ELECTRONICA DE POTENCIA - Revista de la SOBRAEP, vol. 5, núm. 1, 2000, pp. 16-27.
FLYBACK VS. FORWARD CONVERTER TOPOLOGY COMPARISON BASED UPON MAGNETIC DESIGN .
H. E. TACCA.
Cita: H. E. TACCA (2000). FLYBACK VS. FORWARD CONVERTER TOPOLOGY COMPARISON BASED UPON MAGNETIC DESIGN. ELECTRONICA DE POTENCIA - Revista de la SOBRAEP, 5 (1) 16-27.
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FLYBACK VS. FORWARD CONVERTER TOPOLOGY COMPARISON BASED UPON MAGNETIC DESIGN
Hernán Emilio TACCA
Universidad de Buenos Aires , Facultad de Ingeniería LABCATYP - Depto. de Electrónica [email protected]
Abstract - Based on approximative equations for single la air gap step calculation design procedures, a simple method l average mean turn length useful to adopt the most suitable topology in low power eav DC to DC conversion is derived from the volume lem mean turn length calculation of the magnetic cores needed. lFe effective magnetic length
n primary turns Index terms - Low power magnetic components, switching P power converters, flyback and forward magnetics design. nS secondary turns
PCu winding losses NOMENCLATURE P device rated switching power D P core losses Bmax peak value of the induction Fe P converter output power Bmin induction minimum value O D duty-cycle factor Rac a.c. resistance f switching frequency Rdc d.c. resistance Fc window filling factor of the primary/secondary PS/ ℜVFW/ FB total forward core volume to flyback core coil (also known as copper factor) : volume ratio FnSS= cPPS///SCuW PS PS/ Rθ thermal resistance f voltage form factor S inductor wire cross section fV CuL F partition factor of the primary windings : S primary wire cross section P CuP FSSPWW= S secondary wire cross section P CuS f power switch profit factor (or power device pr Sdis heat dissipation surface
utilization factor) SFe minimum core section Fr total winding eddy currents factor SFeef effective core section FW window factor : FSSWWF= e SFe flyback core section required I average value of the inductor current FB Lav SFe inductor core section required I effective value of the inductor current L Lef SFe forward transformer core section required I peak value of the inductor current TFW Lmax SW window area I average value of the primary current Pav S primary window area WP I effective value of the primary current Pef S secondary window area WS I P peak value of the primary current max tC power switch conduction time kC core sizing characteristic converter topology V core volume of the flyback coupled inductor FeFB coefficient (usually named flyback “transformer”) k ES effective core shape factor V total volume occupied by the cores of the FeFW k skin effect factor rS forward converter krX proximity effect factor V core volume of the smoothing inductor FeL kS geometrical core shape factor V core volume of the forward transformer FeTFW ku transformer utilization factor T VP primary supply voltage L inductance of the smoothing inductor Δ penetration depth 2
ΔB maximum induction increment δ iII=Δ (2.2) PPPmax δ i normalized amplitude of the inductor ripple L DtT= C (2.3) current and from the current waveforms it follows : δ i P normalized primary current variation ⎛ δ i ⎞ ID=−I⎜1 P ⎟ (2.4) η converter efficiency PPav max ⎝ 2 ⎠ Φ magnetic flux 1 2 μ o magnetic field constant ID=−I1 δδ ii+ (2.5) PPef max PP3 μ relative static permeability (derived from the rs Then, the output power is : static magnetization curve) ⎛ δ i ⎞ ρ conductor resistivity PPVID==ηη = η⎜1 −P ⎟ VI (2.6) OPPPav⎝ 2 ⎠ PP max σ L inductor current density + Vp D iO σ P primary current density iS
I. INTRODUCTION nP n S Vo RL In DC to DC converters and off-line switching power i P supplies for low power applications, the most used topologies are the flyback and the forward converters [1]. The flyback structure has the advantage of requiring only a single magnetic component. This one, if designed for minimum size, results smaller than the overall volume (a) occupied by both the smoothing inductor plus the power i transformer of the equivalent forward converter. However, P Ip when the flyback coupled inductor size is minimum, the max profit factor of the power switch becomes poor [2][3]. If an Ip improved profit of the flyback power device is required, it min results mandatory to supersize the magnetic component . So, 0 a trade-off arises between the core size and the rated tc = DT T t i switching power (maximum theoretical switching power) of S the power transistor [4]. The better the profit factor, heavier I Smax the magnetic cores become and this apply to both topologies here compared. Nevertheless, the required core volumes I result from different laws, so for a wanted profit factor, one Smin topology will be the most suitable regarding the core weight.
