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A review of subdomain modeling techniques in

electrical machines: performances and applications

Emile Devillers, Jean Le Besnerais, Thierry Lubin, Michel Hecquet, Jean Philippe Lecointe.

Abstract—This paper reviews the recent developments of

semi-analytical subdomains modeling techniques to compute

the ﬂux density distribution in electrical machines by the exact

solving of Maxwell equations.

It is shown that with an appropriate development method-

ology and numerical implementation, these harmonic mod-

els break the traditional compromise between accuracy and

computation time that must be done using ﬁnite element

or other analytical methods. Besides that, subdomains model

development techniques have been improved to overcome its

topological limitations This fact is demonstrated on three

different subdomains models in comparison with ﬁnite element

methods in terms of accuracy and processing time. The ﬁrst

one is a subdomains model of a surface permanent-magnet

synchronous machine, the second one is for an inset permanent-

magnet synchronous machine, and the third one is for a squirrel-

cage induction machine. Thanks to an efﬁcient implementation

method, a very low computation time is obtained. The robust-

ness of the subdomains on the geometrical assumptions is also

demonstrated.

Index Terms—Magnetic ﬁeld, Electric machines, Analytical

model, Harmonic analysis, Performance analysis, Analytical

model, Reviews.

I. INT ROD UC TI ON

It is often necessary to estimate the rated power, ef-

ﬁciency, magnetic losses and even magnetic vibrations of

electrical machines to optimize their design. This estimation

relies on an accurate computation of the machine characteris-

tics, such as electromagnetic torque, magnetic losses and air

gap Maxwell forces. All these quantities can be computed

if the magnetic ﬁeld inside the machine is fully determined,

meaning the space and time distribution for both radial and

tangential components.

For this purpose, different methods have been developed

and can be grouped in four main categories: analytical, semi-

analytical, numerical, and hybrid methods which result from

the combination of the three ﬁrst. Numerical methods are

very ﬂexible to various geometries, include non-linear and

non-homogeneous materials, and enable coupling with other

physics. Yet, this high level of complexity induces very time-

consuming simulations which slows down the design process.

Simpliﬁed analytical models are consequently used for the

ﬁrst design steps as they are very fast and may give more

physical insights, while FEM is more interesting for ﬁnal

validations.

Corresponding author: E.Devillers (email: emile.devillers@phd.ec-lille.fr)

A semi-analytical method named ”Subdomains Model”

(SDM) has been recently developed to compute the magnetic

ﬁeld with great accuracy and much faster than numerical

methods such as Finite Elements Method (FEM), represent-

ing an interesting alternative between analytical and numer-

ical methods.

The ﬁrst part of this paper presents the principles of the

subdomain modeling technique and reviews the works related

to SDM development and applications in electrical machines,

showing how these works have enlarged the application range

of SDMs. Then, a more detailed comparison with FEM is

done to demonstrate the performance of the SDMs despite

their modeling assumptions, especially in terms of accuracy,

computing speed and robustness to geometry. Finally, some

future applications of subdomain models are discussed.

II. STATE OF THE ART

A. Principles

SDM is a semi-analytical method that consists in dividing

the problem into physical regions named subdomains in

which Maxwell governing equations can be solved analyti-

cally. The main processing steps to obtain the magnetic ﬁeld

in each subdomain are presented in Fig. 1. For this purpose,

the subdomains must fulﬁll speciﬁc conditions on geometry

physics.

Fig. 1. Methodology graph of Subdomains Models in 7steps.

In Fig. 2, the problem is composed of one air gap sub-

domain, stator slot subdomains and rotor slots subdomains,

and limited by rotor and stator iron cores.

Fig. 2. Division in polar subdomains for an induction machine.

Then, Maxwell equations are written in each subdomain

for scalar potential or vector potential, from which the

ﬂux density is derived in step (1) in Fig. 1. The vector

potential formulation can always be used whereas the scalar

potential formulation may be used only if the magnetic ﬁeld

is irrotational, meaning there is no current density. Maxwell

equations are ﬁnally solved using the Fourier method consist-

ing in the separation of variables method (2). This analytical

resolution gives in each subdomain the potential in Fourier se-

ries in function of several unknown constants. The analytical

potential solutions may be then reformulated (3) in equivalent

expressions for readability and numerical optimization.

By expressing the boundary conditions of potential and

magnetic ﬁeld continuity at each interface between two sub-

domains, a linear system of analytical independent equations

linking all the unknown constants (4) can be obtained. If

the potential and the magnetic ﬁeld of both subdomains are

expressed in a different Fourier basis, it is necessary to project

one Fourier basis on the other.

Then, assuming a ﬁnite number of Fourier harmonics in

each subdomain, the linear system of equations is put into

matrix form and solved numerically (5).

M∗X=S(1)

The matrix Mis often called ”topological matrix”, and S

the ”source vector”. Solving the linear system gives the nu-

merical value of the unknown constants X, which enables to

compute the magnetic ﬁeld’s spatial and temporal distribution

in each subdomain (6). Because both analytical and numerical

resolutions are successively accomplished, SDM may be

classiﬁed as ”semi-analytical” model. A similar methodology

is also proposed in [1].

