arXiv:2008.13634v3 [physics.app-ph] 20 Oct 2020 op io antclo,rvril nutne amplitu inductance, reversible loop, magnetic inductance minor loop, re ogtidco oesueu o ierneo core of range wide a for extract levels. useful datasheet to induction models magnetic material how inductor and ferrite shown temperatures get a also to from order is parameters It necessary converters. the ope high-frequency power control and electronic design c properly any to in required model employed inductor inductances ferrite-based of design-oriented kinds prehensive several the to related rsne ne utbecnetfrtepwrelectronic power this the In for process. context an suitable design revisited a are inductor under definitions the inductance presented exploit specific enhance better some be to paper, tha can order definitions seems the inductance in core it to of However, same of according variety condition. the [11] this definitions operating [5], At established inductor [4], [4]. actual well inductance [3], many and [2], are permeability [1], there literature time, in treated widely specific core. the the to in employed according b material things, to ferrite needs other defined definitions be among inductance to parametrized, these for need Also, model, properties explained. inductor and and accurate inductances an Th upon certain needed. based is which be level to current has certain method a at inductance specified fidco,amto ofidterqie iiu number minimum required the find to turns electroni of method the a specific of inductor, that operation design of efficiently safe to order the allows In converter. and which power ensured control always that accurate is way the value a in inductance is inductors minimum It these a design current. to low-frequency importance paramount large-amplitude relati relatively a superimposed a filter being con- effectively current DC/DC to high-frequency step-down low-amplitude have or inductors (VSI) Those inverters verters. source voltage of F ne Terms Index h rbe fdsgigsc ido nutr a been has inductors of kind such designing of problem The Abstract olcino entosadfnaetl o a for fundamentals and definitions of collection A autdd inisEats Ingenier Exactas, Ciencias de Facultad r omnyfud o xml,i h Cotu filter output LC the in example, for found, commonly inductor are biased low-frequency-current based Errite-core *GuoSimulaci Grupo ** N Ti ae ensaddvlp sflconcepts useful develops and defines paper —This min nvria ainld oai (UNR) Rosario de Nacional Universidad mgei ici,friecr,mjrmagnetic major core, ferrite circuit, —magnetic n h piu i a length gap air optimum the and eatmnod Electr de Departamento * 1 st [email protected] CIFASIS-CONICET-UNR .I I. nre aqe Sieber Vazquez Andr´es oai,Argentina Rosario, nyCnrld itmsF Sistemas de Control y on NTRODUCTION ´ einoine nutrmodel inductor design-oriented onica ´ ´ ayAgrimensura y ıa g opt ooti a obtain to ´ ısicos hc is which , rated type vely om- on ed de of in is d e c s t g ici osdrdi iue1uiga prahdpnigo depending approach an using 1 Figure in considered circuit eeto rtro fteidcac au swl as well as justify value to inductance the needed of be criterion could propert selection that useful inductance ferri reversible some the the presents of V from to Section permeabilities how datasheet. shows