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Extension of the FundamentalWave Library towards Multi Phase
Electric Machine Models
Christian Kral Anton Haumer Reinhard Wöhrnschimmel
Electric Machines, Drives and Systems Technical Consulting AIT GmbH
1060 Vienna, Austria 3423 St.Andrä-Wördern, Austria 1210 Vienna, Austria
dr.christian.kral@gmail.com a.haumer@haumer.at
Abstract
Electric machine theory and electric machine simulations
models are often limited to three phases. Up to the
Modelica Standard Libray (MSL) version 3.2 the provided
machine models were limited to three phases. Particularly
for large industrial drives and for redundancy reasons in
electric vehicles and aircrafts multi phase electric machines
are demanded. In the MSL 3.2.1 an extension of the
existing FundamentalWave library has been performed to
cope with phase numbers greater than or equal to three.
The developed machine models are fully incorporating
the multi phase electric, magnetic, rotational and thermal
domain. In this publication the theoretical background of
the machines models, Modelica implementation details, the
parametrization of the models and simulation examples are
presented.
Keywords: Modelica Standard Library, multi phase, elec-
tric machine models, induction machine, synchronous ma-
chine, synchronous reluctance machine
1 Introduction
Three phase induction, synchronous and synchronous reluc-
tance machines are state of the art solutions for industrial
applications, traction drives in electric vehicles, railways,
trams, and underground trains, as well as air craft motors
and generators. If a full leg of the supplying three phase
converter fails, the machine cannot be operated any more,
once stopped. In particular, the higher ambient tempera-
tures of traction machines may cause the power electronics
to fail. In order to overcome machine outage due to a single
converter leg failure, phase numbers greater than three may
be used for electric machines and power electronics. In the
following multi phase will indicate phase numbers greater
than three. If it is referred to only three phases, this will be
indicated explicitly.
Multi phase drives consist of the multi phase machine in-
cluding an inverter with power electronics plus control. A
phase number greater than three thus requires a higher num-
ber of power electronic switches, such as IGBTs or MOS-
FETs, etc. The higher phase numbers, however, increase
cost and add complexity to the drive structure.
However, phase numbers equal to 2nwith integer nare ex-
cluded from the actual implementation. The reason for ex-
cluding these phase numbers is that for example four or
eight phase machines have to be handled differently since
two phase are separated by π/2, not π. In general, two dif-
ferent philosophies of multi phase drives exist:
First, the number of phases is divisible by two. In industry,
typically, six phase machines are used to overcome maxi-
mum power limitations of power electronics supplying high
power machines in the Megawatt range [1]. In this case
maximum power of power electronics is doubled by using
two three phase converters supplying a six phase machine.
Usually, the phase winding orientations of the two three
phase are spatially shifted by 30◦in order to additionally
reduce the magnitudes of space harmonics caused by the
winding magneto motive force (MMF) and to reduce the
torque ripple of the machine, respectively. The implemen-
tation of a six phase winding does not significantly increase
cost of the electric machine compared with a three phase
machine with the same power. Solely the additional wind-
ing ends have to be conducted to the terminal box. For six
phase drives, the cost of power electronics increases due the
double number of power electronic switches for the addi-
tional legs. For doubling the power of industrial drives the
double cost of power electronics is in line with doubling the
power. However, for traction machines six or nine phase
machines are used for redundancy reasons [2, 3]. In this
case several state of the art three phase converters can be
used. The redundancy concept is then realized with stan-
dard components which is cheaper than designing the indi-
vidual legs of the inverter. Yet, several three phase inverters
of smaller power rating are usually more expensive than a
three phase inverter of the same total power. Due to the
multiple three phase inverters installation space increases,
too. Yet, state of the art control for three phase drives can
be adapted with relatively low effort due to modularly using
three phase converters.
Second, the number of phases is not divisible by two. In this
case mostly five (or seven) phase drives are used [4–6]. The
drawback over a six phase inverter is that the modularity of
the power electronics design is lower and thus cost may be
higher and more design space may be required.