It is important to mention that the formulation of the

analytical solution (3) may strongly differ from an author

to another. In this review, two main formulations are distin-

guished : the formulation Ain [2], [3] and the formulation

Bin [4], [5]. Formulation Bpresents the interest of giving

dimensionless expressions and a topological matrix Mwith

only 1 on the diagonal and more 0 elsewhere. Speciﬁc

algorithms are dedicated to optimize the inversion of such

matrix.

Hence the choice of formulation is a real matter to

design a SDM as it may compromise the numerical resolution

during (5) if the topological matrix obtained after (4) is ill-

conditioned. Such numerical problems are frequently pointed

out in the SDM literacy, though only few articles such as [6]

analyze the analytical formulation as regards on the numerical

performances.

B. Development history

The main difﬁculties encountered by any analytical mod-

els are how to take into account slotting effect, as air

gap length variations strongly inﬂuence the magnitude and

shape of the magnetic ﬁeld. The ﬁrst methods based on the

formal resolution of Maxwell equations were developed in

the 1980’s for both slotless Permanent Magnet Synchronous

Machine (PMSM) [7] and Induction Machines (IM) [8], and

have been improved by several approaches to better account

for slotting effect and radial and tangential air gap ﬂux

components .

In 1984, [7] used Carter’s coefﬁcients to transform a

slotted stator into an equivalent slotless one. In 1993, [9]

introduced a relative permeance which modulates the radial

air gap ﬂux density previously computed without slotting

effect. Another permeance model was developed in 1997

by [10]. The relative permeance method was extended by

[11] in 1998 to take into account both radial and tangential

components. In 2003, [12] used conformal transformation

and more speciﬁcally Schwarz-Christoffel mapping to model

slotting effect. This method was also adopted by [13] in 2006,

which applied the conformal transformation to the relative

permeance model and deduced a complex permeance model,

giving better accuracy for both components. One can refer

to [6], [14], [15], [16] for their exhaustive history in PMSM

analytical modeling, and to [17] for IM modeling.

The ﬁrst SDM for SPMSM were developed in 2008-

2009 by [14], [18] and [19], although the Fourier projection

between subdomains was already used by [20] and [21]

a few decades ago. Also in 2008, [22] used the same

method for a linear actuator. The term ”Subdomains model”

appeared in 2010 in [15], [23] and was adopted by then

in several contemporary major publications. This method

is also referred as ”exact analytical model” [4], ”semi-

analytical harmonic model” [24] or ”Fourier-based Model”

[25]. Compared to the previous analytical models, SDMs

provide both components of the magnetic ﬁeld by exactly

taking into account slotting effect and the inﬂuence between

slots. In 2010, [23] developed an elementary model to give

a better understanding of slotting effect in SDMs.

C. Available topologies

1) Introduction: Due to the PMSM popularity in the past

decades, most of SDMs deal with them, at the detriment of

IM. This can also be explained because of more complex

physics. Besides, SDMs for other types of machines have

been developed. A complete review on SDMs done until 2014

is presented in [26]. Some examples of existing topologies

are illustrated in Fig. 3.

2) Geometry aspects: The geometry is usually in two-

dimensional (2-D), but have already been extended to 3-D

such as in [27], [28]. In 2-D, axial and radial 3-D end-effects

are neglected.

The problem is either expressed in polar or Cartesian

coordinates. In case of polar coordinates, every subdomain

geometry is approximated by a polar geometry. For example,

the rectangular teeth are supposed to have radial edges with

orthoradial tooth tips, as it is illustrated in [2]. In case of

Cartesian coordinates, the air gap is unrolled by consid-

ering an inﬁnite radius of curvature, giving an equivalent

rectangular topology such as in [29]. The different topology

approximations according to the chosen coordinate systems

are gathered in the aforementioned methodology [6].

Besides, SDMs can be applied to internal or external rotor

[15], [30]. Moreover, semi-closed slots may be used for a

more realistic model [4].

3) Physics aspects: As said previously, the physics is also

approximated. The iron is considered to have inﬁnite relative

permeability, resulting in homogeneous boundary conditions

at the interfaces between subdomains and the iron. The satu-

ration is consequently neglected. For PM machines, magnets

have an isotropic and homogeneous relative permeability

and a linear B(H) curve. For induction machines (IM), the

rotor bars are assumed to have a homogeneous electrical

conductivity.

SDM can model both magnet and current sources. It

accounts for any magnetization shapes such as radial, parallel

or Hallbach magnetization. Concerning the armature reaction

ﬁelds, the windings are usually designed by a connection

matrix which enables to use (non)-overlapping single/double

layer windings. In fact, the magnetic sources are expanded

into Fourier series and injected in Maxwell equations at step

(2). It results in a linear superposition of stator and rotor

ﬁelds, hence the possibility to solve everything at once or

separately.

4) Synchronous Machines (SM): Several models exist for

each topology of PMSM, depending on the chosen modeling

level. For SPMSM with armature reaction ﬁeld and semi-

closed slots, one can refer to [2], [4], [30]. Inset PMSM

(IPMSM) models with armature reaction ﬁeld and semi-

closed slots can be found in [1], [3], [31].

