APEC 2015 Optimal Design of Inductive Components Based on Accurate Loss and Thermal Models Dr. Jonas Mühlethaler APEC 2015 Introduction System Layer (GeckoCIRCUITS) Component Layer (GeckoMAGNETICS) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 2 APEC 2015 Introduction GeckoMAGNETICS How are magnetic components modeled in GeckoMAGNETICS? (Gecko-Simulations; www.gecko-simulations.com) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 3 APEC 2015 Introduction System Layer Component Layer Material Layer www.ferroxcube.com Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 4 APEC 2015 Introduction Application of Inductive Components (1) : Buck Converter (DC Current + HF Ripple) Schematic Current / Flux Waveform Modeling Difficulties Solutions - - - Non-sinusoidal current / flux waveform Current / flux is DC biased - - Mittwoch, 4. März 2015 Gecko-Simulations AG FFT of current waveform for the calculation of winding losses Determine core loss energy for each segment and for each corner point in the piecewise-linear flux waveform Loss Map enables to consider a DC bias Jonas Mühlethaler 5 APEC 2015 Introduction Application of Inductive Components (2) : Inductor of DAB Converter (Non-Sinusoidal AC Current) Current / Flux Waveform Schematic Modeling Difficulties Solutions - - - Non-sinusoidal current / flux waveform Core losses occur in the interval of constant flux Mittwoch, 4. März 2015 Gecko-Simulations AG - FFT of current waveform for the calculation of winding losses Improved core loss equation that considers relaxation effects Jonas Mühlethaler 6 APEC 2015 Introduction Application of Inductive Components (3) : Three-Phase PFC (Sinusoidal Current + HF Ripple) Current / Flux Waveform Schematic Modeling Difficulties Solutions - - - Non-sinusoidal current / flux waveform Major loop and many (DC biased) minor loops - Mittwoch, 4. März 2015 Gecko-Simulations AG FFT of current waveform for the calculation of winding losses Determine core loss energy for each segment and for each corner point in the piecewise-linear flux waveform (-> minor loop losses) Add major loop losses Jonas Mühlethaler 7 APEC 2015 Introduction Overview of Different Flux Waveforms Sinusoidal DC Current + HF Ripple Non-Sinusoidal AC Current Minor Loop Sinusoidal Current + HF Ripple Major Loop Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 8 APEC 2015 Introduction System Layer Component Layer Material Layer www.ferroxcube.com Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 9 APEC 2015 Introduction Overview of Other Modeling Issues Flux Density Distribution Heat Dissipation Core Type Current Distribution Winding Type / Arrangement Mittwoch, 4. März 2015 Air Gap Stray Field Gecko-Simulations AG Jonas Mühlethaler 10 APEC 2015 Introduction Wide Range of Realization Options Inductors / Transformers Core Shapes www.ferroxcube.com Conductor Shapes www.wagnergrimm.ch, www.ferroxcube.com www.pack-feindraehte.de, www.jiricek.de Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 11 APEC 2015 Introduction Modeling Inductive Components (1) Procedure 1) A reluctance model is introduced to describe the electric / magnetic interface, i.e. L = f(i). 2) Core losses are calculated. Reluctance Model 3) Winding losses are calculated. 4) Inductor temperature is calculated. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 12 APEC 2015 Introduction Modeling Inductive Components (2) The following effects are taken into consideration: Magnetic Circuit Model (e.g. for Inductance Calculation): Air gap stray field Non-linearity of core material Core Losses: DC Bias Different flux waveforms (link to circuit simulator) Wide range of flux densities and frequencies Different core shapes Winding Losses: Skin and proximity effect Stray field proximity effect Effect of core on magnetic field distribution Litz, solid, and foil conductors Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 13 APEC 2015 Introduction System Layer Component Layer Material Layer www.ferroxcube.com Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 14 APEC 2015 Introduction Motivation for an Accurate Loss Modeling : Multi-Objective Optimization (1) PFC Rectifier with Input LCL filter Filter Losses vs. Filter Volume Converter Losses vs. Converter Volume Converter = Cooling System + Switches Mittwoch, 4. März 2015 Sometime there are parameters that bring advantages for one subsystem while deteriorating another subsystem (e.g. frequency in above example). Gecko-Simulations AG Jonas Mühlethaler 15 APEC 2015 Introduction Motivation for an Accurate Loss Modeling : Multi-Objective Optimization (2) Losses of a Loss-Optimized Design Optimal Frequency 6 kHz Mittwoch, 4. März 2015 In order to get an optimal system design, an overall system optimization has to be performed. It is (often) not enough to optimize subsystems independent of each other. Gecko-Simulations AG Jonas Mühlethaler 16 APEC 2015 Outline Magnetic Circuit Modeling Core Loss Modeling Winding Loss Modeling Thermal Modeling Multi-Objective Optimization Summary & Conclusion Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 17 APEC 2015 Magnetic Circuit Modeling Reluctance Model Electric Network Conductivity / Permeability Resistance / Reluctance Voltage / MMF R l / ( A) V P2 E ds P1 I J dA Current / Flux A Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler Magnetic Network Rm l / ( A) Vm P2 H ds P1 B dA A 18 APEC 2015 Magnetic Circuit Modeling Why a Reluctance Model is Needed A reluctance model is needed in order to calculate the inductance ( L = N2/Rtot ) calculation the saturation current calculate the air gap stray field calculate the core flux density Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler Accurate loss modeling! 19 APEC 2015 Magnetic Circuit Modeling Core Reluctance Core Reluctance Dimensions Reluctance Calculation Rm Mittwoch, 4. März 2015 li 0 r Ai Mean magnetic length Gecko-Simulations AG Jonas Mühlethaler Mean magnetic cross-sectional area 20 APEC 2015 Magnetic Circuit Modeling Air Gap Reluctance : Different Approaches (1) Assumption of Homogeneous Field Distribution Rm lg 0 Ag lg Air gap length lg Ag Air gap cross-sectional area a Increase of the Air Gap Cross-Sectional Area e.g. [1] (for a cross section with dimension a x t): Rm a a [1] Mittwoch, 4. März 2015 Gecko-Simulations AG lg 0 (a lg )(t lg ) N. Mohan, T. M. Undeland, and W. P. Robbins - “Power Electronics – Converter, Applications, and Design”, John Wiley & Sons, Inc., 2003 Jonas Mühlethaler 21 APEC 2015 Magnetic Circuit Modeling Air Gap Reluctance : Different Approaches (2) Schwarz-Christoffel Transformation Transformation Equation v(t) Transformation Equation z(t) z (t ) l 2 ln 1 1 t ln t 2 1 t v(t ) V ln t (z and t are complex numbers) [2] K. J. Binns, P. J. Lawrenson, and C. W. Trowbridge, «The Analytical and Numerical Solution of Electric and Magnetic Fields», John Wiley & Sons, Inc., 1992 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 22 APEC 2015 Magnetic Circuit Modeling Air Gap Reluctance : Different Approaches (3) Solution to 2-D problems found in literature, e.g. in [3] Can’t be directly applied to 3-D problems. Some 3-D solution to problem found in literature; however, they are complex [4] and/or limited to one air gape shape [5] More simple and universal model desired. [3] [4] [5] A. Balakrishnan, W. T. Joines, and T. G. Wilson - “Air-gap reluctance and inductance calculations for magnetic circuits using a Schwarz-Christoffel transformation”, IEEE Transaction on Power Electronics, vol. 12, pp. 654—663, July 1997. P. Wallmeier, “Automatisierte Optimierung von induktiven Bauelementen für Stromrichteranwendungen”, PhD Thesis, Universität – Gesamthochschule Paderborn, 2001. E. C. Snelling, “Soft Ferrites - Properties and Applications”, 2nd edition, Butterworths, 1988 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 23 APEC 2015 Magnetic Circuit Modeling Aim of New Model Air gap reluctance calculation that - considers the three dimensionality, - is reasonable easy-to-handle, - is capable of modeling different shapes of air gaps, - while still achieving a high accuracy. Illustration of Different Air Gap Shapes: Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 24 APEC 2015 Magnetic Circuit Modeling New Model (1) Basic Structure for the Air Gap Calculation (2-D) [3] Next Core Corner R 'basic Mittwoch, 4. März 2015 1 w 2 h 1 ln 4l 2l 0 Gecko-Simulations AG Jonas Mühlethaler 25 APEC 2015 Magnetic Circuit Modeling New Model (2) Basic Structure for the Air Gap Calculation 2-D (1) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 26 APEC 2015 Magnetic Circuit Modeling Basic Structure for the Air Gap Calculation New Model (3) 2-D (2) Air Gap Type 1 R 'basic Air Gap Type 2 Mittwoch, 4. März 2015 1 w 2 h 1 ln 4l 2l 0 Air Gap Type 3 Gecko-Simulations AG Jonas Mühlethaler 27 APEC 2015 Magnetic Circuit Modeling New Model (4) 2D 3D : Fringing Factor (1) Illustrative Example Air gap per unit length zy-plane ' Rzy ' Rzy y lg 0 a “Idealized” air gap (no fringing flux) zx-plane ' zx R ' Rzx x lg 0t Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 28 APEC 2015 Magnetic Circuit Modeling New Model (5) 2D 3D : Fringing Factor (2) 3-D Fringing Factor: x y Illustrative Example Rg lg 0 at “Idealized” air gap (no fringing flux) Alternative Interpretation: Increase of air gap cross sectional area 1 x 1 y A’ A [6] A’’ 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 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 29 APEC 2015 Magnetic Circuit Modeling FEM Results 3-D FEM Simulation Modeled Example Results a = 40 mm; h = 40 mm Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 30 APEC 2015 Magnetic Circuit Modeling Experimental Results Saturation Calculation EPCOS E55/28/21, N = 80, lg = 1 mm, Bsat = 0.45 T Inductance Calculation EPCOS E55/28/21, N = 80 Measurement Isat = 3.7 A Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 31 APEC 2015 Magnetic Circuit Modeling Non-Linearity of the Core Material Flux and Reluctance Calculation Inductance Calculation This equation must be solved iteratively by using a numerical solving method, e.g. the Newton’s method. Inductance B-H Relation 0.74 1.5 21 1920 18 17 16 15 14 13 12 11 10 0.72 0.7 12 345678 91011213 14 15 Inductance [mL] 0.68 Magentic Flux Density [T] Reluctance Model 16 0.66 17 0.64 18 0.62 19 1 9 8 7 6 0.5 5 4 0.6 20 3 2 0.58 21 0 Mittwoch, 4. März 2015 Gecko-Simulations AG 50 100 150 Current [A] Jonas Mühlethaler 200 0 1 0 500 Magnetic Field [A/m] 1000 32 APEC 2015 Example Introduction Schematic Aim Design PFC rectifier system. Show trade-off between losses and volume. Illustrative example. Modeling of boost inductors (three individual inductors L2a = L2b = L2c) will be step-by-step illustrated in the course of this presentation. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 33 APEC 2015 Example Reluctance Model (1) Reluctance Model Photo & Dimensions Calculation of Core Reluctances (2a ) do 2a Rc1 2 8 (2a ) 0 r at 0 r t 2 (2 20mm) 60mm 2 20mm A 8 2 3654 (2 20mm) 0 20 '000 20mm 28mm Vs 0 20 '000 28mm 2 (2a ) do 2a 2b Rc2 2 8 (2a ) 0 r at 0 r t 2 (2 20mm) 60mm 2 20mm + 2 h A 8 2 12184 (2 20mm) 0 20 '000 20mm 28mm Vs 0 20 '000 28mm 2 Material Grain-oriented steel (M165-35S) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 34 APEC 2015 Example Reluctance Model (2) Calculation Air Gap Reluctances Photo & Dimensions ' Rzy,1 zy-plane 1 a 2 a 0 1 ln l l 2 g 4 g 2 2 ' Rzy,2 1 a 2 (b a ) 0 1 ln l 2 lg 4 g 2 2 1 ' Rzy,3 a 2 b 1 ln 4lg 2lg 0 R ' zy,1 ' ' Rzy,2 Rzy,3 ' ' ' Rzy,1 Rzy,2 Rzy,3 ' Rzy 0.72 lg y zx-plane ' Rzy 0 a ' Rzx,1 1 t 2 a 0 1 ln l l g 2 4 g 2 2 ' Rzx ' Rzx,2 1 t 2 (b a ) 0 1 ln l 2 lg 4 g 2 2 ' ' Rzx,1 Rzx,2 Basic Reluctance x 2 Rg x y ' Rzx 0.84 lg 0t lg 0 at 1.66 MA Wb Inductance Material Grain-oriented steel (M165-35S) Mittwoch, 4. März 2015 R 'basic 1 L w 2 h 1 ln 4l 2l 0 Gecko-Simulations AG Jonas Mühlethaler N2 2.66mH Rc1 Rc2 Rg1 Rg2 meas. 2.69 mH 35 APEC 2015 Outline Magnetic Circuit Modeling Core Loss Modeling Winding Loss Modeling Thermal Modeling Multi-Objective Optimization Summary & Conclusion Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 36 APEC 2015 Core Loss Modeling Overview of Different Core Materials (1) Magnetic Materials Soft Magnetic Materials Hard Magnetic Materials Iron Based Alloys Ferrites - ferromagnetics - - ferrimagnetics - Alloys with some amount of Si, Ni, Cr or Co. Low electric resistivity. High saturation flux density. Ceramic materials, oxide mixtures of iron and Mn, Zn, Ni or Co. High electric resistivity. Small saturation flux density compared to ferromagnetics. Laminated Cores Laminations are electrically isolated to reduce eddy currents. Mittwoch, 4. März 2015 Powder Iron Cores Amorphous Alloys Nanocrystalline Materials Consist of small iron particles, which are electrically isolated to each other. Made of alloys without crystallyine order (cf. glasses, liquids). Ultra fine grain FeSi, which are embedded in an amorphous minority phase. Gecko-Simulations AG Jonas Mühlethaler 37 APEC 2015 Core Loss Modeling Overview of Different Core Materials (2) Selection Criteria Ferrite Powder Iron Core Saturation Flux Density Low Sat. Flux Density (0.45 T) High Sat. Flux Density (1.5 T) Power Loss Density Low Losses Moderate Losses (Frequency Range) Low Price Low Price Price Many Different Shapes Many Different Shapes etc. Very Brittle Distributed Air Gap (low rel. permeability) Laminated Steel Cores Amorphous Alloys Nanocrystalline Materials Very High Sat. Flux Density (2.2T) High Sat. Flux Density (1.5T) High Sat. Flux Density (1.1T) High Losses Low Losses Very Low Losses Low Price High Price Very High Price Many Different Shapes Limited Available Shapes Limited Available Shapes LF: BSAT = Bmax. HF: Bmax is limited by core losses. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 38 APEC 2015 Core Loss Modeling Overview of Different Core Materials (3) [7] [7] M. S. Rylko, K. J. Hartnett, J. G. Hayes, M.G. Egan, “Magnetic Material Selection for High Power High Frequency Inductors in DC-DC Converters”, in Proc. of the APEC 2009. