Thermal analytical winding size optimization for different conductor shapes

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1 ACHIVES OF ELECTICAL ENGINEEING VOL 6(), pp (15) DOI 11515/aee Termal analytical inding size optimization for different conductor sapes AFAL P WOJDA ABB Corporate esearc Center, DMPC &D Team Staroislna 13a, 31-38, Krakó rafalojda@plabbcom (eceived: 191, revised: 5111) Abstract Te aim of tis paper is to derive an analytical equations for te temperature dependent optimum inding size of inductors conducting ig frequency ac sinusoidal currents Derived analytical equations are useful designing tool for researc and development engineers because indings made of foil, square-ire, and solid-round-ire indings are considered Temperature dependent Doell s equation for te ac-to-dc inding resistance ratio is given and approximated Termally dependent analytical equations for te optimum foil tickness, as ell as valley tickness and diameter of te square-ire and solid-round-ire indings are derived from approximated termally dependent ac-to-dc inding resistance ratios Minimum inding ac resistance of te foil inding and local minimum of te inding ac resistance of te solid-round-ire inding are verified it Maxell 3D Finite Element Metod simulations Key ords: Doell s equation, Eddy currents, FEM, inductors, optimization, proximity effect, skin effect, termal effects, inding losses 1 Introduction Poer inductors are important elements in poer conversion process Tis is because teir ability to store te magnetic energy in te magnetic field blocks te flo of te alternating currents Moreover, poer inductors along it te capacitors form a resonant circuit, ic sapes te voltage in te poer amplifier, or as a filter, ic sapes te current aveform in te poer converter Additionally poer inductors and capacitors are used in impedance matcing, ere te active devices are matced to te load or anoter poer stage of te poer system [1-3] Tey are also used in te noise filtering [-6] Common failure mecanisms in inductors are mainly oing to excessive temperature rise [7, 8] Te excessive temperature causes cracks in te ferrite core and ence decreases te inductance of te inductor or te transformer Te inductance reduced due to te cracks of te core results in increased peak of te current in te inding and cause unexpected current stresses in te active devices tat may lead to destruction of te active device Terefore, in te design process of te poer inductors beside magnetic requirements, te termal limitations sould be also taken into te consideration Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

2 198 afal P Wojda Arc Elect Eng Poer losses in inductors are caused by dc and ac currents flo in te inding, ysteresis and eddy currents loss in te magnetic core, as ell as due to te dielectric losses in te insulators of te inding and core For te ig poer inductors, from aforementioned losses, significant role plays te inding losses [9-1, 31-36] Wile core and dielectric loss are decreased by selection of lo-loss materials, te inding losses are significantly reduced by optimization of te inding conductor size [16, 17, 19] To approaces for inding size optimization and inductor inding loss minimization are knon in te ig frequency and ig poer inductor designing process First approac, consider utilization of computer-aided design, ere D/3D model of te inductor is solved directly from Maxell s equations utilizing finite element metods (FEM) [-, 3] In tis metod solution is accurate but adaptation of te solution to te oter cases is a very difficult task Additionally, FEM approac is very time and memory consuming Second approac is utilization of analytical equations in designing process Tis approac gives similar results to FEM simulations and measurements [3-8] and inding optimization is almost instant and can be performed itout using computer Additionally, analytical equations gives a lot of insigt into various dependencies of inding resistances and tey can be easily used in industrial environment Te most commonly used teory considering inding resistance as introduced by Doell in 1966 [9] Doell determined directly from te Maxell s equation te magnetic flux N cutting te inding space, from ic e derived inding impedance based on te voltage induced in te conductor due to magnetic flux N In is teory, Doell derived equations based on te assumption tat te magnetic field in te inductor is parallel to te conductor, and as a result te curvature, end, and edge effects are neglected Moreover, Doell assumed tat te inding operates at room temperature Tis paper consider te analytical optimization of te inductor inding size utilizing temperature dependent Doell s equation for sinusoidal currents Utilization of te derived equations for te optimum conductor size can be applied in design of te magnetic components used in inverters, poer amplifiers, filtering and matcing circuits Te main objectives of tis paper are: adaptation of termal effect into Doell s equation; approximate termally dependent Doell s equation; derive temperature dependent optimum foil tickness; derive temperature dependent valley tickness of te square-ire inding; derive temperature dependent valley diameter of te solid-round-ire inding; verify derived termally dependent Doell s equations for te foil and te solid-roundire inding using Maxell 3D Finite Element Metod (FEM); verify te ac inding resistance of derived equation for te temperature dependent optimum foil tickness using 3D FEM; verify te ac inding resistance of derived equations of te temperature dependent valley diameter of te solid-round-ire inding using 3D FEM Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