DOI
10.3384/ECP14096135
Proceedings of the 10th International ModelicaConference
March 10-12, 2014, Lund, Sweden
135
For the sake of completeness one more redundancy concept
will be discussed here, even though it is not related with
multi phase electric machines. Redundant drives with three
phase machines may use modified topologies which either
use an additional leg or additional switches to operate the
machine in case of a failure. These topologies have the ca-
pability to be reconfigured when a failure occurs [7, 8]. De-
pending on the actual topology even the full power rating
may be provided to the electric machine.
For controlling multi phase electric drives it is advantageous
to control current components which represent the funda-
mental wave MMF and magnetic voltage, respectively [9].
The pulse width modulation scheme for multi phase con-
verters with phase numbers not divisible by three has to be
adapted so that a symmetrical supply can be achieved. A
technical paper dealing in more detail with analysis of the
multi phase drive control is also submitted to the Modelica
2014 conference and will be cited properly, in case it gets
accepted.
In Modelica the first three phase electric machine models
have been introduced with the MSL 2.1 in 2004 [10, 11].
An alternative implementation with magnetic fundamen-
tal wave phasors was introduced in MSL 3.2 in 2010
[12]. Since then in these models copper loss, (eddy cur-
rent) core loss, friction loss, stray load loss, PM loss
and brush loss are taken into account. Multi phase elec-
tric machine models have already been published decades
ago [13, 14]. Yet, in most computer simulations tools
there are currently only three phase machine models avail-
able. However, in the MSL 3.2.1 version of the package
Modelica.Magnetic.FundamentalWave new multi phase
electric machine models are introduced. In general, arbi-
trary phase numbers for stator (and rotor) windings may
be used – excluding phase numbers equal to 2nwith in-
teger n. This article provides the theoretical background,
details about the implementation, parametrization schemes
and some examples.
2 Fundamental Wave Theory
Multi phase electric machine theory often relies on phasor
transformations of currents, voltages and magnetic fluxes
[15, 16]. A typical transformation is the symmetrical com-
ponents of the instantaneous values. In case of fully sym-
metrical supply the machine equations based on the sym-
metrical components of instantaneous components can be
simplified extensively.
The FundamentalWave machine models only consider fun-
damental wave effects so there is also a complex phasor rep-
resentation of the fundamental wave of the magnetic flux
and magnetic potential (difference), respectively. The con-
nector definition of the FundamentalWave library shows:
connector MagneticPort
"Complex magnetic port"
Modelica.SIunits.
ComplexMagneticPotentialDifference V_m
"Complex magnetic potential difference";
flow Modelica.SIunits.
ComplexMagneticFlux Phi
"Complex magnetic flux";
end MagneticPort;
Please note, that the potential and flow variable of the con-
nector represent instantaneous quantities Φ=Φre +jΦim
and Vm=Vm,re +jVm,im. The complex magnetic quantities
represent a spatial distribution of magnetic flux and mag-
netic potential (difference):
Φ(ϕ) = Re[(Φre +jΦim)e−jϕ]
=Φre cos(ϕ) + Φim sin(ϕ)
Vm(ϕ) = Re[(Vm,re +jVm,im)e−jϕ]
=Vm,re cos(ϕ) +Vm,im sin(ϕ)
The complex potential (difference) Vmintroduced in the
connector definition represents the total magnetic poten-
tial (difference) of all poles. This quantity can, thus, also
be seen as the complex magnetic potential difference of an
equivalent two pole machine. Physical interpretations of
the complex magnetic phasors are presented and discussed
in [12].
Voltages and currents are instantaneous quantities, so ar-
bitrary waveforms and operating conditions are covered.
Therefore, the machines can also be supplied with asym-
metric voltages or currents. It is yet assumed that only fun-
damental wave effects due to these asymmetries are consid-
ered. Supply voltage or current imbalance give rise to time
transient magnitudes of the magnetic flux and magnetic po-
tential (difference). Each of the spatial field distributions
can be interpreted as a forward and backward traveling fun-
damental wave component. Those effect of the two waves
is correctly taken in account by the proposed approach.
Particular supply imbalances and certain asymmetries may
cause magnetic flux and magnetic potential (difference)
phasors which are not related with the fundamental wave.
