Extension of the FundamentalWave Library towards Multi … · Extension of the FundamentalWave...

Extension of the FundamentalWave Library towards Multi PhaseElectric Machine Models

Christian Kral Anton Haumer Reinhard WöhrnschimmelElectric Machines, Drives and Systems Technical Consulting AIT GmbH

1060 Vienna, Austria 3423 St.Andrä-Wördern, Austria 1210 Vienna, [email protected] [email protected]

Abstract

Electric machine theory and electric machine simulationsmodels are often limited to three phases. Up to theModelica Standard Libray (MSL) version 3.2 the providedmachine models were limited to three phases. Particularlyfor large industrial drives and for redundancy reasons inelectric vehicles and aircrafts multi phase electric machinesare demanded. In the MSL 3.2.1 an extension of theexisting FundamentalWave library has been performed tocope with phase numbers greater than or equal to three.The developed machine models are fully incorporatingthe multi phase electric, magnetic, rotational and thermaldomain. In this publication the theoretical background ofthe machines models, Modelica implementation details, theparametrization of the models and simulation examples arepresented.

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 industrialapplications, traction drives in electric vehicles, railways,trams, and underground trains, as well as air craft motorsand generators. If a full leg of the supplying three phaseconverter fails, the machine cannot be operated any more,once stopped. In particular, the higher ambient tempera-tures of traction machines may cause the power electronicsto fail. In order to overcome machine outage due to a singleconverter leg failure, phase numbers greater than three maybe used for electric machines and power electronics. In thefollowing multi phase will indicate phase numbers greaterthan three. If it is referred to only three phases, this will beindicated explicitly.Multi phase drives consist of the multi phase machine in-cluding an inverter with power electronics plus control. Aphase 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 2n with integern are ex-cluded from the actual implementation. The reason for ex-cluding these phase numbers is that for example four oreight phase machines have to be handled differently sincetwo phase are separated byπ/2, notπ. 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 highpower machines in the Megawatt range [1]. In this casemaximum power of power electronics is doubled by usingtwo three phase converters supplying a six phase machine.Usually, the phase winding orientations of the two threephase are spatially shifted by 30◦ in order to additionallyreduce the magnitudes of space harmonics caused by thewinding magneto motive force (MMF) and to reduce thetorque ripple of the machine, respectively. The implemen-tation of a six phase winding does not significantly increasecost of the electric machine compared with a three phasemachine with the same power. Solely the additional wind-ing ends have to be conducted to the terminal box. For sixphase drives, the cost of power electronics increases due thedouble number of power electronic switches for the addi-tional legs. For doubling the power of industrial drives thedouble cost of power electronics is in line with doubling thepower. However, for traction machines six or nine phasemachines are used for redundancy reasons [2, 3]. In thiscase several state of the art three phase converters can beused. 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 invertersof smaller power rating are usually more expensive than athree phase inverter of the same total power. Due to themultiple three phase inverters installation space increases,too. Yet, state of the art control for three phase drives canbe adapted with relatively low effort due to modularly usingthree phase converters.

Second, the number of phases is not divisible by two. In thiscase mostly five (or seven) phase drives are used [4–6]. Thedrawback over a six phase inverter is that the modularity ofthe power electronics design is lower and thus cost may behigher and more design space may be required.

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For the sake of completeness one more redundancy conceptwill be discussed here, even though it is not related withmulti phase electric machines. Redundant drives with threephase machines may use modified topologies which eitheruse an additional leg or additional switches to operate themachine 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 ratingmay be provided to the electric machine.For controlling multi phase electric drives it is advantageousto 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 beadapted so that a symmetrical supply can be achieved. Atechnical paper dealing in more detail with analysis of themulti phase drive control is also submitted to the Modelica2014 conference and will be cited properly, in case it getsaccepted.In Modelica the first three phase electric machine modelshave 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 lossand brush loss are taken into account. Multi phase elec-tric machine models have already been published decadesago [13, 14]. Yet, in most computer simulations toolsthere are currently only three phase machine models avail-able. However, in the MSL 3.2.1 version of the packageModelica.Magnetic.FundamentalWave new multi phaseelectric machine models are introduced. In general, arbi-trary phase numbers for stator (and rotor) windings maybe used – excluding phase numbers equal to 2n with in-tegern. This article provides the theoretical background,details about the implementation, parametrization schemesand some examples.