defined material sev IV previously of Section the presented. definitions obtain perme- are III, inductances associated section of its In types and introduced. described are is abilities model core magnetic and contextualized N27 revisited, TDK-EPCOS N87. materials: also TDK-EPCOS ferrite and are specific two paper permeabilit for obtained core this respective in induc- their Those on which method. rely design definitions inductor tance optimized an develop mltd inductance amplitude ntecr antcmdldsrbdi hspprwhich paper inductance this reversible of in concepts described the model employ base to magnetic be allows core can the model inductor on design-oriented A practitioner. autdd inisEats Ingenier Exactas, Ciencias de Facultad opt nti eto,w banamdlfrtegnrlmagnetic general the for model a obtain we section, this In hsppri raie sflos nscinI,ageneral a II, section In follows. as organized is paper This *GuoSimulaci Grupo ** ial,cnlsosaepeetdi eto VI. section in presented are conclusions Finally, . nvria ainld oai (UNR) Rosario de Nacional Universidad eatmnod Electr de Departamento * I M II. [email protected] CIFASIS-CONICET-UNR i.1 eea antccircuit magnetic General 1. Fig. 2 oai,Argentina Rosario, nd GEI ICI MODEL CIRCUIT AGNETIC nyCnrld itmsF Sistemas de Control y on ´ L ´ nc Romero M´onica a n nta inductance initial and onica ´ ´ ayAgrimensura y ıa L ´ ısicos i ofurther to , N min L eral rev and ies, the ˆ ies te n d , integration along the mean magnetic path into the ferrite core, Lc and into the air gap, Lg [5]. Suppose that current i(t) can be easily decomposed into a) a component denoted as iLF (t) at a relatively low frequency fLF and b) a component denoted as iHF (t) at a relatively high frequency fHF , with fLF ≪ fHF . At a certain time tˆLF , iLF reaches its peak value ˆiLF and then we have 1 1 i(t) ≈ ˆiLF + iHF (t) t ∈ tˆLF − , tˆLF + (1)  fHF fHF  In such a situation, Ampere’s law relates the frequency com- ponents of current i(t) with their corresponding magnetic field strength H components as follows

H~ (t,l)dl~ = HˆLF (l)+ HHF (t,l) dl ∪ I ZLc Lg h i

= N ˆiLF + iHF (t) h i where dl~ is the path vector, parallel to H~ . Separating the frequency components yields

ˆ ˆ ˆ HLF (l)dl + HLF (l)dl = NiLF (2) Fig. 2. Ferrite magnetization curve ZLc ZLg

HHF (t,l)dl + HHF (t,l)dl = NiHF (t) Z Z Lc Lg For any of the points pertaining to that LF-commutation curve, the LF-amplitude permeability µLF is defined as ∆HHF (l)dl + ∆HHF (l)dl = N∆iHF (3) a ZLc ZLg 1 Bˆ where ∆HHF is the amplitude of the field strength excursion µLF Bˆ , ∆B = LF a LF HF ˆ due to ∆iHF , the amplitude of the high-frequency current   µ0 HLF ∆BHF excursion during 1 . fHF The magnetic induction B(t,l) = BˆLF (l) + BHF (t,l) because it relates only the low frequency amplitude of the and its peak-to-peak variation ∆BHF determine the peak magnetic induction and field strength in the ferrite material ∆BHF (l) induction Bˆ(l)= BˆLF (l)+ . These are related to their when the low frequency iLF takes also its amplitude value 2 ˆ corresponding field strength Hˆ (l), H (t,l), ∆H (l) iLF . The magnetic induction generated by iLF will not vary LF HF HF ˆ and Hˆ (l) according to the medium permeability. Having the much from BLF while t is into the time span defined in (1). 