Two-Dimensional Fourier-Based Modeling of Electric ...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 55, NO. 10, OCTOBER 2019 8107217

Two-Dimensional Fourier-Based Modeling of ElectricMachines—An Overview

Bert Hannon1,2, Peter Sergeant 1,2, Luc Dupré 1,2, and Pierre-Daniel Pfister 3

1Electrical Energy Laboratory, Department of Electrical Energy, Metals, Mechanical Constructions and Systems,Ghent University, 9000 Ghent, Belgium

2Flanders Make, 3920 Lommel, Belgium3College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China

An increasing need for fast and reliable models has led to a continuous development of Fourier-based (FB) analytical modeling.This paper presents an overview of the techniques that are currently available in FB modeling for electric machines. By couplingthat overview to the most relevant literature related to the subject, an interesting starting point is provided for anyone who wantsto use or improve FB models. The following seven aspects of FB models are discussed in detail: 1) the magnetic potential (scalar orvector potential); 2) the coordinate system and the solution of the partial-differential equations for each magnetic potential and foreach coordinate system; 3) the way in which time dependence is accounted for; 4) the implementation of the source terms; 5) thepossibilities to account for slotted structures; 6) the modeling of eccentricity; and 7) the post-processing computation of physicalquantities, such as flux density, electromotive force, torque, losses, and eddy currents in conductive objects. Furthermore, this papergives the closed form solution of the Laplace, Poisson, and Helmholtz equations in each coordinate system. In addition, this papertackles other important features of FB models such as computational time reduction and coupling the machine model to an electriccircuit.

Index Terms— Analytical models, electric machines, harmonic analysis, subdomain.

I. INTRODUCTION

W ITH evermore strict requirements for electric machines,the need for fast and accurate design tools grows.

In that light, Fourier-based (FB) models have received a lotof attention during the last decades. Today, FB models arecapable of predicting the electromagnetic properties of a broadspectrum of electric machines, ranging from permanent mag-net synchronous machines (PMSMs) to switched reluctancemachines, synchronous reluctance machines [1], and inductionmachines [2], and also from rotating machines, including axialflux machines, to linear actuators. FB models are fast andallow to accurately account for various physical phenomena,such as the slotting effect and the eddy-current reaction field.

There are some interesting alternatives to the FB modelingtechnique. The best-known alternatives are charge or currentmodels [3]–[7] and magnetic equivalent circuit (MEC)models [8], [9].

Like FB models, charge and current models use a potentialformulation to rewrite Maxwell’s equations in the form of asingle partial differential equation (PDE). However, they thensolve that PDE with the help of Green’s functions instead ofseparation of variables.

The major differences between charge and current modelsare which magnetic potential they use and how they accountfor residual magnetism. Charge models use the magnetic scalarpotential and account for permanent magnets (PMs) usingmagnetic charges. Similarly, current models use the magneticvector potential and account for PMs using electric current

Manuscript received December 4, 2018; revised April 9, 2019; acceptedJune 10, 2019. Date of publication July 18, 2019; date of current ver-sion September 18, 2019. Corresponding author: P.-D. Pfister (e-mail:[email protected]).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2019.2923544

densities. Both charge and current models have proven to becapable of modeling 3-D structures. However, they are notvery effective at accounting for materials with permeabilitydifferent than that of an absolute vacuum.

MEC models are widely used. Their main advantages arethat they can account for complex geometrical structures andsaturation. On the other hand, MEC models are not very wellsuited to account for eddy-current reaction fields and theyrequire meshing, which makes them less flexible.

Because of the different advantages of FB modeling,the resulting interest has resulted in a very long list ofpublications. While the fast evolution in FB modeling is agood thing, researchers that are new to the field might get lostin the vast amount of information.

Therefore, an overview of the possibilities and limitations ofFB models and of the available literature is needed. This workaims at providing such an overview. More specifically, the goalis threefold: introducing the basics of FB modeling, evaluatingthe different techniques within FB models, and providing anoverview of the most important literature on the topic. Partof the content of this paper has been presented in a confer-ence [10] and in a PhD [11], but significant contributions wereadded to achieve this in-depth and general overview. Thesecontributions include a synthesis of the extensive literaturereview on the topic and other contributions of the authors ofthis paper’s own research.

Although most of the discussions in this paper also applyto 3-D FB models, the focus is on models with a 2-D approx-imation. This paper is structured as follows. In Section II,an introduction into FB modeling is presented. Sections III– IXgive a broader overview of the different techniques thatcan be used to build a FB model and also give many ref-erences to relevant literature. Finally, Section X concludesthis paper.

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8107217 IEEE TRANSACTIONS ON MAGNETICS, VOL. 55, NO. 10, OCTOBER 2019

II. FOURIER-BASED MODELING

In this section, the FB modeling technique is introduced.First, the technique is defined and briefly compared withits most important alternatives. In a second step, a potentialformulation of the problem is given, and the actual modelingtechnique is discussed. Finally, the most important limitationsof FB modeling are discussed. Note that in this section onlythe most commonly used techniques within FB modelingare considered.

A. Definition

To avoid confusion, an unambiguous definition of FB mod-eling is required. FB models are defined as models that meetfour major requirements as follows.

1) They solve Maxwell’s equations in the entire problemdomain.

2) They use a potential formulation to rewrite Maxwell’sequations as a single PDE.

3) They divide the problem domain in a number of subdo-mains in order to simplify that PDE.

4) They use separation of variables to solve the PDE.Resulting from the use of separation of variables, the final

solution is written as a Fourier series. This explains the termFB modeling. Other names that are often used for FB modelsare harmonic models and subdomain models.

As already mentioned, this work focuses on models thatapply a 2-D approximation. This implies that the problemis assumed invariant along one direction. That directionis referred to as the longitudinal (l) direction. The othertwo directions are the periodic (ρ) direction, along whichthe problem is periodic, and the normal (η) direction. Thegeneralized (η, ρ, l) coordinate system will later be discussedin more detail.

B. Potential Formulation

1) Physical Problem Formulation: As already mentioned,the goal of the FB modeling technique is to solveMaxwell’s equations. More specifically for electric machines,the magneto quasi-static approximation of those equations issolved [12], [13]. In their differential form, these fundamentallaws of electromagnetism are written as

Faraday’s law ∇ × E = −∂B∂ t

(1a)

Ampère’s law ∇ × H = J (1b)

Gauss’s law for magnetism ∇ · B = 0 (1c)

where t is the time and vectors B, H, E, and J are themagnetic flux density, the magnetic field strength, the electricfield strength and the current density, respectively.

Note that many authors further simplify the problem byapplying the magnetostatic approximation, which will be dis-cussed in Section V. However, as it is more general, the mag-neto quasi-static approximation will be used in this section.

From Maxwell’s equations, a mathematical expression forthe behavior of the magnetic field at the boundary betweendomains ν and ν + 1 can be derived

n · (B(ν) − B(ν+1)) = 0 (2a)

n × (H(ν) − H(ν+1)) = Js (2b)

where Js is the current density on the boundary surface, andn is the unity vector normal to that boundary.

To account for material properties, the above is supple-mented with the constitutive relations

J = σ(E + v × B) (3a)

B = μH + B0 (3b)

where B0 is the remanent magnetic flux density, v is the speedvector, μ is the magnetic permeability, and σ is the electricconductivity.

Equations (1)–(3) provide a complete mathematical descrip-tion of the electromagnetic problem that has to be solved.However, as it is difficult to solve analytically, this descriptionis not a suitable formulation to model electrical machines.Therefore, in the following, the problem is reformulated usinga magnetic potential. When studying electric machines, thereare two commonly used magnetic potentials, namely, the mag-netic vector potential and the magnetic scalar potential.

2) Magnetic Scalar Potential Formulation: By definition,the magnetic scalar potential, which is indicated as ϕ, is ascalar quantity. It is defined through its gradient

−∇ϕ = H. (4)

From this definition and Maxwell’s equations, a differentialequation for the magnetic scalar potential can be derived

∇2ϕ = ∇ · B0

μ. (5)

The derivation of (5) can be found in Appendix A.The boundary conditions (2) can be written in terms of the

magnetic scalar potential by accounting for (4) and (3b)

n · [∇(μ(ν+1)ϕ(ν+1) − μ(ν)ϕ(ν))+ (B(ν)0 − B(ν+1)

0

)] = 0

(6a)

n × ∇(ϕ(ν+1) − ϕ(ν)) = Js . (6b)

3) Magnetic Vector Potential Formulation: The magneticvector potential, which is a vector quantity, is defined throughits curl

∇ × A = B. (7)

For a complete description of magneto quasi-static problems,the magnetic vector potential has to be combined with theelectric scalar potential [14], [15]. The electric scalar potentialis indicated by V and defined through its gradient

∇V = −(

E + ∂A∂ t

). (8)

Similar to the magnetic scalar potential, a partial-differentialequation can be obtained from the above-mentioned twodefinitions, Maxwell’s equations and the constitutive relations

∇2A − μσ∂A∂ t

+ μσ(v × (∇ × A)) = −μJext − ∇ × B0.

