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5. Passive Components

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Inductive Elements

• Components - Inductors- Transformers

• Materials- Laminated alloys for example Silicon-steel (High power, low frequency, high Bsat)- Iron powder (Medium power, low to medium frequency, medium Bsat)- Ferrites (Low power, high frequency, low Bsat)

• Designed by power engineering and power electronics designers!

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Inductive Elements- Core Losses

• Core losses depend both on frequency and peak flux density. - Usually specified in loss curves (one curve for certain frequencies)- Also analytical expressions like Steinmetz’s formula:

22 ˆˆ acecaac

ahFe BfkBfkp += 21

• Steinmetz’s formula includes two loss terms - Hysteresis loss- Eddy current loss

• Empirical expressions are provided by some core manufacturers

( )22

65.13.23

ˆ

ˆˆˆac

acacac

Fe Bdf

Bc

Bb

Ba

fp +++

=

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Inductor Design

Figure 5.1: Basic gappediron core inductor.

Inductance

didL ψ

=

δδ lHlHiN FeFe ⋅+⋅=⋅

⎩⎨⎧

==

δδ µµµHB

HB FeFeFe

0

0

δδµµµlBlBiN Fe

Fe

Fe ⋅+⋅=⋅00

δδψ ABABN FeFe ===Φ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=⋅

δ

δµµ

ψAl

Al

NiN Fe

FeFe0

FeFe AABBB =⇔== δδ

δµ

µψ

llNA

iL

Fe +==

Fe

Fe02

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=⋅ δµµ

ψ llNA

iN Fe

FeFe0

FeµδFell >>

δ

µlNAL

2Fe0=

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Inductor Core Size Selection

Figure 5.1: Basic gappediron core inductor.

mFem BNAiL ˆˆˆ ==ψ

Cu

Cuw kNAA =

CuCuCu JAI =

FewmCu

CuCum AABIJkiL ˆˆ =

FewAAAP=

)()( titi Cum =

Area product

For an inductor with a single winding

CuCu

CuCuJBkIiLAP ˆ

ˆ=

For an inductor with several windings

Cuw

Cu

w

ww kN

ANNAA ==1

CumCu

kCukmkw JBk

IiLNAP ˆ

ˆ ,,=

CumCu

CumJBkIiLAP ˆ

ˆ=

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Transformer Core Size SelectionArea product

For sinusoidal voltage

IVS ⋅=

viRvedtd

Cu ≈⋅−==Ψ

vdtdBAN

dtd

Fe ≈⋅⋅=Ψ

2BANV Fe

)

⋅⋅⋅= ω

RMSCuRMS JAII ⋅==

RMSNRMSRMSRMSw

wNwww

JJJJNAAAA

w

w

====

⇔====

,2,1,

,2,1,

K

K

Cu

Cuww k

ANNA ⋅⋅=

RMSw

wCu JNNAkI ⋅

⋅⋅

=

211 BAPJkNdt

dBAAJkN

S RMSCuw

FewRMSCuw

)

⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅= ω

BJkNSAP

RMSCu

w )⋅⋅⋅

⋅⋅=

ω2

For piecewise constant voltage (forward converter)

Fe

swdcdcFe AN

TDVBBVdtdBAN

⋅⋅⋅

+=⇔≈⋅⋅ 0ˆ

IDVP dc ⋅⋅=

BfAPJkN

TBANJ

NNAkDVJ

NNAkP

swRMSCuw

sw

FeRMS

w

wCudcRMS

w

wCu

ˆ1

ˆ

⋅⋅⋅⋅⋅=

=⋅⋅

⋅⋅⋅⋅

=⋅⋅⋅⋅⋅

=

BfJkNPAP

swRMSCu

wˆ⋅⋅⋅

⋅=

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Core Configurations- Main variants

Figure 5.2: EE core (left) and EI core (right) with windings (grey) and geometrical dimensions.

Figure 5.3: Inductive component based on two C core halves with windings (grey) and geometrical dimensions. Note that two windings are used.

Figure 5.4: Toroid core and its geometrical dimensions. Note that the winding is not included.

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Inductor Design Example (I)

Figure 5.5: Inductor based on a C-core. The winding (grey) is split into two parallel connected windings.

Table 5.4: Inductor specification.

