Technical articleThe art of modelling Electrical MachinesAbstract Up to date, numerical techniques for "low frequency"electromagnetic questions have been employed extensively for allpossible types of application. Due to its complicated geometry onthe one hand, the numerical computation of electrical machines wasalways and is still challenging. On the other hand, maximum condi-tions e.g. the aspect ratio of the devices dimensions can be numeri-cally troublesome and can let 3d numerical models grow up to hugesystems which are difficult to handle. Furthermore, the modellingof particular physical effects can be difficult due to the particularboundary conditions in time or space, which must be fulfilled.

In the beginning of Computational Electromagnetics, many au-thors amongst others the well known Peter Silvester [52] introducedto the engineering community the Finite Element Method (FEM) inmathematical theory and by useful Fortran program code. On thisbasis, the fundamental questions of stationary, harmonic or tran-sient behaviour of electromagnetic devices could be tackled at thattime. The FEM was ready for electric motor simulations [43]. Re-finements and extensions of the FEM followed up to day. Withincreasing computational power, model refinements to the class ofcoupled problems appeared [29]. This coupling of physical effectsby numerical models allows to better understand the devices be-haviour. Methods are coupled to methods to e.g. increase the com-putational velocity of the solution process. An attempt for a sys-tematic classification of coupled problems can be found in [28].However, the main goal of such efforts was to explore an overallbehaviour of the device under study including all possible physi-cal side effects simultaneously. As a consequence of this couplingapproach, confusing many parameters describe the device understudy. To unscramble the impact of the parameters, numericalmodels are developed to answer very specific questions [47].

From the numerical magnetic field solution derived parameters,such as e.g. inductivities, representing another class of model, helpto understand the interrelation between physical effect, its modeof operation and its particular source. Therefore, such analyticalapproaches as the lumped parameter model, are still valuable forthe understanding. Numerical solutions deliver the accurate over-all result of a field problem, reflecting the state of energy in theelectromagnetic device. However, the inference from that numeri-cal solution back to the single source of effects is not possible. Thecoupled approach, having simultaneously all sources considered ina large numerical model does not allow to fully understand the de-tailed behaviour of the device and its dependency on the outer pa-rameters. To cope with this dependency of parameters carefullyto the final results, subproblems have to be defined, which employparticular boundary conditions to consider or not single sourcesrespectively physical effects. As a consequence, a lot of simulationswith meaningful parameter variations have to be performed to fullyunderstand the device under study in the physical way. By spendingsuch efforts, it is possible to determine important parameters whichdominate the behaviour of the electromagnetic device. Things aregetting more complicated by considering coupling effects which areof thermal or mechanical origin.

Another way to cope with the mentioned problem of identifyingand evaluating parameters is to develop a methodology to couplethe numerical approach to analytical models. With this approach itis possible to separate a physical source from its effect in the model.This provides the possibility of determining the significant modelparameter and its mutual interaction with others. Due to particularlimitations of analytical models, for example such as non-linearityor complicated geometry, this approach is not in general applicable.However, particular and specific problems can be solved in an effi-cient way. This paper is a contribution which represents a method-ology to better understand an electromagnetic device, such as anelectric motor, and which enables to give an innovative impulse tothe aspects of computation in electromagnetics.

I INTRODUCTION

The modelling of electrical machines is described and presentedin a large pool of scientific publications and its area of science of-fers a large treasure trove of experience. Both, analytical modelsas well as numerical models are used to compute the behaviourof the motor devices.

The main goal of a numerical field simulation is to explorethe overall behaviour of the device under study, including allpossible physical side effects simultaneously.

Numerical field solutions deliver the accurate overall result ofa field problem, reflecting the state of energy in the electromag-netic device. However, the inference from the final numericalsolution back to the single source of effects is not possible. Tounderstand the behaviour of the electric motor and to be able toenhance it, the accurate knowledge of the parameters governingthe physical effect must be known. So called coupled problemscan distinguish between the single effects and may help to betterunderstand the effects and their mode of operation. The couplingof methods plays an important role [28]. As a consequence of acoupled approach, confusing many parameters describe now thedevice under study. To unscramble the impact of the parameter,specific models are developed to answer the very specific ques-tions. Such particular models are discussed in this contribution.

Figure 1: Speed vs. accuracy of various field computation ap-proaches.

Figure 1 shows the computational speed versus the obtainedaccuracy of various field computational approaches. Analyticalmodels are of course fast when compared to the multi physicsapproach. The coupling of the field simulation methods is a pos-sible way to obtain fast and accurate field computation.

In this contribution we want to present exemplarily, in whichway electrical machines can be modelled in an efficient way andwant to reveal that analytical models are still valuable to advo-cate the physical understanding of the device under study. On theother hand, the coupling of analytical and numerical models canreduce the overall computational efforts significantly. Lumpedparameter models e.g. arranged in an iterative loop with a nu-merical FE model shorten the computation time and at the sametime enable the possibility of having a very fast parameter modelof the machine to determine its dynamic behaviour.

This article is therefore organised as follows. The typical de-sign process of an electrical machine, in our case based on anautomated FE-calculation chain, is presented in section II. Theexistence of a valid design candidate is assumed. The result

possibly after a set of design iterations is a machine geome-try and operating figures as e.g. power losses or efficiency. Thisdata can subsequently be used in the analysis of advanced designaspects as for instance a cause-effect dependency analysis, pre-sented in section III. The conjunction between cause and effectthere is built using an extended conformal mapping approach.Equally important to a cause-effect dependency analysis is a de-tailed investigation of the iron losses inside the machine, separat-ing the overall iron losses into eddy current losses and hysteresislosses. A possible approach for this purpose is given in sectionIV. The therein first stated course of action however still is basedon analytical post-processing formulae, neglecting the hystereticmaterial behaviour in the field model and assuming a homoge-neous magnetic field distribution across the lamination. An im-proved approach can be based on a coupled magneto-dynamicvector hysteresis model, as explained in the sections end. Sucha model nevertheless requires a solver providing the magneticscalar potential in 3D for rotating machines. As a consequence,section V presents some ideas for a possible generic solver ap-proach. Until here, all calculations and considerations have beenbased on the assumption of ideal machines, which present geo-metrical and electric symmetric properties. The considered air-gap field of the device under study hence shows a spatial peri-odicity along its circumference and iterates in dependence of acertain angle. Reality however shows that due to tolerances inthe manufacturing process and other deviations the assumptionof an iterating air-gap field is not valid. Section VI accordinglydeals with possibilities to include such stochastic considerationsinto the presented design chain. A typical consequence of thesestochastic deviations is the rise of new harmonic orders, result-ing in torque ripple, losses and noise in particular. On thesegrounds section VII gives some insight in the calculation of ma-chine acoustics. Upon this, section VIII shows in which waythe produced results can be displayed in a virtual reality envi-ronment and how to manipulate them interactively. Section IXconcludes with the determination of lumped parameters, as theyare required for building dynamic control models of the machinein their later use and leads to a final summary.

II DESIGN PROCESS

During the design process of an electrical machine different cal-culation tools must be combined according to the specific ma-chine requirements, the design goals and the requirements of theapplication. Each set of these three design aspects demands fora certain combination of the respective tools.

Figure 2 shows the design approach for calculating a two-dimensional efficiency map of a permanent magnet synchronousmotor (PMSM). After the efficiency map is calculated, furtherdesign aspects can be analysed (sections VI - VII) and the re-sults can be presented by means of virtual reality (section VIII).

To minimise the time required for the design process an au-tomated design approach and parallel computing is introduced.As an example, the automated design process for calculating theefficiency map is presented. Simulations include losses and theevaluation of inductances which are discussed in sections IV andIX in detail.

A Automatisation

Figure 2 contains a flowchart describing an automated processfor the calculation of all relevant machine characteristics to gen-erate a two-dimensional efficiency map. In the following, thefunction of each block from this graph is briefly described. Adetailed explanation is presented in [27] . All simulations wereconducted with the in-house FEM software package iMOOSE.

Virtual Reality

Locked Rotor Module

loss computation

Design Candidate

Load Module and

design aspectsAnalysis of further

No Load Module

Demag. Module

Inductivity Module

Jmax

Ld,Lq (J,)opt(J)

Pcu,Piron,Peddy(M,n)

Ui

(M,n)

(Base Speed Range) (Field Weakening Range)

Tmax(J)

Figure 2: Flowchart of the design process for a PMSM.

Initially, a No-Load Simulation is performed to calculate thestator phase flux-linkage and the back-emf, as well as to evaluatethe occurring higher harmonic field wave. Moreover, this com-putation step detects the position of the d- and q-axis for furtherprocessing steps.

The electromagnetic overload capability of permanent magnet(PM) motors is limited by the demagnetisation strength of thepermanent magnet material. To determine the specific overloadcapability of a given design candidate, a Demagnetisation Testis conducted. Hence the maximum current Imax is determined,still having a working point on the linear part of the demagneti-sation characteristic to avoid an irreversible demagnetisation.

The torque of a PMSM with internal magnets (IPMSM)(Xq > Xd) consists of the synchronous torque Tsyn and thereluctance torque Trel and is given by:

T =3p

UpIq Tsyn

3p IqId (Xq Xd)

Trel

. (1)

where p is the pole pair number, the angular frequency and Upthe back-emf. Inserting Iq = I cos, Id = I sin, where isthe field-weakening angle, the torque can be rewritten as

T = Tsyn cos() Trel sin(2) (2)

i.e. the sum of a fundamental (Tsyn) and the first harmonic (Trel)which are constant for a given current. For operation points inthe base speed range the phase voltage Us is below the maxi-mum voltage, so that the phase current is constrained by the mag-net demagnetization and further thermal limitations or the powerelectronics maximal current. Differentiating (2) with respect to, one sees that the maximum torque Tmax per current (MTPA-control) is realised for the so called optimal field-weakening an-gle opt. The Locked-Rotor Test is thus made to determine the

absolute values of the synchronous torque Tsyn and the reluc-tance torque Trel, as well as to determine the optimal field-weakening angle opt and the maximum torque Tmax. Thiscalculation is performed for a stepwise increasing stator-currentdensity in order to capture the dependency of those quantities onthe load current.

