Extension of the FundamentalWave Library towards Multi Phase Electric Machine Models

Transcription

1 Extension of the FundaentalWave Library towards Multi Phase Electric Machine Models Christian Kral Anton Hauer Reinhard Wöhrnschiel Electric Machines, Drives and Systes Technical Consulting AIT GbH 1060 Vienna, Austria 42 St.Andrä-Wördern, Austria 1210 Vienna, Austria Abstract Electric achine theory and electric achine siulations odels are often liited to three phases. Up to the Modelica Standard Libray (MSL) version.2 the provided achine odels were liited to three phases. Particularly for large industrial drives and for redundancy reasons in electric vehicles and aircrafts ulti phase electric achines are deanded. In the MSL.2.1 an extension of the existing FundaentalWave library has been perfored to cope with phase nubers greater than or equal to three. The developed achine odels are fully incorporating the ulti phase electric, agnetic, rotational and theral doain. In this publication the theoretical background of the achines odels, Modelica ipleentation details, the paraetrization of the odels and siulation exaples are presented. Keywords: Modelica Standard Library, ulti phase, electric achine odels, induction achine, synchronous achine, synchronous reluctance achine 1 Introduction Three phase induction, synchronous and synchronous reluctance achines are state of the art solutions for industrial applications, traction drives in electric vehicles, railways, tras, and underground trains, as well as air craft otors and generators. If a full leg of the supplying three phase converter fails, the achine cannot be operated any ore, once stopped. In particular, the higher abient teperatures of traction achines ay cause the power electronics to fail. In order to overcoe achine outage due to a single converter leg failure, phase nubers greater than three ay be used for electric achines and power electronics. In the following ulti phase will indicate phase nubers greater than three. If it is referred to only three phases, this will be indicated explicitly. Multi phase drives consist of the ulti phase achine including an inverter with power electronics plus control. A phase nuber greater than three thus requires a higher nuber of power electronic switches, such as IGBTs or MOS- FETs, etc. The higher phase nubers, however, increase cost and add coplexity to the drive structure. However, phase nubers equal to 2 n with integer n are excluded fro the actual ipleentation. The reason for excluding these phase nubers is that for exaple four or eight phase achines have to be handled differently since two phase are separated by π/2, not π. In general, two different philosophies of ulti phase drives exist: First, the nuber of phases is divisible by two. In industry, typically, six phase achines are used to overcoe axiu power liitations of power electronics supplying high power achines in the Megawatt range [1]. In this case axiu power of power electronics is doubled by using two three phase converters supplying a six phase achine. Usually, the phase winding orientations of the two three phase are spatially shifted by 0 in order to additionally reduce the agnitudes of space haronics caused by the winding agneto otive force (MMF) and to reduce the torque ripple of the achine, respectively. The ipleentation of a six phase winding does not significantly increase cost of the electric achine copared with a three phase achine with the sae power. Solely the additional winding ends have to be conducted to the terinal box. For six phase drives, the cost of power electronics increases due the double nuber of power electronic switches for the additional legs. For doubling the power of industrial drives the double cost of power electronics is in line with doubling the power. However, for traction achines six or nine phase achines are used for redundancy reasons [2, ]. In this case several state of the art three phase converters can be used. The redundancy concept is then realized with standard coponents which is cheaper than designing the individual legs of the inverter. Yet, several three phase inverters of saller power rating are usually ore expensive than a three phase inverter of the sae total power. Due to the ultiple 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 odularly using three phase converters. Second, the nuber of phases is not divisible by two. In this case ostly five (or seven) phase drives are used [4 6]. The drawback over a six phase inverter is that the odularity of the power electronics design is lower and thus cost ay be higher and ore design space ay be required. 15

