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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

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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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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

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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.

References

[1] B. Stumberger, G. Stumberger, A. Hamler, M. Tr-

lep, M. Jesenik, and V. Gorican, “Increasing of output

power capability in a six-phase ﬂux-weakened perma-

nent magnet synchronous motor with a third harmonic

current injection,” IEEE Transactions on Magnetics,

vol. 39, pp. 3343–3345, 2003.

[2] D. G. Dorrell, C. Y. Leong, and R. A. McMahon,

“Analysis and performance assessment of six-pulse

inverter-fed three-phase and six-phase induction ma-

chines,” IEEE Transactions on Industry Applications,

vol. 42, pp. 1487–1495, November/December 2006.

[3] G. Aroquiadassou, A. Mpanda-Mabwe, F. Betin, and

G.-A. Capolino, “Six-phase induction machine drive

model for fault-tolerant operation,” SDEMPED, 2009.

[4] H. A. Toliyat, M. M. Rahimian, and T. A. Lipo, “dq

modeling of ﬁve phase synchronous reluctance ma-

chines including third harmonic of air-gap mmf,” Con-

ference Record of the 1991 IEEE Industry Applica-

tions Society Annual Meeting, 1991., pp. 231–237,

1991.

[5] D. Dujic, M. Jones, and E. Levi, “Features of

two multi-motor drive schemes supplied from ﬁve-

phase/ﬁve-leg voltage source inverters,” Conference

Proceedings on Power Conversion and Intelligent Mo-

tion, PCIM, Nuremberg, Germany, no. S2d-2, 2008.

[6] D. Dujic, M. Jones, and E. Levi, “Analysis of output

current ripple rms in multiphase drives using space

vector approach,” IEEE Transactions on Power Elec-

tronics, vol. 24, no. 8, pp. 1926–1938, 2009.

[7] R. L. A. Ribeiro, C. B. Jacobina, A. M. N. Lima, and

E. R. C. da Silva, “A strategy for improving reliability

of motor drive systems using a four-leg three-phase

converter,” Sixteenth Annual IEEE Applied Power

Electronic Conference and Expoition. APEC 2001,

vol. 1, pp. 385–391, 2001.

[8] R. Errabelli and P. Mutschler, “Fault-tolerant voltage

source inverter for permanent magnet drives,” IEEE

Transactions on Power Electronics, vol. 27, pp. 500

–508, feb. 2012.

[9] T. Treichl, Regelung von sechssträngigen permanen-

terregten Synchronmaschinen für den mobilen Anwen-

dungsfall. PhD thesis, FernUniversität, 2006.

[10] C. Kral and A. Haumer, “Modelica libraries for DC

machines, three phase and polyphase machines,” In-

ternational Modelica Conference, 4th, Hamburg, Ger-

many, pp. 549–558, 2005.

[11] C. Kral and A. Haumer, Object Oriented Modeling of

Rotating Electrical Machines. INTECH, 2011.

[12] C. Kral and A. Haumer, “The new fundamentalwave

library for modeling rotating electrical three phase

machines,” 8th International Modelica Conference,

2011.

[13] T. A. Lipo, “A d-q model for six phase induction

machines,” Conference Record of International Con-

ference on Electrical Machines, ICEM, pp. 860–867,

1980.

[14] E. Andresen and K. Bieniek, “Der Asynchronmotor

mit drei und sechs Wirkungssträngen am stromein-

prägenden Wechselrichter,” Archiv für Elektrotechnik,

vol. 63, pp. 153–167, 1981.

[15] H.-H. Jahn and R. Kasper, “Koordinatentransforma-

tionen zur Behandlung von Mehrphasensystemen,”

Archiv für Elektrotechnik, vol. 56, pp. 105–111, 1974.

[16] J. Stepina, Einführung in die allgemeine Raumzeiger-

Theorie der elektrischen Maschinen. Kaiserslautern:

Vorlesungsskriptum, Universität Kaiserslautern, 1979.

[17] C. Kral, A. Haumer, M. Bogomolov, and

E. Lomonova, “Harmonic wave model of a per-

manent magnet synchronous machine for modeling

partial demagnetization under short circuit condi-

tions,” XXth International Conference on Electrical

Machines (ICEM), pp. 295 –301, Sept. 2012.

[18] C. Kral and A. Haumer, “Simulation of electrical rotor

asymmetries in squirrel cage induction machines with

the extendedmachines library,” International Model-

ica Conference, 6th, Bielefeld, Germany, no. ID140,

pp. 351–359, 2008.

Session 1D: Electro-Magnetic Models and Libraries 1

DOI

10.3384/ECP14096135

Proceedings of the 10th International ModelicaConference

March 10-12, 2014, Lund, Sweden

143