Besides the above topologies, more singular SM have

been modeled by the subdomain technique. For instance,

SDMs exist for ﬂux switching SM [32], double excitation SM

[33], axial ﬂux SM [29], PMSM with noches [34], pseudo

direct drives SM [35] and Switch Reluctance Motor (SRM)

[36]. The SRM model illustrates the difﬁculties to transform

any geometry into a polar one.

5) Induction Machines (IM): IM SDMs have both com-

mon points and differences with SM ones. Assuming an

internal rotor topology, stator slots subdomains and air gap

subdomain remain the same as for PMSM. Though, the level

complexity is increased because of the the induced current

modeling in the rotor bars, and of the existence of two

asynchronous frequencies as well as the space harmonics.

The former analytical models of IM were designed for

laminated solid rotor, such as in [8], [37], and extended by

the SDM in [38].

The ﬁrst model of Squirrel Cage IM (SCIM) in [39]

accounts for rotor bars with induction and a current sheet

at stator inner bore. A complete SCIM SDM was developed

in [17], [24], using Electrical Equivalent Circuit (EEC) to

drive the feeding current as a function of the slip value.

Fig. 3. Three idealized topologies of existing SDMs. a) SPMSM ; b)

IPMSM ; c) SCIM. The pictures are provided by MANATEE software.

6) Linear or tubular PM machines: SDMS of linear

actuators can be found in [22], [40]. These topologies are

well suited to SDMs because the geometry can be directly

implemented in Cartesian coordinates without simpliﬁcations.

7) Missing machines: To the author’s knowledge, no pa-

per speciﬁcally deals with SDM of Direct-Current Machines

(DCM) but the models used for synchronous machines can

be applied with DC current sources. There is either no Buried

PMSM (BPMSM) SDM, because of the saturation in the

bridges between magnets and the difﬁculty to approximate

interior magnets with polar geometry.

D. Accounting for time parameter

In SDMs, time is differently taken into account according

to the machine type. Its inﬂuence on the topological matrix

Mand the source vector Smust be carefully established.

In fact, sources variations impact on Swhereas reluctance

variations modify M.

Particularly, for SPMSM with internal rotor and magnet

permeability equal to 1, there is no reluctance variation with

rotor rotation. Only the magnetization distribution changes

with the rotation and this results in a source matrix whose

columns are the source vector Sfor each time step. Con-

sequently, a single linear system resolution can give the

constants Xfor every time step.

For IPMSM, both reluctance and magnetization distribu-

tion change over time. In this case, it is the same principle

as for FEM : Mand Shave to be evaluated at each time step

and one linear system resolution may only give one time step.

It is a different method for IM. Every stator quantity,

respectively rotor quantity, is ﬁrst expressed as a phasor of

pulsation ω, respectively slip∗ω. Magnetic potentials are then

solved in terms of complex amplitude and ﬁnally modulated

with their respective pulsation. It implies that one system

resolution gives every time step for one chosen slip value.

E. Current applications

Though it may have been restricted to torque and Back-

ElectroMotive Force (BEMF) at the beginning, SDMs have

been derived for many other applications.

1) Magnetic Forces and Flux linkage computation: Each

local and global electromagnetic quantity may be derived

from the magnetic potential using well-known physics laws.

Electromagnetic forces as well as electromagnetic torque

including torque ripple - such as cogging torque - may be

directly computed from the magnetic ﬁeld using the Maxwell

stress tensor with a great accuracy. The ﬂux linkage is

computed by applying the Stokes theorem to the magnetic

potential of each stator slot. Flux linkage knowledge enables

to deduce the BEMF generated by the rotating ﬁeld.

Thanks to the Fourier series formulation, these previous

temporal integrations are converted in faster and more ac-

curate summations on the magnetic ﬁeld’s harmonic compo-

nents. The limit of any 2-D magnetic models is the fact that

3-D axial end-effects have been neglected so the torque and

the BEMF may be overestimated.

2) Fault simulation: It is possible to simulate unbalanced

magnetic sources distribution by injecting the corresponding

Fourier series at the step (2), such as in [41], and deduce

the Unbalanced Magnetic Force (UMF) - or Pull (UMP).

UMF is also caused by rotor eccentricity. The effect of

rotor eccentricity is modeled in [42] by adding a ﬁrst-order

perturbation component to magnetic potentials, whereas [43]

introduces it with a superposition method. For SCIM, it is

possible to simulate defective bars as in [17] by decreasing

their conductivity.

3) Losses: Eddy-current losses may be computed using

Helmholtz equation in the PM subdomains [5], [44] and in

the windings [45], [46]. In [26], a shielding cylinder is added

at the surface of the PM to reduce the eddy-current losses in

magnets.

4) EEC parameters: The potential solution is used to

compute leakage ﬂux at stator slots [47] and due to end-

windings [48]. A method to compute primary and secondary

impedances for SCIM is developed in [17], [24]. It is also

shown that the estimated EEC parameters enable to compute

and check previous quantities such as torque, back-EMF,

power losses, etc. with another approach.

