INTL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS, 00, VOL. 6, NO., PP. 7-6 Manuscript received July 0, 00: revised September, 00. DOI: 0.78/v077-00-00- Foil Winding Resistance and Power Loss in Individual Layers of Inductors Marian K. Kazimierczuk and Rafal P. Wojda Abstract Tis paper presents an estimation of ig-frequency winding resistance and power loss in individual inductor layers made of foil, taking into account te skin and proximity effects. Approximated equations for power loss in eac layer are given and te optimal values of foil tickness for eac layer are derived. It is sown tat te winding resistance of individual layers significantly increases wit te operating frequency and te layer number, counting from te center of an inductor. Te winding resistance of eac foil layer exibits a minimum value at an optimal layer tickness. Te total winding resistance increases wit te total number of layers. Keywords Eddy currents, individual layer winding resistance, inductors, optimal foil tickness, proximity effect, skin effect, winding power loss. I. INTRODUCTION GENERALLY, te power loss in te winding of an inductor at ig frequencies is caused by two effects of eddy currents: skin effect and te proximity effect []- [7], []-[8], [0]-[0]. Tese effects influence te distribution of te current in te conductor, causing an increase in te winding resistance. Moreover, te winding resistance and te winding power loss increase wit te operating frequency. Te skin effect is caused in te conductor by te magnetic field induced by its own current. Te skin effect is identical in all layers. Te proximity effect is caused by te magnetic field induced by currents flowing in te adjacent conductors. Te proximity effect increases rapidly wen te layer number increases. Inductors made of copper foil ave beneficial properties in designing power circuits. Its termal, mecanical, and electrical properties are muc better tan te properties of round wire inductors. Foil winding are attractive in low profile inductors and transformers. In addition, tey are commonly used in ig current magnetic components. Te purpose of tis paper is to present te analysis of winding resistance of individual layers in multilayer foil inductors wit a magnetic core and compare teir properties wit tose of te uniform layer tickness. II. GENERAL EQUATION FOR RESISTANCE OF INDIVIDUAL LAYERS Inductors made up of straigt, parallel foil conductor are considered. Tere is one winding turn in eac layer. Tis model can be used for low profile flat inductors and inductors Tis work was supported by te Fulbrigt Foundation. M. K. Kazimierczuk and R. P. Wojda are wit te Department of Electrical Engineering, Wrigt State University, 60 Colonel Glenn Higway, Dayton, Oio,, USA (e-mail: marian.kazimierczuk@wrigt.edu; wojda.@wrigt.edu. F Rn Fig.. n. 0 0 7 n 0 / -D plot of ac-to-dc resistance ratio F Rn as a function of / and wound on round magnetic cores wit low radius of curvature. Te magnetic field H in tis kind of inductors can be described by te second-order ordinary differential equation, called te Helmoltz equation, d H dx γ H, ( were γ is te complex propagation constant described by γ jωμ 0 j jωμ 0 σ w +j, ( ρ w ρ w ρ w te skin dept is ωμ 0 σ w ρ w, ( πfμ0 σ w πfμ 0 ρ w /σ w is te conductor resistivity, f is te operating frequency, and μ 0 is te free space permeability. Te solution of ( leads to te distribution of te magnetic field intensity H and te current density J in te n-t winding layer. Te complex power in te n-t layer is [8] P wn ρ w I mγ b [ ( ] γ cot(γ+(n ntan, ( were is te tickness of foil, b is te breadt of te foil and is te mean turn lengt (MTL. Assume tat te current flowing troug te inductor foil winding is sinusoidal