1 Hence during , iHF (t) will produce an approximately air gap paramagnetic properties, along Lg simply hold fHF closed minor magnetic loop of amplitude (∆BHF , ∆HHF ) BˆLF (l) BHF (t,l) ∆BHF (l) Bˆ(l) starting and ending in the neighbourhood of Bˆ , as it is = = = = µ (4) LF ˆ ˆ 0 HLF (l) HHF (t,l) ∆HHF (l) H(l) shown in Figure 2. The incremental permeability µ∆ˆ at that quasi-static induction level BˆLF is then defined as where µ0 is the vacuum permeability. In the magnetic core path Lc, those relationships depend on the shape of the ferrite ˆ 1 ∆BHF magnetization curve which is shown in Figure 2. It is also µ∆ˆ BLF , ∆BHF = µ0 ∆HHF ˆ the specific major loop that characterizes the behaviour of the   BLF ferrite when its magnetic induction evolution spans the two Consequently, on Lc holds extreme points ±Bs, being Bs the saturation induction. In any inductor, although this situation can be reached with a current BˆLF (l) iLF having a sufficiently high ˆiLF , the actual ˆiLF has to be set HˆLF (l)= (5) LF ˆ well below that value since beyond that induction level the core µ0µa BLF , ∆BHF ferrimagnetic properties become severely affected. Starting   ∆BHF (l) from a demagnetized core, as ˆi is gradually increased ∆HHF (l)= (6) LF ˆ from zero towards the maximum value causing saturation, µ0µ∆ˆ BLF , ∆BHF   the tipping points (BˆLF ,HˆLF and −BˆLF ,−HˆLF ) of the ever increasing LF-major loops describe a LF-commutation curve If ∆BHF is made sufficiently small, then µ∆ˆ and the which is also partly shaped by the current magnitude of LF-commutation curve start to be practically independent of ∆BHF , due to the memory properties of the ferrite material. ∆BHF . At this point, on the one hand the existing linear relationship between BHF and HHF is captured by the so- The peak linkage flux Ψˆ and magnetic induction Bˆ(l) in the called reversible permeability at BˆLF , µrevˆ ferrite core are ∆ΨHF µrev BˆLF = lim µ BˆLF , ∆BHF ˆ ˆ ˆ → ∆ˆ Ψ= ΨLF + (11)   ∆BHF 0   2 On the other hand, the LF-commutation curve tends to the Ψˆ Bˆ(l)= regular commutation curve and their respective amplitude NAc(l) permeabilities are related as III.INDUCTANCE DEFINITIONS ˆ ˆ 1 B LF ˆ µa B = = lim µa BLF , ∆BHF Combining (2), (3), (4), (5), (8), (9) and (10) yield the LF- µ0 Hˆ BˆLF →Bˆ LF     amplitude inductance, La and the incremental inductance, ∆BHF →0 L∆ˆ = µa BˆLF   Ψˆ LLF ˆ , LF Now making BˆLF → 0 due to ˆiLF → 0, the initial a ΨLF ∆ΨHF =   ˆiLF permeability µi is defined as ∆ΨHF 2 N µi = lim µa BˆLF = µrevˆ BˆLF =0 = (12) LF BˆLF →0 R ˆ R     ca ΨLF , ∆ΨHF + g Note that in the core, the relationship between B and H   ˆ ∆ΨHF depends not only on the ferrite magnetic characteristics but L∆ˆ ΨLF , ∆ΨHF = ∆iHF ˆ also on the way in which B evolves with time.   ΨLF N 2 In the magnetic circuit of Figure 1, a closed surface = (13) S = Sc ∪ Sg that intersects both Lc and Lg paths will satisfy ˆ Rc∆ˆ ΨLF , ∆ΨHF + Rg according to Gauss’ law that   with B~ (t,l)dS~ = B~ (t,l)dS~ (7) LF dl ZZSc ZZSg R ˆ ca ΨLF , ∆ΨHF = LF Ψˆ LF ∆ΨHF   ZLc µ µ , Ac(l) where Sc is a core cross-section perpendicular to Lc, Sg is 0 a NAc NAc   the remaining surface of S crossing the air gap and dS~ is the (14) area vector of S. The left side of (7) is the core magnetic flux ˆ dl Rc ˆ ΨLF , ∆ΨHF = Φc since the magnetic