(9)

The derivation of (9) can be found in Appendix A.

HANNON et al.: 2-D FB MODELING OF ELECTRIC MACHINES 8107217

Note that, as the magnetic vector potential is not uniquelydefined, gauge fixing is required. As is common in thestatic and quasi-static approximations, the Coulomb gauge(∇ · A = 0) was chosen.

Equation (9) can be interpreted physically by separatingcurrent densities due to eddy currents (Jeddy) and currentdensities that are externally imposed (Jext).

According to Faraday’s law, eddy currents are induced ifa conductive material experiences a varying magnetic field.From a given point of view, this can either be because ofa time-dependent magnetic field in a stationary material orbecause of relative movement of the material with respect to atime-invariant magnetic field. These phenomena are accountedfor by the time-derivative and the speed-dependent termsof (9), respectively.

Note that voltage sources can be accounted for as well. Thiswill be discussed in Section VI.

The substitution of the definition of the magnetic vectorpotential (7) and the constitutive relation for the magnetic fluxdensity (3b) in the boundary conditions (2) gives

n · (∇ × (A(ν) − A(ν+1))) = 0 (10a)

n ×(

∇ ×(

A(ν)

μ(ν)− A(ν+1)

μ(ν+1)

)−

(B(ν)0

μ(ν)− B(ν+1)

0

μ(ν+1)

))= Js .

(10b)

It can be easily seen that (10a) can be simplified by integration.Condition (10a) then imposes continuity of A.

Note that in a 2-D problem, only one component of A isnon-zero. From the definition of the magnetic vector poten-tial (7), it can readily be seen that this is the component per-pendicular to the plane in which the problem is studied, i.e., thel-component in the coordinate system (η, ρ, l) defined inSection II-A. The magnetic vector potential is then written as

A = A(η, ρ, t) · el . (11)

C. Fourier-Based Modeling

The FB modeling technique can solve either of the abovemagnetic potential formulations for a magneto quasi-staticproblem. The following paragraphs explain the subdomains,governing equation, and boundary conditions.

1) Subdomains: The differential equations, (5) and (9), aretoo complex to be solved analytically in the entire problemdomain. To overcome that issue, the studied geometry isdivided into a number of regions, called subdomains, so thatthe PDE is simplified. Two aspects are important when choos-ing the subdomains. First, the differential equation has to besimplified as much as possible. Second, it should be relativelyeasy to impose boundary conditions.

Equations (5) and (9) are greatly simplified if μ and σ areconstants. Therefore, the first point of attention is to makesure that the subdomains are chosen so that the magneticpermeability and the electric conductivity are constant in eachsubdomain. Note that Section VII will show that constantμ and σ are not mandatory. Nevertheless, mostly constantmaterial parameters are assumed.

A second point of attention is to keep the mathematicalexpression for the source terms (Jext) and (B0) as simpleas possible. Usually, the subdomains are chosen so that theyeither have a source term, eddy currents or neither a sourceterm nor eddy currents. This implies that, for a magnetic scalarpotential formulation, only the following governing equationsare to be considered

∇2ϕ = 0 (12a)

∇2ϕ = −∇ · B0

μ. (12b)

For a magnetic vector potential formulation, only the govern-ing equations in (13) are considered

∇2A = 0 (13a)

∇2A = −μJext (13b)

∇2A = −∇ × B0 (13c)

∇2A = μσ∂A∂ t

− μσ(v × (∇ × A)). (13d)

The simplest governing equations, i.e., the Laplace equations,(12a) and (13a), apply in subdomains with no source termsand no eddy currents.

The Poisson equations, (12b), (13b), and (13c) apply insubdomains with a source term, i.e., an externally imposedcurrent density or residual magnetic flux density.

Finally, (13d) applies in subdomains where the effect ofeddy currents is non-negligible. Indeed, the time-derivativeterm accounts for eddy currents due to time-variant magneticfield and the speed-dependent term for eddy currents due tothe movement of conductive materials.

Mostly, it assumed that all of the subdomains withnon-negligible eddy currents are moving at the same speed.This is, for example, the case for the retaining sleeve in asurface-mounted PMSM. It is then possible to fix the spatialreference frame to the conductive subdomains, which impliesthat the speed-dependent term of (9) vanishes [16]. However,it is perfectly possible to account for both time-varying fieldsand movement [15]. Moreover, if the problem contains con-ductive subdomains with various speeds, the movement hasto be accounted for. It will later be shown that (13d) can berewritten as a Helmholtz equation.

Less commonly, some authors do consider subdomainswhere both a source term and eddy currents are present, e.g.,subdomains with both residual magnetic flux density and eddycurrents [17].

The final point of attention relates to easily imposing theboundary conditions. To do that, the subdomains are chosen,so that, for every spatial direction of the chosen coordinatesystem, they have two invariant boundaries. In 2-D problems,this implies two boundaries that are constant along the normaldirection and two boundaries that are constant along the peri-odic direction. These boundaries will, respectively, be referredto as normal and periodic boundaries.

The choice of subdomains is illustrated in Fig. 1 for aproblem in Cartesian coordinates such as a PM in a slottedsoft-magnetic enclosure, for a problem in polar coordinates,such as a PMSM, and for a problem in cylindrical coordinates,


Fig. 1. Choosing of the subdomains and illustration of coordinate systems.(a) Cartesian (η, ρ, l) = (x, y, z). (b) Polar (η, ρ, l) = (r, φ, z). (c) Cylindri-cal (η, ρ, l) = (r, z, φ).

such as a tubular linear machine. More details about coordinatesystems are given in Section IV.

The obtained subdomains may be classified as periodicor non-periodic, depending on whether or not they span anentire period along the ρ-direction. In Fig. 1, for example,subdomains 1 and 2 of the Cartesian problem, the polarproblem and the cylindrical problem are periodic subdomains.The other subdomains are non-periodic.

Note that the periodic boundaries of periodic subdomainscoincide and can be situated at any ρ.

2) Governing Equation and Boundary Conditions: The sec-ond step of FB modeling is solving the governing equa-tions, i.e., (12) or (13), in each of the subdomains. As thisis done using separation of variables, the result containsboth eigenvalues and integration constants. The eigenvalues,which determine the periodicity of the solution, are found byimposing the boundary conditions on the periodic boundaries.As, amongst others, shown in [18], the resulting solution is

written as

ϕ(ν)(η, ρ, t) =∞∑

n=−∞

∞∑k=−∞

ϕ(ν)n,k(η)e

j(

2kπT (ν)

(ρ−ρ(ν)0

)−nωt

)(14)

for the magnetic scalar potential and

A(ν)(η, ρ, t) =∞∑

n=−∞

∞∑k=−∞

A(ν)n,k(η)ej(

2kπT (ν)

(ρ−ρ(ν)0

)−nωt

)(15)

for the magnetic vector potential.Here, ν refers to the subdomain in which (14) and (15)

apply, n is the time-harmonic order, k is the spatial-harmonicorder, ρ = ρ

(ν)0 is the position of subdomain ν’s first periodic

boundary, and ω is the time-based rotational speed of the stud-ied machine. T (ν) is the subdomain ν’s periodicity along theρ-direction. It is one time its width in periodic subdomains andtwo times its width in non-periodic subdomains. An exampleof a periodic subdomain is the air gap of the PM machineshown in Fig. 1(c), and an example of the non-periodicsubdomain is a slot of that PM machine. The reason for thisfactor of two is explained further below. The form of theη-dependent part A(ν)n,k(η) depends on the coordinate system

and the subdomain’s PDE. A(ν)n,k(η) will be discussed in moredetail in Section IV.

Note that (14) and (15) are exponential Fourier series. Evi-dently, trigonometric Fourier series apply as well. In this paper,the exponential form is preferred because of its more compactnotation and because it is easier to use in the quasi-staticapproximation. Indeed, applying exponential Fourier seriesand assuming that v = veρ allows to easily rewrite (13d) forevery time- and spatial-harmonic combination as (16), i.e., inthe form of a Helmholtz equation

∇2An,k = τ 2n,kAn,k (16)

where τn,k is defined in Section IV-B.As already mentioned, in periodic subdomains, the periodic

boundaries coincide and can be situated at any ρ. Therefore,imposing the periodic boundary conditions implies imposinga periodicity of T (ν) in the ρ-direction. It can readily be seenthat (14) and (15) indeed satisfy this periodicity.

