Eddy. - Core losses, number of turns etc: not discussed here. - Copper losses eddy current losses are troublesome in inductive components in power electronics.
Inductive Components in Power Electronics
Alex Van den Bossche Ghent University, EELAB Electrical Energy Laboratory Sint –Pietersnieuwstraat 41 Gent Belgium
Intelec 2011 Keynote 12th October 2011
I: Introduction
I. Introduction II. Background of eddy currents in round wires III. Application to transformers IV. Application to inductors V. Special shapes for larger power components Conclusion
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I: Introduction
Losses
Core
Copper
Ohmic
Eddy
- Core losses, number of turns etc: not discussed here. - Copper losses eddy current losses are troublesome in inductive components in power electronics - Presented for round wires here - Thermal aspects: at the end
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II: Background Need? Using finite elements the number of elements may be very high: 1000 elements/strand, 100 strands/wire 100 turns = 107 elements in 2D Even more in 3D. Many designs can be done without finite elements or with static field finite elements using the method. One graph in a few seconds… Proposed method for eddy currents in Round wires - Satisfies the analytic equations of free current carrying wires - Satisfies the analytic equations of a wire in a transverse field - Effect of wires close to each other - Effect of layers - Effect of air gaps - Low frequency approximation extended to high frequency Compatible with the book “Inductors and transformers for power electronics”
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II: Background
a) Rotational field Homogenous
b) Homogenous transverse field
c) Radial increasing field
Fig. 1. Three common different field types across a wire, at low frequency. When “Low frequency”? d< 1.8 δ
δ(f ) =
2ρ
ω µ0
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II: Background
α) Rotational tangential field
β) Homogenous transverse field
γ) Hyperbolic field type increasing field
Fig. 2. A set of three orthogonal field types across a wire, at low frequency.
Fig. 3. Pattern of hyperbolic field type.
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II: Background E
E
E
r
α) Rotational tangential field
x,y
y
β) Homogenous transverse field radial increasing
γ) Hyperbolic field type y-direction dashed
Fig 4. Low frequency E-field distribution in the conductor corresponding to the three field types.
Peddy, LF
2 π R 4 ω 2 µ 02 1 H 2 H hym 2 rom + H =l + trm ρ 4 12 6
Remaining other field types results in very a low loss contribution
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II: Background Same low frequency equation expressed in diameter, (rms value for H)
PeddyLF
2 π d 4ω 2 µ 02 H 2 H hym rom + H 2 + =l trm 12 64 ρ 6
Low frequency eddy current in an equivalent scheme:
Fig. 5. Equivalent loss resistance of the low frequency model for a transformer.
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II: Background Effect of layers
Fig. 6. Layer arrangement showing horizontal filling factor η and vertical filling factor λ
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II: Background Effect of filling factors and number of layers averaged shift σ removes λ if η≥λ (rather usual case) M: number of layers η: horizontal filling
d 2ω 2 µ02 I 2 Peddy, LF = lw 64 ρ
1 + 1.3537η 4 4M 2 − 1 1 2 + η 12 3 4
It is a quadratic equation in η2 M= 0.5 is a half layer Same with DC resistance R0 and relative height ζ = copper diameter/penetration depth
Reddy, LF =
ζ 4 Ro 1 + 1.3537η 4 64
12
4M 2 − 1 1 2 + η 3 4
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II: Background High frequency, e.g. Effect of layers for example effect of field reducing effect of adjacent wires Fig. 7. The eddy currents in the adjacent wires tends to reduce the field in the considered wire in a layer, only free wires solutions are analytically known
Complex impedance of a current carrying free wire, using Bessel functions…
ρz Z (ω ) = 2πro 2
3 j2
ζ
ber0 (
ζ 2
) + j bei 0 (
ζ 2
)
2 ber ( ζ ) + j bei ( ζ ) 1 1 2 2
Bessel °1784 +1846
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II: Background Approximation instead of Bessel? High frequency, current carrying free wire, compared with Bessel functions….. 1 1 4 Rwf = R0 1 + ζ 64 *12 G A (ζ ) 1 + 36864 G A (ζ ) = ζ 6 + 6.1ζ 5 + 32ζ 4 + 13ζ 3 + 90ζ 2 + 110ζ Wide frequency correction factor rotational and hyperbolic field including other wires in layers η 2.5 + 0.3 λ10
FA =
Is tuned by finite elements
1
(1 + 1.3537η )
4 −2
(
)
G (ζ ) π 2.5 10 + A 1 − η + 0.3 λ 36864 12
4
Fig. 8. Eddy current losses approximation compared to the exact Bessel equation for a free wire.
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II: Background Approximation instead of Bessel? High frequency, transverse field compared with Bessel functions….. Correction factor for from low to high frequency transverse field Single wire lc π d 2 ρ 4 B Pwf = ζ 16 µ0
2
1 1+
1 (GT (ζ ))6 1024
GT (ζ ) = ζ 6 + 2.7 ζ 5 − 1.3ζ 4 − 17ζ 3 + 85ζ 2 − 43ζ Fig. 9 Eddy current losses approximation compared to the exact Bessel equation for a free wire.