For operation points in the field-weakening range, the con-trol strategy of the power converter limits the phase voltage toits maximum by shifting . For the simulation of this con-trol strategy, the direct- and quadrature-axis reactances (Xd andXq), which are both functions of the load current and the fieldweakening angle , are calculated by means of the XdXq-Computation. By knowing the reactances Xd and Xq and a cor-responding field weakening angle the electromagnetic que foreach operation point is given by (1).

In order to evaluate and rate the motors energy consump-tion and efficiency, all losses are required to be determinedfor all operation points. This computation is conducted by theOperation Point Simulation. At this step, the optimal field-weakening angle opt is used to set the maximum torque in thebase speed range, whereas has to be set by means of controlstrategies in the field-weakening range. Ohmic losses Pcu areestimated considering end winding effects. Iron losses are com-puted by means of quasi-static numerical FE simulations and animproved post-processing formula based on the loss-separationprinciple considering rotational hysteresis losses as presented insection IV. Eddy-current losses in permanent magnets are calcu-lated for each considered operation point by means of a transient3D-FE approach, as described in [39] and [14].

By performing the loss calculation it is possible to determinethe total losses Ptot for each operation point. Resulting fromthe total losses and the input power Pin, the efficiency can becalculated in function of speed and torque. The results can bevisualised as two-dimensional colour maps, e.g. efficiency mapsas depicted in figure 3. In order to generate such a map, the basespeed region is discretised by N1Map operation points, whereasthe field-weakening region is sampled by N2Map points.

Figure 3: Exemplary result of the efficiency map of a PMSM.

B The Aspect of Parallelisation

The primary requirement of the development process for electri-cal machines is to simulate each design candidate over its entiretorque-speed range. The proposed design process, figure 2, iscapable of this by computing the non-linear machine character-istic as well as the T-n diagram by numerous 2D transient, quasi-static FE computations. As a consequence, the determination onthe overall machine behaviour requires a high computational ef-fort. This yields, in the case of a sequential processing, to along simulation time. To limit and minimise this time delay inthe design procedure, the necessary FE simulations can be pro-cessed in parallel. Reorganizing the flowchart of figure 2 into atimeline diagram shows that the response time can be shortenedto a minimal duration of four FE simulations (figure 4). Since

Number of

Parallel Processes

1

0

Inductivity M.

time

Demag. M. Locked Rotor M.

Map (Region 1)

Base Range

Map (Region 2+3)

NoLoad M.

Weakening Range

D/QAxis

max. Current

Machine

Chracteristic

NJ

FE Simulation

2 NJ

TFE 2 TFE 3 TFE 4 TFE

N1

Map

Fre

quen

cySca

ling

N1

Map + N2

Map

Figure 4: Parallelisation of the automated design process astimeline diagram, showing the necessary number of parallel pro-cesses in function of time.

the computational blocks, such as No-Load Simulation and De-magnetization Test are interdependent, the load is unbalancedgrowing with each timestep (TFe), where all modules whichidentify the machine characteristics are carried out by a currentdiscretisation of NJ .

The automated virtual prototyping described in section II-Aand II-B has been applied in several projects. Application ex-amples are the PMSMs developed for a parallel hybrid elec-tric vehicle in the context of the Federal Ministry of Economicsand Technology cooperative project Europa Hybrid [12], [13],[57] or for a battery electric vehicle within the framework of theFederal Ministry of Education and Research research project e-performance [1].

The next section will discuss a hybridisation of a classical con-formed mapping approach for an air gap field computation ofelectrical machines by several FE-Reparameterations. The im-plementaion of such hybrid models is conducted as part of theabove discussed automated tool chain for FE-Computations. Byapplying this, each level of hybridisation can be applied withoutan additional effort. By this approach, the increase of complexityand control strategy of each step does not constrain the applica-bility and simplicity of the model.

III CONFORMAL MAPPING AND NUMERICPARAMETERISATION

Numerical methods such as the FEM are usually applied for thefield calculation in electrical machines. This method is charac-terised by high level of detail the modelling, such as non-linearpermeability of iron and exact modelling of the geometry in-cluding uncertainties due to the manufacturing. However, thisapproach does not allow for a detailed inside to causes-effect de-pendency. A well-known approach to perform the field calcula-tion of electrical machines is based on conformal mapping (CM).This approach is based on different Ansatz-functions for differ-ent physical effects, such as slotting effects of rotor and statorfield or eccentricity. In the classical approaches non-linear mate-rial properties and uncertainties can not be described. Therefore,we propose a deduction of Ansatz-functions based on of FEMsimulations. The basics of the CM approach and the deductionof additional Ansatz-functions are described.

A Standard Conformal Mapping

The air gap field computation by conformal mapping is gener-ally obtained from the solution of a linear Laplace problem, as-

suming the magnetic core has an infinite permeability. Since thissystem is linear, the field excitation by magnets and coils, as wellthe influence of the slotting, can be modelled individually.

Assuming a slotless stator, the field ~B () at a certain coordi-nate angle in the air gap, [0, 2pi[, consists of a radial fluxdensity Br () and a tangential flux density B ()

~B () = Br () ~er +B () ~e. (3)The angle dependent quantities Br () and B () can be ex-panded into a Fourier Series

~B () =

n=0

(Br,n ~er +B,n ~e) enp, (4)

where n is the frequency order and p the number of pole pairs.In this representation of the air gap field, the Fourier coefficientsBr,n and B,n are the solution of a linear Laplace problem withmagnets and a slotless stator depending on the magnetizationconfiguration [63], [64], [30]. The field at a certain instance oftime t due to rotor movement is given by

~B (t) = ~B ert (5)where r is the angular speed of the rotor.

Stator slotting significantly influences the magnetic field dis-tribution. It is modelled by "permeance functions". Thesepermeance functions ~ represent the impact of slotting on theslotless field distribution and can be obtained by Schwarz-Christoffel transformations [60], [61]. Correlating the field dis-tribution with slotting, s ~B (t), with the field without slotting (5),yields the vectorial permeance ~

s ~B (t) = ~ ~B (t) (6)~ =

(r r

). (7)

The air gap field excited by the armature current is given by

a ~B (t, I, ) =

p ~B

(2Ie(st+0

))

p ~B(

2Ie(st+120))

p ~B(

2Ie(st+240))

e(q++0

)

e(q++120)

e(q++240)

,(8)

where p ~B is the normalised field due to one phase and also ob-tained by conformal mapping [4]. In (8) the angle q definesthe relative phase orientation to the quadrature axis of the ma-chine, is the flux weakening angle and s is the stator currentangular frequency. We shall in the sequel systematically omitthe coordinate and the time argument t, and only indicate thedependency in I and when applicable. We shall also label thequantities obtained by the conformal mapping approach with aCM exponent. The overall air gap field is thus defined as

g ~BCM (I, ) = a ~BCM (I, ) + s ~BCM . (9)

B Leakage and Non-linearity

The main idea of rewriting the CM governing equations sketchedin section A is to obtain a CM formulation in which each para-meter represents a physically motivated quantity in order to dis-tinguish their origin within the electromagnetic field computa-tion. The definition of the armature field a ~B in (8) includes animplicit formulation of ~ in the Ansatz-function of the field p ~B.For further purposes, ~ must be factorised

g ~BCM (I, ) = ~ (~BCM + au

~BCM (I, ))

, (10)

where an arbitrary permeance state ~ is identical for bothfield fraction of g ~BCM (I, ). In (10) the auxiliary fieldau~BCM (I, ) is defined by

au~BCM (I, ) =

(~)1

a ~BCM (I, ). (11)

C Ideal Case

The comparative study [59] for modern analytical models for theelectromagnetic field prediction concludes that the relative errorof the air gap flux density obtained by CM with respect to FEprediction increases strongly in function of the slot-opening toslot-pitch-ratio. In order to analyse this with CM field computa-tion, a PMSM with a slot-opening-factor of 43% is studied. Thecross-section of the motor is depicted in Fig 5. All parameters

Figure 5: PMSM Cross-Section.

Table 1: PMSM Parameters.

3 Number of Pole Pairs18 Number of Stator Teeth

0.73 Pole Pitch Factor3 mm Permanent Magnet Height

24.5 mm Rotor Radius (inc. PM)0.8 mm Air Gap Height54.2 mm Outer Stator Radius

43 % Slot Opening Factor1.35 T Remanence Flux Density

101 mm Length4 mm Yoke Width

of the geometry and the electromagnetic evaluation are given inTable 1. The torque computation by Maxwell stress theory incase of CM is given in [23]. Figure 6(a) compares the coggingtorque in function of rotor position and the corresponding timestep obtained by (10) and by a FE computation where the sta-tor is modelled as infinite permeable iron (IFEA), utilizing Neu-mann boundary condition. One observes that in several rotor po-sitions the torque results are close to each other, but in other an-gle ranges CM overestimates the integral quantity torque, whichis in agreement to the statements of [59]. The maximum devia-tion occurs at step three, where the edge of the magnet is alignedwith the center of the slot. The corresponding flux density dis-tribution, depicted in figure 7, shows a significant increase in itstangential component evoked by a flux path deformation in di-rection of the stator teeth. This effect is also known as slot leak-age [46], and cannot be covered by the scalar vector quantity ~in (6). Adding this effect as optional correction, (10) becomes

g ~BCMMod (I, ) = ~ (~BCM + au ~B

CM (I, ) + ~B)

,

(12)where ~B is the slot leakage of the rotor field. ~B can bedetermined by a single IFEA in no-load case. A comparisonof the torque results in rated operation in case of IFEA and theapplication of (10) and (12) shows that the deviation of standardCM vanishes, see figure 6(b) and figure 7.