induction there is mainly concentrated ∆ ˆ ZL ΨLF ∆ΨHF   c µ0µ∆ˆ NA , NA Ac(l) into Sc because µa,µ ˆ ≫ 1. All the magnetic induction in c c ∆   (15) the air gap will then pass through Sg, thus the right side of dl (7) is the air gap magnetic flux Φg. Consequently, Φc =Φg = R ˆ g = (16) ΦLF +ΦHF (t). Lc passes perpendicular through the center of ZLg µ0Age(l) S , so the magnetic induction along L will be approximately c c LF R , Rc ˆ and Rg are the core LF-amplitude reluctance, core an average of that existing inside Sc and equal to ca ∆ incremental reluctance and the air gap reluctance respectively. Φˆ +Φ (t) LF HF Considering the situation where ∆ΨHF → 0, µa and µrevˆ BˆLF (l)+ BHF (t,l)= (8) Ac(l) define the amplitude and reversible inductances at Ψˆ LF , La where Ac(l) is the area of Sc at a certain point l ∈ Lc. Around and Lrevˆ , as the air gap, the magnetic induction is far more nonuniform ˆ LF ˆ La ΨLF = lim La ΨLF , ∆ΨHF in S than in S due to the fringing flux. Thus, the mean ˆ → ˆ g c   ΨLF Ψ   ∆Ψ →0 induction on Lg can be quite different from the actual values HF at the edges of the gap, but being a paramagnetic region, it N 2 = (17) suffices to propose an effective gap area Age(l) with l ∈ Lg, ˆ Rca ΨLF + Rg as if all the induction were there concentrated.   ˆ ˆ ˆ t Lrevˆ ΨLF = lim L ˆ ΨLF , ∆ΨHF ΦLF +ΦHF ( ) ∆Ψ →0 ∆ BˆLF (l)+ BHF (t,l)= (9)   HF   Age(l) N 2 = Note that Age(l) is approximately equal to Acg, the core cross- ˆ Rcrevˆ ΨLF + Rg section in contact with the air gap, if its length is much smaller   dl than the linear dimensions characterizing Acg. Given that the R Ψˆ = (18) ca LF ˆ winding turns N embrace practically all Φc, it follows that the ZL ΨLF   c µ0µa NA Ac(l) linkage flux Ψ is  c  dl R Ψˆ = (19) Ψˆ LF +ΨHF (t) crevˆ LF ˆ ˆ ZL ΨLF ΦLF +ΦHF (t)= (10)   c µ0µrevˆ NA Ac(l) N  c  Rca and Rcrevˆ are the core amplitude reluctance and the core (B-H) curve of Figure 2 [7], reversible reluctance respectively. B 1 H B − H (22) La and Lrevˆ usually have dissimilar values at a same u( )= au c µ0µc 1 − B Ψˆ and vary differently as Ψˆ increases from zero to Bs LF LF   relatively high values. It is then important to find a common B 1 H (B)= + H (23) situation to relate and relativize their current values with. l al c µ0µc − B 1 B In a demagnetized material, µa and µrevˆ coincide at the  s  origin which means that L and L converge to the initial a revˆ with the positive parameters: coercive field strength Hc, coer- inductance L i cive permeability µc and squareness coefficients au and al for each branch. Supposing that al ≈ au and being Bˆ = BˆLF , 2 N ˆ Li = lim La Ψˆ LF = Lrevˆ Ψˆ LF =0 = µrevˆ BLF ,Tc can be expressed as [7] ˆ → R R ΨLF 0     ci + g   (20) al BˆLF 1 + (al − 1) dl  Bs 1 R µ Bˆ ,T   ci = (21) revˆ LF c =  a 2 Z µ µiAc(l)  Bˆ l µc Lc 0    − LF 1 B h  s  i R  −1 being ci the core initial reluctance. Note that only µi does  not vary with the core cross-sectional area A (l) along the c bo  + magnetic path. However, µi as well as µa and µrevˆ do depend ˆ ˆ ao  − BLF − − BLF  heavily on the core temperature, as is modeled in the next 1 B 2 1 B  s  h  s  i section.   (24) boBs 1 1 ao = bo = − IV. PERMEABILITY MODELS µ0Hc µi µc