2 Extension of the FundaentalWave Library towards Multi Phase Electric Machine Models For the sake of copleteness one ore redundancy concept will be discussed here, even though it is not related with ulti phase electric achines. Redundant drives with three phase achines ay use odified topologies which either use an additional leg or additional switches to operate the achine in case of a failure. These topologies have the capability to be reconfigured when a failure occurs [7, 8]. Depending on the actual topology even the full power rating ay be provided to the electric achine. For controlling ulti phase electric drives it is advantageous to control current coponents which represent the fundaental wave MMF and agnetic voltage, respectively [9]. The pulse width odulation schee for ulti phase converters with phase nubers not divisible by three has to be adapted so that a syetrical supply can be achieved. A technical paper dealing in ore detail with analysis of the ulti phase drive control is also subitted to the Modelica 2014 conference and will be cited properly, in case it gets accepted. In Modelica the first three phase electric achine odels have been introduced with the MSL 2.1 in 2004 [10, 11]. An alternative ipleentation with agnetic fundaental wave phasors was introduced in MSL.2 in 2010 [12]. Since then in these odels copper loss, (eddy current) core loss, friction loss, stray load loss, PM loss and brush loss are taken into account. Multi phase electric achine odels have already been published decades ago [1, 14]. Yet, in ost coputer siulations tools there are currently only three phase achine odels available. However, in the MSL.2.1 version of the package Modelica.Magnetic.FundaentalWave new ulti phase electric achine odels are introduced. In general, arbitrary phase nubers for stator (and rotor) windings ay be used excluding phase nubers equal to 2 n with integer n. This article provides the theoretical background, details about the ipleentation, paraetrization schees and soe exaples. 2 Fundaental Wave Theory Multi phase electric achine theory often relies on phasor transforations of currents, voltages and agnetic fluxes [15, 16]. A typical transforation is the syetrical coponents of the instantaneous values. In case of fully syetrical supply the achine equations based on the syetrical coponents of instantaneous coponents can be siplified extensively. The FundaentalWave achine odels only consider fundaental wave effects so there is also a coplex phasor representation of the fundaental wave of the agnetic flux and agnetic potential (difference), respectively. The connector definition of the FundaentalWave library shows: connector MagneticPort "Coplex agnetic port" Modelica.SIunits. CoplexMagneticPotentialDifference V_ "Coplex agnetic potential difference"; flow Modelica.SIunits. CoplexMagneticFlux Phi "Coplex agnetic flux"; end MagneticPort; Please note, that the potential and flow variable of the connector represent instantaneous quantities Φ = Φ re + jφ i and V = V,re + jv,i. The coplex agnetic quantities represent a spatial distribution of agnetic flux and agnetic potential (difference): Φ(ϕ) = Re[(Φ re + jφ i )e jϕ ] = Φ re cos(ϕ) + Φ i sin(ϕ) V (ϕ) = Re[(V,re + jv,i )e jϕ ] = V,re cos(ϕ) +V,i sin(ϕ) The coplex potential (difference) V introduced in the connector definition represents the total agnetic potential (difference) of all poles. This quantity can, thus, also be seen as the coplex agnetic potential difference of an equivalent two pole achine. Physical interpretations of the coplex agnetic phasors are presented and discussed in [12]. Voltages and currents are instantaneous quantities, so arbitrary wavefors and operating conditions are covered. Therefore, the achines can also be supplied with asyetric voltages or currents. It is yet assued that only fundaental wave effects due to these asyetries are considered. Supply voltage or current ibalance give rise to tie transient agnitudes of the agnetic flux and agnetic potential (difference). Each of the spatial field distributions can be interpreted as a forward and backward traveling fundaental wave coponent. Those effect of the two waves is correctly taken in account by the proposed approach. Particular supply ibalances and certain asyetries ay cause agnetic flux and agnetic potential (difference) phasors which are not related with the fundaental wave. Those higher haronic waves are not covered by the FundaentalWave library. It is therefore the user s responsibility to consider these odel liitations with particular focus on the supply conditions. Any higher haronic wave effects are also not taken into account by the FundaentalWave library. In case of higher haronic waves an alternative ipleentation has to be considered