Working PaperPDF Available
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
Index Terms—Magnetic field, Electric machines, Analytical
model, Harmonic analysis, Performance analysis, Analytical
model, Reviews.
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
Corresponding author: E.Devillers (email:
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.
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
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).
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
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
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
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
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 slipω. 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
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.
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.
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.
Computation time [s] 0.109 0.199 0.7337
Air gap harmonics 150 250 300
Number of unknowns 1572 1909 2298
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.
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-
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].
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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.
... Methodologies for a mathematical model can be roughly categorized as numerical method and analytical approach. Numerical methods have better accuracy and are capable of handling machines with complex geometry, non-homogeneous and non-isotropic materials, and saturation effect [6], while analytical approaches owns better efficiency [7] and give more physical insights. Therefore, an initial design is usually assisted by analytical model and refined by numerical method [8]. ...
... where coefficients (6 ) ...
Full-text available
Most subdomain models do not solve the magnetic vector potential in the iron part of the machine, this paper proposes a complete subdomain model for (PM) machines with toroidal windings and mainly solves the magnetic vector potential in iron part. The proposed model can accurately predict the field distribution under load or no-load conditions. The electromagnetic parameters of the motor (including magnetic flux density, back-emf, electromagnetic torque, and cogging torque) are calculated using the proposed analytical model, and validated by a FEM model. This paper presents a technique to handle with the stator of machines with outer teeth and the very specific techniques to solve the magnetic field in the iron part.
... T HE "current sheet (CS) model" is a semi-analytical model for the calculation of the magnetic field in an electrical machine. It belongs to the family of "subdomain models," i.e., it is based on the formal resolution of Maxwell's equations in each subdomain [2]. Its distinctive feature is to divide the machine into annular subdomains and to model the windings as cylindrical CSs. ...
Full-text available
An incremental improvement of the classical semi-analytical current sheet (CS) model of electrical machines is proposed. First, it provides a better description of the CSs that allows considering, at the same time, the rotation of the rotor field winding and the time-dependent stator armature windings currents. This derivation is kept generic, and the system to be solved is explicitly written in order to facilitate the implementation. Second, a refined iterative scheme that permits to account effectively for the nonlinearity of iron cores is introduced. It is demonstrated that the nonlinear CS model is particularly suitable for slotless wound rotor machines, being able to represent both space harmonics and saturation of the machine with a fair accuracy and computing speed compared with the nonlinear finite-element model.
... This model can directly solve the governing equations of every region, and determine every coefficient by applying boundary conditions. A review of subdomain modeling techniques was provided in [24]. However, the three analytical models mentioned above have a basic assumption that the permeability of stator and rotor iron is infinite. ...
Full-text available
A new subdomain and magnetic circuit hybrid model (SMCHM) is proposed for on-load field prediction in surface-mounted permanent-magnet (SPM) machines. Equivalent current sheets are introduced to represent the nonlinearity effect, whose values are obtained by magnetic circuit and correlated to boundary conditions in subdomain model. The number of reluctances in the magnetic circuit (MC) of proposed model can be selected flexibly according to nonlinearity effect. Instead of sectorial tooth in the conventional subdomain model, parallel tooth is considered in the proposed model to improve the accuracy. The SMCHM can accurately calculate the flux density distributions and electromagnetic performance considering the heavy nonlinearity effect under load conditions with fast computation speed. The finite element (FE) analysis is performed to validate the proposed model, which shows excellent agreement between them. A prototype machine is manufactured to further prove these predictions.
... In this case, (15) can be redefined by where k is a positive integer, and ν k = kπ/a and λ k = kπ/ln(R 2 /R 1 ) are, respectively, the periodicity of A θ z and A r z . For (2) and (3), representative of Poisson's equation, the particular solution should be added to (15). Thus, the magnetic potential vector can be defined as follows [23]: ...
... Indeed, a very small harmonic of magnetic force may induce large acoustic noise and vibrations due to a resonance with a structural mode of the stator. The magnetic flux can be determined everywhere in the machine with a Finite Element Analysis (FEA), only in the air-gap and windings with semi-analytical methods such as Sub-Domain Model (SDM) [1], or only in the middle of the air-gap using the permeance magneto-motive force (PMMF) [2]. ...