2 Fundamental Wave Theory

Multi phase electric machine theory often relies on phasortransformations 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 besimplified 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 fluxand 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 quantitiesΦ = Φre + jΦim

andVm = Vm,re+ jVm,im. The complex magnetic quantitiesrepresent a spatial distribution of magnetic flux and mag-netic potential (difference):

Φ(ϕ) = Re[(Φre+ jΦim)e−jϕ]

= Φrecos(ϕ)+ Φim sin(ϕ)

Vm(ϕ) = Re[(Vm,re+ jVm,im)e−jϕ]

= Vm,recos(ϕ)+Vm,im sin(ϕ)

The complex potential (difference)Vm introduced in theconnector definition represents the total magnetic poten-tial (difference) of all poles. This quantity can, thus, alsobe seen as the complex magnetic potential difference of anequivalent two pole machine. Physical interpretations ofthe complex magnetic phasors are presented and discussedin [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 timetransient magnitudes of the magnetic flux and magnetic po-tential (difference). Each of the spatial field distributionscan be interpreted as a forward and backward traveling fun-damental wave component. Those effect of the two wavesis correctly taken in account by the proposed approach.

Particular supply imbalances and certain asymmetries maycause 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 particularfocus on the supply conditions.

Any higher harmonic wave effects are also not taken intoaccount by the FundamentalWave library. In case of higherharmonic waves an alternative implementation has to beconsidered as presented in [17]. The impact of rotorsaliency on the fundamental wave components of magneticflux 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 dueto better field weakening capabilities, higher pole numbersand higher power density. Such fractional slot windings canbe considered in the FundamentalWave library, as long asthe main power exchanging harmonic component is inter-preted as fundamental wave. All higher harmonic wavescause by fractional slot windings are not explicitly consid-ered, but the total effect of those higher harmonics can betaken into account by the total leakage inductance of thestator winding.

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Figure 1: Coil of an equivalent two pole machine with ori-entation; complex magnetic potential (difference) phasorsVm; complex magnetic flux phasorΦ

3 Electromagnetic Coupling

3.1 Single Phase Electromagnetic Coupling

The coupling of fundamental wave magnetic fluxand magnetic potential (difference) interacting withinstantaneous voltages and currents is elementarymodeled in the electro magnetic coupling modelsSinglePhaseElectroMagneticConverter . Figure 1shows the magnetic phasors and a winding of an equivalenttwo pole machine. In this case one magnetic pole coversa spatial angle equal toπ. The displayed coil is skewedand coil span is smaller thanπ. The coil can actually alsobe seen as a distributed winding. Skewing and distributedwindings with respect to the fundamental wave are con-sidered by the effective number of turns, represented bythe parametereffectiveTurns. The effective number ofturns of a real machine is determined by the real numberof turns, multiplied by the skewing factor and the chordingfactor. A more detail investigation on common windingsand the determination of winding factors is publishedin [10, 18]. A current through the investigated windingsgives rise to magnetic potential (difference) distributionwhich magnitude is equal effective number of turns timesthe current. The peak of accessory sinusoidal magneticpotential (difference) caused by the current is in line withtheorientation of the winding axis.In an electric machine several windings of stator and rotorwindings and permanent magnets contribute the total mag-netic potential difference – depending on the type of ma-chine. In the electromagnetic coupling model two physicallaws are implemented, i.e., Ampere’s law and the inductionlaw.

3.2 Ampere’s Law

Ampere’s law states that the total exciting magneto motiveforce is equal to the magnetic potential difference. For theinvestigated single phase winding it is useful to define thecomplex number of turns:

final parameter ComplexN=effectiveTurns*Modelica.ComplexMath.exp(

Complex(0, orientation))"Complex number of turns";

