductance, respectively.the leakage resistance and inductance, respectively.

ppthth layer layer


A generic layer A generic layer pp (p=1(p=1m) is m) is considered, and considered, and inside it, an inside it, an infinitesimal layer infinitesimal layer xx at position x (as at position x (as respect to the edge of respect to the edge of ppthth layer closer to the layer closer to the position at zero position at zero m.m.f.)m.m.f.)

a

h u g

b

x

0 x

0




Flux linking elementary Flux linking elementary layer layer x is x is bb++cc::

a

h u g

b

b c

pth layer

xdx

d cb

Bx T

x

0

dyybJ1pINb

1

b

iH

Magnetic field at position x:Magnetic field at position x:


a

h u g

b

b c

pth layer

NxjxJNV cbT

Voltage across pVoltage across pthth layer: layer:

VV is independent of x: is independent of x:

Ndx

dj

dx

dJN0

dx

dV cbT

dx

dj

dx

dJ cb

T


a

h u g

b

b c

pth layer

x

0

0 dyyJb

1pINj

dx

dJ

JJj

dx

Jd 202

2

j1D

j

PEN

0

xsinhQxcoshPJ Solution:Solution:


1pb

INQ

a

h u g

b

b c

pth layer

2

htanh1p

hsinh

1IN

bP

Coefficients:Coefficients:

Current Density DistributionCurrent Density Distribution

Current density inside pCurrent density inside pthth layer layer

xsinh1pxcosh2

htanh1p

hsinh

xcoshIN

bxJ

j1D

j

PEN

0

DDPENPEN = skin depth = skin depth

Normalized Current DensityNormalized Current Density

0

5

10

15

20

0 h

1°1° layer layer 2°2° layer layer 3°3° layer layer

ffss = 100kHz = 100kHz

JN(x) JN(x) JN(x)

ah

I

xJxJ

N

JN(x)30

10

20

00 h

Normalized Current DensityNormalized Current Density

3°3° layer layer







NxjxJNdt

dNJNV

cbT

cbTp

VVpp is independent of x. is independent of x.

Thus, it is calculated at Thus, it is calculated at x = h:x = h:

NjJNdt

dNJNV chT

chTp

a

h u g

b

b c

pth layer


a

h u g

b

b c

pth layer

m

1pnnc

cc is the flux in all the is the flux in all the

winding layers beyond winding layers beyond the pthe pthth layer edge at layer edge at position x = h:position x = h:

x

0

0Tbcb dyyJ

b

1pIN

dx

d

dx

d

Let’s calculate the flux in Let’s calculate the flux in the generic pthe generic pthth layer: layer:


h

0

x

0

0T0T

h

0

x

0

0T

0

h

x

0

0T

0

bp

dxdyyJhb

1pIN

dxdyyJb

1pIN

dxdyyJb

1pINd

p


D2

1p

hb

IN20Tp

Total flux crossing Total flux crossing

ppthth layer: layer:

2

pmD

bh

IN

2

1nD

bh

IN

22

20T

m

1pn20T

m

1pnnc

Total flux linking pTotal flux linking pthth layer: layer:

2

htanhh2jDDD ir


2

Dpm1pM

bh

INVVV 22T

2

iprpp


Current density at the edge of pCurrent density at the edge of pthth layer far from layer far from the position at zero m.m.f.:the position at zero m.m.f.:

D2

1pM

bh

INhJJh

hcothhjMMM ir


1mm3

DmM

bh

INVV 2T

2m

1pp

Total voltage across the winding Total voltage across the winding portion:portion:

Associated leakage impedance:Associated leakage impedance:

ww2T

2

w LjR1mm3

DmM

bh

N

I

VZ


R0w2r

r0ww FR1m3

DMRR

AC resistance:AC resistance:

L0w222

2ii

0ww FLhm

1mDM3LL

AC inductance:AC inductance:


m.m.f.

a

h u g

b

x

0 x

0

Layer (p+ )12

h/2

Layers 1 to m

Half layer

Integer number Integer number mm of of layers layers ++ halfhalf layer layer


• The overall leakage impedance associated to the integer The overall leakage impedance associated to the integer number m of layers is calculated.number m of layers is calculated.

