prof. s. ben-yaakov , dc-dc converters [3- 1] magnetics designdcdc/slides/dc-dc part 3_double...

1

Prof. S. Ben-Yaakov , DC-DC Converters [3- 1]

Magnetics Design

3.1 Important magnetic equations3.2 Magnetic losses3.3 Transformer

3.3.1 Ideal transformer (voltages and currents)3.3.2 Equivalent circuit of transformer

(coupling, magnetization current)3.3.3 Design of transformer

3.4 Inductor design


;dtdBnA

dtdnV e=Φ=

HB

∆∆=µ

Φ - magnetic flux Weber [ Wb ]

B - flux density ]T[TeslamWb

2 =

V - voltage [ V ]

Also : Gauss [ G ] 1T = 10,000 G

B

H

BDC

HDC

ΦAe

Faraday’s law

2


Ampere’s law

H - magnetic field [ A/m ]

∫ ⋅= InHdl

eHIn l⋅=⋅

e

InHl

⋅= [ A/m ]

In

el


Magnetic losses ~∆B

“Good number” = 100mW/cm3 =100KW/m3

P3cmmW

B∆

Magnetic losses

B

H

BDC

HDC

3


“Good number”=100mW/cm3=100 kW/m3

Magnetic Losses


Magnetic losses ~∆B

“Good number” = 100mW/cm3 =100KW/m3

P3cmmW

B∆

Magnetic losses

B

H

BDC

HDC

4


Magnetic Losses


Curves for constant loss: 500mW/cm3

Figure of merit B*fEach material has optimum operating temperature (minimum loss)

Magnetic Losses

5


Transformer currents

n1 n2

I1I2

For ideal transformer 2211 InIn =1

2

2

1nn

II=

At any given moment 2211 InIn =

I1,I2 opposite direction.

No magnetic energy stored due to useful currents I1, I2 (they cancel each other)

n1 n2

I1 I2


Transformer voltages

φ

dtdnV 1

11φ=

dtdnV 2

22φ=

21gminAssu φ=φ

dtd

dtd 21 φ=φ

2

1

2

1nn

VV

=

n1 n2V1 V2

6


Since each winding also represents an inductance, therefore for any winding 0=nVPermissible voltages: AC only on any winding

S1

V

S1

S2 S2

S1

V

S1

V

S2S2

S1S2

S1S2

t

t

t

A

B

C

Voltages


Equivalent circuit (preliminary)

2mm nLL

12=

∞→LIdeal transformer

Ideal

1:n

Lm1

Llkg1

Ideal

1:n

Lm2

Llkg1

Ideal

1:n

Lm1

Llkg2

7


Leakage inductance

I2V2

I1V1 n2

n1

Leakage inductance is theuncoupled magnetic flux

1:nLkg1

Lm1

ideal

Lkg2

Relationship between Llkg, M and k (coupling coefficient).