Imposing the periodic boundary conditions in non-periodicsubdomains is not as evident. Moreover, (14) and (15) areonly valid in non-periodic subdomains if they are enclosed byinfinitely permeable material in the ρ-direction. The reason forthis is that, in regular non-periodic subdomains, the periodicboundary conditions do not result in a periodicity constraintfor ϕ(ν)(η, ρ, t) or A(ν)(η, ρ, t), i.e., they do not determine theeigenvalues of the PDE. If, however, that subdomain bordersinfinitely permeable material, the demand for a continuoustangential component of the magnetic field, i.e., (6b) or (10b),imposes a periodicity constraint for the magnetic scalar poten-tial’s gradient or the magnetic vector potential’s curl. Indeed,H(ν)η has to equal zero at both of the periodic boundaries. This

imposes a periodicity of twice the subdomain’s width. It caneasily be verified that (14) and (15) satisfy this requirement.

Much like the PDE’s eigenvalues are determined by the peri-odic boundary conditions, the integration constants, which arecomprised in ϕ(ν)(η) or A(ν)n,k(η), are determined by the normal


boundary conditions. Combining the boundary conditions at allof the problem’s normal boundaries results in a system thatcan be written as[

C1n

] · [Xn] = [S1

n

][Jextn] + [S2

n

][B0n] (17)

where [C1n ], [S1

n ], and [S2n ] are coefficient matrices, [Xn]

contains the integration constants, [Jextn] the current densities,and [B0n] the residual magnetic flux densities.

The problem’s magnetic field is now determined by solv-ing (17) and inputting the results in (14) or (15).

Note that the system of boundary conditions and integrationconstants is generally solved numerically. This may be thereason why the FB modeling technique is referred by some asa semi-analytical modeling technique. It should also be notedthat the computational time is largely determined by the sizeof the system and therefore by the number of subdomains andharmonics that are taken into account.

D. Computational Considerations

The computational time of a FB model mainly depends onthe size of the system (17) that has to be solved, i.e., onthe amount of integrations constants that have to be calcu-lated. This implies that the computational time is primarilydetermined by the number of subdomains and the amount ofharmonic combinations. In order to reduce the computationaltime, Hannon et al. [19] proposed three measures.

The first was to simplify the representation of the slots,i.e., modeling a single slot instead of both a slot and a slotopening. This reduces the number of subdomains and thereforethe number of integration constants.

A second measure is to use a preliminary harmonic analysisof the studied machine [20]. This allows to exclude a numberof harmonic combinations from the system, which reduces thecomputational time.

Third, that same harmonic analysis can be used to findrelations between the integration constants of different slots.This again implies that not all of the integration constants haveto be calculated, which further reduces the computational time.

In addition to the above efforts to reduce the computationaltime, Gysen et al. [18] discussed how numerical problems,that might occur when solving (17), can be avoided.

It should be noted that the number of harmonics that need tobe computed depends also on the output quantity that needs tobe calculated. For the same machine, the number of harmonicsthat need to be computed for torque calculation may not bethe same as the one needed for calculation of eddy currentlosses.

In finite elements, the size of the mesh that will provide agiven accuracy does depend on the structure that is modeled.In the same way, the accuracy of FB models depends not onlyon the number of harmonics that are taken into account butalso on the structure itself and at this stage there is no generalrule of the number of harmonics that need to be taken intoaccount to obtain a given accuracy.

In machines where magnetic saturation is not taken intoaccount and where the FB technique is used for preoptimiza-tion, increasing dramatically the number of harmonics that are

calculated may not help at all as there is anyway an error dueto the initial assumption on the materials [21].

E. Limitations of FB Modeling

As mentioned, the FB modeling technique’s main advantageis that it combines a low computational time with very highaccuracy. However, it has some important limitations as well.First and foremost, the technique can only study machinesin steady-state operation. Second, despite recent efforts toaccount for subdomains with a variable permeability and avariable conductivity, the technique is not very efficient ataccounting for a finite permeability in slotted geometries orfor eddy-current reaction field in segmented structures. Whenthe magnetic saturation effect is taken into account, this typeof technique requires high computational time [22]. Finally,the need for subdomains with simple boundaries makes thatthe FB modeling is not very well suited to study machineswith very complex geometry.

Note that combinations of FB models and MEC modelshave been proposed as well [23], [24]. These hybrid modelsare promising, but they add the complexity of the meshingwhich makes them less flexible.

F. Aspects of Fourier-Based Models

Based on the existing literature, a number of interestingaspects related to FB modeling can be identified. In the nextsections, those aspects are discussed and references to someof their most interesting publications are provided. The resultmay be used as a guideline to decide which techniques haveto be used to model a given situation.

The first and probably the most important choice whenconstructing a FB model is which magnetic potential willbe used. It goes without saying that this choice is one ofthe aspects that needs a more detailed discussion. A secondinteresting aspect is the choice of a well-suited coordinatesystem and the explicit general solution of the PDE in thegiven coordinate system. Third, the way in which the field’stime dependence is accounted for is very important as well.The fourth aspect of interest relates to the representationand implementation of the source terms and the particularsolution of the PDE as a function of the source terms. A fifthinteresting aspect is how slotted structures are accounted for.Sixth, the way FB models can account for eccentricity of therotor is an interesting aspect. Finally, the obtained magneticpotential has to be translated into meaningful values, such asmagnetic flux density, torque, and losses. The following sevensections each tackle one of the above aspects in detail andgive for each aspect a conclusion.

III. CHOOSING A MAGNETIC POTENTIAL

As mentioned in Section II, the two most commonly usedmagnetic potentials are the magnetic scalar potential and themagnetic vector potential. The goal of this section is to discussthe advantages and disadvantages of both potentials.

A. Magnetic Scalar Potential

Due to its scalar nature, the magnetic scalar potential isvery easy to use. This is the reason why it has long been a


commonly used magnetic potential in FB models. Nowadays,it is still often used in 3-D problems [25], [26], where theuse of the magnetic vector potential requires extensive vectorcalculus. The magnetic scalar potential’s main drawback,however, is that it can only be defined in current-free regions.This can easily be seen from the magnetic scalar potential’sgoverning equation (5). There are two major workarounds toavoid this need for current-free regions.

The first and most commonly used workaround is to replacethe current density in the studied region with a current sheeton its boundary [27]. This current sheet is then accounted forthrough the boundary conditions (6). However, using currentsheets may imply a loss of accuracy [28], especially in slotlessmachines.

The second workaround is to use a combination of thereduced magnetic scalar potential and the electric vectorpotential. However, as the analytical derivation of the electricvector potential is often devious, the use of this workaround isvery rare in analytical models. Nevertheless, interested readersmay find an extensive discussion on the subject in the bookof Kuczmann and Iványi [14].

B. Magnetic Vector Potential

In models that cannot be reduced to 2-D problems, the mag-netic vector potential’s vectorial nature implies a more exten-sive calculus. In those problems, this complexity is the mainreason to not use the magnetic vector potential. However,in 2-D problems, where the magnetic vector potential reducesto a scalar, the magnetic vector potential is by far the mostpopular magnetic potential [29]–[32]. The primary reason forthat is the magnetic vector potential’s ability to account forsubdomains with a current density.

Note that some authors do use the magnetic vector potentialto study 3-D problems. This can either be done directly [33]or by using the second-order magnetic vector potential [34].

C. Conclusion

Due to its flexibility, the magnetic vector potential is usuallythe best-suited magnetic potential to study electric machines.Only in current-free problems or problems where the magneticvector potential’s vector calculus is too complex, the magneticscalar potential may be preferred.

The mathematical framework, necessary to use the magneticscalar potential or the magnetic vector potential, has beenprovided in Section II. The actual implementation of a 2-D FBmodel using the magnetic scalar potential is clearly describedin [35], an equally interesting paper that focuses on the mag-netic vector potential is Gysen’s, very generally formulated,work [32]. For 3-D models, Meessen [26], Gerling [33], andJumayev [34] present a comprehensive implementation of themagnetic scalar potential, the magnetic vector potential, andthe second-order magnetic vector potential, respectively.

IV. GENERAL SOLUTION OF THE PDES IN THE DIFFERENT

COORDINATE SYSTEMS

In Section II, the generalized (η, ρ, l) coordinate systemwas used. In order to model an actual machine, an appropriate

TABLE I

COORDINATE SYSTEMS

coordinate system has to be chosen. The earliest publicationson FB modeling tended to use a Cartesian coordinate system.However, for most geometries, this implies simplifying thegeometry while only resulting in a slightly less complexmodel. Nowadays, the choice is mostly determined by thegeometry of the studied problem. For example, when the goalis to model a radial-flux rotational machine in two dimen-sions [15], [27], [28], [30], [31], [35]–[41], a polar coordinatesystem (r, φ, z) may be used. Two-dimensional models oftubular linear machines [42]–[45] use a cylindrical coordi-nate system (r, φ, z). Also, 2-D approximations of axial-fluxmachines are usually modeled in a cartesian coordinate system(x, y, z) [46]–[48]. The use of different coordinate systems forvarious problems is illustrated in Fig. 1.