III. ADVANTAG ES A ND D RAWB ACK S IN C OM PARISON

WI TH FEM

A. Introduction

In most of SDMs papers, the model is validated by

comparing its accuracy with a parallel FEM analysis using

the same approximations. For the same modeling level, SDM

is naturally as accurate as FEM, since the former is an exact

resolution of Maxwell equations. It is actually more exhaus-

tive than FEM because the solution is continuously deﬁned

in each subdomain and not only at the mesh points. Tab. I

shows the qualitative performances criteria in comparison

with FEM.

TABLE I

QUALITATIVE COMPARISON ON PERFORMANCES CRITERIA.

SDM FEM

Geometry Complexity - +

Non homogeneity, non isotropy - +

Saturation - +

Mesh sensitivity + -

Computation time + -

B. Comparison on the model limitations

1) Sensitivity to mesh in the FEM: Signiﬁcant problems

due to the meshing quality exist, as regards on computing

derived magnetic quantities. This has been studied a lot

for the evaluation of vibrations due to magnetic forces and

cogging torque. For example, Fig 2.38 in [49] states a

vibration variation up to 4dB below 10 kHz between different

meshing methods in Flux3D [50]. These problems may be

solved by reﬁning the mesh in the air gap, but it signiﬁcantly

slows down the computation.

2) Robustness to geometry in SDMs: One drawback of

SDM is the geometry simpliﬁcation. However by deﬁning

an equivalent polar geometry as in Fig. 4, the air gap ﬂux

computation is still accurate. In Fig. 5, a comparison has been

done using MANATEE simulation environment [51] on the

SPMSM 6s/4p presented in [4].

Fig. 4. Polar approximation of constant tooth width and curved magnet.

Fig. 5. Magnetic ﬁeld comparison between idealized and real geometries.

3) Saturation in SDMs: Saturation is still a limitation

that prevents from modeling several machine topologies. The

ﬁrst challenge is to compute the magnetic ﬁeld assuming a

ﬁnite iron permeability. This is done for example in [38]

where the magnetic ﬁeld is computed in the rotor iron core. In

[52], a method is proposed to account for linear soft-magnetic

iron with ﬁnite permeability, but there exists no SDMs which

account for non-linear materials to the authors’ knowledge.

However, it is possible to couple SDM with Magnetic

Equivalent Circuit (MEC) [53], [54], [55] or FEM [56] in an

iterative way to account for saturation.

C. Comparison on computing performance of SDMs

1) SDM numerical optimization: An efﬁcient implemen-

tation is necessary to take advantage of SDM performances.

To illustrate this, the SDM recently presented in [17], a SCIM

36s/4p at no-load condition has been developed and opti-

mized. However, the analytical resolution at (3) is done with

reformulation Bto improve numerical performances, and the

numerical implementation uses only vectorized operations.

Besides, the choice of the harmonics number in each sub-

domain at step (4) relies on a compromise between accuracy

and computation time [57]. Fig. 6 shows the variation of

the Mean Squared Error (MSE) between the magnetic ﬁeld

and the reference obtained with 1000 air gap harmonics, the

computation time and the memory used by Min function

of the air gap harmonics number. Harmonics numbers in the

other subdomains are the same as in [17].

Fig. 6 illustrates the fast convergence of the Fourier series,

as 150 harmonics gives a MSE of 0.2% compared with

1000 harmonics, within 0.34 seconds. In [17], the SDM

computation time is 3.24 minutes and very close to the FEM

computation time.

Fig. 6. MSE, computation time and memory used in function of the air

gap harmonics number.

Furthermore, a few SDMs for SPMSM accounts for

periodicity and one slot per pole approximation [15] [44]

to reduce computation time.

The air gap harmonics number can be reduced by a

previous analytical study of the magnetic ﬁeld’s harmonic

content. For instance, in a SPMSM with Zsstator teeth and

2ppoles, the fundamental at no load is at 2pand harmonics

are linear combinations of 2pand Zs. Hence any other

harmonics ranks can be ignored to improve performances.

2) Comparison with FEM: An efﬁcient implementation

of SDMs can lead to much lower computation time than

FEM, as presented in [58]. Tab. II shows a quantitative

comparison on performances between FEM using Femm

[59] and three particular SDMs : the SPMSM 6s/4p without

armature load presented in [4], the IPMSM 15s/4p without

armature load in [31], and the previous SCIM 36s/4p in [17].

For the SDM, the computation time includes building

the linear system, solving it, and reconstructing the air gap

magnetic ﬁeld in the air gap subdomain. For the FEM,

it includes building the topology, meshing it, solving the

problem in the entire topology and extracting the air gap

ﬁeld. Both models are linear and without any symmetry or

periodicity simpliﬁcations.

TABLE II

COM PARI SON O N TH E ELA PSE D SI MUL ATIO N TI ME FO R ON E TIM E STE P.

SPMSM IPMSM SCIM

SDM

Computation time [s] 0.109 0.199 0.7337

Air gap harmonics 150 250 300

Number of unknowns 1572 1909 2298

FEM

Computation time [s] 6.152 10.11 15.27

Number of nodes 45888 64605 73003

Number of elements 91054 128488 145284

When computing temporal quantities such as cogging

torque, FEM computation time values in tab. II are multiplied

by the number of time steps. As said previously, it is not

the case for an SPMSM model whose computation time

is very less sensitive to the number of time steps. For

instance, considering the same topology of SPMSM 6s/4p,

the computation time for the SDM is 0.139 seconds whereas

it takes around 347 seconds for the FEM because of the 50

time steps, meaning 50 meshings and resolutions.

