abstract - psma...abstract 2 simple equations of various loss components in inductive components...

A simple analytical method to calculate air gap induced eddy current losses in inductive components

Baumann, Grübl, Malcolm

Sumida-Europe Components, Germany

Abstract

2

Simple equations of various loss components in inductive components such as ohmic losses, core losses, proximity losses and skin effect losses are well known and are presented by literature. To calculate air gap induced losses in the winding, time-consuming computer simulation programs with the numerical finite element analysis are commonly used. On the other hand, the existing accurate analytical calculation methods of several authors are mostly not easy to handle for the user and have to be implemented in a computer design tool which doesn´t allow an understanding of what is really going on. We are proposing a simple analytical method which enables us to estimate the so-called gap losses which are proportional to the proximity losses. The model is based on a simple geometrical consideration which also takes into account the number of distributed air gaps. The derivated equations work in a closed form. First, the eddy current losses according to Butterworth are explained. Then the general stray field influence is modeled by using the fringing factor as a stray field cross section equivalent. Afterwards, the model is expanded to a location-dependent stray field which considers the influence of the distance of the winding to the air gap. The evaluation of the equations is done with the finite element analysis by using several examples. The comparison shows a satisfying correlation. In the end, the algorithm is simplified to a rule of thumb which enables first approximation of gap losses by hand. The rule can be used for a quick and simple double-check of the inductor layout, for double-checking computer calculation, or it can be used as a design rule.

Confidential © SUMIDA. All rights reserved.

In today‘s presentation

3

1. Current Situation 2. Fringing factor 3. Eddy current losses according to Butterworth 4. Model for gap losses 5. Determine gap losses with the help of proximity losses 6. Simplification 7. Evaluation via simulation 8. Limits and weakness 9. Rule of thump


Current Situation

4

• To calculate air gap induced losses in the winding, time-consuming numerical computer

simulation programs with the numerical finite element analysis are commonly used [1]

1. http://www.femm.info/wiki/HomePage 2. Albach M, Roßmanith H (2001 The influence of air gap size and winding position on the proximity losses in high frequency transformers. PESC, Vancouver, Canada, S 1485-1490 3. Jensen RA, Sullivan CR (2003) Optimal core dimensional ratios for minimizing winding loss in high-frequency gapped-inductor windings. APEC, Bd 2, S 1164–1169 4. Dowell PL (1966) Effects of eddy currents in transformer windings. PROC IEE, Bd 113, Nr 8, S 1387-1394 5. Albach M (2017) Induktivitäten in der Leistungselektronik. Springer Vieweg

)cos()cosh()sin()sinh(2)(

xxxxxxD

+−

⋅=

nhb

PP

w

w

prox

gap

43

≈

[5]

• A simple analytical equation which enables first approximation of gap losses by hand is to our knowledge still missing

• On the other hand existing accurate analytical calculation methods are not easy to handle for the user or have to be implemented in a computer design tool [2, 3]

• Simple equations of various loss components in inductive components such as Ohmic losses, core losses, proximity losses and skin effect losses are well known [4]


Definition of the fringing factor

5

The fringing factor F describes the increase of the effective magnetic cross section in the area of the air gap [1]:

L

et A

Aswith ⋅≈ 0: µ

ep AA /

Ae: effective cross section area AL: normalized inductivity Vp: fringing volume Vgap: volume of air gap

We have the integral information about the stray field volume Vw:

( )1−= FVV gapw

The fringing volume Vw is the key for the following model

1. Snelling EC (1988) Soft Ferrites – Properties and Applications.. Butterworths, second edition

gap

p

e

p

t

p

VV

AA

ss

F ===

wV

gapp VV /

tp ss /

pepw sAAV )( −=


Calculation of the fringing factor

6

Following is a simple equation based on ds=sp/2 from Mohan [1]:

Expanding to rectangular cross section with Uk as circumference of Ae [3] :

1. N. Mohan, T. M. Undeland, and W. P. Robbins -“Power Electronics –Converter, Applications, and Design”, John Wiley & Sons, Inc., 2003 2. McLyman WT (2004) Transformer and Inductor Design Handbook. Marcel Dekker, New York 3. Albach M (2017) Induktivitäten in der Leistungselektronik. Springer Vieweg

bw: length of winding space n: number of distributed air gaps

Ans

hbnshnsb

F p

kk

pkpk /5,21

)/)(/(+≈

++=

McLyman gives the most popular correlation with sp as air gap and Ae as magnetic cross section [2]:

bk: with of center leg hk: depth of center leg

1

7,0

3,07,0

3,01/

2ln2/

1−

−≈

+=

e

kwt

p

w

e

kp

AnUbs

nsb

AnUs

F

Ans

Anbsns

bA

nsF

t

e

wtp

w

e

p

5,21

1

2,11

1/

2ln/

1

7,0

3,07,0−

≈−

≈+= ps

Ae pA


ts

Eddy current losses according to Butterworth

7

The electron compensation due to the conductivity in longitudinal direction results in power losses [1,2] :

