DC Boost Stage in Photovoltaic Systems

The performance of Grid connected Photovoltaic System working with DCBoost stage is investigated. The DC-Boost Converter topology is deduced from three phase half controlled bridge and controlled by Sliding Mode Control. Due to the fact that Grid connected Photovoltaic System includes Solar cells as a DC source and inverter for grid connection, those are under the scope of this research as well. The advantages of using MPPT are analyzed. The system is simulated in Matlab-Simulink™ environment.

Permanent Magnet Synchronous Machine for Parallel Hybrid Vehicle

The aim of this thesis is to describe hybrid drive design problems, the advantages and difficulties related to the drive. A review of possible hybrid constructions, benefits of parallel, series and series-parallel hybrids is done. In the thesis analytical and finite element calculations of permanent magnet synchronous machines with embedded magnets were done. The finite element calculations were done using Cedrat’s Flux 2D software. This machine is planned to be used as a motor-generator in a low power parallel hybrid vehicle. The boundary conditions for the design were found from Lucas-TVS Ltd., India. Design Requirements, briefly: • The system DC voltage level is 120 V, which implies Uphase = 49 V (RMS) in a three phase system. • The power output of 10 kW at base speed 1500 rpm (Torque of 65 Nm) is desired. • The maximum outer diameter should not be more than 250 mm, and the maximum core length should not exceed 40 mm. The main difficulties which the author met were the dimensional restrictions. After having designed and analyzed several possible constructions they were compared and the final design selected. Dimensioned and detailed design is performed. Effects of different parameters, such as the number of poles, number of turns and magnetic geometry are discussed. The best modification offers considerable reduction of volume.

Tasasähköverkon modulaarisen vaihtosuuntaajan rakenne- ja kustannusanalyysi

Diplomityössä tarkastellaan tapausta, jossa tasasähköverkon vaihtosuuntaaja toteutetaan kolmella yksivaiheisella suuntaajalla yhden kolmivaiheisen sijaan. Modulaarinen rakenne mahdollistaa sen, että osa vaiheista voitaisiin sammuttaa kun kuormitus on erittäin pientä eikä kuorma ole kolmivaiheista. Galvaaninen erotus oletetaan tehtävän erottavalla DC-DC -konvertterilla, joka mahdollistaa samalla vaihtosuuntaajan DC-syöttöjännitteen muuttamisen. Työssä tutkitaan, mitkä tekijät vaikuttavat DC-syöttöjännitteen valintaan sekä tehdään valinta kyseisen jännitteen arvolle. Valittua jännitetasoa käytetään vaihtosuuntaajan suunnittelemiseen. Vaihtosuuntaajan rakennetta tarkastellaan vaihtosuuntaussillan kytkinkomponenttien sekä lähtöjännitteen suotimen osalta. Yksivaiheiselle moduulille määritetään optimaalinen teholuokka investointi- ja käyttökustannusten kannalta. Tarkastelu pitää sisällään erilaisten transistoritekniikkojen, kytkentätaajuuksien sekä kahden suodintyypin vaikutuksen kustannustehokkuuteen.

Internet-based breast cancer patient’s pathway as an empowering patient educational tool

This three-phase study was conducted to examine the effect of the Breast Cancer Patient’s Pathway program (BCPP) on breast cancer patients’ empowering process from the viewpoint of the difference between knowledge expectations and perceptions of received knowledge, knowledge level, quality of life, anxiety and treatment-related side effects during the breast cancer treatment process. The BCPP is an Internet-based patient education tool describing a flow chart of the patient pathway during the breast treatment process, from breast cancer diagnostic tests to the follow-up after treatments. The ultimate goal of this study was to evaluate the effect of the BCPP to the breast cancer patient’s empowerment by using the patient pathway as a patient education tool. In phase I, a systematic literature review was carried out to chart the solutions and outcomes of Internet-based educational programs for breast cancer patients. In phase II, a Delphi study was conducted to evaluate the usability of web pages and adequacy of their content. In phase III, the BCPP program was piloted with 10 patients and patients were randomised to an intervention group (n=50) and control group (n=48). According to the results of this study, the Internet is an effective patient education tool for increasing knowledge, and BCPP can be used as a patient education method supporting other education methods. However, breast cancer patients’ perceptions of received knowledge were not fulfilled; their knowledge expectations exceed the perceived amount of received knowledge. Although control group patients’ knowledge expectations were met better with the knowledge they received in hospital compared to the patients in the intervention group, no statistical differences were found between the groups in terms of quality of life, anxiety and treatment-related side effects. However, anxiety decreased faster in the intervention group when looking at internal differences between the groups at different measurement times. In the intervention group the relationship between the difference between knowledge expectations and perceptions of received knowledge correlated significantly with quality of life and anxiety. Their knowledge level was also significant higher than in the control group. These results support the theory that the empowering process requires patient’s awareness of knowledge expectations and perceptions of received knowledge. There is a need to develop patient education to meet patients’ perceptions of received knowledge, including oral and written education and BCPP, to fulfil patient’s knowledge expectations and facilitate the empowering process. Further research is needed on the process of cognitive empowerment with breast cancer patients. There is a need for new patient education methods to increase breast cancer patients’ awareness of knowing.