21 LLkM ⋅=)k1(LL 1m1lkg −≅

21lkg2lkg nLL ⋅≅

Leakage


1:nLlkg1

Lm1

ideal

Llkg2

Vo

Llkg1

Lm1

L'lkg2

V'o 22lkg

2lkg nL

L =′

2o

o nVV =′

Leakage

8


Magnetization Current

Vint

Vo

t

I2t

Imt

I1t

Ideal

1:n

Lm1

Llkg2

RI2

Vo

I1

ImVin


Transformer

1. Bmax ( could be symmetrical or asymmetrical )2. Bmax < Bsat

3. In most case ( high frequency ) Bmax limit by magnetic losses.

dtdBAn

dtdnV e111 =Φ

=

I2V2

I1V1 n2

n1

H

B

B sat-

Bsat+

B max+

∆ΒBmax-

9


Symmetrical operation

∫= VdtAn1B

e1

−+ = maxmax BB

e1

onmmax An

tVB2B ==∆

{ }emax

onm1 AB2

t,Vn =

B

Vm

Ts

Bmax-

Bmax+

ton

V1

s2Ttone1 f

1~t~An Son→


Skin effect

depthskin−δ

f72)mm( =δ

Hzinf

δ 1RR

DC

AC >

DC FrequencyHigh

10


Skin Effect Solutions

Litz wire

Tape


Proximity effect

I

I

Current crowding due to magnetic fields

11


[ ]k

nwnwA 22A1A

w1

⋅+⋅=

k - filling factor k<1

JI

w rms1A1=

J - current density A/m2

J ≅ 4.5 A/mm2

12

12 I

nnI =

Aw - winding area

AwwA

Aw


2JkIn

A rms11w ⋅=

rms1

w1 I2

JkAn⋅

=

{ }emax

on11 AB2

t,Vn =

{ }emax

on1

rms1

w

AB2t,V

2IJkA

=⋅

{ }{ }JkB2

I2t,VAAAmax

rms1on1ewp

⋅==

{ }JkB

I2t,VA rms1on1

p ⋅∆⋅

=

{ }JkBfI2D,V

As

1on1p

rms

⋅∆⋅⋅

=

Ap

12


Transformer design stages

1. Calculate Ap

2. Look for core3. Calculate n1 by:4. Calculate n2

{ }JkBfI2D,V

As

1on1p

rms

⋅∆⋅⋅

=

In symmetrical operation∆B = Bmax

+ - Bmax-

In asymmetrical operation∆B = Bmax - 0

{ }emax

onm1 AB2

t,Vn =


Inductor design

Need to store energy( in transformer n1·I1= n2·I2 )

roµµ=µµo - air (vacuum) permeabilityµr - relative permeability

I

L

B

Hµ

13


µo = 1.26·10-6m

Henry

If µ is high B will reach quickly Bsat

Need to slower µ

B

HHo

<2rµ

2rµ

1rµ

1rµBo

µr of ferrites ∼ 2000 - 4000

B = µH

Permeability


Same Φ magnetic lines in ferromagnetic material and in air.

eg ll <<

eeg lll ≅+

Discreteair gap

Φµo

µr

el

gl

Distributedair gap

Gaps

14


Current crowding due to magnetic fields

RAC high around gap

Current Crowding


Φ = constant B ≅ constant

og

BHµ

=m

mBHµ

=

ggeme HHHnI lll +==ogB

meB

eHµ

+µ

=ll

l

eg ll <<

eeg lll ≅+

el

gl

Inductance with Gap

15


g

eo

a

m

m

e

BBBH

l

lµ

+µ

=µ

=

o

g

m

ee

BBHµ

+µ

=ll

l

Dividing out le and defining HB

e =µ

Inductance with Gap


µ

+µ

=µ

g

eo

me

111

l

l

µ

+µµ

=µµ

g

eo

ormore

111

l

l

+

µ=

µ

g

ermre

111

l

l

µ

µ+

=µ

g

erm

rmg

e

re

1

l

l

l

l

rmg

e

g

erm

re

µ+

µ

=µ

l

l

l

l

rmg

eIf µ<l

l

≈µ

g

ere

l

l

Gap Calculation

16


dtdILV =

dtdnV Φ=

dtdInnA

dtdHnA

dtdBnA

dtdn

eeeel

µ=µ==Φ

dtdn

dtdIL Φ

=L-?

dtdIAn

dtdIL

e

e2

l

µ=

e

e2AnLl

µ=

Inductance


2

2

1

2

1nn

LL

=

Inductor design

B

H

Bmax

Two windings on same core

17


dtdn

dtdIL Φ

=

∫∫

=

maxpk B

0e

I

0dt

dtdBnAdt

dtdIL

L Ipk = nAeBmax

maxe

pk

BALI

n =

max

pke nB

LIA =

rms

wI

JkAn =

quick design and check

Saturation Limits


JkBILI

AAAmax

rmspkwep ==

2rmspk LIILI ≈

2LIstoredEnergy

2=

Air gapped core Design1. Calculate Ap

2. Choose a core3. Iterate4. Calculate ( or increase gap until L is as required ) gl

Ap

18


Cores

Transformer core Inductor core

air gap


1. E - core

2. TOROID

3. ARENCO 4. POT

Cores

19


Commercial cores


Distributed gap core

turnH

A yL = )

turns1000H

sometime( y

L for n turns: L2 AnL ⋅=

The concept of AL

Distributedair gap

20


AL


Toroid Data

21


1 Amp/m =79.5 Oe

L decreases with DC current !

Permeability change


These curves are measured by feeding ac signals.If the current is composed of DC + ripple, core loss is due only to ripple component !

DC bias tend to increase loss

“Good number”=100mW/cm3

Misleading notations !∆B NOT B

Losses

22


“Hot Spot” - Critical parameter

Temp. Rize


Hanna Curve

orHmaxB

eV

2LImaxHB

maxB1

eV

2LIH

µµ=µ=

=

=

maxBeAelLInI

Hn

maxBeApkLIH

Hn

maxBeApkLI

n

=

=

=

23


Hanna Curve


Core Size Selection

24


Basic Design of Distributed Gap Core

)nI(fel

=µ

1. Calculate LI2

2. Look up manufacturer data 3. Select Core

4. Calculate

5. Check Lmin

6. Calculate losses. Temp rise and and

7. Iterate

)(LALn1000

1000=

prof. s. ben-yaakov , dc-dc converters [3- 1] magnetics designdcdc/slides/dc-dc part 3_double...

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