In the following, the above coordinate systems are linked tothe general coordinate system, and the homogeneous solutionsof their η-dependent parts are introduced.

A. Coordinate Systems

Table I links the generalized coordinates to the Cartesian,polar, and cylindrical systems.

Note that the only difference between problems that applya polar coordinate system and problems that apply a cylindri-cal coordinate system is the direction along which they areperiodic and invariant.

B. General Solution of the PDEs

As discussed above, the general form of the magneticpotentials’ solutions can be written as in (14) or (15). Theη-dependent part of (14) and (15) depends on the PDE’ssimplification in subdomain ν. As discussed above, the dif-ferential equation for the magnetic scalar potential can eitherbe a Laplace or a Poisson equation [49], [50]. The differentialequation for the magnetic vector potential, on the other hand,can either be a Laplace, a Poisson, or a Helmholtz equation.Knowing this, η-dependent parts of (14) and (15) can bewritten in full for each of the coordinate systems. The Poissonequations are made of a solution that is the sum of the Laplaceequation solution, which is called the general solution, andthe particular solution. The particular solutions are given inSection VI-D.

1) Cartesian Coordinates System: In Cartesian coordinates,the solution for the magnetic scalar potential in subdomain νis written as

ϕ(ν)(x, y, t) =∞∑

n=−∞

∞∑k=−∞

ϕ(ν)n,k(x)e

j(

2kπT (ν)

(y−y(ν)0

)−nωt

)(18)


where, when k �= 0

ϕ(ν)n,k(x)

=⎧⎨⎩C(ν)

n,ke

∣∣ 2kπT (ν)

∣∣x +D(ν)n,ke

−∣∣ 2kπ

T (ν)

∣∣x, Laplace

C(ν)n,ke

∣∣ 2kπT (ν)

∣∣x +D(ν)n,ke

−∣∣ 2kπ

T (ν)

∣∣x + P(ν)n,k (x), Poisson(19)

and when k = 0

ϕ(ν)n,k(x) = C(ν)

n,k + D(ν)n,k x (20)

where C(ν)n,k and D(ν)

n,k are integration constants, and P(ν)n,k (x)is the particular solution of the Poisson equation, the form ofwhich depends on the subdomain’s source term.

Similarly, the solution for the magnetic vector potential iswritten as

A(ν)(x, y, t) =∞∑

n=−∞

∞∑k=−∞

A(ν)n,k(x)ej(

2kπT (ν)

(y−y(ν)0

)−nωt

)(21)

where, when k �= 0 and n �= 0

A(ν)n,k(x)

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

C(ν)n,ke

∣∣ 2kπT (ν)

∣∣x + D(ν)n,ke

−∣∣ 2kπ

T (ν)

∣∣x, Laplace

C(ν)n,ke

∣∣ 2kπT (ν)

∣∣x + D(ν)n,ke

−∣∣ 2kπ

T (ν)

∣∣x + P(ν)n,k (x), Poisson

C(ν)n,ke

√(2kπT (ν)

)2+τ 2n,k x +D(ν)

n,ke−

√(2kπT (ν)

)2+τ 2n,k x,

Helmholtz

(22)

with τ 2n,k = jμ(ν)σ (ν)((2kπ/T (ν))v(ν) − nω), v(ν) being the

speed of subdomain ν. When k = 0 and n �= 0, we have

A(ν)n,k(x) =

⎧⎪⎨⎪⎩

C(ν)n,k + D(ν)

n,k x, Laplace

C(ν)n,k + D(ν)

n,k x + P(ν)n,k (x), Poisson

C(ν)n,keτn,k x + D(ν)

n,ke−τn,k x , Helmholtz

(23)

and when k = 0 and n = 0

A(ν)n,k(x) =

⎧⎪⎨⎪⎩

C(ν)n,k + D(ν)

n,k x, Laplace

C(ν)n,k + D(ν)

n,k x + P(ν)n,k (x), Poisson

C(ν)n,k + D(ν)

n,k x, Helmholtz.

(24)

2) Polar Coordinates System: In polar coordinates, the solu-tion for the magnetic scalar potential in subdomain ν is writtenas

ϕ(ν)(r, φ, t) =∞∑

n=−∞

∞∑k=−∞

ϕ(ν)n,k(r)e

j(

2kπT (ν)

(φ−φ(ν)0

)−nωt

)(25)


ϕ(ν)n,k(r)=

⎧⎨⎩C(ν)

n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣, Laplace

C(ν)n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣ + P(ν)n,k (r), Poisson

(26)

and when k = 0

ϕ(ν)n,k(r) =

{C(ν)

n,k + D(ν)n,k ln r, Laplace

C(ν)n,k + D(ν)

n,k ln r + P(ν)n,k (r), Poisson.(27)

For the magnetic vector potential, we have

A(ν)(r, φ, t) =∞∑

n=−∞

∞∑k=−∞

A(ν)n,k(r)ej(

2kπT (ν)

(φ−φ(ν)0

)−nωt

)(28)

where, when k �= 0 and and τn,k �= 0

A(ν)n,k(r)

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

C(ν)n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣, Laplace

C(ν)n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣ + P(ν)n,k (r), Poisson

C(ν)n,k I 2kπ

T (ν)(τn,kr)+D(ν)

n,k K 2kπT (ν)

(τn,kr), Helmholtz

(29)

where I(2kπ/T (ν)) is the (2kπ/T (ν))-order modified Besselfunction of the first kind, and, similarly, K(2kπ/T (ν)) is the(2kπ/T (ν))-order modified Bessel function of the second kind,and where τ 2

n,k = jμ(ν)σ (ν)((2kπ/T (ν)ω(ν)−nω)), ω(ν) beingthe mechanical rotational speed of subdomain ν. When k = 0and τn,k �= 0, we have

A(ν)n,k(r)

=

⎧⎪⎪⎨⎪⎪⎩

C(ν)n,k + D(ν)

n,k ln r, Laplace

C(ν)n,k + D(ν)

n,k ln r + P(ν)n,k (r), Poisson

C(ν)n,k I 2kπ

T (ν)(τn,kr)+ D(ν)

n,k K 2kπT (ν)

(τn,kr), Helmholtz

(30)

when k �= 0 and τn,k = 0

A(ν)n,k(r)

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

C(ν)n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣Laplace

C(ν)n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣ + P(ν)n,k (r) Poisson

C(ν)n,kr

∣∣ 2kπT (ν)

∣∣ + D(ν)n,kr

−∣∣ 2kπ

T (ν)

∣∣Helmholtz

(31)

and when k = 0 and τn,k = 0

A(ν)n,k(r) =

⎧⎪⎨⎪⎩

C(ν)n,k + D(ν)

n,k ln r, Laplace

C(ν)n,k + D(ν)

n,k ln r + P(ν)n,k (r), Poisson

C(ν)n,k + D(ν)

n,k ln r, Helmholtz.

(32)

3) Cylindrical Coordinates System: In cylindrical coordi-nates, the solution for the magnetic scalar potential in subdo-main ν is written as

ϕ(ν)(r, z, t) =∞∑

n=−∞

∞∑k=−∞

ϕ(ν)n,k(r)e

j(

2kπT (ν)

(z−z(ν)0

)−nωt

)(33)


ϕ(ν)n,k(r)

=⎧⎨⎩

C(ν)n,k I0

(2kπT (ν)

r)

+ D(ν)n,kK0

(2kπT (ν)

r)

Laplace

C(ν)n,k I0

(2kπT (ν)

r)

+ D(ν)n,kK0

(2kπT (ν)

r)

+ P(ν)n,k (r) Poisson

(34)


where I0 is the zeroth-order modified Bessel function of thefirst kind, and K0 is the zeroth-order modified Bessel functionof the second kind, and when k = 0

ϕ(ν)n,k(r) =

{C(ν)

n,k + D(ν)n,k ln r, Laplace

C(ν)n,k + D(ν)

n,k ln r + P(ν)n,k (r), Poisson.(35)

For the magnetic vector potential, we have

A(ν)(r, z, t) =∞∑

n=−∞

∞∑k=−∞

A(ν)n,k(r)ej(

2kπT (ν)

(z−z(ν)0

)−nωt

)(36)


A(ν)n,k(r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

C(ν)n,k I1

(2kπT (ν)

r)

+ D(ν)n,kK1

(2kπT (ν)

r), Laplace

C(ν)n,k I1

(2kπT (ν)

r)

+ D(ν)n,kK1

(2kπT (ν)

r)

+ P(ν)n,k (r),

Poisson

C(ν)n,k I1

(r√( 2kπ

T (ν))2 + τ 2

τn,k

),

+D(ν)n,kK1

(r√( 2kπ

T (ν))2 + τ 2

τn,k

), Helmholtz

(37)

where τ 2n,k = jμ(ν)σ (ν)((2kπ/T (ν)v(ν) + nω)). When k = 0

and τn,k �= 0, we have

A(ν)n,k(r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

C(ν)n,kr + D(ν)

n,k1r , Laplace

C(ν)n,kr + D(ν)

n,k1r + P(ν)n,k (r), Poisson

C(ν)n,kI1(τn,kr)+ D(ν)

n,kK1(τn,kr), Helmholtz

(38)

and when k = 0 and τn,k = 0

A(ν)n,k(r) =

⎧⎪⎨⎪⎩

C(ν)n,kr + D(ν)

n,k1r , Laplace

C(ν)n,kr + D(ν)

n,k1r + P(ν)n,k (r), Poisson

C(ν)n,kr + D(ν)

n,k1r , Helmholtz.