SDMs are implemented in the Matlab scientiﬁc environ-

ment -version R2014b. The computer is equipped with a CPU

Intel Xeon E5 1620 v2@3.70GHz and 24GB RAM DDR3

@797 MHz. All results in this paper have been repeated 10

times and then averaged for robustness purpose.

IV. CONCLUSION

The paper presents a review of the semi-analytical subdo-

mains models applied to electrical machines. The methodol-

ogy is ﬁrst explained step by step, including both analytical

and numerical aspects such as model assumptions and the

choice of formulation. Then, a state of the art presents the

main existing models and their numerous applications. Ad-

vantages and drawbacks of the method are ﬁnally illustrated

in comparison with FEM for three machines topologies.

The results validate the model assumptions and show great

computation time performances especially when accounting

for rotor rotation.

Furthermore, an efﬁcient implementation of subdomains

models strongly reduces the computation time and allows to

increase the electromagnetic model complexity. This can be

done to include:

•the effect of magnet shaping using harmonic superposi-

tion

•the effect of winding space harmonics in induction

machines using ﬁeld superposition

•3-D effects such as fringing ﬂux and skewing

•strong coupling with electrical circuit (calculation of

equivalent circuit parameters iteratively)

•integration in complex multiphysics models, especially

for vibroacoustic analysis [60].

REF ER EN CE S

[1] S. Teymoori, A. Rahideh, H. Moayed Jahromi, and M. Mar-

daneh, “Two-Dimensional Analytical Magnetic Field Prediction for

Consequent-Pole Permanent Magnet Synchronous Machines,” IEEE

Trans. Mag., vol. 9464, no. c, pp. 1–1, 2016.

[2] L. J. Wu, Z. Q. Zhu, D. Staton, M. Popescu, and D. Hawkins, “An

Improved Subdomain Model for Predicting Magnetic Field of Surface-

Mounted Permanent Magnet Machines Accounting for Tooth-Tips,”

IEEE Trans. Mag., vol. 47, no. 6, pp. 1693–1704, 2011.

[3] A. Rahideh and T. Korakianitis, “Analytical magnetic ﬁeld calculation

of slotted brushless permanent-magnet machines with surface inset

magnets,” IEEE Trans. Mag., vol. 48, no. 10, pp. 2633–2649, 2012.

[4] T. Lubin, S. Mezani, and A. Rezzoug, “2-D exact analytical model

for surface-mounted permanent-magnet motors with semi-closed slots,”

IEEE Trans. Mag., vol. 47, no. 2 PART 2, pp. 479–492, 2011.

[5] F. Dubas and A. Rahideh, “Two-dimensional analytical permanent-

magnet eddy-current loss calculations in slotless PMSM equipped with

surface-inset magnets,” IEEE Trans. Mag., vol. 50, no. c, pp. 54–73,

2014.

[6] B. L. J. Gysen, K. J. Meessen, J. J. H. Paulides, and E. a. Lomonova,

“General Formulation of the Electromagnetic Field Distribution in

Machines and Devices Using Fourier Analysis,” Magnetics, IEEE

Transactions on, vol. 46, no. 1, pp. 39–52, 2010.

[7] N. Boules, “Two-dimensional ﬁeld analysis of cylindrical machines

with permanent magnet excitation,” Industry Applications, IEEE Trans-

actions on, vol. IA-20, no. 5, pp. 1267–1277, 1984.

[8] E. Bolte and K. Oberretl, “Three-dimensional analysis of linear motor

with solid iron secondary,” International Conference on Electrical

Machines (ICEM), pp. 68–75, 1980.

[9] Z. Zhu and D. Howe, “Instantaneous magnetic ﬁeld distribution in

brushless permanent\nmagnet DC motors. III. Effect of stator slotting,”

IEEE Trans. Mag., vol. 29, no. 1, 1993.

[10] J. F. Brudny, “Modelisation de la denture des machines asynchrones :

phenomenes de resonances,” Journal of Physics III, vol. 37, no. 7, pp.

1009–1023, 1997.

[11] U. Kim and D. K. Lieu, “Magnetic ﬁeld calculation in permanent mag-

net motors with rotor eccentricity: Without slotting effect considered,”

IEEE Trans. Mag., vol. 34, no. 4, pp. 2243–2252, 1998.

[12] X. Wang, Q. Li, S. Wang, and Q. Li, “Analytical calculation of

air-gap magnetic ﬁeld distribution and instantaneous characteristics

of brushless DC motors,” IEEE Transactions on Energy Conversion,

vol. 18, no. 3, pp. 424–432, 2003.

[13] D. ˇ

Zarko, D. Ban, and T. A. Lipo, “Analytical calculation of magnetic

ﬁeld distribution in the slotted air gap of a surface permanent-magnet

motor using complex relative air-gap permeance,” IEEE Trans. Mag.,

vol. 42, no. 7, pp. 1828–1837, 2006.

