Modeling, Implementation, and Simulation of Plate-Core Coupled Inductors

Transcription

1 Modeling, Implementation, and Simulation of Plate-Core Coupled Inductors Han (Helen) Cui Committee members: Dr. Khai Ngo (Chair) Dr.GQLu Dr. Dong Ha Dr. Qiang Li Dr. Louis Guido Final defense date: Apr. 8, 17 1

2 Bias Power Supply for Isolated Power Module In telecom application, saving board area allows significant cost reduction Discrete Bias Supply 14 x 1 mm on top side Integrated Bias Supply: 1 x 6 x.5 mm Discrete Bias Supply occupies board area on both sides. Even the smallest discrete transformer for bias supply is quite large: 5 x 5 x.9 mm. In some cases transformer is the tallest part on board. Challenges for integrated magnetics: Limited power loss Limited size Inductance requirement Isolation requirement Integrating magnetics allows 75% expensive board area saving

3 Power Density of Commercial Products Power Density (W/in 3 ) CUI Linear Tech Traco Power Delta Dc/dc converter for bias power supply: To power controller IC or gate driver Power: 1 W Input voltage: 75 V Output voltage: 5 1 V Current:.1. A RECOM Murata Efficiency (%) 3

4 Integrated Magnetics in Power Supply Application: Isolated power supply for telecom Ratings: 1 - W, V 1 V,.1 A. A Most bulky, limits height reduction Linear Technology s Recognized Isolated µmodule Converters (LTM858) Coilcraft smallest coupled inductor: 5 x 5 x.9 mm Planar magnetic with plate-core structure.3 mm core plate 1 mm Winding layers To reduce thickness Compared to surface-mount structure Simple for fabrication Low profile Pcb winding for integration 4

5 Challenges of Magnetic Modeling Flux density distribution of E-core structures with air gap: Flux density distribution of plate-core structure: A l c z Flux flow direction Φ l g Φ relatively uniform flux distribution Predictable flux path Calculate inductance: L N A l g Non-uniform flux distribution Large fringing How to find A and l for calculation? How to model fringing effect? 5

6 Current Image Method for Field/Inductance Concept: reflecting current to many images across the boundaries Top core R c Inductance is the sum of A from all the images: N N 1 L ds A m I m1 n1 n (r, z) H c l g Bottom core images current images Indcutance (µh) Current image method Normalized core radius Rc/Rwo Limitations: R c > Rw, H c > 5l g H c = 5.3 l g H c =.7 l g H c = 1.7 l g H c =.5 l g Roshen, W.A., "Analysis of planar sandwich inductors by current images," in Magnetics, IEEE Transactions on, vol.6, no.5, pp , Sep 199 6

7 Hurley s Method for Field/Inductance Region I Region II Region III Region IV Indcutance (µh) Hurley s method 35% error Hc =.3 mm (~l g ) Simulated L Normalized core radius Rc/Rwo Solution based on Maxwell s equation: Z j M Z p sw j r a p Z sw ln( ) ln( ) S( kr, kr1) S( ka, ka1)[ f ( ) g( )] Q( kh1, kh) dk hhl 1 r1 a 1 Limitations: R c > Rwo Hurley, W.G.; Duffy, M.C., "Calculation of self- and mutual impedances in planar sandwich inductors," in Magnetics, IEEE Transactions on, vol.33, no.3, pp.8-9, May

8 Derivation of Conventional Equivalent Circuit d V R I N N dt p p p p p p d dt m d V R I N N dt s s s s s s d dt m Considering non-ideality R p, R s are winding resistors, R fe is core loss resistor [Ref] W. G. Hurley and W. Wölfle, Transformers and Inductors for Power Electronics, New York: Wiley, 13. 8

9 Challenges of Equivalent Circuit An fly-buck converter for isolated power supply: I 1 I 1 I - Current waveform: V i V o + I t Conventional equivalent circuit t I 1 R p L lk1 N 1 : N L lk R s I + + V R c L m 1 V Difference from conventional transformer: Primary and secondary not conduct at the same time - - Phase shift not always 18 9

10 Dissertation Outline Plate-core coupled inductor (dimensions, material, current) for plate-core Field Distribution by PREF PREF: Proportional-Reluctance, Equal-Flux Magnetizing Inductance from Energy Distribution Leakage L from Leakage Energy Distribution Ac Winding Loss with Additional Eddy Current Core Loss From Core- Loss Density Distribution Impedance matrix R jl R jl R jl R jl Transient core-loss subcircuit Equivalent circuit SPICE model Predicts open-circuit winding loss Includes phase-shift impact on winding loss Dynamic core loss for any structure 1