as presented in [17]. The ipact of rotor saliency on the fundaental wave coponents of agnetic flux and agnetic potential (difference) is, however, considered in the presented ipleentation. Concentrated windings and fractional slot windings, respectively, are very coon in PM synchronous achines due to better field weakening capabilities, higher pole nubers and higher power density. Such fractional slot windings can be considered in the FundaentalWave library, as long as the ain power exchanging haronic coponent is interpreted as fundaental wave. All higher haronic waves cause by fractional slot windings are not explicitly considered, but the total effect of those higher haronics can be taken into account by the total leakage inductance of the stator winding. 16

3 Session 1D: Electro-Magnetic Models and Libraries 1 Coplex(0, orientation)) "Coplex nuber of turns"; This coplex quantity has the agnitude of the effective nuber of turns and the phase angle orientation. The agnetic potential (difference) of the coupling odel, V_, and the current of the investigated winding, i, are then related by: V_ = (2.0/pi)*N*i; The factor 2.0/pi is the consequence of averaging the sinusoidal fundaental wave flux wavefor over one pole pair. Figure 1: Coil of an equivalent two pole achine with orientation; coplex agnetic potential (difference) phasors V ; coplex agnetic flux phasor Φ Electroagnetic Coupling.1 Single Phase Electroagnetic Coupling The coupling of fundaental wave agnetic flux and agnetic potential (difference) interacting with instantaneous voltages and currents is eleentary odeled in the electro agnetic coupling odels SinglePhaseElectroMagneticConverter. Figure 1 shows the agnetic phasors and a winding of an equivalent two pole achine. In this case one agnetic pole covers a spatial angle equal to π. The displayed coil is skewed and coil span is saller than π. The coil can actually also be seen as a distributed winding. Skewing and distributed windings with respect to the fundaental wave are considered by the effective nuber of turns, represented by the paraeter effectiveturns. The effective nuber of turns of a real achine is deterined by the real nuber of turns, ultiplied by the skewing factor and the chording factor. A ore detail investigation on coon windings and the deterination of winding factors is published in [10, 18]. A current through the investigated windings gives rise to agnetic potential (difference) distribution which agnitude is equal effective nuber of turns ties the current. The peak of accessory sinusoidal agnetic potential (difference) caused by the current is in line with the orientation of the winding axis. In an electric achine several windings of stator and rotor windings and peranent agnets contribute the total agnetic potential difference depending on the type of achine. In the electroagnetic coupling odel two physical laws are ipleented, i.e., Apere s law and the induction law..2 Apere s Law Apere s law states that the total exciting agneto otive force is equal to the agnetic potential difference. For the investigated single phase winding it is useful to define the coplex nuber of turns: final paraeter Coplex N=effectiveTurns*Modelica.CoplexMath.exp(. Induction Law Induction law describes the the relationship between the tie derivative of the agnetic flux and the induced voltage of the investigated winding. The projection of the coplex agnetic flux onto the orientation ties the effective nuber of turns is equal to the negative terinal voltage. -v = Modelica.CoplexMath.real( Modelica.CoplexMath.conj(N)* Coplex(der(Phi.re),der(Phi.i)));.4 Multi Phase Electroagnetic Coupling The ulti phase electroagnetic coupling odel is coposed of a vector of single phase electroagnetic coupling odels, where is the nuber of phases. The electrical pins of the single phase electroagnetic coupling odel are connected to the pins of the electrical ulti phases connector used by the ulti phase coupling odel. The agnetic fundaental wave ports of the single phase electroagnetic coupling odels are connected in series. The agnetic series connection is a consequence of, first, each winding being exposed to the sae agnetic flux wave but being located spatially on different locations. Second, the total agnetic potential (difference) excited by all windings is deterined by the su of the agnetic potential (differences) of all individual windings. 4 Phase Orientations Winding Axes In the FundaentalWave library only syetrical phase windings are considered. For ulti phase systes and windings with phase nubers greater than three two different cases are distinguished, first, the nuber of phases is divisible by three and second, the nuber of phases is not divisible by three. The general function syetricorientation for deterining the orientations of windings of an phase electric achine is located in the package Modelica.Electrical.MultiPhase.Functions: function syetricorientation extends Modelica.Icons.Function; input Integer "Nuber of phases"; output Modelica.SIunits.Angle orientation[] "Orientation of the resulting fundaental wave field phasors"; iport Modelica.Constants.pi; 17