D Non-linear Case

The formalism in C to obtain (12) can also be applied in orderto represent saturation. In this case, the permeance state ~ turnsfrom a constant quantity into a function of the magnitude of thecurrent I and the flux weakening angle , yielding

g ~BCM (I, ) = ~ (I, ) (~BCM + au

~BCM (I, ) + ~B)

.

(13)

-2.5-2

-1.5-1

-0.5 0

0.5 1

1.5 2

2.5

0 2 4 6 8 10

0 10 20 30 40 50 60To

rque

[Nm]

Computational Step

Rotor Position in Electrical Degree

IFEACM

CM-Mod

(a) Cogging Torque.

22

23

24

25

26

27

28

29

30

0 2 4 6 8 10

0 10 20 30 40 50 60

Torq

ue [N

m]

Computational Step

Rotor Position in Electrical Degree

IFEACM

CM-Mod

(b) Torque for J = 6 Amm2

.

Figure 6: Comparison of cogging and rated torque obtained byIFEA, CM and CM-Mod.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

90 100 110 120 130 140 150

Rad

ial F

lux

Dens

ity [T

]

Circumferential Position (degree)

IFEACM

CM-ModRotor

Slot

(a) Br

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

90 100 110 120 130 140 150

Rad

ial F

lux

Dens

ity [T

]

Circumferential Position (degree)

IFEACM

CM-ModRotor

Slot

(b) B

Figure 7: Comparison of radial and tangential flux density ob-tained by IFEA, CM and CM-Mod in time step three. (Ro-tor/Slot denote the relative geometry position).

For the evaluation of single points of interest, e.g. rated oper-ation, ~ (I, ) can be identified from a single non-linear FE sim-ulation g ~BFEA (I, ) by

~ (I, ) =(~BCM + au ~B

CM (I, ) + ~B)

g ~BFEA (I, )(14)

to further improve the analytic field computation.

E Evaluation of cause-effect dependency

The classical CM approach lacks in the modelling of non-linearand additional parasitic effects. Still, the CM formalism, theassociated time and space-harmonic analysis and the separationof different effects in Ansatz-functions deliver the possibility toseparate the cause-effect dependency. Therefore, the idea is toestimate each non-ideal phenomenon by means of FEM modelsand deduct a corresponding Ansatz-function. Taking all Ansatz-functions into account, identical results compared to FEM areachieved. When only considering specific Ansatz-function adeeper inside to the causes of particular effects can be reached.Therefore, this method is of particular interest in the optimisa-tion of electrical machines.

IV CALCULATION OF LOSSES IN ELECTRICAL MACHINES

The knowledge of the occurring electrical losses in an electricalmachine in each operating point is eminent for an optimal de-sign. This knowledge allows to consider the thermal conditionsinside the machine for possible cooling requirements as well asthe calculation of the input power and complete efficiency maps.The overall losses Ploss of a PM-machine are a compound ofohmic losses Pcu in the copper windings of the stator, the eddycurrent losses inside the permanent magnet volume Ppm as wellas the iron losses Pfe due to varying magnetisation (in time & inspace) inside the rotor and stator core:

Ploss = Pcu + Ppm + Pfe. (15)

A Estimation of copper losses

The ohmic losses without skin effect for one phase can be calcu-lated by the knowledge of the resistance of an electrical phase ofthe machine Rph and the phase current I having stranded con-ductors by:

Pcu,ph = Rph I2. (16)

B Estimation of eddy current losses in permanent magnets

According to [14], the eddy current losses in the permanent mag-nets Ppm of the electrical machine can be simulated in line witha transient 3-D FE-simulation. The simulations output data ofeddy current density Jec is used by an integration over the per-manent magnets volume Vpm

Ppm =

1

pm

J 2ecdVpm (17)

with the specific electrical conductivity pm of the permanentmagnet material.

C Estimation of iron losses

In the application field of efficient electrical machines there is astrong need for the accurate estimation of iron losses Pfe andtheir sources in a wide operational range of frequency f andmagnetic flux densityB - in particular for the electrical machineswith elevated operating frequencies such as the ones incorpo-rated into hybrid or full-electric drive trains of vehicles and theones under power electronic supply (e.g. pulse-width modula-tion). Such estimation of iron losses occurring in the machinesstator and rotor parts is indispensable in order to effectively per-form electromagnetic and thermal design of these electrical ma-chines. This forms the basis for the selection of the most appro-priate electrical steel grade that suits the specific working condi-tions in the rotating electrical machine under study. Moreover,such a development enables a deeper understanding of the spe-cific trade-offs made during the machine design process in or-der to identify the particular specifications of electrical steels,which could be tailor made for specific applications. Two dif-ferent approaches to estimate the iron losses are presented in thefollowing two subsections: On the one hand, the commonly ap-plied iron-loss is calculated in the post-processing using formula(C1.), on the other hand, an advanced approach aiming at theincorporation of the interdependence of eddy currents and hys-teresis in electrical steel laminations using a magneto-dynamichysteresis model directly in the processing step (C2.) can be ap-plied.

C1. Post-processing iron-loss estimation

For the iron-loss modelling a 5-parameter-formula for high mag-netic flux densities and high frequencies considering the non-linear material behaviour is developed by modifying the Bertotti[3] loss equation [32]. This formula utilizes the well-knownloss-separation principle, that splits up the total iron losses inhysteresis losses, classical eddy current losses and excess losses,combined with an additional term for the increased eddy currentloss component due to the saturated material using two addi-tional parameters a3 and a4 (22). This equation is extended tobe applicable to vector fields and arbitrary magnetic flux den-sity waveforms, since in rotating electrical machines higher har-monics (in time) due to iron saturation, skin effect, stator yokeslots and the use of a power electronics supply (inverter, PWM)can occur, as well as vector magnetic fields (in space), the lat-ter giving rise to so-called rotational losses. A Fourier-Analysisof the magnetic flux density during one electrical period serves

for the identification of the harmonic content. As a consequencethe eddy current and excess loss description are extended witha summation over all harmonics [15] (20, 21). The locus of themagnetic flux density vector over one electrical period is char-acterized by the minimum Bmin and maximum magnetic fluxdensity magnitude Bmax during this. This enables to identifythe level of flux distortion. The hysteresis and excess loss termsare enhanced to model the influence of rotational and flux distor-tion effects (19, 21). This results in the IEM-Formula

PIEM = Physt + Peddy + Pexcess + Psat (18)with

Physt = a1

(1 +

Bmin

Bmax (rhyst 1)

)B2max f (19)

Peddy = a2

n=1

(B2n (nf)2

) (20)Pexcess = a5

(1 +

Bmin

Bmax (rexcess 1)

)

n=1

(B1.5n (nf)1.5

) (21)Psat = a2a3B

a4+2max f2 (22)

and the magnetising frequency f (Hz), the maximum value ofthe magnetic flux density Bmax (T), the material specific pa-rameters a1 a5, the rotational loss factors rhyst, rexcess, theorder of the harmonic n and the amplitude of the n-th harmoniccomponent of the magnetic flux density Bn (T). The materialspecific parameters a1 a5 used in (19, 22) are identified eitherby a pure mathematical fitting procedure performed on measureddata sets or by a semi-physical identification procedure. Figure8 shows exemplarily the separated loss components calculatedin the post-processing (19) - (22), which are used for the electro-magnetic design process of an induction machine with a statorcurrent frequency of f = 400Hz and a mechanical rotor speedof n = 5925min1 and 4 pole pairs. The main contribution toeach fraction of total specific losses occurs in the magneticallyhigh exploited stator teeth. The areas of rotating magnetic fluxescan be identified by the ratio BminBmax in the regions of the statorteeth back, that leads to increased specific losses in this area for(19) and (21), see figure 8(a), 8(c).

P_hyst [W/kg]P_hyst [W/kg]

0.00 0.00

4.94 4.94

9.88 9.88

14.8 14.8

19.8 19.8

24.7 24.7

29.7 29.7

(a) Hysteresis losses

P_classic [W/kg]P_classic [W/kg]

0.00 0.00

15.0 15.0

30.0 30.0

45.0 45.0

60.0 60.0

75.0 75.0

90.0 90.0

(b) Eddy current losses

P_excess [W/kg]P_excess [W/kg]

0.00 0.00

15.0 15.0

30.0 30.0

45.0 45.0

60.0 60.0

75.0 75.0

90.0 90.0

(c) Excess losses

P_sat [W/kg]P_sat [W/kg]

0.00 0.00

4.17 4.17

8.33 8.33

12.5 12.5

16.7 16.7

20.8 20.8

25.0 25.0

(d) Saturation losses

Figure 8: Visualisation of the separated loss contributions of(18).