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 39 APEC 2015 Core Loss Modeling Physical Origin of Core Losses (1) Weiss Domains / Domain Walls - Spontaneous magnetization. Material is divided to saturated domains (Weiss domains). In case an external field is applied, the domain walls are shifted or the magnetic moments within the domains change their direction. The net magnetization becomes greater than zero. B-H-Loop - - Mittwoch, 4. März 2015 Gecko-Simulations AG The flux change is partly irreversible, i.e. energy is dissipated as heat. The reason for this are the so called Barkhausen jumps, that lead to local eddy current losses. In case the loop is traversed very slowly, these Barkhausen jumps lead to the static hysteresis losses. Jonas Mühlethaler 40 APEC 2015 Core Loss Modeling Physical Origin of Core Losses (2) B-H-Loop Mittwoch, 4. März 2015 Gecko-Simulations AG - If the process would be fully reversible, going from B1 to B2 would store potential energy in the magnetic material that is later released (i.e. the area of the closed loop would be zero). - Since the process is partly irreversible, the area of the closed loop represents the energy loss per cycle Jonas Mühlethaler 41 APEC 2015 Core Loss Modeling Classification of Losses (1) (Static) hysteresis loss Rate-independent BH Loop. Loss energy per cycle is constant. Irreversible changes each within a small region of the lattice (Barkhausen jumps). These rapid, irreversible changes are produced by relatively strong local fields within the material. Eddy current losses Residual Losses – Relaxation losses Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 42 APEC 2015 Core Loss Modeling Classification of Losses (2) (Static) hysteresis loss Eddy current losses Depend on material conductivity and core shape. Affect BH loop. Residual Losses – Relaxation losses Measurements VITROPERM 500F 0.4 0.3 0.2 i [A] 0.1 0 -0.1 -0.2 -0.3 -0.4 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 0 2 4 6 8 10 t [s] 12 14 16 18 20 43 APEC 2015 Core Loss Modeling Classification of Losses (3) (Static) hysteresis loss Eddy current losses Residual losses – Relaxation losses Rate-dependent BH Loop. Reestablishment of a thermal equilibrium is governed by relaxation processes. Restricted domain wall motion. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 44 APEC 2015 Core Loss Modeling Typical Flux Waveforms Sinusoidal e.g. 50/60 Hz isolation transformer Triangular e.g. Buck / Boost converter Trapezoidal e.g. boost inductor of Dual Active Bridge Combination e.g. Boost inductor in PFC Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 45 APEC 2015 Core Loss Modeling Outline of Different Modeling Approaches Steinmetz Approach Loss Map Approach P k f α Bβ (Loss Database) - Simple Steinmetz parameter are valid only in a limited flux density and frequency range DC Bias not considered (Only for sinusoidal flux waveforms) - Loss Separation Hysteresis Model (e.g. Preisach Model, Jiles-Atherton Model) P Physt Peddy Presidual - Needed parameters often unknown Model is widely applicable Increases physical understanding of loss mechanisms Mittwoch, 4. März 2015 Measuring core losses is indispensable to overcome limits of Steinmetz approach wwww.epcos.com - Gecko-Simulations AG - Jonas Mühlethaler Difficult to parameterize Increases physical understanding of loss mechanisms 46 APEC 2015 Core Loss Modeling Overview of Hybrid Modeling Approach “The best of both worlds” (Steinmetz & Loss Map approach) Outline of Discussion (2)/(3) Loss Map (Loss Material Database) - - - Derivation of the i2GSE. (1) How to measure core losses in order to build loss map. (2) Use of loss map. (3) How to calculate core losses for cores of different shapes? (4) k , ki , , , r , r , qr , T 1 dB Pv ki T 0 dt Mittwoch, 4. März 2015 Gecko-Simulations AG B n dt Qrl Prl l 1 Jonas Mühlethaler (1) P V (4) 47 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Motivation (1) Steinmetz Equation SE Pv k f α Bˆ β - Pv : Only sinusoidal waveforms ( iGSE). iGSE T 1 dB Pv ki T 0 dt ki B time-average power loss per unit volume dt k (2 ) 1 2 0 cos 2 d - DC bias not considered - Relaxation effect not considered ( i2GSE) - Steinmetz parameter are valid only for a limited flux density and frequency range Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 48 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Motivation (2) iGSE [8] T 1 dB Pv ki T 0 dt ki B Idea - Generalized formula that is applicable for different flux waveforms - Losses depend on dB/dt dt k (2 ) 1 2 0 cos 2 d For Sinusoidal Waveforms T 1 dB Pv ki T 0 dt How to apply the formula? k Pv i T B B dt kf 2 B B DT B ... B (1 D)T DT (1 D)T [8] K. Venkatachalam, C. R. Sullivan, T. Abdallah, and H. Tacca, “Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters”, in Proc. of IEEE Workshop on Computers in Power Electronics, pp. 36-41, 2002. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 49 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Motivation (3) Waveform Results iGSE Conclusion T 1 dB Pv ki T 0 dt Mittwoch, 4. März 2015 B dt Gecko-Simulations AG Losses in the phase of constant flux! Jonas Mühlethaler 50 APEC 2015 Core Loss Modeling Derivation of the i2GSE – B-H-Loop Relaxation Losses Rate-dependent BH Loop. Reestablishment of a thermal equilibrium is governed by relaxation processes. Restricted domain wall motion. Current Waveform Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 51 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Model Derivation 1 (1) Derivation (1) Waveform Relaxation loss energy can be described with t 1 E E 1 e Loss Energy per Cycle is independent of operating point. How to determine E? Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 52 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Model Derivation 1 (2) E – Measurements Waveform Conclusion E follows a power function! d E kr B(t ) dt Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler r B r 53 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Model Derivation 1 (3) Model Part 1 T 1 dB Pv ki T 0 dt B 1 d Prl k r B(t ) T dt Mittwoch, 4. März 2015 Gecko-Simulations AG n dt Prl l 1 r B r t 1 1 e Jonas Mühlethaler 54 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Model Derivation 2 (1) Waveform Power Loss Explanation 1) 2) 3) 4) For values of D close to 0 or close to 1 a loss underestimation is expected when calculating losses with iGSE (no relaxation losses included). For values of D close to 0.5 the iGSE is expected to be accurate. Adding the relaxation term leads to the upper loss limit, while the iGSE represents the lower loss limit. Losses are expected to be in between the two limits, as has been confirmed with measurements. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 55 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Model Derivation 2 (2) Waveform Model Adaption T 1 dB Pv ki T 0 dt B n dt Qrl Prl l 1 Qrl should be 1 for D = 0 Qrl should be 0 for D = 0.5 Power Loss Qrl should be such that calculation fits a triangular waveform measurement. Qrl e Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler qr dB ( t ) / dt dB ( t ) / dt qr 1DD e 56 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Model Derivation 2 (3) Waveform Mittwoch, 4. März 2015 Power Loss Gecko-Simulations AG Jonas Mühlethaler 57 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Summary The improved-improved Generalized Steinmetz Equation (i2GSE) [9] T 1 dB Pv ki T 0 dt B with 1 d Prl k r B(t ) T dt and Qrl e [9] qr n dt Qrl Prl l 1 r B r t 1 1 e dB ( t ) / dt dB ( t ) / dt J. Mühlethaler, J. Biela, J.W. Kolar, and A. Ecklebe, “Improved Core Loss Calculation for Magnetic Components Employed in Power Electronic Systems”, in Proc. of the APEC, Ft. Worth, TX, USA, 2011. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 58 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Example i2GSE Evaluated for each piecewise-linear flux segment T 1 dB Pv ki T 0 dt Example Pv Evaluated for each voltage step, i.e. for each corner point in a piecewise-linear flux waveform. T 2t T Mittwoch, 4. März 2015 B n dt Qrl Prl l 1 B T 2 t dB 0 B dt T 2 t 0 B ki T 2 t B for t 0 and t T 2 t for t T 2 t and t T 2 for t T 2 and t T t for t T t and t T 2 Qrl Prl with l 1 Gecko-Simulations AG Qr1 = Qr2 = 0 1 B Pr1 Pr2 kr T T 2 t Jonas Mühlethaler r B r t 1 e 59 APEC 2015 Core Loss Modeling Derivation of the i2GSE – Conclusion Evaluated for each piecewise-linear flux segment i2GSE T 1 dB Pv ki T 0 dt B n dt Qrl Prl l 1 Evaluated for each voltage step, i.e. for each corner point in a piecewise-linear flux waveform. 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. Measuring core losses is indispensable! Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 60 APEC 2015 Core Loss Modeling Overview of Hybrid Modeling Approach “The best of both worlds” (Steinmetz & Loss Map approach) Outline of Discussion (2)/(3) Loss Map (Loss Material Database) - - - Derivation of the i2GSE. (1) How to measure core losses in order to build loss map. (2) Use of loss map. (3) How to calculate core losses for cores of different shapes? (4) k , ki , , , r , r , qr , T 1 dB Pv ki T 0 dt Mittwoch, 4. März 2015 Gecko-Simulations AG B n dt Qrl Prl l 1 Jonas Mühlethaler (1) P V (4) 61 APEC 2015 Core Loss Modeling Core Loss Measurement – Measurement Principle Waveforms Excitation System Schematic Sinusoidal Triangular Trapezoidal Voltage 0 … 450 V Current 0 … 25 A Frequency 0 … 200 kHz Loss Extraction 1 B(t ) N 2 Ae t u ( ) d 0 P[W ] V [ m3 ] N i (t ) H (t ) 1 le Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 62 APEC 2015 Core Loss Modeling Core Loss Measurement - Overview System Overview P/V T N f i1 (t ) 1 v2 (t ) dt f N P 2 0 V Aele Mittwoch, 4. März 2015 T 0 H (t )le N1 dB(t ) N 2 Ae dt N1 N 2 dt f Aele Gecko-Simulations AG Jonas Mühlethaler B (T ) B (0) H ( B)dB 63 APEC 2015 Core Loss Modeling Overview of Hybrid Modeling Approach “The best of both worlds” (Steinmetz & Loss Map approach) Outline of Discussion (2)/(3) Loss Map (Loss Material Database) - - - Derivation of the i2GSE. (1) How to measure core losses in order to build loss map. (2) Use of loss map. (3) How to calculate core losses for cores of different shapes? (4) k , ki , , , r , r , qr , T 1 dB Pv ki T 0 dt Mittwoch, 4. März 2015 Gecko-Simulations AG B n dt Qrl Prl l 1 Jonas Mühlethaler (1) P V (4) 64 APEC 2015 Core Loss Modeling Needed Loss Map Structure Typical flux waveform Mittwoch, 4. März 2015 Content of Loss Map Gecko-Simulations AG Jonas Mühlethaler 65 APEC 2015 Core Loss Modeling Minor and Major Loops Idea Minor Loop Major Loop Implementation P V Major P Piecewise Linear V Segment (+ Turning Point) Losses due to Minor and Major Loops are calculated independent of each other and summed up. Actually, it is not considered how the minor loop closes: each piecewise linear segment is modeled as having half the losses of its corresponding closed loop (cf. next slides). P P P Piecewise Linear V V Major V Segment (+ Turning Point) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 66 APEC 2015 Core Loss Modeling Hybrid Loss Modeling Approach (1) HDC / BDC (I) (II) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 67 APEC 2015 Core Loss Modeling Evaluated for the according corner point in a piecewise-linear flux waveform. Hybrid Loss Modeling Approach (2) Evaluated for the according piecewiselinear flux segment Loss Map T 1 dB Pv ki T 0 dt B n dt Qrl Prl l 1 (I) (II) Three loss map operating points are required in order to extract the parameters , , and k (or ki). Pv1 ki 2 f1 B1 Pv2 ki 2 f 2 B2 Pv3 ki 2 f3 B3 P V These equations arise when one evaluates the i2GSE for symmetric triangular waveforms. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 68 APEC 2015 Core Loss Modeling Hybrid Loss Modeling Approach (3) Interpolation and Extrapolation (HDC*, T*, B*, f*) HDC and T B and f Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 69 APEC 2015 Core Loss Modeling Hybrid Loss Modeling Approach (4) Advantages of Hybrid 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. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 70 APEC 2015 Core Losses Summary of Loss Density Calculation Sinusoidal P k f α Bβ T 1 dB Pv ki T 0 dt Triangular B n dt Qrl Prl l 1 with Q 0, n = 1 T Trapezoidal 1 dB Pv ki T 0 dt B n dt Qrl Prl l 1 with Q = 1, n = 2 Combination T 1 dB Pv ki T 0 dt B n dt Qrl Prl l 1 with different Q’s and n >> 0. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 71 APEC 2015 Core Loss Modeling Overview of Hybrid Modeling Approach “The best of both worlds” (Steinmetz & Loss Map approach) Outline of Discussion (2)/(3) Loss Map (Loss Material Database) - - - Derivation of the i2GSE. (1) How to measure core losses in order to build loss map. (2) Use of loss map. (3) How to calculate core losses for cores of different shapes? (4) k , ki , , , r , r , qr , T 1 dB Pv ki T 0 dt Mittwoch, 4. März 2015 Gecko-Simulations AG B n dt Qrl Prl l 1 Jonas Mühlethaler (1) P V (4) 72 APEC 2015 Core Loss Modeling Effect of Core Shape Procedure Reluctance Model 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. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 73 APEC 2015 Core Loss Modeling Effective Core Dimensions of Toroid Motivation for Effective Core Dimensions Core loss densities are needed to model core losses. It is difficult to determine these loss densities from a toroid, since the flux density is not distributed homogeneously in a toroid. Illustration Definition: Ideal Toroid A toroid is ideal when he has a homogenous flux density distribution over the radius (r1 r2). Idea for Real Toroid Find effective core magnetic length and cross section, so one can calculate as if it were an ideal toroid, i.e. as if the flux density distribution were homogenous. Effective magnetic length Effective magnetic cross-section Mittwoch, 4. März 2015 le 2 ln r2 / r1 1/ r1 1/ r2 h ln 2 r2 / r1 Ae 1 / r1 1 / r2 Gecko-Simulations AG Jonas Mühlethaler 74 APEC 2015 Core Loss Modeling Impact of Core Shape on Eddy Current Losses Eddy current loss density can be determined as [5] Peddy ˆ )2 ( Bfd kec Peddy ˆ )2 ( Bfd 6 For a laminated core it is The eddy current losses per unit volume depend not on the shape of the bulk material, but on the size and geometry of the insulated regions. In case of laminated iron cores, it is still appropriate to calculate with core loss densities that have been measured on a sample core with a geometrically different bulk material, but with the same lamination or tape thickness. [5] E. C. Snelling, “Soft Ferrites - Properties and Applications”, 2nd edition, Butterworths, 1988 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 75 APEC 2015 Core Loss Modeling Effect in Tape Wound Cores www.vacuumschmelze.de Thin ribbons (approx. 20m) Wound as toroid or as double C core. Amorphous or nanocrystalline materials. Losses in gapped tape wound cores higher than expected! Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 76 APEC 2015 Core Loss Modeling Effect in Tape Wound Cores - Cause 1 : Interlamination Short Circuits Machining process Surface short circuits introduced by machining (particular a problem in in-house production). After treatment may reduce this effect. At ETH, a core was put in an 40% ferric chloride FeCl3 solution after cutting, which substantially (more than 50%) decreased the core losses. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 77 APEC 2015 Core Loss Modeling Effect in Tape Wound Cores - Cause 2 : Orthogonal Flux Lines (1) Eddy current Fringing flux Main flux Eddy current A flux orthogonal to the ribbons leads to very high eddy current losses! Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 78 APEC 2015 Core Loss Modeling Effect in Tape Wound Cores - Cause 2 : Orthogonal Flux Lines (2) An experiment that illustrates well the loss increase due to an orthogonal flux is given here. Displacements Horizontal Displacement Vertical Displacement Core Loss Results Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 79 APEC 2015 Core Loss Modeling Effect in Tape Wound Cores - Cause 2 : Orthogonal Flux Lines (3) Core loss increase due to leakage flux in transformers. Results Secondary Winding Primary Winding Core Losses (W) Measurement Set Up 12 10 A higher load current leads to higher orthogonal flux! 8 6 4 0 1 2 3 Secondary Current (A) 4 FEM Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 80 APEC 2015 Core Loss Modeling Effect in Tape Wound Cores - Cause 2 : Orthogonal Flux Lines (4) In [10] a core loss increase with increasing air gap length has been observed. Figures from [10] [10] H. Fukunaga, T. Eguchi, K. Koga, Y. Ohta, and H. Kakehashi, “High Performance Cut Cores Prepared From Crystallized Fe-Based Amorphous Ribbon”, in IEEE Transactions on Magnetics, vol. 26, no. 5, 1990. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 81 APEC 2015 Example Core Loss Modeling (1) Photo & Dimensions Flux Density Distribution Reluctance Model An approximately homogeneous flux density distribution inside the core. Flux Density Waveform GeckoMAGNETICS Presentation Material Grain-oriented steel (M165-35S) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 82 APEC 2015 Example Core Loss Modeling (2) SiFe vs. Ferrite 2 kHz vs. 20 kHz Modeling with GeckoMAGNETICS APEC 2015 Example Core Loss Modeling (3) 2 kHz / Tmax = 65 °C / L = 2.5 mH / Solid Round Wires SiFe (M165-35S) Ferrite (N87) Sat. Limit APEC 2015 Example Core Loss Modeling (4) 20 kHz / Tmax = 65°C / L = 1 mH / Litz Wires SiFe (M165-35S) Ferrite (N87) Therm. Limit APEC 2015 Example Core Loss Modeling (5) 19.02.14 Gecko-Simulations AG Jonas Mühlethaler 86 APEC 2015 Outline Magnetic Circuit Modeling Core Loss Modeling Winding Loss Modeling Thermal Modeling Multi-Objective Optimization Summary & Conclusion Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 87 APEC 2015 Winding Loss Modeling Skin Effect (1) H-field in conductor Ampere’s Law Induced Eddy Currents Faraday’s Law Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 88 APEC 2015 Winding Loss Modeling Skin Effect (2) FEM Simulation 50 Hz 5 kHz 20 kHz 100 kHz Current Distribution Outer radius of conductor Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 89 APEC 2015 Winding Loss Modeling Skin Effect (3) Skin Depth (, where the current density has 1/e of surface value) 1 [19] 0 f Power Loss Increase with Frequency PS FR ( f ) RDC Iˆ2 FR ( f ) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler d 2 90 APEC 2015 Winding Loss Modeling Skin Effect (4) Current Distributions Parallel-connected (not twisted) conductors! Figure from [19] Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 91 APEC 2015 Winding Loss Modeling Proximity Effect (1) H-field of neighboring conductor induces eddy currents Ampere’s Law Faraday’s Law Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 92 APEC 2015 Winding Loss Modeling Proximity Effect (2) Eddy Currents in Conductor 50 Hz 5 kHz 15 kHz 100 kHz Induced Eddy Currents (He,rms = 35 A/m parallel to conductor) Current Concentration I Mittwoch, 4. März 2015 I I Gecko-Simulations AG Jonas Mühlethaler I Figures from [19] 93 APEC 2015 Winding Loss Modeling Skin vs. Proximity Effect Situation Results (f = 100 kHz, Ipeak = 1 A, He,peak = 1000 A/m) Conductor length l = 1 m PTotal PProx PSkin Definition Skin Effect Losses PSkin Losses due to current I, including loss increase due to self-induced eddy currents. Proximity Effect Losses PProx Losses due to eddy currents induced by external magnetic field He. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 94 APEC 2015 Winding Loss Modeling Litz Wire (1) - What are Litz wires? Idea Advantages of Litz wires HF losses can be reduced substantially Disadvantages of Litz wires High price Heat dissipation difficult Higher RDC www.wikipedia.org Implementation Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 95 APEC 2015 Winding Loss Modeling Litz Wire (2) - Why Litz Wires Have to be Twisted? (1) Bundle-Level Skin Effect Current Distributions Internal Proximity Effect (will be explained later) (Ampere’s Law) (Faraday’s Law) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 96 APEC 2015 Winding Loss Modeling Litz Wire (3) - Why Litz Wires Have to be Twisted? (2) Bundle-Level Proximity Effect (Ampere’s Law) Figures from [19] (Faraday’s Law) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 97 APEC 2015 Winding Loss Modeling Litz Wire (4) – Strand-Level Effects Internal and External Fields lead to Internal and External Proximity Effects Losses in Litz Wires Losses in Solid Wires l=1m l=1m Internal prox. effect Skin effect Skin effect Ext. prox. effect Prox. effect (25 x di = 0.5 mm, Ipeak = 5 A, He,peak = 300 A/m) Mittwoch, 4. März 2015 Gecko-Simulations AG (d = 2.5 mm, Ipeak = 5 A, He,peak = 300 A/m) Jonas Mühlethaler 98 APEC 2015 Winding Loss Modeling Litz Wire (5) – Types of Eddy-Current Effects in Litz Wire Figure from [11] Mittwoch, 4. März 2015 