Apart from µi, µrevˆ depends on Tc through Bs, Hc, al and µc The dependence of Rca in (18) and Rci in (21) from core and thus (23) has to be numerically fitted for each particular ˆ ˆ temperature Tc and magnetic induction BLF = B, in each Tc. The fitting data is obtained from the 3-D lookup table specific part of the core, is addressed when the corresponding H(B,T = Tc) based on the corresponding curves from the functions µa BˆLF ,Tc and µi (Tc) are extracted from the ferrite material datasheet [12] [13].   ferrite material datasheet [12] [13]. Permeability µa is a func- The starting guess points for the fitting process are ex- ˆ ∗ ∗ ∗ tion of magnetic induction amplitude B and core temperature tracted from 2-D lookup tables Bs (T ), Hc (T ), al (T ) and ∗ ∗ ∗ (3-D lookup table) and permeability µi is a function of core µc (T ). Bs (T ) and Hc (T ) are built to linearly interpolate the temperature (2-D lookup table). To get the best accuracy in the two saturation induction and coercive field strength values inductor model, the µa curve given by the ferrite manufacturer (Bs1, Bs2, Hc1 and Hc2 respectively), that are stated at should have been obtained at a frequency close to fLF . the two corresponding temperatures T1, T2, in the datasheet of the magnetic material. Lookup tables a∗(T ) and µ∗(T ) The temperature and induction dependence of Rcrevˆ in l c are conformed in the following way. Let H11(B11,T1) and (19) is subjected to find µrevˆ BˆLF ,Tc . In [6] it is con- H (B ,T ) be the field strength at two different induction cluded that the commutation curve coincides with the so-called 12 12 1 levels from the lower branch of the B-H curve at temperature initial magnetization curve for soft ferrite materials, that is T given by the datasheet. The value B could be from the (B ,H ) = (B,ˆ Hˆ ). This means that µ is equal to 1 11 DC DC revˆ ”linear” region of the curve, while B could be taken from DC-biased µ which can be extracted from a graph or as a 12 rev the ”knee” between ”linear” and ”saturation” regions of the function of DC-bias field strength H . That curve may not DC curve at temperature T . The estimations of coefficients a be given in datasheets for a particular ferrite material or for 1 l and µ from (23) at temperature T , a∗ and µ∗ respectively, the core temperatures at which µ has to be obtained, but c 1 l1 c1 revˆ are found numerically solving even if it were available it should be put in terms of BˆLF to ∗ be employed in (19). To overcome these limitations, we use a al1 − B11 1 B 1 − permeability model directly relating DC-biased µrev with DC- s H12 Hc1 B11  a∗ = l1 H − H B bias magnetic induction BDC [7], where all its parameters at − B12 11 c1 12 1 B 1 the desired core temperature can be entirely obtained from  s  any ferrite datasheet, in the way it is next explained. This ∗ 1 B11 1 µc1 = a∗ µ H − H l1 approach has been experimentally validated for many ferrite 0 11 c1 − B11 1 B 1 materials operating at different temperatures [8] and it is  s  currently employed by major ferrite manufacturers [9]. ∗ ∗ Using the B-H curve at temperature T2, al2 and µc2 can ∗ Let us first consider the empirical models that curvefit the be also obtained following a similar reasoning. Finally, al (T ) ∗ ∗ ∗ ∗ ∗ upper (u) and lower (l) branches of the dynamic magnetization and µc (T ) are built to linearly interpolate al1,al2 and µc1,µc2 and its first derivative, the inverse of the so-called differential 0.5 permeability µ [4] N27 100oC d 0.45 N27 25oC LF o dHu 1 N87 100 C 0.4 = N87 25oC dB µd

