PSMA...pi Select A pi +1 Calculate Turns Calculate J o Select Wires Calculate Copper Loss Calculate Core Loss A c W a MLT m Select A p Calculate High Frequency Losses B max ≤ B sat

PSMA

1Power Electronics Research Centre, NUI Galway

High Frequency Effects in the Core

Losses in Magnetic Components


Copper losses

Core losses

Hysteresis loss

Eddy current loss

Skin effect loss

Proximity effect loss

Ferromagnetic Materials


(a) Hard magnetic materials (b) Soft magnetic materials

B

Br

HHc-Hc

-Br

(a)

B

H

(b)

Br

Hc-Hc

-Br

Core Loss


Hysteresis loss in a ferromagnetic material Eddy current loss in a ferromagnetic material

dB

0

a

H

B

bc

die ie/n

t

ˆfe cP K f Bα β=

Hysteresis loss is the area inside the B-H loop Eddy current loss is reduced by laminations in steel Eddy current loss is reduced by higher resistivity in ferrites

Ferromagnetic Materials


Soft magnetic materials are classified as:

Ferrites

Laminated iron alloys

Powered iron

Amorphous alloys

Nanocrystalline materials

Core Shapes


Toroid core PQ core Pot core RS/DS core RM core

EP core

EE core EI core ER core EFD core ETD core

U coreUR core C core Planar core

Core Loss Density vs Frequency


B=0.1 T

Performance Factor


Fringing (Flux)


Gap in the centre leg Gap in the outer leg


Transformer Design

Basic Equations


Kv =4.44 for a sinewave=4.00 for a squarewave

Voltage equation

Power equation

rmsˆ v cV K f NBA=

Vrms: the rms value of the applied voltageKv: the voltage waveform factorf: the frequency of the applied voltageJo: the current density in each winding

: the maximum flux density in the coreIi: the current in winding iNi: the number of turns in winding iAwi: the conductor area in winding iku: the window utilisation factor

ˆVA v i i c

i o wi

K fB N I AI J A

= ⋅ ⋅∑ ∑=

Window utilisation factor

ˆVA v o u a cK fBJ k W A=∑

wi1

n

ii

u

a

N Ak

W=∑

=

secp a cA W A

Windowarea cross tional area= ×

× −

B

ˆv u p oVA K fBk A J=∑

Transformer Losses


Winding losses

Total resistive losses

is window utilization factor

is volume of the windings

Core losses

Typical layout of a transformer

22 wi

cu1

wi

( )ni o

wi

N MLT J AP RIA

ρ=

= =∑ ∑

2

cu w w u oP V k Jρ=

wi1

n

ii

u

a

N Ak

W=∑

=

w aV MLT W= ×

feˆ=Vc cP K f Bα β

Volume ofcore,VC

Cross-sectionalarea, AC

Mean Length of a Turn, MLT

Volume ofwindings,VW

Window area,Wa

2ro

Heat loss by convection

total cu fe= c tP P P h A T+ = ∆

Dimensional Analysis


kw=10, kc =5.6 and kt=40 are typical values

Typical layout of a transformer

3/4 2

cu w w p u oP k A k Jρ=3/4

feˆ=k c p cP A K f Bα β

Volume ofcore,VC

Cross-sectionalarea, AC

Mean Length of a Turn, MLT

Volume ofwindings,VW

Window area,Wa

2ro

3/4w w pV k A=

3/4c c pV k A=

1/2t t pA k A=

(14)

(15)

1/2

total =h c t pP k A T∆ˆ

v u p oVA K fBk A J=∑

Losses Optimization


Winding losses

Total losses

Core losses

At a given operation frequency,

The minimum losses occur when

2

cu 2 2

VAˆ ˆw w u

v f u p

aP V kK fBk k A f B

ρ ∑= =

feˆ ˆ= c cP V K f B bf Bα β α β=

2 2ˆ

ˆaP bf B

f Bα β= +

1

2 3

2 ˆ 0ˆ ˆP a bf BB f B

α β∂ β∂

−= − + =

cu fe2P Pβ

= total cu

2P Pββ+

=

Losses Optimization


Winding, core and total losses at different frequencies

The first step in the design is to establish whether the optimum flux density given by the optimization criterion is greater or less than the saturation flux density.

A

B

C

D

Losses

Flux densityBoptD BoptBBsat

P

Pfe

Pcu

P

Pfe

Pcu

50Hz50kHz

Losses Optimization

Power Electronics Research Centre, NUI Galway 16

( )8 27

7 2 7( )8 7 2

[ ]2( 2) [ ][ ] VA

t v uo o o

w w c c

hk T K kf B fk k K

β α β ββ ρ

− − ∆

= + ∑

8/74/7 VA2 1ˆ

w wp

t u v o

kAhk k T K fBρ β

β +

= ∆

∑

1/41

2t

ow u p

hk TJk k A

ββ ρ

∆=

+

3/4 2

cu w w p u oP k A k Jρ=3/4

feˆ=k c p cP A K f Bα β

1/2

total =h c t pP k A T∆

ˆv u p oVA K fBk A J=∑

Core size

Current density

Optimum flux density

total cu

2P Pββ+

=

total fe

22

P Pβ +=

Core Losses Correction


ˆ( ) 1cr

V ccr

fP K fB ff

αβ α β−

= +

feˆ=Vc cP K f Bα β

Core size v Frequency


27 127

47 2 8( )22 2 2

7 22( 2) ( 2)7

VA( ) ( 2)( )