3 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 199 Doell s inding resistance and its approximation For te inding conductor conducting sinusoidal current i l ( ω t) I sin( πf t), I sin (1) m m te inding resistance of te inductor at room temperature is described by Doell s equation [9] sin(a) sin(a) ( N sin( A) sin( A) F A, cos( ) cos( ) 3 cos( ) cos( ) () dc A A A A ere I m is te current amplitude, ω is te angular frequency, t is time, f is te current frequency A is te effective tickness or diameter of te inding conductor at room temperature, N l is te number of layers, is te inding ac resistance, and dc is te inding dc resistance From () it can be seen tat te inding resistance ratio is a complex nonlinear equation consisting of yperbolic and trigonometric functions, and terefore, analytical optimization is impossible In general, inding resistance depends on te inding size, operating frequency, and temperature Doell s equation as been derived for te inding operating at te room temperature Hoever, most of industrial devices consisting of magnetic components operates at temperatures muc iger tan te room temperature Terefore, for te efficient inductor design, tere is a uge need for adaptation of Doell s equation and optimization of te inding sizes including termal effects Termally dependent skin dept of te inding conductor is given by * 1 (3) f ρ δ, ωμσ ( T ) πμ σ πμ f ere σ (T) is te temperature dependent inding conductor conductivity, μ is te free space permeability, and ρ (T) is te temperature dependent conductor resistivity Te resistivity of te conductor is strongly dependent on te temperature and is described by ρ ρ[ α( T )], () 1 T ere ρ(t ) is te resistivity of te conductor at te room temperature, T 9315 K is te room temperature, T is te conductor temperature, and α is te temperature coefficient of te inding conductor For te copper inding, α 393 (K!1 ) and ρ(t ) 17 1!9 (Ω/m) [7, 8, 16] Substituting () into (3) one obtains te temperature dependent skin dept of te inding conductor ρ( T T T )] δ πμ f (5) Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

4 afal P Wojda Arc Elect Eng Due to complexity of Doell s equation te analytical optimization is impossible and terefore, approximation of Doell s equation is required Expanding yperbolic and trigonometric functions into Maclaurin s series te first term of Doell s equation is approximated by sin( A) sin(a) 1 A cos(a) cos(a) A 3 (6) Fig 1 Foil inding ac resistance as functions of temperature and foil tickness at b 8 mm, f 1 khz, N l 16, and l 5 m Similarly, second term of Doell s equation can be expanded into Maclaurin s series yielding approximated expression sin( A) sin( A) 1 3 A (7) cos( A) cos( A) 6 Substituting (6) and (7) into (), one obtains te approximate ac-to-dc inding resistance ratio 5N l 1 F 1 A for A (8) 5 3 Termal optimization of te foil inding For te foil inding inductor, te effective tickness of te conductor is given by [9, 1, 16] A (9) δ Substituting (5) into (9) one obtains te temperature dependent effective tickness of te conductor Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