8 MARIAN K. KAZIMIERCZUK AND RAFAL P. WOJDA 0 0 n 7 0 0 F Rn F Rn, F R n N l, and F R 0 / 0 / Fig.. Individual layers ac-to-dc resistance ratio F Rn as a function of / for eac of te first several layers. Fig.. Plots of F Rn and F R as a function of / for tree layer foil winding inductor (N l and for total resistance of tree layers. i L I m sin(ωt. ( Te time-average real power loss in te n-t layer is P wn R wn I rms [R skin(n +R prox(n ]I rms [R skin +R prox(n ]I rms, (6 were R skin(n R skin is te resistance of eac layer due to te skin effect and is te same for eac layer and R prox(n is te resistance of te n-t layer due to te proximity effect and appreciably increases from te innermost layer to te outermost layer. If te RMS current is equal to te dc current troug te inductor, ten te time-average real power loss in te n-t layer of te winding P wn, normalized wit respect to te dc power loss P wdcn, is equal to te ac-to-dc resistance ratio of te n-t layer R wn /R dcn. Hence, te acto-dc resistance ratio in te n-t layer is given by [] F Rn P wn R wn P wdcn R wdcn ( [(n n+ sin( +sin( cos( cos( (n n sin( cos( +cos( sin( cos( cos( ]. (7 Fig. sows a -D plot of ac resistance ratio F Rn as a function of / and n. Fig. sows plots of F Rn as a function of / for several individual layers. It can be seen tat te normalized ac-to-dc resistance ratio F Rn significantly increases as te ratio/ increases and as te layer numbern increases, counting from te innermost layer to te outermost layer. At a fixed foil tickness, tree frequency ranges can be distinguised: low-frequency range, medium-frequency range, and ig-frequency range. In te low-frequency range, <<, te skin and te proximity effects are negligible, te current density is uniform, R w R wdc, and terefore F Rn. In te medium-frequency range, te current density is no longer uniform, and tereby F Rn increases wit frequency. Te boundary between te low and medium frequency ranges decreases as te layer number n increases. In te igfrequency range, te current flows only near bot foil surfaces andf Rn increases wit frequency. Te rate of increase off Rn in te ig-frequency range is lower tan tat in te-medium frequency range for n. Te sum of te ac-to-dc resistance ratios of all layers is given by F RNS R w + R w + R w... R N l wn l F Rn. R wdc R wdc R wdc R wdcnl n (8 were N l is te number of foil winding layers. Fig. sows plots of F Rn and F RNS as functions of / for tree-layer foil winding inductor. Te ac-to-dc resistance ratio F Rn can be expressed as F Rn F S +F Pn, (9 were te skin effect ac-to-dc resistance ratio is identical for eac layer and is expressed by F S R ( skin sin( δ w +sin( R wdcn cos( cos( (0 and te proximity effect ac-to-dc resistance ratio of te n-t layer is given by F Pn R ( prox(n sin( δ n(n w sin( R wdcn cos( +cos(. ( Te skin effect factor F S is identical for all te winding layers. Te proximity effect factor F Pn is zero for te first layer and rapidly increases wit te layer number n. For multilayer inductors, te proximity effect becomes dominant. Fig. sows te skin effect factorf S as a function of/ for eac layer. It can be seen tat te skin effect is negligible for / <. For/ >,F S increases rapidly wit/. Te
FOIL WINDING RESISTANCE AND POWER LOSS IN INDIVIDUAL LAYERS OF INDUCTORS 9 0 0 0 n 0 F S F Pn 0 0 / 0 0 0 / Fig.. Skin effect factor F S as a function / for eac layer. Fig. 6. Proximity effect factor F Pn as a function / for n-t layer in log-log scale. proximity effect factor F Pn as a function of / is sown in Figs. and 6 in linear-log and log-log scales, respectively. It can be seen from Fig. tat te proximity effect is negligible for / < and does not exist for te first layer. It can be observed from Fig. 6 tat te proximity effect factor F Pn increases rapidly wit / for te range < / < and increases wit / at a lower rate for / >. III. OPTIMUM THICKNESS OF INDIVIDUAL LAYERS Te effective widt of te current flow is approximately equal to te skin dept. Terefore, te winding resistance and te power loss in te innermost layer at ig frequencies are, respectively, R w(hf ρ w b ( and P w(hf ρ w I Lm b. ( Te dc resistance of a single layer is R wdc ρ w b. ( Te normalized winding resistance of te n-t layer is F rn R wn ρ w b R wn F Rn R w(hf (n n+ sin( +sin( cos( cos( (n n sin( cos( +cos( sin( cos( cos(. ( Fig. 7 sows a -D plot of normalized ac resistance R wn /(ρ w /b as a function of / and n. Fig. 8 sows plots of R wn /(ρ w /b as a function of / for several 00 90 n 80 70 0 0 F Pn 60 0 0 R wn /(ρ w /(b 0 0 0 0 0 0 0 0 0 0 0 / n / Fig.. Proximity effect factor F Pn as a function / for n-t layer in linear-log scale. Fig. 7. -D plot of normalized ac resistance R wn/(ρ w /b as a function of / and n.
