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Modellierung von Kern- und Wicklungsverlusten
Jonas Mühlethaler, Johann W. Kolar
Power Electronic Systems Laboratory, ETH Zurich
Montag, 10. Oktober 2011 2D-ITET/PES Jonas Mühlethaler
Motivation
Employing best state-of-the-art approaches and
embedding newly developed approaches
into a novel loss calculation framework.
Modeling Inductive Components
Montag, 10. Oktober 2011 3D-ITET/PES Jonas Mühlethaler
Loss Calculation Framework
Overview
1) A reluctance model is introduced to describe the electric / magnetic interface, i.e. L = f(i).
2) Core losses are calculated.
3) Winding losses are calculated.
Agenda
Montag, 10. Oktober 2011 4D-ITET/PES Jonas Mühlethaler
Motivation
Reluctance Model
Core Losses
Winding Losses
Experimental Results
Montag, 10. Oktober 2011 5D-ITET/PES Jonas Mühlethaler
Motivation
Modeling Inductive Components
Calculation of the inductance L with a reluctance model
Inductance calculation Reluctance definition
Montag, 10. Oktober 2011 6D-ITET/PES Jonas Mühlethaler
Existing Calculation Approaches
Assumption of homogeneous field distribution
gm
0 g
lR
A
Air gap reluctance calculation
gl
gAAir gap length
Air gap cross-sectional area
Montag, 10. Oktober 2011 7D-ITET/PES Jonas Mühlethaler
Existing Calculation Approaches
Increase of the Air Gap Cross-Sectional Area
gm *
0 g
lR
A
gm
0 g g( )( )l
Ra l d l
e.g. [4] (for a cross section with dimension a x d):
Air gap reluctance calculation
gl*gA
Air gap length
“New” air gap cross-sectional area
Montag, 10. Oktober 2011 8D-ITET/PES Jonas Mühlethaler
Aim of new model
Air gap reluctance calculation that
- considers the three dimensionality,
- is reasonable easy-to-handle,
- is capable of modeling different shape of air gaps,
- while still achieving a high accuracy.
Illustration of different air gap shapes:
Montag, 10. Oktober 2011 9D-ITET/PES Jonas Mühlethaler
Derivation of new model
Capacitance-To-Reluctance Analogy
If air is the dielectric, capacitance C can be expressed as
The reluctance of an air gap with the same geometry is then
0 ( )C F g
Represents geometry
m,airgap0
1( )
RF g
0ACd
m,airgap0
dRA
Montag, 10. Oktober 2011 10D-ITET/PES Jonas Mühlethaler
Derivation of new model
The Schwarz-Christoffel Transformation for Air Gaps
Schwarz-Christoffel Transformation
Derivation of new model
Montag, 10. Oktober 2011 11D-ITET/PES Jonas Mühlethaler
basic
0
1'2 1 ln
2 4
Rw hl l
Basic Structure for the Air Gap Calculation (2-D)
Basic structure for the air gap calculation:
Montag, 10. Oktober 2011 12D-ITET/PES Jonas Mühlethaler
Derivation of new model2-D (1)
Montag, 10. Oktober 2011 13D-ITET/PES Jonas Mühlethaler
Derivation of new model2-D (2)
Air gap type 1
Air gap type 2
Air gap type 3
Montag, 10. Oktober 2011 14D-ITET/PES Jonas Mühlethaler
Derivation of new model2D 3D : Fringing Factor (1)
Illustrative Example zy-plane
zx-plane
'zyR
'zxR
Montag, 10. Oktober 2011 15D-ITET/PES Jonas Mühlethaler
Derivation of new model2D 3D : Fringing Factor (2)
Illustrative Example zy-plane
zx-plane
'zy
0
y
Ra
b
“Idealized” air gap (no fringing flux)
'zx
0
xRa
t
Montag, 10. Oktober 2011 16D-ITET/PES Jonas Mühlethaler
Derivation of new model2D 3D
Fringing Factor
Illustrative Example
x y 3-D Fringing Factor:
“Idealized” air gap (no fringing flux)
g0
aRbt
Montag, 10. Oktober 2011 17D-ITET/PES Jonas Mühlethaler
w = 40 mm; h = 40 mm
Results3-D FEM Simulation
Modeled Example
Results
Montag, 10. Oktober 2011 18D-ITET/PES Jonas Mühlethaler
Inductance CalculationEPCOS E55/28/21, N = 80
ResultsExperimental Results
Montag, 10. Oktober 2011 19D-ITET/PES Jonas Mühlethaler
ResultsExperimental Results
Saturation CalculationEPCOS E55/28/21, N = 80,lg = 1 mm, Bsat = 0.45 T
Isat = 3.7 AMeasurementCalculations
Agenda
Montag, 10. Oktober 2011 20D-ITET/PES Jonas Mühlethaler
Motivation
Reluctance Model
Core Losses
Winding Losses
Experimental Results
Steinmetz Equation SE
- Only sinusoidal waveforms ( iGSE).
