A Novel Magnetic Equivalent Circuit Model for the Corner of Inductor Core

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Article A Novel Magnetic Equivalent Circuit Model for the Corner of Inductor Core

Xiaodong Wang 1,2 , Hui Chen 1, Mingquan Shi 1,* and Lyes Douadji 1

1 Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Beibei District, Chongqing 400714, China; [email protected] (X.W.); [email protected] (H.C.); [email protected] (L.D.) 2 University of Chinese Academy of Sciences, Shijingshan District, Beijing 100049, China * Correspondence: [email protected]

Received: 7 November 2019; Accepted: 3 December 2019; Published: 6 December 2019

Abstract: The accurate calculation of inductance is the most basic problem of inductor design. Based on finite element method (FEM) analysis of the flux density distribution in the corner of the core, a local corner magnetic equivalent circuit (MEC) model is proposed for a core-type inductor. In this model, the corner is divided into three parallel branches, and the joint part of the corner and the core limb are divided into two branches, one of which is continuously divided into three branches. In addition, a flux leakage branch of the core window inner corner is modeled. These branches form a local corner MEC by series and parallel to describe the flux distribution of the corner. The calculation results show that, in the local corner MEC model, the core flux density distribution is consistent with the FEM simulation. The maximum relative error of the inductance is less than 6% and the average relative error is less than 4% compared to the physical prototype measured data with the range of current from 20 A to 1000 A, which shows that the local corner MEC model has a high accuracy inductance calculation.

Keywords: modeling; inductance; corner; ﬁnite element methods; magnetic equivalent circuit; magnetic analysis

1. Introduction Inductance calculation is one of the most basic problems in inductor design. The most accurate method is to use the electromagnetic finite element method (FEM) [1], which requires a geometry model and a long calculation time so that it cannot be calculated online in real-time [2–4]. Another method is to use the magnetic equivalent circuit (MEC) model which has the advantages of fast calculation speed and acceptable accuracy. At present, the MEC model can be used to accurately calculate the inductance of the operating point below the saturation point. However, the inductance of multiple operating points or the full current range cannot be well predicted, especially when the core is saturated. The main factor affecting the calculation accuracy of the MEC model is whether the reluctance calculation of each part is correct. Recently, for the power inductor, some researchers focused on the model of the air gap [5,6], and leakage flux reluctance such as leakage flux of the core window [7] and fringing leakage [8]. Some researchers are concerned about the MEC model’s net forms of different types of inductor. Liu [9] establishes a MEC model of the Y-shape, -shape to design an inductor by 4 the magnetic integration method. Duppalli [10] and Cale [11] established a standard UI-shape (where an “UI-shape” core is composed of an U-shaped core and an I-shaped core) and EI-shape (where an “EI-shape” core is composed of an E-shaped core and an I-shaped core) MEC model. These models make great contributions to improving reluctance accuracy. On the other hand, a few institutions and individuals aimed at the main reluctance of the core, especially the corner. Except for the equivalent corner reluctance model established by the finite

Appl. Sci. 2019, 9, 5341; doi:10.3390/app9245341 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 5341 2 of 12 element method [12], the common method is to simplify the corner as a quarter-circle that has a fixed radius, replacing the original rectangular geometry, such as the IEC (International Electrotechnical Commission) 60205 [13] model, which has a radius equal to half the average length of the corners and is adopted in [9]. In the Snelling [14,15] model, the authors choose a quarter of the average length of the corner as the radius and this corner model is adopted in [5,6]. In the Sudhoff [16] model, the corner flux path length is equal to half the rectangular side beside the limb and is adopted in [11]. However, these corner models use a single fixed flux path length to calculate the reluctance without considering the nonlinearity of reluctance. In real cases, the corner flux density distribution is non-uniform where the inner corner is high, and the outer corner flux density is low [17]. The magnetic flux mainly flows around the inner corner. As the strength of the magnetic field increases, the inner corner reaches saturation and leads to a significant increase in local reluctance and the flux density (B) distribution also changes. Therefore, the simplified reluctance equivalent length model of the corner is not consistent with the real condition and contributes to inductance error, especially under the saturation condition. In this paper, we focus on the power inductor, especially with UI/EI or other core structures with rectangular corners. In order to describe the flux density distribution at the corner and improve the accuracy of the core corner reluctance, so as to improve the accuracy of inductance calculation, we propose a local MEC model of a corner. Based on the finite element analysis, the corner and the joint part between the corner and the core limb enclosed by winding are divided into several reluctance branches to describe the flux density distribution. In addition, a reluctance branch describes the leakage flux of how the core window inner corner is modeled, which could contribute to improving the accuracy of the inductance especially when the core is saturated. By the series and parallel combination of the reluctance branches, the flux density distribution of the corner is fully described and accurately calculated, and the inductance can also be accurately calculated. The rest of this paper organized as follows. In Section2, the local MEC model of the corner is established and the corresponding reluctance calculation approach is demonstrated. Section3 models the square core-type inductor MEC and gives the methods to calculate the inductance. Section4 describes the experiments and analysis, where the core branch flux density B distribution of the local corner MEC model and FEM simulation are compared. Then, the inductance with the current of the local corner MEC model and IEC model are compared to physical prototype measured results, whereas the relative errors are analyzed. The number of corner branches is also discussed. Section4 concludes the study and gives further research points.