In this work, the total magnetic material volume required T t tc T - tc for both alternatives are computed and related. The mm... f comparison criterion is stated as the ratio of core sizing mm... f max approximative equations, which become justified by the niS S ni closed matching with results obtained using manufacturer P P magnetomotive force mm... f min design data and recommended procedures. mm... f= nPP i+ n SS i Derivation of some equations involved on design procedures are presented in appendices (including some T t tc T - tc application examples agreeing with manufacturer data). DT In order to compare topologies, first, the required core sections are obtained and then, the core volumes are (b) computed assuming identical core shapes for all the magnetic components involved. Figure 1 : Flyback converter, (a) basic circuit, (b) basic Finally, the volume rate is plotted to bring an easy method waveforms. to know wich topology will be the lightest. B. Core sizing II. FLYBACK CONVERTER The primary current density is : I P σ = ef (2.7) A. Basic flyback circuit P S Fig. 1.a depicts the basic circuit of a flyback converter and CuP where S is the cross section of the primary wire that fig. 1.b shows the corresponding waveforms. From fig. 1.b it CuP may be defined : must verify : ΔIIPP=− I P (2.1) SF= FSn (2.8) max min CuPP P c W P 3
In accordance with the Faraday law : G ≅ 1 δ i ΔB ()p Vn= S (2.9) PPt Fe as it was done with flyback converters. C The equations 3.1 and 3.2 are particular cases of where, ΔBB=−max B min and generalized expressions valid for many other converter μ μ topologies (Apendix II). B = o nI ≅ o nI (2.10) max PPmaxPP max L l D Fe la FW iO la + μ rs + Vp because usually : i L llaF>> erμ s Vo nP DL nS R where μ rs is the relative static permeability. L i By a similar way, P
μ o v ΔΔB ≅ nIPP (2.11) CE Dm la Relating 2.10 and 2.11 yields : ΔB ΔI (a) P i ==δ P (2.12) i B I P max Pmax using this equation (with the simplified notation Ipmax BBmm= ax) the expression 2.9 yields : Ip f min VniB= δ S (2.13) PPPmFD e 0 DT T t With eqs. 2.5 , 2.6 , 2.7 , 2.8 and 2.13 , using the window i i S L factor definition FSSWWF= / e, it results : Is = I max Lmax DP i L SG= O (2.14.a) FeFB i ()δ ip ηδiBf σ F FF S Pm PcP PW I Lmin where,
1 2 DT T 1 −+δδii v mm... f t PP3 CE G = (2.14.b) V δ i CEmax v ()p ⎛ δ i P ⎞ CE ⎜1 − ⎟ ⎝ 2 ⎠
For all the possible values 01≤≤δ i it results P mm... f Vp 11≤ To complete the design, it is necessary to adopt the (b) partition factor of the primary windings (see section IV), the primary current density (section V) and the maximum value Figure 2 : Forward converter, (a) basic circuit, (b) basic of the induction (Appendix I). waveforms. III. FORWARD CONVERTER B. Inductor core sizing The effective value of the inductor current can be A. Basic circuit and transformer core sizing expressed as : The typical circuit of a forward converter is presented in S S WLLFe fig. 2.a , while its waveforms are depicted in fig. 2.b . The ISLL==σσCuL Fc = σLcWFF (3.3) ef L n n equation of the primary voltage becomes : L L B 1 On the other hand, Vn==m S nBfS (3.1) nBS PP Fe P m Fe nL Φ LmFeL tC D L == (3.4) i I from which, the sizing equation results : L Lmax Using eqs. 3.3 and 3.4 : DPO SGFe = (3.2) LILL I TFW ()δ ip ησBf F FF ef max mPcPWP S = (3.5) FeL σ FF B where it may be approximated LcW m From fig. 2.b it follows : 4 1 and secondary coils, while FP is the partition factor of the II=−1 δδ i+ i2 (3.6) LLef max L3 L primary windings defined as the ratio between the window area occupied by the primary winding and the total window ⎛ δ i ⎞ II=−⎜1 L ⎟ (3.7) area , FSS= . LLav max ⎝ 2 ⎠ PWWP Substituting the eqs. 4.3 into eqs. 4.2 , then the result into and then, the output power results : 4.1, and considering both nS ⎛ δ iL ⎞ nI= nI and SFS= , PVI==D⎜1 −⎟ VI (3.8) PPef SS ef WWFe OOLav n ⎝ 2 ⎠ PLmax P yields : During the interval 0 ≤≤tDTit must be 22 In⎡ l lFF⎤ PPef emP emSS() r r P nS ΔI L P =+ρ ⎢ ⎥ (4.4) VV−= L Cu eqP PO FSWFe⎢ FFcP FFcP()1− ⎥ nP DT ⎣ P S ⎦ that may be expressed as On the other hand, the current densities in each winding n Lf are : 1−=D S Iiδ (3.9) () LLmax σ = IS and σ = IS, nPPDV PPCefu P SSef Cu S Substituting eqs. 3.6 , 3.8 and 3.9 in 3.5 , yields : that related give : F ()1− DP σ P cS ⎛ 1 ⎞ SG= O (3.10) =−⎜ 1⎟ (4.5) FeL ()δ iL σ FF δσiBfLm Lc FF W S cPP ⎝ ⎠ where , as previously stated , it may be aproximated Considering the common winding techniques, two alternatives will be studied. G ≅ 1 . Also, it can be assumed that δ iiLP≅ δ ()δ iL provided that the magnetizing inductance be large enough. 