5 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 1 ( ) A T T δ ( ) πμ f, ρ T T T )] ( (1) ere is te foil tickness Te temperature dependent foil inding dc resistance is given by dc ρ l ρ( T T T )] l, b ere l is te inding conductor lengt and b is te foil idt Substituting (1) into () ones obtains temperature dependent Doell s Equation [9] A T F sin(a( T )) sin(a( T )) cos(a( T )) cos(a( T )) dc l ( ) b ( N 1) sin( A( T )) sin( A( T )) 3 cos( A( T )) cos( A( T )) Te temperature dependent ac inding resistance of te foil inding is given by l b πμ fρ( T T T )] sin(a( T )) sin(a( T )) ( N sin( A( T )) sin( A( T )) cos( ( )) cos( ( )) 3 cos( ( )) cos( ( )) A T A T A T A T Figure 1 sos 3D plot of foil inding ac resistance as functions of temperature and foil tickness at b 8 mm, f 1 khz, N l 16, and l 5 m It can be seen tat for te certain tickness of te foil, ie optimum tickness, te inding ac resistance exibit a global minimum Moreover, it can be seen tat as te temperature increases, te optimum tickness increases Additionally, as te temperature increases, te inding resistance increases Substituting (1) into (8) te temperature dependent approximated ac-to-dc inding resistance ratio for te foil inding is given by F 5N l 1 (5N πμ 1 A ρ( T T T )] Te approximated temperature dependent ac inding resistance of te foil inding is given by F dc (11) (1) (13) (1) ρ( T T T )] l b (5N 1 5 πμ ρ( T T T )] (15) ρ( T T T )] l b 1 (5N 5 3 πμ ρ( T T T )] Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

6 afal P Wojda Arc Elect Eng Taking te derivative of (15) it respect to te foil tickness and equating te result to zero one obtains d d ρ( T T T )] l b 1 (5N 15 Tis yields te temperature dependent optimum foil tickness πμ ρ( T T T )] (16) opt [1 ( T T )] 15 5N 1 ρ α πμ f l (17) Substituting (17) into (15) one obtains ac inding resistance at global minimum min l πμ ρ( T T T )] f 3b 15 5N 1 l (18) Termal optimization of te square-ire inding For te square-ire inding inductor, te effective tickness of te conductor is given by [1, 16] A η (19) δ Substituting (5) into (19), one obtains te temperature dependent effective tickness of te square-ire inding conductor Fig Normalized square-ire inding ac resistance as functions of temperature and square ire tickness at η 8, f 1 khz, N l 1, and l 1 m Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

7 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 3 ( ) A T T δ ( ) η πμη f ρ T T T )] ( () Te temperature dependent square-ire inding dc resistance is given by ρ l ρ( T T T )] l (1) and te temperature dependent normalized ac inding resistance of te square-ire is given by F r F η ( ) δ δ () sin(a( T )) sin(a( T )) ( N sin( A( T )) sin( A( T )) cos( ( )) cos( ( )) 3 cos( ( )) cos( ( )) A T A T A T A T Figure sos te 3D plot of normalized ac inding resistance of te square inding as functions of square-ire tickness and temperature at η 8, f 1 khz, N l 1, and l 1 m It can be seen tat te beaviour of te square-ire inding resistance at constant frequency is different tan for te foil inding As te tickness of te square-ire inding increases, te inding resistance decreases, reaces local minimum, ten increases, reaces local maximum, and ten decreases, ile for te foil inding it remains constant Tis is because for te foil inding, idt and tickness of te foil are independent variables, ile for te square-ire, tese variables are self-dependent Similarly, as for te foil inding, as te temperature increases, te valley tickness of te square-ire inding increases Hoever, as te temperature increases te local minimum of te inding ac resistance remains constant Te approximated temperature dependent ac-to-dc inding resistance ratio for te square-ire inding is F 5N l 1 (5N πμη 1 A ρ( T T T )] (3) Te approximated temperature dependent ac inding resistance of te square-ire inding is given by F dc ρ( T T T )] l (5N 1 5 πμη ρ( T T T )] () ρ( T T T )] l 1 (5N 5 πμη ρ( T T T )] Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