11 Summary of Accomplishments Challenges Improvements Outcomes Flux penetrates surface winding loss model? PREF. m Total current density: J J J () x j self prox y H Jy() x eddy x Additional Eddy current losses Winding loss impacted by phase shift Even A still has loss LTSPICE model for coupled windings: Non-linear core loss: P C f B c m ac P V R c m / c Only works for β = > 1% error 11

12 Outline 1

13 Motivation of PREF Model Generate contours with: Equal Flux flux Φ flowing through each path is constant NI i path _ i Proportional Reluctance reluctance of each path proportional to Ampere-turns NI z R c µ r Φ i Φ i+1 Φ Φ H c l g r Φ Φ Φ path _ i wind _ i core _ i fring _ i Reluctance determined by contour shape and boundary conditions 3D view of a path R wi R wo Shape of contours: Fringing Schwartz-Christoffel transformation Core elliptical + hyperbolic tangent Winding elliptical Ampere-turns NI enclosed by a path Φ 13

14 Fringing Region Z Finite core r θ f Input: core radius, core height, air gap length Method: SC transformation toolbox in Matlab Output 1: contours in fringing region Physical domain P 1 Infinite large core P 1 P 1 Conformal map Finite core infinite domain Modeled flux matches simulation P P P Infinite large core 14

15 Core Region z ( r1.4e3) z (.86e3) (.57e3) 1 Ellipse Hyperbolic z e e r tanh(1 (1.8-3 )) µ r (R x(i), Z p ) R c From SC transform r (.7mm,. mm) R Hy Rc=1.7mm, Rwo=1.65mm, Hc=.3mm, s/w=5um, lg=.5mm, Sps=.1mm, Slayer=35um, Rwi=.9mm, h=18um First half of contour is modeled by ellipse: ( ) r R Hy z acore _ i bcore _ i 1 Secondary half of contour is modeled by hyperbolic tangent: z tanh( ( r)) i core_ i i Each contour uniquely determined by boundary conditions and (R x(i), Z p ) 15

16 Winding Region z ( r1.4e3) z (.86e3) (.57e3) 1 z e e r tanh(1 (1.8-3 )) µ r circular r (r1.4 e 3) z (.9e 3) (R x(i), Z p ) (.7mm,. mm) R Hy R c From SC transform Rc=1.7mm, Rwo=1.65mm, Hc=.3mm, s/w=5um, lg=.5mm, Sps=.1mm, Slayer=35um, Rwi=.9mm, h=18um Modeled by circular function: ( rh ) z R wind _ i wind _ i Each contour uniquely determined from (R x(i), Z p ) and boundary conditions 16

17 Methodology: Obtaining the Contours z Elliptical Hyperbolic r Circular R x(3) R x() R x(1) Φ Φi Φ SC transform NI i path _ i Flow chart: Initialize R x(i) Contour segment in the fringing Contour segment in the core Contour segment in the winding path _ i wind _ i core _ i fring _ i If Φ i Φ i> i= 1, Adjust R x(i) Calculate reluctance Calculate initialized _ and Φ i reluctance _ and Φ 17

18 Modeling Result of Magnetic Field z Modeled core Path winding fringing r Path 3 Path 1 core Total energy integrated from field error < 1% B (mt) in the core region Simulated path 3 Simulated path Normalized distance Simulated path Path 1 Path Path 3 B (mt) in the fringing region Simulated path 1 Simulated path 3 Simulated path Normalized distance Path 1 Path Path 3 B (mt) in the winding region Simulated path 1 Simulated path Normalized distance Path 1 Path Path 3 Simulated path 18

19 Outline 19

20 Extracting the Impedance Matrix V1 Z11 Z1 I1 R11 jl11 R1 jl1 I1 V Z Z I R jl R jl I I 1 I V 1 V Resistance matrix from winding loss Inductance matrix from energy * * R11 R1 I 1 L11 L1 I 1 P loss I1 I * R1 R E I1 I * I L1 L I 3 unknowns 3 unknowns Solved from three field conditions: CM : i 1 A, i Ni P R i R N i R Ni, E L i L N i L Ni DM : i 1 A, i Ni P R i R N i R Ni, E L i L N i L Ni CM DM : i A, i A P Ri, E Li Energy E calculated from field integration Winding loss P calculated from boundary field around conductors

