New Scientific Contribution on the 2-D Subdomain Technique in Cartesian Coordinates: Taking into Account of Iron Parts

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1 Mathematical and Computational pplications rticle New Scientific Contribution on the 2-D Subdomain Technique in Cartesian Coordinates: Taking into ccount of Iron Parts Frédéric Dubas 1, * and Kamel Boughrara 2 1 Département ENERGIE, FEMTO-ST, CNRS, Univ. Bourgogne Franche-Comté, F90000 Belfort, France 2 Laboratoire de Recherche en Electrotechnique LRE-ENP, lgiers, lgeria; kamel.boughrara@g.enp.edu.dz * Correspondence: frederic.dubas@univ-fcomte.fr; Tel.: cademic Editor: Fazal M. Mahomed Received: 17 Januar 2017; ccepted: 1 Februar 2017; Published: 10 Februar 2017 bstract: The most significant assumptions in the subdomain technique i.e., based on the formal resolution of Mawell s equations applied in subdomain is defined b: The iron parts i.e., the teeth and the back-iron are considered to be infinitel permeable, i.e., µ iron +, so that the saturation effect is neglected. In this paper, the authors present a new scientific contribution on improving of this method in two-dimensional 2-D and in Cartesian coordinates b focusing on the consideration of iron. The subdomains connection is carried out in the two directions i.e., - and -edges. For eample, the improvement was performed b solving magnetostatic Mawell s equations for an air- or iron-cored coil supplied b a direct current. To evaluate the efficac of the proposed technique, the magnetic flu densit distributions have been compared with those obtained b the 2-D finite-element analsis FE. The semi-analtical results are in quite satisfing agreement with those obtained b the 2-D FE, considering both amplitude and waveform. Kewords: air- or iron-cored coil; Cartesian coordinates; Fourier analsis; two-dimensional; saturation effect; subdomain technique 1. Introduction 1.1. Contet of this Paper Generall, the modeling of the electromagnetic field distribution is a ke step in the design process for developing electromechanical sstems. lthough there are man papers in this scientific domain, the modeling approach is still a challenging and attractive research topic. Some comprehensive reviews on the models of electrical machines for magnetic field calculations along with their advantages and disadvantages can be found in 1 6 and the accompaning references. The modeling techniques can thus be classified in various categories: Graphical method of Lehmann ; Numerical methods i.e., the finite-element, finite-difference or boundar-element analsis 8 12; Electrical/Thermal/Magnetic equivalent circuit EEC/TEC/MEC 13 16; Schwarz-Christoffel SC mapping method 17 19; Mawell-Fourier methods 10,18 22: i Multi-laers models; ii Eigenvalues model; and iii Subdomain technique. The graphical method of Lehmann, which determines the magnetic field distribution in all parts of an electrical machine even when the machine is saturated, has been forgotten to the detriment of other methods, mainl numerical. In the past few decades, numerical modeling techniques Math. Comput. ppl. 2017, 22, 17; doi: /mca

2 Math. Comput. ppl. 2017, 22, 17 2 of 39 have been applied to electromechanical sstems analsis. These methods are precise and take into account the eact/simplified geometr, the nonlinear BH curve, the rotor motion, etc. The most accurate models are the three-dimensional numerical methods. Nevertheless, these approaches are time-consuming and not suitable for the optimization problems. In 23,24, it is possible to optimize electromagnetic sstems from numerical methods. Nowadas, in order to reduce the computation time, hbrid numerical methods can be developed The actual design works are mainl based on semi-analtical models i.e., EEC/TEC/MEC, SC mapping and Mawell-Fourier methods. This tpe of model consists of N interrelated analtical equations which must be solved numericall a.k.a. semi-numerical models. For eample, in Mawell-Fourier methods, the unknown coefficients of the series are computed b solving a nonlinear matri sstem. Indeed, under certain assumptions, these models have the advantage to be eplicit/accurate/fast. Moreover, the allow us to take into account rigorousl the slotting effect in the electrical machines as well as various electromagnetic domains without the current penetration effect in the conductive materials. Ecept in the numerical methods and nonlinear MEC, the saturation effect remains one of the scientific challenges in the modeling. Tiegna et al. 5 p.168 wrote: No eamples of analtical models based on the formal solution of Mawell s equations which take into account local magnetic saturation are available to date. Thus, in this paper, the main scientific focus will be on the consideration of iron in Mawell-Fourier methods with the local/global saturation State-of-the-rt: Saturation in Mawell-Fourier Methods Ver few works have included the iron in Mawell-Fourier methods with the local/global saturation due to variation of the material properties e.g., in case of stator and/or rotor slotting, buried magnets, etc.. The most significant assumptions is defined b: The iron parts i.e., the teeth and the back-iron are considered to the infinitel permeable, i.e., µ iron +, so that the saturation effect is neglected. It results in an overestimation of the magnetic flu and, consequentl, the electromagnetic performances e.g., the back electromotive force EMF, the electromagnetic torque, the efficienc. Thus, consideration of iron in the modeling is a mandator task in order to have a reliable estimation of the electromechanical sstems behavior. Eisting models in the electrical machines, based on Mawell s electromagnetic field equations, taking into account the iron parts without the nonlinear BH curve are: Multi-laers models onl the global saturation: Carter s coefficient: The slotted machine is transformed into a slotless equivalent structure b appling the usual Carter s coefficient 28. Generall, the armature slotting is taken into account through the SC mapping method. The analtical magnetic field distribution is determined in five or si homogeneous laers i.e., eterior, slotless stator, winding/air-gap, magnets, and rotor In 29, the magnetic permeabilities in stator/rotor iron cores have a constant value corresponding to linear zone of the BH curve. n iterative technique to include the nonlinear properties of core material has been developed in 30 for a no-load operation and 31 for a load operation whose the source term in the slot caused b the armature currents is represented b a winding current region over the stator slot-isthmus. In this tpe of modeling, the local distribution of flu densities in the teeth and slots is neglected. However, b calculating the flu entering the stator surface from the air-gap magnetic field and thus assuming uniform distribution of flu, the flu densit in middle of the stator teeth can be obtained. Saturation coefficient: It represents the ratio between the total magnetomotive force MMF required for the entire magnetic circuit and the air-gap MMF 32. The main magnetic saturation is included in the saturation factor, in an iterative wa, b using the nonlinear BH curve. The saturation effect is accounted for b modifing the air-gap length or b changing the phsical properties of magnets in this case, the saturated load operation is

3 Math. Comput. ppl. 2017, 22, 17 3 of 39 calculated b considering an equivalent no-load operation with a fictitious magnet having a remanent flu densit that creates the same MMF as the one created b both real magnet and stator MMF 35. The analtical magnetic field distribution is mainl determined in one or two regions viz., air-gap or air-gap/magnets of slotless machines b appling the Carter s coefficient 32. The slotting effect can be neglected 32,35 or taken into account through the SC mapping method 33,34. The magnetic flues in the stator/rotor iron cores are obtained from the air-gap magnetic field 32,33,35 or/and with a simple MEC 34. This technique has been applied to surface-mounted/-inset magnets machines 32 35, surface-inset magnet machines 33, and others electrical machines. Concept wave impedance: The are based on a direct solution of Mawell s field equations in homogeneous multi-laers of magnetic material properties, viz., the magnetic permeabilit and the electrical conductivit. This approach, developed b Mishkin , was first applied to squirrel-cage induction machine in Cartesian coordinates with three-laers i.e., stator slotting, air-gap, and rotor slotting. It was used and enhanced b man authors, viz., simplification of the electromagnetic theor 37; etended with an infinite number of laers 38; converted into equivalent circuits and terminal impedance 39; included the curvature effect with the magnetizing current 40; incorporated spatial harmonics in the multi-laers theor b considering isotropic and anisotropic e.g., laminated, composite, and toothed regions 41,42; introduced the nonlinear BH curve in homogenous laers b an iterative procedure 43,44; taking account of the slot-opening effect 45, i.e., the multi-laers model is combined with the subdomain technique for slotted structures b assuming infinitel permeable tooth-tips; included the current penetration effect in conductive laers 43,46. The analtical solution for the electromagnetic field in conductive laers is then defined b Bessel functions. Convolution theorem: The electrical machine is divided into an infinite number of inhomogeneous laers. The permeabilit in the stator and/or rotor slotting is represented b a comple Fourier series along the direction of permeabilit variation The permeabilit variation in the direction of the periodicit is directl included into the solution of the magnetic field equation. The resulting formulation, based on a direct solution of Mawell s field equations using the Cauch s product theorem i.e., the discrete convolution of two infinite series, is completel defined in terms of comple Fourier series 47. Recentl, Djelloul et al etended this modeling taking into account the nonlinear BH curve in each soft-magnetic section b an iterative procedure. For the moment, this technique has been applied to a switched reluctance machine 48 and a snchronous reluctance machine 49. Eigenvalues model onl the global saturation: The electromagnetic field can be solved directl b appling the method of Truncation Region Eigenfunction Epansions TREE 50. The iron cores have finite magnetic permeabilit and finite height/width. The studied domains can be nonconductive. The boundar value problem is formulated in terms of the magnetic vector potential, which is epanded in a series of appropriate eigenfunctions. The unknown coefficients of the series are computed b solving a matri sstem b using a standard method such as the lower upper decomposition, which is formed b appling the usual interface conditions. The corresponding eigenvalues are the real roots of a function with the geometrical and the magnetic permeabilit of the core as parameters. Nevertheless, an iterative numerical method e.g., the bisection 50 and Newton-Raphson 51 method is alwas adopted to compute the discrete eigenvalues in both the odd and even parit solutions. For the moment, this technique has been