8 afal P Wojda Arc Elect Eng Fig 3 Normalized inding ac resistance of te solid-round-ire inding as functions of temperature and round ire diameter Taking te derivative of () it respect to te square-ire tickness and equating te result to zero one obtains d d 1 (5N 1) l πμ η ρ( T T T )] l 3 5 ρ( T T T )] (5) Tis yields te temperature dependent valley tickness of te square-ire inding v ρ( T T T )] 5 πμ η f 5N l 1 (6) 5 Termal optimization of te solid-round-ire inding For te solid-round-ire inding inductor, te effective diameter of te inding conductor is given by [16] 75 π d A η (7) δ ere d is te diameter of te solid-round-ire inding Substituting (5) into (7) one obtains te temperature dependent effective diameter of te solid-round-ire inding conductor 75 d f ( T )[1 ( T T )] ( ) π π πμη η d δ ρ α A T 75 (8) Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

9 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 5 Te temperature dependent solid-round-ire inding dc resistance is given by dc ρ l ρ( T π π T T )] l (9) d d and te temperature dependent normalized solid-round-ire ac inding resistance is given by F r π 75 ( ) η F d d ( ) δ δ (3) sin(a( T )) sin(a( T )) ( N sin( A( T )) sin( A( T )) cos( ( )) cos( ( )) 3 cos( ( )) cos( ( )) A T A T A T A T Figure 3 sos 3D plot of te normalized solid-round-ire inding ac resistance as functions of temperature and ire diameter at η 8, N l 6, f 1 khz, and l 8 m It can be seen tat te beavior of te normalized ac inding resistance for te solid-round-ire indings is similar to te beaviour of normalized inding ac resistance for te square-ire indings Moreover, it can be seen tat te minimum of te inding ac resistance for te solid-round-ire inding is independent on te temperature Hoever, te diameter of te ire at ic te local minimum occurs increases as te temperature increase Approximated temperature dependent ac-to-dc inding resistance ratio for te solidround-ire inding is F 3 5N l 1 (5 1) ( ) 1 ( ) 1 π N l d πμη T A T 5 5 ρ( T T T )] (31) Te approximated temperature dependent ac inding resistance of te solid-round-ire inding is given by F dc ρ( T T T )] l π d π 1 3 (5N d 5 πμη ρ( T T T )] (3) ρ( T T T )] l π d π 1 3 (5N d 5 πμη ρ( T T T )] Taking te derivative of (3) it respect to te solid-round-ire diameter and equating te result to zero one obtains Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

10 6 afal P Wojda Arc Elect Eng d dd ρ( T T T )] l π 3 1 π 3 d (5N d 5 πμη ρ( T T T )] (33) Fig Valley diameter as a function of frequency for te solid-round-ire inding as a function of te temperature at η 9, f khz, and N l Tis yields te temperature dependent valley tickness of te solid-round-ire inding d v [1 ( T T )] ρ α 3 πμη f π 5 (5N l 1) (3) Table 1 Parameters of simulated inductors Inductor number I II III Inductance (:H) Temperature (EC) Frequency (khz) I Lm (A) 15 Turns N Layers N l or d (mm) Figure sos te valley diameter as a function of te temperature at η 9, f khz, and N l It can be seen tat as te temperature increases te valley diameter increases nonlinearly Substituting (3) into (3) one obtains te minimum inding resistance of te solid-roundire inding Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

11 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 7 (min) 75 π (5N 8μ η fl (35) 5 Note tat for te solid-round ire inding, te local minimum of te inding ac resistance is independent on te temperature and terefore, te ac resistance at minimum is similar for different temperatures 6 Simulation verification In tis section verification of te derived equations utilizing 3D Maxell environment is done Te 3D FEM is used in order to take into te consideration of te curvature end and edge effect tat are not considered in te 1D model especially en solid-round-ire indings are analyzed Tree inductors, one it te foil inding (Fig 5) and to it solid-round-ire indings (Fig 6 and Fig 7) are analyzed and simulated Fig 5 3D Model of te four layer foil inding inductor Fig 6 3D Model of te to layer solid-round-ire inding inductor Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