21 Energy from Field z Modeled r z r i+1 ϕ r z i+1 B z B z r i B(r, z ) r i Enclosed Ampere-turns (ra) i = Φ i Magnetic field calculation from flux contours: 1 B r z r z B r B z (, ) ( ) r z r zi zi 1 ri ri 1 Inductance is calculated from energy: 1 E E B( r, z) dv, L I Φ = Φ i Φ i+1 (ra) i+1 = Φ i+1 1

22 Review of Winding Loss Models 1D 1D model by Dowell: F ac sinh sin (p 1) sinh sin cosh cos 3 cosh cos Proximity effect factor for sine excitation 5 1D solution 4.5 Simulation 4 H y H yh h 4% error Fac p 8 (Number of layers) h Wf W Assumptions: 1. The magnetic field in the core is (µ r ~ inf). Only vertical field exists, parallel to surface Freq(Hz) x 1 6 P. L. Dowell, "Effects of eddy currents in transformer windings," in Electrical Engineers, Proceedings of the Institution of, vol. 113, no. 8, pp , August 1966.

23 Improved 1D Winding Loss Model with D Effects Edge effect A correction factor is added to 1D solution: or Rac R1 D CF( b'/ a') where X * is corrected by curve fitting Assumptions: 1. Assumes the field is mostly 1D, only models D at the edge. The correction factor is obtained from numerical regression Robert, F., P. Mathys and J. P. Schauwers. "A closed-form formula for -D ohmic losses calculation in smps transformer foils." IEEE Trans. Power Electron., on 16, no. 3 (1): N. H. Kutkut, "A simple technique to evaluate winding losses including two-dimensional edge effects," IEEE Trans. Power Electron., vol. 13, no. 5, pp , Sep

24 D Winding Loss Model from D Field H x (a) H y () H y (h) H x () H cosh( ) x H Hyh Hy kh xa k j, 1, P1 k, Q1 khy a sinh( kh) Hy Hyh Hxa Hx cosh( ka) I, P k, Q khx, Jdc h sinh( ka) a h J1( x) 1 P1cosh( kx) Q1sinh( kx) J( y) Pcosh( ky) Qsinh( ky) J J1J Jdc 1 Pac J Jdxdy Assumption: D fields are parallel to conductor surface H y () H y () H x (a) J 1 (x) H x () H x (a) J (y) H x () H y (h) H y (h) N. Wang, T. O'Donnell, and C. O'Mathuna, "An improved calculation of copper losses in integrated power inductors on silicon." IEEE Trans. Power Electron., vol. 8, no. 8, pp , 13. Lotfi, A. W. and F. C. Lee. "Two dimensional field solutions for high frequency transformer windings." In Proc. Power Electronics Specialists Conference, PESC '93 Record., 4th Annual IEEE, ,

25 Eddy Current Loss Calculation Total winding loss [1]: 4 ldc db Pt () ( ) 64 dt P c di1 di di dt D dt dt di dt 1 total (Maxtrix D is extracted from FEA) Total current density []: J ( xy, ) J1( x) J( y) [ Bx y By x ] B y B x P loss 1 dxdy J( x, y) Both assumes uniform B field [1] C. R. Sullivan, "Computationally efficient winding loss calculation with multiple windings, arbitrary waveforms, and two-dimensional or three-dimensional field geometry," in IEEE Transactions on Power Electronics, vol. 16, no. 1, pp , Jan 1. [] Roshen, W. A. "Fringing Field Formulas and Winding Loss Due to an Air Gap." IEEE Trans. Magnetics., on 43, no. 8 (7):

26 D Field in Plate-Core Structure Plate-core inductor z Field vector plot at cross section: r Field magnitude around windings: H x distribution H y distribution Need to use D winding loss model for loss calculation 6

27 Improved D Winding loss from Field Conventional D loss model: Plate-core structure: H eddy winding Boundary field Flux is parallel Ac current density for winding loss: Total winding loss (W) J J J x j Additional Eddy J y () skin prox y e ( ) H x ddy x.5 P current losses Frequency dependent 1 * J J dxdy Simulated.5 Conventional Modeled Winding width (um) H H field (A/m) eddy Flux not parallel windings boundary field under dc MHz - 1 MHz Distance w 4 w 3 w w / H.( ).788( ) 1.363( ).331( ) dc Additional eddy loss Frequency impact Normalized Validated over wide range 7