2. Modeling and Analysis of Corner

2.1. Geometry of Square Core-Type Inductor In this paper, a symmetric square core-type inductor is designed, as shown in Figure1, which uses Fe-Si power 26µ [18] material and contains four blocks of 23 mm 30 mm 90 mm in the core. Since the × × power core material is used, the core does not contain functional air gaps as a steel core inductor, in which the reluctance of the core is the main factor of the inductance. The geometry parameters are shown in Table1. In the square inductor, there are four windings, a total of 40 turns, and each winding has 10 turns evenly distributed over four limbs, such that the four core limbs have even and identical flux distribution, and the corner has symmetrical flux distribution. This makes it easy to study the flux density distribution of the corner. In addition, the corners and the joints between the corner and limb account for 40% of the entire core volume, of which reluctance significantly contributes to the inductance calculation. Combined with the above geometric characteristics, the square inductor is a good model for studying the features of the corner. Appl. Sci. 2019, 9, x 5341 FOR PEER REVIEW 3 of 1213 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 13

(a) (b) (a) (b) Figure 1. A square core-type inductor: (a) structure parameters; (b) electrical schematic. FigureFigure 1. 1. AA square square core core-type-type inductor: inductor: (a (a) )s structuretructure parameters; parameters; ( (bb)) e electricallectrical schematic. schematic. Table 1. Parameters of square core-type inductor geometry. Table 1. Parameters of square core-type inductor geometry. Table 1. Parameters of square core-type inductor geometry. Symbol Description Value Symbol Description Value Symbolws width Descriptionof the core window Value67 mm w width of the core window 67 mm wwss i widthwidth of the of core the limbwindow 6723 mmmm wi width of the limb 23 mm wiw widthheight of of the winding limb 2356 mmmm ww height of winding 56 mm we wwe distancedistance of of theheight the end end of of winding thethe windingwinding to to limb limb 565.5 5.5 mm mm mm dwdwew distance of thewidthwidth end ofof winding windingthe winding to limb 5.54 4 mmmm dlclwc widthdepthdepth of of winding thethe corecore 234 23 mm mm mm N number of turns of each limb 10 lNc numberdepth of turns of the of core each limb 23 10mm N number of turns of each limb 10 2.2. Finite Element Analysis of Flux Density Distribution 2.2. Finite Element Analysis of Flux Density Distribution The electromagnetic finite finite element analysis of of the inductor is carried out to analyze the flux flux densityThe distribution electromagnetic when finite the the current element ii isisanalysis from 20 of A the to 1000inductor A. At is 20carried A, the out average to analyze fluxflux density the flux of densitythe core distribution is 0.085 T. Atwhen 1000 the A, current the average i is from fluxflux 20 density A to 1000 of the A. core At 20 is A, 1.58 the T, average which hasflux reached density the of thesaturation core is 0.085 range. T. The At flux flux1000 density A, the distributionaverage flux is density shown ofin theFig Figure urecore2 2.. is 1.58 T, which has reached the saturation range. The flux density distribution is shown in Figure 2.

(a) (b) (c) (d) (a) (b) (c) (d) Figure 2. Flux density distribution: ( (aa)) 50 50 A; (b)) 100 100 A; ( c) 500 500 A; ( d) 1000 A. Figure 2. Flux density distribution: (a) 50 A; (b) 100 A; (c) 500 A; (d) 1000 A. The fluxflux densitydensity distribution distribution is is symmetric symmetric due due to theto the structural structural symmetry symmetry and and uniform uniform in the in limb the limbenclosedThe enclosed flux by the density by winding. the winding.distribution In the In corner the is symmetriccorner of the of core, the due core, the to flux thethe structural densityflux density is non-uniformlysymmetry is non-uniformly and distributeduniform distributed in andthe limbandthe non-uniformity theenclosed non-uniformity by the increases winding. increases with In the with current corner current significantly. of the significantly. core, the In the flux In joint thedensity joint part, ispart, between non between-uniformly the corner the distributed corner and limband andlimbsurrounded the surrounded non-uniformity by winding, by winding, increases the flux thewith density flux current densitydistribution significantly. distribution of the In area the of joint near the part, areathe geometrically between near the the geometrically c symmetricorner and limbplanesymmetric surrounded is more plane uniform, is by more winding, and uniform, the flux the and density flux the density flux distribution density distribution ofdistribution the region of the of near areathe the region near corner thenear is geometrically morethe corner uneven. is symmetricmoreIn addition, uneven. plane as In the isaddition, excitationmore uniform, as is the enhanced, excitation and the the flux is leakage enhanc density fluxed, distribution the of theleakage core of windowflux the ofregion the inner core near corner window the increases.corner inner is morecorner Takinguneven. increases. into In addition, account the as characteristicsthe excitation is of enhanc flux densityed, the distribution, leakage flux in of this the paper,core window we propose inner a cornerlocalTaking model increases. into of these account three the parts characteristics of the core, of flux as shown density in distribution, Figure3, which in this are paper, corner, we propose the limb a jointlocal modelTaking of these into three account parts the of thecharacteristics core, as shown of flux in Figure density 3, distribution, which are corner, in this the paper, limb wejoint propose parts between a local model of these three parts of the core, as shown in Figure 3, which are corner, the limb joint parts between Appl. Sci. 2019, 9, 5341 4 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 13 Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 13 windingparts between end and winding corner named end and Part corner B and named Part C, Part and Bthe and core Part window C, and inner the core corner window area named inner Part corner D. Sinceareawinding namedthe model end Part andstructure D. corner Since of named thethis modelpaper Part is B structure symmetrical,and Part ofC, thisand Part paperthe B andcore is thewindow symmetrical, Part C inner are identical. corner Part B areaand named the Part Part C D. areSince identical. the model structure of this paper is symmetrical, Part B and the Part C are identical.