4 Extension of the FundaentalWave Library towards Multi Phase Electric Machine Models Figure : Winding axes of syetrical (a) three phase winding and (b) five phase winding Figure 2: Peranent agnet synchronous achine with optional daper cage Figure 4: Syetrical phase angles of voltages and currents, respectively, of (a) three phase winding and (b) five phase winding algorith if od(, 2) == 0 then // Even nuber of phases if == 2 then // Special case two phase achine orientation[1] := 0; orientation[2] := +pi/2; else orientation[1:integer(/2)] := syetricorientation(integer(/2)); orientation[integer(/2) + 1:] := syetricorientation(integer(/2)) - fill(pi/, integer(/2)); end if; else // Odd nuber of phases orientation := {(k - 1)*2*pi/ for k in 1:}; end if; syetricorientation; So the function is designed recursively so that subsystes are odularly designed. In the following soe exaples of phase nubers will be discussed. Two, four, eight, sixteen, thirty-two, etc. phase windings are currently not supported, as in general phase nubers equal to 2 n with integer n are not considered. In order to suarize atheatical equations for different phase nubers in the following, abbreviation orientation k will be used to indicate the angles of the orientation of the winding axes of a syetrical phase winding. So orientation k is the k-th eleent (phase index) of the result vector returned by function syetricorientation, called with arguent. The syetrically supply voltages and currents, respectively, have the phase angles φ k = orientation k The winding orientations and the phase shifts of the supplying syste have the sae agnitudes for each phase index k, but different signs. This is a general property of syetrical systes supplying syetrical windings, see Figs. 6. In the following only syetrical winding axes will be assued. The phase angles of syetrical voltage and current supply are also presented, even though syetric supply is not assued in the FundaentalWave library. 4.1 Odd Phase Nubers For all odd phase nubers (not divisible by 2) the syetrical orientations of the winding axes are orientation k = 2π(k 1). So this applies for the case =, = 5, = 7, = 9, = 11, = 1, = 15, etc., see Fig Even Phase Nubers For even phase nubers unequal to 2 n with integer n, the phase syste is separated into two subsystes with /2 phases. The winding orientations of the second sub syste lags the first sub syste by π. This is then the point where the recursive deterination of phase angle is initiated. For each of the two sub systes functions syetricorientation is called, considering the lag angle π. It is iportant to explicitly note that the phase shift be- tween the two sub systes is not 2π, since in this case the 18

5 Session 1D: Electro-Magnetic Models and Libraries 1 Figure 5: Winding axes of syetrical (a) six phase winding and (b) ten phase winding Figure 7: Nubering of nine phase syetrical winding according to (a) an engineering approach using three three phase inverters and (b) the schee of the FundaentalWave library Figure 6: Syetrical phase angles of voltages and currents, respectively, of (a) six phase winding and (b) ten phase winding two sub systes where aligned at ±π which does not ake sense in a technical syste for redundancy reasons. A six phase syste is then separated into two three phase systes. The winding orientations of the second sub systes lags the first sub syste by π/6, see Figs The phase sequences (1-2-) and (4-5-6) of the two sub systes are equal. A ten phase syste consists of two five phase systes with the phase sequences ( ) and ( ). The second sub systes lags the first sub syste by π, see Figs Phase Nubers Divisible by Three Phase nubers divisible by three are either covered by subsection 4.1 and 4.2. Therefore, fro a foral point of view no additional explanations are required to handle, for exaple, nine phase achines. Yet, a typical engineering approach and function syetricorientation for nubering the phase nubers are different and ay require soe additional coents: Practically, in ost technical cases, electrical achines with phase nubers divisible by three will be supplied by an appropriate nuber of three phase inverters. For six phase systes this has already been deonstrated in subsection 4.2. In