This complex quantity has the magnitude of the effectivenumber of turns and the phase angleorientation. Themagnetic 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 factor2.0/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 thetime derivative of the magnetic flux and the induced volt-age of the investigated winding. The projection of the com-plex magnetic flux onto theorientation times the effectivenumber 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 ofm single phase electromagnetic couplingmodels, wherem is the number of phases. Them electricalpins of the single phase electromagnetic coupling model areconnected to them pins of the electrical multi phases con-nector used by the multi phase coupling model.Them magnetic fundamental wave ports of the single phaseelectromagnetic 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 allwindings 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 symmetricalm phasewindings are considered. For multi phase systems andwindings with phase numbers greater than three twodifferent cases are distinguished, first, the number ofphases is divisible by three and second, the number ofphases is not divisible by three. The general functionsymmetricOrientation for determining the orientations ofwindings of anm phase electric machine is located in thepackageModelica.Electrical.MultiPhase.Functions:

function symmetricOrientationextends Modelica.Icons.Function;input Integer m "Number of phases";output Modelica.SIunits.Angleorientation[m]"Orientation of the resultingfundamental wave field phasors";

import Modelica.Constants.pi;

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Figure 2: Permanent magnet synchronous machine with op-tional damper cage

algorithmif mod(m, 2) == 0 then// Even number of phases

if m == 2 then// Special case two phase machineorientation[1] := 0;orientation[2] := +pi/2;

elseorientation[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 phasesorientation := {(k - 1)*2*pi/m

for k in 1:m};end if;

symmetricOrientation;

So the function is designed recursively so that subsystemsare modularly designed. In the following some examples ofphase numbers will be discussed. Two, four, eight, sixteen,thirty-two, etc. phase windings are currently not supported,as in general phase numbers equal to 2n with integern arenot considered.In order to summarize mathematical equations for dif-ferent phase numbersm in the following, abbreviationorientationk will be used to indicate the angles ofthe orientation of the winding axes of a symmetricalmphase winding. Soorientationk is the k-th element(phase index) of the result vector returned by functionsymmetricOrientation, called with argumentm.The symmetrically supply voltages and currents, respec-tively, have the phase angles

φk = −orientationk

Figure 3: Winding axes of symmetrical (a) three phasewinding and (b) five phase winding

Figure 4: Symmetrical phase angles of voltages and cur-rents, respectively, of (a) three phase winding and (b) fivephase winding

The winding orientations and the phase shifts of the supply-ing system have the same magnitudes for each phase indexk, 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 numbersm (not divisible by 2) the sym-metrical orientations of the winding axes are

orientationk =2π(k−1)

m.

So this applies for the casem = 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 2n with integern, them phase system is separated into two subsystems withm/2phases. The winding orientations of the second sub sys-

tem lags the first sub system byπm

. This is then the

point where the recursive determination of phase angleis initiated. For each of the two sub systems functionssymmetricOrientation is called, considering the lag angleπm

. It is important to explicitly note that the phase shift be-

tween the two sub systems is not2πm

, since in this case the



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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) tenphase winding

two sub systems where aligned at±π which does not makesense in a technical system for redundancy reasons.A six phase systemis then separated into two three phasesystems. The winding orientations of the second sub sys-tems lags the first sub system byπ/6, see Figs. 5–6. Thephase sequences (1-2-3) and (4-5-6) of the two sub systemsare equal.A ten phase systemconsists of two five phase systems withthe phase sequences (1-2-3-4-5) and (6-7-8-9-10). The sec-

ond sub systems lags the first sub system byπ10

, 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 viewno additional explanations are required to handle, for ex-ample, nine phase machines. Yet, a typical engineering ap-proach and functionsymmetricOrientation for numberingthe phase numbers are different and may require some ad-ditional comments:Practically, in most technical cases, electrical machineswith phase numbers divisible by three will be supplied by anappropriate number of three phase inverters. For six phasesystems this has already been demonstrated in subsection4.2. In the FundamentalWave library nine phase systems arehandled differently only in that sense, the sequence num-bering the phase windings is most likely different from anengineer who uses three three phase inverters. In the engi-neering phases (1-2-3) are most likely assigned to the firstinverter, phases (4-5-6) are assigned to the second inverterand phases (7-8-9) are assigned to the third inverters, seeFig. 7(a). In the FundamentalWave library the phase are