For a sinus similar stray field a voltage Ueff is induced [1,2]:

By integration dx we get the power losses [5,6]:

cu

d

cuprox

lbdBdxxlbBPρ

ωρ

ω24

ˆˆ 3222/

0

222

== ∫

The well-known equation for a round wire results with b=3π /16 ∙d. The flux density B² is perpendicular to the axis in longitudinal direction [1,2]:

cuprox

ldBPρ

πω128

ˆ 422

=

tB ωsinˆ

Flux density due to stray field of neighbored windings or air gap

b

l

AR I

AB

dx d

dtdNU ind

φ−=Excess of electrons

x

1. Snelling EC (1988) Soft Ferrites – Properties and Applications.. Butterworths, second edition 2. Butterworths, s.: 'Eddy-current losses in cylindrical conductors with special application to the alternating current resistance of short coils', Phil. Trans., [A], 1921, 222, p. 57

bdxlRwithdxlbxB

RU

dP cu

cu

effprox

ρρ

ω 2:ˆ 2222

===

22ˆ

2

ˆ xlBABNUdtdNU B

effindωωφ

==→−=


Proximity losses proportional to the winding volume

8

Fwcu

gprox ABfdl

pP ⋅⋅⋅⋅⋅= 222

0

2ˆ

8 ρπ

For a round wire diameter d0 we get the proximity losses specified as a function of B²AF:

π2

4:d

lAplwith wFG ⋅⋅⋅=

Fcu

gprox VBfd

pP ⋅⋅⋅⋅= 222

0

2ˆ

8 ρπ

cuprox

ldBPρ

πω128

ˆ 422

=

Substitution of the total length l of wire in the proximity losses with the help of the filling factor pG=Acu/AF :

Adapted to the volume of winding space VF we get the losses proportional to B²VF. This equation is only valid for a small resistance factor Fr<2 :


Induced losses in copper wire by the fringing field

Additional proximity loses due to the air gap

9

Additional proximity losses results from fringing flux density B(x,y)gap of the air gap:

( ) Fgapwirecu

gprox VyxByxBfd

pP

2220

2

),(),(8

+⋅=

ρπ

distribution of proximity losses Pprox=1,3mW; constant current [1]

1. Albach M (2017) Induktivitäten in der Leistungselektronik. Springer Vieweg

a) The classical proximity losses are high in outer winding space due to missing field compensation due to missing proximity of a wire

distribution of proximity losses Pprox=9,5mW including a air gap [1]

After solving the bracket we get an expression containing the scalar product:

b) The additional losses due to the air gap are high in the center of the component

Location of the wire field and gap field:

( ) Fgapwiregapwirecu

gprox VyxByxByxByxBfd

pP ⋅++⋅= α

ρπ cos),(ˆ),(ˆ2),(ˆ),(ˆ8

22220

2

α

2xB

2yB


Definition of the gap losses

10

( ) Fgapwirewcu

gprox VBBfdl

pP ⋅+⋅≈ 2222

0

2ˆˆ

8 ρπ

0),(ˆ),(ˆ: >>− gapwire yxByxBwith

As a result classical proximity losses not appear in the same position as the additional proximity losses due to air gap. This results that the scalar product can be neglected.

We get two field components which are independent from each other. Therefore the gap losses can be defined in first approximation as a separate loss component:

( )Fgapwcu

ggap VBfdl

pP 222

0

2ˆ

8 ρπ

≈


Location and direction of the wire field and gap field

Model for the gap losses

Generating a type of function for B(V) with the requirement B(V)V is constant according to a constant magnetic moment from a magnetic dipole [1].

Now we insert the initial condition B(0)=Bw and we get:

( )VV

VBVBw

ww

+=

ˆˆ

( )VB̂

wB̂

V

( ) wwVBVVB ˆˆ =( ) constVVB =⋅ˆ

In order to get simple equations, the basic idea is to solve in a bulk and not wire by wire. Now we take into account that the local B is not constant within the winding volume V.