4 Math. Comput. ppl. 2017, 22, 17 4 of 39 widel applied to the non-destructive testing of conductive materials e.g., for the I-cored 50 and E-cored 52,53 probes, for a long coil with a slot in a conductive plate 51, etc.. Hbrid models the local/global saturation: The analtical solution can be combined with numerical methods or nonlinear MEC Usuall, the analtical solution is established in uniform regions of ver low permeabilit e.g., air-gap, and magnets and other methods are sought in regions where magnetic saturation cannot be neglected i.e., the stator and/or rotor iron cores Objectives of this Paper To the best of the author s knowledge, in the literature, there is no semi-analtical model based on the subdomain technique that taking into account of iron parts without the nonlinear BH curve. Thus, the work in this paper takes part in the development and improvement of the subdomain technique on this scientific topic. The disadvantage of multi-laers models, ecept those with the convolution theorem, is that the do not give a ver accurate description of the local magnetic field distribution in the iron parts with a global saturation. In the harmonic modeling technique using the convolution theorem, convergence problems due to the truncated Fourier series around the soft-magnetic material discontinuities ma eist Ecept in multi-laers models using the conception wave impedance and in the TREE method, the electrical conductivit is assumed to be zero. It is interesting to note that the TREE method is not similar to the novel method proposed in this paper. The difference is that TREE method imposes a term-b-term field continuit on one direction and a weak continuit on the other direction, while the 2-D subdomain technique imposes a weak continuit on both directions. Furthermore, the latter method does not need to find an special eigenvalues b using iterative numerical schemes. Contrar to the TREE method, the new approach proposed in this paper allows to decompose the analtical solution in Fourier series into two solutions according to the two directions and to respect the boundar conditions b appling the principle of superposition on the magnetic quantities. Moreover, it also allows evaluation of the local distribution of flu densities in the iron parts with a global saturation, does not have numerical convergence problems, and would easil introduce the current penetration effect in the conductive materials. Section 2 presents this new scientific contribution based on the subdomain technique. For eample, it was performed b solving 2-D magnetostatic Mawell s equations in Cartesian coordinates, for an air- or iron-cored coil supplied b a direct current. The subdomains connection is carried out in the two directions i.e., - and -edges. The iron magnetic permeabilit is constant corresponding to linear zone of the initial magnetization curve. Nevertheless, as in 48, it should be mentioned that the material properties could be updated iterativel to take the nonlinear BH curve of the material into account. However, this is beond the scope of the paper. In Section 3, in order to evaluate the efficac of the proposed technique, the magnetic flu densit distributions have been compared with those obtained b the 2-D finite-element analsis FE 8. The comparisons are ver satisfing in amplitudes and waveforms. This major scientific contribution could be applied to rotating and/or linear electrical machines without magnets supplied b a direct or alternate current with an waveforms whose the analsis would be based on a 2-D semi-analtical model in Cartesian coordinates e.g., plane linear machines, aial-flu machines, etc D Subdomain Technique of Magnetic Field 2.1. Problem Description and ssumptions The application eample, namel an air- or iron-cored coil, with the geometrical and phsical parameters is illustrated in Figure 1. The sstem consists of a coil with N t turns of the copper wire which is suppl b a direct current I. The direction of current in the conductor is defined b for the forward conductor and for return conductor. The material in the middle of the coil can be air or iron. The sstem is surrounded b the vacuum via an infinite bo.

5 Math. Comput. ppl. 2017, 22, 17 5 of 39 Coil N I; S ; 4 t c c 0 Vacuum v z ir or Iron v 0 iron 0 ri Figure 1. ir- or iron-cored coil see Table 1 for the various parameters. The 2-D magnetic field distribution in the air- or iron-cored coil has been studied in Cartesian coordinates, b solving magnetostatic Mawell s equations from subdomain technique. In this analsis, the magnetic field solution is based on the following simplifing assumptions: The end-effects are neglected i.e., that the magnetic variables are independent of z; The electrical conductivities of materials are assumed to be null i.e., the edd-currents induced in the copper/iron are neglected; The magnetic materials are considered as isotropic i.e., the permeabilit can be assumed the same in the two directions; The effect of global saturation is taken into account with a constant magnetic permeabilit corresponding to linear zone of the BH curve i.e., the initial magnetization curve Problem Discretization in Subdomains s shown in Figure 2, the problem domain is divided into 7 subdomains with µ = C st. The vacuum around to the air- or iron-cored coil is defined b 4 regions, i.e., Region 1 1, 2 with µ 1 = µ v ; Region 2 3, 4 with µ 2 = µ v ; Region 3 1, 2 2, 3 with µ 3 = µ v ; Region 4 5, 6 2, 3 with µ 4 = µ v. The air or iron in the middle of the coil is defined b the Region 5 2, 3 2, 3 with µ 5 = µ v for the air or µ 5 = µ iron for the iron. The coil i.e., forward and return conductors is defined b 2 regions, i.e., Region 6 2, 3 2, 3 with µ 6 = µ c ; Region 7 4, 5 2, 3 with µ 7 = µ c Governing Partial Differential Equations in Cartesian Coordinates ccording to 4 see ppendi, the 2-D magnetic vector potential distribution in Cartesian coordinates, is governed b the Laplace s equation in regions j with j = 1,..., 5, i.e., zj = 2 zj zj 2 = 0, 1

6 Region 6 Region 7 Math. Comput. ppl. 2017, 22, 17 6 of 39 Region 2 Boundar Condition i.e., Dirichlet Region 3 Region 5 Region 4 Region 1 z Figure 2. Subdomains in the air- or iron-cored coil. and the Poisson s equation in regions k with k = 6, 7, i.e., zk = 2 zk zk 2 = µ k J zk, 2a where J zk represents the current densit due to suppl currents which is defined b J zk = C k Nt I S c, 2b in which S c is the conductor surface, and C k is the coefficient for the direction of current in the conductor e.g., with C 6 = 1 for the forward conductor and C 7 = 1 for return conductor. ccording to the method of separation of variables, it is interesting to note that z can be decomposed into two potentials according to the two directions see ppendi, i.e., z for the -edges 5b and z for the -edges 5c. The periodicit of z and z are respectivel defined b β h and λ n with h and n the spatial harmonic orders Boundar Conditions Reminder on the Boundar Conditions at the Interface of Two Surfaces In electromagnetic, as shown in Figure 3, the magnetic field H obes mpère s continuit condition, H n a H b = K, 3a where n is the unit vector normal to the boundar between two surfaces, H the parallel component of H on one side of the interface, and K the current densit at the surface of the interface. t this same surface, the magnetic flu continuit condition also applies B n a B b = 0 or a b = 0, 3b where B is the perpendicular component of B on one side of the interface. The Dirichlet condition on one surface is defined b a = 0 or b = 0. 3c

7 Math. Comput. ppl. 2017, 22, 17 7 of 39 H b Bb Surface b Current densit K n H a Ba Surface a Figure 3. Boundar conditions at the interface of two surfaces pplication to the ir- or Iron-Cored Coil On the outer boundaries for 1 6, and, 1 4 see Figure 2, the component of the magnetic vector potential satisfies the Dirichlet boundar condition, i.e., 3c. B appling 3 and using 2 see ppendi, the respective boundaries at the interface between the various regions are illustrated in Figure General Solutions Region 1 The general solution of z1, B 1 and B 1 are determined b the particular case of the case-stud no 1 i.e., z imposed on all edges of a region in ppendi B. The boundar conditions on the -edges of the region see Figure 4a are met b posing ch = 0 in B4. Therefore, the magnetic vector potential z1, which is a solution of 1 satisfing the boundar conditions of Figure 4a, is defined b z1 = the components of B 1 = B 1 ; B 1 ; 0 b B 1 = B 1 = d1h1 sh β1 h1 1 β1 h1=1 h1 ch sin β1 h1 1, 4 β1 h1 τ 1 d1h1 ch β1 h1 1 h1=1 ch sin β1 h1 1, 5 β1 h1 τ 1 h1=1 d1 h1 sh β1 h1 1 ch β1 h1 τ 1 cos β1 h1 1, 6 where h1 is the spatial harmonic orders in Region 1, d1 h1 the integration constant, β1 h1 = h1 π / τ 1 with τ 1 = 6 1, and τ 1 = 2 1. The coefficient d1 h1 is determined using a Fourier series epansion of F 1 see Figure 4a over the interval = 1, 6 = 1, 1 + τ 1 : d1h1 = 2 1 +τ 1 F τ 1 sin β1 h1 1 d The epression of d1h1 is developed in ppendi C.

8 Region 3 Region 5 Region 5 Region 4 Region 6 Region 7 Region 6 Region 7 Region 6 Region 7 Region 6 Region 7 Math. Comput. ppl. 2017, 22, 17 8 of z Region 3 0 z1 1 Region 5 Region B3 1, B 6 2, 3 2 B1 F B 5 2 3, B7 4, B4 5, 6 2 Region 1 z z Region 3 Region 2 z B3 1, B 6 2, 3 3 B2 G B 5 3 3, B7 4, B4 5, z2 6 Region 5 Region a b Region 2 Region 2 Region z3 z2 3 3 z4 z2 3 3 z5 z2 3 3 z3 z6 2 2 z4 z7 5 5 z5 z6 3 3 Region 3 Region 4 Region 5 0 z3 1 z3 z1 2 2 z4 z z4 6 z5 z5 z1 2 2 z Region 1 Region 1 Region 1 c d e Region 2 Region z6 z2 3 3 z7 z2 3 3 B B B B Region 6 Region 7 B B B B z6 z1 2 2 z7 z Region 1 Region 1 f g Figure 4. Boundar conditions in subdomains: a Region 1; b Region 2; c Region 3; d Region 4; e Region 5; f Region 6; and g Region Region 2 The same method than Region 1 is used to find the solution in Region 2. B posing dh = 0 in B4 see ppendi B, the magnetic vector potential z2, which is a solution of 1 satisfing the boundar conditions of Figure 4b, is defined b z2 = h2=1 the components of B 2 = B 2 ; B 2 ; 0 b B 2 = c2 h2 β2 h2 sh β2 h2 4 ch β2 h2 τ 2 sin β2 h2 1, 8 c2h2 ch β2 h2 4 h2=1 ch sin β2 h2 1, 9 β2 h2 τ 2