0 MARIAN K. KAZIMIERCZUK AND RAFAL P. WOJDA individual layers. It can be seen tat te ac resistance reaces a fixed value at iger values of /. It can be also seen tat te plots exibit minimum values. Fig. 9 sows tese plots in te vicinity of te minimum values in more detail. IV. APPROXIMATION OF R wn /R w(hf An exact analytical expression for te minimum winding resistance of individual layers cannot be found from (. For low and medium foil ticknesses, te winding resistance of te first layer, ( can be approximated by F rn R wn ρ w b for and for large foil ticknesses, F rn R wn ρ w b for R wn R w(hf P wn P w(hf < and n (6 R wn R w(hf P wn P w(hf > and n. (7 Fig. 0 sows te exact and approximate plots of R w /(ρ w /b as functions of / for te first layer. For low and medium foil ticknesses, te normalized winding resistance and normalized winding power loss in ten-t layer can be approximated by F rn R wn ρ w b + n(n ( δ w or R wn R w(hf P wn P w(hf for F Rn R wn R wdc P wn P wdc <. and n (8 R wn / (b 8 7 6 n 7 0 0. 0. 0. 0. 0. 0.6 0.7 0.8 0.9 / Fig. 9. Normalized ac resistance R wn/(ρ w /b as a function of / for eac of te first several layers in enlarged scale. + n(n ( for 6 <. and n. (9 Fig. sows exact and approximate plots of R w /(ρ w /b as functions of / at n for low and medium foil ticknesses. It can be seen tat te approximation is excellent in vicinity of te minimum value of te layer resistance. For ig tickness of te foil, te normalized winding resistance and normalized winding power loss in te n-t layer is approximately given by or F rn R wn R w(hf P wn P w(hf n +(n for (0 00 90 80 7 Exact Approximate R wn / (b 70 60 0 0 6 R w / (b 0 0 0 0 n 6 / / Fig. 8. Normalized ac resistance R wn/(ρ w /b as a function of / for eac of te first several layers. Fig. 0. Exact and approximated plots of R w /(ρ w /b as a function of / for n.
FOIL WINDING RESISTANCE AND POWER LOSS IN INDIVIDUAL LAYERS OF INDUCTORS R w / (b 0 Exact Approximate 0 Fig.. Exact and approximate plots of R w /(ρ w /b as functions of / at n for low and medium tickness. / F Rn R wn P wn R wdc P wdc ( [n +(n ] for. ( Fig. sows exact and approximate plots of R w /(ρ w /b as functions of / at n for ig foil ticknesses. Taking te derivative of (7 wit respect to /, we obtain ( cos n n cos (. ( For n, ( becomes ( cos 0, ( TABLE I EXACT AND APROXIMATE OPTIMUM FOIL THICKNESS FOR INDIVIDUAL INDUCTOR LAYERS Layer Number Exact Approximate n optn/ optn/ π/.707 0.8767 0.809 0.6 0.689 0.7 0.7 0.788 0.79 6 0.6676 0.7 7 0.9 0.98 8 0.67 0.66 9 0.089 0.. 0.7 wic gives te optimum tickness of te first layer, subjected only to te skin effect opt π for n. ( For n, ( as no closed-form solution and was solved numerically; te exact results are given in Table I. In order to obtain analytical expression for opt /, we will use (8. Te minimum values of te ac resistance R wn(min and te winding power loss P wn(min in te n-t layer for n are obtained by taking te derivative of (8 and setting te result to zero ( ( ( + n(n 0, ( d d( Rwn R w(hf yielding te optimum tickness of te n-t layer optn for n. (6 n(n R w / (b 0 Exact Approximate Te approximated results of optn /, are listed in Table