Ch. R. Sullivan, “Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding”, in IEEE Transactions on Power Electronics, vol. 14, no. 2, 1999. Gecko-Simulations AG Jonas Mühlethaler 99 APEC 2015 Winding Loss Modeling Litz Wire (6) – Real Litz Wire Worst Case Litz Wire (parallel connected strands, no twisting) Ideal Litz Wire (ideally twisted strands) Operating Point f = 20 kHz / n = 130 / di = 0.4 mm How do “real” Litz wires behave? [12] Skin Effect / Internal Proximity Effect External Proximity Effect Litz Wire Type 1: Litz Wire Type 2: Rskin,λ skin Rskin,ideal (1 skin ) Rskin,parallel Rprox,λ prox Rprox,ideal (1 prox ) Rprox,parallel 7 bundles with 35 strands each: 4 bundles with 61/62 strands each: skin 0.5 / prox 0.99 skin 0.9 / prox 0.99 [12] H. Rossmanith, M. Doebroenti, M. Albach, and D. Exner, “Measurement and Characterization of High Frequency Losses in Nonideal Litz Wires”, IEEE Transactions on Power Electronics, vol. 26, no. 11, November 2011 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 100 APEC 2015 Winding Loss Modeling Litz Wire (7) - Are Litz Wires Better than Solid Conductors? Skin and Internal Proximity Effect l=1m Solid Wire (d = 2.5 mm, Ipeak = 5 A) Litz Wire (25 x di = 0.5 mm, Ipeak = 5 A) Litz Wire (50 x di = 0.35 mm, Ipeak = 5 A) Litz Wire (100 x di = 0.25 mm, Ipeak = 5 A) External Proximity Effect l=1m Litz Wire of 25 strands with d = 0.5mm Solid Wire with d = 2.5 mm Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 101 APEC 2015 Winding Loss Modeling Foil Windings Enclosed by Magnetic Material Conductor length l = 1 m d = 1.95 mm Ipeak = 1 A Same cross section! 10 mm x 0.3 mm Ipeak = 1 A Advantages of foil windings HF losses can be reduced Lower price compared to Litz wire High filling factor “Skin” of foil conductor larger than of round conductor with same cross section; hence, skin effect losses lower in foil conductor. Disadvantages of foil windings Increased winding capacitance Risk of orthogonal flux Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 102 APEC 2015 Winding Loss Modeling Foil Windings Not Enclosed by Magnetic Material (1) Single Conductor 10 mm x 0.3 mm Ipeak = 1 A Three Conductors l=1m d = 1.95 mm Ipeak = 1 A 10 mm x 0.3 mm Ipeak = 1 A l=1m d = 1.95 mm Ipeak = 1 A FEM @ 8 kHz Orthogonal flux leads to increased skin and proximity effect. Mittwoch, 4. März 2015 Orthogonal Flux! Gecko-Simulations AG Jonas Mühlethaler 103 APEC 2015 Winding Loss Modeling Foil Windings Not Enclosed by Magnetic Material (2) (Foil) Windings with Return Conductors l=1m d = 1.95 mm Ipeak = 1 A 10 mm x 0.3 mm Ipeak = 1 A FEM @ 8 kHz Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 104 APEC 2015 Winding Loss Modeling Overview About Different Winding Types Type Price Skin & Proximity Filling Factor Heat Dissipation Round Solid Wire ++ -- + + Litz Wire -- ++ - -- Foil Winding + + ++ + Rectangular Wire ++ - ++ + Rectangular Wire Foil windings are better only in this region! Round conductor P0 … round conductor Table and Figure from [13] M. Albach, “Induktive Komponenten in der Leistungselektronik”, VDE Fachtagung - ETG Fachbereich Q1 "Leistungselektronik und Systemintegration", Bad Nauheim, 14.04.2011 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 105 APEC 2015 Winding Loss Modeling Skin Effect of Foil Conductor PS FF ( f ) RDC Iˆ2 Geometry Considered (Loss per unit length) with v sinh v sin v 4 cosh v cos v 1 RDC bh h v FF Current Distribution 1 0 f FF evaluated [19] Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 106 APEC 2015 Winding Loss Modeling Proximity Effect of Foil Conductor ˆ2 PP GF ( f ) RDC H S Geometry Considered (Loss per unit length) with sinh v sin v cosh v cos v 1 RDC bh h v GF b 2 v 1 0 f Current Distribution GF evaluated [19] Mittwoch, 4. März 2015 Gecko-Simulations AG (b = 1 m) Jonas Mühlethaler 107 APEC 2015 Winding Loss Modeling Skin Effect of Solid Round Conductor Geometry Considered PS FR ( f ) RDC Iˆ2 (Loss per unit length) with FR evaluated 4 d 2 d 2 1 0 f RDC FR ber0 ( )bei1 ( ) ber0 ( )ber1 ( ) bei 0 ( )ber1 ( ) bei 0 ( )bei1 ( ) 2 2 ber ( ) bei ( ) ber1 ( ) 2 bei1 ( ) 2 4 2 1 1 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 108 APEC 2015 Winding Loss Modeling Proximity Effect of Solid Round Conductor Geometry Considered PP GR ( f ) RDC Hˆ S2 (Loss per unit length) with GR evaluated 4 d 2 d 2 1 0 f RDC GR (d = 1 m) 2 d 2 ber2 ( )ber1 ( ) ber2 ( )bei1 ( ) bei 2 ( )bei1 ( ) bei 2 ( )ber1 ( ) ber0 ( ) 2 bei 0 ( ) 2 ber0 ( ) 2 bei 0 ( ) 2 2 2 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 109 APEC 2015 Winding Loss Modeling Skin and Proximity Effect of Litz Wire Iˆ PS n RDC FR ( f ) n Skin Effect 2 (Loss per unit length) Losses in Litz Wires Proximity Effect l=1m PP PP,e PP,i Iˆ n RDC GR ( f ) H e2 2 2 2 da Internal prox. effect 2 Skin effect Ext. prox. effect (Loss per unit length) Average internal field Hi under the assumption of a homogeneous current distribution inside the Litz wire. Mittwoch, 4. März 2015 Gecko-Simulations AG (25 x di = 0.5 mm, Ipeak = 5 A, He,peak = 300 A/m) Jonas Mühlethaler 110 APEC 2015 Winding Loss Modeling Orthogonality of Winding Losses It is valid to calculate the losses for each frequency component independently and total them up. P ( PS,i PP,i ) i 0 It is valid to calculate the skin and proximity losses independently and total them up. [14] J. A. Ferreira, “Improved analytical modeling of conductive losses in magnetic components”, in IEEE Transactions on Power Electronics, vol. 9, no. 1, 1994. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 111 APEC 2015 Winding Loss Modeling Calculation of External Field He (1D - Approach) Un-Gapped Transformer Cores M 2 2 ˆ P RDC FR/F I NM NGR/F Hˆ avg,m lm m 1 with H avg 1 H left H right 2 it is 4M 2 1 2 3 ˆ P RDC I FR/F NM N MGR/F lm 2 12 b F where H left H avg N … the number of conductors per layer (i.e. N = 1 for foil windings) H right M … the number of layers. (Ampere’s Law) Mittwoch, 4. März 2015 Figure from [19] Gecko-Simulations AG Jonas Mühlethaler 112 APEC 2015 Winding Loss Modeling Short Foil Conductors “Porosity Factor” NbL bF Redefinition of Parameters ' ' 1 f ' 0 ' Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler h ' 113 APEC 2015 Winding Loss Modeling FEM Simulations : Foil Windings N = 2 x 10 h = 0.3 mm bL = 33 mm bF = 37 mm Ipeak = 1 A N=2x7 h = 0.4 mm bL = 37 mm bF = 37 mm Ipeak = 1 A Error < 6.5% Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 114 APEC 2015 Winding Loss Modeling Calculation of External Field He (2D - Approach) Gapped cores: 2D approach is necessary ! Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 115 APEC 2015 Winding Loss Modeling Effect of the Air Gap Fringing Field The air gap is replaced by a fictitious current, which … … has the value equal to the magneto-motive force (mmf) across the air gap. P2 P2 I P1 P1 Vm,air P2 Hdl R air P1 Vm,air P2 Hdl I P1 An accurate air gap reluctance model is needed! Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 116 APEC 2015 Winding Loss Modeling Effect of the Core Material The method of images (mirroring) “Pushing the walls away” Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 117 APEC 2015 Winding Loss Modeling Calculation of External Field He (2D - Approach) Gapped cores: 2D approach Winding Arrangement External field vector across conductor qxi;yk Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 118 APEC 2015 Winding Loss Modeling Different Winding Sections Section 1 Many mirroring steps necessary in order to push the walls away. Section 2 Only one mirroring step necessary (only one wall). Normally, higher proximity losses in Section 1. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 119 APEC 2015 Winding Loss Modeling FEM Simulations : Round Windings (Including Litz Wire Windings) (1) Major Simplification - magnetic field of the induced eddy currents neglected. - This can be problematic at frequencies above (rule-of-thumb) [15] f max 2.56 0 d 2 Results of considered winding arrangements [15] f-range f < fmax f > fmax Error < 5% > 5% (always < 25%) A. Van den Bossche, V. C. Valchev, “Inductors and Transformers for Power Electronics”, CRC Press. Taylor & Francis Group, 2005 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 120 APEC 2015 Winding Loss Modeling FEM Simulations : Round Windings (Including Litz Wire Windings) (2) Results FEM Simulation f max f max f max Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 121 APEC 2015 Winding Loss Modeling Methods to Decrease Winding Losses (1) Interleaving Optimal Solid Wire Thickness l=1m PTotal PProx PSkin [19] Optimal Foil Thickness l=1m PTotal (f = 100 kHz, Ipeak = 1 A, He,peak = 1000 A/m) PProx Litz Wire PSkin l=1m Increase number of strands. (resp.: find optimal number of strands) (f = 20 kHz, Ipeak = 1 A, b = 20 cm, He,peak = 1000 A/m) Avoid Orthogonal Flux in Foil Windings Mittwoch, 4. März 2015 Gecko-Simulations AG Push this point to higher frequencies! Jonas Mühlethaler 122 APEC 2015 Winding Loss Modeling Methods to Decrease Winding Losses (2) Arrangement of Windings Proximity losses increase in more compact winding arrangements. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 123 APEC 2015 Winding Loss Modeling Methods to Decrease Winding Losses (3) Aluminum vs. Copper [13] Proximity losses are lower in aluminum conductors over a wide frequency range. Figure shows a comparison of single round solid conductors in external field. d = 0.25 mm d = 0.50 mm d = 0.75 mm d = 1.00 mm [13] M. Albach, “Induktive Komponenten in der Leistungselektronik”, VDE Fachtagung - ETG Fachbereich Q1 "Leistungselektronik und Systemintegration", Bad Nauheim, 14.04.2011 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 124 APEC 2015 Example Winding Loss Modeling Photo & Dimensions Current Waveform Demonstration in GeckoMAGNETICS Material Grain-oriented steel (M165-35S) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 125 APEC 2015 Outline Magnetic Circuit Modeling Core Loss Modeling Winding Loss Modeling Thermal Modeling Multi-Objective Optimization Summary & Conclusion Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 126 APEC 2015 Thermal Modeling Motivation & Model (1) Ambient Temperature Motivation Thermal modeling is important to … … avoid overheating. … improve loss calculation (since the losses depend on temperature). Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 127 APEC 2015 Thermal Modeling Motivation & Model (2) Model Inductor Transformer Determination of thermal resistors is challenging! Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 128 APEC 2015 Thermal Modeling Heat Transfer Mechanisms Rth T f (T) P Conduction Independent of temperature T for most materials Difficult to determine interfaces between materials Rth T l P A Convection Combined effect of conduction and fluid flow Changes with changing absolute temperature (nonlinear) Good empirical calculation approach available Rth T 1 P A Radiation Small compared to other mechanisms Modeling the system is demanding (nonlinear eq. / to describe which components “sees” the other component). P eff A1 (Tb4 Ta4 ) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 129 APEC 2015 Thermal Modeling Thermal Resistance Calculation : (Natural) Convection (1) Rth T 1 P A is a coefficient that is influenced by … … the absolute temperature, … the fluid property, … the flow rate of the fluid, … the dimensions of the considered surface, … orientation of the considered surface, … and the surface texture. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 130 APEC 2015 Thermal Modeling Thermal Resistance Calculation : (Natural) Convection (2) Empirical solutions known for … vertical plane horizontal plane gap -top: - horizontal: - vertical: - bottom : and more … Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 131 APEC 2015 Thermal Modeling Thermal Resistance Calculation : (Natural) Convection (3) g l Structure of Empirical Solutions - Theory v gravity of earth: 9.81 m/s2 characteristic length volumetric thermal expansion coefficient : 1/T (air) kinematic viscosity: 162.6 ·10-7 m2/s (air) Name Measure of … Nusselt number Nu … improvement of heat transfer compared to the case with hypothetical static fluid. Grashof number Gr … ration between buoyancy and frictional force of fluid. Prandtl number Pr … ratio between viscosity and heat conductivity of fluid. Pr 0.7 (for air) Rayleigh number Ra … flow condition (laminar or turbulent) of fluid. Ra Pr Gr l 1 l f (Gr, Pr ) A A gl 3 Gr 3 T v Nu is the heat conductivity of the fluid air = 25.873 mW/(m K) @ 20C [16] Procedure Nu(Gr, Pr ) Nu (Gr, Pr ) l Rth 1 A [16] VDI Heat Atlas, Springer-Verlag, Berlin, 2010 Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 132 APEC 2015 Thermal Modeling Thermal Resistance Calculation : (Natural) Convection (4) Structure of Empirical Solutions - Example Vertical Plane Nu 0.825 0.387Ra f1 Pr 1/ 6 2 0.492 f1 1 Pr Nu (Gr, Pr ) l (l = h) 9 / 16 16/ 9 Reference: [16] VDI Heat Atlas, Springer-Verlag, Berlin, 2010 Tp Rth T 1 P A Ta Example (h = 10 cm, Tp = 60 C, Ta = 20 C, A = h · h) Rth = 16.6 K/W Increase of Winding Surface Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 133 APEC 2015 Thermal Modeling Thermal Resistance Calculation : Conduction Overview Normally, with natural convection it is Rth,W << Rth,WA Foil Conductor Round Conductor Rth,W is difficult to determine. One difficulty is, e.g., to model the influence of pressure on the thermal resistance. Mittwoch, 4. März 2015 Gecko-Simulations AG Litz wire shows low thermal conductivity. Jonas Mühlethaler 134 APEC 2015 Example Thermal Modeling (1) (1) Horizontal Plane - Top g l v gravity of earth: 9.81 m/s2 characteristic length : l = a*b / ( 2 ( a+b ) ) volumetric thermal expansion coefficient : 1/T (air) kinematic viscosity: 162.6 ·10-7 m2/s (air) gl 3 Gr 3 T v Pr 0.7 (for air) Nu (Gr, Pr ) l Ra Pr Gr 0.766 ( Ra f 2 ( Pr ))1/5 for Ra f 2 ( Pr ) 7 104 Nu 1/3 for Ra f 2 ( Pr ) 7 104 0.15 ( Ra f 2 ( Pr )) with 0.322 f 2 1 Pr 11/20 Mittwoch, 4. März 2015 (turbulent) Rth T 1 P A Resistor Rth is now calculated for one operating point! ( more iterations are necessary!) 