Those higher harmonic waves are not covered by the Fun-
damentalWave library. It is therefore the user’s responsi-
bility to consider these model limitations – with particular
focus on the supply conditions.
Any higher harmonic wave effects are also not taken into
account by the FundamentalWave library. In case of higher
harmonic waves an alternative implementation has to be
considered as presented in [17]. The impact of rotor
saliency on the fundamental wave components of magnetic
flux and magnetic potential (difference) is, however, con-
sidered in the presented implementation.
Concentrated windings and fractional slot windings, respec-
tively, are very common in PM synchronous machines due
to better field weakening capabilities, higher pole numbers
and higher power density. Such fractional slot windings can
be considered in the FundamentalWave library, as long as
the main power exchanging harmonic component is inter-
preted as fundamental wave. All higher harmonic waves
cause by fractional slot windings are not explicitly consid-
ered, but the total effect of those higher harmonics can be
taken into account by the total leakage inductance of the
stator winding.
Extension of the FundamentalWave Library towards Multi Phase Electric Machine Models
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Figure 1: Coil of an equivalent two pole machine with ori-
entation; complex magnetic potential (difference) phasors
Vm; complex magnetic flux phasor Φ
3 Electromagnetic Coupling
3.1 Single Phase Electromagnetic Coupling
The coupling of fundamental wave magnetic flux
and magnetic potential (difference) interacting with
instantaneous voltages and currents is elementary
modeled in the electro magnetic coupling models
SinglePhaseElectroMagneticConverter . Figure 1
shows the magnetic phasors and a winding of an equivalent
two pole machine. In this case one magnetic pole covers
a spatial angle equal to π. The displayed coil is skewed
and coil span is smaller than π. The coil can actually also
be seen as a distributed winding. Skewing and distributed
windings with respect to the fundamental wave are con-
sidered by the effective number of turns, represented by
the parameter effectiveTurns. The effective number of
turns of a real machine is determined by the real number
of turns, multiplied by the skewing factor and the chording
factor. A more detail investigation on common windings
and the determination of winding factors is published
in [10, 18]. A current through the investigated windings
gives rise to magnetic potential (difference) distribution
which magnitude is equal effective number of turns times
the current. The peak of accessory sinusoidal magnetic
potential (difference) caused by the current is in line with
the orientation of the winding axis.
In an electric machine several windings of stator and rotor
windings and permanent magnets contribute the total mag-
netic potential difference – depending on the type of ma-
chine. In the electromagnetic coupling model two physical
laws are implemented, i.e., Ampere’s law and the induction
law.
3.2 Ampere’s Law
Ampere’s law states that the total exciting magneto motive
force is equal to the magnetic potential difference. For the
investigated single phase winding it is useful to define the
complex number of turns:
final parameter Complex
N=effectiveTurns*Modelica.ComplexMath.exp(
Complex(0, orientation))
"Complex number of turns";
This complex quantity has the magnitude of the effective
number of turns and the phase angle orientation. The
magnetic potential (difference) of the coupling model, V_m,
and the current of the investigated winding, i, are then re-
lated by:
V_m = (2.0/pi)*N*i;
The factor 2.0/pi is the consequence of averaging the sinu-
soidal fundamental wave flux waveform over one pole pair.
3.3 Induction Law
Induction law describes the the relationship between the
time derivative of the magnetic flux and the induced volt-
age of the investigated winding. The projection of the com-
plex magnetic flux onto the orientation times the effective
number of turns is equal to the negative terminal voltage.
-v = Modelica.ComplexMath.real(
Modelica.ComplexMath.conj(N)*
Complex(der(Phi.re),der(Phi.im)));
3.4 Multi Phase Electromagnetic Coupling
The multi phase electromagnetic coupling model is com-
posed of a vector of msingle phase electromagnetic coupling
models, where mis the number of phases. The melectrical
pins of the single phase electromagnetic coupling model are
connected to the mpins of the electrical multi phases con-
nector used by the multi phase coupling model.
The mmagnetic fundamental wave ports of the single phase
electromagnetic coupling models are connected in series.
