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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 =