• The leakage impedance of the half layer is calculated as The leakage impedance of the half layer is calculated as the ratio between its voltage Vthe ratio between its voltage V1/21/2 and current I. Being the and current I. Being the calculated impedances of each winding portion summed calculated impedances of each winding portion summed together, the contribution of the layer that is splitted into together, the contribution of the layer that is splitted into two parts (each half layer belonging to adjacent portions) two parts (each half layer belonging to adjacent portions) must be equally subdivided into the two portions.must be equally subdivided into the two portions.

• Consequently, the contribution of the half layer to the Consequently, the contribution of the half layer to the overall impedance of the winding portion is half the overall impedance of the winding portion is half the impedance of the full layer whose the considered half impedance of the full layer whose the considered half layer belongs to.layer belongs to.



ww

22/1T

2

w

LjR

12

D1m6m4mmM12M6

bh

INZ

2

hcoth

2

h

2

hMjMMM i2/1r2/12/1



AC resistance:AC resistance:

R0w

r2

rr2/10ww FR

6m12

D1m6m4mmM12M6RR

AC inductance:AC inductance:

L0w3

22

i2

ii2/10ww FL

2

1mh4

D1m6m4mmM12M6LL


Such parameter represents the thickness of Such parameter represents the thickness of the winding layer, corrected by a form factor the winding layer, corrected by a form factor that depends on the turn separation in that that depends on the turn separation in that layer, normalized to the skin depthlayer, normalized to the skin depth

j1qj1D

hh

PEN

2

h

D

hq

PEN

hFF RR hFF LL

Normalized resistance and inductance:Normalized resistance and inductance:

Dowell CurvesDowell Curves

0.1 1 101

10

100

FR

q

1

1+1/2

2+1/2

3+1/2

4+1/2

2

3

4

Dowell CurvesDowell Curves

0.1 1 100.1

1

FL

q

1+1/2

1

Leakage Inductance CoefficientLeakage Inductance Coefficient

3

22

i2

ii2/1

222

2ii

L

2

1mh4

D1m6m4mmM12M6

hm

1mDM3

F

mm layers layers

mm ++ halfhalf layerslayers

22

iL

m h

DFLim

Example #1: Simple WindingExample #1: Simple Winding


b

DD11

uu1010 gg00



• Primary conductor diameter AWG17: Primary conductor diameter AWG17: DD11 = 1.15mm = 1.15mm• Number of turns per primary layer:Number of turns per primary layer: NN11 = 12 = 12• Gap width between primary layers:Gap width between primary layers: uu1010= 0.2mm= 0.2mm• Gap width between prim. and sec.:Gap width between prim. and sec.: gg00 = 0.5mm = 0.5mm• Gap width between secondary layers:Gap width between secondary layers: uu22= 0.15mm= 0.15mm• Height of secondary turn:Height of secondary turn: aa22= 10mm= 10mm• Thickness of secondary layer:Thickness of secondary layer: hh22 = 0.6mm = 0.6mm• Core window height:Core window height: b = 14mmb = 14mm• Internal diameter of bobbin:Internal diameter of bobbin: DDcoil coil = 13.4mm= 13.4mm• Operating frequency:Operating frequency: ffss = 100kHz = 100kHz


bb

aa11

aa22

hh11

uu11 gg hh22

uu22

Primary winding substituted by an equivalent squared Primary winding substituted by an equivalent squared cross section conductorcross section conductor


aa11

aa22

hh11

uu11

gg hh22

uu22

2Da 11

= 1.019mm= 1.019mm

uu1 1 = u= u1010+D+D11-a-a11= 0.33mm= 0.33mm g = gg = g00+(D+(D11-a-a11)/2= 0.565mm)/2= 0.565mm


• LLgg = 8.74 = 8.74HH• LLUU = 4.48 = 4.48HH• LLw0w0 = 21.01 = 21.01HH• FFLL = 0.26 = 0.26• LLww = 5.49 = 5.49HH• LLdd = L = Lgg+L+LUU+L+Lacac = 18.7 = 18.7HH• RRw0w0 = 80m = 80m • FFRR = 45.6 = 45.6• RRww = 3.63 = 3.63

PrimaryPrimary winding winding portion: portion:

• LLUU = 3.28nH = 3.28nH• LLw0w0 = 23.6nH = 23.6nH• FFLL = 0.65 = 0.65• LLww = 15.3nH = 15.3nH• LLdd = L = LUU+L+Lacac = 18.6nH = 18.6nH• RRw0w0 = 0.56m = 0.56m • FFRR = 11.7 = 11.7• RRww = 6.58m = 6.58m

SecondarySecondary winding winding portion: portion:

Example #2: Interleaved Example #2: Interleaved WindingsWindings




m.m.f.(dc)

0

P1 P2 P3 P4


• LLgg = 2.48 = 2.48HH

• LLUU = 0.36 = 0.36HH

• LLw0w0 = 2.98 = 2.98HH

• FFLL = 0.28 = 0.28

• LLww = 0.85 = 0.85HH

• LLdd = L = Lgg+L+LUU+L+Lacac = 3.69 = 3.69HH

• RRw0w0 = 45m = 45m

• FFRR = 12.3 = 12.3

• RRww = 0.553 = 0.553

PrimaryPrimary winding winding portion portion PP11: :

SecondarySecondary winding winding portion portion PP22::



• LLgg = 1.45 = 1.45HH

• LLUU = 0.22 = 0.22HH

• LLw0w0 = 1.74 = 1.74HH

• FFLL = 0.28 = 0.28

• LLww = 0.50 = 0.50HH

• LLdd = L = Lgg+L+LUU+L+Lacac = 2.16 = 2.16HH

• RRw0w0 = 26m = 26m

• FFRR = 12.3 = 12.3

• RRww = 0.324 = 0.324

PrimaryPrimary winding winding portion portion PP44::

SecondarySecondary winding winding portion portion PP33::



• LLgg = 3.93 = 3.93HH

• LLUU = 0.58 = 0.58HH

• LLw0w0 = 4.72 = 4.72HH

• LLww = 1.35 = 1.35HH

• LLdd = 5.85 = 5.85HH

• RRw0w0 = 71m = 71m

• RRww = 0.877 = 0.877

PrimaryPrimary winding: winding: Secondary Secondary winding: winding:

• LLUU = 0.46nH = 0.46nH• LLw0w0 = 8.14nH = 8.14nH• LLww = 5.51nH = 5.51nH• LLdd = 5.95nH = 5.95nH• RRw0w0 = 0.78m = 0.78m • RRww = 2.73m = 2.73m

Comparison between Simple and Comparison between Simple and Interleaved Windings Interleaved Windings

Interleaved windingInterleaved winding

• LLgg = 3.93 = 3.93HH

• LLUU = 0.58 = 0.58HH

• LLw0w0 = 4.72 = 4.72HH

• LLww = 1.35 = 1.35HH

• LLdd = 5.85 = 5.85HH

• RRw0w0 = 71m = 71m

• RRww = 0.877 = 0.877

Simple windingSimple winding

• LLgg = 8.74 = 8.74HH

• LLUU = 4.48 = 4.48HH

• LLw0w0 = 21.01 = 21.01HH

• LLww = 5.49 = 5.49HH

• LLdd = 18.7 = 18.7HH

• RRw0w0 = 80m = 80m

• RRww = 3.63 = 3.63

PrimaryPrimary winding: winding:

Comparison between Simple and Comparison between Simple and Interleaved WindingsInterleaved Windings

SecondarySecondary winding: winding:

Interleaved windingInterleaved winding• LLUU = 0.46nH = 0.46nH• LLw0w0 = 8.14nH = 8.14nH• LLww = 5.51nH = 5.51nH• LLdd = 5.95nH = 5.95nH• RRw0w0 = 0.78m = 0.78m • RRww = 2.73m = 2.73m

Simple windingSimple winding• LLUU = 3.28nH = 3.28nH• LLw0w0 = 23.6nH = 23.6nH• LLww = 15.3nH = 15.3nH• LLdd = 18.6nH = 18.6nH• RRw0w0 = 0.56m = 0.56m • RRww = 6.58m = 6.58m

leakage impedance of transformer windings

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winding layer

single winding

winding portions

secondary winding portion

primary winding portion

leakage flux distribution

themlf current distribution

current concentrates