I. Te minimum normalized power loss in te n-t layer is R wn(min P wn(min n(n for n. (7 R w(hf P w(hf Dividing (6 by (, one obtains te ratio of te optimum tickness of te n-t layer to te optimum tickness of te first layer as optn opt π n(n for n. (8 0 / Fig.. Exact and approximate plots of R w /(ρ w /b as functions of / at n for ig tickness. V. EXAMPLE FOR OPTIMUM WINDING RESISTANCE Te minimum winding resistance can be acieved wen te tickness of eac layer is different and equal to te optimum value given by ( and (6. For tree-layer copper inductor and conducting sinusoidal current at frequency khz, te optimum tickness of te bare foil of te first layer is
MARIAN K. KAZIMIERCZUK AND RAFAL P. WOJDA 0 Te total dc winding resistance is a sum of dc winding resistance of eac layer R wdc R wdc +R wdc +R wdc 0.97 mω. (7 R wn (min, (mω n Assuming an RMS current of 0 A, te dc and low-frequency power loss in eac layer of te inductor is and P wdc R wdc I rms 0.7 0 0. W, (8 P wdc R wdc I rms 0. 0 0.80 W, (9 P wdc R wdc I rms 0. 0.07 W. (0 0 0 0 0 0 6 0 7 f (Hz Fig.. Plots of R wn(min and as a function of frequency f for te foil inductor. opt π π ρw πfµ 0 0. mm. (9 From (8, te optimum tickness of te bare conductor of te second layer n is opt π opt 0.6 0. 0.67 mm, (0 and te optimum tickness of te bare conductor of te tird layer n is opt π 6 opt 0.06 0. 0.0 mm. ( Te ac winding resistance for n-t layer is given by R wn F Rn R wdcn. ( Terefore, te overall ac resistance of te foil inductor is N l R w F Rn R wdcn. ( n Te breadt of te inductor, wic is equal to te foil widt is b cm. Te lengt of eac turn is 0 cm. Te resistivity of copper at room temperature is ρ Cu.7 0 8 Ωm. Te dc resistances of eac layer is R wdc ρ Cu b opt R wdc ρ Cu b opt R wdc ρ Cu b opt.7 0 8 0. 0. 0 0.7 mω, 0 0 (.7 0 8 0. 0.67 0 0. mω, 0 0 (.7 0 8 0. 0.0 0 0. mω. 0 0 (6 Since te optimum tickness optn of te subsequent layers decreases, te dc resistance of te individual layers increases wit increasing layer number n. Te total dc winding power loss is a sum of dc power loss of eac layer P wdc P wdc +P wdc +P wdc 0.+0.80+.07.9 W. ( It can be seen tat te dc winding power loss of te subsequent layers increases wit te layer number. Substituting te optimum layer tickness given by ( and (6 into (7, te minimum values of te ac-to-dc resistance of n-t layer F Rn(min were calculated numerically. Te results are F R(min.07, F R(min.70, F R(min.8. Hence, te ac resistances in te subsequent layers are and R w(min F R(min R wdc.07 0.7 0 0.78 mω, ( R w(min F R(min R wdc.70 0. 0 0. mω, ( R w(min F R(min R wdc.8 0. 0 0.69 mω. ( Te total ac winding resistance of an inductor wit te optimum layer ticknesses is R w(min +R w(min +R w(min 0.78+0.+.8.8 mω. ( Fig. sows te ac winding resistance R wn(min of eac layer and te total ac winding resistance as functions of frequencyf for tree-layer winding (N l. Te ac power losses in te individual layers for a sinusoidal inductor current of RMS value I rms 0 A are P w(min R w(min I rms 0.78 0 0.69 W, (6 P w(min R w(min I rms 0. 0.0W, (7 P w(min R w(min I rms 0.69 0.W. (8 It can be seen tat te ac power loss in eac layer increases wit te layer number n.