1. Shared coil-former windings In order to complete the design, the air-gap and the In this case : lll= = so , the eq. 4.4 becomes : winding turns must be determined, which may be done by emPS em em several graphical [5][6] or analytical methods [7]-[9][11]. ⎡ FF ⎤ 22l 1 ( rrSP) PI=+ρ nem ⎢ ⎥ Cu eqPef P P FS⎢ F F FF()1− ⎥ IV. PARTITION FACTOR OF THE WINDING AREA WFec⎣ P P cPS ⎦ (4.6) Optimal partition factor : The optimal partition factor will be the one which minimizes The optimum is defined as that value which minimizes the the copper losses given by eq. 4.6 . That is : winding losses, given by 1 22 FP = (4.7) PPCu=+= Cu P Cu IR P Cu + IR S Cu (4.1) opt PSefPefS Fc Fr 1 + P S where, the winding resistances are F F l c S rP emP RnCu= ρ eq P (4.2.a) and from eq. 4.5 , the current densities become related by : PPS CuP Fc F l σ P S rS emS = (4.8) RnCu= ρ eq S (4.2.b) σ F F SSS S cP rP CuS For the particular case when FF= and FF= and, l are the primary and secondary mean turn ccPS rrPS emPS/ lengths, while ρ are the equivalent resistivities of the it results : eqPS/ 1 primary and secondary conductors, taken into account skin FP = (4.9.a) opt 2 and proximity effects [7][8][12][13] : and ρρeq= F r Cu (4.2.c) PS// PS σ PS= σ (4.9.b) Fk= + k (4.2.d) rrPS///Sr PSX PS In such a case, the Joule losses result : being k the primary and secondary skin effect factors 22 4 lem rSPS/ PI= ρ n (4.10) Cu eqPef P P FFS and k the proximity factors (Appendix III). cWFeP rX PS/ Defining the equivalent turn factor : The conductor sections are : ⎡ F F ⎤ SW 1 cP FP rS SF= F (4.3.a) Fle =+⎢1 ⎥ (4.11) CuPP P c n 2 ⎢ F 1 − F F ⎥ P ⎣ c S ()P rP ⎦ S SF=−()1 FW (4.3.b) the eq. 4.6 may be rewritten as : CuSS P c n Fl S PI= 2ρ 22nle em (4.12) where, F are respectively, the fill factors of the primary Cu eqPef P P FF FS cPS/ PcP WFe 5 where, according to eqs. 4.2.c and 4.2.d , it is 1 lllee=+mem (4.14) ρρ=+kk av2 ( P S ) eqPPP() rS rX Cu and using the eq. 4.13 , the definition 4.14 may be expressed as : 2. Stacked windings π In case of superposed winding sections (for example lA=+++2()BCCCπ − (4.15) ePav ()S secondary over primary) the mean turn lengths become 2 different and will depend on the the primary partition factor For CCPS= the average mean turn length coincides adopted. For the core of fig. 3.a , according to fig. 3.b , the with the single section mean turn length, defined by : sectional mean turn lengths will be : ll= =++2(AB ) π C (4.16) em eav ] lA=++2(BC ) π (4.13.a) CCPS= emP P and from fig. 3.a , the following geometrical coefficients are ⎛ C ⎞ lA=++22()BCπ ⎜ −S ⎟ (4.13.b) defined : emS ⎝ 2 ⎠ FHCh = (4.17.a) where : and CC=+ C (4.13.c) PS FBAs = (4.17.b) The primary partition factor is : SW C F ==P P (4.18) A/2 P SW C where C may be expressed by , H FW C = SFe (4.19) Fh Using eqs. 4.13.c , 4.16 , 4.18 and 4.19 , the eqs. 4.13.a and 4.13.b may be expressed as : B F ll=−π W 1 −FS (4.20.a) AAC /2 emP em ()PFe Fh (a) F ll=+π W FS (4.20.b) C emS em PFe l Fh emP (primary) Substituting 4.20.a and b into 4.4 yields : l l PI=⋅ρ 22nem emS Cu eqPef P P FSWFe (secondary) ⎡ ⎤ (4.21.a) B 11−−MF()1+ MF ⎛ Fr ⎞ ⎢ P P ⎜ S ⎟⎥ ⋅ + ⎜ ⎟ ⎢ FFcP FFcP()1− ⎝ Fr ⎠⎥ ⎣ PS P ⎦ CP where : C CS π FW M = SFe (4.21.b) lem Fh A The expression 4.21.a has a minimum for : (b) 1 F = (4.22) Popt FM1 + ⎛ F ⎞ cP ()rS S 1+ ⎜ ⎟ disFe FM1− ⎜ F ⎟ c S ()⎝ rP ⎠ S =+++2 π CAHC CA disCu []()() therefore , from eq. 4.5 , the optimal current density ratio is : Sdis =+22{}()AB[]() AC +++ H AB FMc ()1+ ⎛ F ⎞ S Fe σ P S rS disCu = ⎜ ⎟ (4.23) σ FM1− ⎜ F ⎟ S c P ()⎝ rP ⎠ For square section scrapless lamination cores, with (c) FF= and FF= the optimal values are : ccSP rrSP Figure 3 : E-type core, (a) dimensions, (b) mean turn lenghts, FP = 043. and σ PSσ = 134. . (c) dissipation surfaces. opt Adopting these values PCu becomes 28 % lower with Defining the average mean turn length as : respect to the case corresponding to stacked windings with FP = 05. and σ PS= σ . Nevertheless, adopting 6 FP for minimal losses implies , as disadvantage, one composed by one part corresponding to the coil windings S and other one S concerning to the core. For higher current density in the inner winding , which has the disCu disFe worst heat transfer capability. By this reason , equal current example, for square section scrapless laminated cores , from densities are often adopted , even if this selection leads away fig. 3.a and c , one obtains : of the optimum value from the winding losses point of view. SA=+(52π ) 2 , SA= 14. 5 2 In a similar fashion as done with the shared coil-former disCu disFe and the total dissipation surface becomes : case an equivalent turn factor ( Fle ) may be defined SSS=+=+19.. 5 2π A22 = 258 A (5.4) keeping the Joule losses still given by eq. 4.12 . For this distot dis Cu dis Fe () purpose it must be : Defining the thermal resistance as : ⎡ F ⎤ Δθ 1 c P Fr FMPP()1+ F S Rθ = (5.5) FMle=−−+⎢11()F P ⎥ (4.24) 2 F F 1− F Pt ⎢ c S rP ()P ⎥ ⎣ ⎦ from eqs. 5.3 and 5.4 it results : M (where is defined by 4.21.b). 