0.35 dµd Since µrev < µd [4] it is sufficient to show that dB < 0 0.3 to conclude that Lrev is decreasing when B takes values 0.25 from zero to Bˆ and hence L is the absolute minimum. B [T] LF revˆ

0.2 Accordingly, 2 LF 0.15 dµ d H d −µ 2 u = d 2 0.1 dB dB au au au B B B 0.05 2 LF au 1 − + au + au d H Bs Bs Bs u =   h     i 0 dB2 au 3 0 50 100 150 200 − B H [A/m] µ0µc 1 B B h  s  i ˆ − ˆ 2 β ln Hl(BLF ) HLF Hˆ − H (Bˆ ) Fig. 3. Curve-fitting of the lower branch of the magnetization loops (first Hˆ −H Bˆ LF u LF + LF u( LF ) h i quadrant) for materials N27 and N87. Dashed lines are interpolated data from  ˆ  α datasheet 2BLF Hl(BˆLF ) − HˆLF   h i (26) respectively. Figure 3 shows the fitting goodness of (23) HˆLF is inside the area delimited by the largest major loop, for TDK-EPCOS ferrite materials N27 and N87 while their the magnetization curve, that is described by (22)-(23). Con- corresponding parameters for using (24) are summarized in sequently, (26) is positive for B ∈ [0, BˆLF ] and thus µd is Table I. To get the best accuracy in the inductor model, the ever decreasing for increasing values of B > 0. magnetization curve given by the ferrite manufacturer should Now suppose that gradually ∆ΨHF is increased and Ψˆ LF is have been obtained at a frequency close to fHF . decreased in such a way that Ψˆ remains unchanged, implying that Bˆ and Hˆ are unmodified in all parts of the core. This TABLE I scenario brings into existence increasingly asymmetric minor REVERSIBLEPERMEABILITY µrevˆ MODEL PARAMETERS (EQUATION (24)) loops in the B − H plane with tipping points

o ∆BHF Material Tc [ C] al Hc [A/m] µc µi Bs [T ] Bˆ = BˆLF + N27 100 1.25 18.12 14079 3231 0.4165 2 N27 25 2.00 24.35 11154 1700 0.4895 Hˆ = HˆLF + kH ∆HHF N87 100 8.00 10.94 4330 3976 0.3925 ∆B N87 25 3.78 21.17 6014 2210 0.4803 Bˆ − ∆B = Bˆ − HF HF LF 2 Hˆ − ∆HHF = HˆLF − (1 − kH )∆HHF being k ∈ [0.5, 1) the magnetic field symmetry factor. V. CONSIDERATIONS ON THE REVERSIBLE INDUCTANCE H Considering that along a general magnetic loop µd increases Recall that if ∆ΨHF → 0 we can consider as B decreases, it can be stated that dB ˆ ˆ ˆ Bˆ − ∆HHF ≥ Bˆ − ∆BHF Lrevˆ Ψ = L∆ˆ ΨLF , ∆ΨHF = Lrevˆ ΨLF dH Bˆ       and hence It is important to remark that in this situation Lrevˆ is the ˆ ≤ ˆ ≤ ˆ minimum value of reversible inductance arising along the µrevˆ B µd B µ∆ˆ BLF , ∆BHF whole symmetric major loop having ± Hˆ , Bˆ as tip-       LF LF Inside the minor loop, we can define the minor-loop amplitude ping points. This is in fact proved by taking into considerat ion MN permeability µa as the upper branch of the particular ± HˆLF , BˆLF major loop ˆ − ˆ which can be described in terms of (22)-(23) as [6] MN 1 B BLF 1 ∆BHF 1 µa = = = µ∆ˆ µ0 Hˆ − HˆLF µ0 2kH ∆HHF 2kH β HˆLF − Hu(BˆLF ) ˆ ˆ LF and to note that it holds µrevˆ B < µrevˆ BLF , since Hu (B)= Hu(B)+ h iα (25)     when B is far from the origin, µrevˆ decreases as B increases. Hl(BˆLF ) − HˆLF h i Consequently, B − BˆLF B + BˆLF α = β = ˆ ˆ ˆ ∆BHF ˆ ˆ BLF + µ0µrevˆ BLF kH ∆HHF ≥ BLF + 2BLF 2BLF   2 and hence ACKNOWLEDGMENT ˆ ≥ MN ≤ The first author wants to thank