(39)

Note that, both in polar and cylindrical coordinates, solu-tions with ordinary instead of modified Bessel functions arepossible as well.

C. Conclusion

In the above, the three coordinate systems that are mostcommonly used in literature are introduced. Their coordinatesare linked to those of the generalized coordinate system.Moreover, general solutions for each of the coordinate systemswere discussed as well.

V. TIME DEPENDENCE

In the above, the magneto quasi-static approximation ofMaxwell’s equations was used. However, many authors furthersimplify the problem by using the magnetostatic approxima-tion. This implies that they do not consider the physical effectsof time dependence. This choice between the magnetostaticand the magneto quasi-static approximation is the first aspectof time dependence that is discussed in this section.

However, even if the physical effects of time dependenceare neglected, the actual magnetic field is still time-dependent.

If not accounted for directly, this indirect approach can beused to calculate the field at various time instances. Thechoice between multiple static calculations and direct timedependence is the second aspect of time dependence that willbe addressed in this section.

A. Static Versus Quasi-Static Problems

The physical consequence of assuming static conditions isthat the eddy-current reaction field is neglected. In the potentialformulation, this means that the time-derivative and speed-dependent terms of (9) are disregarded. The magnetostaticapproximation is hence simpler than the magneto quasi-staticone: it may, therefore, be preferred if the effect of eddy-currents is small.

Note that, as the magnetic scalar potential assumescurrent-free regions, it cannot account for the eddy-currentreaction field, and its use implies the assumption of magneto-static conditions.

B. Multiple Static Calculations Versus Direct Time Dependence

Regardless of whether the magnetostatic or themagneto quasi-static approximation is used, the magneticfield in the studied problem will almost always vary in time.This time dependence can either be accounted for directly orthrough multiple static calculations. Direct time dependenceimplies that the equations for the magnetic potential arewritten as a function of time. This approach was used inSection II. Another approach is to divide the problem’speriod into a number of time instances. The time-independentproblem is then solved for each of these instances of time,i.e., with updated values of the rotor position and currents.

Indirect time dependence has one major advantage; it is sim-pler than directly accounting for the time dependence. Indeed,neglecting the time dependence allows disregarding thetime-harmonic orders. This, in turn, results in a single Fourierseries instead of the double Fourier series of (14) and (15).

Directly accounting for the field’s time dependence, on theother hand, allows for a more accurate solution. Indeed,allowing for a time-dependent solution implies that themagneto quasi-static approximation can be used.

Considering the computational time related to bothapproaches, the technique that uses indirect time dependencecan very quickly compute a single time instance. However,direct time dependence will generally have a better accuracy-to-computational-time ratio. Indeed, assuming that the systemof boundary conditions can be solved independently for everytime-harmonic order, the time required to compute the inte-gration constants of a single time instance is equal to that ofa single harmonic order. At the same time, the calculation ofone extra harmonic order results in a larger accuracy increasethan the calculation of one extra time instance. Note that inmany cases such as [15]–[17], [51] the time-harmonic ordersare considered independently.

Because of its simplicity, most authors of magneto-static models use indirect time dependence. Authors ofmagneto quasi-static models tend to directly account forthe time dependence. However, if only the fundamental


time-harmonic is considered, the eddy-current reaction fieldcan be accounted for in models that use an indirect timedependence. This was, a.o. done by Boughrara et al. [52],[53] to model induction machines.

C. Conclusion

Two aspects of time dependence were discussed in theabove.

First, it was noted that, depending on the importance ofthe eddy-current reaction field, either the simpler magnetosta-tic approximation or the more complete magneto quasi-staticapproximation can be used.

Next, models with multiple static calculations and modelswith direct time dependence were compared. It was concludedthat models with an indirect time dependence are simpler toimplement and may be preferred in magnetostatic models.Models with a direct time dependence are to be preferred inmagneto quasi-static models, especially if the effect of highertime-harmonic orders is relevant. Generally, directly account-ing for the time dependence results in a more computationallyefficient model.

Interesting examples of magnetostatic models that apply thetechnique of multiple static calculations have been publishedin [18], [54]–[56]. Magneto quasi-static models that directlyaccount for time dependence may be found in [15]–[17], [51],and finally, publications on quasi-static models with an indirecttime dependence can be found in [52].

VI. SOURCE TERMS

Obviously, there can only be a magnetic field if the studiedproblem contains source terms, i.e., materials with a residualmagnetic flux density and/or externally imposed current densi-ties. The latter can either be imposed directly or by applyinga voltage source. There are thus three types of sources thathave to be discussed: residual magnetic flux densities, externalcurrent sources, and external voltage sources. The followingpresents a discussion about the way each of these sources isaccounted for.

A. Residual Magnetization

There are two ways in which materials with residual mag-netic flux densities can be implemented, directly or throughequivalent currents. In both techniques, a Fourier representa-tion of the residual magnetization has to be available. In itsmost general form, that representation is

B(ν)0 =∞∑

n=−∞

∞∑k=−∞

(B(ν)0,η,n,k(η)eη + B(ν)0,ρ,n,k(η)eρ

) ·

ej(

2kπT (ν)

(ρ−ρ(ν)0

)−nωt

)(40)

where B(ν)0,η,n,k(η) and B(ν)0,ρ,n,k(η) depend on the magnetizationpattern. In order to find particular solutions, we assume thatthe residual magnetization does not depend on η, so it can be

expressed as

B(ν)0 =∞∑

n=−∞

∞∑k=−∞

(B(ν)0,η,n,keη + B(ν)0,ρ,n,keρ

) ·

ej(

2kπT (ν)

(ρ−ρ(ν)0

)−nωt

). (41)

The PDEs for both the magnetic scalar potential and themagnetic vector potential can directly account for permanentmagnetic materials. To do that, a particular solution for theappropriate Poisson equation has to be determined. Next, thatsolution has to be substituted in the appropriate potentialequation, as presented in Section IV. Directly accountingfor residual magnetic flux densities is definitely the moststraightforward approach. Moreover, mostly it is not morecomplex than using current sheets. Therefore, nowadays thevast majority of authors directly accounts for permanent mag-netic materials [41], [57]–[60].

As already mentioned, the alternative is to represent the PMsby equivalent current densities. Two types of current densitieshave to be considered: the equivalent volume current densities(Jm), which manifest in the entire magnet volume, and theequivalent surface current densities (jm), which manifest onthe magnet’s boundaries. They are calculated as

Jm = ∇ × M0

jm = M0 × n (42)

where M0 = μ0B0 is the residual magnetization vector, andn is the outward facing unit vector, normal to the magnet’sboundary.

Evidently, in a 2-D approximation, the volume currentdensities are reduced to surface current densities. They aretherefore implemented in the same way as the externallyimposed current densities. Similarly, the equivalent surfacecurrent densities are reduced to current sheets. These areaccounted for through the boundary conditions: (6) or (10).

Whereas working with equivalent current densities usedto be widespread in FB modeling, nowadays, it is onlyused in some specific cases. For example, in 3-D models,where analytical solutions of the Poisson equation may becomplicated [34] or in models that apply Schwarz–Christoffeltransformations to account for slotting [61], [62].

B. Externally Imposed Current Densities

Similar to accounting for residual magnetization, accountingfor externally imposed current densities can either be donedirectly or by considering a current sheet instead. The exter-nally imposed current density is therefore again expressed asa Fourier series

J(ν)ext =∞∑

n=−∞

∞∑k=−∞

J (ν)ext,n,kej(

2kπT (ν)

(ρ−ρ(ν)0

)−nωt

)el . (43)

As the magnetic scalar potential assumes current-free regions,directly accounting for current densities is only possible forproblems that are formulated in the magnetic vector potential.To do so, the particular solution of the Poisson equation isagain substituted in the appropriate solution for the magneticvector potential.


In slotted machines, the current density is then oftenassumed uniform in every subdomain [17], [18], [28], [41].However, some authors do consider spatial dependence ofthe current density within a single slot [56], [63]. Evidently,in slotless machines, where the entire slotting region is a singlesubdomain, the spatial dependence of the current density isaccounted for as well [15], [31], [64].