12 8 afal P Wojda Arc Elect Eng Table 1 lists parameters of te modeled inductors Te designed inductors are based on te Magnetics Inc 616 ferrite ungapped pot core Te ungapped core as selected in order to eliminate te influence of te induced eddy currents in te inding from te air gap fringing flux Te initial permeability of te core material is μ r 3, te mean turn lengt (MTL) is l T 53 cm, and te breadt of te bobbin is b b 11 mm Simulations ave been performed at different conductor temperatures using Maxell 3D eddy current solver at selected frequencies Table lists analytical and numerical calculations of te ac inding resistances as ell as relative errors at temperatures T, 7, 15EC for foil inding inductor Te analytical and numerical calculations are also soed in Figures 8-1, ere te solid line represents te analytical predictions of ac inding resistance (calculated from Doell s equation) and bullets represents numerical predictions of te ac inding resistance (obtained from te FEM eddycurrent analysis) It can be seen tat te analytical calculations of ac inding resistance for te foil inding tracks te numerical calculations of ac inding resistance it ig accuracy Moreover, it can be seen tat above certain frequency, te ac inding resistance of te foil at T EC is iger tan for te foil at T 7 and 15EC Tis cause excessive poer loss in te inding and excessive temperature rise of te inductor Terefore, it is substantial to take into account te influence of te temperature in designing process of te inductor indings Note tat for te foil inding Doell s equation at T EC overestimates te numerical ac inding resistance on average by 36% and maximum by 16% At T 7EC, Doell s equation overestimates te numerical ac inding resistance on average by % and maximum by 11% On average at T 15EC, Doell s equation underestimates ac inding resistance by 6% Tables 3 and list te analytical and numerical calculations of te ac inding resistances for te solid-round-ire indings at temperatures T, 7, 15EC for to- and four-layer inding inductors, respectively Note tat for bot inductors calculations of solid-round-ire inding ac resistances are similar Toug Doell s equation disregards curvature, end, and edge effects, for solid-round-ire indings it underestimates te simulated ac inding resistance on average by 9% Moreover, calculated inding resistances at te local minimum are also similar to te simulated values Tis proves tat analytical Equations (17), (6), and (3) are a very good and ease to use equations for first step of inductor inding design process Fig 7 3D Model of te to layer solid-round-ire inding inductor Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

13 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 9 Table Analytical and numerical calculations of AC inding resistances of te four-layer foil inding inductor Frequency f (khz) Numerical (ms) EC 7EC 15EC Analytical (ms) Error, (%) ! Numerical (ms) Analytical (ms) Error, (%) 3 5!73! Numerical (ms) Analytical (ms) Error, (%)!!18! ! Fig 8 Simulated (bullets) and calculated (solid-line) ac inding resistances of foil inding as function of frequency at T EC Table 3 Analytical and numerical calculations of AC inding resistances of te solid-round-ire inding inductor at 9, N t Frequency f (khz) Numerical (ms) EC 7EC 1EC Analytical (ms) Error, (%) 31!!196!9!8 63 9!71!911 Numerical (ms) Analytical (ms) Error, (%) 331!196!193!37!5 197!3!35!183 Numerical (ms) Analytical (ms) Error, (%)!19!188!65 1!33!17 15!355 Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

14 1 afal P Wojda Arc Elect Eng Fig 9 Simulated (bullets) and calculated (solid-line) ac inding resistances of foil inding as function of frequency at T 7EC Fig 1 Simulated (bullets) and calculated (solid-line) ac inding resistances of foil inding as a function of frequency at T 15EC Table Analytical and numerical calculations of AC inding resistances of te solid-round-ire inding inductor at 9, N t Frequency f (khz) Numerical (ms) EC 7EC 1EC Analytical (ms) Error, (%)!196!198! ! Numerical (ms) Analytical (ms) Error, (%) 5!13!179!5!1! Numerical (ms) Analytical (ms) Error, (%) 8!196!196 3! Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