FOIL WINDING RESISTANCE AND POWER LOSS IN INDIVIDUAL LAYERS OF INDUCTORS R w l w / b Exact Approximate 0 / N l Fig.. Exact and approximate plots of R w/(l w/b as functions of / for tree-layer inductor N l for low and medium uniform tickness. Te total minimum ac power loss in te inductor winding is given by P wmin P w(min +P w(min +P w(min 0.69+.0+.. W, (9 wic gives te ratio of te ac-to-dc winding resistance and ac-to-dc winding power loss P wmin..7. (0 R wdc P wdc.9 VI. MINIMUM WINDING RESISTANCE FOR INDUCTORS WITH UNIFORM FOIL THICKNESS For low and medium foil ticknesses, te normalized resistance of te inductor wit fixed foil tickness and any number of layers N l can be approximated by [8] R w ( ρwl w b ( + (N l 7 δ w for <., ( were l w N l is te total foil winding lengt. Fig. sows exact and approximate plots of R w /( l w /b as functions of / for tree-layer inductor (N l wit uniform foil tickness for low and medium foil ticknesses. Te minimum values of te ac winding resistance R wopt and te winding power loss P wopt of an inductor wit uniform foil tickness are determined by taking te derivative of ( and setting te result to zero ( d R w ρwlw bδw ( ( + d 6(N l 7 ( 0, ( yielding te optimum value of te uniform foil tickness in te inductor opt 6(N l 7 for N l. ( Te approximate results of opt /, are listed in Table II. For N l, te optimum foil tickness is defined by (. In te subsequent analysis, te properties of winding wit non-uniform tickness will be compared wit tose of te winding wit uniform tickness. For a tree-layer copper inductor (N l wit an uniform foil tickness and conducting a sinusoidal current at frequency f khz, te optimum tickness of te bare foil is opt 0.77 0.77 ρw πfµ 0 0. mm. ( Te dc and low frequency winding resistance is [8] R wdc ρ Cul w A wopt ρ Cu N l b opt.0 mω, ( were A wopt is te cross-sectional area of te foil. Assuming an RMS current of 0 A, te dc and low-frequency power loss in all tree layers (N l of te inductor is given by 0 Optimized tickness Constant tickness TABLE II APROXIMATE OPTIMUM UNIFORM FOIL THICKNESS FOR MULTILAYER INDUCTOR Layer Number Approximate N l opt/.707 0.988 0.77 0.69 0.86 6 0. 7 0.99 8 0.60 9 0.8., R w (mω 0 0 0 0 6 0 7 f (Hz Fig.. Plots of and R w as functions of frequency f for te inductor wit optimized tickness of eac layer opt 0. mm, opt 0.67 mm, opt 0.0 mm and for te inductor wit a constant layer of tickness opt 0. mm.
MARIAN K. KAZIMIERCZUK AND RAFAL P. WOJDA 0 Optimized tickness Constant tickness 0 Optimized tickness Constant tickness, R w (mω, R w (mω 0 0 0 0 6 0 7 f (Hz 0 0 0 0 6 0 7 f (Hz Fig. 6. Plots of and R w as functions of frequency f for te inductor wit optimized tickness of eac layer opt 0. mm, opt 0.67 mm, opt 0.0 mm and for te inductor wit a constant layer of tickness opt 0.67 mm. Fig. 7. Plots of and R w as functions of frequency f for te inductor wit optimized tickness of eac layer opt 0. mm, opt 0.67 mm, opt 0.0 mm and for te inductor wit a constant layer of tickness opt 0.0 mm. P dc R dc I rms.0 0 0.6 W. (6 Te ac-to-dc total winding resistance ratio of tree-layer inductor wit an uniform optimum winding tickness [8] was calculated numerically and is given by F R R wopt R dc P wopt P dc.. (7 Hence, te optimum ac winding resistance of te inductor wit uniform foil tickness is R wopt F R R dc..0 0. mω. (8 Te total ac winding power loss is of an inductor wit an uniform foil tickness equal to te optimum tickness of te second layer opt for tree layers. It can be seen tat for te ig-frequency range te ac winding resistance of te inductor wit optimized foil ticknesses is approximately equal to te ac winding resistance R w of te inductor wit an uniform foil tickness equal to te optimum tickness of te second layer. Fig. 7 compares te ac winding resistance of an inductor wit te optimum individual layer ticknesses and te ac winding resistance R w of an inductor wit an uniform foil tickness equal to te optimum tickness of te tird layer opt for tree layers. Fig. 8 compares te ac winding resistance of an inductor wit te optimum individual layer ticknesses P wopt R wopt I rms. 0 0. W. (9 Te ratio of te ac winding resistance R wopt of te inductor wit te optimum uniform foil tickness to te ac winding resistance of te inductor wit optimum foil tickness for eac layer is ǫ R wopt..6. (60.8 Fig. compares te ac winding resistance of an inductor wit te optimum individual layer ticknesses and te ac winding resistance R w of an inductor wit uniform foil tickness equal to te optimum tickness of te first layer opt for tree layers. It can be seen tat for te ig-frequency range te ac winding resistance of te inductor wit te optimum individual layer ticknesses is significantly lower tan te ac winding resistance R w of te inductor wit an uniform foil tickness. Fig. 6 compares te ac winding resistance of an inductor wit te optimum individual layer ticknesses and te ac winding resistance R w, R wopt (mω 0 Optimized tickness Constant tickness 0 0 0 0 6 0 7 f (Hz Fig. 8. Plots of and R w as functions of frequency f for te inductor wit optimized tickness of eac layer opt 0. mm, opt 0.67 mm, opt 0.0 mm and for te inductor wit a constant layer of tickness opt 0. mm for tree-layer inductor (N l.