30 For the particular case with FF= and FF= , R ≅ (5.6) ccSP rrSP θ ⎡ºC/W⎤ 2 tot ⎣⎢ ⎦⎥ ()A[]cm adopting F = 05. yields : P This expression may be used to estimate the transformer Fle = 1 (4.25) temperature rise when the copper and iron losses cause For square section scrapless laminations with similar rises. Notice that Sdis and Sdis are similar areas. FF= it results : F = 084. . Cu Fe PPopt le Therefore, when the core and winding losses are similar, the For E ferrite cores with the following typical dimensions : temperature rise will be comparable , and the approximation AC= ; BA= 15. ; HA= 15. made will be acceptable. one obtains , In other circumstances , it will be suitable to verify the F = 04. and F = 077. coil and core temperature rise separately, assuming isolated Popt le ]F Popt dissipations paths and using each kind of loss with its own (always with FFcc= and FFrr= ). dissipation area. It should be ensured that the highest SP SP temperature rise computed be lower than the maximum specified. V. CURRENT DENSITY ADOPTION The thermal conduction resistance between coil and core is usually high because the coil former is plastic made and an A. Thermal considerations. Dissipation of losses air filled gap lies between the coil former and the core central For temperature rises ranging about 50 ºC , assuming an leg. ambient temperature of 40 ºC , the power dissipated by As usually P and P are similar, so they are S radiation may be estimated as : Cu Fe disCu and S , the core and coil temperature rise become Pr[]W disFe = 708. Δθ (5.1) S []ºC similar. Therefore, stated the thermal resistance is high, the dis[]m2 heat conduction exchange between coil and core must be (linear approximation obtained from the Stefan-Boltzmann neglectible in a first trial approximation. [15] equation assuming an emissivity coefficient of 0.8). On the other hand , the power dissipated by natural Example : convection may be approximated through [15] : From the transformer thermal model [16] depicted in fig. P 4 , one concludes : c[]W 125. PR− PR = 217. Δθ []ºC (5.2) CuθθCu Fe Fe S PCu Fe = (5.7) dis m2 RRR++ [] θθθCu/ Fe Cu Fe (valid for bodies with dimensions smaller than 0.5 m). Utilizing a core E42-15 , with 230 turns of 0.60 mm wire Therefore , the total dissipated power is : the D.C. resistance results : RDC = 13. Ω . Pt[]W Applying a direct current of 1.13 A the temperature rise =+708.. 217ΔΔθθ025. S [][]ºCC[]º measured was Δθ = 26 º C when vertical mounted, and dis m2 [] 33 ºC if horizontal mounted. Therefore, the average and always assuming Δθ = 50 º C , the above equation leads value Δθ = 29.ºC 5 will be adopted for calculation to the estimative expression : purposes. P P t[]W t[]W From equations deduced for Sdis and Sdis : Δθ =≅0. 0778 780 (5.3) Cu Fe []ºC S S R = 24.ºCW 2 and R = 17.ºCW 6 . dis[]m22dis[]cm θ Cu θ Fe (linear equation valid only in the near range of From the model shown in fig. 4 (with PFe = 0 ) the total Δθ = 50 º C , rising over 40 ºC ambient temperature). equivalent thermal resistance becomes : In case of E type cores the surface of heat dissipation is 7 1 Δθ β KCu Rθ = = (5.8) PPPKB=+= + (5.13) eq tot 11 2 tot Fe Cu Fe m 2 + IRDC DC Bm RR+ R θθCu Cu/ Fe θ Fe expression that has a minimum for : from which : 2 β 2 KBCu m= β KB Fe m (5.14) 1 which leads to the optimal condition : Rθ = − Rθ (5.9) Cu/ Fe IR2 1 Fe β DC DC − PP= (5.15) Δθ R Cu Fe θ Cu 2 and using the above estimated values for R and R , Notice that only if β = 2 then PPCu= Fe which is the θ Cu θ Fe optimal condition. However, even if β = 26. , the optimal this gives R = 49. 7 º C W . θ Cu/ Fe values of PCu and PFe are not quite different. Ptot Assuming a typical case where PPCu== Fe , eq. 5.7 2 C. Current density adoption as function of temperature rise yields : 1. Transformers: General expression PCu Fe 1 ⎛ RRθθ− ⎞ Assuming winding and core losses of the same order , and = ⎜ Cu Fe ⎟ (5.10) P 2 ⎜ RRR++⎟ provided similar dissipation surfaces , the final temperatures tot ⎝ θθθCu/ Fe Cu Fe ⎠ will resemble. thus, for the above experimental values, eq. 5.10 gives : Therefore, given a high winding-core thermal resistance , PPCu Fe= 0036. tot which allows, in first instance, to the heat exchange will be small enough to assume that the neglect the thermal exchange between the core and the coil. windings will dissipate only through the air-exposed coil surface , this should be the area used for calculations ( S ). According to the nomenclature of fig. 3.a : PCu/Fe disCu ⎛ π ⎞ SC=++24()π AHC⎜ 2 +AC⎟ (5.16) disCu ⎝ 2 ⎠ R θ Cu/ Fe P With definitions 4.17 and eq. 4.19 : Cu PFe Rθ Rθ Cu Fe Sdis ⎡ 1 2 + F ⎤ Cu =+21π F ⎢ +h ⎥ (5.17) S W F Fe ⎣⎢ h π FFFWsh⎦⎥ Figure 4 : Thermal model of an open-core transformer The effective value of the primary current may be (C , E , EC or RM type cores). expressed as function of the current density by eqs. 2.7 and 2.8 as : S B. Loss balance between core and coil [3] [7] [8] [14] IS==σσ FFFFe (5.18) PPCefuPcP P P W The iron losses may be estimated by [6][12][18] : nP ξ β V PkfBFe= Fe m Fe (5.11) that substituted into eq. 4.12 gives : PF= 2ρσ2 FFFlS (5.19) where for most ferrites : ξ = 13. and 22≤≤β .7 being Cu eqPP P c P W le em Fe β ≅ 2 for high permeability materials aimed for switching where Fle depends on the the winding structure (given by frequencies ranging from 20 to 40 kHz , and β ≅ 25. for low eqs. 4.11 or 4.24) , but in most cases adopting Fle ≅ 1 may permeability ferrites suitable for higher frequencies. be an acceptable first trial approximation. Depending