15 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 11 7 Conclusion Tere are many publications considering analytical optimization of te inding conductor size for te conductor temperature equal to room temperature [1, 16-19] Hoever, commercially available poer devices, ie converters, poer amplifiers, rectifiers, filtering and matcing circuits operates at temperatures muc iger tan te room temperature, and terefore, optimization of te inding conductor taking into consideration termal effect is crucial in tese devices design process Current literature lacks an analytical optimization of te inding conductor including termal effects In tis paper, termal analytical optimization of te inding conductor sizes, tickness and diameter, ave been performed Main advantage of derived equations is tat tey are easy to implement in te design process of te inding conductors conducting sinusoidal currents Temperature dependent 1D ac inding resistance of te inductor inding as been derived and approximated Moreover, te conductor temperature dependent analytical optimization of te foil inding tickness as been performed as ell as temperature dependent valley tickness and diameter of te square-ire and solid-round-ire inding ave been derived Te derived equations for te optimum foil tickness and valley diameter of te solid-round-ire indings ave been validated by te 3D FEM simulations Te simulations soed tat te temperature dependent ac inding resistance calculated from Doell s equation tracks te FEM results it ig accuracy Moreover, it as been son tat te equations for te optimum foil tickness (17) and te optimum diameter of te solid-round-ire indings (3) are a very good designing tool for indings operating at different temperatures Te folloing conclusions can be dran from te analysis of tis paper: $ For te foil inding, te ac resistance at te global minimum is temperature dependent and it increases as temperature increase; Fig 11 Simulated (bullets) and calculated (solid-line) ac inding resistances of te four-layer solid-round-ire inding inductor as a function of frequency at T o C $ As te temperature increases te optimum foil tickens increases non-linearly $ For te foil inding inductors, Doell s equation overestimates te inding resistance on average by 13% Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

16 1 afal P Wojda Arc Elect Eng Fig 1 Simulated (bullets) and calculated (solid-line) ac inding resistances of te four-layer solid-round-ire inding inductor as a function of frequency at T 7EC Fig 13 Simulated (bullets) and calculated (solid-line) ac inding resistances of te four-layer solid-round-ire inding inductor as a function of frequency at T 1EC $ For te square-ire and solid-round-ire indings, te ac resistance at te local minimum is independent on temperature $ As te temperature increases, te valley tickness and valley diameter increases nonlinearly $ For te solid-round-ire inding inductors, Doell s equation underestimates te inding resistance on average by 9% eferences [1] Kazimierczuk MK, F Poer Amplifiers Jon Wiley & Sons, Cicester, UK (8) [] Kazimierczuk MK, Czarkoski D, esonant Poer Converters nd ed IEEE Press/Jon Wiley & Sons, Ne York, NY (11) [3] Kazimierczuk MK, Pulesidt Modulated DC-DC poer converters Jon Wiley & Sons, Cicester, UK (9) Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