The magnetic series connection is a consequence of, first,
each winding being exposed to the same magnetic flux wave
– but being located spatially on different locations. Sec-
ond, the total magnetic potential (difference) excited by all
windings is determined by the sum of the magnetic poten-
tial (differences) of all individual windings.
4 Phase Orientations – Winding Axes
In the FundamentalWave library only symmetrical mphase
windings are considered. For multi phase systems and
windings with phase numbers greater than three two
different cases are distinguished, first, the number of
phases is divisible by three and second, the number of
phases is not divisible by three. The general function
symmetricOrientation for determining the orientations of
windings of an mphase electric machine is located in the
package Modelica.Electrical.MultiPhase.Functions:
function symmetricOrientation
extends Modelica.Icons.Function;
input Integer m "Number of phases";
output Modelica.SIunits.Angle
orientation[m]
"Orientation of the resulting
fundamental wave field phasors";
import Modelica.Constants.pi;
Session 1D: Electro-Magnetic Models and Libraries 1
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Figure 2: Permanent magnet synchronous machine with op-
tional damper cage
algorithm
if mod(m, 2) == 0 then
// Even number of phases
if m == 2 then
// Special case two phase machine
orientation[1] := 0;
orientation[2] := +pi/2;
else
orientation[1:integer(m/2)] :=
symmetricOrientation(integer(m/2));
orientation[integer(m/2) + 1:m] :=
symmetricOrientation(integer(m/2))
- fill(pi/m, integer(m/2));
end if;
else
// Odd number of phases
orientation := {(k - 1)*2*pi/m
for kin 1:m};
end if;
symmetricOrientation;
So the function is designed recursively so that subsystems
are modularly designed. In the following some examples of
phase numbers will be discussed. Two, four, eight, sixteen,
thirty-two, etc. phase windings are currently not supported,
as in general phase numbers equal to 2nwith integer nare
not considered.
In order to summarize mathematical equations for dif-
ferent phase numbers min the following, abbreviation
orientationkwill be used to indicate the angles of
the orientation of the winding axes of a symmetrical m
phase winding. So orientationkis the k-th element
(phase index) of the result vector returned by function
symmetricOrientation, called with argument m.
The symmetrically supply voltages and currents, respec-
tively, have the phase angles
φk=−orientationk
Figure 3: Winding axes of symmetrical (a) three phase
winding and (b) five phase winding
Figure 4: Symmetrical phase angles of voltages and cur-
rents, respectively, of (a) three phase winding and (b) five
phase winding
The winding orientations and the phase shifts of the supply-
ing system have the same magnitudes for each phase index
k, but different signs. This is a general property of symmet-
rical systems supplying symmetrical windings, see Figs. 3–
6.
In the following only symmetrical winding axes will be as-
sumed. The phase angles of symmetrical voltage and cur-
rent supply are also presented, even though symmetric sup-
ply is not assumed in the FundamentalWave library.
4.1 Odd Phase Numbers
For all odd phase numbers m(not divisible by 2) the sym-
metrical orientations of the winding axes are
orientationk=2π(k−1)
m.
So this applies for the case m=3, m=5, m=7, m=9,
m=11, m=13, m=15, etc., see Fig. 3–4.
4.2 Even Phase Numbers
For even phase numbers unequal to 2nwith integer n, the
mphase system is separated into two subsystems with m/2
phases. The winding orientations of the second sub sys-
tem lags the first sub system by π
m. This is then the
point where the recursive determination of phase angle
is initiated. For each of the two sub systems functions
symmetricOrientation is called, considering the lag angle
π
m. It is important to explicitly note that the phase shift be-
tween the two sub systems is not 2π
m, since in this case the
Extension of the FundamentalWave Library towards Multi Phase Electric Machine Models
138 Proceedings of the 10th International ModelicaConference
March 10-12, 2014, Lund, Sweden
DOI
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Figure 5: Winding axes of symmetrical (a) six phase wind-
ing and (b) ten phase winding
Figure 6: Symmetrical phase angles of voltages and cur-
rents, respectively, of (a) six phase winding and (b) ten
phase winding
two sub systems where aligned at ±πwhich does not make
sense in a technical system for redundancy reasons.