C2. Coupled magneto-dynamic hysteresis modelling

As discussed earlier, the iron losses in electrical machines arecommonly estimated using empirical iron-loss models during thepost-processing due to their computational advances and theirsatisfactory accuracy. Nevertheless, these approaches apply sim-plifications such as the assumption of homogeneous magneticfield distribution across the lamination and no influence of theskin effect. Additionally no minor hysteresis loops and dc-biased hysteresis effects are considered. In numerical modelse.g., a non-conductive and non-dissipative soft magnetic mate-rial is assumed. The material is described by a reversible mag-netisation curve. This assumption yields the entire negligenceof the iron losses in the field model. However, an important as-pect for the calculation of iron losses in soft magnetic materialsis the consideration of the influence of non-local eddy currents,non-linear material behaviour on the field distribution and lossesin electrical steel laminations as well as the interdependence be-tween the dynamic magnetic hysteresis and the non-local eddycurrents. The latter are directly induced by the externally ap-plied time-varying magnetic field H(t) in the conductive mate-rial (i.e. electrical steel sheet). These non-local eddy currents areof macroscopic nature and determined by the sample geometry.Due to the Joule effect this results in a further portion of the ironlosses and the total losses resulting from the sum of both con-tributions (dynamic magnetic hysteresis + non-local eddy cur-rents). The problem of modelling these non-local eddy currentsis directly linked with the laminated structure of soft magneticcores. This is the reason why the eddy current distribution inthe individual laminations cannot be explicitly simulated in theFE-simulation. The measurable quantities Hmeas and Bmeasrepresent only the intrinsic material behaviour when the fieldis distributed homogeneously over the lamination, i.e. at lowfrequencies where the skin depth is larger than the sheet thick-ness. To overcome the aforementioned shortcomings we are de-veloping a coupled magneto-dynamic vector-hysteresis model,which enables the modelling of dynamic vector-hysteresis loopswith consideration of non-local eddy currents and the laminatedstructure of electrical steel grades. Therefore, an energy-basedvector-hysteresis model is coupled to an one-dimensional modelof half the sheet thickness of the lamination, to study the re-lationship between the externally applied magnetic field on thesurfaceHsurface(t) and the internal magnetic fieldHmaterial(t)more accurately and to consider the influence of macroscopiceddy currents on the field distribution. In the following, bothmodels will be briefly introduced. The used dynamic hysteresismodel is based on the first law of thermodynamics, resulting inan energy-based modelling of magnetic materials [31], [33], [53]in order to characterise the non-linear behaviour of magnetic ma-terials as well as the associated energy losses for any instant oftime. This enables to go beyond the limitations of currently usedmodels. The first law of thermodynamics (23) states that everysystem has an internal energy that can only be changed by thetransport of work and/or heat beyond the boundaries of the sys-tem.

= W + Q (23)A change in internal energy density corresponds to the workdone on the system W plus the emitted or absorbed heat Q.The internal energy corresponds to a reversible magnetic fieldstrength ~hr and the dissipated work within the system to an irre-versible magnetic field strength

h irr =

h i +

h j . Therewith

the energy densities are described by

=h r M (24)

Q =h irr M. (25)

Deriving the energy dissipation functional (25) with respect toM enables to represent the energy balance as a function of themagnetic field strength

h =

h r +

h irr. At the macroscopic

level the microscopic distribution of the pinning points, hinder-ing the domain wall motion, cannot be modelled explicitly. Thepinning force can be modelled as an analogon by a dry frictionforce as in the J-A model [36]. This force counteracts anychange in magnetisation and the corresponding energy density isconverted into heat. Considering the dynamics of the magneti-sation process, the attenuation by microscopic eddy currents canbe represented as a mechanical analogue by a movement withviscous friction with the friction constant .

h irr =

M

( M

)+

1

2

M

( M

2)

(26)

Since the energy dissipation functional is not differentiable, itmodels the memory effect. This enables to specify the macro-scopic magnetisation with consideration of hysteresis

B (h ) =

M(

h r) + 0 h . (27)

The magnetisation M(h r) is described by a parametric satu-ration curve, whose parameters are identified by measurements.To validate the identified parameters, the response of the hys-teresis model is compared to measured material characteristics.A comparison of the measured losses as well as the magnetichysteresis loops is conducted. The magnetic fieldHmeas(t) usedon the Epstein frame or single sheet tester serves as the modelinput. The model response Bmod(Hmeas) obtained from (27) iscompared to the measurement Bmeas(Hmeas)(figure 9, 10).

-1.5

-1

-0.5

0

0.5

1

1.5

-200 -150 -100 -50 0 50 100 150 200

Maic fie e A

Ma

ic f

nsi

ty B

0.7T0.9T1.1T1.3T

-1.5

-1

-0.5

0

0.5

1

1.5

-200 -150 -100 -50 0 50 100 150 200

0.7T

0.9T1.1T1.3T

Magnetic field strenght H [A/m]

Mag

net

ic f

lux d

ensi

ty B

[T

]

200Hz

Figure 9: Comparison of modelled (left) and measured (right)hysteresis loops for M270-35A at 200Hz.

-1.5

-1

-0.5

0

0.5

1

1.5

-250 -200 -150 -100 -50 0 50 100 150 200 250

Maic fie e! " #A$%&

Ma'

(

)

*

ic f

+

,

-

.

)

nsi

ty B

/

0

1

0.7T0.9T1.1T1.3T

400Hz

-1.5

-1

-0.5

0

0.5

1

1.5

-250 -200 -150 -100 -50 0 50 100 150 200 250

Ma2

3

4

5

ic f

lux d

ensi

ty B

[T

]

Magnetic field strength H [A/m]

0.7T

1.3T

0.9T1.1T

400Hz

Figure 10: Comparison of modelled (left) and measured (right)hysteresis loops for M270-35A at 400Hz.

The utilisation of the 1D cross lamination model, the eddycurrent model of half the thickness of an individual laminationsheet can enable the nearly exact determination of the field dis-tribution in steel laminations and improves the iron loss calcula-tion by considering the influence of eddy currents. The relation-ship between the externally applied magnetic field at the surfaceHsurface(t) and the internal magnetic fieldHmaterial(t) is moreprecisely studied taking the influence of macroscopic eddy cur-rents into account. In combination with the dynamic vector hys-teresis model a tightly coupled transient problem is obtained thatcan enable nearly the exact determination of the magnetic fields

and losses under the special conditions of an Epstein frame orsingle sheet tester. Therewith, the initial mentioned shortcom-ing will be reversed. With this coupled vector-hysteresis modelthe estimation of iron losses in a 3-dimensional FE-problem canbe done in an accurate and reliable way. The input data of thevector-hysteresis model has to be the applied magnetic field H ,but FE-problems for electrical machines are commonly solvedby a magnetic vector potential (A ) approach. The resulting datafor each time step is the magnetic flux density B. At the mo-ment, the magnetic field is calculated by H = B0r with the inthe FE-simulation selected material data of the reluctivity (B2)of the relative permeability r = 1 . For the future we planto implement a magnetic scalar potential solver to provide thevector-hysteresis model directly with the resulting data (H).

V 3D ELECTRICAL MACHINE FE MODELLING WITHMOTION

Static, transient and especially field coupling simulations ofelectrical machines as described in section IX require the flex-ible displacement of the rotor by an arbitrary angle in rotating ora distance in translational electric machines. Recent studies [40]on the applicability of biorthogonal shape functions in the FiniteElement analysis (FEA) of electrical machines have shown con-siderable potential towards a generic approach for 2D and 3Dproblems with motion. A standard formulation for motion prob-lems is the moving band (MB) technique [9], which, for practicalreasons, is only applicable to 2D rotating machines.

The presented nonconforming approach belongs to the cate-gory of Lagrange multiplier (LM) methods, but instead of us-ing standard basis functions for the Lagrange multiplier, thespecial biorthogonal basis functions proposed in [58] are used.The biorthogonality property makes it possible to eliminate alge-braically the Lagrange multiplier, turning the saddle point prob-lem (which is typical of LM approaches) into a symmetric, pos-itive definite system of equations. However, biorthogonal edgebased Whitney functions could not be constructed in a canonicalway. In 3D, the technique can thus be applied to magnetic scalarpotential T- formulations, but not to magnetic vector poten-tial A formulations. The implementation of the T- formula-tion will additionally allow the estimation of iron losses by thevector-hysteresis model as described in section C2..

A Variational Formulation

Let m and s be two complementary domains called masterand slave, m s = , e.g. the stator and rotor of an electricmachine. Let m m and s s be their common inter-face and p : s m be a smooth mapping, which may accountfor a relative sliding between the master and the slave domain.In a magnetic scalar potential T- formulation [7], the magneticfield Hk = Tk gradk, k {m, s}, is expressed in termsof an electric vector potential Tk such that Jk = curlTk and asinglevalued scalar magnetic potential k. The variational cal-culus applied to the energy balance of the system leads to theweak formulation

k=m,s

k

Bk grad k d

+

s (s m p) d

+

s (s m p) d = 0,

(28)

which must be verified for all k and fulfilling the homo-geneous boundary conditions where the unknown field is theLagrange multiplier.