( ) VVBfdp

Pcu

ggap

2220

2ˆ

8⋅⋅⋅=

ρπ

Fringing flux in the winding volume

Fringing flux volume equivalent

Das Bild kann zurzeit nicht angezeigt werden.

11 Confidential © SUMIDA. All rights reserved.

FB

AABBwith core

p

ecorew ==:

1. Kohlrausch (1996) Praktische Physik, Band2. B.G. Teubner Stuttgart. 24. Auflage

Fringing flux in dependence of winding volume

Model for the gap losses

12

Only regarding the volume of windings which are lying in the field B(V) we get via integration the product B²(V)V:

wFw

w

w

w

VVVVV

VVVk

σσ +≈

+

−+

=1

1 Vσ : volume of air gap upholstery VF: effective winding space σw: distance to the air gap Uk : circumference of Ae

We get a winding trans flux number 0<k<1:

( ) ( )

+

−+

⋅=

+==⋅ ∫∫

Fw

w

w

www

V

V w

wwV

V VVV

VVVVBdV

VVVBdVVBVVB

FF

σσσ

2

2

22 ˆˆˆˆ

( )1:

−=

≈

FVVUsVwith

gapw

kpwσσ

wσ

wV σV

σV

( )VB2ˆ

2ˆwB

FV


Now we get a simple equation for the gap losses including the volume of the air gap Vgap:

( ) kF

FVBfdp

P gapcorecu

ggap ⋅

−⋅⋅= 2

2220

2 1ˆ8 ρ

π

(only valid for low resistance factor Fr<2)

FBBwith core

w =:

Determine gap losses with the help of proximity losses

13

The middle flux density in the winding space is according [1] : As a result we get the B-Field in the winding space to:

;ˆˆ:

L

ecore

NAABIwith =

The fringing flux density Bw is the corresponding flux density equal to the cross section of fringing air gap Ap:

Simplification via setting both B-equations into the rescaled relation above: p

ecorew A

ABBˆˆ =

2

02ˆ

31ˆ

=

Lw

ecorewire Ab

ABB µ

22 2ˆ: BBwith =

The gap losses Pgap can be determined with the proximity losses Pprox by dividing both equations. This rescaling is useful because the accurarity of the Pgap improves according to the accuratiry of the Pprox.

1. Snelling EC (1988) Soft Ferrites – Properties and Applications.. Butterworths, second edition

kVAµbAV

PP

Fp

wLw

prox

gap22

0

223=

2

02

2

ˆ

31

=

wbINB µ

kVB

VBVBVB

PP

Fwire

ww

Fwire

Fgap

prox

gap2

2

2

2

ˆˆ

ˆˆ

== kF

FV

VBB

F

gapcoregap 2

22 )1(ˆ

ˆ −=

fringing flux volume equivalent

wwVB̂

the increase of effective cross section due to fringing

Ae pA


Simplification with substitution of some parameters

14

The effective air gap area is proportional to the fringing factor:

kVAµVbA

PP

Fp

wwL

prox

gap22

0

223=Regarding the volume ratio we get:

Fringing field volume Vw can be described by the effective air gap area Ap and the effective air gap sp : ( ) pepw sAAV −=

ep FAA =

The effective air gap sp can be described by fringing: tp Fss =

The air gap st without fringing is:

ksA

µA

Vb

PP

orkF

FVµ

bAPP

p

eL

F

w

prox

gap

F

wL

prox

gap

−=

−

=0

2

0

2 313

1

5,21−

−=

AnsF t

This results in:

With modificated fringing from McLyman we get the simplification:

Let AAs /0 ⋅= µ

11

)1(11

−−

−

⋅+=

+=

FAU

VVk

e

kw

w

σσ

klnh

AbPP

ww

ew

prox

gap 5,7≈

The winding trans flux number:


Simplification based on the area ratio via Mohan

15

Regarding the area relation we get:

Including the number of distributed air gap n we get : n

sds p

2=

The stray field window Aw with the extension ds and the length of gap sp is: pw sdsA ⋅=

The actual cross section of fringing air gap Ap: ep FAA =

The actual total air gap sp can be describe by the fringing factor: tp Fss =

This results in: knh

bPP

w

w

prox

gap

23

≈

The air gap st without fringing is: L

et A

As ⋅= 0µ

sp

Ap

AσAw

The stray field extension ds is according [1]: 2ps

ds =

pw sn

AAkwith σσ 21

1

1

1:+

=+

=

1. N. Mohan, T. M. Undeland, and W. P. Robbins -“Power Electronics –Converter, Applications, and Design”, John Wiley & Sons, Inc., 2003

kAAµbAA

PP

kVAµVbA

PP

Fp

wLw

prox

gap

Fp

wwL

prox

gap22

0

22

220

22 33→=


σwn=1

µi

bw

Ae

Af

hw

Simulation tools for evaluation at different conditions [1;2]

16

Different air gaps Different distance to air gap Different frequency

Number of air gaps Different depth Different window shapes 1. http://www.femm.info/wiki/HomePage 2. Martin Grübl, Sumdia Components & Modules GmbH Confidential © SUMIDA. All rights reserved.