on the switching frequency the material choice From fig. 3.b , using eq. 4.16 yields lem , and utilizing the should be done using the loss charts available from ferrite definitions 4.17 and eq. 4.19 it results : manufacturers [6][9][10]. ⎡ (1+ F ) F ⎤ The copper losses may be expressed by eq. 4.12 l = ⎢2 s + π W ⎥ S (5.20) em F Fe (with Fle defined by eq. 4.11 or 4.24 , depending on the ⎣⎢ Fs h ⎦⎥ winding structure). Substituting eq. 5.20 into 5.19 : 2 Obtaining nP from eq. AII.2 (see Appendix II) and PF=⋅2ρσ FFF Cu eqPP P c P W le substituting into eq. 4.12 , yields: 2 ⎡ 1+ F ⎤ (5.21) VI ()s FW 32/ ⎛ PPef ef ⎞ Fl 1 K ⋅ ⎢2 + π ⎥ S P = 2ρ ⎜ ⎟ le em = Cu F Fe Cu eqP ⎜ ⎟ 32 2 ⎣⎢ Fs h ⎦⎥ ⎝ kfCf f⎠ FFPc FS WFem B Bm V P Substituting eqs. 5.21 and 5.17 into 5.3 yields : (5.12) ⎡ ()1+ Fs 1 F ⎤ The total losses are the sum of the core losses plus the ⎢ + W ⎥ windings ones , becoming a function of B which should be ⎢ π F 2 Fh ⎥ m Δθρ= 0078, σ2 FFF s S eqPP P c P le ⎢ 1 2 + F ⎥ Fe adopted looking for minimal total losses [14]. ⎢1++ h ⎥ According to eqs. 5.11 and 5.12 : ⎢ F π FFF⎥ ⎣ h Wsh⎦ 8 (5.22) VI. VOLUMETRIC CORE COMPARISON BETWEEN from which one obtains : FLYBACK AND FORWARD CONVERTERS Δθ []ºC Fgm 1 A. Volumetric ratio σ ⎡ ⎤ = 0358, P A ⎢ 2 ⎥ ρ eq FFPc F le S The volume of a magnetic core is related to the effective ⎣ mm ⎦ P μΩ cm P 4 Fe ⎡cm2 ⎤ [] ⎣⎢ ⎦⎥ length by , (5.23.a) ⎛ SFe ⎞ V ==Sl ⎜ ef ⎟ Sl= kSl (6.1) where Fgm is a core geometry dependent factor : Fe Feef Fe ⎜ ⎟ Fe Fe E Fe Fe ⎝ S Fe ⎠ 1 2 + F 1++ h and the magnetic length is related to the geometric minimal Fh π FFFWsh core section by, Fgm = (5.23.b) lkSFe= S Fe (6.2) ()1+ Fs 1 FW + which yields : π Fs 2 Fh V ==kkS32 k S 32 (6.3) For example, for square section scrapless lamination Fe E S Fe SE Fe cores, assuming a temperature rise of 50 ºC and where : ρμ= 20 Ω cm , the following estimative expression is 3 eqP k = k k = S l S SE SE Feef Fe Fe obtained : For scrapless laminations k = 1 and k = 6 so 5 E S σ ≅ ⎡A ⎤ kSE = 6 , while for ferrite E cores kSE usually lies P ⎢ 2 ⎥ A ⎣ mm ⎦ []cm between 5 and 7 [6][10] (as it may be calculated from core classical empirical formula well known by craftsmen, where manufacturer data). A is the central leg width. The required core volumes will be compared assuming equal shapes and proportions, so with the same kSE . 2. Ferrite made chokes The total volume occupied by the cores of the magnetic In this particular case, it is possible to find from components of a forward converter is : manufacturer tables AR such as : VVFe=+ Fe V Fe RAn= 2 (5.24) FW TFW L Cu R Then, the volumetric ratio for converter comparison may Usually AR ]05. is specified as the AR value be defined as : corresponding to F = 05. [6]. Therefore : VFe VVFe+ Fe c ℜ==V FW TFW L (6.4) FW/ FB V V AR ] FeFB FeFB A = 05. (5.25) R 2 F Substituting the aproximative form of the sizing equations c 2.14 , 3.2 and 3.10 in the expression 6.3 and replacing the The copper losses may be estimated by : results in 6.4 yields : 2 2 2 2 PIR==σ SAn (5.26) 34 Cu ef Cu Cu R V ⎡ σ ⎤ FeFW 34 ()1− D I P where : ℜ==+VFW/ FB δηi PP⎢ F ⎥ (6.5) V D σ S S FeFB ⎣⎢ I L ⎦⎥ SF==W FF Fe (5.27) Cu c n cWn The most commonly partition factor adopted is Substituting 5.25 and 5.27 into 5.26 yields : FP = 05. . Assuming this partition value and σ PL= σ , 2 22Fc the volumetric ratio is shown in figs. 5.a and 5.b with PSCu= σ FeF W AR ] (5.28) 2 05. parameters D and δ iP for η = 1 . Obviously, when Neglecting the iron losses (respect to the copper ones) it ℜVFW/ FB < 1 the forward topology must be preferred. may be assumed that : PR=Δθ (5.29) Cu θ tot B. Power switch sizing considerations Equating expressions 5.28 and 5.29 yields : The profit factor (or power utilization ratio) of the power device is defined as : 2 Δθ 10 []ºC P σ = O ⎡ A ⎤ f pr = ⎢ ⎥ FSW ⎤ P 2 Fe ⎡cm2 ⎤ RFA D ⎣⎢ mm ⎦⎥ ⎢ ⎥ θ ºC W cR⎥ ⎣ ⎦ tot [] ⎦05. []μ Ω where PO is the maximum available output power of the (5.30) converter and PD is the rated switching power for the Notice that eq. 5.30 do not consider the resistance rise due power device. The profit factor becomes better in continuous to both the skin and proximity effects. In most choke operation mode and it may be easily demonstrated that for application cases, this is not important because the D.C. both topologies it results : component is the main harmonic current component. ⎛ δ i ⎞ fDD=−−η ()11⎜ P ⎟ (6.6) pr ⎝ 2 ⎠ 9 The profit factor becomes maximum when D = 05. . In When the selected operation duty cycle is near the optimal this case, if η = 1 the former expression yields : value 0.5 , adopting σ L higher than σ P does not change the boundary value obtained. 1 ⎛ δ i P ⎞ f pr =−⎜1 ⎟ (6.7) Assuming D as the maximum duty cycle for nominal 4 ⎝ 2 ⎠ output power, the maximum voltage over the power switch Substituting 6.7 in 6.5 , for FP = 05. : will be : VVD=−1 expression valable for both CEmax P ( ) 34 − 98 topologies here involved. Using this equations with 6.5 and ℜ=−+VFW/ FB21() 4f pr 2 (6.8) [] 6.6 , ℜV is plotted in fig. 5.c , as function of In accordance with 6.6 it must be : FW/ FB f with the parameter VV. There, the most 1 1 pr CEmax P ≤≤f (6.9) 8 pr 4 suitable operation area for each topology are marked. so from 6.8 it results : V ℜ η = 1 2,5 FW / FB 0458..