Dr. Hernan Haimovich for µrevˆ BLF µa µ∆ˆ   his guidance and constructive suggestions. If the minor loop keeps some degree of symmetry around REFERENCES ˆ ˆ ≈ MN ≈ BLF , HLF , i.e. kH 0.5, then µa µ∆ˆ and hence   [1] C. Wm. T. McLyman, Transformer and Inductor Design Handbook, 3rd ed. New York, USA: Marcel-Dekker, 2004. ˆ ≤ ˆ ≤ ˆ [2] N. Mohan, T. M. Undeland and W. P. Robbins, Power Electronics. µrevˆ B µ∆ˆ BLF , ∆BHF µrevˆ BLF       Converters, Applications, and Design, 3rd ed. New York, USA: John Wiley & Sons, 2003. However, that minor loop with tipping points [3] M. K. Kazimierczuk, High-Frequency Magnetic Components, 2nd ed. West Sussex, England: John Wiley & Sons, 2014. B,ˆ Hˆ ; Bˆ − ∆BHF , Hˆ − ∆HHF [4] A. Van den Bossche and V. C. Valchev, Inductors and Transformers for     Power Electronics, 1st ed. Boca Raton, USA: Taylor & Francis, 2005. and an enclosing symmetric major loop with tipping points [5] E. C. Snelling, Soft Ferrites. Properties and Applications, 1st ed. London, England: Iliffe Books Ltd, 1969. ˆ ˆ ˆ ˆ [6] M. Esguerra, Computation of minor hysteresis loops from measured B, H ; −B, −H major loops, Journal of Magnetism and Magnetic Materials, no. 157/158,     pp. 366-368, Elsevier Science B.V., 1996. coincide in the vicinity of their uppermost tipping point [7] M. Esguerra, Modelling Hysteresis loops of Soft Ferrite Materials, in ˆ ˆ ˆ Proc. of the International Conference on Ferrites ICF 8, Kyoto, Japan, B, H [10] [7]. Hence, µrevˆ (B) at the existing minor loop pp. 220-220, Sep. 2000.   [8] M. Esguerra, M. Rottner and S. Goswami, Calculating Major Hysteresis is equal to µrevˆ (Bˆ) at that hypothetical major loop. In fact, loops from DC-biased Permeability, in Proc. of the International Con- µrev (BDC ) [7], from which (24) is particularly derived, ference on Ferrites ICF 9, San Francisco, USA, Aug. 2004. depends on the current magnetic induction value regardless [9] M. Esguerra, Magnetics Design Tool for Power Applications, its previous evolution [11]. Note that from the minor loop Bodo’s Power Systems, pp. 42-46, Apr. 2015. [Online]. Available: ˆ www.bodospower.com/restricted/downloads/bp 2015 04.pdf. standpoint, Ψ is given by (11), but it can be also put in terms [10] R. G. Harrison, Modeling High-Order Ferromagnetic Hysteretic Minor of the amplitude permeability as Loops and Spirals Using a Generalized Positive-Feedback Theory, in IEEE Transactions on Magnetics, vol. 48, no. 3, pp. 1115-1129, March 2012. Ψ=ˆ La Ψˆ ˆi   [11] C. Heck, Magnetic Materials and their Applications, 1st ed. London, England: Butterworth, 1974. ˆi = max ˆiLF + iHF (t) t [12] EPCOS AG, SIFERRIT material N27, Ferrite and   Accessories, May 2017. [Online]. Available: www.tdk- and thus it is valid electronics.tdk.com/download/528850/d7dcd087c9a2dbd3a81365841d4aa9a5/pdf- n27.pdf ∆ΨHF LF ∆ΨHF [13] EPCOS AG, SIFERRIT material N87, Ferrite and La Ψˆ LF + ˆi = L ˆiLF +  2  a 2 Accessories, Sep 2017. [Online]. Available: www.tdk- electronics.tdk.com/download/528882/71e02c7b9384de1331b3f625ce4b2123/pdf- LF LF ˆ n87.pdf La = La ΨLF , ∆ΨHF   On that enclosing symmetric major loop, characterized by ˆ MJ ˆMJ ˆMJ parameters Ψ , iLF and i , we have that

MJ MJ MJ Ψˆ = La Ψˆ ˆi = Ψˆ   ˆMJ ˆMJ ˆ i = iLF = i

Consequently, for ∆ΨHF > 0 we get ˆ MJ ˆ ˆ Lrevˆ Ψ = Lrevˆ Ψ