[14] F. Dubas and C. Espanet, “Analytical solution of the magnetic ﬁeld

in permanent-magnet motor taking into account slotting effect,” IEEE

Trans. Mag., vol. 45, no. 5, pp. 2097–2109, 2009.

[15] Z. Q. Zhu, L. J. Wu, and Z. P. Xia, “An Accurate Subdomain

Model forMagnetic Field Computation in Slotted Surface-Mounted

Permanent-Magnet Machines,” IEEE Trans. Mag., vol. 46, no. 4, pp.

1100–1115, 2010.

[16] P. D. Pﬁster and Y. Perriard, “Slotless permanent-magnet machines:

General analytical magnetic ﬁeld calculation,” IEEE Trans. Mag.,

vol. 47, no. 6, pp. 1739–1752, 2011.

[17] K. Boughrara, N. Takorabet, R. Ibtiouen, O. Touhami, and F. Dubas,

“Analytical Analysis of Cage Rotor Induction Motors in Healthy,

Defective and Broken Bars Conditions,” IEEE Trans. Magn., vol. 02,

no. c, pp. 1–1, 2014.

[18] Z. J. Liu and J. T. Li, “Accurate Prediction of Magnetic Field and

Magnetic Forces in Permanent Magnet Motors Using an Analytical

Solution,” IEEE Transactions on Energy Conversion, vol. 23, no. 3,

pp. 717–726, 2008.

[19] A. Bellara, Y. Amara, G. Barakat, and B. Dakyo, “Two-Dimensional

exact analytical solution of armature reaction ﬁeld in slotted surface

mounted PM radial ﬂux synchronous machines,” IEEE Trans. Mag.,

vol. 45, no. 10, pp. 4534–4538, 2009.

[20] B. Ackermann and R. Sottek, “Analytical modeling of the cogging

torque in permanent magnet motors,” Electrical Engineering, vol. 78,

no. 2, pp. 117–125, 1995.

[21] E. Bolte, “Analytical calculation of the two-dimensional ﬁeld in the air

gap and the slots of electrical machines,” IEEE Trans. Mag., vol. 20,

no. 5, pp. 1783–1785, 1984.

[22] B. L. J. Gysen, K. J. Meessen, J. J. H. Paulides, and E. a. Lomonova,

“Semi-analytical calculation of the armature reaction in slotted tubular

permanent magnet actuators,” IEEE Trans. Mag., vol. 44, no. 11 PART

2, pp. 3213–3216, 2008.

[23] T. Lubin, S. Mezani, and A. Rezzoug, “Exact Analytical Method for

Magnetic Field Computation in the Air Gap of Cylindrical Electrical

Machines Considering Slotting Effects,” IEEE Trans. Mag., vol. 46,

no. 4, pp. 1092–1099, 2010.

[24] R. L. J. Sprangers, J. J. H. Paulides, B. L. J. Gysen, E. a. Lomonova,

and J. Waarma, “Electric circuit coupling of a slotted semi-analytical

model for induction motors based on harmonic modeling,” 2014 IEEE

Energy Conversion Congress and Exposition (ECCE), pp. 1301–1308,

2014.

[25] B. Hannon, P. Sergeant, and L. Dupre, “2-D Analytical Subdomain

Model of a Slotted PMSM With Shielding Cylinder,” IEEE Trans.

Mag., vol. 50, no. 7, pp. 1–10, 2014.

[26] P.-D. Pﬁster, X. Yin, and Y. Fang, “Slotted Permanent-Magnet Ma-

chines: General Analytical Model of Magnetic Fields, Torque, Eddy

Currents and Permanent Magnet Power Losses including the Diffusion

Effect,” IEEE Trans. Mag., vol. 9464, no. c, pp. 1–1, 2015.

[27] O. De La Barri`

ere, S. Hlioui, H. Ben Ahmed, M. Gabsi, and M. LoBue,

“3-D formal resolution of maxwell equations for the computation of the

no-load ﬂux in an axial ﬂux permanent-magnet synchronous machine,”

IEEE Trans. Mag., vol. 48, no. 1, pp. 128–136, 2012.

[28] K. J. Meessen, B. L. J. Gysen, J. J. H. Paulides, and E. a. Lomonova,

“General formulation of fringing ﬁelds in 3-D cylindrical structures

using fourier analysis,” IEEE Trans. Mag., vol. 48, no. 8, pp. 2307–

2323, 2012.

[29] H. Tiegna, A. Bellara, Y. Amara, and G. Barakat, “Analytical Modeling

of the Open-Circuit Magnetic Field in Axial Flux Permanent-Magnet

Machines With Semi-Closed Slots,” IEEE Trans. Mag., vol. 48, no. 3,

pp. 1212–1226, 2012.

[30] H. Moayed Jahromi, A. Rahideh, and M. Mardaneh, “2-D Analytical

Model for External Rotor Brushless PM Machines,” IEEE Trans.

Energy Conv., pp. 1–10, 2016.

[31] T. Lubin, S. Mezani, and A. Rezzoug, “Two-Dimensional Analytical

Calculation of Magnetic Field and Electromagnetic Torque for Surface-

Inset Permanent-Magnet Motors,” IEEE Trans. Mag., vol. 48, no. 6,

pp. 2080–2091, 2012.