FOIL WINDING RESISTANCE AND POWER LOSS IN INDIVIDUAL LAYERS OF INDUCTORS ε.....0 0.9 0 0 0 6 0 8 f (Hz Fig. 9. Ratio of te winding resistance wit uniform optimum foil tickness opt 0. mm, opt 0.67 mm, opt 0.0 mm to te winding resistance wit optimum individual layer ticknesses opt 0. mm for tree-layer inductor (N l. and te ac winding resistance R wopt of an inductor wit an uniform optimum foil tickness opt for tree layers. It can be seen tat te resistance for inductor wit te optimized tickness for eac layer is lower tan tat of te inductor wit te uniform optimum tickness. Fig. 9 sows te ratio of te ac winding resistance R wopt wit uniform optimum foil tickness to te ac winding resistance wit te optimum individual layer ticknesses. It can be seen tat te resistance of te inductor wit te optimum uniform foil tickness for te low-frequency range is % iger tan tat of te inductor wit te optimized tickness of eac layer. In te medium-frequency range, te resistance of uniform inductor winding tickness increases. At a frequency of 00 khz, te winding resistance of te inductor wit te optimum uniform tickness is.8% greater tan te winding resistance of te inductor wit te optimum foil tickness of individual layers. However, in te ig-frequency range, te winding resistances of bot inductors are te same. Te inductance of te foil wound inductor is expressed by L µ rcµ 0 A c N l b 800 π 0 7 0 0 07 µh, (6 were b cm, µ rc 800 is te core permeability, N l is te number of layers, and A c cm is te cross-sectional area of te core. VII. CONCLUSIONS Te equation for te winding resistance of individual layers for inductors made of foil conductor as been analysed and illustrated. Tis equation as been approximated to derive an expression for te optimum tickness of individual layers. Te comparison of winding resistances at various values of foil tickness as been presented. It as been sown tat te minimum value of te winding resistance of eac individual layer at a fixed frequency occurs at different values of te normalized layer tickness optn /. Te optimum normalized layer tickness optn / decreases wit increasing layer number n. In addition, te resistance of eac layer appreciably increases as te layer number n increases from te innermost to te outermost layer. Moreover, te approximated equation for low-frequency resistance of inductors wit a uniform foil tickness as been given. Te optimum normalized value of te uniform foil tickness as been derived. It as been sown tat te winding resistance of te inductor wit an optimum uniform foil tickness for low-frequency range is % iger tan tat of te inductor wit an optimized tickness of eac layer. In te medium-frequency range, te ratio of te winding resistance wit a uniform optimum foil tickness to te winding resistance wit te optimum tickness of eac individual layer first increases, reaces a maximum value, and ten rapidly decreases wit frequency. At a frequency of 00 khz, te winding resistance of uniform optimum foil tickness was.8% greater tan te winding resistance of te inductor wit te optimum foil tickness of eac layer, in te given example. For te ig-frequency range, te winding resistances of bot inductors were identical. Hig-quality power inductors are used in ig-frequency applications, suc as pulse-widt-modulated (PWM DC-to-DC power converters [], [9], [0], resonant DC-to-DC power converters [], radio-frequency power amplifiers []-[7], and LC oscillators []. REFERENCES [] A. Aminian and M. Kazimierczuk, Electronic Devices. A Design Approac. Upper Saddle, NJ: Prentice Hall, 00. [] M. Bartoli, N. Noferi, A. Reatti, and M. K. 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