28 Outline Predicts open-circuit winding loss Includes phase-shift impact on winding loss Dynamic core loss 8

29 Prediction of Loss in Open-Circuit Winding When I = A, I 1 A: Inductor with plate cores I 1 R p L lk1 N 1 : N L lk R s I + + V R c L m 1 V - primary secondary - All the losses are on R p Predicted loss in conventional model: Loss distribution in open winding I1 I Primary Secondary 1 A 1 A A 1.8 W W A + V - Simulated loss with secondary open: by proximity effect I1 I Primary Secondary How to include open-circuit 1 A A 1.33 W.56 W Open-circuit loss simulated: loss in circuit model? 9 (5% of primary)

30 Prediction of Winding Losses vs. Phase Shift Conventional transformer model: o phase shift I 1 R p L lk1 N 1 : N L lk R s I + + V R c L 1 m V Ac winding loss distribution 1 A 1 A - - P loss = I 1 R 1 + I R no phase infomation between I 1 and I 18 o phase shift Total winding loss (W) Conventional model 81% difference Phase shift (degree) Ac winding loss distribution 1 A -1 A How to model phase impact on winding loss? 3

31 Prediction of Transient Core Loss Conventional transformer model: I 1 R p L lk1 N 1 : N L lk R s I Material MHz + + V R c L 1 m V constant - over time - β =.5 β =.64 β = 3.1 Loss on R c represents core loss: Core loss from Steinmetz equation: P C f P ~ V P ~ V B Core loss (mw) Conventional Simulation β =.8 Use fixed R c gives large error when β > 1% error Current excitation (A) c m c m c m ac Transient core loss not correct How to model transient core loss? 31

32 Improved Equivalent Circuit for Coupled Windings I 1 R p L lk1 N 1 : N L lk R s I + + Conventional model: V 1 R c L m V - - R cp _ lk R cs _ lk I 1 I Improved model: (based on field distribution) L lk1 L k = 1 R lk R lk1 lk R cp _ m L m1 L ms 1 R cs _ m Two mutual inductors k = 1 Mutual core-loss resistor R wp _ m L m L m s R ws _ m Mutual winding-loss resistor primary secondary 3

33 Model Derivation from Flux Lines Flux line distribution with one side conducting: Reluctance model: Simplified one-turn structure 1w w pw w pw 1w N 1 I 1 A NI 1 1 lk1 m1 m Symmetric equivalent circuit: Duality to circuit model R lk1 L lk1 L lk R lk 1w w ' + I 1 R wp L m1 L m L ms 1 L ms Define R ws I N 1 I 1 primary + w - + pw w ' pw ' secondary 33

34 Parameters Extraction I R L 1 lk1 lk 1 L lk R lk I Rwp Rwp L m1 L m k = 1 k = 1 Input: Impedance matrix L ms 1 L m s Z Z I Rws Rws Z Z Method: Solve (1) (8) equations Output: SPICE model with 8 parameters R lk1, L lk1, L m1, R wp, R lk, L lk, L m1s, R ws I From voltage equations: Lm Rws slm Lm s Z11 Zlk1 slm 1 Lm Rws R sl sl L R Z s L L 1 m1 m1s ws m m ms wp Lm RwssLm Lm s L R R sl sl From winding loss on primary and secondary: P I R I R wp1 1 lk1 Rwp wp P I R I R m ws ws m m Lm s Rwp RwssLm Z Zlk slm 1s L R R sl sl m ws ws m m Lm s Rwp ws1 lk Rws ws (1) () (3) (4) (5) (6) (7) (8) 34

35 Prediction of Loss in Open-Circuit Winding Simulation: SPICE model: I 1 I R L lk1 L k = 1 lk lk1 R lk L m1 L ms 1 Loss distribution in open winding 1 A A + V - by proximity effect primary R wp _ m P open L m k = 1 V R open ws _ m L m s R ws _ m + V open - secondary Simulated loss with secondary open: I1 I Primary Secondary 1 A A 1.33 W.56 W Open-circuit loss modeled: I1 I Primary Secondary 1 A A W.6 W 35

36 Prediction of Winding Losses vs. Phase Shift o phase shift I 1 SPICE model: I Ac winding loss distribution R L lk1 L k = 1 lk lk1 R lk 1 A 1 A L m1 k = 1 L ms o phase shift Ac winding loss distribution 1 A -1 A Winding loss (W) primary Model captures phase impact by mutual resistors R wp _ m L m L m s R ws _ m Conventional Improved Maxwell V open secondary Phase-shift angle (degree) - 36