[32] K. Boughrara, T. Lubin, and R. Ibtiouen, “General Subdomain Model

for Predicting Magnetic Field in Internal and External Rotor Mul-

tiphase Flux-Switching Machines Topologies,” IEEE Trans. Mag.,

vol. 49, no. 10, pp. 5310–5325, 2013.

[33] H. Bali, Y. Amara, G. Barakat, R. Ibtiouen, and M. Gabsi, “Analytical

Modeling of Open Circuit Magnetic Field in Wound Field and Series

Double Excitation Synchronous Machines,” IEEE Trans. Mag., vol. 46,

no. 10, pp. 3802–3815, oct 2010.

[34] Y. Oner, Z. Zhu, L. Wu, X. Ge, H. Zhan, and J. Chen, “Analytical

On-Load Sub-domain Field Model of Permanent Magnet Vernier

Machines,” IEEE Transactions on Industrial Electronics, vol. 34220,

no. c, pp. 1–1, 2016.

[35] A. Penzkofer and K. Atallah, “Analytical Modeling and Optimization

of Pseudo-Direct Drive Permanent Magnet Machines for Large Wind

Turbines,” IEEE Trans. Mag., vol. 51, no. 12, pp. 1–14, 2015.

[36] R. L. J. Sprangers, J. J. H. Paulides, B. L. J. Gysen, J. Waarma,

and E. A. Lomonova, “Semi-Analytical Framework for Synchronous

Reluctance Motor Analysis including Finite Soft-Magnetic Material

Permeability.” IEEE Trans. Mag., vol. 51, no. 11, p. 3, 2014.

[37] D. Gerling and G. Dajaku, “Three-dimensional analytical calculation

of induction machines with multilayer rotor structure in cylindrical

coordinates,” Electrical Engineering, vol. 86, no. 4, pp. 199–211, 2003.

[38] K. Boughrara, F. Dubas, and R. Ibtiouen, “2-D Analytical Prediction

of Eddy Currents, Circuit Model Parameters, and Steady-State Perfor-

mances in Solid Rotor Induction Motors,” IEEE Trans. Mag., vol. 50,

no. 12, pp. 1–14, 2014.

[39] T. Lubin, S. Mezani, and A. Rezzoug, “Analytic Calculation of Eddy

Currents in the Slots of Electrical Machines: Application to Cage Rotor

Induction Motors,” IEEE Trans. Mag., vol. 47, no. 11, pp. 4650–4659,

2011.

[40] K.-H. Shin, S.-H. Lee, H.-W. Cho, C.-H. Park, J.-Y. Choi, and

K. Kim, “Analysis on the Pitching Moment in Permanent Magnet

Linear Synchronous Motor for Linear Motion Stage Systems,” IEEE

Trans. Mag., vol. 9464, no. c, pp. 1–1, 2016.

[41] L. J. Wu, Z. Q. Zhu, J. T. Chen, and Z. P. Xia, “An Analytical Model

of Unbalanced Magnetic Force in Fractional-Slot Surface-Mounted

Permanent Magnet Machines,” IEEE Trans. Mag., vol. 46, no. 7, pp.

2686–2700, jul 2010.

[42] J. Fu and C. Zhu, “Subdomain model for predicting magnetic ﬁeld

in slotted surface mounted permanent-magnet machines with rotor

eccentricity,” IEEE Trans. Mag., vol. 48, no. 5, pp. 1906–1917, 2012.

[43] Y. Li, Q. Lu, Z. Q. Zhu, L. Wu, G. Li, and D. Wu, “Analytical

Synthesis of Air-gap Field Distribution in Permanent Magnet Machines

with Rotor Eccentricity by Superposition Method,” IEEE Trans. Mag.,

vol. 2, no. c, pp. 1–1, 2015.

[44] L. J. Wu and Z. Q. Zhu, “Simpliﬁed analytical model and investigation

of open-circuit AC winding loss of permanent-magnet machines,” IEEE

Transactions on Industrial Electronics, vol. 61, no. 9, pp. 4990–4999,

2014.

[45] Y. Amara, P. Reghem, and G. Barakat, “Analytical Prediction of Eddy-

Current Loss in Armature Windings of Permanent Magnet Brushless

AC Machines,” IEEE Trans. Mag., vol. 46, no. 8, pp. 3481–3484, 2010.

[46] P. Arumugam, T. Hamiti, and C. Gerada, “Estimation of eddy current

loss in semi-closed slot vertical conductor permanent magnet syn-

chronous machines considering eddy current reaction effect,” IEEE

Trans. Mag., vol. 49, no. 10, pp. 5326–5335, 2013.

[47] B. Prieto, M. Mart´

ınez-Iturralde, L. Font´

an, and I. Elosegui, “Ana-

lytical Calculation of the Slot Leakage Inductance in Fractional-Slot

Concentrated-Winding Machines,” IEEE Transactions on Industrial

Electronics, vol. 62, no. 5, pp. 2742–2752, 2015.

[48] X. Liu, H. Hu, J. Zhao, A. Belahcen, L. Tang, and L. Yang, “Analytical

Solution of the Magnetic Field and EMF Calculation in Ironless BLDC

Motor,” IEEE Trans. Mag., vol. 52, no. 2, 2016.