9 Math. Comput. ppl. 2017, 22, 17 9 of 39 B 2 = c2h2 sh β2 h2 4 h2=1 ch cos β2 h2 1, 10 β2 h2 τ 2 where h2 is the spatial harmonic orders in Region 2, c2 h2 the integration constant, β2 h2 = h2 π / τ 2 with τ 2 = 6 1, and τ 2 = 4 3. The coefficient c2 h2 is determined using a Fourier series epansion of G 2 see Figure 4b over the interval = 1, 6 = 1, 1 + τ 2 : c2h2 = 2 1 +τ 2 G 2 sin β2 τ h2 1 d The epression of c2h2 is developed in ppendi C Region 3 The general solution of z3, B 3 and B 3 are determined b the case-stud no 1 i.e., z imposed on all edges of a region in ppendi B. The boundar conditions on the -edges of the region see Figure 4c are met b posing e n = 0 in B1 B3. Therefore, the magnetic vector potential z3, which is a solution of 1 satisfing the boundar conditions of Figure 4c, is defined b z3 = z3 = z3 + z3, 12a c3 h3 sh β3 h3 3 β3 h3=1 h3 sh + d3 h3 sh β3 h3 2 β3 h3 τ 3 β3 h3 sh sin β3 h3 1, 12b β3 h3 τ 3 z3 = n3=1 f 3 n3 sh λ3 n3 1 sin λ3 λ3 n3 sh λ3 n3 n3 2, 12c τ 3 the -component of B 3 b B 3 = B3 + B 3, 13a B3 = c3h3 ch β3 h3 3 h3=1 sh + d3h3 β3 h3 τ ch β3 h3 2 3 sh sin β3 h3 1, β3 h3 τ 3 13b B 3 = f 3 n3 sh λ3 n3 1 cos λ3 sh λ3 n3=1 n3 n3 2, 13c τ 3 the -component of B 3 b B 3 = B3 + B 3, 14a B3 = c3h3 sh β3 h3 3 h3=1 sh + d3h3 β3 h3 τ sh β3 h3 2 3 sh cos β3 h3 1, 14b β3 h3 τ 3 B 3 = f 3 n3 ch λ3 n3 1 sin λ3 sh λ3 n3=1 n3 n3 2, 14c τ 3 where h3 and n3 are the spatial harmonic orders in Region 3; c3h3, d3 h3 and f 3 n3 the integration constants; β3 h3 = h3 π / τ 3 with τ 3 = 2 1 ; and λ3 n3 = n3 π / τ 3 with τ 3 = 3 2. The coefficients c3h3 and d3 h3 are respectivel determined using Fourier series epansion of z1 =2 and z2 =3 see Figure 4c over the interval = 1, 2 = 1, 1 + τ 3 : c3 h3 = 2 τ 3 1 +τ 3 1 β3 h3 z1 =2 sin β3 h3 1 d, 15a

10 Math. Comput. ppl. 2017, 22, of 39 d3h3 = 2 1 +τ 3 β3 τ h3 z2 =3 sin β3 h3 1 d. 15b 3 1 The coefficient f 3 n3 is determined using a Fourier series epansion of z6 =2 see Figure 4c over the interval = 2, 3 = 2, 2 + τ 3 : f 3 n3 = 2 2 +τ 3 λ3 n3 z6 τ =2 sin λ3 n3 2 d The epression of c3h3, d3 h3 and f 3 n3 are developed in ppendi C Region 4 The same method than Region 3 is used to find the solution in Region 4. B posing f n = 0 in B1 B3 see ppendi B, the magnetic vector potential z4, which is a solution of 1 satisfing the boundar conditions of Figure 4d, is defined b z4 = z4 = z4 + z4, 17a c4 h4 sh β4 h4 3 β4 h4=1 h4 sh + d4 h4 sh β4 h4 2 β4 h4 τ 4 β4 h4 sh sin β4 h4 5, 17b β4 h4 τ 4 z4 = e4 n4 sh λ4 n4 6 sin λ4 λ4 n4=1 n4 sh λ4 n4 τ 4 n4 2, 17c the -component of B 4 b B 4 = B4 + B 4, 18a B4 = c4h4 ch β4 h4 3 h4=1 sh + d4h4 β4 h4 τ ch β4 h4 2 4 sh sin β4 h4 5, β4 h4 τ 4 18b B 4 = e4 n4 sh λ4 n4 6 cos λ4 sh λ4 n4=1 n4 τ 4 n4 2, 18c the -component of B 4 b B 4 = B4 + B 4, 19a B4 = c4h4 sh β4 h4 3 h4=1 sh + d4h4 β4 h4 τ sh β4 h4 2 4 sh cos β4 h4 5, 19b β4 h4 τ 4 B 4 = e4 n4 ch λ4 n4 6 sin λ4 sh λ4 n4=1 n4 τ 4 n4 2, 19c where h4 and n4 are the spatial harmonic orders in Region 4; c4h4, d4 h4 and e4 n4 the integration constants; β4 h4 = h4 π / τ 4 with τ 4 = 6 5 ; and λ4 n4 = n4 π / τ 4 with τ 4 = 3 2. The coefficients c4h4 and d4 h4 are respectivel determined using Fourier series epansion of z1 =2 and z2 =3 see Figure 4d over the interval = 5, 6 = 5, 5 + τ 4 : c4 h4 = 2 τ 4 5 +τ 4 5 β4 h4 z1 =2 sin β4 h4 5 d, 20a

11 Math. Comput. ppl. 2017, 22, of 39 d4h4 = 2 5 +τ 4 β4 τ h4 z2 =3 sin β4 h4 5 d. 20b 4 5 The coefficient e4 n4 is determined using a Fourier series epansion of z7 =5 see Figure 4d over the interval = 2, 3 = 2, 2 + τ 4 : e4 n4 = 2 2 +τ 4 λ4 τ n4 z7 =5 sin λ4 n4 2 d The epression of c4h4, d4 h4 and e4 n4 are developed in ppendi C Region 5 ccording to case-stud no 1 i.e., z imposed on all edges of a region in ppendi B, the magnetic vector potential z5, which is a solution of 1 satisfing the boundar conditions of Figure 4e, is defined b z5 = z5 + z5, 22a z5 = z5 = h5=1 n5=1 n5=1 c5 h5 sh β5 h5 3 β5 h5 sh β5 h5 τ 5 + d5 h5 β5 h5 sh β5 h5 2 sh β5 h5 τ 5 e5 n5 sh λ5 n5 4 + f 5 n5 sh λ5 n5 3 λ5 n5 sh λ5 n5 τ 5 λ5 n5 sh λ5 n5 τ 5 sin β5 h5 3, 22b sin λ5 n5 2, 22c the -component of B 5 b B 5 = B5 + B 5, 23a B5 = c5h5 ch β5 h5 3 h5=1 sh + d5h5 β5 h5 τ ch β5 h5 2 5 sh sin β5 h5 3, β5 h5 τ 5 23b B 5 = e5 n5 sh λ5 n5 4 + f 5 sh λ5 n5 τ 5 n5 sh λ5 n5 3 cos λ5 sh λ5 n5 n5 2, τ 5 23c the -component of B 5 b B 5 = B5 + B 5, 24a B5 = c5h5 sh β5 h5 3 h5=1 sh + d5h5 β5 h5 τ sh β5 h5 2 5 sh cos β5 h5 3, 24b β5 h5 τ 5 B 5 = e5 n5 ch λ5 n5 4 + f 5 sh λ5 n5=1 n5 τ 5 n5 ch λ5 n5 3 sin λ5 sh λ5 n5 n5 2, 24c τ 5 where h5 and n5 are the spatial harmonic orders in Region 5; c5h5, d5 h5, e5 n5 and f 5 n5 the integration constants; β5 h5 = h5 π / τ 5 with τ 5 = 4 3 ; and λ5 n5 = n5 π / τ 5 with τ 5 = 3 2. The coefficients c5h5 and d5 h5 are respectivel determined using Fourier series epansion of z1 =2 and z2 =3 see Figure 4e over the interval = 3, 5 = 3, 3 + τ 5 : c5 h5 = 2 τ 5 3 +τ 5 3 β5 h5 z1 =2 sin β5 h5 3 d, 25a

12 Math. Comput. ppl. 2017, 22, of 39 d5h5 = 2 3 +τ 5 β5 τ h5 z2 =3 sin β5 h5 3 d. 25b 5 3 The coefficient e5 n5 and f 5 n5 are respectivel determined using a Fourier series epansion of z6 =3 and z7 =4 see Figure 4e over the interval = 2, 3 = 2, 2 + τ 5 : e5 n5 = 2 2 +τ 5 λ5 n5 z6 τ =3 sin λ5 n5 2 d, 26a 5 2 f 5 n5 = 2 2 +τ 5 λ5 n5 z7 τ =4 sin λ5 n5 2 d. 26b 5 2 The epression of c5h5, d5 h5, e5 n5 and f 5 n5 are developed in ppendi C Region 6 ccording to case-stud no 2 i.e., B and z are respectivel imposed on - and -edges of a region in ppendi B, the magnetic vector potential z6, which is a solution of 2 satisfing the boundar conditions of Figure 4f, is defined b z6 = z6 + z6 + zp6, 27a 3 c6 z6 = d60 + c6h6 h6=1 β6 shβ6 h6 3 h6 shβ6 h6τ 6 + d6 h6 β6 shβ6 h6 2 cos β6 h6 shβ6 h6τ 6 h6 2 e6 z6 = n6 ch λ6 n6 2 f 6 n6 ch λ6 n6 3 λ6 n6=1 n6 sh λ6 n6 τ 6 λ6 n6 sh λ6 n6 τ 6, 27b sin λ6 n c Considering 27b and 27c as well as the form of the current densit distribution, i.e., 2b, a particular solution zp6 can be found. The following quadratic form can be proposed as a particular solution: zp6 = 1 2 µ 6 J z d The -component of B 6 is defined b B 6 = B 6 + B 6 + B P6, 28a c6 B6 = 0 + d6 0 + h6=1 c6 h6 chβ6 h6 3 shβ6 h6τ 6 B 6 = e6 n6 ch λ6 n6 2 sh λ6 n6=1 n6 τ 6 + d6h6 chβ6 h6 2 cos β6 shβ6 h6τ 6 h6 2, 28b f 6 n6 ch λ6 n6 3 cos λ6 sh λ6 n6 n6 2, 28c τ 6 B P6 = zp6 = µ 6 J z6, 28d and the -component of B 6 b B 6 = B 6 + B 6 + B P6, 29a