the FundaentalWave library nine phase systes are handled differently only in that sense, the sequence nubering the phase windings is ost likely different fro an engineer who uses three three phase inverters. In the engineering phases (1-2-) are ost 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 FundaentalWave library the phase are nubered according to Fig. 7(b). Even though the nubering is different the angles of the orientations are fully identical. For a fifteen phase a design engineer could always argue whether such syste can be seen as five three phase systes or as three five phase systes. However, fro operating conditions point of view, there is no difference between these two cases. The nubering schee of the FundaentalWave follows a foral schee and the user decides how the achine phases are supplied. 5 Magnetic Coponents All the existing agnetic coponents of the FundaentalWave library can be re-used for the ulti-phase achine odels, since the agnetic port representation did not change. In the current ipleentation only linear agnetic aterials are considered. Saturation effects are not taken into account. In all electric achine odels of the FundaentalWave library the total agnetic reluctance is concentrated in the air gap odel. An exaple of peranent agnet synchronous achine with optional daper cage is displayed in Fig. 2. In the actual ipleentation of the FundaentalWave library the agnetic reluctances of the stator, rotor and air gap are not individually odeled. Even the linear characteristic of the peranent agnet is represent by the total air gap reluctance of the achine. The total reluctance takes the variable reluctance of the air gap length δ into account. A sketch of the air gap and the reciprocal function 1/δ are shown in Fig. 8. The effect of variable agnetic reluctance due to the uneven shape of the air gap and the arrangeent of agnets, respectively, is called saliency. The effect of saliency on fundaental wave fors is fully considered by unequal direct (d) and quadrature (q) axis reluctances. For rotor fixed agnetic flux Phi and total agnetic potential difference V_ the following relationship applies: (pi/2.0)*v_.re = Phi.re * R_.d; (pi/2.0)*v_.i = Phi_i * R_.q; The d and q axis are, however, fixed with the rotor structure. Magnetic rotor excitation of synchronous achines is, however, always aligned with the d axis. 19

6 Extension of the FundaentalWave Library towards Multi Phase Electric Machine Models Figure 8: (a) Variable air gap length δ of a synchronous achine and (b) reciprocal air gap function 1/δ versus spatial angle ϕ of an equivalent two pole achine Quantity s = s Noinal stator phase voltage V sn V sn = V sn Noinal stator phase current I sn I sn = I sn Noinal stator frequency f sn f sn = f sn Noinal electrical torque τ N τ N = τ N Noinal electrical stator power P sn P sn = P sn Table 1: Paraeters of achines with phases nubers equal and greater than three In case that the saturation characteristics of the different regions of the achine shall be considered in the future, the agnetic equivalent circuit has to be adapted such way that each region is then represented by one non-linear reluctance. 6 Paraetrization Where do the paraeters of a achine with s stator phases coe fro? First, in the design stage of the achine, an engineer deterines these paraeters fro finite eleent analysis or any other electroagnetic design software. Second, the paraeters of a three phase achine are known or estiated and the users wants to deterine the paraeters of an equivalent phase achine. 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 following: Assue, the noinal paraeter of a three phase and arbitrary phase achine as listed in Tab. 1. All the paraeters of a three phase achine are indicated with. According to the relationship between the phase noinal phase voltage and current, all resistances and inductances of an phase achine are scaled with /. A list of relevant paraeters is suarized in Tab. 2. The rotor winding of squirrel cage induction achines are ipleented as equivalent s phase windings where s is the nuber of stator phases. Slip ring induction achines ay have different phases nubers of stator and rotor where r is the nuber of rotor phases. For synchronous achines with peranent agnets, electrical excitation and reluctance rotor, the optional daper cage is ipleented Quantity = Stator resistance R s R s = R s Stator stray inductance L sσ L sσ = L sσ Main field inductance L L = L Main field inductance, d-axis L d L d = L d Main field inductance, q-axis L q L q = L q Table 2: Stator paraeters of three and s phase achines Quantity = Induction achine with squirrel cage Rotor cage resistance R r R r = R s r Rotor