<ℜVFW/ FB < 146 (6.10) Fp = 0.5 σσ= The boundary value for the profit factor is 2 PL δ i = 1 f pr =≅0195.. 02 . P lim 1,5 Actually, the adoption σ PL= σ may be too 1 conservative, because the core for the inductor usually results δ i = 05. smaller than the one required for the transformer. Therefore, 0,5 δ i = 01. P the inductor can dissipate heat better than the transformer P due to its higher surface/volume ratio. Also, the inductor core 0 0 0,10,20,30,40,50,60,70,80,91 D power losses are lower, since the hysteresis loop there is smaller than the one performed in the transformer. (a) Consequently, higher copper losses (per volume) should be admissible, allowing higher current density on inductor V η = 1 ℜ FW / FB Fp = 0.5 windings (according to eq. 5.30). 2,5 σσPL= The inductor current density is usually adopted between 2 D = 01. σ PL≤≤σ 15. σ P. A higher current density reduces the inductor volume but degrades the efficiency and complicates 1,5 the close loop converter operation making the voltage D = 04. transfer ratio a function of the output load. D = 08. 1 Using σ LP= 15. σ and recalculating the boundary value D = 1 for the profit factor yields, f = 0209. , so the 0,5 prlim aproximative value 0.2 is again valid. 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 δ i P VII. CONCLUSIONS (b) V The widely used stacked winding technique allows ℜ FW / FB η = 1 partition factors others than 1/2 if required , but usually , the 2,5 Fp = 0.5 V V = 11. σσ= inner winding density current has to be limited to values such CEmax P PL as the optimal efficiency cannot be achieved. However, these 2 V V = 167. kind of windings have smaller leakage inductance than 1,5 flyback area CEmax P shared coil-former made, but they present a bigger interwinding capacity and poorer isolation features. 1 Therefore, even if the secondary was allocated in a forward V V = 8 V V = 2 0,5 CEmax P CEmax P separated coil-section, the demagnetizing coil should be area placed over the primary winding to ensure a good magnetic 0 0 0,05 0,1 0,15 0,2 0,25 coupling. In battery powered converters, bifilar winding will fpr improve the magnetic coupling. In off-line SMPS bifilar (c) windings are not reliable enough [7] and a good practice should be to interpose the primary between both the halved Figure 5 : Volume ratio as function of (a) duty-cycle, demagnetizing winding sections . This increases the parasite (b) primary current increment and (c) device profit factor. interwinding capacity [12] but an appropriate connection of the demagnetizing diode may overcome this drawback [7]. APPENDIX I : Magnetic flux density adoption For ffpr> pr the forward topology requires small lim A. Maximum efficiency selection core volume than the flyback one and vice versa, so for For the optimal condition stated by eq. 5.15 the total ff< a flyback implementation should be preferred. pr prlim losses become : 10 ⎛ ββ⎞ ⎛ ⎞ values , through the equation deduced keeping constant the PPP=+=+⎜1 ⎟ PkfB=+ξ β ⎜1 ⎟ V tot Cu Fe⎝ 2 ⎠ Fe Fe m⎝ 2 ⎠ Fe core density losses : ξ (AI.1) ⎛ f ⎞ β and the temperature rise may be estimated through : ref Bmf()= ⎜ ⎟ Bref (AI.9) Δθ = PR (AI.2) ⎝ f ⎠ tot θ tot From eqs. AI.1 and AI.2 it results : For high permeability ferrites used from 10 to 50 kHz , the 1 references may be BBref= sat and ffref= min , where ⎡ Δθ V ⎤ β ⎢ Fe ⎥ f min is the minimum recommendable operating frequency, Rθ B = ⎢ tot ⎥ (AI.3) that is, the maximal switching frequency which allows using m ]opt ⎢ ξ ⎛ β ⎞ ⎥ BB= . Under f it should be preferable to adopt ⎢ kfFe ⎜1+ ⎟ ⎥ msat min ⎝ 2 ⎠ ⎣ ⎦ other material , while for ff> min it is necessary to reduce For most ferrite cores , this value is smaller than the B according to eq. AI.9 . maximum allowable (which depends on the switching m frequency adopted). Therefore, a higher flux density might Usually ξ β ≅ 05. so, the eq. AI.9 may be approximated be adopted to increase the available output power at price of as : an efficiency degradation. f ref B = B (AI.10) mf() f ref B. Maximum output power selection In this case, the current density will be adopted as function For low permeability ferrites suitable for high frequency of the temperature rise (see section V - C ). switching , the reference values may be determined as For a given core, once the current density was function of temperature rise from manufacturer issued curves adopted, P becomes determinated from eq. 5.19 . For (or tables). Cu For example, suitable reference values for N27 and N47 these copper losses , it should be avoided that the core materials are : temperature rise surpasses the maximum specified for the N27 : f = 20 kHz ; B = 0.2 T coil , in order to prevent an additional heat transfer to the ref ref windings. Consequently , it must be : N47 : f ref = 100 kHz ; Bref = 0.1 T . P P