2

c w w cp

t v u

k k KA fhk K k T

ββ

βα βββ ββ

β β β

ρ β

β+

−−+−

+ +

+ = ∆

∑

Optimum Core Size


27 127

48

7 2 8( ) 7 222 2 27 2

2( 2) ( 2)7

VA( ) ( 2) 1( )

2

cr

c w w cp o

t crv u

k k K fA fhk fK k T

ββ

β αα β βββ ββ

β β β

ρ β

β+

−− −+−

+ +

+ = + ∆

∑

Design Methodology


Specifications : ∑VA,K,f,ku,ΔT

Select Material : Bsat,ρc,Kc,α,β

Calculate Bo

AcWaMLTm

Bo < Bsat

Yes No

Calculate Ap

Select Ap

Calculate Api

Select Api+1

Calculate Turns

Calculate Jo

Select Wires

Calculate Copper Loss

Calculate Core Loss

AcWa

MLTm

Calculate High Frequency LossesSelect Ap

Bmax ≤ Bsat

Calculate Efficiency, η


Push-pull Converter TransformerCircuit Waveforms

+

_

vp2

is1

+

_

vs1

Lo

Co

+

_

Vo

is2

+

_

vs2

+

_

vp1

D1

D2

Np : Ns

Vs

+

_

S2 S1

t0

Vp,Vs

t0

ΦVs

Io

ip1 ip2

t0

Io

2oI

t0

Io

2oI

is1

is2

TDT’ T’

τ


Push-pull Converter: Specifications

Input 36 → 72 V

Output 24 V, 300 W

Frequency, f 50 kHz

Temperature Rise, ΔT 35 ºC

Ambient Temperature, Ta 45 ºC

Kc 9.12

α 1.24

β 2.0

Bsat 0.4 T

Design specifications Core data: EPCOS N67 Mn-Zn

fe c mP K f Bα β=

Core loss

Push-pull Converter: Voltage factor


Calculations:

(1) Voltage waveform factor KvVp,Vs

t0

ΦVs

τTDT’ T’

Push-pull converter voltage and flux waveforms

4 4.88vKD

= =

24 0.6736

D = =

max maxs max'

4 4V = = = = / 2p p c p c p c p c

B Bd dBN N A N A N A fN A Bdt dt DT DT Dφ

=

rms max max4V = = K s p c v p cDV fN B A fN B AD

=

Push-pull Converter: Power factor


Calculations:

(2) Power factor kpp, kps

( )rms rms; 12

o os s s

V IV DV I DD

= = = +

'

0

1( ) ( ) '2

DT

s s s s

Dp v t i t dt V I DT V IT

< >= = =∫

rms rms rms rms

1 ; 12pp ps

p p s s

p p Dk kV I V I D< > < >

= = = =+

rms rms

1 1 12 2

os s o o

D D PV I V ID D+ +

= =

rms rms; ( / 2) ; p s p sV DV I D I= =

t0

Vp,Vs

t0

ΦVs

Io

ip1 ip2

t0

Io

2oI

t0

Io

2oI

is1

is2

TDT’ T’

τ


Push-pull Converter: Core sizeCalculations:

(4) Optimum Ap

The optimum flux density is less than Bsat

(3) VA ratings of the windings

1 1 122 2 2 2

1 0.672 (300) 898.6 VA0.67

o o o oo

pp ps

P P P P DVA Pk k D

+ = + + + = +∑

+= + =

[ ][ ]

18 7 2.07 2 (7 1.24 2)7 2

7 2.0 278 28

(10)(40)(35)2 2.0 4.899 (0.4)ˆ 50000(2.0 2) 898.6[(1.72 10 )(10)] (5.6)(9.12)

0.127T

oB× − − × −

× −−

× = • + × =

4/7 8/784(1.72 10 )(10) 2.0 2 1 898.6 2.54cm

(10)(40) 2.0 0.4 35 (4.899)(50000)(0.127)pA− × +

= = ×

ETD44 Core Data

Power Electronics Research Centre, NUI Galway 26

Ac 1.73 cm2

Wa 2.10 cm2

Ap 3.63 cm4

Vc 17.70 cm3

kf 1.0

ku 0.4

MLT 7.77 cm

ρ20 1.72 µΩ-cm

α20 0.00393

Fringing (Flux)


Gap in the centre leg Gap in the outer leg

Fringing (Different Frequencies)


Frequency 1kHz Frequency 100kHz

Width of conductor: 0.2mm Core: Magnetics® port core

Magnetic Field Intensity




Magnetic Flux




Current Density


High Frequency Effects in the Windings

High Frequency Effects


High frequency

effects

Windings optimization

Skin effect

Proximity effect

Core optimization Eddy current

Windings arrangement

Thickness optimization

Design Issues for High Frequency


High frequency winding loss

Core loss: Steinmetz equation, iGSE.