α βP k f B
Montag, 10. Oktober 2011 21D-ITET/PES Jonas Mühlethaler
iGSE
- DC bias not considered- Relaxation effect not considered- Steinmetz parameter are valid only in a limited flux density and
frequency range
v0
1 d dd
T
iBP k B t
T t
Core LossesDifferent Modeling Approaches (1)
Relaxation Losses
Montag, 10. Oktober 2011 22D-ITET/PES Jonas Mühlethaler
Loss increase in the zero voltage phases (where dB/dt = 0)!
Core LossesDifferent Modeling Approaches (2)
Ferrite N87 from EPCOS
iGSE
v0
1 d dd
T
iBP k B t
T t
i2GSE
n
lll
T
i PQtBtBk
TP
1rr
0v d
dd1
Montag, 10. Oktober 2011 23D-ITET/PES Jonas Mühlethaler
Remaining Problems
Steinmetz parameter are valid only in a limited flux density and frequency range.
Core Losses vary under DC bias condition.
Modeling relaxation and DC bias effects need parameters that are not given by core material manufacturers.
Core Losses Different Modeling Approaches (3)
i2GSE
n
lll
T
i PQtBtBk
TP
1rr
0v d
dd1
Measuring core losses is indispensable!
Montag, 10. Oktober 2011 24D-ITET/PES Jonas Mühlethaler
Core LossesDifferent Modeling Approaches (4)
4th dimension: temperature.
Loss Map (Loss Material Database)
Question What loss map structure is needed to take all loss effects into consideration?
Montag, 10. Oktober 2011 25D-ITET/PES Jonas Mühlethaler
Core LossesNeeded Loss Map Structure
Content of Loss MapTypical flux waveform
Montag, 10. Oktober 2011 26D-ITET/PES Jonas Mühlethaler
Core LossesHybrid Loss Modeling Approach (1)
Loss Map (Loss Material Database)
n
lll
T
i PQtBtBk
TP
1rr
0v d
dd1
r r r, , , , , , ,ik k q
PV
“The best of both worlds”
Montag, 10. Oktober 2011 27D-ITET/PES Jonas Mühlethaler
Core LossesHybrid Loss Modeling Approach (2)
Montag, 10. Oktober 2011 28D-ITET/PES Jonas Mühlethaler
Loss Map
n
lll
T
i PQtBtBk
TP
1rr
0v d
dd1
PV
Core LossesHybrid Loss Modeling Approach (3)
Montag, 10. Oktober 2011 29D-ITET/PES Jonas Mühlethaler
Interpolation and Extrapolation(HDC*, T*, B*, f*)
HDC and T
B and f
Core LossesHybrid Loss Modeling Approach (4)
Montag, 10. Oktober 2011 30D-ITET/PES Jonas Mühlethaler
Core LossesHybrid Loss Modeling Approach (5)
Advantages of combined approach (loss map and i2GSE):
Relaxation effects are considered (i2GSE).
A good interpolation and extrapolation between premeasured operating points is achieved.
Loss map provides accurate i2GSE parameters for a wide frequency and flux density range.
A DC bias is considered as the loss map stores premeasured operating points at different DC bias levels.
Montag, 10. Oktober 2011 31D-ITET/PES Jonas Mühlethaler
Core Losses Effect of Core Shape
1) The flux density in every core section of (approximately) homogenous flux density is calculated.
2) The losses of each section are calculated.
3) The core losses of each section are then summed-up to obtain the total core losses.
Reluctance ModelProcedure
Agenda
Montag, 10. Oktober 2011 32D-ITET/PES Jonas Mühlethaler
Motivation
Reluctance Model
Core Losses
Winding Losses
Experimental Results
Winding Losses Skin Effect in Solid Round Conductors
Montag, 10. Oktober 2011 33D-ITET/PES Jonas Mühlethaler
2S R DC
ˆ( )P F f R I
with
DC 2
0
0 1 0 1 0 1 0 1R 2 2 2 2
1 1 1 1
4
21
ber ( )bei ( ) ber ( )ber ( ) bei ( )ber ( ) bei ( )bei ( )ber ( ) bei ( ) ber ( ) bei ( )4 2
Rd
d
f
F
(Loss per unit length)
Winding Losses Proximity Effect in Solid Round Conductors
Montag, 10. Oktober 2011 34D-ITET/PES Jonas Mühlethaler
2P R DC e
ˆ( )P G f R H
with
DC 2
0
2 22 1 2 1 2 1 2 1
R 2 2 2 20 0 0 0
4
21
ber ( )ber ( ) ber ( )bei ( ) bei ( )bei ( ) bei ( )ber ( )ber ( ) bei ( ) ber ( ) bei ( )2 2
Rd
d
f
dG
(Loss per unit length)
Winding Losses Calculation of external field He (1D - approach)
Montag, 10. Oktober 2011 35D-ITET/PES Jonas Mühlethaler
with
avg left right12
H H H
Un-gapped cores (e.g. in transformers)
2 2DC R R avg,m m
1
ˆ ˆM
m
P R F I NM NG H l
Winding Losses Calculation of external field He (2D - approach)
Montag, 10. Oktober 2011 36D-ITET/PES Jonas Mühlethaler
Gapped cores: 2D approach is necessary !