Asix phase system is then separated into two three phase
systems. The winding orientations of the second sub sys-
tems lags the first sub system by π/6, see Figs. 5–6. The
phase sequences (1-2-3) and (4-5-6) of the two sub systems
are equal.
Aten phase system consists of two five phase systems with
the phase sequences (1-2-3-4-5) and (6-7-8-9-10). The sec-
ond sub systems lags the first sub system by π
10, see Figs. 5–
6.
4.3 Phase Numbers Divisible by Three
Phase numbers divisible by three are either covered by sub-
section 4.1 and 4.2. Therefore, from a formal point of view
no additional explanations are required to handle, for ex-
ample, nine phase machines. Yet, a typical engineering ap-
proach and function symmetricOrientation for numbering
the phase numbers are different and may require some ad-
ditional comments:
Practically, in most technical cases, electrical machines
with phase numbers divisible by three will be supplied by an
appropriate number of three phase inverters. For six phase
systems this has already been demonstrated in subsection
4.2. In the FundamentalWavelibrary nine phase systems are
handled differently only in that sense, the sequence num-
bering the phase windings is most likely different from an
engineer who uses three three phase inverters. In the engi-
neering phases (1-2-3) are most likely assigned to the first
inverter, phases (4-5-6) are assigned to the second inverter
and phases (7-8-9) are assigned to the third inverters, see
Fig. 7(a). In the FundamentalWave library the phase are
Figure 7: Numbering of nine phase symmetrical winding
according to (a) an engineering approach using three three
phase inverters and (b) the scheme of the FundamentalWave
library
numbered according to Fig. 7(b). Even though the num-
bering is different the angles of the orientations are fully
identical.
For a fifteen phase a design engineer could always argue
whether such system can be seen as five three phase sys-
tems or as three five phase systems. However, from operat-
ing conditions point of view, there is no difference between
these two cases. The numbering scheme of the Fundamen-
talWave follows a formal scheme and the user decides how
the machine phases are supplied.
5 Magnetic Components
All the existing magnetic components of the Fundamen-
talWave library can be re-used for the multi-phase ma-
chine models, since the magnetic port representation did not
change. In the current implementation only linear magnetic
materials are considered. Saturation effects are not taken
into account.
In all electric machine models of the FundamentalWave li-
brary the total magnetic reluctance is concentrated in the air
gap model. An example of permanent magnet synchronous
machine with optional damper cage is displayed in Fig. 2.
In the actual implementation of the FundamentalWave li-
brary the magnetic reluctances of the stator, rotor and air
gap are not individually modeled. Even the linear charac-
teristic of the permanent magnet is represent by the total air
gap reluctance of the machine. The total reluctance takes
the variable reluctance of the air gap length δinto account.
A sketch of the air gap and the reciprocal function 1/δare
shown in Fig. 8.
The effect of variable magnetic reluctance due to the un-
even shape of the air gap and the arrangement of magnets,
respectively, is called saliency. The effect of saliency on
fundamental wave forms is fully considered by unequal di-
rect (d) and quadrature (q) axis reluctances. For rotor fixed
magnetic flux Phi and total magnetic potential difference
V_m the following relationship applies:
(pi/2.0)*V_m.re = Phi.re *R_m.d;
(pi/2.0)*V_m.im = Phi_im *R_m.q;
The dand qaxis are, however, fixed with the rotor struc-
ture. Magnetic rotor excitation of synchronous machines is,
however, always aligned with the daxis.
Session 1D: Electro-Magnetic Models and Libraries 1
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139
Figure 8: (a) Variable air gap length δof a synchronous ma-
chine and (b) reciprocal air gap function 1/δversus spatial
angle ϕof an equivalent two pole machine
Quantity ms=3ms≥3
Nominal stator phase voltage V′
sN VsN =V′
sN
Nominal stator phase current I′
sN IsN =I′
sN 3
m
Nominal stator frequency f′
sN fsN =f′
sN
Nominal electrical torque τ′
NτN=τ′
N
Nominal electrical stator power P′
sN PsN =P′
sN
Table 1: Parameters of machines with phases numbers equal
and greater than three
In case that the saturation characteristics of the different
regions of the machine shall be considered in the future,
the magnetic equivalent circuit has to be adapted such way
that each region is then represented by one non-linear reluc-
tance.