13 Math. Comput. ppl. 2017, 22, of 39 B 6 = B 6 = c6h6 sh β6 h6 3 h6=1 n6=1 sh + d6h6 β6 h6 τ sh β6 h6 2 6 sh sin β6 h6 2, 29b β6 h6 τ 6 e6 n6 sh λ6 n6 2 sh λ6 n6 τ 6 + f 6 n6 sh λ6 n6 3 sin λ6 sh λ6 n6 n6 2, 29c τ 6 B P6 = zp6 = 0, 29d where h6 and n6 are the spatial harmonic orders in Region 6; c60, d6 0, c6 h6, d6 h6, e6 n6 and f 6 n6 the integration constants; β6 h6 = h6 π / τ 6 with τ 6 = 3 2 ; and λ6 n6 = n6 π / τ 6 with τ 6 = 3 2. The coefficients c60 & c6 h6 and d6 0 & d6 h6 are respectivel determined using Fourier series epansion of z1 =2 and z2 =3 see Figure 4f over the interval = 2, 3 = 2, 2 + τ 6 : c60 = 1 2 +τ 6 τ τ 6 z1 =2 zp6 =2 d, 30a c6h6 = 2 2 +τ 6 β6 τ h6 z1 =2 zp6 =2 cos β6 h6 2 d, 30b 6 2 d60 = 1 2 +τ 6 τ τ 6 z2 =3 zp6 =3 d, 30c d6h6 = 2 2 +τ 6 β6 τ h6 z2 =3 zp6 =3 cos β6 h6 2 d. 30d 6 2 The coefficient e6 n6 and f 6 n6 are respectivel determined using a Fourier series epansion of µ 6/ µ5 B 5 =3 and µ 6/ µ3 B 3 =2 see Figure 4f over the interval = 2, 3 = 2, 2 + τ 6 : e6 n6 = 2 τ 6 f 6 n6 = 2 τ 6 2 +τ τ 6 2 µ6 µ 5 B 5 =3 B P6 =3 µ6 µ 3 B 3 =2 B P6 =2 sin λ6 n6 2 d, 31a sin λ6 n6 2 d. 31b The epression of c60, d6 0, c6 h6, d6 h6, e6 n6 and f 6 n6 are developed in ppendi C Region 7 ccording to case-stud no 2 i.e., B and z are respectivel imposed on - and -edges of a region in ppendi B, the magnetic vector potential z7, which is a solution of 2 satisfing the boundar conditions of Figure 4g, is defined b z7 = z7 + z7 + zp7, 32a 3 c7 z7 = d70 + c7h7 shβ7 h7=1 h7τ 7 β7 h7 shβ7 h7 3 + d7 h7 β7 shβ7 h7 2 cos β7 h7 shβ7 h7τ 7 h7 4, 32b

14 Math. Comput. ppl. 2017, 22, of 39 e7 z7 = n7 ch λ7 n7 4 f 7 n7 ch λ7 n7 5 sin λ7 λ7 n7=1 n7 sh λ7 n7 τ 7 λ7 n7 sh λ7 n7 n c τ 7 Considering 32b and 32c as well as the form of the current densit distribution, i.e., 2b, a particular solution zp7 can be found. The following quadratic form can be proposed as a particular solution: zp7 = 1 2 µ 7 J z d The -component of B 7 is defined b B 7 = B 7 + B 7 + B P7, 33a c7 B7 = 0 + d7 0 + c7h7 chβ7 h7 3 shβ7 h7τ 7 h7=1 B 7 = e7 n7 ch λ7 n7 4 sh λ7 n7=1 n7 τ 7 + d7h7 chβ7 h7 2 cos β7 shβ7 h7τ 7 h7 4, 33b f 6 n7 ch λ7 n7 5 cos λ7 sh λ7 n7 n7 2, 33c τ 7 B P7 = zp7 = µ 7 J z7, 33d and the -component of B 7 b B 7 = B7 + B 7 + B P7, 34a B7 = c7h7 sh β7 h7 3 h7=1 sh + d7h7 β7 h7 τ sh β7 h7 2 7 sh sin β7 h7 4, β7 h7 τ 7 34b B 7 = e7 n7 sh λ7 n7 4 + f 7 sh λ7 n7 τ 7 n7 sh λ7 n7 5 sin λ7 sh λ7 n7 n7 2, τ 7 34c n7=1 B P7 = zp7 = 0, 34d where h7 and n7 are the spatial harmonic orders in Region 7; c70, d7 0, c7 h7, d7 h7, e7 n7 and f 7 n7 the integration constants; β7 h7 = h7 π / τ 7 with τ 7 = 5 4 ; and λ7 n7 = n7 π / τ 7 with τ 7 = 3 2. The coefficients c70 & c7 h7 and d7 0 & d7 h7 are respectivel determined using Fourier series epansion of z1 =2 and z2 =3 see Figure 4g over the interval = 4, 5 = 4, 4 + τ 7 : c70 = 1 4 +τ 7 τ τ 7 z1 =2 zp7 =2 d, 35a c7h7 = 2 4 +τ 7 β7 τ h7 z1 =2 zp7 =2 cos β7 h7 4 d, 35b 7 4 d70 = 1 4 +τ 7 τ τ 7 z2 =3 zp7 =3 d, 35c d7h7 = 2 4 +τ 7 β7 τ h7 z2 =3 zp7 =3 cos β7 h7 4 d. 35d 7 4

15 Math. Comput. ppl. 2017, 22, of 39 The coefficient e7 n7 and f 7 n7 are respectivel determined using a Fourier series epansion of µ 7/ µ4 B 4 =5 and µ 7/ µ5 B 5 =4 see Figure 4g over the interval = 2, 3 = 2, 2 + τ 7 : e7 n7 = 2 τ 7 f 7 n7 = 2 τ 7 2 +τ τ 7 2 µ7 µ 4 B 4 =5 B P7 =5 µ7 µ 5 B 5 =4 B P7 =4 sin λ7 n7 2 d, 36a sin λ7 n7 2 d. 36b The epression of c70, d7 0, c7 h7, d7 h7, e7 n7 and f 7 n7 are developed in ppendi C Solving of Linear Sstem The integration constants can be determined b solving the following linear equations which can be written in matri form as 68 IC = BC 1 ES, 37 where IC is the integration constants vector of dimension X ma 1, IC = IC1 IC2 IC3 IC4 IC5 IC6 IC7 IC1 = d1h1, T, 38a 38b IC2 = c2h2, IC3 = c3h3 d3h3 f 3 n3 IC4 = c4h4 d4h4 e4 n4 IC5 = c5h5 d5h5 e5 n5 f 5 n5 IC6 = c60 c6h6 d60 d6h6 e6 n6 f 6 n6 IC7 = c70 c7h7 d70 d7h7 e7 n7 f 7 n7 38c, 38d, 38e, 38f, 38g, 38h ES the electromagnetic sources vector of dimension X ma 1, ES = ES1 ES2 ES3 ES4 ES5 ES6 ES7 ES1 = ES16 h1 + ES17 h1, T, 39a 39b ES2 = ES26 h2 + ES27 h2, 39c ES3 = 0 0 ES36 n3, 39d ES4 = 0 0 ES47 n4, 39e ES5 = 0 0 ES56 n5 ES57 n5, 39f ES6 = ES ES , 39g ES7 = ES ES , 39h

16 Math. Comput. ppl. 2017, 22, of 39 and BC the boundar conditions matri of dimension X ma X ma BC = I 0 BC13 BC14 BC15 BC16 BC17 0 I BC23 BC24 BC25 BC26 BC27 BC31 BC32 I 0 0 BC36 0 BC41 BC42 0 I 0 0 BC47 BC51 BC I BC56 BC57 BC61 BC62 BC63 0 BC65 I 0 BC71 BC72 0 BC74 BC75 0 I, 40a in which I is identit matri, and BC13 = Q13c h1,h3 Q13d h1,h3 Q13 f h1,n3 BC14 = Q14c h1,h4 Q14d h1,h4 Q14e h1,n4 BC15 = Q15c h1,h5 Q15d h1,h5 Q15e h1,n5 Q15 f h1,n5 BC16 = Q16c h1,0 Q16c h1,h6 Q16d h1,0 Q16d h1,h6 Q16e h1,n6 Q16 f h1,n6 BC17 = Q17c h1,0 Q17c h1,h7 Q17d h1,0 Q17d h1,h7 Q17e h1,n7 Q17 f h1,n7 40b for Region 1, BC23 = BC24 = BC25 = BC26 = BC27 = Q23c h2,h3 Q23d h2,h3 Q23 f h2,n3 Q24c h2,h4 Q24d h2,h4 Q24e h2,n4 Q25c h2,h5 Q25d h2,h5 Q25e h2,n5 Q25 f h2,n5 Q26c h2,0 Q26c h2,h6 Q26d h2,0 Q26d h2,h6 Q26e h2,n6 Q26 f h2,n6 Q27c h2,0 Q27c h2,h7 Q27d h2,0 Q27d h2,h7 Q27e h2,n7 Q27 f h2,n7 40c for Region 2, BC31 = Q31d h3,h1 0 0 BC32 = 0 Q32c h3,h2 0 BC36 = for Region 3, T T Q36c n3,0 Q36c n3,h6 Q36d n3,0 Q36d n3,h6 Q36e n3,n6 Q36 f n3,n6 BC41 = Q41d h4,h1 0 0 BC42 = 0 Q42c h4,h2 0 BC47 = T T Q47c n4,0 Q47c n4,h7 Q47d n4,0 Q47d n4,h7 Q47e n4,n7 Q47 f n4,n7 40d 40e

17 Math. Comput. ppl. 2017, 22, of 39 for Region 4, T BC51 = Q51d h5,h T BC52 = 0 Q52c h5,h BC56 = Q56c n5,0 Q56c n5,h6 Q56d n5,0 Q56d n5,h6 Q56e n5,n6 Q56 f n5,n BC57 = Q57c n5,0 Q57c n5,h7 Q57d n5,0 Q57c n5,h7 Q57e n5,n7 Q57 f n5,n7 40f for Region 5, for Region 6, for Region 7. T BC61 = Q61d 0,h1 Q61d h6,h T BC62 = 0 0 Q62c 0,h2 Q62c h6,h BC63 = Q63c n6,h3 Q63d n6,h3 Q63 f n6,n BC65 = Q65c n6,h5 Q65d n6,h5 Q65e n6,n5 Q65 f n6,n T BC71 = Q71d 0,h1 Q71d h7,h T BC72 = 0 0 Q72c 0,h2 Q72c h7,h BC74 = Q74c n7,h4 Q74d n7,h4 Q74e n7,n BC75 = Q75c n7,h5 Q75d n7,h5 Q75e n7,n5 Q75 f n7,n5 40g 40h