As an alternative to directly accounting for external currentdensities, a surface current density (Js) can be imposedthrough the boundary conditions; (6) or (10). Js then has tobe calculated so that the totally imposed current is constant.In 2-D problems, this implies that its line integral has to equalthe surface integral of the original current density.

Although models with equivalent current sheets mayyield accurate results for some machine topologies,Atallah et al. [28] showed that using equivalent currentsheets can reduce the model’s accuracy. This is especiallytrue in slotless machines, where the winding’s thickness isnot negligible with respect to the pole pitch. For that reason,equivalent current sheets are usually avoided nowadays.In recent literature, the use of equivalent current sheets islimited to models that use the magnetic scalar potential [65],3-D models [34] or models in which slotting is accounted forthrough Schwarz–Christoffel transformation [60], [66].

C. Voltage Sources and Coupling to External Circuits

Most publications on FB models assume idealized currentwaveforms in the machine’s coils. However, as the vast major-ity of modern machines is powered with a voltage source,this assumption might not hold. It is therefore often moreinteresting to apply a realistic voltage signal, or to couplethe machine to an external circuit. However, as discussed inAppendix X, directly imposing voltage sources through thePDE is too complex. To avoid that complexity, the magneticcalculations (17) can be coupled with the classical equation forthe terminal voltage in an electric machine. The latter can berewritten as a function of the FB model’s integration constants,the stator’s currents densities, the remanent magnetic fluxdensities and the terminal voltages[

C2n

] · [Xn] + [C3

n

] · [Jextn] = [S3

n

][B0n] + [S4

n

][Vn] (44)

where [C2n ], [C3

n ], [S3n ] and [S4

n ] are coefficient matrices and[Vn] contains the voltage sources.

The combination of (17) and (44) results in a new set ofequations[[

C1n

] −[S1

n

][C2

n

] [C3

n

]]

·[ [Xn][Jextn]

]=

[ [S2

n

] · [B0n][S3

n

] · [B0n] + [S4

n

] · [Vn]

].

(45)

The above technique was discussed by Hannon et al. [67]and by Sultan et al. [68]. In order to clarify the theory, thispaper [67] gives a completely elaborated example of how todetermine the coefficient matrices for a PMSM. All data of themachines and resulting waveforms are given, so that interestedresearchers can use the example to learn how to implement theFB technique coupled with voltage sources.

D. Particular Solutions

The particular solutions are solutions of the PDE that arerelated to the source terms. In order to be able to finda relatively simple particular solution, we assume that theremanent magnetic flux density is constant as a function ofη. Explicit particular solutions are given here in the threecoordinate systems.

1) Cartesian Coordinate System: The particular solutionfor the Poisson equation for the magnetic scalar potential inCartesian coordinates is given by

P(ν)n,k (x) =⎧⎨⎩− j B(ν)0,y,n,k T (ν)

2πkμ(ν), for k �= 0

0, for k = 0.(46)

The particular solution for the Poisson equation for the mag-netic vector potential in Cartesian coordinates is given by

P(ν)n,k (x) =⎧⎨⎩

T (ν)(μ(ν) J (ν)ext,n,k T (ν)−2πk B(ν)0,x,n,k

)4π2k2 , for k �= 0

−μ(ν) J (ν)ext,n,02 x2, for k = 0.

(47)

2) Polar Coordinate System: The particular solution for thePoisson equation for the magnetic scalar potential in polarcoordinates, when the source term is remanent magnetic fluxdensity, is given by

P(ν)n,k (r) =

⎧⎪⎪⎨⎪⎪⎩

B(ν)0,r,n,k + 2 jπkT (ν)

B(ν)0,φ,n,k

μ(ν)(

1−(

2πkT (ν)

)2) r, for 2kπ �= ±T (ν)

B(ν)0,r,n,k + 2 jkπT (ν)

B(ν)0,φ,n,k

2μ(ν)r ln r, for 2kπ = ±T (ν).

(48)

The particular solution for the Poisson equation for the mag-netic vector potential in polar coordinates, when the sourceterm is current, is given by

P(ν)n,k (r) =⎧⎨⎩μ(ν)J (ν)ext,n,k

1(2kπT (ν)

)2−4r2, for kπ �= ±T (ν)

− 14μ

(ν) J (ν)ext,n,kr2 ln r, for kπ = ±T (ν).(49)

The particular solution for the Poisson equation for themagnetic vector potential in polar coordinates, when the sourceterm is remanent magnetic flux density, is given by

P(ν)n,k (r) =

⎧⎪⎪⎨⎪⎪⎩

(B(ν)0,r,n,k

2 jkπT (ν)

−B(ν)0,φ,n,k

)r

1−(

2kπT (ν)

)2 , for 2kπ �= ±T (ν)(B(ν)0,r,n,k

2 jkπT (ν)

−B(ν)0,φ,n,k T (ν))

r ln r

2 , for 2kπ = ±T (ν).

(50)

3) Cylindrical Coordinate System: The particular solutionfor the Poisson equation for the magnetic scalar potential incylindrical coordinates is given by

P(ν)n,k (r)=

⎧⎪⎨⎪⎩

T (ν)(πB(ν)0,r,n,k L0

(2kπrT (ν)

)−2 j B(ν)0,z,n,k

)4πkμ(ν)

, for k �= 0

B(ν)0,r,n,kr

μ(ν), for k = 0

(51)

where L0 is the zeroth-order modified Struve function.


For k �= 0, the particular solution for the Poisson equationfor the magnetic vector potential in cylindrical coordinates isgiven by

P(ν)n,k (r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−(

2πB(ν)0,r,n,k k− j J (ν)ext,n,kμ(ν)T (ν)

)12(T (ν))2

·(2π2kr3Y1

( − 2 j kπrT (ν)

)1 F2

( 32 ; 2, 5

2 ; k2π2r2

(T (ν))2)

+3 jπr2T (ν)J1( 2 j kπr

T (ν))�(r)

) (52)

with

�(r) = G2,12,4

(− jkπr

T (ν),

1

2| 0,−1− 1

2 ,12 ,−1,−1

)

G2,12,4 being a Meijer G function, 1 F2 a generalized hyper-

geometric function, Y1 the first order Bessel function ofthe second kind, and J1 the first order Bessel function of thefirst kind. For k = 0 the particular function is given by

P(ν)n,k (r) = −1

3J (ν)ext,n,kμ

(ν)r2. (53)

E. Conclusion

Three types of sources were considered: residual magnetiza-tion, externally imposed current density, and voltage sources.It was discussed that both residual magnetization and externalcurrent densities can either be imposed directly through thePDE or, with the help of equivalent currents, through theboundary conditions. As the direct technique is more straight-forward and usually does not add to the complexity, it shouldbe preferred. Accounting for sources through the boundaryconditions can be interesting in some specific cases. It was alsodiscussed that, although it increases the model’s complexity,accounting for voltage sources may result in more realisticresults.

VII. SLOTTING

Accounting for slotted geometries is probably one of themost challenging aspects of FB modeling. It is thereforenot surprising that literature describes a large number oftechniques to model slotting effects. These techniques can bedivided into three categories: techniques that use conformalmapping, techniques that use exact subdomain modeling andtechniques that allow for subdomains with variable perme-ability. All of these techniques will be discussed in thefollowing.

A. Conformal Mapping

Models that apply conformal mapping do not directly cal-culate the field in a slotted geometry. Instead, the solutionfor a slotless machine is calculated, and the slotting effect isaccounted for a posteriori, by multiplication with a permeancefunction. The determination of that permeance function isperformed by consecutive conformal transformations, whichmap the slotted geometry in its slotless equivalent. One ofthese transformations is the Schwarz–Christoffel transforma-tion. Therefore, models that apply conformal mapping areoften referred to as Schwarz–Christoffel models. A detaileddiscussion on conformal transformations is beyond the scope

of this work, but papers that clearly describe the mappingprocess may be found in [37], [62], [69], [70].

The simplest permeance functions are 1-D, i.e., they onlyvary in the ρ-direction. This is, of course, a very coarseapproximation; the effect of slotting on the magnetic fielddiminishes when moving away from the slots.

Zhu et al. [37] proposed a 2-D relative permeance function.By accounting for the permeance function’s η-dependence,a more accurate result was obtained. Because of their sim-plicity, 2-D relative permeance functions have been widelyused in the past. However, as was shown in [70], a relativepermeance function cannot accurately predict the tangentialcomponent of the magnetic field. The inaccuracy of the field’stangential component may result in a significant error in somepost-processing calculations, such as the torque [71]. To avoiderrors in the field’s tangential component, Žarko et al. [70]proposed a more complete solution of the conformal transfor-mations by considering their complex nature. Although thistechnique is more complex, it results in a much more accuratemodel.