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III: Application to transformers Wide frequency for transformers, transverse field losses, tuned by finite elements. l ρ Ptr = w (ζ )4 π FT (ζ, η, λ )(µ 0 H rms )2 16
FT : Correction factor for from low to high frequency for transverse field also with effect of layers and other wires and tuned with FE: 1
FT =
π2 G (ζ ) π 2 1+ T 1+ Fi (λ ,η ) χ (ζ ) 2 − 1 − 1024 12 12
Surface effect of induced field χ( f ) =
1 1.5 1+ ζ(f)
(
)
λ10 + η 10 χ (ζ )10
Induced field by layer factor
Fi (η , λ ) =
η −λ 2
(η − λ + η + λ ) + η λ
4
HF finite element tuning at high f (10MHz) and high filling (>0.9) …10
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III: Application to transformers Graphical way for transformers
Peddy =
(
2 Ro I ac
)k
c , tr ( mE , ζ(
f , d ), η, λ)
kc,tr ≈ m E p ktf 2
Note that mE = 0.5 for a half layer (additional factor 4 in loss) p: parallel wires But feq increases with p f eq
dp = f ap 0 . 5
2
20 ×10 − 9 ρc
100
ktf ( 0.5 , ff k , d , 0.9 , 0.5)
100
10
ktf ( 1 , ff k , d , 0.9 , 0.5) ktf ( 2 , ff k , d , 0.9 , 0.5)
1
ktf ( 10 , ff k , d , 0.9 , 0.5) Tlf ( ff k , d , 0.9) 0.1
0.01 0.01 4 1 .10 4
10
1 .10
1 .10
5
ff k
6
1 .10 10
7
7
Fig. 10, ktf for transformers, for a horizontal filling factor 0.9 and a vertical of 0.5, for a wire of 0.5mm diameter of resistivity 20*10-9 Ωm, the full line is the low frequency model.
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III: Application to inductors Graphical method for Inductors
Additional loss factor for air gap KF:
KF =
κ=
3.5(0.5 − κ ) 2 + 0.69
Fig. 10, KF air gap effect for inductors
κ
d wl + t / 3 w/ K
Fig. 11. Relevant dimensions for κ
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IV: Application to inductors Graphical method for Inductors Also fields not parallel to layers 2
100
( )
pNd kc,in = k F kin f eq w
10
Tlf ( ff k , d , 1) kina( ff k , d , 0.9 , 0.1 , KFx , KFy ) kina( ff k , d , 0.9 , 0.3 , KFx , KFy )
Peddy =
2 Ro I ac
100
1
kina( ff k , d , 0.9 , 0.9 , KFx , KFy )
kc,in
kinb( ff k , d , KF ) 0.1
Red line: no layer effect no local field considered
FT =
1 G (ζ ) 1+ T 1024
0.01 0.01 4 1 .10 10
4
1 .10
1 .10
5
6
ff k
1 .10
7
7
10
Fig.15. Inductor cases, kin as function of feq for λ=0.7, d=0.5mm, high mE , Dotted line: η=0.1 0.3, 0.9; Red solid line simplified with FT only
ρ=23×10-9,
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IV: Simplified thermal calculation
Simplified heat transfer with natural convection a and b: the largest outer dimensions of the component, including copper
Ph = k A a b kA =2500 W/m2
Fig. 16. Allowed copper loss as a part of total heat transfer if copper loss is more than 50% of loss.
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V: Special shapes for larger power components
Fig 17 Litz wire optimized for transverse field
Advantage: - Low losses by transverse field - easy low frequency loss calculation Disadvantages: -Lower temperature class of isolation -Low heat conduction -5% more length and lower the copper filling factor
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V: Special shapes for larger power components
Fig. 18. Inductor EE55 shape, with a hole under the coil end for increased inner convection and lower eddy current losses. 1500W PFC choke for battery charging, natural convection.
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V: Special shapes for larger power components
Fig 19. Improved heat drain by aluminum plates between ferrites Example: 6 UU 93/76/16 ferrite cores No aluminium in the air gap itself.
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V: Special shapes for larger power components Playing with “building blocks”, multiple air gap and versatile?
Fig. 20a. Ferrite disc with hole 50/15/10, low frequency high induction ferrite (unknown brand)
Fig. 20b. Iron powder yokes 60x30x15 mm bricks (Magnetics
Fig. 20c. Four leg inductor using 50mm ferrite discs and iron powder yokes for A 3-phase 10kVA filter 30kHz PWM, natural cooling
Spang)
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Conclusion Eddy currents in round wires Three field types are the main components of low frequency eddy current losses. The transition from low to high frequency is done in a consistent way with known analytic equations for free wires, and tuned by FE. Incorporating the effects of air gaps in inductors.
Larger power components, still with natural convection? Litz wire? A hole under the coil end for better copper cooling Multiple gap. Ferrite discs φ = 50mm as building blocs? More heat drain: alu plates in parallel with ferrite
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Thank you for your attention
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