18 Math. Comput. ppl. 2017, 22, of 39 The corresponding elements in 39 and 40 are defined in ppendi C. One can note that 37 consists of H1 X ma = ma + H2 ma + 2 H3 ma + N3 ma + 2 H4 ma + N4 ma H5 ma + N5 ma + 2 H6 ma + N6 ma H7 ma + N7 ma + 1 equations and unknowns. n mathematical software can quickl give the numerical solution of 37. This set is implemented in Matlab R R2015a, Mathworks, Natick, M, US b using the sparse matri/vectors according to the method described in Section 2. The analtical solutions of z and B = B ; B ; 0 in the various regions have been computed with a finite number of spatial harmonics terms H1 ma H7 ma for the -edges and N3 ma N7 ma for the -edges. Usuall, the two reasons for the possibilit of including a finite number of harmonics is a limiting computational time and numerical accurac Numerical Problems: Harmonics and Ill-Conditioned Sstem discussion on the numerical limitations of such semi-analtical models has been presented in 69,70. Numerical methods, which use a meshed geometr, will have a limited accurac related to the densit of the mesh. The Mawell-Fourier methods ehibit a similar problem due to the periodicit of Fourier series, and consequentl to the finite number of harmonics. The size of the model, or more specificall, the size of the matri BC, as defined in 40, depends on the number of: i subdomains; ii boundar conditions; and iii spatial harmonics terms. Consequentl, an electromagnetic device with a high number of teeth/slots results in large model and, hence, in high computational time 70. The numerical accurac of magnetic field solution and the computational time depend on the highest spatial harmonic orders considered in the different subdomains. It is interesting to note that the maimum number of harmonics also depends on the available memor of computer. Beond a certain number of harmonics, the linear sstem becomes ill-conditioned and the results inaccurate 69. Therefore, the number of harmonics has to be carefull selected to obtain a correctl converged solution. n etensive discussion on the effect of the harmonics number taken into account is given in 71,72. However, owing to the different sizes of the regions e.g.,the finite height/width, etc., such series could be truncated at different points. Considering an optimal ratio between the numbers of harmonic terms taken into account in each region might lead to a lower calculation error and a higher rate of convergence 72. Limiting the number of harmonics will lead to inaccurate field solutions at discontinuous points in the geometr, especiall at the corner points of magnets, current regions, or soft-magnetic material 69. Moreover, the Gibbs phenomenon can become dominant at these positions at interfaces between region with unequal width Comparison of the Semi-naltic and Finite-Element Calculations 3.1. Introduction The objective of this section is to show the effectiveness of 2-D subdomain model on the magnetic field distribution. The main parameters of the air- and iron-cored coil are given in Table 1. For the comparison, the sstem has been set up using Cedrat s Flu2D Version , ltair Engineering, Melan Cede, France software package i.e., an advanced finite-element method based numeric field analsis program 8. The finite-element computations are done under same assumptions on which the semi-analtical model is based see 2.1. Problem Description and ssumptions. The spatial harmonics terms in each subdomain, given in Table 1 rounded to 0 decimal, have been imposed according to an optinal ratio as indicated in 71,72, i.e., for H1 ma given, H ma = H1 ma τ τ 1 and N ma = H ma τ τ. 42

19 Math. Comput. ppl. 2017, 22, of 39 The linear sstem 37 consists of 3,346 elements representing the size of linear sstem to solve which is much smaller than the 2-D FE mesh having 8,566 surfaces elements of second order viz., the triangles number of sstem. The 2-D FE mesh for an air- or iron-cored coil is illustrated in Figure 5. The personal computer used for this comparison has the following characteristics: HP Z800 IntelR XeonR GHz with 2 processors RM 16 Go 64 bits. The computation time of 2-D subdomain model is equal to 0.01 sec for 2-D subdomain model and 1 sec for the 2-D FE. The proposed design approach can thus reduce the computation time b approimatel 100-fold versus to 2-D FE. Parameters, Smbols Units Table 1. Parameters of the ir- or Iron-cored Coil. Values Number of coils turns, N t 1600 Maimum direct current, I 5 Surface of conductors, S c mm Current densit due to suppl currents, J zk /mm 2 ±10 Effective aial length, L z cm 4 Geometrical parameters in the -ais, 1 ; 2 ; 3 ; 4 ; 5 ; 6 cm 0; 10; 12; 16; 18; 28 Geometrical parameters in the -ais, 1 ; 2 ; 3 ; 4 cm 0; 10; 14; 24 Relative magnetic permeabilit of the iron, µ iron 1500 Number of spatial harmonics for Region 1, H1 ma 300 Number of spatial harmonics for Region 2, H2 ma 300 Number of spatial harmonics for Region 3, H3 ma ; N3 ma 107; 268 Number of spatial harmonics for Region 4, H4 ma ; N4 ma 107; 268 Number of spatial harmonics for Region 5, H5 ma ; N5 ma 43; 268 Number of spatial harmonics for Region 6, H6 ma ; N6 ma 21; 268 Number of spatial harmonics for Region 7, H7 ma ; N7 ma 21; 268 Coil ir or Iron Figure 5. 2-D finite-element analsis FE mesh for an air- or iron-cored coil Results Discussion The 2-D subdomain model is implemented so that it is possible to get values of z in the airand iron-cored coil. Figure 6 present the equipotential lines 30 lines of z in the sstem with the 2-D subdomain model and 2-D FE. s can be seen, a good evaluation is obtained, comparing those results with 2-D FE, for both air- and iron-core.

20 Path 3 Path 4 Path 5 Math. Comput. ppl. 2017, 22, of 39 2-D FE 2-D FE r 2-D Subdomain model 2-D Subdomain model a z b Figure 6. Equipotential lines of z with the 2-D subdomain model and FE for an a air- and b iron-cored coil. The paths of the magnetic flu densit validation for the comparison are given in Figure 7. The waveforms of the components of B = B ; B ; 0 are represented on the various paths in Figures The solid lines represent the magnetic flu densit computed b the 2-D FE and the circles correspond to 2-D subdomain model. It can be seen that a ver good agreement is obtained for the components of B, whatever the paths, for both air- and iron-core. This confirms that the effect of global saturation, with a constant magnetic permeabilit corresponding to linear zone of BH curve, is taken into account accuratel. Nevertheless, the numerical accurac of magnetic field solution is reduced as the number of considered is lowered. The relative error is 1.5% for the different components of magnetic flu densit. Some slight discrepancies are observed between numerical and analtical results see Figures 11 and 12b which can be caused b the finite number of spatial harmonic taken into account in the semi-analtical model according to the - and -edges see 2.7. Numerical Problems: Harmonics and Ill-conditioned Sstem. The increase of harmonics number can resolve these deviations, however, at the epense of the computation time. 4 3 Path 2 2 Path 1 z Figure 7. Paths of the magnetic flu densit validation for the comparison.

21 The -component of the magnetic flu densit for Path 3 T The -component of the magnetic flu densit for Path 3 T The -component of the magnetic flu densit for Path 2 T The -component of the magnetic flu densit for Path 2 T The -component of the magnetic flu densit for Path 1 T The -component of the magnetic flu densit for Path 1 T Math. Comput. ppl. 2017, 22, of FE Subdomain model FE Subdomain model Iron-cored coil Iron-cored coil ir-cored coil ir-cored coil Length of Path 1 mm a Length of Path 1 mm b Figure 8. Waveform of the magnetic flu densit for Path 1: a - and b -component; 2-D FE FE Subdomain model FE Subdomain model Iron-cored coil Iron-cored coil 0.05 ir-cored coil 0 ir-cored coil Length of Path 2 mm a Length of Path 2 mm b Figure 9. Waveform of the magnetic flu densit for Path 2: a - and b -component FE Subdomain model FE Subdomain model Iron-cored coil ir-cored coil ir-cored coil Iron-cored coil Length of Path 3 mm a Length of Path 3 mm b Figure 10. Waveform of the magnetic flu densit for Path 3: a - and b -component.

22 The -component of the magnetic flu densit for Path 5 T The -component of the magnetic flu densit for Path 5 T The -component of the magnetic flu densit for Path 4 T The -component of the magnetic flu densit for Path 4 T Math. Comput. ppl. 2017, 22, of FE Subdomain model ir-cored coil FE Subdomain model Iron-cored coil ir-cored coil Iron-cored coil Length of Path 4 mm a Length of Path 4 mm b Figure 11. Waveform of the magnetic flu densit for Path 4: a - and b -component FE Subdomain model FE Subdomain model Iron-cored coil ir-cored coil Iron-cored coil Length of Path 5 mm a ir-cored coil Length of Path 5 mm b Figure 12. Waveform of the magnetic flu densit for Path 5: a - and b -component. 4. Conclusions n overview on the eisting semi-analtical models in Mawell-Fourier methods i.e., multi-laers models and subdomain technique with the saturation effect has been realized. It has been demonstrated that there is no semi-analtical model based on the subdomain technique taking into account the iron parts without the nonlinear BH curve. Then, the new scientific contribution on the 2-D subdomain technique in Cartesian coordinates to stud the local magnetic field distribution in the iron parts with a global saturation is presented in this paper. For eample, it was performed b solving 2-D magnetostatic Mawell s equations in Cartesian coordinates, for an air- or iron-cored coil supplied b a direct current. The subdomains connection is carried out in the two directions i.e., - and -edges. The iron magnetic permeabilit is constant, corresponding to linear zone of the initial magnetization curve. However, nonlinear magnetic materials could be accounted for b means of an iterative algorithm as in 48. This major scientific contribution will be applied to rotating and/or linear electrical machines without magnets supplied b a direct current or alternate current with an waveforms whose the analsis would be based on a 2-D semi-analtical model in Cartesian coordinates e.g., plane linear machines, aial-flu machines, etc.. n etension of the 2-D subdomain technique in polar coordinates as well as various electrical machines viz., radial-/aial-/transverse-flu machines, linear machines, U-/E-cored electromagnetic device, etc. will be made in the net studies.

23 Math. Comput. ppl. 2017, 22, of 39 This new approach to account for the effect of global saturation is semi-analticall based and takes significantl less computing time than the FE approimatel 100-fold versus to FE; it is eminentl suitable for design and optimization of the electromechanical sstems. Predicted results from the eact semi-analtical model have been compared finite-element predictions, and good agreement has been achieved, in both amplitudes and waveforms. uthor Contributions: The work presented here was carried out in cooperation among all authors, which have written the paper and have gave advice for the manuscripts. Conflicts of Interest: The authors declare no conflict of interest. ppendi. The 2-D Magnetostatic General Solution in Cartesian Coordinates ppendi.1. Governing Partial Differential Equations EDPs B assuming that the term / D t is negligible, the magnetostatic Mawell s equations are represented b Mawell-mpère and Mawell-Thomson H rot = J with J = 0 for the no-load operation, 1a B div = 0 Magnetic flu conservation, 1b B = rot with div = 0 Coulomb s gauge, 1c where, B, H, and J are respectivel the magnetic vector potential, the magnetic flu densit, magnetic field, and the current densit due to suppl currents vectors. The field vectors B and H are coupled b the magnetic material equation B = µ H + µ0 M, 2 where M is the magnetization vector with M = 0 for the vacuum/iron or M = 0 for the magnets according to the magnetization direction 4, and µ = µ 0 µ r the absolute magnetic permeabilit of the magnetic material in which µ 0 and µ r are respectivel the vacuum permeabilit and the relative permeabilit of the magnetic material with µ r = 1 for the vacuum or µ r = 1 for the magnets/iron. B using 1 and 2, the general EDPs of magnetostatic are defined b 74: X ν = J + X B, 3a X = grad ν rot, 3b X B = µ 0 grad ν M M + ν rot, 3c where ν = 1 / µ is the absolute magnetic reluctivit of the magnetic material. B neglecting the end-effects i.e., the sstem is infinitel long which leads to = 0; 0; z : the magnetic variables are independent of z, 3 in Cartesian coordinates, with µ = C st can be epressed b: z = 2 z z 2 = ES, 4a M ES = µ J z + µ 0 M. 4b

