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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
To cite this version:
Emile Devillers, Jean Le Besnerais, Thierry Lubin, Michel Hecquet, Jean-Philippe Lecointe. A review
of subdomain modeling techniques in electrical machines: performances and applications. Interna-
tional Conference on Electrical Machines ICEM 2016, 2016, Lausanne, Switzerland. �hal-01910609�
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. INTRODUCTION
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 7 steps.
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 M is 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 A in [2], [3] and the formulation
B in [4], [5]. Formulation B presents the interest of giving
dimensionless expressions and a topological matrix M with
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
M and the source vector S must be carefully established.
In fact, sources variations impact on S whereas 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 S for each time step. Con-
sequently, a single linear system resolution can give the
constants X for 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 : M and S have 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. ADVANTAGES AND DRAWBACKS IN COMPARISON
WITH 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 4 dB 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.