Figure 7: Numbering of nine phase symmetrical windingaccording to (a) an engineering approach using three threephase inverters and (b) the scheme of the FundamentalWavelibrary

numbered according to Fig. 7(b). Even though the num-bering is different the angles of the orientations are fullyidentical.For a fifteen phase a design engineer could always arguewhether 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 betweenthese two cases. The numbering scheme of the Fundamen-talWave follows a formal scheme and the user decides howthe 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 notchange. In the current implementation only linear magneticmaterials are considered. Saturation effects are not takeninto account.In all electric machine models of the FundamentalWave li-brary the total magnetic reluctance is concentrated in the airgap model. An example of permanent magnet synchronousmachine 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 airgap are not individually modeled. Even the linear charac-teristic of the permanent magnet is represent by the total airgap reluctance of the machine. The total reluctance takesthe variable reluctance of the air gap lengthδ into account.A sketch of the air gap and the reciprocal function 1/δ areshown 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 onfundamental wave forms is fully considered by unequal di-rect (d) and quadrature (q) axis reluctances. For rotor fixedmagnetic fluxPhi and total magnetic potential differenceV_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 d andq axis are, however, fixed with the rotor struc-ture. Magnetic rotor excitation of synchronous machines is,however, always aligned with thed axis.


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Figure 8: (a) Variable air gap lengthδ of a synchronous ma-chine and (b) reciprocal air gap function 1/δ versus spatialangleϕ of an equivalent two pole machine

Quantity ms = 3 ms ≥ 3Nominal stator phase voltage V ′

sN VsN = V ′sN

Nominal stator phase current I ′sN IsN = I ′sN3m

Nominal stator frequency f ′sN fsN = f ′

sNNominal electrical torque τ′

N τN = τ′N

Nominal electrical stator power P′sN PsN = P′

sN

Table 1: Parameters of machines with phases numbers equaland greater than three

In case that the saturation characteristics of the differentregions of the machine shall be considered in the future,the magnetic equivalent circuit has to be adapted such waythat each region is then represented by one non-linear reluc-tance.

6 Parametrization

Where do the parameters of a machine withms stator phasescome from? First, in the design stage of the machine, anengineer determines these parameters from finite elementanalysis or any other electromagnetic design software. Sec-ond, the parameters of a three phase machine are known orestimated and the users wants to determine the parametersof an equivalentm phase machine. The equivalence thenoften refers to equivalent speed, frequency, torque, power,phase voltage, power factor and efficiency. For the secondcase the exact calculations will be provided in the follow-ing:Assume, the nominal parameter of a three phase and arbi-trarymphase machine as listed in Tab. 1. All the parametersof a three phase machine are indicated with′. According tothe relationship between themphase nominal phase voltageand current, all resistances and inductances of anm phasemachine are scaled withm/3. A list of relevant parametersis summarized in Tab. 2.The rotor winding of squirrel cage induction machines areimplemented as equivalentms phase windings – wherems

is the number of stator phases. Slip ring induction machinesmay have different phases numbers of stator and rotor –wheremr is the number of rotor phases. For synchronousmachines with permanent magnets, electrical excitation andreluctance rotor, the optional damper cage is implemented

Quantity m= 3 m≥ 3

Stator resistance R′s Rs= R′

sm3

Stator stray inductance L′sσ Lsσ = L′

sσm3

Main field inductance L′m Lm = L′

mm3

Main field inductance,d-axis L′md Lmd = L′

mdm3

Main field inductance,q-axis L′mq Lmq= L′

mqm3

Table 2: Stator parameters of three andms ≥ 3 phase ma-chines

Quantity m= 3 m≥ 3Induction machine with squirrel cage

Rotor cage resistance R′r Rr = R′

rms

3Rotor stray inductance L′

rσ Lrσ = L′rσ

ms

3Induction machine with slip ring rotor

Rotor cage resistance R′r Rr = R′

rmr

3Rotor stray inductance L′

rσ Lrσ = L′rσ

mr

3All synchronous machinesDamper cage resistance,d-axis R′

rd Rrd = R′rd

Damper cage resistance,q-axis R′rq Rrq = R′

rqDamper cage

stray inductance,d-axis L′rσ,d Lrσ,d = L′

rσ,d

Damper cagestray inductance,q-axis L′

rσ,q Lrσ,q = L′rσ,q

Table 3: Rotor parameters of machines with three andms ≥3 andmr ≥ 3 phases

with salient rotor parameters with respect to the direct (d)and quadrature (q) axis. So there is no difference betweenthree 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 alltypes of machines, comparing three phase and multi phase(m= 5) machines. The three and five phase machines areoperated with equal nominal phase voltages. The parame-ters of the five phase machine are parameterized such wayautomatically 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 bechanged 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 twoasynchronous induction machines with squirrel cage rotorare started directly on line (DOL) by means of an idealswitch; see Figure 9. The machines start from standstill.