Figure 3. The divided parts of a corner. Figure 3. The divided parts of a corner. Figure 3. The divided parts of a corner. 2.3. Modeling Modeling of of Corner Corner 2.3. Modeling of Corner The flux flux density distribution of the diagonal line of a corner, corner, La as shown in Fig Figureure 33,, isis analyzedanalyzed and presented The flux in density Fig Figureure 4distribution4.. We cancan seesee of that the thediagonal fluxflux density line of aBB corner, inin the the corner cornerLa as shown has has nonlinear nonlinear in Figure distributio distribution. 3, is analyzedn. Specifically,Specifically,and presented the fluxfluxin Fig isure concentrated 4. We can see around that the the flux inner density corner B inbecause the corner the flux fluxhas nonlinearselects a minimum distributio n. reluctanceSpecifically, path, path, the where where flux the theis concentrated reluctance is is aroundequal to to the the innerratio of corner length because and the the product flux selects of permeability a minimum andreluctance cross cross-section-section path, area where [19]. [19 the]. With With reluctance the the current current is equal rising, rising, to the the the ratio flux flux of density length increases and the product and the of distribution permeability changechangesand scross because -section of the area nonlinear [19]. With permeability the current of rising, the core the material. flux density increases and the distribution changes because of the nonlinear permeability of the core material.

Figure 4. TheThe flux flux density distribution of line La.La. Figure 4. The flux density distribution of line La. On the other hand, most of of the flux flux line shape is approximately a a circle and the flux flux density is is concentratedOn the inin other thethe inner innerhand, corner. corner. most Aof A quarter-circlethe quarter flux line-circle shape area area is is namedis approximately named Part Part A, inA, a which incircle which theand radiusthe the radius flux is thedensity is limb the is limbwidth.concentrated width. The energyThe inenergy the in Part inner in APart andcorner. A energyand A energy quarter in the in- cornercircle the corner area are compared, isare named compared, asPart shown A,as shownin inwhich Figure in theFig5. ureradius 5. is the limbIt can width. be seen The thatenergy the in ratio Part of A the and energy energy in Partin the A corner and the are corner compared, is higher as thanshown 93% in inFig theure range 5. of 20 A to 1000 A, and it slightly decreases as the current increases. Therefore, we chose Part A to replace the corner and use the circle path to study the corner with a negligible error. Considering the non-uniform flux density B at the corner, we selected n parallel reluctance branches to replace the traditional single corner reluctance. Here we divide Part A into n equal divisions along line Lab and make n quarter circles by the inner corner and divided points, which divides Part A into n-1 quarter rings and 1 quarter circle, as shown in Figure6a. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 13

Appl. Sci. 2019, 9, 5341 5 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 13

Figure 5. The energy comparison of Part A and the corner.

It can be seen that the ratio of the energy in Part A and the corner is higher than 93% in the range of 20 A to 1000 A, and it slightly decreases as the current increases. Therefore, we chose Part A to replace the corner and use the circle path to study the corner with a negligible error. Considering the non-uniform flux density B at the corner, we selected n parallel reluctance branches to replace the traditional single corner reluctance. Here we divide Part A into n equal divisions along line Lab and make n quarter circles by the inner corner and divided points, which divides Part A into n-1 quarter rings and 1 quarter circle, as shown in Figure 6a. Figure 5. TheThe energy energy comparison comparison of Part A and the corner.

(a) (b)

FigureFigure 6. 6.TheThe magnetic magnetic equivalent equivalent circuit circuit (MEC (MEC)) branches branches of of the the corner corner and and joint joint part: part: ( (aa)) the the branches branchesof the corner; of the (corner;b) the branches(b) the branches of the corner of the and corner joint and part. joint part.