37 Outline 37

38 Core Loss Modeling in Frequency Domain The most commonly used is Steinmetz equation: P k f B c ac Limitations: Only sinusoidal No H dc considered Modified Steinmetz Equation: Relates to averaged db/dt in one cycle B ac Flux waveform in a Buck converter P k f B f 1 c eq ac f eq 1 T db dt B dt ac Use same Steinmetz parameters Equivalent frequency is critical Arbitrary waveform Frequency domain Averaged field information Equivalent circuit not available Current waveform information [Ref] Reinert, J.; Brockmeyer, A.; De Doncker, R.W.A.A., "Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation," Industry Applications, IEEE Transactions on, vol.37, no.4, pp.155,161, 1. 38

39 Core Loss Modeling in Time Domain B m H m Equivalent elliptical loop(eel) method for dynamic core loss: B Bm H H sin cos m According to parametric equation of an ellipse The transient B becomes the y-coordinate of an ellipse The transient H becomes the x-coordinate of an ellipse The core loss for ferrite material in time domain: db pv () t K dt 1 K Cm Bmcos C C ( ) cos d C m, α, β are the steinmetz parameters B m is the maximum of one minor loop Constant after knowing α, β D. Lin, P. Zhou, W. N. Fu, Z. Badics, Z. J. Cendes, "A dynamic core loss model for soft ferromagnetic and power ferrite materials in transient finite element analysis," IEEE Trans. Magnetics,, vol. 4, no., pp , Mar

40 Field Factor for Core Loss Sub-Circuit Modeled B field in core plates: 8, 37.3, 1.195,.6 C m B eff used for core loss : B eff ( t ) 1 Brzt (,, ) V 1 core _ tube _ i dv Loss uniform = Loss non-uniform 4F1 material 4 B ( t )/ I t, eff 1 1 db dt di dt : Relates current to the field The core loss for ferrite material in time domain: dbeff () t pv ( t) K dt 1 K Cm Bm_ eff cos C C ( ) c os d

41 Outline 41

42 Core Loss Resistors Extraction Core loss sub-circuit Use of core loss sub-circuit: Input attributes: Cm, α, β, C αβ, I 1 R L lk1 L k = 1 lk lk1 R lk primary R cp1 R cp Rwp L m1 L m k = 1 L m 1 s L m s R cs1 R cs Rws secondary I Core loss (W) Current I(t) Core loss resistance value determined from V /core loss R cp V P Maxwell SPICE Time (us) cp core 4

43 Prediction of Transient Core Loss Material MHz R cp _ lk I 1 (t) R cs _ lk (t) I β =.5 R L lk1 L k = 1 lk lk1 R lk R cp _ m (t) L m1 L ms 1 R cs _ m (t) β =.64 β = 3.1 R wp _ m L m k = 1 L m s R ws _ m Pc ~ V m Pc ~ when β Transient core loss V m Improved Variable resistance from core loss sub-circuit Dynamically calculates R c Core loss (mw) primary Conventional Simulation Maxwell Improved β =.8 secondary Current excitation (A) 43

44 Test in a FlyBuck Converter Conventional model Structure with :1 turn ratio Rc=1.7mm, Rwo=1.65mm, Hc=.3mm, s/w=5um, lg=.5mm, Sps=.1mm, Slayer=35um, Rwi=.9mm, h=18um Impedance matrix R(Ω), L(µH) Improved model I1 I I , ,.664 I.744, ,.379 Material used LTCC41 C m α β C αβ LTCC

45 Testing Results of Voltage and Loss Core loss (mw) Maxwell Improved Conv Time (us) Primary Secondary Voltage (V) Time (us) 45

46 Testing Results of Losses Total winding loss (mw) Improved Conventional Duty cycle 7 Maxwell 6 Improved 5 Conventional Total core loss (mw) Winding loss (mw) D =.6 loss on primary loss on secondary Maxwell Improved Conv Vin (V) 46

47 Outline Experimental verification 47

48 Prototype of Coupled Inductors 1 st Row - vary air gap: N = 8, R wi = 3 mm Prototype of core and winding: L g =.4 mm L g =.5 mm L g =.75 mm L g = 1 mm OD = 1 mm Thickness =.3 mm nd Row - vary R wi : N = 8, l g =.5 mm 4F1 material from Ferroxcube Double-layer flexible circuit 1 mm mm 3 mm R wi 4 mm 3 rd Row - vary number of turns: lg =.5 mm N = 6 N = 4 N = 6 N = 4 R wi = 3 mm R wi = mm 48