Full-text available
This paper presents a comparison of several methods to compute the magnetic forces experienced by the stator teeth of electrical machines. In particular, the comparison focuses on the virtual work principle (VWP)-based nodal forces and the Maxwell tensor (MT) applied on different surfaces. The VWP is set as the reference. The magnetic field is computed either with finite element analysis or with the semi-analytical subdomain method (SDM). First, the magnetic saturation in iron cores is neglected (linear B-H curve). Then, the saturation effect is discussed in a second part. Homogeneous media are considered and all simulations are performed in 2-D. The link between the slot's magnetic flux and the tangential force harmonics is also highlighted. The comparison is performed on the stator of a surface-mounted permanent-magnet synchronous machine. While the different methods disagree on the local distribution of the magnetic forces at the stator surface, they give similar results concerning the integrated forces per tooth, referred as lumped forces. This conclusion is mitigated for saturated cases: the time harmonics are correctly computed with any of the presented lumped force methods but the amplitude of each harmonic is different between methods. Nonetheless, the use of the SDM remains accurate with MT in the air gap even with saturation for design and diagnostic of electromagnetic noise in electrical machines. However, for more accurate studies based on the local magnetic pressure, the VWP is strongly recommended.
Conference Paper
Full-text available
This paper presents an analytical modeling of electromagnetic noise in spoke-type permanent-magnet (PM) machines (STPMMs). The two-dimensional (2-D) elementary subdomain (E-SD) technique in polar coordinates is used to predict the magnetic pressure, which is transferred to frequency domain and used in harmonic response mechanical analysis to predict generated electromagnetic noise. An electro-vibro-acoustic coupling based on a 2-D finite-element method (FEM) has been realized in order to validate the obtained results by the analytical modeling.
Full-text available
The most significant assumption in the subdomain technique (i.e., based on the formal resolution of Maxwell's equations applied in subdomain) is that the iron parts are considered to be infinitely permeable, so that the saturation effect is neglected. In this paper, the author presents a novel contribution on improving of this two-dimensional (2-D) technique in polar coordinates by focusing on the consideration of iron relative permeability. In non-periodic regions, magnetostatic Maxwell's equations are solved considering nonhomogeneous Neumann boundary conditions (BCs) and the general solution is obtained by applying the interfaces conditions (ICs) in both directions (i.e., r-and í µí»‰-edges ICs). The proposed model is relevant for different types of radial-flux electrical machines with(out) permanent-magnets (PMs) supplied by a direct or alternate current (with any waveforms). For example, the semi-analytical model has been implemented for spoke-type PM machines (STPMM). The magnetic field calculations have been performed for two different values of iron core relative permeability (viz, 100 and 600), and compared with those obtained by the 2-D finite-element method (FEM). The semi-analytic results are in very good agreement with those obtained numerically, considering both amplitude and waveform.
Full-text available
This paper proposes an improved subdomain model of squirrel-cage induction machines (SCIMs) with imposed stator currents. This new model enables to accurately compute electromagnetic quantities such as air-gap flux density, instantaneous torque and forces, and electromotive force including all harmonic contents. The first improvement is to explicitly account for rotor motion with time-stepping technique. The second improvement consists in modeling the skin effect in rotor bars by considering each space harmonic of the stator magnetomotive force separately. Eddy currents in rotor bars are therefore "skin limited" and not "resistance limited." The results are then validated with linear transient finite-element analysis for both no-load and load cases, taking a topology of squirrel-cage machine already used in the previous references. The time and spatial harmonic content of all electromagnetic quantities is also validated by comparison with analytical expressions. The computation time is a hundred times lower than finite element as it does not require achieving a numerical transient first, and the resolution for a time step is shorter. Thanks to its computational efficiency and intrinsic mesh insensitivity, this method is particularly suited to magnetic forces computation and vibroacoustic analysis of SCIMs.
Conference Paper
Full-text available
This paper presents a new multiphysic model and simulation environment for the fast calculation and analysis of acoustic noise and vibration levels due to Maxwell forces in variable-speed rotating electrical machines. In the first part, some numerical methods for the prediction of electromagnetic noise are analyzed and compared to analytical or semi analytical techniques. In the second part, a new coupling of electrical, electromagnetic and vibro-acoustic models based on analytical and semi-analytical modeling techniques is presented. This model is validated by comparing simulation results to experimental results on several electrical machines at variable speed, including surface permanent magnet (SPMSM), interior permanent magnet (IPMSM) and squirrel cage induction machines (SCIM). The main resonances and noise levels are correctly estimated by the models implemented in MANATEE® simulation software, and the calculation time at variable speed varies from one second to a few minutes including harmonics up to 20 kHz.