Cu = Fe (AI.4) S S disCudis Fe APPENDIX II : Transformer core sizing equation Being both temperature rises identical : Δθ =+()PPR (AI.5) The Faraday law applied to a transformer magnetic circuit Cu Fe θ tot yields : From eqs. AI.4 and AI.5 it results : ΔB VnS= (AII.1) ⎛ S ⎞ PPFe Δθ disFe Δt PFe = ⎜ ⎟ (AI.6) R ⎜ S ⎟ For symmetrical converters ΔBB= 2 max and θ tot ⎝ distot ⎠ Using eqs. 5.11 and AI.6 , one obtains : ΔtDT= 2 , while for continuous operating mode forward 1 converters ΔBB= − B≅ B and ΔtDT= , where β max r max ⎡ Δθ ⎛ Sdis ⎞ ⎤ ⎜ Fe ⎟ B is the residual flux density (assumed that ⎢ ⎜ ⎟ ⎥ r ⎢ Rθ ⎝ Sdis ⎠ ⎥ tot tot BBrm<< axin soft magnetic materials). Bm ] = ⎢ ⎥ (AI.7) Pomax ξ V ⎢ kfFe Fe ⎥ On the other hand , in flyback converters : ⎢ ⎥ ΔBB= − B= Bδ i and Δt = TD . ⎣⎢ ⎦⎥ max min max P In symmetrical converters operating in continuous mode : Relating the expressions AI.3 and AI.7 : VD= V and VD= V , 1 PPef PPav B β m ]Po ⎡⎛ Sdis ⎞ β ⎤ max ⎜ Fe ⎟ ⎛ ⎞ so the voltage form factor results : = ⎢⎜ ⎟ ⎜1+ ⎟⎥ (AI.8) Bm ] ⎢⎝ Sdis ⎠ ⎝ 2 ⎠⎥ fD= 1 . opt ⎣ tot ⎦ f V For example, for a core E55-21 made with a material such For asymmetrical converters operating in continuous as β = 25. , Bm ] becomes approximately 14 % mode : Pomax D greater than Bm ] . V = V and VD= 2 V opt PPef 1− D PPav Using the Bm adoption criteria stated by eqs. AI.3 or therefore , AI.7 requires a previously made core selection, but it is just 1 f = for selecting the core (as output power function) that B is f V m 21DD()− first needed. To overcome this obstacle , one estimation a which allows expressing the eq. AII.1 as : priori of Bm can be made based upon known reference 11 VkfnSBf= (AII.2) If the foil conductor is immersed into the magnetic field PCfPFef V em due to other conductors (proximity field) the resistance will where : rise even more [12][13]. kC = 4 for symmetrical converters Assuming the proximity field uniform over the conductor kC = 2 for forward converters cross section, orthogonality appears between skin and kiCP= 2 δ for flyback converters proximity effects [20]. This decouples both effects and and the notation was simplified writing BB= . simplifies calculations [21]. mmax For the foil conductor of the m layer the increase of Using eq. 5.18 the primary current may be expressed as density current function. Then , using eqs. AII.2 and 5.18 the resistance is given by : primary apparent power is obtained : Rac ξ ⎡sinhξξ+ sin 2 sinhξξ− sin ⎤ = ⎢ +−()21m ⎥ (AIII.2) VI= kfBfσ FFFS2 (AII.3) R 2 coshξξ− cos coshξξ+ cos PPef ef Cfm V PcPWFe P dc ⎣ ⎦ from which the minimal required core section ( S ) may The first term in eq. AIII.2 is identical to eq. AIII.1 and Fe describes the skin effect. be found. For multilayer windings : The output power may be expressed as function of the apparent primary power, yielding : R ξ ⎡sinhξξ+ sin ⎛ 41p2 − ⎞ sinhξξ− sin ⎤ F ==ac ⎢ + ⎜ ⎟ ⎥ Pk= η VI (AII.4) r R 2 coshξξ− cos ⎜ 3 ⎟ coshξξ+ cos OuPPTefef dc ⎣⎢ ⎝ ⎠ ⎦⎥ where k is the transformer utilization factor , defined as uT (AIII.3) the ratio between the active primary power (or D.C. primary where p is the number of layers. supply power) and the apparent primary power adopted for Therefore Fr may be expressed as : transformer design. Suitables design values are : Fkrr= Sr+ kX (AIII.4) kDu =−1 for asymmetrical converters and T where krS is given by eq. AIII.1 and krX results : k = 1 for symmetrical ones. 2 uT ξ ⎛ 41p − ⎞ sinhξξ− sin krX = ⎜ ⎟ (AIII.5) Substituting AII.4 into AII.3 and rearranging , the required 2 ⎝ 3 ⎠ coshξξ+ cos minimal core section is obtained as output power function : For ξ ≤ 2 the following approximations apply [12] : P S = O (AII.5) Fe krS ≅ 1 (AIII.6.a) ησkkfuCf Bf m PcPW FFF TV P and , 2 51p − 4 Examples: krX ≅ ξ (AIII.6.b) For a ferrite core E42-15 made with material N27 , 45 In order to extend this one-dimensional approach to round assuming Δθ = 30 º C yields σ = 31./Amm2 (see P wire windings, Dowell [13] introduces an equivalent square section 4). Adopting the typical values : D = 0.4 , conductor thickness from : Fb = 036. , FP = 05. , η = 09. , f = 20 kHz and 2 2 ⎛ d ⎞ π B = 02.T , for the forward topology , from eqs. AII.4 Sh==π⎜ ⎟ therefore , hd= (AIII.7) m Cu ⎝ 2 ⎠ 2 and AII.5 , it results : PO = 100 W , while the empirical Each layer is supposed formed by nl turns of square manufacturer graphic [6] yields PO = 110 W . section equivalent conductors (fig. A3). For an E55-21 core with the same material , adopting the Since the square section conductors are separated by a gap same topology and working conditions, for equal s , a one-dimensional layer copper factor has to be defined 2 temperature rise it should be σ P = 27./Amm , which as: n yields PO = 272 W , while from manufacturer data , F = l h (AIII.8) l b PO = 275 W . W where, APPENDIX III : Skin and proximity effects bW : overall winding breadth h : equivalent conductor thickness For a single foil conductor subjected only to the skin n : number of turns per layer effect, the increase of resistance is given by [20] : l In order to