L 0.3 mH

IRMS @ 1 kHz 120 A

IPEAK @ 5 kHz 10 A

IPEAK @ 10 kHz 5 A

A 120≈CuI

A 185A 5102120ˆ =++=Cui

T 35.0ˆ =B

4cm 2379ˆˆ

==CuCu

CuCuJBkIiLAP

The C-core TELMAG Su 150b (Figure 5.5), have geometrical properties according Table 5.5.

a 255.6 mmb 150.2 mmc 49.4 mmd 76.2 mme 154.0 mmg 50.0 mm

Table 5.5: Geometry of the core Su 150b.

Inductor specification

Area product

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Inductor Design Example (II)

TELMAG Su 150b

22 cm 8.32cm 9.33968.0 =⋅=FeA

22 cm 0.77cm 0.54.15 =⋅≈wA

4cm 2527== FewAAAP

turns48ˆˆ

==BAiLNFe

Cu

mm 162mm 322

⋅===LNAl Fe0µ

δ

265.22ln1 =⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+=

δ

δle

AlkFe

FF

turns32==FFFekALlN

0µδ

T 53.0ˆˆ ==NAiLBFe

Cu

⎪⎩

⎪⎨

⎧

===

⇒⎪⎩

⎪⎨

⎧

===

W6 W10 W141

T 014.0ˆT 029.0ˆT 486.0ˆ

kHz ,10

kHz ,5

kHz ,1

kHz 10

kHz 5

kHz 1

Fe

Fe

Fe

PPP

BBB

W157, ==∑i

iFeFe PP

2,

ˆ388 iii BfldP δδ =

⎪⎩

⎪⎨

⎧

===

W2 W4

W212

kHz ,10

kHz ,5

kHz ,1

δ

δ

δ

PPP

W218, ==∑i

iPP δδ

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Inductor Design Example (III)

TELMAG Su 150b

mm 351222 =++= gdcMLT

Ω=⋅

= m 04.4Cu

CuCu AMLTNR ρ

W582 == CuCuCu IRP

2, m 1276.02)(4)(2)(4 =++++++= dggcggdegceA CuT

2, m 0676.0424 =++= cfbdbcA FeT

2

,, W/m27821

=⎟⎠⎞

⎜⎝⎛ ++

+=Ψ CuFe

CuTCuT PPP

ebe

A δ

2

,, W/m11541

=⎟⎠⎞

⎜⎝⎛

+=Ψ Fe

CuTFeT P

ebb

A

( )448, 1070.5 asradT TT −⋅=Ψ − ε

( ) pTTF asconvTη−=Ψ 17.2,

convTradTT ,, Ψ+Ψ=Ψ

a

Tas TkTT

αα +Ψ

+=0

C 174,o=− aCus TT

C 72,o=− aFes TT

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Inductor Design Example (IV)

102

103

100

101

102

ΨT [W/m2]

Ts-T

a [°C]

Figure 5.6: Calculated temperature rise at an ambient temperature of 40 °C, based on radiated heat (black) and an approximate method (grey).

2

,, W/m10741

=⎟⎠⎞

⎜⎝⎛ +

+=Ψ CuFe

CuTCuT PP

ebe

A

Air-gap losses not included ⇒

C 67,o=− aCus TT

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Inductor Design Example (V)

Figure 5.7: Magnetic flux lines (left) of the entire inductor, and (right) of the region around one of the air gaps. Note the component of the fringing flux that is perpendicular to the surface of the steel tape.

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Capacitors

DesignMetallized film polypropylene capacitors have a thin plastic film to support the metal layer of the electrodes. The plastic used for the film can for example be polyester. If the plastic film has electrodes (of the same polarity) on both sides it is referred to as double metallized film. The dielectric consists of a polypropylene film. To avoid air pockets resulting in a locally high electric field strength, the polypropylene film should be somewhat porous to be able to absorb oil.

Wet aluminium electrolytic capacitors contain a fluid, the electrolyte, between the aluminium electrodes. The electrolyte is absorbed by paper in between the aluminium electrodes, in order to avoid air pockets. Since the electrolyte is conductive, the aluminium electrodes are electrically close together, only separated by the dielectric of the capacitor. The dielectric constitutes of a thin aluminium oxide layer on the positive electrode.

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Capacitors- Equivalent circuit

Figure 5.8: Capacitor simulation model.

RESR

iC

LESL

C

fC20

πδtan)( += sESR RfR

)()()( 2 fIfRfP CESRESR ⋅=

2C

ESRESR I

PR =