B Discrete Formulation

In order to establish the FE equations in matrix form, the vectorsof unknowns uk, k {m, s} are divided into two blocks each.The block uk contains the unknowns lying on the sliding inter-faces k, whereas the block uki contains the unknowns lying inthe interior of the domains k. The magnetic scalar potential and are both approximated with nodal shape functions:

k =l

lkl ,

k = {kl }, (29)

=l

ll, = {l}. (30)

The superscript s is omitted for because the shape functionsof are defined on s only. From the weak formulation (28)one obtains the saddle-point problemSmi,i S

mi, 0 0 0

Sm,i Sm, 0 0 MT

0 0 Ss, Ss,i D

T

0 0 Ssi, Ssi,i 0

0 M D 0 0

umiumususi

=

bm

00bs

0

(31)

with

Skln =

k

gradkl gradkn d, (32)

bkl =

k

Tk gradkl d, (33)

Djl =

sj

sl d, Mjl =

sj

ml p d. (34)

In order to obtain a symmetric positive definite system, the de-grees of freedom us associated to the slave side s of the slidinginterface and the Lagrange multiplier are eliminated [41]:

us = D1Mum Qum , (35)

Q D1M (MTDT )T (36)Smi,i Smi, 0Sm,i Sm, +QTSs,Q QTSs,i

0 Ssi,Q Ssi,i

umium

usi

=

bm0

bs

(37)

This system of equations is symmetric, positive definite and can,contrary to (31), be solved efficiently by standard Krylov sub-space methods. However, to obtain (37) it is necessary to evalu-ate D1, as seen in (35) and (36).

If the matrix D is diagonal, the evaluation reduces to a sim-ple matrix product. This is the case when the shape functionsof the Lagrange multiplier verify the following biorthogonalityrelation [58, 16]:

Djl =

sj

sl d

= jl

ssl d, with jl =

{1 if j = l,0 if j 6= l, (38)

Previous studies have indicated that it is not feasible to constructsuch biorthogonal basis function for the magnetic vector poten-tial formulation in a canonical way but for magnetic scalar po-tential formulation [41]. Therefore, the presented approach isapplied to magnetic scalar potential T- formulations.

Figure 11: Isoplanes of scalar potential across the interface in a PM-machine.

-1.5

-1

-0.5

0

0.5

1

1.5

0 20 40 60 80 100 120 140 160 180

no

rma

lized

torq

ue ()

angular displacement (degrees)

3D sliding interfacereference simulation

Figure 12: Comparison of resulting cogging torque with refer-ence.

C Example application

A permanent magnet excited synchronous motor is presented asbenchmark problem with scalar unknown fields. The formula-tions have been implemented within the institutes in-house FE-package iMOOSE [www.iem.rwth-aachen.de]. The 3D modelof the motor is extruded from a 2D geometry, so that a refer-ence solution is available. The sliding interface is a concentriccylinder in the air gap with nonconforming discretisation onthe master and slave sides. The source field Tk of the T formulation is determined by the permanent magnets in this ap-plication [5]. The proposed approach would however work inthe same way with a coil system. Neumann boundary conditions(no flux) apply on all external surfaces of the model. The slid-ing interfaces m and s intersect thus the Neumann boundarysurface at the front and back ends of the model in axial direc-tion. The restored continuity of the potential is controlled infigure 11 by plotting isovalue surfaces of and checking thatthey are continuos across the nonconformally discretised inter-face . The cogging torque has been calculated and is comparedwith the 2D reference solution in Fig 12. It shows a very goodagreement.

VI CONSIDERATION OF NON-IDEAL MANUFACTURING

The previous studies are performed assuming ideal machines.An ideal machine presents geometrical and electric symmetricproperties. Depending on the number of pole pairs p, the num-ber of slots N and the winding configuration, the air-gap fieldof an ideal machine shows a spatial periodicity along its circum-ference. The function iterates in dependence of a certain angle.However, these characteristics will never be achieved within areal machine. The cause of this behaviour is the manufacturingprocess which is subjected to tolerances and therefore leads to

deviations from the ideal machine. Material dependant failures,geometrical or shape deviations may occur causing asymmetries.The assumption of an iterating air-gap field is not valid. Instead,new harmonic orders arise whereby undesired parasitic effectssuch as torque ripple, losses, vibrations and noise are influencedin particular.

A Causes and effects

The most studied manufacturing tolerance regarding its influ-ence on parasitic effects in electrical machines is eccentricity[10]. Eccentriticity means that rotor and stator axis do not lie inthe same location. This can be caused by bearing tolerances ordeflection of the shaft. Above all, eccentricity presents a crucialinfluence onto the acoustical behaviour of electrical machines[55].

The properties of the applied materials are significant con-cerning the machines characteristics. Due to mechanical pro-cessing of electrical steel during fabrication of electrical ma-chines, the crystal structure of the steel is being modified [11].This finally leads to increased iron losses.

For permanent magnet excited machines the magnets qualityis important. Not only the dimensions, but also the magnetic fluxdensity of the magnets is subjected to tolerances. This results invaryingly strong evolved magnetic poles. This asymmetry influ-ences the cogging torque in particular [19].

Because of their strong influence on parasitic effects, a con-sideration of deviations caused by manufacturing is required. Itis advantageous to study this problem by use of simulations tocover a large state space. Moreover, an experimental setup isvery difficult to realise because it is hardly possible to imple-ment accurately defined faults.

As a consequence, the consideration of manufacturing toler-ances in electromagnetic modelling is exemplarily presented inthe following.

B Impact estimation

For the illustration of a stochastic analysis the aspect of magnetvariations with magnets similar to those which are used in thegiven machine design is considered. The applied approach [54]is depicted in figure 13:

2Estimation

of inputparameters

1Systemmodel

4calculation

of outputparameters

3propagation

of uncertainties

sensitivity analysis

Figure 13: Chosen approach for uncertainty propagation as pro-posed in [54].

The beginning in stochastic considerations is the selection ofa suitable model. In general all kinds of models, including ana-lytical formulations as well as FEM models are applicable. Thisselection typically is influenced by the models accuracy for theoutput size of interest and boundary constraints as allowed com-putational time. It has to be anticipated that the models choicewill strongly influence the uncertainty propagation of step 3. In

this case, we decide to use a 2D-FEM model to simulate the im-pact of the two in figure 14 presented magnet error types ontothe magnets air field:

(a) Error type A: deviation of radialmagnetisation towards undirectionalmagnetisation.

(b) Error type B: deviation of localmagnetisation strength, weakening to-wards the magnet edge.

Figure 14: Considered variations (black) in magnet in contrastto ideal magnetisation (grey).

1. Magnetisation errors tending from radial magnetisation to-wards an unidirectional magnetisation as illustrated in fig-ure 14(a). The implementation allows for an arbitrary errorbetween both extremes, = 0 representing complete uni-directional magnetisation, = 1 representing ideal radialmagnetisation.

2. Spatial changing remanence induction magnitude, shapeddecreasingly from the magnet middle to the magnet borderas depicted in figure 14(b), = 0 representing a sinusoidalshaped remanence induction, = 1 representing an idealuniform value for the remanence induction over the entiremagnet surface.

In step 2, the input distributions for both error cases were es-timated. Both error cases have been chosen to be normal dis-tributed in a way, that the maximum error of the radial compo-nent (located at a spread of 3) has been allowed to be 1.5%.

For the propagation of uncertainties in step 3, Monte-Carlosimulation based on a polynomial-chaos meta-model has beenexecuted. At this point analytical models would offer the possi-bility of a direct uncertainty propagation, which is not possiblewith the FEM. Possible approaches to mitigate this problem arepresented approach as well as the intrusive methods presented in[51] and [18].

Figure 15 finally shows the calculated flux-densitys cumula-tive distribution function at the magnets middle and enables togive error probabilities for any defined failure critera.

C Utilisation of the results

With the presented approach it is possible to evaluate the influ-ence of manufacturing tolerances on the later produced machine.This information is valuable for the mass production of the ma-chine and its subsequent application. Depending on the applica-tion, there are certain requirements on the machines character-istics. Employing this methodology, it is possible to prove if or for which percentage of the production these characteristicscan be achieved with respect to the allowed tolerance ranges.

If the requirements on the machine can not be reached, it isfirst of all obvious, to consider an adjustment of the allowedtolerances. However, this might be impossible or expensive torealise. The other possibility is to apply a robust design of themachine, for instance with the aim to achieve a cogging torqueminimisation [35]. The general proceeding would be to definethe problem at first. The parasitic effect which shall be min-imised and the relevant design parameters are chosen. In a sec-ond step, it is important to employ a model which allows an ac-curate calculation of the interaction between the chosen design

Figure 15: Cumulative distribution function (CDF) of the flux-density at the evaluation point at the magnets middle (angle 0).

parameters and the target values. The third step would be tochose a convenient optimisation strategy, for instance differen-tial evolution [8]. After performing the optimisation, the resultsneed to be analysed and finally verified by an experimental setup.

For the described robust design, modelling is the most impor-tant factor, as the optimised machine can only be as good as themodel which qualifies it. Once a good model has been devel-oped, the resulting electromagnetic forces as well as the there-from arising noise can be calculated, which is presented in thefollowing section.

VII AURALISATION OF ELECTROMAGNETIC EXCITEDAUDIBLE NOISE

For the evaluation of the radiated acoustic noise of electrical ma-chines in variable operating conditions, a real-time auralisationprocedure applicable in virtual reality environments is devel-oped. Electromagnetic forces, structural dynamics and acous-tic radiation as well as room acoustic aspects are considered.This overall task represents a multi physics problem formu-lation which usually consumes extreme computational efforts.The combination of electromagnetic simulation with a unit-waveresponse-based approach and a room acoustic virtual environ-ment software allows for an efficient implementation. Simu-lation results are presented for two different types of electricalmachines, an induction machine and a permanent magnet ex-cited synchronous machine. Practical experiments are used tofine tune and validate the numerical models. A detailed descrip-tion of the simulation chain can be found in [21].