TheThe area area of of each each branch branch AAc,i c,iis isexpressed expressed in in Eq Equationuation (1). (1).

w푤i푖 ∙l푙c푐 퐴A푐,푖 = · ,, (1) (1) c,i n푛 TakingTaking the the middle middle line line of of geometry geometry as as the the length length of of the the magnetic magnetic path, path, the the length length of of the the ithith (a) (b) magneticmagnetic path path lc,il c,iis isexpressed expressed in inEq Equationuation (2). (2). Figure 6. The magnetic equivalent circuit (MEC푤) branches∙ 휋 of the corner and joint part: (a) the w푖 π branches of the corner; (b) the branches푙 푐lof,푖 the= corneri · (( 2andi푖 − joint11)),, part. (2) (2) c,i 4n푛 − c,i ThusTheThus thearea the ith ofith reluctance each reluctance branch RR A c,iisc,i is calculatedis calculated expressed in ininEq EquationEquationuation (3). (3).(1). 휋 ∙ (2푤푖 −∙ 푙1) 푅 = π (2i 푖 1푐) , 푐,푖 퐴푐,푖 = , (3)(1) Rc,i = 4푙·푐 ∙ 휇−푖푛(퐵), (3) 4lc µ (B) · i where Takingui(B) is the imiddleth branch line permeability. of geometry as the length of the magnetic path, the length of the ith wheremagneticIn theui( Bpathcorner,) is thelc,i istheith expressed branchflux density permeability. in Eq distributionuation (2). can be accurately reflected with more parallel Rc,i In the corner, the flux density distribution can be accurately reflected with more parallel Rc,i branches. However, too many parallel flux paths푤 can푖 ∙ 휋 cause the magnetic equivalent circuit of the 푙 = (2푖 − 1), (2) cornerbranches. to be too However, complicated too many and affect parallel the fluxconvergence푐,푖 paths4푛 can of causethe calculation, the magnetic especially equivalent when circuit the branch of the iscorner deeply to saturated. be too complicated In this paper, and we aff ectcho these three convergence parallel ofbranches the calculation,, the inner especially section, middle when the section branch, is deeplyThus saturated.the ith reluctance In this paper, Rc,i is calculated we chose three in Eq paralleluation (3). branches, the inner section, middle section, and the outer section of the corner, which can balance휋 ∙ ( the2푖 − accuracy1) of the solution and keep the model 푅푐,푖 = , (3) simple, as shown in Figure6b. 4푙푐 ∙ 휇푖(퐵)

w2.4.here Modeling ui(B) is ofthe the ith Limb branch Joint permeability. Part In the corner, the flux density distribution can be accurately reflected with more parallel Rc,i branches.The distribution However, too of the many ﬂux parallel density Bfluxon thepaths line canLyb causeat the the position magnetic of winding equivalent end andcircuitLab ofat the cornerjunction to ofbe atoo corner, complicated shown inand Figure affect3 ,the were convergence analyzed andof the shown calculation, in Figure especially7. It can when be clearly the branch seen is deeply saturated. In this paper, we chose three parallel branches, the inner section, middle section, Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 13 and the outer section of the corner, which can balance the accuracy of the solution and keep the model simple, as shown in Figure 6b.

2.4. Modeling of the Limb Joint Part The distribution of the flux density B on the line Lyb at the position of winding end and Lab at theAppl. junction Sci. 2019, 9of, 5341 a corner, shown in Figure 3, were analyzed and shown in Figure 7. It can be clearly6 of 12 seen that the flux density distribution limb joint part is similar to the corner, which is high around thethat inner the flux corner density and distributionlow near the limb outer joint side. part On is the similar other to hand, the corner, the distribution which is high of the around limb joint the inner part closedcorner andto the low winding near the is outer uniform, side. which On the is other similar hand, to the the distribution of the the limb limb joint surrounded part closed by windings.to the winding The flux is uniform, density distribution which is similar of line to Lyb the distributionis more uniform of the than limb the surrounded line Lab, and by the windings. ratio of BThe maximum flux density and distribution B minimum of is line lessLyb thanis more 1.52 uniform at different than currents. the line Lab However,, and the in ratio line of LabB maximum, the flux densityand B minimum is high at is the less area than close 1.52 to at ditheff erentinner currents. corner of However, the core, inand line theLab ratio, the of flux B maximum density is highand atB minimumthe area close reaches to the 8.67. inner corner of the core, and the ratio of B maximum and B minimum reaches 8.67.

(a) (b)

Figure 7. TheThe flux ﬂux density density distribution distribution comparison comparison of ofline line LybLyb andand La:La (a:() thea) the B distributionB distribution of line; of line; (b) the(b) themaximum, maximum, minimum minimum,, and and ratio ratio of B of. B.

To calculate the B distribution, the joint part waswas divided into two equal sections, as shown in FigFigureure 66bb;; one section close close to to the the limb limb enclosed enclosed by by winding winding is isexpressed expressed by by a reluctance a reluctance Rpy Randpy and the otherthe other section section is continuously is continuously divided divided into into n nreluctancesreluctances RRpc,ipc,i seriesseries with with the the neighboring neighboring corner branches Rc,i toto form form a a sub sub-branch.-branch. The The sub sub-branches-branches connect connecteded in in parallel parallel and then series with Rpy.. The reluctance can be calculated with EquationsEquations (4) andand (5).(5).