24 Math. Comput. ppl. 2017, 22, of 39 ppendi.2. General Solution It is interesting to note that z is governed b Poisson s equation, when there is one or more electromagnetic sources i.e., ES = 0, or Laplace s equation, when there is no electromagnetic sources i.e., ES = 0. ccording to the method of separation of variables, the 2-D magnetostatic general solution of z in Cartesian coordinates, can be written as z = z = z = z + z + zp, C 0 + D0 E 0 + F 0 + Ch ch β h h=1 + Dh sh β h C 0 + D 0 E 0 + F 0 + C n cos λ n n=1 + Dn sin λ n E h cos β h + F h sin β h E n ch λ n + F n sh λ n 5a, 5b, 5c where zp are the particular solution of z respecting the second member ES in 4, C 0 F h & C 0 F h the integration constants, β h & λ n the periodicit of z & z, and h and n the spatial harmonic orders. ccording to 1c, the components of B = B ; B ; 0 can be deduced from z b B = z and B = z 6 which leads to and B = B + B + zp, D B 0 E 0 + F 0 = + C β h h sh β h h=1 + Dh ch β h D B 0 E 0 + F 0 = + C n sin λ λ n n n=1 + Dn cos λ n B = B + B zp, F B 0 C 0 + D 0 = + C β h h ch β h h=1 + Dh sh β h F B 0 C 0 + D 0 = + C n cos λ λ n n n=1 + Dn sin λ n E h cos β h + F h sin β h E n ch λ n + F n sh λ n E h sin β h + F h cos β h E n sh λ n + F n ch λ n 7a, 7b, 7c 8a, 8b. 8c ppendi B. Simplification of Laplace s Equations ccording to Imposed Boundar Conditions ppendi B.1. Case-Stud No 1: z Imposed on ll Edges of a Region Figure B1a shows a region for l, r and l, t whose the magnetic vector potentials are imposed on all edges. B appling the principle of superposition on the magnetic quantities, Figure B1a is redefined b Figure B1b.

25 Math. Comput. ppl. 2017, 22, of 39 t z & t F z & l L z 0 z & r R l z & l G 0 l r a t z & z & t t 0 F z & z & l l 0 L z z 0 z & r z & r 0 R l z & z & l l 0 G 0 l r b Figure B1. z imposed on all edges of a region: a General and b Principle of superposition. In the case-stud no 1, the magnetic vector potential z = z + z, i.e., 5, is redefined b z = z = c h sh β h t β h=1 h sh + d h sh β h l β h τ β h sh sin β h l, B1a β h τ en sh λ n r + f n sh λ n l sin λ n λ n=1 n sh λ n τ λ n sh λ n τ l, B1b the component B = B + B of B, i.e., 7, b B = B = ch ch β h t h=1 sh + dh β h τ ch β h l sh sin β h l, B2a β h τ en sh λ n r sh λ n=1 n τ + fn sh λ n l cos λ n sh λ n τ l, B2b

26 Math. Comput. ppl. 2017, 22, of 39 and the component B = B + B of B, i.e., 8, b B = B = h=1 n=1 ch sh β h t sh + dh β h τ sh β h l sh β h τ e n ch λ n r sh λ n τ cos β h l, B3a + fn ch λ n l sin λ n sh λ n τ l, B3b where c h, d h, e n and f n are new integration constants; β h = h π / τ with τ = r l ; and λ n = n π / τ with τ = t l. For the particular case illustrated in Figure B2 whose the magnetic vector potentials are zero on -edges and imposed on -edges, the magnetic vector potential z, according to B1 with z = 0, is epressed b z = c h sh β h t β h=1 h sh + d h sh β h l β h τ β h sh sin β h l, B4a β h τ the -component of B, according to B6 with B = 0, b B = ch ch β h t h=1 sh + dh β h τ ch β h l sh sin β h l, B4b β h τ the -component of B, according to B3 with B = 0, b B = h=1 ch sh β h t sh + dh β h τ sh β h l sh β h τ cos β h l. B4c t z & t F 0 z & l z 0 0 z & r l z & l G 0 l r Figure B2. Particular case: z = 0 on -edges and z imposed on -edges of a region. ppendi B.2. Case-Stud No 2: B and z re Respectivel Imposed on X- and Y-Edges of a Region Figure B3a shows a region for l, r and l, t whose the magnetic flu densities and vector potentials are respectivel imposed on - and -edges. B appling the principle of superposition on the magnetic quantities, Figure B3a is redefined b Figure B3b.

27 Math. Comput. ppl. 2017, 22, of 39 t z & t F B & l L 0 z z B & r R l z & l G 0 l r a t z & z & t t 0 F B B & l & l 0 L z z 0 B B & r & r 0 R l z & z & l l 0 G 0 l r b Figure B3. B imposed on -edges and z imposed on -edges of a region: a General and b Principle of superposition. In the case-stud no 2, the magnetic vector potential z = z + z, i.e., 5, is redefined b z = z t c0 + = l d0 + h=1 n=1 ch β shβ h t + d h h shβ hτ β shβ h l h shβ hτ en ch λ n l f n ch λ n r λ n sh λ n τ λ n sh λ n τ the component B = B + B of B, i.e., 7, b B c0 + = d 0 + h=1 c h chβ h t shβ hτ cos β h l, B5a sin λ n l, B5b + dh chβ h l cos β shβ hτ h l, B6a

28 Math. Comput. ppl. 2017, 22, of 39 B = n=1 e n ch λ n l sh λ n τ and the component B = B + B of B, i.e., 8, b B = B = fn ch λ n r cos λ n sh λ n τ l, B6b ch sh β h t h=1 sh + dh β h τ sh β h l sh sin β h l, B7a β h τ en sh λ n l sh λ n=1 n τ where c 0, d 0, c h, d h, e n and f n are new integration constants. ppendi C. Elements of Linear Sstems ppendi C.1. Simplifing Function of General Integrals + fn sh λ n r sin λ n sh λ n τ l, B7b For the determination of the integral constants, it is required to calculate general integrals of the form F s = l l +w l l sin α s l l s dl, C1a F cs = l l +w l l cos α c l l c sin α s l l s dl, C1b F ss = l l +w l l sin α s1 l l s1 sin α s2 l l s2 dl, C1c F ls = F l2s = l l +w l l l sin α s l l s dl, C1d l l +w l l l 2 sin α s l l s dl, C1e F chs = l l +w l l ch α ch l l ch sin α s l l s dl, C1f F shs = l l +w l l sh α sh l l sh sin α s l l s dl. C1g The functions C1 will be used in the epression of the integration constants. The epressions of C1a C1c have given in 68. The development of C1d C1g gives F ls α s, l s, l l, w = sin α s l l + w l s sin α s l l l s α s l l + w cos α s l l + w l s l l cos α s l l l s α 2 s, C2a

29 Math. Comput. ppl. 2017, 22, of 39 F l2s α s, l s, l l, w = F chs α ch, α s, l ch, l s, l l, w = F shs α sh, α s, l sh, l s, l l, w = 2 cos α s l l + w l s cos α s l l l s α 2 s l l + w 2 cos α s l l + w l s ll 2 cos α s l l l s ppendi C.2. Epression of d1h1 for Region α s l l + w sin α s l l + w l s l l sin α s l l l s α 3 s sh α α ch ch l l l ch sin α s l l l s sh α ch l l + w l ch sin α s l l + w l s ch α + α s ch l l l ch cos α s l l l s ch α ch l l + w l ch cos α s l l + w l s α 2 ch +α2 s ch α α sh sh l l l sh sin α s l l l s ch α sh l l + w l sh sin α s l l + w l s sh α + α s sh l l l sh cos α s l l l s sh α sh l l + w l sh cos α s l l + w l s α 2 sh +α2 s, C2b, C2c. C2d B incorporating F 1 see Figure 4a into 7 and b using 13, 18, 23, 28 and 33, the development of 7 gives h4=1 d1h1 = + h5=1 + h6=0 + h7=0 ES1 h1 h3=1 c3 h3 Q13c h1,h3 + d3h3 Q13d h1,h3 + + c4 h4 Q14 h1,h4 + d4h4 Q14d h1,h4 n3=1 + f 3 n3 Q13 f h1,n3 n4=1 c5 h5 Q15c h1,h5 + d5h5 Q15d h1,h5 + n5=1 c6 h6 Q16c h1,h6 + d6h6 Q16d h1,h6 + n6=1 c7 h7 Q17c h1,h7 + d7h7 Q17d h1,h7 + n7=1 e4 n4 Q14e h1,n4 e5 n5 Q15e h1,n5 + f 5 n5 Q15 f h1,n5 e6 n6 Q16e h1,n6 + f 6 n6 Q16 f h1,n6 e7 n7 Q17e h1,n7 + f 6 n7 Q17 f h1,n7, C3a Q13c h1,h3 = 2 τ 1 µ1 µ 3 coth β3 h3 τ 3 Fss β3 h3, β1 h1, 1, 1, 1, τ 3, C3b Q13d h1,h3 = 2 τ 1 µ1 µ 3 csch β3 h3 τ 3 Fss β3 h3, β1 h1, 1, 1, 1, τ 3, C3c Q13 f h1,n3 = 2 τ 1 µ1 µ 3 csch λ3 n3 τ 3 F shs λ3 n3, β1 h1, 1, 1, 1, τ 3, C3d Q14c h1,h4 = 2 τ 1 µ1 µ 4 coth β4 h4 τ 4 Fss β4 h4, β1 h1, 5, 1, 5, τ 4, C3e Q14d h1,h4 = 2 τ 1 µ1 µ 4 csch β4 h4 τ 4 Fss β4 h4, β1 h1, 5, 1, 5, τ 4, C3f Q14e h1,n4 = 2 τ 1 µ1 µ 4 csch λ4 n4 τ 4 F shs λ4 n4, β1 h1, 6, 1, 5, τ 4, C3g Q15c h1,h5 = 2 τ 1 µ1 µ 5 coth β5 h5 τ 5 Fss β5 h5, β1 h1, 3, 1, 3, τ 5, C3h Q15d h1,h5 = 2 τ 1 µ1 µ 5 csch β5 h5 τ 5 Fss β5 h5, β1 h1, 3, 1, 3, τ 5, C3i Q15e h1,n5 = 2 τ 1 µ1 µ 5 csch λ5 n5 τ 5 F shs λ5 n5, β1 h1, 4, 1, 3, τ 5, C3j