[49] J. Hallal, “Etudes des vibrations d’origine ´

electromagn´

etique d’une

machine ´

electrique : conception optimis´

ee et variabilit´

e du comporte-

ment vibratoire,” Ph.D. dissertation, Universit´

e de Technologie de

Compi`

egne (UTC), 2014.

[50] Flux3D, “Electromagnetic and thermal ﬁnite element analysis,”

http://www.cedrat.com.

[51] MANATEE, “Magnetic acoustic noise analysis tool for electrical

engineering,” http://www.eomys.com.

[52] R. L. J. Sprangers, J. J. H. Paulides, B. L. J. Gysen, and E. A.

Lomonova, “Magnetic Saturation in Semi-Analytical Harmonic Mod-

eling for Electric Machine Analysis,” IEEE Trans. Mag., vol. 52, no. 2,

2016.

[53] H. Gholizad, M. Mirsalim, and M. Mirzayee, “Dynamic analysis of

highly saturated switched reluctance motors using coupled magnetic

equivalent circuit and the analytical solution,” 6th International Confer-

ence on Computational Electromagnetics (CEM), pp. 1–2, April 2006.

[54] E. Ilhan, B. L. J. Gysen, J. J. H. Paulides, and E. a. Lomonova, “Ana-

lytical hybrid model for ﬂux switching permanent magnet machines,”

IEEE Trans. Magn., vol. 46, no. 6, pp. 1762–1765, 2010.

[55] Y. Laoubi, M. Dhiﬂi, G. Barakat, and Y. Amara, “Hybrid analytical

modeling of a ﬂux switching permanent magnet machines,” Interna-

tional Conference on Electrical Machines (ICEM), pp. 1018–1023,

Sept 2014.

[56] Z. Liu, C. Bi, H. Tan, and T.-s. Low, “A combined numerical and

analytical approach for magnetic ﬁeld analysis of permanent magnet

machines,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1372–1375, 1995.

[57] H. Tiegna, Y. Amara, and G. Barakat, “Overview of analytical models

of permanent magnet electrical machines for analysis and design

purposes,” Mathematics and Computers in Simulation, vol. 90, pp.

162–177, 2013.

[58] A. Gilson, S. Tavernier, F. Dubas, D. Depernet, C. Espanet, and

M. M. Technologies, “2-D Analytical Subdomain Model for High-

Speed Permanent-Magnet Machines,” ICEM International Conference

on Electrical Machines, pp. 1–7, 2015.

[59] Femm, “Finite element method magnetics,” http://www.femm.info.

[60] J. Le Besnerais, “Fast prediction of acoustic noise and vibrations due to

magnetic forces in electrical machines,” Submitted to the International

Conference on Electrical Machines (ICEM), 2016.

V. BIOGRAPHIES

Emile Devillers is currently doing an industrial PhD thesis at EOMYS

ENGINEERING (Lille, France) and L2EP laboratory of the Ecole Centrale

de Lille, North of France. His research interests are the modeling of noise

and vibrations in electrical machines due to magnetic forces for reduction

purposes. He obtained a M.Sc. specialized in Energy Systems in 2015 from

the Grenoble Institute of Technology, France.

Jean Le Besnerais currently works in EOMYS ENGINEERING as an R &

D engineer on the analysis and reduction of acoustic noise and vibrations

in electrical systems. He made an industrial PhD thesis in Electrical

Engineering at the L2EP laboratory of the Ecole Centrale de Lille, North of

France, on the reduction of electromagnetic noise and vibrations in traction

induction machines with ALSTOM Transport. In 2013, he founded EOMYS

ENGINEERING, a company providing applied research and development

services including modeling and simulation, scientiﬁc software development

and experimental measurements.

Thierry Lubin received the M.Sc. degree from the University Pierre et

Marie Curie, Paris 6, France in 1994 and the Ph.D. degree from the Univer-

sity Henri Poincar, Nancy, France, in 2003. He is currently an associate pro-

fessor of electrical engineering at the University of Lorraine, at the Groupe

de Recherche en Electrotechnique et Electronique de Nancy (GREEN).

His interests include analytical modeling of electrical devices, contactless

torque transmission systems, modeling and control of synchronous reluctance

motors, and applied superconductivity in electrical engineering.

Michel Hecquet received the Ph.D degree from the University of Lille,

France, in 1995. His Ph.D dissertation presented a 3D permeance network

of a claw-pole alternator, used for the simulation and the determination of the

electromagnetic forces. Since 2008, he is full professor at Ecole Centrale de

Lille in L2EP laboratory (Electrotechnic and Power Electronic Laboratory).

His main interests are the development of multi-physics models of electrical

machines (electromagnetic, mechanic and acoustic) and the optimal design

of electrical machines.

Jean-Philippe Lecointe , DSc, received the MSc degree in Electrical

Engineering from the Universit des Sciences et Technologies de Lille,

France, in 2000. He received the PhD degree from the Universit dArtois,

France, in 2003. He is currently Full Professor at the Artois University and

he is the head of the LSEE (Electrical Systems and Environment Research

Laboratory), France. His research interests focus on efﬁciency, noise and

vibrations of electrical machines.