0.50.5w∙e푤푒 R푅py푝푦= = · ,, (4) µ(휇B()퐵)lc∙ 푙푐w∙ 푤푖 · · i

0.50.5w∙e푤푒n∙ 푛 R푅pc푝푐,i,푖= = · · ,, (5) µ(휇B()퐵)lc∙ 푙w∙ 푤 · · 푐 i 푖 2.5. Modeling the Leakage of Core Window Inner Corner 2.5. Modeling the Leakage of Core Window Inner Corner The core window inner corner area, Part D as shown in Figure3, is square and its side length is The core window inner corner area, Part D as shown in Figure 3, is square and its side length is the distance between the winding end and the core. The energy in Part D accounts for about 20% of the distance between the winding end and the core. The energy in Part D accounts for about 20% of the total leakage energy. With the increased current, the leakage of the core window inner corner rises, the total leakage energy. With the increased current, the leakage of the core window inner corner especially when the core is saturated. Therefore, in order to calculate the inductance accurately in the rises, especially when the core is saturated. Therefore, in order to calculate the inductance accurately saturation situation, the leakage flux of this part cannot be ignored. in the saturation situation, the leakage flux of this part cannot be ignored. In this work, we chose the Sudhoff method [20] to establish the equivalent magnetic reluctance of In this work, we chose the Sudhoff method [20] to establish the equivalent magnetic reluctance the core window inner corner which can be calculated in Equation (6). of the core window inner corner which can be calculated in Equation (6). π 휋 R푅wi = = ,, (6) 푤푖 + πw휋e 푤 (6) µ휇0 l∙c푙 ln∙ ln1(1 +4e 푒) 0· ·푐 4푒 wwherehere e is a small value to reflectreflect thethe processprocess airair gapgap oror thethe demoldingdemolding processprocess gapgap atat thethe corners.corners.

2.6. Local Corner MEC Model Combining the branch reluctance of the limb joint part, corner, and core window inner corner leakage, a local corner MEC model is proposed, as shown in Figure8. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 13 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 13 2.6. Local Corner MEC Model 2.6. Local Corner MEC Model Combining the branch reluctance of the limb joint part, corner, and core window inner corner Appl. Sci.Combining2019, 9, 5341 the branch reluctance of the limb joint part, corner, and core window inner corner7 of 12 leakage, a local corner MEC model is proposed, as shown in Figure 8. leakage, a local corner MEC model is proposed, as shown in Figure 8.

Figure 8. The local MEC of a corner. Figure 8. The local MEC of a corner.

Rpc,i and Rc,i are connected in series as a sub-branch and then in parallel with Rwi. The Rpy is Rpc,i andand RRc,ic,i areare connected connected in in series series as a sub-branchsub-branch and then then in in parallel parallel with with RRwiwi.. The The RRpypy is connected in series with the parallel branch. There are three meshes in the MEC network, which make connected in series with the parallel branch. There There are are three three meshes in the MEC MEC network, network, which make the flux density of the corner different in four branches. During the iterative calculation, the flux will the flux flux density of the corner different different in four branches. During the ite iterativerative calculation, the flux flux will automatically calculate the reluctance based on the branch length and permeability. The local corner automatically calculate the reluctance based on the branch length and permeability. The local corner MEC model fully describes the flux paths at the corner and the flux density distribution. It makes the MEC model fully describes the flux flux paths at the corner and the flux flux density distribution. It makes the flux density distribution consistent with the real distribution to improve the calculation accuracy. fluxflux density distributiondistribution consistent with the real distribution to improve thethe calculationcalculation accuracy.accuracy. 3. MEC of Square Inductor 3. MEC MEC of of Square Square Inductor Inductor 3.1. MEC of Square Inductor 3.1. MEC of Square Inductor By using the local corner MEC model, the MEC model of the square core-typecore-type inductor is By using the local corner MEC model, the MEC model of the square core-type inductor is established, as shown in FigFigureure 99,, where where the the core core limb limb branch branch and and winding winding leakage leakage flux flux branch branch established, as shown in Figure 9, where the core limb branch and winding leakage flux branch reluctance areareR Re eand andR Rlimblimb,, respectively.respectively. The The main main flux flux path path flows flows through through the the limb limb and and is divided is divided into reluctance are Re and Rlimb, respectively. The main flux path flows through the limb and is divided fourinto sub-fluxesfour sub-flux intoes the into corner the corner sub-branches. sub-branches. The winding The winding leakage leakage flux of eachflux windingof each winding flows through flows into four sub-fluxes into the corner sub-branches. The winding leakage flux of each winding flows adjacentthrough adjacent core limb core and limb winding. and winding. These flux These paths flux assure paths that assure the fluxthat densitythe fluxB densityof MEC B is of correctly MEC is through adjacent core limb and winding. These flux paths assure that the flux density B of MEC is distributedcorrectly distributed with a simple with modela simple and model high precision.and high precision. correctly distributed with a simple model and high precision.