2D simulation tools for evaluation wire losses at different air gaps [1;2]

17

Gap losses are in first approximation independent from the distance of the air gap (step function)

1. http://www.femm.info/wiki/HomePage 2. Martin Grübl, Sumdia Components & Modules GmbH

Component: Wire: Air gap: Curent: L-value:

JMAG: 0,0743 W Model: 0,0623 W FEMM : 0,0122 W


E36 58Wdg, 1,25mm

1 mm 0,2 A, 100kHz

0,556 mH

E36 58Wdg, 1,25mm

2 mm 0,2 A, 100kHz

0,296 mH

E36 58Wdg, 1,25mm

4 mm 0,2 A, 100kHz

0.157 mH


JMAG: 0,2996 W Model : 0,3091 W FEMM : 0,0936 W

E36 58Wdg, 1,25mm

0 mm 0,2 A, 100kHz

19,35mH


2D simulation for evaluation wire losses at different number of gaps [1;2]

18

Gap losses are inversely proportional to number of air gaps: Pgap~1/n


Component: Wire: Air gap: Curent: L-value:


E36 58Wdg, 1,25mm

1x4 mm 0,2 A, 100kHz

0.157 mH

JMAG: 0,0757 W Model: 0,1168 W FEMM: 0,0211 W

E36 58Wdg, 1,25mm

4x1 mm 0,2 A, 100kHz

0.139 mH


E36 58Wdg, 1,25mm

2x2 mm 0,2 A, 100kHz

0.147 mH


2D simulation tools for evaluation at different distance to gap [1;2]

19

This increasing distance the losses decrease with: Pgap~1/(1+2nσ/sp)


Component: Wire: Distance to gap: Air gap: Current: L-value:


E36 58Wdg, 1,25mm

2mm 1 mm

0,2 A, 100kHz 0,556 mH

E36 58Wdg, 1,25mm

1mm 1 mm

0,2 A, 100kHz 0,556 mH



2D simulation result of FEMM [1]

20 1. http://www.femm.info/wiki/HomePage 2. Dowell PL (1966) Effects of eddy currents in transformer windings. PROC IEE, Bd 113, Nr 8, S 1387-1394

y = 1,1948xR² = 0,4237

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1

Mod

el (P

tota

l/W)

Simulation (Ptotal/W)

The proximity losses are calculated via Dowell [2]

y = 1,3188xR² = 0,507

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

1 1,05 1,1 1,15 1,2 1,25 1,3

F (M

cLym

an)

F (Simulation)

Worse correlation because 2D fringing does not correlate with reality (F-Lyman ≠ F-Simulation)


y = 1,0856xR² = 0,9467

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1

Mod

el (P

tota

l/W)


2D simulation of “JMAG transformer module” with 3D correction [1]

21

y = 1,0661xR² = 0,904

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8

F (M

cLym

an)

F (Simulation)

Better correlation because 3D correction improves fringing calculation (F-Lyman ≈ F-Simulation)

1. Manfred Wohlstreicher, Sumdia Components & Modules GmbH 2. Dowell PL (1966) Effects of eddy currents in transformer windings. PROC IEE, Bd 113, Nr 8, S 1387-1394

kF

FVµ

bAPP

F

wL

prox

gap

−

=13

0

2

−

⋅+=

)1(1/1

FAUk

e

kwσ


The proximity losses are calculated via Dowell [2]

Limits and weakness

22

• The overlay of wire field with the air gap fringing field is neglected • The different directions of the air gap field are not taken into account • The model was only tested on round wires


Rule of thumb

23

knh

bPP

w

w

prox

gap

23

≈

Pgap: gap losses Pgap: gap losses bw: length of winding bulk hw: height of winding bulk n: number of air gaps sp: distance of air gap σw: distance to the air gap

5,0/21

1: ≈+

=pw sn

kwithσ

y = 1,1046xR² = 0,8636

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1

Mod

el (P

tota

l/W)



σwn=1

µi

bw

Ae

Af

hw

Thank you for your attention!

abstract - psma...abstract 2 simple equations of various loss components in inductive components...

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