30 Math. Comput. ppl. 2017, 22, of 39 Q15 f h1,n5 = 2 µ1 csch λ5 n5 τ 5 F τ 1 µ shs λ5 n5, β1 h1, 3, 1, 3, τ 5, C3k 5 Q16c h1,h6 = 2 µ1 F s β1 h1, 1, 2, τ 6 for h6 = 0 τ 1 µ 6 coth C3l β6 h6 τ 6 Fcs β6 h6, β1 h1, 2, 1, 2, τ 6 for h6 = 0 Q16d h1,h6 = 2 τ 1 µ1 µ 6 F s β1 h1, 1, 2, τ 6 for h6 = 0 csch β6 h6 τ 6 Fcs β6 h6, β1 h1, 2, 1, 2, τ 6 for h6 = 0 C3m Q16e h1,n6 = 2 τ 1 µ1 µ 6 csch λ6 n6 τ 6 F chs λ6 n6, β1 h1, 2, 1, 2, τ 6, C3n Q16 f h1,n6 = 2 µ1 csch λ6 n6 τ 6 F τ 1 µ chs λ6 n6, β1 h1, 3, 1, 2, τ 6, C3o 6 Q17c h1,h7 = 2 µ1 F s β1 h1, 1, 4, τ 7 for h7 = 0 τ 1 µ 7 coth C3p β7 h7 τ 7 Fcs β7 h7, β1 h1, 4, 1, 4, τ 7 for h7 = 0 Q17d h1,h7 = 2 τ 1 µ1 µ 7 F s β1 h1, 1, 4, τ 7 for h7 = 0 csch β7 h7 τ 7 Fcs β7 h7, β1 h1, 4, 1, 4, τ 7 for h7 = 0 C3q Q17e h1,n7 = 2 τ 1 µ1 µ 7 csch λ7 n7 τ 7 F chs λ7 n7, β1 h1, 4, 1, 4, τ 7, C3r Q17 f h1,n7 = 2 τ 1 µ1 µ 7 csch λ7 n7 τ 7 F chs λ7 n7, β1 h1, 5, 1, 4, τ 7, C3s ppendi C.3. Epression of c2h2 for Region 2 ES16 h1 = 2 τ 1 µ1 µ 6 B P6 =2 F s β1 h1, 1, 2, τ 6, C3t ES17 h1 = 2 τ 1 µ1 µ 7 B P7 =2 F s β1 h1, 1, 4, τ 7. C3u B incorporating G 2 see Figure 4b into 11 and b using 13, 18, 23, 28 and 33, the development of 11 gives h4=1 c2h2 = + h5=1 + h6=0 + h7=0 ES26 h2 + ES27 h2 h3=1 c3 h3 Q23c h2,h3 + d3h3 Q23d h2,h3 + + c4 h4 Q24c h2,h4 + d4h4 Q24d h2,h4 n3=1 f 3 n3 Q23 f h2,n3 + n4=1 c5 h5 Q25c h2,h5 + d5h5 Q25d h2,h5 + n5=1 c6 h6 Q26c h2,h6 + d6h6 Q26d h2,h6 + n6=1 c7 h7 Q27c h2,h7 + d7h7 Q27d h2,h7 + n7=1 e4 n4 Q24e h2,n4 e5 n5 Q25e h2,n5 + f 5 n5 Q25 f h2,n5 e6 n6 Q26e h2,n6 + f 6 n6 Q26 f h2,n6 e7 n7 Q27e h2,n7 + f 6 n7 Q27 f h2,n7, C4a Q23c h2,h3 = 2 τ 2 µ2 µ 3 csch β3 h3 τ 3 Fss β3 h3, β2 h2, 1, 1, 1, τ 3, C4b Q23d h2,h3 = 2 µ2 coth β3 τ 2 µ h3 τ 3 Fss β3 h3, β2 h2, 1, 1, 1, τ 3, C4c 3 λ3n3 τ 3 Q23 f h2,n3 = 2 µ2 cos τ 2 µ 3 sh λ n3 τ 3 F shs λ3 n3, β2 h2, 1, 1, 1, τ 3, C4d Q24c h2,h4 = 2 τ 2 µ2 µ 4 csch β4 h4 τ 4 Fss β4 h4, β2 h2, 5, 1, 5, τ 4, C4e

31 Math. Comput. ppl. 2017, 22, of 39 Q24d h2,h4 = 2 µ2 coth β4 τ 2 µ h4 τ 4 Fss β4 h4, β2 h2, 5, 1, 5, τ 4, C4f 4 λ4n4 τ 4 Q24e h2,n4 = 2 µ2 cos τ 2 µ 4 sh λ4 n4 τ 4 F shs λ4 n4, β2 h2, 6, 1, 5, τ 4, C4g Q25c h2,h5 = 2 τ 2 µ2 µ 5 csch β5 h5 τ 5 Fss β5 h5, β2 h2, 3, 1, 3, τ 5, C4h Q25d h2,h5 = 2 µ2 coth β5 τ 2 µ h5 τ 5 Fss β5 h5, β2 h2, 3, 1, 3, τ 5, C4i 5 λ5n5 τ 5 Q25e h2,n5 = 2 µ2 cos τ 2 µ 5 sh λ5 n5 τ 5 F shs λ5 n5, β2 h2, 4, 1, 3, τ 5, λ5n5 τ 5 Q25 f h2,n5 = 2 µ2 τ 2 Q26c h2,h6 = 2 µ2 τ 2 µ 6 Q26d h2,h6 = 2 τ 2 µ2 µ 6 Q26e h2,n6 = 2 τ 2 µ2 Q26 f h2,n6 = 2 µ2 τ 2 Q27c h2,h7 = 2 µ2 τ 2 µ 7 Q27d h2,h7 = 2 τ 2 µ2 µ 7 Q27e h2,n7 = 2 τ 2 µ2 Q27 f h2,n7 = 2 τ 2 µ2 cos µ 5 sh λ5 n5 τ 5 F shs λ5 n5, β2 h2, 3, 1, 3, τ 5, F s β2 h2, 1, 2, τ 6 for h6 = 0 csch β6 h6 τ 6 Fcs β6 h6, β2 h2, 2, 1, 2, τ 6 for h6 = 0 F s β2 h2, 1, 2, τ 6 for h6 = 0 coth β6 h6 τ 6 Fcs β6 h6, β2 h2, 2, 1, 2, τ 6 for h6 = 0 cos λ6n6 τ 6 µ 6 sh λ6 n6 τ 6 F chs λ6 n6, β2 h2, 2, 1, 2, τ 6, cos λ6n6 τ 6 µ 6 sh λ6 n6 τ 6 F chs λ6 n6, β2 h2, 3, 1, 2, τ 6, F s β2 h2, 1, 4, τ 7 for h7 = 0 csch β7 h7 τ 7 Fcs β7 h7, β2 h2, 4, 1, 4, τ 7 for h7 = 0 F s β2 h2, 1, 4, τ 7 for h7 = 0 coth β7 h7 τ 7 Fcs β7 h7, β2 h2, 4, 1, 4, τ 7 for h7 = 0 cos λ7n7 τ 7 µ 7 sh λ7 n7 τ 7 F chs λ7 n7, β2 h2, 4, 1, 4, τ 7, cos λ7n7 τ 7 µ 7 sh λ7 n7 τ 7 F chs λ7 n7, β2 h2, 5, 1, 4, τ 7, C4j C4k C4l C4m C4n C4o C4p C4q C4r C4s ES26 h2 = 2 τ 2 µ2 µ 6 B P6 =3 F s β2 h2, 1, 2, τ 6, C4t ES27 h2 = 2 τ 2 µ2 µ 7 B P7 =3 F s β2 h2, 1, 4, τ 7. C4u ppendi C.4. Epression of c3h3, d3 h3 and f 3 n3 for Region 3 B using 4, the development of 15a gives c3h3 = d1h1 Q31d h3,h1, h1=1 Q31d h3,h1 = 2 τ 3 β3 h3 β1 h1 th β1 h1 τ 1 Fss β1 h1, β3 h3, 1, 1, 1, τ 3. C5a C5b

32 Math. Comput. ppl. 2017, 22, of 39 B using 8, the development of 15b gives d3h3 = c2h2 Q32c h3,h2, h2=1 C6a Q32c h3,h2 = 2 τ 3 β3 h3 β2 h2 th β2 h2 τ 2 Fss β2 h2, β3 h3, 1, 1, 1, τ 3. B using 27, the development of 16 gives Q36c n3,h6 = 2 τ 3 λ3 n3 Q36d n3,h6 = 2 τ 3 λ3 n3 f 3 n3 = ES36 n3 c6 h6 Q36c n3,h6 + d6h6 Q36d n3,h6 h6=0 + n6=1 C6b, C7a e6 n6 Q36e n3,n6 + f 6 n6 Q36 f n3,n6 3 F s λ3n3, 2, 2, τ 3 Fls λ3n3, 2, 2, τ 3 1 β6 csch β6 h6 h6 τ 6 Fshs β6h6, λ3 n3, 3, 2, 2, τ 3 2 F s λ3n3, 2, 2, τ 3 Fls λ3n3, 2, 2, τ 3 1 β6 csch β6 h6 h6 τ 6 Fshs β6h6, λ3 n3, 2, 2, 2, τ 3 for h6 = 0 for h6 = 0 Q36e n3,n6 = 2 τ 3 λ3 n3 λ6 n6 csch λ6 n6 τ 6 F ss λ6n6, λ3 n3, 2, 2, 2, τ 3, for h6 = 0 for h6 = 0 Q36 f n3,n6 = 2 τ 3 λ3 n3 λ6 n6 coth λ6 n6 τ 6 F ss λ6n6, λ3 n3, 2, 2, 2, τ 3, C7b C7c C7d C7e ES36 n3 = µ 6 J z6 λ3 n3 τ 3 F l2s λ3n3, 2, 2, τ 3. C7f ppendi C.5. Epression of c4h4, d4 h4 and e4 n4 for Region 4 B using 4, the development of 20a gives c4h4 = d1h1 Q41d h4,h1, h1=1 Q41d h4,h1 = 2 τ 4 β4 h4 β1 h1 th β1 h1 τ 1 Fss β1 h1, β4 h4, 1, 5, 5, τ 4. B using 8, the development of 20b gives d4h4 = c2h2 Q42c h4,h2, h2=1 C8a C8b C9a Q42c h4,h2 = 2 τ 4 β4 h4 β2 h2 th β2 h2 τ 2 Fss β2 h2, β4 h4, 1, 5, 5, τ 4. B using 32, the development of 21 gives Q47c n4,h7 = 2 τ 4 λ4 n4 e4 n4 = ES47 n4 c7 h7 Q47c n4,h7 + d7h7 Q47d n4,h7 h7=0 + n7=1 C9b, C10a e7 n7 Q47e n4,n7 + f 7 n7 Q47 f n4,n7 3 F s λ4n4, 2, 2, τ 4 Fls λ4n4, 2, 2, τ 4 1 β cosβ7 h7τ 7 F h7 shβ7 h7τ 7 shs β7h7, λ4 n4, 3, 2, 2, τ 4 for h7 = 0 for h7 = 0 C10b