A Simulationconcept

A transfer-function-based approach is applied in this work.Amplitudes of force density waves are directly linked to thefree-field sound pressure on an evaluation sphere surroundingthe electrical machine. In this way, a separation into off-linepre-calculation of structural and radiation data and an online-auralisation becomes possible. As shown in figure 16, the pro-posed concept for the auralisation of electrical machines con-sists of four simulation steps: The electromagnetic forces andthe structural dynamic behaviour are simulated by means of theFEM. The transfer function of the mechanical structure leads toa relation between the electromagnetic force f and the surfacevelocity v. The sound radiation simulation yields the transferfunction from surface velocity v to sound pressure p at the evalu-ation points in the free-field. The room acoustic transfer function

Figure 16: Schematic of transfer function-based simulation con-cept.

Figure 17: Structure dynamic model of an induction machinetogether with normal surface velocity on the housing at 533 Hz.

describes the sound propagation inside a room or car compart-ment from the source to a receiving position. This work onlyaccounts for airborne sound transmission and shows the basicsteps to simulate time signals for electrical drives in automotiveapplications. In order to include structure-borne sound transmis-sion transfer paths have to be simulated or measured and inte-grated into the above-mentioned model. In the same way as theforce-to-velocity transfer function is expanded in a circumferen-tial Fourier series called force modes r, there is also one transferfunction from velocity to acoustic pressure per mode. The totaltransfer function from force to acoustic pressure is thus denotedHr () for each force mode r. For each observation point andfor each mode r, a transfer function from force to acoustic pres-sure is calculated.

B Structural dynamics

A structural dynamics simulation connects simulated electro-magnetic forces and the surface velocity on the electrical ma-chine. Commercial software packages can be used to processthis simulation task by means of FEM simulations. Changes inthe geometry directly require a re-run of the computationally ex-tensive simulations. The determination of the material parame-ters has been found to be most critical in this step. The struc-tural model of the electrical machine is shown in figure 17. Inthis case the machine is air-cooled. The results are organised ina way that the surface velocities are normalised to applied unitforces for each force mode r.

C Operational Transfer Function

As mentioned before, each block of the transfer function chaincan either be studied by analytical calculation, by numerical sim-ulation or by measurements. In this sense, there is a signal (andenergy) flow from current/PM excitation, via flux density and re-luctance forces, through surface velocity to sound pressure andparticle velocity. For each of the three parts a transfer functioncan be defined, where the transfer function of the latter two typ-ically can be considered being linear. For example, the mechan-

ical transfer function can be determined by means of numericalmodal analysis (using FEM) or by means of experimental modalanalysis (using shaker, accelerometer and dual-spectrum anal-yser), for a point excitation.

Alternatively to determining the transfer function purely frommeasurement, or purely from simulation and under the assump-tion that the operating conditions are approximately equal, wecan define a mixed formulation for the transfer function as

H =Bmeas

Asimu, (39)

where Bmeas is the measured output of the transfer function, andAsimul is the simulated input. Using magnetic forces as Asimu isa good choice for two reasons: First they can be simulated com-paratively accurate using 2D-FEM and they are very difficult tomeasure. In [50], Bmeas is chosen to be the surface accelerationand Asimu is indeed the magnetic force. In this case, the transferfunction from the electromagnetic forces to sound pressure de-termined, thus the sound pressure is used as Bmeas. The idea offormulation of the transfer function has been deduced from thewell known operational modal analysis, and is therefore consid-ered to be a valid approximation.

The starting point for the determination of transfer functionsis a microphone measurement of sound pressure pmeas(t) and asynchronised speed measurement n(t) during an unloaded run-up of the PMSM with sufficiently slow slew rate. This is mappedto a so called spectrogram given by pmeas(, 2pin), which showsdominant lines due to harmonic force excitations, see Fig. 20.Using a peak-picking technique along these lines, allows for thedefinition of lines of constant order k as pmeas(, k), k = 2pin .

As a second step, it is essential to trace back each order lineto a specific harmonic. This can either be done using standardtable works [37], or more sophisticated even tracing back to in-dividual field harmonics [22], the latter detailed approach is notnecessary for the method proposed in this paper, it however mayreveal more insight and may help trouble shooting the computerroutines. For the unloaded run-up, the space vector diagrams areflat, i.e. there is no imaginary component of the force waves, dueto the absence of stator currents. The Fourier decomposition ofthe reluctance-force-density waves reads

(x, t) =

Kk=1

Rr=R

rk cos(rx+ k 2pin t+ ) . (40)

Now the assumption is made, that one harmonic force wavegiven by frequency harmonic number k and by wave number ris dominant and solely accounted for at one line of constant or-der. Then, the FEM-to-measurement transfer function is definedas

Hr() =pmeas(, k)

rk

k=k

, (41)

where rk is determined from a FEM simulation of the very ge-ometry as the prototype delivering sound pressure measurementspmeas(, k).

The sound pressure level (SPL) for a given force excitationrk can be calculated by means of superposition:

Lp(, n) = 20 log10

Kk=

2pin

Rr=R

rk Hr()pref

A() ,

(42)where pref = 20Pa, however all results presented in this paperare referred to a slightly different undisclosed reference pressureleading to an undisclosed offset in all level plots.

Figure 18: Sound-pressure level distribution per unit-force (dBre 20 Pa/Pa) for the force modes r = 0 . . . 4 [21].

To compensate for the frequency dependency of the hu-man ear, measured or simulated audible signals are passedthrough a filter, of which the A filter is the most common one.Therefore, (42) gives the unweighted pressure level in dB, ifA () = 0, and it gives SPL in dB(A), if the A filtervalues are used for A().

The quadratic pressure levels of incoherent signals may besummed up immediately. Therefore, the (A-weighted) total SPLis then given by

Lp,tot(n) = 10 log10

Nl=1

10Lp(l0,n)/10 (43)

in dB(A), where 0 corresponds to the window, which was ini-tially used to determine pmeas(, 2pin).

D Sound Radiation and Auralisation

The free-field sound radiation of the electrical machine can becalculated by means of the Boundary Element Method. Thismethod requires very long computation times if higher frequen-cies are considered [45]. Due to the geometry of electrical ma-chines an analytic approach exploiting cylindrical harmonics cansignificantly reduce the computational costs as published by theauthors [20]. However, we use a different approach using mea-surements and only electromagnetic simulations - structural dy-namics and sound radiation are not simulated explicitly as de-scribed in the section about the operational transfer function.

A PM synchronous machine is used as an example: The sta-tor consists of a laminated sheet stack (M250-AP) with slots anda rotor made from the same material, which is equipped withsurface-mounted permanent magnets (r 1, Brem = 1.17T).The number of stator slots is N1 = 6 and the number of mag-nets is 2p = 4. The transfer functions relating force densities

-40

-30

-20

-10

0

10

20

30

40

0 1000 2000 3000 4000 5000

LW

(dB

re1

pW

/Pa2

)

Frequency f (Hz)

r=0r=6

-40

-30

-20

-10

0

10

20

30

40

0 1000 2000 3000 4000 5000

LW

(dB

re1

pW

/Pa2

)

Frequency f (Hz)

r=1r=5r=7

-40

-30

-20

-10

0

10

20

30

40

0 1000 2000 3000 4000 5000

LW

(dB

re1

pW

/Pa2

)

Frequency f (Hz)

r=2r=4r=8

-40

-30

-20

-10

0

10

20

30

40

0 1000 2000 3000 4000 5000

LW

(dB

re1

pW

/Pa2

)

Frequency f (Hz)

r=3

Figure 19: Sound-power level per unit-force [21].

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00510152025303540455055606570

Fre

quen

cyf

(Hz)

Time t (s)

Sound-pressure level Lp (dB re 20 Pa)

Figure 20: Spectrogram of a simulated run-up from 600 to6000min1 [21].

to sound pressure are calculated. The resulting sound pressuredistributions per unit-force are shown in figure 18. It can beseen that even force orders cause a rotating velocity distributionwhere the radiation pattern of force order 3 indicates standingvelocity distributions almost independent on frequency. Figure19 shows the results of the unit-wave response-based approach:the sound power radiated by each force mode r. In order to sim-ulate a run-up, sweep signals are generated up to an order of100. Thereby, the length of the signal is set to 5 s, start and endspeeds are chosen to be 600 and 6000min1. After the multipli-cation with the actual electromagnetic force density amplitudes,the resulting free-field sound pressure signal is obtained in 5 mdistance. It is presented in form of a spectrogram shown in figure20.

In the vicinity of certain eigen-frequencies of structural modesthat are likely to be excited, the sound pressure increases signifi-cantly. In particular, those natural frequencies are located around1, 2 and 4.5 kHz.

In figure 20, several order lines are visible. Each line is ex-cited by a certain time-harmonic order, which is determined bythe slope in the spectrogram, and by multiples of space orders.Despite this superposition of different space orders, one timeharmonic order line is most commonly dominated by a singleforce space order.

VIII VIRTUAL REALITY

Efficient methods for the visualisation of finite element solutionsare essential for the evaluation of electromagnetic devices underresearch and development. Important decisions are made on ba-sis of solution visualisations and further design steps are planned

on basis of the ongoing understanding of the device under re-search. Therefore, effective post-processing algorithms beingable of handling large amounts of finite element data and theusability of such methods in an interactive way allow a fasterdesign process.

The advantages of a graphical visualisation of the finite ele-ment solution include the ability to intuitively evaluate the nu-merical data regardless of data size and complexity. Further-more, an integrated environment allows various interactive post-processing methods to obtain graphical representations, e.g. fluxlines, as well as to compute integral quantities such as torque orflux.