Winding Losses Effect of the air gap fringing field
Montag, 10. Oktober 2011 37D-ITET/PES Jonas Mühlethaler
The air gap is replaced by a fictitious current, which …
… has the value equal to the magneto-motive force (mmf) across the air gap.
Winding Losses Effect of the core material
Montag, 10. Oktober 2011 38D-ITET/PES Jonas Mühlethaler
“Pushing the walls away”
The method of images (mirroring)
Winding Losses Calculation of external field He (2D - approach)
Montag, 10. Oktober 2011 39D-ITET/PES Jonas Mühlethaler
Gapped cores: 2D approach Winding Arrangement
External field vector across conductor qxi;yk
Winding Losses Different Winding Sections
Montag, 10. Oktober 2011 40D-ITET/PES Jonas Mühlethaler
Major Simplification
- magnetic field of the induced eddy currents neglected.
- This can be problematic at frequencies above (rule-of-thumb)
Winding Losses FEM Simulations
Montag, 10. Oktober 2011 41D-ITET/PES Jonas Mühlethaler
max 20
2.56fd
Results of considered winding arrangements
f-range f < fmax f > fmax
Error < 5% > 5% (always < 25%)
Agenda
Montag, 10. Oktober 2011 42D-ITET/PES Jonas Mühlethaler
Motivation
Reluctance Model
Core Losses
Winding Losses
Experimental Results
Montag, 10. Oktober 2011 43D-ITET/PES Jonas Mühlethaler
Experimental ResultsMeasurement 1
Flux Waveform
Results (measured with power analyzer Yokogawa WT3000)
InductorEPCOS E55/28/21, N = 18,d = 1.7 mm, lg = 1 mm
Montag, 10. Oktober 2011 44D-ITET/PES Jonas Mühlethaler
Experimental ResultsMeasurement 2
Flux Waveform
Results (measured with power analyzer Yokogawa WT3000)
InductorEPCOS E55/28/21, N = 18,d = 1.7 mm, lg = 1 mm
fLF = 100 HzfHF = 10 kHz
Montag, 10. Oktober 2011 45Power Electronic Systems Laboratory
Experimental Results 3Comparison Circuit Simulated Modeling vs. Measurements on a Boost Inductor Employed in a Three-Phase PFC Rectifiers [3]
Photo
Loss modeling very accurate.
Results
Conclusion
Conclusion & Outlook
Montag, 10. Oktober 2011 46D-ITET/PES Jonas Mühlethaler
A high accuracy in modeling inductive components has been achieved.
The following effects have been considered:Reluctance model [1]:
Air gap stray field.Non-linearity of core material considered.
Core Losses [2]:DC Bias (Loss Map)Relaxation Effects (i2GSE)Different flux waveforms (iGSE / i2GSE)Wide range of flux density and frequencies (Loss Map)
Winding Losses [2]:Skin and proximity effectStray field proximity effectEffect of core material
Thank you !
Montag, 10. Oktober 2011 47D-ITET/PES Jonas Mühlethaler
Do you have any questions ?
References
Montag, 10. Oktober 2011 48D-ITET/PES Jonas Mühlethaler
[1] J. Mühlethaler, J.W. Kolar, and A. Ecklebe, “A Novel Approach for 3D Air Gap Reluctance Calculations”, in Proc. of the ICPE - ECCE Asia, Jeju, Korea, 2011
[2] J. Mühlethaler, J.W. Kolar, and A. Ecklebe, “Loss Modeling of Inductive Components Employed in Power Electronic Systems”, in Proc. of the ICPE - ECCE Asia, Jeju, Korea, 2011
[3] J. Mühlethaler, M. Schweizer, R. Blattmann, J.W. Kolar, and A. Ecklebe, “Optimal Design of LCL Harmonic Filters for Three-Phase PFC Rectifiers”, in Proc. of the IECON, Melbourne, 2011
[4] N. Mohan, T. M. Undeland, and W. P. Robbins - “Power Electronics – Converter, Applications, and Design”, John Wiley & Sons, Inc., 2003