All of the above techniques provide closed-form solutionsfor the magnetic field. This makes those models very interest-ing in terms of computational time and ease of use. However,the transformations introduce distortions of the geometry.Especially the magnet edges and paths with a constant η areaffected. The latter is important because such paths are oftenused as integration paths, e.g., to compute the torque. More-over, the above permeance functions are determined assuminga single slot. This implies that the influence of neighboringslots is neglected. To avoid errors due to deformation ofthe geometry and/or neglection of neighboring slots, variousauthors have recently used numerical techniques to performthe mapping process [62], [72], [73]. An additional benefit ofthat technique is that it allows for more complex geometries.

B. Exact Subdomain

Whereas models that apply conformal mapping accountfor slotting a posteriori, exact subdomain models directlyaccount for the slotting effect [16], [17], [57], [74]–[78].This is achieved by considering each slot as a non-periodicsubdomain.

The advantage of that approach is that it is much morestraightforward than models that are based on conformal map-ping. Moreover, it was shown in [48] and [79] that the exactsubdomain method is more accurate than the technique withcomplex permeance functions. More specifically, [48] com-pared three analytical methods to the finite-elements method(FEM) solution: the method with simple permeance functions,the method with complex permeance functions, and the exactsubdomain method. Only the subdomain method was capableto give the correct results.

The most important downfall of the exact subdomainmethod is that its computational time is rather high. The reasonfor that is twofold. First, by introducing every slot as a separatesubdomain, the number of subdomains rises significantly.As this also implies more integration constants, the timerequired to solve the system of boundary conditions increases.


Second, whereas models with permeance functions will onlycontain periodic subdomains, exact subdomain models haveboth periodic and non-periodic subdomains. This implies thatboundary conditions have to be imposed between subdomainswith different periodicities, which in turn implies that theboundary conditions cannot be imposed for every time- andspatial-harmonic combination separately. As a result, the sizeof the system that has to be solved may become very large.

Despite these downfalls, the technique’s accuracy and itsstraightforward approach have made it the most widely usedtechnique to account for slotting. To counter the relatively highcomputational times, Ackermann and Sottek [80] presentedan approximation that does allow for a closed-form solution.In addition, the techniques that were discussed in Section II-Dalso reduce the computational time.

Until recently, non-periodic subdomains could only be con-sidered if they are enclosed by infinitely permeable materialat their periodic boundaries. However, Dubas et al. [81] havepresented a technique that does allow for regular non-periodicboundaries. They do this by rewriting the magnetic vectorpotential as the superposition of a part that is periodic in theρ-direction (A(νp)(η, ρ, t)) and a part that is periodic in theη-direction (A(νη)(η, ρ, t))

A(ν)(η, ρ, t) = A(νρ)(η, ρ, t) + A(νη)(η, ρ, t) (54)

where

A(νρ)(η, ρ, t) =∞∑

n=−∞

∞∑k=−∞

A(νρ)n,k (η)ej(

2kπT (ν)

(ρ−ρ(ν)0

)−nωt

)

A(νη)(η, ρ, t) =∞∑

n=−∞

∞∑k=−∞

A(νη)n,k (ρ)ej(

2kπT (ν)

(η−η(ν)0

)−nωt

).

(55)

The periodic boundary conditions are then imposed onA(νη)(η, ρ, t) while at these boundaries A(νρ)(η, ρ, t) is forcedto be zero and similarly, the normal boundary conditions areimposed on A(νρ)(η, ρ, t) while A(νη)(η, ρ, t) is kept zero.

This superposition technique is a relatively simple way toaccount for any non-periodic subdomain. This does not onlyimply that teeth with a finite permeability can be accountedfor, but it also means that the magnetic field can be calculatedin the soft-magnetic parts of a slotted geometry. This wasnot possible with earlier techniques. However, the amount ofintegration constants is again increased, which will furtherincrease the computational time.

C. Variable Permeability

Recently, Sprangers et al. [82] have proposed a techniquethat allows for subdomains in which the permeability varies.To achieve this, a Fourier series is used to express thepermeability’s variation along the ρ-direction and the con-stitutive relation between the magnetic flux density and themagnetic field strength is written using convolution matrices.The differential equation is then formulated in its matrix form,so that it can be solved with variable permeability.

Spranger’s approach is very interesting since, like Dubas’superpositions technique, it enables calculation of the field in

the soft magnetic parts of a slotted structure. Moreover, it isnot restricted to subdomains with a variable permeability; thesame principles can also be used to account for subdomainswith a variable conductivity [83]. Apart from its complexity,the main downfall of this technique is that it suffers fromGibb’s phenomenon, i.e., the Fourier series representing thepermeability does not converge at the boundary between a slotand a tooth. This introduces inaccuracies in the computationof the field and results in higher computational times.

D. Combination of an Analytical Method With a MagneticEquivalent Circuit

As modeling slots in electrical machines is not very easyusing analytical methods, another approach is based on acombined solution of two models. The first model solves ananalytical model on a simplified (slotless) geometry to producean exact solution of Maxwell’s equations. The second modelgives the solution in the parts of the machine with complicatedgeometry, e.g., the teeth and the slots. This is typically a MEC.Both models are coupled, typically via the flux density patternin the air gap. This method was applied in [24] for axial fluxPM machines, but it is generally applicable.

The method has another advantage compared to conven-tional analytic models: it can easily account for non-linearityand saturation effects. Compared to FEMs, the combinedanalytical-MEC approach is much faster and much easier toparametrize. Compared to a pure MEC model, the combinedmodel does not need varying reluctances in the air gap of therotating machine, which are usually the most difficult to obtainin a MEC.

E. Conclusion

Four techniques to account for slotting were introduced.1) using a permeance function that alters the magnetic

field’s computation a posteriori [37], [62], [69], [70];2) considering each slot, and possibly each tooth, as a

separate subdomain [16]–[18], [74], [81], [84];3) allowing for subdomains with a variable

permeability [79], [82];4) combining an analytical method with a MEC [24].

Among purely analytical techniques, only Dubas’ superposi-tion technique and Spranger’s technique with variable perme-abilities are capable of accounting for soft-magnetic materialswith a finite permeability and computing the field in thesoft-magnetic parts of a slotted geometry.

The advantage of using permeance functions is that rel-atively good accuracy can be achieved with a closed-formsolution, especially when using complex permeance functions.However, as discussed in [85], using complex permeance func-tions implies that the results cannot be integrated analytically.This may be unwanted, for example, if the torque has to bestudied.

Because of their straightforward approach and very highaccuracy, exact subdomain models are nowadays the mostwidely used models that account for slotting. Using thesuperposition technique adds another advantage to this type of


models, the possibility to compute the field in the soft-magnetic parts of a slotted structure.

Despite their complexity and problems with the Gibb’sphenomenon, models that account for subdomains with avariable permeability are definitely an interesting option tocalculate the field in slotted structures.

Note that quite some comparative studies of differenttechniques to account for slotting have been published inliterature [48], [79], [86].

VIII. ECCENTRICITY

Although not a lot of FB models account for eccentricity,it is worth mentioning that there are two techniquescapable of doing so. The first technique uses perturbationfunctions, the second technique accounts for eccentricity viasuperposition.

A. Perturbation Functions

To account for eccentricity of the rotor, Kim and Lieu [87],[88] used two coordinate systems, one in the center of thestator and the other in the center of the rotor. The relationbetween both coordinate systems was expressed using a per-turbation function. Although Kim obtained very good results,his technique is rather complex.

B. Superposition

Li et al. [89] avoided the complexity of perturbation func-tions by using superposition instead. Their approach is todivide the eccentric machine in a number of sections. For eachof these sections, the equivalent air gap length is determined.Next, a non-eccentric machine is studied for each of these airgap lengths. The obtained results are then put together so thateach section of the original eccentric machine corresponds tothe correct section of a non-eccentric machine. Clearly, thistechnique is much simpler than Kim’s perturbation technique.However, it will introduce errors, especially at the boundariesbetween sections. Moreover, as different computations arerequired to study a single machine, a higher computationaltime may be expected. Nevertheless, Li presented very goodresults for the magnetic flux density and the back electromotiveforce (EMF).

C. Conclusion

This section discussed two techniques to study the effectof rotor eccentricity on the machine’s magnetic field. Themajor advantage of Li’s superposition technique is its sim-plicity. However, it is expected to have a non-negligible error.Especially if quantities that are sensitive to errors in themagnetic field, such as torque, have to be studied. Despitethe perturbation technique’s higher complexity, it is morestraightforward.

IX. PHYSICAL OUTPUT QUANTITIES

All of the above aspects are very specific to FB modeling,i.e., they apply to the calculation of the magnetic scalar

potential or the magnetic vector potential. However, thesepotentials are only of interest if they can be translated to themachine’s physical output quantities, such as the flux density,back EMF, torque, and eddy currents. As the computation ofthese quantities is not specific for FB models, they will onlyvery briefly be discussed here.