Since many years we develope the software tooliMOOSE.trinity for visualisation of finite element field so-lutions, which has undergone multiple development cycles overthe years and has been extended by various post-processingmethods. The current version applies the Open-Source Visu-alization Toolkit (VTK) for the graphical representation andis designed to add pre-processing and processing facilities aslong term research goals. Beside the pure visualisation of scalarand vector fields iMOOSE.trinity contains amongst others thefollowing features and interactive post-processing capabilities:

Flux line computation in 3D [25] Interactive plane cutting [24] Navigation through n-dimensional parameter spaces [26] Coupling to virtual reality (VR) systems [6] Support for various input devices like 3D mouse with six

degrees-of-freedom

Two of these interesting features, namely computation of fluxlines and coupling to professional VR systems are described inthe following sections.

A Flux line computation in 3D

The design process of electrical machines benefits from the visu-alisation of flux lines during the post-processing of finite elementdata. The example chosen here is an interior permanent magnetsynchronous machine (IPMSM) with rotor staggering, since inthis rotor configuration stray fluxes with axial flux componentare expected which are hard to locate in the solution data, andalmost impractical to adequately being visualised by standardmethods in 3D.

For the visualisation given in figure 21(a) an interactivelyplaceable point widget is used as seed point for the flux linecomputation. By this, the user has the possibility to explorethe regions of interest and locate the corresponding coordinatesquickly. The placement of the seed points in the 3D machinemodel is done intuitively by using a 3D mouse, which offers sixdegrees-of-freedom. In this example, a chosen starting point inthe vincinity of the permanent magnet edge, shows a stray fluxline which closes through the other magnet angular to the z-axis.The same mechanism can be used to start a number of streamsfrom a placed line, as exemplified in figure 21(b) for the samemachine configuration without rotor skewing.

B Coupling to virtual reality systems

To improve the interactive possibilities of iMOOSE.trinity abidirectional coupling to the VR software framework ViSTA[49] has been developed [6]. ViSTA has a scalable interfacethat allows its deployment in desktop workstations, small andlarge VR systems. The purpose of this coupling is to offer thepossibility to link a simulation package with the whole range of

(a) Stray flux stream line between the skewed permanent magnets.

(b) Seeding points along a line below a unskewed rotors pole.

Figure 21: Interactive streamline computation and visualisationexemplified on a flux density distribution of an IPMSM with andwithout rotor skewing [25].

immersive visualisation systems, from 3D office systems up tocave-like systems (figure 22(a)).

The underlying idea, which is in the focus of this work, isto implement such a linkage between software packages by ageneralised network-based bidirectional coupling. Since tech-niques such as making interactive cuts or seeding particles forflow (or flux line) representation are common in both systemssuch a mechanism mirrors all performed actions from one sys-tem to the other, so that one ends up with a consistent represen-tation on both environments.

The benefit in such an additional effort is to provide an ex-tended electromagnetic processing and post-processing frame-work, where users employ the advantages of immersive VR-Systems, e.g. the interactive point or surface selection in 3D,by switching from the standard GUI-based finite-element envi-ronment directly to such professional VR environments. For in-stance, in the case of leakage flux visualisation described in theprevious section, the interaction for navigating to the points ofinterest takes place in the VISTA driven application feeding theiMOOSE.trinity algorithm, which computes the correspondingclosed magnetic flux lines.

In the future the bidirectional coupling to ViSTA can be im-proved by integrating VR input devices into the mirroring mech-anism, so that VR controllers such as a flystick or instrumentedgloves (figure 22(b)) can be utilised for electromagnetic post-processing methods.

IX FIELD CIRCUIT COUPLING

Within the design process of electrical drive trains the systemsimulation represents an important step. Modern drives do notonly consist of the electrical machine but also of the powerelectronics and the controls, it is necessary to consider all cor-

(a) Holodeck.

(b) Instrumented gloves.

Figure 22: VR devices.

responding parts to achieve a proper operation in all workingpoints. Especially for complex control strategies such as self-sensing where saturation and inverter influences have to be takeninto account this becomes very important [48, 42]. With cir-cuit simulation software it is possible to combine the converterand the controls with an analytical representation of the ma-chine. Since the magnetic circuit of the machine is typically de-signed with the help of Finite Element Analysis (FEA) in orderto develop high power, highly efficient, or low noise machines[62, 56, 17] it is reasonable to take the FE model into consider-ation, which can substitute the analytical model completely (di-rect coupling) on the one hand. On the other hand it is possibleto extract the linear parameters of the machine for several pointsof operation off-line and store them in lookup tables, that areused by the model during the simulation [44, 34]. Especially formachines that operate in magnetic saturation and for those withconcentrated windings that do not entirely fulfill the assumptionof a sinusoidal winding distribution [56] this simulation methodcan improve accuracy compared to a dq-model based simulation.

This section focusses on the extraction of lumped parametersby FEA, representing an electrical machine for the use inside asystem simulation environment. A full system simulation of aPMSM servo drive using the direct coupling is applied in [56].

A Lumped parameter representation of electrical machines

Rotating electrical machines, e.g. three phase PMSMs, can berepresented by an inductance matrix Lkl of self and mutual in-ductances with dimension 3 3 and a vector of motion inducedvoltages ek with dimension three. The flux induced voltage ofphase k is given by the time derivative of the flux linkage k(with implicit summation over l) as the difference of the termi-nal voltage vk and the ohmic voltage drop Rik:

tk = vk Rik = tkl + tf,k . (44)

Herein, f,k = f() is the remanence flux embraced by thepermanent magnets being a function of the angular position of

the rotor = f(t), and kl = f(, il) is the flux linkage inphase k depending on the current il = f(t) carried by phase land the rotor position . Applying the differential operator in(44) yields:

tk = t (i,k + f,k)

= tiii,k + t,k + tf,k

= (til)Lkl + k = (til)L

kl + e

k .

(45)

The first term of (45) expresses the induced voltage bythe flux linkage described by the inductance matrix Lkl andthe second term the motion induced voltage with the speednormalised electromotive force (emf) ek.

B Extraction of the inductance matrix from FEA

LetMij(a) aj = bi, (46)

with the right handside (implicit summation over l)

bi =

j i = il

wl i := ilWil, (47)

be the non-linear 2D FE equations describing the PMSM withpermanent magnet excitation and stator currents. Herein, j is thecurrent density and i are the shape functions of the degrees offreedom i.e. the nodes. The magnetic vector potential is givenby a and M is the non-linear system matrix arising from theGalerkin scheme, see [2]. In 2D, the current shape functionsbecome wk = wlAl ez with wl being the turns of phase l and Althe corresponding turn area. Following (47), Wil is defined asthe current shape vector with respect to phase l.

Now, let il be the currents at time t, and bi = ilWil thecorresponding righthand sides. Solving (46) with bi bi anda fixed rotor angular position = 0 gives aj and a first orderlinearisation around this particular solution writes

Mij(a

j +aj) = Mija

j + Jijaj = b

i +bi (48)

with the Jacobian matrix Jij (ajMin(a

j ))an. Since

Mija

j = b

i , one has

Jij(a

j ) aj |=0 = bi. (49)

One can now repeatedly solve (49) with the right-hand sidesbi = ilWil obtained by perturbing one after the other mphase currents il and obtain m solution vectors for aj |=0.Since (49) is linear, the magnitude of the perturbations il isarbitrary. One can so define by inspection the inductance matrixLkl of the electrical machine seen from terminals as

k|=0 = Wkj aj |=0= WkjJ

1ji (a

j )Wil il, Lkl il (50)

withLkl = WkjJ

1ji (a

j )Wil. (51)

Beside the extraction of the tangent inductances Lkl it is alsopossible to extract the secant inductances Lkl.

C Extraction of the motion induced voltage

One can now complement (45) to account for the emf:

k = Lklil + e

k (52)

Static FEA at t = t 0 = 0

Evaluation of L r s

Magnetostatic FEA at t = t kF E

Set excitation currents at t = t kF E

Solve ODE system

[t < te nd ]

[t < t k1F E

+TF E ]

[t t k1F E

+TF E ][t te nd ]

Figure 23: Transient coupling scheme FE-model and power elec-tric circuit.

with ek k. The direct computation of the derivativerequires to slightly shift the rotor, remesh, solve the FE problem,evaluate new fluxes and calculate a finite difference. In orderto avoid this tedious process, one can again call on the energyprinciples. One has

ek = k = ikM = ikM = ikT (53)

where T is the torque and M is the magnetic energy of thesystem. During the identification process described above, it ispossible to calculate additionally the torque corresponding to theperturbed solutions aj |=0, and to evaluate the motion in-duced voltage ek of each phase k as the variation of torque withthe perturbation of the corresponding phase current ik.

However, as the torque is a non-linear function of the fields,the perturbations need in this case to be small. Because of thelinearity of (49), one may scale the perturbation currents in (53)which yields:

ek =T (a) T (a + a|=0)

ikwith = ||a

||2||a||2 . (54)

Herein, the scale factor is chosen between 0.01 0.05.The current and position dependent excitation flux is extractableas well. This is useful if the modelling of the machine followsthe approach described in [34].

D Coupling to a circuit simulator

The decoupled solution of the field and the circuit problems re-quire a time stepping scheme to coordinate the interaction of theutilised FE-solver and the circuit simulator.The basic scheme is shown in figure 23. The time step width of

the magnetostatic FE simulation is constant while the time stepwidth of the circuit simulator is freely chosen by the simulator.The cosimulation starts with a static FE analysis to evaluate theinitial lumped parameters, i.e., the tangent inductance matrix Land the motion induced voltages e.For the calculation of the motor currents two designs are feasi-ble. The first one is the determination of the currents outside thecircuit simulator generating a signal given to signal controlledcurrent sources within the circuit simulator. Therefore, the linecurrents of the machine are calculated by (44) and (45) after be-ing transformed to

i = (L)1(v Ri e)dt . (55)

Figure 24: Reducing of circuit elements to solve overdetermina-tion.