Figure 9. The MEC of a square core core-type-type inductor. Figure 9. The MEC of a square core-type inductor.

The limb reluctance Rlimblimb isis calculated calculated by by the the core core material material and and geometry [18],[18], as shown in The limb reluctance Rlimb is calculated by the core material and geometry [18], as shown in EquationEquation (7). Equation (7). llimb Rlimb(Φlimb) = , (7) Φlimb Alimb µB · Alimb Appl. Sci. 2019, 9, 5341 8 of 12

where Alimb and limb are the cross-sectional area and the length of the core limb branch, respectively. µ ( ) is the branch apparent permeability, which is represented by the function of the branch ﬂux due B · to the nonlinear characteristics of the magnetic core material. The leakage reluctance Re is calculated in Equation (8) [19].

p 2 2 2 we(ww + we) + 2dw 256wwdw Re = + , (8) µ0welc 4 3 2 2 3 4 2 √2k2 µ0lc 16k + 16 √2k1k + 4k k 2 √2k k2 + k ln 1 + 2 2 1 2 − 1 1 k1 where

k1 = ww 2dw , | 1 − | (9) k2 = min(2dw, ww), √2

3.2. Inductor Calculation Method Since the core reluctance is a function of the branch flux, the mesh method is used to iteratively calculate the MEC branch flux until convergence, which is described in [20,21]. The inductance can be obtained by the magnetic flux linkage method and the incremental inductance [22] can be calculated by using the differential method in Equation (10). P P P P λn,i λn,i 1 Φn,i Nn,i Φn,i 1 Nn,i 1 L = − − = · − − · − , (10) inc ∆i ∆i where λn,i is the flux linkage of branch n at current i, and λn,i 1 is the flux linkage of branch n at current − i 1. − 4. Results and Discussion

The average ﬂux density of the corner branch Rc1, Rc2, and Rc3 regions is calculated by the FEM, andAppl. comparedSci. 2019, 9, x withFOR PEER the ﬂux REVIEW density of the local corner MEC model, as shown in Figure 10. 9 of 13

Figure 10. The average fluxflux density ofof thethe cornercorner branchbranchR Rc1c1,, Rc2c2,, and and Rc3 regionsregions obtained obtained by by the the finite finite element method (FEM)(FEM) compared withwith thethe fluxflux densitydensity ofof MEC.MEC. 4.1. Flux Density Distribution of the Corner The three branches flux density of the corner by the MEC method is obviously different and increasesBy using from theRc3 to local Rc1. The corner corner MEC branch model, Rc2 theand MECthe Rc3 model branch of flux the density square are core-type consistent inductor with the is established,calculated average as shown value in of Figure the FEM.9, where Their the average core limbrelative branch errors and of windingthe currents leakage from flux20 A branchto 1000 reluctanceA are 0.93% are andRe and 4.05%,Rlimb respectively., respectively. The The relative main flux error path of flows the R throughc1 branch the is limb 8.85% and because is divided the intosaturation four sub-fluxes of the inner into corner the corner affects sub-branches.the accuracy of The the windingcalculation. leakage flux of each winding flows In summary, the local corner MEC model can effectively illustrate the inner and outer flux density distribution of the corner with high calculation accuracy.

4.2. Inductance Comparison of MEC and Prototype

The physical prototype inductor was made and the incremental inductance with the current (Linc- i) curve was measured using the DPG10 Power Choke Tester, which has an accuracy of 1.9%, as shown in Figure 11. The Linc-i curve corresponding to the MEC with the local corner model and IEC corner is solved by Equation (11) and the Linc-i curves are shown in Figure 12a.

(a) (b)

Figure 11. The physical prototype and tester: (a) physical prototype; (b) DPG10 Power Choke Tester.

The initial inductance of the IEC corner model was greater than the measured value and the descent rate was high. After 150 A, the inductance was already less than the measured value. In the local corner model, the initial inductance was slightly less than the measured value and fit with the prototype measured data very well with the current 20 A to 1000 A. Both of the MEC Linc-i curves were compared with the measured data to calculate the relative error of each point, as shown in Figure 12b. The maximum relative error emax and average error eavg were calculated by Equations (11) and (12), and presented in Table 2.