17 Vol 6 (15) Termal analytical inding size optimization for different conductor sapes 13 [] Yu Q, Holmes TW, Naisadam K, F equivalent circuit modeling of ferrite core inductors and caracterization of core materials IEEE Trans Electromagn Compat (1): 58-6 () [5] Huang F, Zang DM, Tseng K-J, Determination of dimension independent magnetic and dielectric properties for MnZn ferrite cores and its EMI applications IEEE Trans Electromagn Compat 5(3): (8) [6] Naisadam K, Closed-form design formulas for te equivalent circuit caracterization of ferrite inductors IEEE Trans Electromagnet Compat 53(): (11) [7] Wrobel, Mellor PH, Termal design of a ig-energy-density ound components IEEE Trans Ind Electron 58(9): 96-1 (1) [8] Wrobel, Mlot A, Mellor PH, Contribution of end-inding proximity losses to temperature variation in electromagnetic devices IEEE Trans Ind Electron 59(): (11) [9] Doell PL, Effects of eddy currents in transformer inding Proc IEE 113(8): (1966) [1] Snelling EC, Soft Ferrites, Properties and Applications nd ed London, UK:Butterort (1988) [11] Kutkut NH, A simple tecnique to evaluate inding losses including to-dimensional edge effect IEEE Applied Poer Electronics Conference, Atlanta, Feb 1997 [1] Kutkut NH, Divan DM, Optimal air-gap design in ig-frequency foil indings IEEE Applied Poer Electronic Conference, Atlanta (1997) [13] Bartoli M, eatti A, Kazimierczuk MK, Modelling iron-poder inductors at ig frequencies IEEE Industry Applications Society Annual Meeting, pp (199) [1] Murty-Bellur D, Kazimierczuk MK, Harmonic inding loss in buck DC-DC converter for discontinuous conduction mode IET, Poer Electron 3(5): 7-75 (1) [15] Kondrat N, Kazimierczuk MK, Inductor inding loss oing to skin and proximity effects including armonics in non-isolated pulse-idt modulated dc-dc converters operating in continuous conduction mode IET, Poer Electron 3(6): (1) [16] Kazimierczuk MK, Hig-Frequency Magnetic Components Jon Wiley & Sons, Cicester, UK (1) [17] Wojda P, Kazimierczuk MK, Winding resistance of litz-ire and multi-strand inductors IET, Poer Electron 5(): (1) [18] Wojda P, Kazimierczuk MK, Analytical optimization of solid-round-ire indings IEEE Trans Ind Electron 6(3): , 13 [19] Wojda P, Kazimierczuk MK, Magnetic field distribution and analytical optimization of foil indings conducting sinusoidal current IEEE Magnetics Letters (13) [] Koteras D, Calculation of eddy current losses using te electrodynamic similarity las Arcives of Electrical Engineering 63(1): (1) [1] Budnik K, Macczynski W, Magnetic field of complex elical conductors Arcives of Electrical Engineering 6(): (13) [] Strouboulis T, Zang L, Babuska I, Assessment of te cost and accuracy of te generalized FEM International Journal for Numerical Metods in Engineering 69(): 5-83 (7) [3] Gradzki PM, Jovanovic MM, Lee FC, Computer-aided design for ig-frequency poer transformer Annual Applied Poer Electronics Conference and Exposition, Los Angeles, CA, USA, 199, pp (1989) [] Lotfi AW, Gradzki PM, Lee FC, Proximity effects in coils for ig frequency poer applications IEEE Trans Magn 8(5): (199) [5] Ceng KWE, Evans PD, Calculation of inding losses in ig frequency toroidal inductors using single strand conductors IEE Electr Poer Appl 11(): 5-6 (199) [6] Ceng KWE, Evans PD, Calculation of inding losses in ig frequency toroidal inductors using multi-strand conductors IEE Electr Poer Appl 1(5): (1995) [7] obert F, Matys P, Scauers J-P, Omic losses calculation in SMPS transformers numerical study of Doells approac accuracy IEEE Trans Magn 3(): (1998) [8] Pernia AM, Nuno F, Lopera JM, 1D/D transforer electric model for simulation in poer converters 6t Annual IEEE Poer Electronics Specialists Conf, Atlanta, GA, USA, June 1995, vol, pp (1995) Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM

18 1 afal P Wojda Arc Elect Eng [9] Ayacit A, Kazimierczuk MK, Termal effects on inductor inding resistance at ig frequencies IEEE Magnetic Letters (13) [3] de Gersem H, Hameyer K, A finite element model for foil inding simulation IEEE Transactions on Magnetics 37(5): (1) [31] Wojda P, Kazimierczuk MK, Analytical inding foil tickness optimisation of inductors conducting armonic currents IET Poer Electronics 6(5): (13) [3] Wojda P Kazimierczuk MK, Analytical inding size optimisation for different conductor sapes using Ampre s La IET Poer Electronics 6(6): (13) [33] Wojda P, Kazimierczuk MK, Analytical optimisation of solid-round-ire indings conducting dc and ac non-sinusoidal periodic currents IET Poer Electronics 6(7): (13) [3] Wojda P, Kazimierczuk MK, Optimum foil tickness of inductors conducting DC and nonsinusoidal periodic currents IET Poer Electronics 5(6): (1) [35] Wojda P, Kazimierczuk MK, Proximity-effect inding loss in different conductors using magnetic field averaging COMPEL:Te International Journal for Computation and Matematics in Electrical and Electronic Engineering [36] Kazimierczuk MK, Wojda P, Foil inding resistance and poer loss in individual layers of inductors Int J Electron Telecommun 56(3): 37-6 (1) Brougt to you by Biblioteka Glóna Zacodniopomorskiego Uniersytetu Tecnologicznego Szczecinie Donload Date //16 1:37 PM