-
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.
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