1000 100 |퐿푡푒푠푡,푖 − 퐿푀퐸퐶,푖| 푒푎푣푔 = × ∑ , (11) (1000 − 20) 푖=20 퐿푡푒푠푡,푖 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 13

Appl. Sci. 2019, 9, 5341 9 of 12

Figure 10. The average flux density of the corner branch Rc1, Rc2, and Rc3 regions obtained by the finite throughelement the method adjacent (FEM core) compared limb and with winding. the flux These density flux of MEC. paths assure that the flux density B of the MEC is correctly distributed with a simple model and high precision. The three branches flux flux density of the corner by the MEC method is obviously different different and increases fromfrom RRc3c3 to Rc1.. The The corner corner branch branch RRc2c2 andand the the RRc3c3 branchbranch flux flux density density are are consistent consistent with the calculated average valuevalue ofof thethe FEM.FEM. TheirTheir average average relative relative errors errors of of the the currents currents from from 20 20 A A to to 1000 1000 A areA are 0.93% 0.93% and and 4.05%, 4.05%, respectively. respectively. The relative The relative error of error the R ofc1 branch the Rc1 isbranch 8.85% because is 8.85% the because saturation the ofsaturation the inner of corner the inner affects corner the accuracyaffects the of accuracy the calculation. of the calculation. In summary, summary, the the local local corner corner MEC MEC model model can caneffectively effectively illustrate illustrate the inner the and inner outer and flux outer density flux distributiondensity distribution of the corner of the with corner high with calculation high calculation accuracy. accuracy.

4.2. Inductance Comparison of MEC and Prototype

The physical prototype inductorinductor waswas mademade and and the the incremental incremental inductance inductance with with the the current current (L (inc-iLinc)- curvei) curve was wa measureds measured using using the the DPG10 DPG10 Power Power Choke Choke Tester, Tester, which which has an has accuracy an accuracy of 1.9%, of as 1.9%, shown as inshown Figure in 11Fig. Theure 11.Linc-i Thecurve Linc-i corresponding curve corresponding to the MEC to the with MEC the with local the corner local model corner and model IEC cornerand IEC is solvedcorner is by solved Equation by Eq (11)uation and the(11)L inc-iand curvesthe Linc are-i curves shown are in shown Figure in 12 Figa. ure 12a.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 13

|퐿푡푒푠푡,푖 − 퐿푀퐸퐶,푖| 푒푚푎푥 = 100 × max , (12) 푖=20~1000 퐿푡푒푠푡,푖

where Ltest,i is the measured inductance and LMEC,i is the calculated inductance of the MEC method at (a) (b) each current i from 20 A to 1000 A. Figure 11. TheThe physical physical prototype prototype and tester: ( (aa)) physical physical p prototype;rototype; ( (bb)) DPG10 DPG10 P Powerower C Chokehoke T Tester.ester.

1000 100 |퐿푡푒푠푡,푖 − 퐿푀퐸퐶,푖| 푒푎푣푔 = × ∑ , (11) (1000 − 20) 푖=20 퐿 (a) 푡푒푠푡,푖 (b)

FigureFigure 12. 12. TheThe incremental incremental inductance inductance and and relative relative error error with with the the current: current: ( (aa)) the the incremental incremental inductance;inductance; ( b(b)) relative relative error error of of MEC MEC with with IEC IEC corner corner and and local local corner corner model. model.

The initial inductance ofTable the IEC 2. Relative corner error model statistics was greater of inductance. than the measured value and the descent rate was high. After 150 A, the inductance was already less than the measured value. In the local corner model, the initial inductanceModel was slightly less thanemax the measuredeavg value and ﬁt with the Local corner model 6.0% 3.6% prototype measured data very well with the current 20 A to 1000 A. Both of the MEC Linc-i curves were compared with the measuredIEC data corner to calculate model the relative error14.7% of each point,8.6% as shown in Figure 12b.

Compared to the IEC corner model, the emax of the local corner model decreased by 8.7%. The eavg of the local corner decreased by 5.0%. This shows the local corner model has higher accuracy than the IEC corner model. Compared to the measured data, the maximum relative error of the local corner MEC was < 6%, and the average relative error was < 4%. The local corner MEC model provided a Linc- i curve calculation with high accuracy and a high fit degree.

4.3. Discuss the Number of Corner Branches Rc,i

We divided the corner Rc,i into two, three, four, and five branches and calculated the inductance from 1–1000A which compared with the inductance of the measured square inductor. The average relative error was calculated by Equation (11) and the program running time in MATLAB under the computer configuration of Intel Xeon E5-2670 CPUs @2.6 GHz 48 GB RAM is shown in Table 3.

Table 3. Inductance Comparison of different corner branch numbers.

Model eavg Time Two branches 3.4% 26’30” Three branches 3.3% 30’44” Four branches 3.3% 46’48” Five branches 3.2% 66’47”

It can be seen that as the number of branches increases, the average relative error decreases slightly. This is because when the corner has two branches, the distribution of the corner branch flux can be automatically changed by the reluctance of the inner branch and the outer branch due to the change in the flux density. Multiple corner branching can better describe the magnetic density distribution in different regions of the corner, but the accuracy improvement of the flux calculation is not obvious. However, the calculation time of iteration increases significantly when a branch is added. The calculation time is 26’30” at two branches, and the time doubles when it reaches five Appl. Sci. 2019, 9, 5341 10 of 12

The maximum relative error emax and average error eavg were calculated by Equations (11) and (12), and presented in Table2.