A. Magnetic Flux Density and Magnetic Field

The magnetic flux density and the magnetic field canreadily be calculated from the constitutive relation (3b) andthe definition of the magnetic potentials, i.e., (4) or (7).In most of the publications on FB modeling, results forthe magnetic flux density are used to validate the presentedmodel [28], [70], [82].

B. Back Electromotive Force

The back EMF of a coil can be found from the integralform of Faraday’s law, which can be written in terms of theflux linkage (ψ) as

e(t) = −∂ψ(t)∂ t

. (56)

The physical flux is found as the integration of the magneticflux density over a surface of the coil.

Note that the exact position of each coil is usually notavailable. Therefore, the boundaries of the integration surfaceare not exactly known. To cope with that, most authors ofslotless machines assume the conductors to be located in thecenter of each slot [84], [90]. In machines with stator slots,the magnetic potential is often averaged over the slot in orderto calculate the flux [55], [67], [91].

C. Torque and Forces

The most commonly used way to calculate forces is byintegrating Maxwell’s stress tensor over the surface of thevolume on which the force acts

F =∫∫

S�ds (57)

where � is Maxwell’s stress tensor, which can be calculated as

� = μ0(n · H)H − μ0

2(H · H)n (58)

where, in turn, n is the unit vector, normal to the integrationsurface.

The torque is then calculated by multiplying with theradius [17]. Note that, alternatively, the torque could be calcu-lated using Lorentz force or Poynting’s theorem. It is beyondthe scope of this work to elaborate on that, but more informa-tion can be found in [92]. Also, comparison of cogging torquecalculated through different FB methods can be found in [93].

D. Eddy Currents and Eddy-Current Losses

From Faraday’s law (1a) and the constitutive relations (3a),the current density in any subdomain can easily be written interms of the magnetic potential. Note that it is not necessaryto have considered the eddy-current reaction field to do an aposteriori calculation of the eddy currents.


The eddy-current loss is calculated as the volume integralof the eddy current divided by the electric conductivity [44],[57], [94]–[96]

Pec =∫∫∫

V

J · Jσ

dv. (59)

Evidently, (59) reduces to the multiplication of the stack lengthand a surface integral in 2-D models.

The problem with (59) is that, in the polar and cylin-drical coordinate system, the equation for J may includemodified Bessel functions. As the analytical integration ofthese functions is difficult as it often involves hypergeometricfunctions, the Poynting theorem is often used to calculatethe eddy-current losses [17], [97], [98]. By integrating thePoynting vector (S) along a surface, the power passing throughthat surface is calculated

PS =∫∫

SS · ds

=∫∫

SE × H · ds. (60)

If the integration surface encloses the rotor, integration of Sgives the total power, i.e., the sum of the mechanical powerand the power losses, that goes from the stator to the rotor.By subtracting the mechanical power, which can be calculatedfrom the torque, the eddy-current losses are isolated. As anexample, for the design shown in Fig. 1(b) it is shown thatthe knowledge of the field in the air gap of the machine issufficient to know the power losses in the rotor [17].

Note that there are other techniques to isolate theeddy-current losses from the mechanical power; in [98],Joule’s equation (59) is used, and in [92], calculation of PS

in rotor coordinates is combined with an interpretation of theslip of individual harmonic components of the magnetic field.

Note as well that traditional FB models cannot accountfor segmentation of conductive subdomains in the l-direction.To overcome that issue, Nair et al. [99] coupled a classical2-D FB model with the current vector potential in. In addition,Custers et al. [83] used a technique similar to Spranger’s tech-nique of variable permeabilities to account for segmentationof conductive subdomains.

As the Poynting theorem directly applies the solution ofthe magnetic field, it can only account for eddy-current lossesof subdomains in which the eddy-current reaction field isconsidered.

E. Conclusion

The above is a very brief introduction on some of the mostimportant quantities that can be calculated from a FB model.The way in which the magnetic flux density, the back EMF,the forces, the torque, the eddy currents, and the eddy-currentlosses are calculated was introduced. One important remarkwith respect to FB modeling is that the classical way ofcalculating eddy-current losses, i.e., using Joule’s equation,cannot always be used. An alternative that uses Poynting’stheorem was proposed.

X. CONCLUSION

The goal of this work was to provide an overview of theFB modeling technique. This was done in two steps.

First, Section II discussed how to formulate and build a FBmodel, i.e., how to choose its subdomains, solve the PDEs,and impose the boundary conditions. In addition, typicallimitations and computational considerations were discussedin Section II as well.

Second, all of the most important aspects of FB modelswere discussed in Sections II-F– IX, including a comparison ofdifferent techniques to cope with those aspects. These sectionsalso contain many references to the most relevant publicationson each of those aspects. They make it possible to obtain botha general overview and a sufficient level of detail concerningFB modeling of electrical machines.

As shown in this paper, FB modeling techniques providean in-depth understanding of electrical machines and alsoprovide an efficient method to calculate machines with lowcomputation time. Nevertheless, when magnetic saturation istaken into account this type of modeling technique may take animportant computation time. Although some work has alreadybeen done, we think that there is much research still to be donein coupling fast FB techniques and techniques that can takesaturation into account, such as reluctance network techniques.

Also, some publications involving 3-D calculation havealready been published, much is still to be done in order toobtain more general 3-D models.

APPENDIX

MAGNETIC POTENTIAL FORMULATIONS

FB models extensively use potential formulations to sim-plify the mathematical solution of Maxwell’s equations. In thisappendix, the derivation of those potentials is presented.

A. Magnetic Scalar Potential

The magnetic scalar potential’s governing equation caneasily be found by substituting its definition (4) in Gauss’slaw (1c) while accounting for the magnetic constitutiverelation

∇ (μ (−∇ϕ)+ B0) = 0 (61)

which is then rewritten as

∇2ϕ = ∇ · B0

μ. (62)

Note that, as mentioned in Section II, the magnetic scalarpotential only applies to current-free regions. Indeed, fromthe magnetic scalar potential’s definition, it can be seen thatusing the magnetic scalar potential implies that H is assumedto be a gradient vector field. Considering Ampère’s law, thismeans that the current density has to be zero.

B. Magnetic Vector Potential

The derivation of the governing equation for the magneticvector potential requires some more calculus. In a first step,


the definition of the magnetic vector potential (7) is substitutedin Faraday’s law

∇ × E = − ∂

∂ t∇ × A. (63)

The integration of the above gives

E = −(∂A∂ t

+ ∇V

)(64)

where the integration constant is the gradient of the electricscalar potential. Indeed, (64) matches the definition of V (8).

A second step in the derivation is choosing a gauge. Thisis important because the above potential formulation is notuniquely defined. Indeed, if f is an arbitrary scalar field,an alternative formulation can be defined as{

Aalt = A + ∇ f

Valt = V − ∂ f∂t .

(65)

The lack of uniqueness can now be illustrated by consideringthe magnetic flux density

B = ∇ × A

= ∇ × (A + ∇ f )

= ∇ × Aalt (66)

and the electric field strength

E = −(∂

∂ tA + ∇V

)

= −(∂

∂ tA + ∂∇ f

∂ t+ ∇V − ∇ ∂ f

∂ t

)

= −(∂

∂ tAalt + ∇Valt

). (67)

This redundant degree of freedom is tackled by introducinga gauge. When using the magneto quasi-static approximation,the Coulomb gauge (∇ · A = 0) is most commonly used.

For the third and final step of the derivation, Ampère’slaw (1b) is combined with the definition of the magnetic vectorpotential (7), the constitutive relations (3) and the equation forthe electric field strength (64)

∇ × ∇ × A=−μσ(∂A∂ t

+ ∇V − v × (∇ × A))

+∇ × B0.

(68)

Using the identity for the curl of the curl and Coulomb’sgauge, the governing equation for the magnetic vector poten-tial can finally be written as

∇2A − μσ∂A∂ t

+ μσ(v × (∇ × A)) = μσ∇V − ∇ × B0.

(69)

In (69), externally imposed current densities are accountedfor by the μσ∇V term in (69). Here, V is the electric scalarpotential in the considered problem. In the stator windingsof an electrical machine for example, V is the terminalvoltage. However, directly accounting for V would imply thatevery conductor has to be modeled separately. Instead, Jextis imposed directly. Assuming a generator reference, this is

done by substituting σ∇V by −Jext in the governing equationfor the magnetic vector potential. The result is an alternativegoverning equation

∇2A − μσ∂A∂ t

+ μσ(v × (∇ × A)) = −μJext − ∇ × B0.

(70)

Note that most modern electric drives are powered with thehelp of a voltage-source inverter. This fact advocates the useof (69) as a governing equation. However, as mentioned,the implementation of V as a source term is complex.Section VI discusses a technique that accounts for voltagesources without having to use (69).

ACKNOWLEDGMENT

This work was supported in part by the National NaturalScience Foundation of China under Grant 51650110505 andGrant 51851110761.

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