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Figure 9: Comparing a three and a multi phase (m = 5)permanent magnet synchronous machine, operated on anidealized voltage inverter

Figure 10: Simulation result of electrical torque of a threeand five phase squirrel cage induction machines; the torquesare equal

Figure 11: Simulation result of the quasi RMS currents ofa three and five phase squirrel cage induction machines; thecurrent ratio is five to three

Figure 12: Comparing a three and a multi phase (m = 5)induction machine started directly on line , operated on anidealized voltage inverter

The mechanical load is modeled by means of a quadraticspeed dependent load torque and an additional load inertia.This example demonstrates equivalent dynamic behavior ofthe three and five phase machine. Particularly, the electricaltorque, speed, and the particular losses are equal.

Both machines have the same nominal phase voltage, butdifferent nominal phase currents according to Tab. 1. InFig. 10 the two identical electric torques of the two ma-chines are shown. The different quasi RMS currents of twomachines are displayed in Fig. 11. The current ratio is equalto 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 ringinduction machine, operating the stator direct on line; seeFig. 12 The number of stator phasesms = 5 and the numberof rotor phases,mr = 5, are equal. The multi phase ro-tor windings of each machine are connected with a rheostatwhich is shorted after a give time periodtRheostat =1.0 second. The rheostat enables a greater starting torque– but worse efficiency. Therefore, after one second, therheostats are shortened to achieve a higher efficiency andspeed of the machines. The user can copy the example andchange the rotor phase numbermr such way that it differsfrom the stator phase number,ms. This case is also sup-ported the FundamentalWave library. In Fig. 13 and 14 theelectromagnetic torques and the quasi RMS currents of thetwo machines are compared. The torques are identical andthe current ratio is five to three according to Tab. 1.


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Figure 13: Simulation result of electrical torque of a threeand five phase slip ring induction machines; the torques areequal

Figure 14: Simulation result of the quasi RMS currents ofa three and five phase slip ring induction machines; the cur-rent ratio is five to three

7.3 Synchronous Generator with ElectricalExcitation

In exampleExamples.SMPM_Generator two mainssupplied electrical excited synchronous machine with threeand five stator phases are compared; see Fig. 16. For eachmachine shaft speed is constant and slightly different thansynchronous speed. In this experiment each rotor is forcedto make a full revolution relative to the magnetic field. InFig. 16 the generated torques versus load angle are shownfor a fixed level of excitation. In addition to the sinusoidalwaveform of the torque a second harmonic component issuperimposed due to the saliency of the rotor. However, forthe investigated machines, the saliency effect is very smallso that the torque waveform almost appears as a pure sinewave. The quasi RMS currents of the two machines arecompared in Fig. 17. The current ratios of the three and fivephase machine is five to three.

Figure 15: Comparing a three and a multi phase (m = 5)permanent magnet synchronous machine, operated on anidealized voltage inverter

Figure 16: Simulation result of electrical torque, comparinga three and five phase

Figure 17: Simulation result of the quasi RMS currents ofa three and five phase synchronous machine with electricexcitation; the current ratio is five to three



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8 Conclusions

The paper presents an extension of the FundamentalWavelibrary towards multi phase stator (and rotor) windings withphase numbers greater or equal than three. This library isincluded in the MSL 3.2.1. Assumptions and limitationsof the presented implementation are explained. In the newFundamentalWave library only symmetrical windings aresupported. 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 phasemachines 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 fundingfrom the ENIAC Joint Undertaking under grant agreementno. 270693-2 and from the Österreichis- che Forschungs-förderungsgesellschaft mbH under project no. 829420.

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[18] C. Kral and A. Haumer, “Simulation of electrical rotorasymmetries in squirrel cage induction machines withthe extendedmachines library,”International Model-ica Conference, 6th, Bielefeld, Germany, no. ID140,pp. 351–359, 2008.


DOI10.3384/ECP14096135


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