stray inductance L rσ L rσ = L s rσ Induction achine with slip ring rotor Rotor cage resistance R r R r = R r r Rotor stray inductance L rσ L rσ = L r rσ All synchronous achines Daper cage resistance, d-axis Daper cage resistance, q-axis R rd R rq R rd = R rd R rq = R rq Daper cage stray inductance, d-axis L rσ,d L rσ,d = L rσ,d Daper cage stray inductance, q-axis L rσ,q L rσ,q = L rσ,q Table : Rotor paraeters of achines with three and s and r phases with salient rotor paraeters with respect to the direct (d) and quadrature (q) axis. So there is no difference between three and ulti phase daper cage odels and paraeters. The rotor cage paraeters of all achine types are suarized in Tab.. 7 Exaples In the FundaentalWave library there are exaples for all types of achines, coparing three phase and ulti phase ( = 5) achines. The three and five phase achines are operated with equal noinal phase voltages. The paraeters of the five phase achine are paraeterized such way autoatically that the phase nuber can be increased without changing torques and powers for the ulti phase achines. In a duplicate exaple the phase nubers can be changed for fro five to higher nubers. The following exaples are included in the Fundaental- Wave library: 7.1 Induction Machine with Squirrel Cage In odel Exaples.AIMC_DOL_MultiPhase two asynchronous induction achines with squirrel cage rotor are started directly on line (DOL) by eans of an ideal switch; see Figure 9. The achines start fro standstill. 140

7 Session 1D: Electro-Magnetic Models and Libraries 1 Figure 9: Coparing a three and a ulti phase ( = 5) peranent agnet synchronous achine, operated on an idealized voltage inverter Figure 12: Coparing a three and a ulti phase ( = 5) induction achine started directly on line, operated on an idealized voltage inverter The echanical load is odeled by eans of a quadratic speed dependent load torque and an additional load inertia. This exaple deonstrates equivalent dynaic behavior of the three and five phase achine. Particularly, the electrical torque, speed, and the particular losses are equal. Both achines have the sae noinal phase voltage, but different noinal phase currents according to Tab. 1. In Fig. 10 the two identical electric torques of the two achines are shown. The different quasi RMS currents of two achines are displayed in Fig. 11. The current ratio is equal to five over three. Figure 10: Siulation result of electrical torque of a three and five phase squirrel cage induction achines; the torques are equal Figure 11: Siulation result of the quasi RMS currents of a three and five phase squirrel cage induction achines; the current ratio is five to three 7.2 Induction Machine with Slip Ring Rotor Model Exaples.BasicMachines.AIMS_Start- _MultiPhase copares a three and a five phase slip ring induction achine, operating the stator direct on line; see Fig. 12 The nuber of stator phases s = 5 and the nuber of rotor phases, r = 5, are equal. The ulti phase rotor windings of each achine are connected with a rheostat which is shorted after a give tie 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 achines. The user can copy the exaple and change the rotor phase nuber r such way that it differs fro the stator phase nuber, s. This case is also supported the FundaentalWave library. In Fig. 1 and 14 the electroagnetic torques and the quasi RMS currents of the two achines are copared. The torques are identical and the current ratio is five to three according to Tab

8 Extension of the FundaentalWave Library towards Multi Phase Electric Machine Models Figure 1: Siulation result of electrical torque of a three and five phase slip ring induction achines; the torques are equal Figure 15: Coparing a three and a ulti phase ( = 5) peranent agnet synchronous achine, operated on an idealized voltage inverter Figure 14: Siulation result of the quasi RMS currents of a three and five phase slip ring induction achines; the current ratio is five to three 7. Synchronous Generator with Electrical Excitation Figure 16: Siulation result of electrical torque, coparing a three and five phase In exaple Exaples.SMPM_Generator two ains supplied electrical excited synchronous achine with three and five stator phases are copared; see Fig. 16. For each achine shaft speed is constant and slightly different than synchronous speed. In this experient each rotor is forced to ake a full revolution relative to the agnetic field. In Fig. 16 the generated torques versus load angle are shown for a fixed level of excitation. In addition to the sinusoidal wavefor of the torque a second haronic coponent is superiposed due to the saliency of the rotor. However, for the investigated achines, the saliency effect is very sall so that the torque wavefor alost appears as a pure sine wave. The quasi RMS currents of the two achines are copared in Fig. 17. The current ratios of the three and five phase achine is five to three. Figure 17: Siulation result of the quasi RMS currents of a three and five phase synchronous achine with electric excitation; the current ratio is five to three 142