100 X1000 Ltest,i LMEC,i eavg = − , (11) (1000 20) × i=20 Ltest,i −

Ltest,i LMEC,i emax = 100 max − , (12) × i=20 1000 Ltest i ∼ , where Ltest,i is the measured inductance and LMEC,i is the calculated inductance of the MEC method at each current i from 20 A to 1000 A.

Table 2. Relative error statistics of inductance.

Model emax eavg Local corner model 6.0% 3.6% IEC corner model 14.7% 8.6%

Compared to the IEC corner model, the emax of the local corner model decreased by 8.7%. The eavg of the local corner decreased by 5.0%. This shows the local corner model has higher accuracy than the IEC corner model. Compared to the measured data, the maximum relative error of the local corner MEC was < 6%, and the average relative error was < 4%. The local corner MEC model provided a Linc-i curve calculation with high accuracy and a high ﬁt degree.

4.3. Discuss the Number of Corner Branches Rc,i

We divided the corner Rc,i into two, three, four, and ﬁve branches and calculated the inductance from 1–1000A which compared with the inductance of the measured square inductor. The average relative error was calculated by Equation (11) and the program running time in MATLAB under the computer conﬁguration of Intel Xeon E5-2670 CPUs @2.6 GHz 48 GB RAM is shown in Table3.

Table 3. Inductance Comparison of diﬀerent corner branch numbers.

Model eavg Time

Two branches 3.4% 26030” Three branches 3.3% 30044” Four branches 3.3% 46048” Five branches 3.2% 66047”

It can be seen that as the number of branches increases, the average relative error decreases slightly. This is because when the corner has two branches, the distribution of the corner branch flux can be automatically changed by the reluctance of the inner branch and the outer branch due to the change in the flux density. Multiple corner branching can better describe the magnetic density distribution in different regions of the corner, but the accuracy improvement of the flux calculation is not obvious. However, the calculation time of iteration increases significantly when a branch is added. The calculation time is 26030” at two branches, and the time doubles when it reaches five branches. Based on the above experimental data, we chose three corner branches, which can effectively describe the magnetic density distribution between the corners, and show a high calculation accuracy of magnetic flux and acceptable calculation time.

5. Conclusions In this work, in order to improve the inductance accuracy and describe the flux density distribution of the corner, a novel local corner MEC model is proposed. In this model, a corner is divided into Appl. Sci. 2019, 9, 5341 11 of 12 three branches to make the corner flux density different from the inner corner to the outer corner, and automatic calculation of the reluctance and flux of each branch by iterative calculation. The uneven flux distribution of the joint part between the corner and the limb surrounded by winding is discussed. Based on this, the joint part is innovatively divided into several branches in series with the corner branches. In addition, a branch reluctance of leakage in the inner corner of the core window is modeled and added in parallel with the corner branches, contributing to the accuracy of the inductance calculation especially when the core is under saturation. The local corner MEC model fully describes the flux path and flux density distribution of the corner, which has a high accuracy of corner branch flux density B calculation and provides a ground for judging the magnetic core saturation. In general practice, the core loss is calculated by the Steinmetz equation for sinusoidal current waveform or the improved Steinmetz equation and general Steinmetz equation for other current waveforms, which are a function of flux density amplitude. The power loss of the corner accounts for a considerable portion of the total core loss because the flux density is uniformly distributed, and the inner corner flux density is high. Through the local MEC model, the flux density distribution of the corner is described, and the flux density can be accurately calculated, which can improve the accuracy of the core corner loss calculation. Furthermore, the core hotspot calculation depends on the loss dissipation and can benefit from the accurate core loss. Furthermore, the accurate flux calculation improves the inductance accuracy. A wide range from 20 A to 1000 A inductance of the local MEC model is compared to the measured inductance of the prototype square inductor, and the average relative error is less than 4%. Considering the measurement error of the test equipment, the average relative error upper limit of the local MEC corner model and the real value is less than 6%, which shows a highly accurate inductance calculation of this model. By analyzing the number of corner branches, at least two parallel corner branches can be used to model the local corner MEC model and ensure calculation accuracy. In addition, compared with the MEC models with the fixed single reluctance branch of the corner, the local corner MEC model can describe the flux distribution of the corner and improve the inductance calculation accuracy. Compared with the finite element method, the local corner MEC model is simple and does not need to mesh or regenerate the model after the geometry is modified. The local corner MEC model has great advantages of rapid parametric modeling and high computational efficiency with an acceptable calculation accuracy and flux density description, which makes it ideal for performing inductance curve calculations, optimizing design, and as an online digital twin lumped parameter model. Although the local MEC model is developed by the symmetric powder core, it can also be applied in the symmetric steel laminated core, because both have the same geometry structure and only differ in permeability. Since most of the power inductors have the same width of the yoke and limb, the local MEC corner is suitable for common type inductors. In further research, a study of the local corner MEC model of the inductor with complicated structures will be established, which will expand the application range of the local corner model and further improve the accuracy.

Author Contributions: Conceptualization, X.W. and M.S.; methodology, X.W.; formal analysis, H.C.; writing—original draft preparation, X.W.; writing—review and editing, L.D. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂicts of interest.

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