20/11 Gecko-Simulations AG (laminar) Jonas Mühlethaler 135 APEC 2015 Example Thermal Modeling (2) (2) Core to Winding RCF l K 1.05 A W RW RWA Drawing represents the cross-section of one coil former side (there are total 2 x 4 coil former sides). Heat flow via mechanical attachment has not been considered. Quantity Values Rth,CA 17.4 K/W Rth,WA 5.8 K/W Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 136 APEC 2015 Outline Magnetic Circuit Modeling Core Loss Modeling Winding Loss Modeling Thermal Modeling Multi-Objective Optimization Summary & Conclusion Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 137 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Introduction to the PFC Rectifier Simplified Schematic Converter Specifications Photo of Converter Goal: Design boost inductor such that the max. current ripple is 4A. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 138 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Boost Inductor Photo of Inductor Selected Inductor Shape Material Grain-oriented steel (M165-35S, lam. thickness 0.35 mm) Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 139 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Pareto-Optimized Design of Boost Inductor Simplified Schematic Boost Inductance Constraints concerning boost inductors max. current ripple IHF,pp,max = 4 A max. temperature Tmax = 120 ºC Mittwoch, 4. März 2015 Gecko-Simulations AG L2,min can be calculated based on the constraint IHF,pp,max.. The maximum current ripple IHF,pp,max occurs when the fundamental current peaks. Jonas Mühlethaler 140 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Pareto-Optimized Design of Boost Inductor - Simplifications Simplified current / voltage waveforms for optimization procedure Expectations Loss overestimation in L2 expected. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 141 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Pareto-Optimized Design of Boost Inductor - Design Space in GeckoMAGNETICS Start LBoost = 2.53 mH Tmax = 120 ºC Core UI 39 – UI 90 cores of DIN41302 Material M165-35S gap 2.0 … 3.0 mm (by 0.25 mm) 70 – 90 stacked sheets (by 10) Winding Solid round wire with fill factor 0.3 Split windings Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 142 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Pareto-Optimized Design of Boost Inductor - Waveform/Cooling in GeckoMAGNETICS Waveform Cooling Forced convection (-> 3 m/s) ILF,peak = 21.8 A Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 143 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Pareto-Optimized Design of Boost Inductor - Performing Calculation Considered effects Procedure A reluctance model is introduced to describe the electric / magnetic interface, i.e. L = f(i). Air gap stray field Non-linearity of core material 2) Core losses are calculated. DC Bias Different flux waveforms Wide range of flux densities and frequencies 3) Winding losses are calculated. 4) Inductor temperature is calculated. 1) Mittwoch, 4. März 2015 Gecko-Simulations AG Skin and proximity effect Stray field proximity effect Effect of core to magnetic field distribution Jonas Mühlethaler 144 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Pareto-Optimized Design of Boost Inductor - Results Filter Losses vs. Filter Volume Pareto Front Prototype built Pareto-front Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 145 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Detailed Modeling in GeckoMAGNETICS GeckoCIRCUITS Mittwoch, 4. März 2015 GeckoMAGNETICS Gecko-Simulations AG Jonas Mühlethaler 146 APEC 2015 Multi-Objective Optimization – Volume vs. Losses Results Photo Conclusion Loss modeling very accurate. Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 147 APEC 2015 Outline Magnetic Circuit Modeling Core Loss Modeling Winding Loss Modeling Thermal Modeling Multi-Objective Optimization Summary & Conclusion Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 148 APEC 2015 Summary & Conclusion Magnetic Circuit Modeling Air Gap Reluctance Calculation Air gap per unit length zy-plane ' zy R ' Rzy y lg 0 a zx-plane ' zx R Results ' Rzx x lg 0t Rg x y Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler lg 0 at 149 APEC 2015 Summary & Conclusion Core Loss Modeling “The best of both worlds” (Steinmetz & Loss Map approach) Loss Map (Loss Material Database) k , ki , , , r , r , qr , T 1 dB Pv ki T 0 dt Mittwoch, 4. März 2015 Gecko-Simulations AG B n dt Qrl Prl l 1 Jonas Mühlethaler P V 150 APEC 2015 Summary & Conclusion Winding Loss Modeling Optimal Solid Wire Thickness Foil vs. Round Conductors l=1m 10 mm x 0.3 mm Ipeak = 1 A PTotal l=1m d = 1.95 mm Ipeak = 1 A PProx PSkin (f = 100 kHz, Ipeak = 1 A, He,peak = 1000 A/m) Gapped cores: 2D approach Losses in Litz Wires l=1m Internal prox. effect Skin effect Mittwoch, 4. März 2015 Ext. prox. effect Gecko-Simulations AG Jonas Mühlethaler 151 APEC 2015 Summary & Conclusion Magnetic Design Environment Magnetics Design Software GeckoMAGNETICS Core Material Database GeckoDB Circuit Simulator GeckoCIRCUITS Automated Measurement System Prototype Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler Verification (e.g. with Calorimeter) 152 APEC 2015 Summary & Conclusion Multi-Objective Optimization PFC Rectifier Inductor Pareto Front Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 153 APEC 2015 GeckoMAGNETICS Save Time Fast and accurate design of magnetic components Easy-to-use for non-expert Increase Flexibility Tool shows more than one realization possibility In-house design of magnetics crucial for optimal designs. Most Loss Effects are Considered Skin- and proximity losses in litz, round and foil windings, air gap stray field losses, DC bias core losses, thermal model, ... Mittwoch, 4. März 2015 Gecko-Simulations AG Jonas Mühlethaler 154 APEC 2015 Additional References [6] 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 [9] J. Mühlethaler, J. Biela, J.W. Kolar, and A. Ecklebe, “Improved Core Loss Calculation for Magnetic Components Employed in Power Electronic Systems”, in Proc. of the APEC, Ft. Worth, TX, USA, 2011. [19] Jürgen Biela, “Wirbelstromverluste in Wicklungen induktiver Bauelemente”, Part of script to lecture Power Electronic Systems 1, ETH Zurich, 2007 [20] 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 [21] J. Mühlethaler, J. Biela, J.W. Kolar, and A. Ecklebe, “Core Losses under DC Bias Condition based on Steinmetz Parameters”, in Proc. of the IPEC - ECCE Asia, Sapporo, Japan, 2010. [22] B. Cougo, A. Tüysüz, J. Mühlethaler, J.W. Kolar, “Increase of Tape Wound Core Losses Due to Interlamination Short Circuits and Orthogonal Flux Components”, in Proc. of the IECON, Melbourne, 2011. [23] 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 Contact Mittwoch, 4. März 2015 Jonas Mühlethaler jonas.muehlethaler@gecko-simulations.com www.gecko-simulations.com Gecko-Simulations AG Jonas Mühlethaler 155