9 Session 1D: Electro-Magnetic Models and Libraries 1 8 Conclusions The paper presents an extension of the FundaentalWave library towards ulti phase stator (and rotor) windings with phase nubers greater or equal than three. This library is included in the MSL.2.1. Assuptions and liitations of the presented ipleentation are explained. In the new FundaentalWave library only syetrical windings are supported. The structures of syetrical ulti phase windings and supplies are introduced. The paraetrization of the ulti phase achines is discussed. Conversion tables for paraeterizing ulti phase achines equivalent to three phase achines are presented. Siulation exaples of three and equivalent five phase induction and synchronous achines are presented and copared. 9 Acknowledgeent The research leading to these results has received funding fro the ENIAC Joint Undertaking under grant agreeent no and fro the Österreichis- che Forschungsförderungsgesellschaft bh under project no References [1] B. Stuberger, G. Stuberger, A. Haler, M. Trlep, M. Jesenik, and V. Gorican, Increasing of output power capability in a six-phase flux-weakened peranent agnet synchronous otor with a third haronic current injection, IEEE Transactions on Magnetics, vol. 9, pp. 4 45, 200. [2] D. G. Dorrell, C. Y. Leong, and R. A. McMahon, Analysis and perforance assessent of six-pulse inverter-fed three-phase and six-phase induction achines, IEEE Transactions on Industry Applications, vol. 42, pp , Noveber/Deceber [] G. Aroquiadassou, A. Mpanda-Mabwe, F. Betin, and G.-A. Capolino, Six-phase induction achine drive odel for fault-tolerant operation, SDEMPED, [4] H. A. Toliyat, M. M. Rahiian, and T. A. Lipo, dq odeling of five phase synchronous reluctance achines including third haronic of air-gap f, Conference Record of the 1991 IEEE Industry Applications Society Annual Meeting, 1991., pp , [5] D. Dujic, M. Jones, and E. Levi, Features of two ulti-otor drive schees supplied fro fivephase/five-leg voltage source inverters, Conference Proceedings on Power Conversion and Intelligent Motion, PCIM, Nureberg, Gerany, no. S2d-2, [6] D. Dujic, M. Jones, and E. Levi, Analysis of output current ripple rs in ultiphase drives using space vector approach, IEEE Transactions on Power Electronics, vol. 24, no. 8, pp , [7] R. L. A. Ribeiro, C. B. Jacobina, A. M. N. Lia, and E. R. C. da Silva, A strategy for iproving reliability of otor drive systes using a four-leg three-phase converter, Sixteenth Annual IEEE Applied Power Electronic Conference and Expoition. APEC 2001, vol. 1, pp , [8] R. Errabelli and P. Mutschler, Fault-tolerant voltage source inverter for peranent agnet drives, IEEE Transactions on Power Electronics, vol. 27, pp , feb [9] T. Treichl, Regelung von sechssträngigen peranenterregten Synchronaschinen für den obilen Anwendungsfall. PhD thesis, FernUniversität, [10] C. Kral and A. Hauer, Modelica libraries for DC achines, three phase and polyphase achines, International Modelica Conference, 4th, Haburg, Gerany, pp , [11] C. Kral and A. Hauer, Object Oriented Modeling of Rotating Electrical Machines. INTECH, [12] C. Kral and A. Hauer, The new fundaentalwave library for odeling rotating electrical three phase achines, 8th International Modelica Conference, [1] T. A. Lipo, A d-q odel for six phase induction achines, Conference Record of International Conference on Electrical Machines, ICEM, pp , [14] E. Andresen and K. Bieniek, Der Asynchronotor it drei und sechs Wirkungssträngen a stroeinprägenden Wechselrichter, Archiv für Elektrotechnik, vol. 6, pp , [15] H.-H. Jahn and R. Kasper, Koordinatentransforationen zur Behandlung von Mehrphasensysteen, Archiv für Elektrotechnik, vol. 56, pp , [16] J. Stepina, Einführung in die allgeeine Rauzeiger- Theorie der elektrischen Maschinen. Kaiserslautern: Vorlesungsskriptu, Universität Kaiserslautern, [17] C. Kral, A. Hauer, M. Bogoolov, and E. Loonova, Haronic wave odel of a peranent agnet synchronous achine for odeling partial deagnetization under short circuit conditions, XXth International Conference on Electrical Machines (ICEM), pp , Sept [18] C. Kral and A. Hauer, Siulation of electrical rotor asyetries in squirrel cage induction achines with the extendedachines library, International Modelica Conference, 6th, Bielefeld, Gerany, no. ID140, pp ,