33 Math. Comput. ppl. 2017, 22, of 39 Q47d n4,h7 = 2 τ 4 λ4 n4 2 F s λ4n4, 2, 2, τ 4 Fls λ4n4, 2, 2, τ 4 1 β cosβ7 h7τ 7 F h7 shβ7 h7τ 7 shs β7h7, λ4 n4, 2, 2, 2, τ 4 for h7 = 0 for h7 = 0 Q47e n4,n7 = 2 τ 4 λ4 n4 λ7 n7 coth λ7 n7 τ 7 F ss λ7n7, λ4 n4, 2, 2, 2, τ 4, Q47 f n4,n7 = 2 τ 4 λ4 n4 λ7 n7 csch λ7 n7 τ 7 F ss λ7n7, λ4 n4, 2, 2, 2, τ 4, C10c C10d C10e ES47 n4 = µ 7 J z7 λ4 n4 τ 4 F l2s λ4n4, 2, 2, τ 4. C10f ppendi C.6. Epression of c5h5, d5 h5, e5 n5 and f 5 n5 for Region 5 B using 4, the development of 25a gives c5h5 = d1h1 Q51d h5,h1, h1=1 Q51d h5,h1 = 2 τ 5 β5 h5 β1 h1 th β1 h1 τ 1 Fss β1 h1, β5 h5, 1, 3, 3, τ 5. B using 8, the development of 25b gives d5h5 = c2h2 Q52c h5,h2, h2=1 C11a C11b C12a Q52c h5,h2 = 2 τ 5 β5 h5 β2 h2 th β2 h2 τ 2 Fss β2 h2, β5 h5, 1, 3, 3, τ 5. B using 27, the development of 26a gives Q56c n5,h6 = 2 τ 5 λ5 n5 e5 n5 = ES56 n5 Q56d n5,h6 = 2 τ 5 λ5 n5 c6 h6 Q56c n5,h6 + d6h6 Q56d n5,h6 h6=0 + n6=1 C12b, C13a e6 n6 Q56e n5,n6 + f 6 n6 Q56 f n5,n6 3 F s λ5n5, 2, 2, τ 5 Fls λ5n5, 2, 2, τ 5 1 β6 cosβ6 h6τ 6 F h6 shβ6 h6τ 6 shs β6h6, λ5 n5, 3, 2, 2, τ 5 2 F s λ5n5, 2, 2, τ 5 Fls λ5n5, 2, 2, τ 5 β6 1 cosβ6 h6τ 6 F h6 shβ6 h6τ 6 shs β6h6, λ5 n5, 2, 2, 2, τ 5 for h6 = 0 for h6 = 0 for h6 = 0 for h6 = 0 Q56e n5,n6 = 2 τ 5 λ5 n5 λ6 n6 coth λ6 n6 τ 6 F ss λ6n6, λ5 n5, 2, 2, 2, τ 5, Q56 f n5,n6 = 2 τ 5 λ5 n5 λ6 n6 csch λ6 n6 τ 6 F ss λ6n6, λ5 n5, 2, 2, 2, τ 5, C13b C13c C13d C13e ES56 n5 = µ 6 J z6 λ5 n5 τ 5 F l2s λ5n5, 2, 2, τ 5. C13f B using 32, the development of 26b gives f 5 n5 = ES57 n5 c7 h7 Q57c n5,h7 + d7h7 Q57d n5,h7 h7=0 + n7=1, C14a e7 n7 Q57e n5,n7 + f 7 n7 Q57 f n5,n7

34 Math. Comput. ppl. 2017, 22, of 39 Q57c n5,h7 = τ 2 3 F s λ5n5, 2, 2, τ 5 Fls λ5n5, 2, 2, τ 5 5 λ5 n5 1 β7 csch β7 h7 h7 τ 7 Fshs β7h7, λ5 n5, 3, 2, 2, τ 5 Q57d n5,h7 = τ 2 2 F s λ5n5, 2, 2, τ 5 Fls λ5n5, 2, 2, τ 5 5 λ5 n5 β7 1 csch β7 h7 h7 τ 7 Fshs β7h7, λ5 n5, 2, 2, 2, τ 5 for h7 = 0 for h7 = 0 for h7 = 0 for h7 = 0 Q57e n5,n7 = 2 τ 5 λ5 n5 λ7 n7 csch λ7 n7 τ 7 F ss λ7n7, λ5 n5, 2, 2, 2, τ 5, Q57 f n5,n7 = 2 τ 5 λ5 n5 λ7 n7 coth λ7 n7 τ 7 F ss λ7n7, λ5 n5, 2, 2, 2, τ 5, C14b C14c C14d C14e ES57 n5 = µ 7 J z7 λ5 n5 τ 5 F l2s λ5n5, 2, 2, τ 5. C14f ppendi C.7. Epression of c60, d6 0, c6 h6, d6 h6, e6 n6 and f 6 n6 for Region 6 B using 4 and 27d, the development of 30a and 30b gives c6 h6 = ES61 h6 h1=1 d1 h1 Q61d h6,h1, Q61d h6,h1 = τ β1 th 1 β1 h1 h1 τ 1 τ6 F s β1 h1, 1, 2, τ 6 for h6 = 0 2 β6 h6 F cs β6 h6, β1 h1, 2, 1, 2, τ 6 for h6 = 0 1 ES61 h6 = τ6 zp6 =2 for h6 = 0 0 for h6 = 0 C15a C15b C15c B using 8 and 27d, the development of 30c and 30d gives Q62c h6,h2 = 1 τ 6 1 β2 h2 th β2 h2 τ 2 d6 h6 = ES62 h6 ES62 h6 = h2=1 c2 h2 Q62c h6,h2, 1 τ6 F s β2 h2, 1, 2, τ 6 for h6 = 0 2 β6 h6 F cs β6 h6, β2 h2, 2, 1, 2, τ 6 for h6 = 0 1 τ6 zp6 =3 for h6 = 0 0 for h6 = 0 C16a C16b C16c B using 24, the development of 31a gives e6 n6 = c5 h5 Q65c n6,h5 + d5h5 Q65d n6,h5 h5=1 + n5=1, C17a e5 n5 Q65e n6,n5 + f 5 n5 Q65 f n6,n5 Q65c n6,h5 = 2 τ 6 µ6 µ 5 csch β5 h5 τ 5 Fshs β5h5, λ6 n6, 3, 2, 2, τ 6, C17b Q65d n6,h5 = 2 τ 6 µ6 µ 5 csch β5 h5 τ 5 Fshs β5h5, λ6 n6, 2, 2, 2, τ 6, C17c Q65e n6,n5 = 2 τ 6 µ6 µ 5 coth λ5 n5 τ 5 F ss λ5n5, λ6 n6, 2, 2, 2, τ 6, C17d Q65 f n6,n5 = 2 τ 6 µ6 µ 5 csch λ5 n5 τ 5 F ss λ5n5, λ6 n6, 2, 2, 2, τ 6, C17e

35 Math. Comput. ppl. 2017, 22, of 39 B using 14, the development of 31b gives f 6 n6 = c3 h3 Q63c n6,h3 + d3h3 Q63d n6,h3 h3=1 + f 3 n3 Q63 f n6,n3 n3=1, C18a Q63c n6,h3 = 2 τ 6 µ6 µ 3 cos β3 h3 τ 3 sh β3 h3 τ 3 F shs β3h3, λ6 n6, 3, 2, 2, τ 6, Q63d n6,h3 = 2 τ 6 µ6 µ 3 cos β3 h3 τ 3 sh β3 h3 τ 3 F shs β3h3, λ6 n6, 2, 2, 2, τ 6, C18b C18c Q63 f n6,n3 = 2 τ 6 µ6 µ 3 coth λ3 n3 τ 3 F ss λ3n3, λ6 n6, 2, 2, 2, τ 6. C18d ppendi C.8. Epression of c70, d7 0, c7 h7, d7 h7, e7 n7 and f 7 n7 for Region 7 B using 4 and 32d, the development of 35a and 35b gives c7 h7 = ES71 h7 h1=1 d1 h1 Q71d h7,h1, Q71d h7,h1 = τ β1 th 1 β1 h1 h1 τ 1 τ7 F s β1 h1, 1, 4, τ 7 for h7 = 0 2 β7 h7 F cs β7 h7, β1 h1, 4, 1, 4, τ 7 for h7 = 0 1 ES71 h7 = τ7 zp7 =2 for h7 = 0 0 for h7 = 0 C19a C19b C19c B using 8 and 32d, the development of 35c and 35d gives Q72c h7,h2 = 1 τ 7 1 β2 h2 th β2 h2 τ 2 d7 h7 = ES72 h7 ES72 h7 = h2=1 c2 h2 Q72c h7,h2, 1 τ7 F s β2 h2, 1, 4, τ 7 for h7 = 0 2 β7 h7 F cs β7 h7, β2 h2, 4, 1, 4, τ 7 for h7 = 0 1 τ7 zp7 =3 for h7 = 0 0 for h7 = 0 C20a C20b C20c B using 24, the development of 36a gives f 7 n7 = c5 h5 Q75c n7,h5 + d5h5 Q75d n7,h5 h5=1 + n5=1, C21a e5 n5 Q75e n7,n5 + f 5 n5 Q75 f n7,n5 Q75c n7,h5 = 2 τ 7 µ7 µ 5 cos β5 h5 τ 5 sh β5 h5 τ 5 F shs β5h5, λ7 n7, 3, 2, 2, τ 7, Q75d n7,h5 = 2 τ 7 µ7 µ 5 cos β5 h5 τ 5 sh β5 h5 τ 5 F shs β5h5, λ7 n7, 2, 2, 2, τ 7, C21b C21c Q75e n7,n5 = 2 τ 7 µ7 µ 5 csch λ5 n5 τ 5 F ss λ5n5, λ7 n7, 2, 2, 2, τ 7, C21d

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