The terminal voltages of the machine are the voltages over thesecurrent sources. The second design is the modelling of the mo-tor inductance, resistance, and induced voltage directly withinthe circuit simulator. For this purpose the simulation softwarehas to provide controllable mutual inductances.If the global simulation time reaches t tk1FE + TFE , thephase currents of the circuit simulator are transfered to the FE-solver and step k of the magnetostatic FE-system is calculatedfollowed by the identification of the lumped parameters Lkl andek. Returning the new set of parameters to the circuit simulatorthe transient circuit analysis proceeds until t tend is reached.The variable time step width of the circuit simulator allows it toadapt the step width according to the requirements of the highfrequency switching components of the circuit domain.Since there is no zero phase-sequence system, the current equa-tion above is overdetermined if solved independently for eachphase. To solve this over-determination the system is trans-formed as shown in figure 24. Now, only two phase currentsare calculated whereas the third is given by Kirchhoffs law.

E Use of the parameter extraction for control design

By the reason that the capability of a machine control is stronglydependent on its parametrisation the extraction methods that areused for the field circuit coupling described above can help toidentify the required lumped parameters of the machine for thecontrol design. Since the control design needs both, the secantand the tangent inductances, as well as the induced voltages orexcitation fluxes, the extraction method delivers all required pa-rameters for a current and position dependent control design. Es-pecially advanced control strategies like the sensorless controlprofit by an accurate determination of the machine parameters[38]. For this purpose the parameters can be transformed intothe d-q reference frame.

X CONCLUSIONS

Numerical computation of electrical machines still is challeng-ing. In order to give an innovative impulse to the aspects of com-putation in electromagnetics, a state-of-the-art design method-ology for electrical machines has been presented, starting withthe presentation of an automated calculation chain. Extendedconformal mapping for faster simulations as well as cause-effectanalysis has been presented, leading to a detailed investigation

methodology for loss computation. Since the advanced dynamichysteresis approach requires the magnetic scalar potential, solverconsiderations have been presented subsequently. As all modelsuntil here assumed a symmetric machine behaviour, stochasticanalysis of electrical machines has been discussed, followed bynotions about the calculation of acoustics and the machine pre-sentation in VR. The paper concludes with the calculation oflumped parameters, allowing finally to build models for the ma-chine controls. Following this design chain enables a better un-derstanding of the computation and analysis of electromagneticdevices such as an eletrical machine.

XI REFERENCES

[1] Audi AG: Forschungsprojekt e-performance.http://www.audi.de/eperf/brand/de.html. version: May2012

[2] Bastos, J. P. a. ; Sadowski, N: Electromagnetic Model-ing by Finite Element Methods. Taylor & Francis, 2003. ISBN 0824742699

[3] Bertotti, G: General Properties of Power Losses in SoftFerromagnetic Materials. Magnetics, IEEE Transactionson 24 (1989), vol. 1, p. 621630

[4] Binns, K. J.: Analysis and computaion of electric and mag-netic field problems,. Pergamon Press, 1963

[5] Biro, O ; Preis, K ; Vrisk, G ; Richter, K. R. ; Ticar,I: Computation of 3-D magnetostatic fields using a re-duced scalar potential. IEEE Transactions on Magnetics29 (1993), March, vol. 2, p. 13291332. ISSN 00189464

[6] Buendgens, D ; Hamacher, A ; Hafner, M ; Kuhlen, T ;Hameyer, K: Bidirectional Coupling Between 3-D FieldSimulation and Immersive Visualization Systems. Mag-netics, IEEE Transactions on 48 (2012), Feb, vol. 2, p. 547550. ISSN 00189464

[7] Carpenter, C ; Wyatt, E: Efficiency of numerical tech-niques for computing eddy currents in two and three di-mensions. Proc. COMPUMAG. Oxford, April 1976,p.242250

[8] Das, S ; Suganthan, P: Differential Evolution: A Surveyof the State-of-the-Art. Evolutionary Computation, IEEETransactions on 15 (2011), Feb., vol. 1, p. 4 31. ISSN1089778X

[9] Davat, B ; Ren, Z ; Lajoie-Mazenc, M: The movementin field modeling. Magnetics, IEEE Transactions on 21(1985), vol. 6, p. 22962298. ISSN 00189464

[10] Ebrahimi, B ; Faiz, J: Diagnosis and performance analysisof threephase permanent magnet synchronous motors withstatic, dynamic and mixed eccentricity. Electric PowerApplications, IET 4 (2010), january, vol. 1, p. 53 66. ISSN 17518660

[11] Emura, M ; Landgraf, F ; Ross, W ; Barreta, J: The in-fluence of cutting technique on the magnetic properties ofelectrical steels. Journal of Magnetism and Magnetic Ma-terials 254 (2003), jan, p. 358 360

[12] Finken, T ; Hameyer, K: Design and optimization ofan IPMSM with fixed outer dimensions for application inHEVs. IEEE International Conference on Electric Ma-chines and Drives, IEMDC, 2009, p. 17431748

[13] Finken, T ; Hafner, M ; Felden, M ; Hameyer, K: De-sign rules for energy efficient IPM motors in HEV appli-cations. ELECTROMOTION 17 (2010), July, vol. 3, p.143154. ISSN 1223057X

[14] Finken, T ; Hameyer, K: Computation of Iron- and Eddy-Current Losses in IPM Motors depending on the Field-Weakening Angle and Current Waveform. Przeglad Elek-trotchniczny 5 (2010), May, p. 123128. ISSN 00332097

[15] Fiorillo, F ; Novikov, A: Power losses under sinusoidal,trapezoidal and distorted induction waveform. Magnetics,IEEE Transactions on 26 (1990), vol. 5, p. 25592561

[16] Flemisch, B ; Wohlmuth, B: Nonconforming Discretiza-tion Techniques for Coupled Problems. Helmig, R (Hrsg.); Mielke, A (Hrsg.) ; Wohlmuth, B (Hrsg.): MultifieldProblems in Solid and Fluid Mechanics Bd. 28. SpringerBerlin / Heidelberg, 2006, p. 531560

[17] Franck, D ; Giet, M van d. ; Burdet, L ; Coleman, R; Hameyer, K: Simulation of Acoustic Radiation of anAC Servo Drive Verified by Measurements. InternationalSymposium on electromagnetic fields, ISEF. Arras, France,September 2009

[18] Gaignaire, R ; Clenet, S ; Moreau, O ; Sudret, B: CurrentCalculation in Electrokinetics Using a Spectral StochasticFinite Element Method. Magnetics, IEEE Transactionson 44 (2008), June, vol. 6, p. 754 757. ISSN 00189464

[19] Gasparin, L ; Cernigoj, A ; Markic, S ; Fiser, R: Addi-tional Cogging Torque Components in Permanent-MagnetMotors Due to Manufacturing Imperfections. Magnetics,IEEE Transactions on 45 (2009), march, vol. 3, p. 12101213. ISSN 00189464

[20] Giet, M van d. ; Mller-Trapet, M ; Dietrich, P ; Pollow,M ; Blum, J ; Hameyer, K ; Vorlnder, M: Comparisonof acoustic single-value parameters for the design processof electrical machines. 39th International Congress onNoise Control Engineering INTER-NOISE. Lisbon, Portu-gal, June 2010

[21] Giet, M van d.: Analysis of electromagnetic acoustic noiseexcitations:: A contribution to low-noise design and to theauralization of electrical machines. 1., Aufl. Shaker, 2011. ISBN 3832299734. Dissertation, Institute of electricalMachines, RWTH-Aachen University

[22] Giet, M van d. ; Rothe, R ; Herranz Gracia, M ; Hameyer,K: Analysis of noise exciting magnetic force waves bymeans of numerical simulation and a space vector defini-tion. 18th International Conference on Electrical Ma-chines, ICEM 2008. Vilamoura, Portugal, September 2008

[23] Hafner, M ; Franck, D ; Hameyer, K: Static Electromag-netic Field Computation by Conformal Mapping in Perma-nent Magnet Synchronous Machines. Magnetics, IEEETransactions on 46 (2010), vol. 8, p.31053108. ISSN00189464

[24] Hafner, M ; Schning, M ; Antczak, M ; Demenko, A ;Hameyer, K: Interactive Postprocessing in 3D Electro-magnetics. Magnetics, IEEE Transactions on 46 (2010),vol. 8, p. 34373440. ISSN 00189464

[25] Hafner, M ; Schning, M ; Antczak, M ; Demenko, A ;Hameyer, K: Methods for Computation and Visualiza-tion of Magnetic Flux Lines in 3-D. Magnetics, IEEE

Transactions on 46 (2010), vol. 8, p. 33493352. ISSN00189464

[26] Hafner, M ; Bhmer, S ; Henrotte, F ; Hameyer, K: Inter-active visualization of transient 3D electromagnetic and n-dimensional parameter spaces in virtual reality. COMPEL30 (2011), May, vol. 3, p. 906915. ISSN 03321649

[27] Hafner, M ; Finken, T ; Felden, M ; Hameyer, K: Au-tomated Virtual Prototyping of Permanent Magnet Syn-chronous Machines for HEVs. IEEE Transactions onMagnetics 47 (2011), May, vol. 5, p. 10181021. ISSN00189464

[28] Hameyer, K ; Driesen, J ; De Gersem, H ; Belmans, R:The classification of coupled field problems. Magnetics,IEEE Transactions on 35 (1999), May, vol. 3, p. 1618 1621. ISSN 00189464

[29] Hameyer, K ; Belmanns, R: Numerical modelling and de-sign of electrical machines and devi