6 Parametrization
Where do the parameters of a machine with msstator phases
come from? First, in the design stage of the machine, an
engineer determines these parameters from finite element
analysis or any other electromagnetic design software. Sec-
ond, the parameters of a three phase machine are known or
estimated and the users wants to determine the parameters
of an equivalent mphase machine. The equivalence then
often refers to equivalent speed, frequency, torque, power,
phase voltage, power factor and efficiency. For the second
case the exact calculations will be provided in the follow-
ing:
Assume, the nominal parameter of a three phase and arbi-
trary mphase machine as listed in Tab. 1. All the parameters
of a three phase machine are indicated with ′. According to
the relationship between the mphase nominal phase voltage
and current, all resistances and inductances of an mphase
machine are scaled with m/3. A list of relevant parameters
is summarized in Tab. 2.
The rotor winding of squirrel cage induction machines are
implemented as equivalent msphase windings – where ms
is the number of stator phases. Slip ring induction machines
may have different phases numbers of stator and rotor –
where mris the number of rotor phases. For synchronous
machines with permanent magnets, electrical excitation and
reluctance rotor, the optional damper cage is implemented
Quantity m=3m≥3
Stator resistance R′
sRs=R′
sm
3
Stator stray inductance L′
sσLsσ=L′
sσm
3
Main field inductance L′
mLm=L′
mm
3
Main field inductance, d-axis L′
md Lmd =L′
md m
3
Main field inductance, q-axis L′
mq Lmq =L′
mq m
3
Table 2: Stator parameters of three and ms≥3 phase ma-
chines
Quantity m=3m≥3
Induction machine with squirrel cage
Rotor cage resistance R′
rRr=R′
rms
3
Rotor stray inductance L′
rσLrσ=L′
rσms
3
Induction machine with slip ring rotor
Rotor cage resistance R′
rRr=R′
rmr
3
Rotor stray inductance L′
rσLrσ=L′
rσmr
3
All synchronous machines
Damper cage resistance, d-axis R′
rd Rrd =R′
rd
Damper cage resistance, q-axis R′
rq Rrq =R′
rq
Damper cage
stray inductance, d-axis L′
rσ,dLrσ,d=L′
rσ,d
Damper cage
stray inductance, q-axis L′
rσ,qLrσ,q=L′
rσ,q
Table 3: Rotor parameters of machines with three and ms≥
3 and mr≥3 phases
with salient rotor parameters with respect to the direct (d)
and quadrature (q) axis. So there is no difference between
three and multi phase damper cage models and parameters.
The rotor cage parameters of all machine types are summa-
rized in Tab. 3.
7 Examples
In the FundamentalWave library there are examples for all
types of machines, comparing three phase and multi phase
(m=5) machines. The three and five phase machines are
operated with equal nominal phase voltages. The parame-
ters of the five phase machine are parameterized such way
automatically that the phase number can be increased with-
out changing torques and powers for the multi phase ma-
chines. In a duplicate example the phase numbers can be
changed for from five to higher numbers.
The following examples are included in the Fundamental-
Wave library:
7.1 Induction Machine with Squirrel Cage
In model Examples.AIMC_DOL_MultiPhase two
asynchronous induction machines with squirrel cage rotor
are started directly on line (DOL) by means of an ideal
switch; see Figure 9. The machines start from standstill.
Extension of the FundamentalWave Library towards Multi Phase Electric Machine Models
140 Proceedings of the 10th International ModelicaConference
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DOI
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Figure 9: Comparing a three and a multi phase (m=5)
permanent magnet synchronous machine, operated on an
idealized voltage inverter
Figure 10: Simulation result of electrical torque of a three
and five phase squirrel cage induction machines; the torques
are equal
Figure 11: Simulation result of the quasi RMS currents of
a three and five phase squirrel cage induction machines; the
current ratio is five to three
Figure 12: Comparing a three and a multi phase (m=5)
induction machine started directly on line , operated on an
idealized voltage inverter
The mechanical load is modeled by means of a quadratic
speed dependent load torque and an additional load inertia.