adjust this model to the one-dimensional Rac ξ sinhξξ+ sin krS == (AIII.1) approach of the single-foil winding , the penetration depth Rdc 2 coshξξ− cos must be modified due to the porosity of the winding layer where: which increases the effective resistivity. ξ = h Δ Therefore, defining : ρρef= F l (AIII.9) Δ : penetration depth , Δ= ρπf μo h : foil thickness it yields, 12 Δ [6] Siemens, Ferrites and Accesories. Data Book 90/91 , Berlin, Δ ef==ρπμ eff o (AIII.10) Germany : Siemens Matsushita Components, 1990. Fl [7] K. H. Billings, Switchmode Power Supply Handbook , N. Y. : which leads to the effective value of ξ to be applied in McGraw-Hill, 1989. Fr estimations in case of round wire windings : [8] A. I. Pressman, Switching Power Supply Design , N. Y.: McGraw-Hill, 1991. Fhl π d ξ ==Fl (AIII.11) [9] C. W. T. McLyman, Transformer and Inductor Design ΔΔ2 Handbook , N. Y. U.S.A. : M. Dekker, 1988. If the increase in resistance due to eddy currents is [10] C. W. T. McLyman, Magnetic Core Selection for excessive, one alternative is to use bunched conductors or litz Transformer and Inductors , N. Y. U.S.A. : M. Dekker, 1997. wires. Then : [11] H. E. Tacca, Amélioration des caractéristiques des π hD= convertisseurs quasi-résonnants à transfert direct d’énergie , 2 st Thèse doctorale, Univ. des Sciences et Technologies de Lille, France, avril 1993. (where Dst is the strand diameter) and FFlW≅ . However, when the number of turns is small, the adoption of [12] E. C. Snelling, Soft Ferrites, Properties and Applications , foil windings usually gives lower ac resistances. The use of London U. K. : London Iliffe Books, 1969. litz wire may be considered for multilayer winding [13] P. L. Dowell, “Effects of Eddy Currents in transformer applications ranging over 500 kHz . Windings”, Proc. of the IEE , vol. 113 , no. 8 , August 1966 , For inductors carrying DC (choke applications) p. 1387 . adopting d Δ≤2 is suitable enough in discontinuous mode [14] A. Urling, V. Niemela, G. Skutt and T. Wilson, “Characterizing High-Frequency Effects in Transformer operating converters, while d Δ≤4 is acceptable in Windings - A Guide to Several Significant Articles”, included continuous mode operation. in Power Electronics Technology and Applications , N. Y. s U.S.A. : IEEE Press , 1993. [15] M.I.T. E.E. staff, Magnetic Circuits and Transformers, d Cambridge, U.S.A. : The MIT Press, 1943 , Chap. 8. [16] R. Petkov, “Optimun Design of a High-Power, High- h 12 34 nl layer m Frequency Transformer”, IEEE Trans. on Power Electronics, vol. 11 , No. 1 , Jan. 1996 , p. 33. [17] W. G. Hurley , W. H. Wölfle, and J. G. Breslin , “Optimized h Transformer Design : Inclusive of High-Frecuency Effects”, IEEE Trans. on Power Electronics, vol. 13 , No. 4 , July bW 1998 , p. 651. [18] R. Boll (editor), Soft Magnetics Materials: Fundamentals, Figure A3 : A layer of square section conductors equivalent to Alloys, Properties, Products and Applications, London, one of round section conductors. U.K. : Heyden, 1978. [19] R. M. Bozorth, Ferromagnetism , N.Y. : IEEE Press, 1993. ACKNOWLEDGEMENT [20] J. A. Ferreira, “Improved Analytical Modeling of Conductive Losses in Magnetic Components”, IEEE Trans. on Power The author would like to thank Prof. Carlos Christiansen Electronics, vol. 9 , No. 1 , Jan. 1994 , p. 127. for his interest and support that constantly encouraged this [21] W. G. Hurley, W. H. Wölfle, and J. G. Breslin, “Optimized work. Transformer Design: Inclusive of High-Frequency Effects”, IEEE Trans. on Power Electronics, vol. 13 , No. 4 , July REFERENCES 1998 , p. 651. [1] R. Bausière , F. Labrique and G. Séguier, Les convertisseurs Hernán Emilio Tacca was born in Argentina on December de l’électronique de puissance. Tome 3: La conversion continu-continu, Paris Fr. : Techn. et Doc. Lavoisier , 1987, 15, 1954. He received the B.E. degree in electrical Chap. 6 . engineering from the University of Buenos Aires (Argentina), in 1981 and the M.S. and Ph.D. degrees from [2] J. P. Ferrieux , F. Forest , Alimentations à découpage, convertisseurs à résonance , Paris Fr. : Ed. Masson , 1987, the University of Sciences and Technologies of Lille Chap. 1 and 2. (France) in 1988 and 1993 , respectively. In 1998 he received the Doctorate degree from the University of Buenos [3] R. Tarter , Solid-State Power Conversion Handbook , N. York : Wiley-Interscience publ. , 1993 , Chap. 3 and 9. Aires. Since 1984 , he has been with the Faculty of Engineering of the University of Buenos Aires, where he is [4] H. Tacca , Convertidores asimétricos con doble transferencia currently Assistant Professor and engaged in teaching and directa e indirecta , Tesis doctoral, Univ. de Buenos Aires, Fac. de Ingeniería, abril , 1998. research in the areas of industrial electronics. His research interest are in the field of SMPS, UPS, battery chargers, soft- [5] C. R. Hanna, “Design of Reactances and Transformers Which switching techniques and microcontroller control of power Carry Direct Current”, AIEE Journal , vol.46 , Febr. 1927 , p. 128 . converters.