49 Measurement Results of L and R Open Inductance verification: Inductance (uh) Air gap length (mm) Inductance (uh) Winding inner radius (mm) Inductance (uh) Number of turns Ac resistance verification: Maxwell PREF model Measurement ACR (Ω) 6 4 ACR (Ω) 3 1 ACR (Ω) Inner winding radius (mm) Air gap length (mm) Number of turns 49

50 Measurement Results of L and R Short Inductance verification: Llk (nh) Inner winding radius (mm) Air gap length (mm) Llk (nh) Llk (nh) Number of turns per layer Ac resistance verification: Maxwell PREF model Measurement ACR_short (Ω) Inner winding radius (mm) ACR_short (Ω) ACR_short (Ω) Air gap length (mm) Number of turns 5

51 Summary of Accomplishments Challenges Improvements Outcomes Flux penetrates surface winding loss model? PREF. m Total current density: J J J () x j self prox y H Jy() x eddy x Additional Eddy current losses Winding loss impacted by phase shift Even A still has loss LTSPICE model for coupled windings: Non-linear core loss: P C f B c m ac P V R c m / c Only works for β = > 1% error 51

52 Reasons for Modeling beyond Simulation 1. Modeling results can be integrated by an equivalent circuit All the parameters are based on modeling of field, inductance, winding loss, core loss. Provides more insights towards the problem Reluctance in the air is dominant for inductance Penetrating field gives rise to winding loss J J J x j () skin prox y Core loss is related to db/dt and Bm 3. Provides opportunity for further optimization Inductance reduce reluctance in the air region Winding loss use foil windings, variable winding width for each turn Core loss avoid large db/dt and Bm, reduce field factor 5

53 Publications Journals: H. Cui, K. D. T. Ngo, J. Moss, M. H. Lim, and E. Rey, "Inductor geometry with improved energy density," IEEE Trans. Power Electron., vol. 9, no. 1, pp , Oct. 14. H. Cui and K. D. T. Ngo, "Constant-flux inductor with enclosed winding for high-density energy storage," Electronics Letters, vol. 49, no. 13, pp , June. 13. H. Cui and K. D. T. Ngo, "Synthesis and design of a distributed inductor," IEEE Trans. Industrial Electron., accepted, 16. H. Cui and K. D. T. Ngo, "Field modeling for plate-core inductor with significant fringing using equal-flux contours," IEEE Trans. Magn., under review, 17. H. Cui and K. D. T. Ngo, Equivalent circuit for gapped coupled-windings from flux distribution," IEEE Trans. Power Electron., under preparation, 17. Conference: Cui, Han; Ngo, Khai D.T.; Moss, J.; Lim, M.; Rey, E., "Design and evaluation of the constant-flux inductor with enclosed-winding," in Energy Conversion Congress and Exposition (ECCE), 14 IEEE, vol., no., pp , Sept. 14 Cui, Han; Ngo, Khai D.T., "Inductor geometries with significantly reduced height," in Applied Power Electronics Conference and Exposition (APEC), 13 Twenty-Eighth Annual IEEE, vol., no., pp , 17-1 March 13 Patent: Khai Ngo and Han Cui, Compact inductor employing redistributed magnetic flux, VTIP No Khai Ngo, Han Cui, and Chi-Ming Wang, Cooling with EMI-limiting inductor, VTIP No

54 Dissertation Outline (Part I) Chapter 1 Introduction 1.1 Background 1. Modeling methods for plate-core inductor 1.3 Equivalent circuit for two-winding inductors 1.4 Contributions of this dissertation Chapter Proportional-Reluctance, Equal-Flux Model.1 Concept of PREF model. Functions for the flux lines.3 Reluctance calculation.4 Modeling results Chapter 3 Inductance Matrix of Coupled Inductor CM field from PREF - 3. DM field Inductance matrix from energy Example utilization of model 54

55 Dissertation Outline (Part II) Chapter 4 Winding Loss Modeling D magnetic field around winding - 4. Winding loss calculation with additional eddy current loss Experimental verification Example utilization Chapter 5 Time-Domain Core Loss Modeling Effective magnetic flux density in the core - 5. Sub-circuit modeling of core loss in LTSPICE Example utilization Chapter 6 Equivalent Circuit for Coupled Windings Limitations with conventional equivalent circuit - 6. Equivalent circuit from flux distribution Extraction of model parameters from impedance matrix Results and example utilization Chapter 7 Conclusion and Future Work 55

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