This example demonstrates equivalent dynamic behavior of
the three and five phase machine. Particularly, the electrical
torque, speed, and the particular losses are equal.
Both machines have the same nominal phase voltage, but
different nominal phase currents according to Tab. 1. In
Fig. 10 the two identical electric torques of the two ma-
chines are shown. The different quasi RMS currents of two
machines are displayed in Fig. 11. The current ratio is equal
to five over three.
7.2 Induction Machine with Slip Ring Rotor
Model Examples.BasicMachines.AIMS_Start-
_MultiPhase compares a three and a five phase slip ring
induction machine, operating the stator direct on line; see
Fig. 12 The number of stator phases ms=5 and the number
of rotor phases, mr=5, are equal. The multi phase ro-
tor windings of each machine are connected with a rheostat
which is shorted after a give time period tRheostat =
1.0 second. The rheostat enables a greater starting torque
– but worse efficiency. Therefore, after one second, the
rheostats are shortened to achieve a higher efficiency and
speed of the machines. The user can copy the example and
change the rotor phase number mrsuch way that it differs
from the stator phase number, ms. This case is also sup-
ported the FundamentalWave library. In Fig. 13 and 14 the
electromagnetic torques and the quasi RMS currents of the
two machines are compared. The torques are identical and
the current ratio is five to three according to Tab. 1.
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Figure 13: Simulation result of electrical torque of a three
and five phase slip ring induction machines; the torques are
equal
Figure 14: Simulation result of the quasi RMS currents of
a three and five phase slip ring induction machines; the cur-
rent ratio is five to three
7.3 Synchronous Generator with Electrical
Excitation
In example Examples.SMPM_Generator two mains
supplied electrical excited synchronous machine with three
and five stator phases are compared; see Fig. 16. For each
machine shaft speed is constant and slightly different than
synchronous speed. In this experiment each rotor is forced
to make a full revolution relative to the magnetic field. In
Fig. 16 the generated torques versus load angle are shown
for a fixed level of excitation. In addition to the sinusoidal
waveform of the torque a second harmonic component is
superimposed due to the saliency of the rotor. However, for
the investigated machines, the saliency effect is very small
so that the torque waveform almost appears as a pure sine
wave. The quasi RMS currents of the two machines are
compared in Fig. 17. The current ratios of the three and five
phase machine is five to three.
Figure 15: Comparing a three and a multi phase (m=5)
permanent magnet synchronous machine, operated on an
idealized voltage inverter
Figure 16: Simulation result of electrical torque, comparing
a three and five phase
Figure 17: Simulation result of the quasi RMS currents of
a three and five phase synchronous machine with electric
excitation; the current ratio is five to three
Extension of the FundamentalWave Library towards Multi Phase Electric Machine Models
142 Proceedings of the 10th International ModelicaConference
March 10-12, 2014, Lund, Sweden
DOI
10.3384/ECP14096135
8 Conclusions
The paper presents an extension of the FundamentalWave
library towards multi phase stator (and rotor) windings with
phase numbers greater or equal than three. This library is
included in the MSL 3.2.1. Assumptions and limitations
of the presented implementation are explained. In the new
FundamentalWave library only symmetrical windings are
supported. The structures of symmetrical multi phase wind-
ings and supplies are introduced.
The parametrization of the multi phase machines is dis-
cussed. Conversion tables for parameterizing multi phase
machines equivalent to three phase machines are presented.
Simulation examples of three and equivalent five phase in-
duction and synchronous machines are presented and com-
pared.
9 Acknowledgement
The research leading to these results has received funding
from the ENIAC Joint Undertaking under grant agreement
no. 270693-2 and from the Österreichis- che Forschungs-
förderungsgesellschaft mbH under project no. 829420.
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Session 1D: Electro-Magnetic Models and Libraries 1
DOI
10.3384/ECP14096135
Proceedings of the 10th International ModelicaConference
March 10-12, 2014, Lund, Sweden
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