Full-text available
The use of empirically determined coefficients to include the effects of leakage and fringing flux is a large drawback of traditional induction motor (IM) models, such as lumped parameter, magnetic equivalent circuit and anisotropic layer models. As an alternative, Finite Element Analysis (FEA) is often used to determine the magnetic field distribution accurately, although at the cost of a much longer computation time. A good alternative to FEA, both in terms of computation time and accuracy, is harmonic modeling. Therefore, in this work, a previously presented magnetic model for slotted IMs based on harmonic modeling, is extended. The main contribution is the implementation of a direct coupling between the magnetic model and the stator and rotor electric circuit models, both for steady-state and time-stepping approximation of the time dependence. Except for end winding leakage flux, no additional empirical coefficients are required to include leakage and fringing flux effects. The results of the models are validated against FEA results and measurements on a prototype. It is shown that good agreement is obtained for torque prediction, whereas the predicted stator current shows some discrepancy. This discrepancy is caused by saturation of the main magnetic flux path and can be accounted for by hybrid coupling to a model that includes the effect of non-linear soft-magnetic material on the main magnetic flux.
This paper presents a 2-dimensional analytical model for external rotor brushless machines with surface mounted magnets. The subdomain technique is used to divide the problem into magnets, airgap, slot-openings, and slots regions. The magnetic flux density distributions are initially calculated, and then the important quantities of the machine such as back electromotive force, cogging torque, electromagnetic torque, self-and mutual-inductances, and unbalanced magnetic forces are computed. The finite-element method and experimental results are used to confirm the validity of the proposed model.
Permanent-magnet vernier machine (PMVM) is a relatively new type of PM machines. An analytical subdomain model accounting for tooth-tips and flux modulation poles is developed to accurately predict on-load field distributions in PMVMs. Based on two-dimensional (2-D) polar coordinate and magnetic vector potential, this method solves the Maxwell's equations in slot, air-gap, flux modulation pole slot (FMPS), and PM regions. Consequently, the electromagnetic performance such as cogging torque, back-electromotive force (EMF), electromagnetic torque, power factor, and magnet loss are calculated. In addition, the model can also be used for the evaluation of demagnetization withstand capability. The finite-element analysis (FEA) and experimental results validate the accuracy of the developed analytical model.
This paper presents the characteristic analysis and experiment of pitching moment in a high-precision permanent-magnet linear synchronous motor. The pitching moment is analyzed according to suggested strategy; therefore, analytical solutions for magnetic fields generated by PMs are derived based on the analytical magnetic field calculation in terms of a 2-D polar coordinate system. The analytical solution of each subdomain (PM, air gap, slot, slot opening, and end region) is derived, and the field solution is obtained by applying the boundary and interface conditions between the subdomains. The magnetic force is determined based on the magnetic field analysis results. Finally, the predictions are compared with the measured data to confirm the validity of the analysis methods presented in this paper.
Consequent-pole permanent magnet (PM) synchronous machines (PMSMs) provide especial features by the use of alternate PM and ferromagnetic poles. The design of these types of electric machines requires an accurate model due to the asymmetrical flux distribution in the air gap under adjacent poles. The equivalent magnetic circuit technique can hardly offer an accurate model for consequent-pole PMSMs. To this end, a 2-D analytical model is presented for consequent-pole slotted stator PMSMs to accurately compute the magnetic field distribution due to PMs and armature reaction. The slotting effects and the tooth-tip effects are taken into consideration by using the subdomain technique. The proposed model is used to calculate the magnetic flux density distributions of three consequent-pole PMSMs: an 8-pole, 9-slot machine with a non-overlapping winding; a 4-pole, 15-slot machine with an overlapping winding; and a 10-pole, 12-slot machine with a non-overlapping winding, each with three different magnetization patterns, i.e., radial, parallel, and Halbach. Based on the magnetic flux distribution, the electromagnetic torque, the self- and mutual-inductances, and the unbalanced magnetic forces have been analytically calculated. The analytical results are compared with those obtained from the finite-element method to show the accuracy and efficacy of the proposed 2-D analytical model.
This paper presents an exact analytical subdomain model of slotted permanent magnet (PM) machines, taking into account the diffusion effect. The model gives the magnetic fields, both the eddy currents and the power losses in the PM, the instantaneous and average torque. In the model, eddy currents are not assumed to be resistance limited, contrary to what is done in conventional exact subdomain models of slotted machines. The model is hence general and valid both at low and high frequencies. It shows that it is possible to calculate the power losses in the PMs, only knowing the fields in the airgap. The model takes into account the presence of tooth-tips. The comparison with conventional exact subdomain models shows that the model presented in this paper is more accurate. The model is validated using finite element methods.
A field harmonic theory is given which takes the longitudinal and transverse edge effect as well as the winding distribution into account. The variable permeability is replaced by an effective permeability which has been obtained by comparison of the eddy current losses with a one-dimensional nonlinear exact solution. Field, currents, losses and forces have been calculated and compared with tests.