Parasitic parameters: leakage inductance, stray capacitance

Proximity effect

I I I I

H0

Primary SecondaryH1

Core Eddy currents

Skin effect

2r

Jz

Eddy current

r

Fringing effect

Gap

Core

Ohm loss

Eddy Current in the Core


Eddy current losses in a toroidal core Equivalent core inductance versus frequency

The inductance terms of the core impendence is

coileddy

current

ac flux

magnetic material with ferrite conductivityand relativepermeability

σ

rμ- +0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

SLL

The inductance of the toroid under the lower frequency 2

00

r c

c

N AL µ µ=

4

0 4

0 3 4

1 2.112 1.43

1 1 1 2.116 16

sL L

L

∆ = − ∆ < + ∆ = + + ∆ > ∆ ∆ ∆

core radiusskin depth

bδ

∆ = =

Eddy Current in the Core


Fig. 7.7 Resistivity of P type ferrite Inductance and rel. permeability v frequency and

Complex permeability 4

4

3 4

' 1 2.112 1.43

1 1 1 2.116 16

rs r

r

µ µ

µ

∆= − ∆ < + ∆ = + + ∆ > ∆ ∆ ∆

core radiusskin depth

bδ

∆ = =

Eddy Current Core Losses


The equivalent core resistance:

The core losses may be reduced by increasing the electrical resistivity or reducing the electrical conductivity of the core material.

The use of a smaller core cross-section to reduce eddy current losses suggests the use of laminations.

The average power loss in the core due to eddy currents is

22 2

2 00 4 2

r cs

c

N l bR L fl

µ µ πω π σ ∆

= =

2 2 2

max

4f B bp π σ π

=

Eddy Current Core Losses


Performance Factor:

Flux Distribution in the Core


Assumptions:Homogeneous coreConstant resistivityConstant permeabilityNo dielectric effects

Axial Flux Distribution in the Core


1. Flux bunches to the surface2. Flux is higher at the surface

Axial Flux Distribution in the Core


Core Losses (GSE, iGSE)


Steinmetz equation:

The time-average power loss with non-sinusoidal excitation using the iGSE

fe maxcP K f Bα β=

0 0

1 ( ) 1 ( )

( )

T T

v i i

i

dB t dB tP k B dt k B dtT dt T dt

dB tk Bdt

α αβ α β α

αβ α

− −

−

= ∆ = ∆∫ ∫

= ∆

21 1

02 cosc

i

Kkdαπβ απ θ θ− −

=∫

where

A useful approximation is

1 1 6.82442 1.10441.354

ci

Kkβ απ

α− −

= + +

Push-pull Converter Transformer


Core data: ETD44

Kc 9.12

α 1.24

β 2.0

Bsat 0.4 T

Core data: EPCOS N67 Mn-Zn

Ac 1.73 cm2

Wa 2.78 cm2

Ap 4.81 cm4

Vc 17.70 cm3

kf 1.0

ku 0.4

MLT 7.77 cm

ρ20 1.72 µΩ-cm

α20 0.00393

Push-pull Converter Transformer


Flux waveform for the push-pull converter

Calculations:

(1) power loss per unit volume

-Bmax

t

v

Bmax

0 DT/2 T/2 (1+D)T/2 T

ΔB

p

/2 (1 ) /2 1

0 /2

1 1 2 ( )/ 2 / 2

DT D T

Tv i i

B BP k B dt dt k B B DTT DT DT T

α αβ α β α α α− −+ − ∆ ∆ = ∆ + ≈ ∆ ∆∫ ∫

1 1

2.0 1 1.24 1

6.82442 1.10441.354

9.12 0.92756.82442 1.1044

1.24 1.354

ci

Kkβ απ

α

π

− −

− −

= + +

= = + +

Compare iGSE and GSE


Calculations:

(2) ΔB

(3) The core loss per unit volume

(5) The total core loss by GSE = 1.466 W

[ ]

1

2.0 1.24 1.24 1 1.24

5 3

1 2 ( )

(0.9275)(0.232) (50000) (2 0.232) (0.67 / (50000))0.871 10 W/m

v iP k B B DTT

β α α α− −

− −

= ∆ ∆

= × ×

= ×

(4) The total core loss

max 4

0.67(36) 0.116 T(4.88)(50000)(6)(1.73 10 )

d

p cv

cDVBK fN A −

= = =×

max2 0.232 TB B∆ = =

6 517.71 10 0.871 10 1.543 W−× × × =

Documents

PSMA...pi Select A pi +1 Calculate Turns Calculate J o Select Wires Calculate Copper Loss Calculate Core Loss A c W a MLT m Select A p Calculate High Frequency Losses B max ≤ B sat