Taajuusmuuttaja aktiivisesti säädetyn magneettilaakerin virtalähteenä

Työssä tutkitaan mahdollisuutta hyödyntää sähkömoottorien nopeudensäätöön suunniteltuja kaupallisia taajuusmuuttajia osana aktiivisesti säädetyn magneettilaakeroinnin säätöjärjestelmää. Magneettilaakerijärjestelmän ohjaamiseksi tarvitaan vahvistin, jonka tehtävänä on muuntaa paikkasäätöjärjestelmältä tuleva ohjearvo virraksi, jännitteeksi tai magneettivuoksi voiman tuottamiseksi laakerikäämityksellä. Nykyaikaiset modernit taajuusmuuttajat mahdollistavat säätöalgoritmien suorittamisen sekä liitynnän muuhun automaatiojärjestelmään kenttäväylien kautta. Tämän järjestelmäintegraation myötä olisi mahdollista rakentaa modulaarinen säätöjärjestelmä hyödyntäen luotettavaksi todettuja teollisuusautomaatiotuotteita vain muutamalla itse magneettilaakerijärjestelmään liittyvällä tuotteella. Haluttaessa hyödyntää kolmivaiheinen taajuusmuuttaja mahdollisimman tehokkaasti magneettilaakerin teholähteenä tulee laakerikäämityksien kytkeytymistä taajuusmuuttajaan tarkastella tarkemmin. Kirjallisuustutkimuksessa keskitytään taajuusmuuttajan tehokytkimien muodostaman vaihtosuuntaajan eri rakennevaihtoehtojen sekä virtasäädön dynaamisten ominaisuuksien tarkasteluun. Soveltuvien rakennevaihtoehtojen sekä virtasäädön suorituskyky todennetaan simuloinnein ja lopuksi todellisella koelaitteistolla. Magneettilaakeroinnilla varustetun sähkökoneen roottorin onnistunut leijuttaminen viiden vapausasteen suhteen paikkasäädettynä sekä mittauksien tulokset osoittavat, ettei taajuusmuuttajien järjestelmäarkkitehtuurista löydy merkittäviä esteitä muuttajien hyödyntämiseksi magneettilaakerisovelluksissa.

Aktiivisen verkkosillan tuottaman yhteismuotoisen jännitteen vaimentaminen kaksitasoisessa, jännitevälipiirillisessä, taajuusmuuttajatopologiassa

Pulssinleveysmoduloidun vaihtosuuntaajan hyötysuhteen parantaminen ja kytkentätaajuuden suurentaminen ovat johtaneet lähtöjännitteen suuritaajuiseen taajuussisältöön kaksitasoisessa, jännitevälipiirillisessä taajuusmuuttajatopologiassa. Kasvava tarve siirtää tehoa myös verkkoon päin on lisännyt aktiivisen verkkosillan käyttöä. Kaksitasoisen aktiivisen verkkosillan vaikutuksesta DC-välipiirin keskipisteen ja kolmivaiheisen kuorman tähtipisteen välinen jännite on nollasta poikkeava aiheuttaen suurentuneen yhteismuotoisen jännitteen taajuusmuuttajan lähtöön ja verkon puolelle. Lisäksi yhteismuotoisten jännitteiden aiheuttamat kytkentätaajuiset häiriövirrat voivat aiheuttaa vikavirtasuojien tahatonta laukeamista, vaikeuttaa EMC-standardien vaatimusten täyttämistä, lisätä moottorin käämieristyksien rasitusta ja mahdollisuutta moottorin laakerivaurioille. Diplomityössä tutkitaan aktiivisen ja passiivisen verkkosillan tuottamaa yhteismuotoista jännitettä simuloinneilla. Esitellään aikaisempaa tutkimustietoa yhteismuotoisen jännitteen ja virran vaimennusratkaisuista aktiivista verkkosiltaa käytettäessä. Tutkimustiedon pohjalta suunnitellaan koelaitteistolle soveltuva suodin. Suotimen toiminta testataan simuloinnein sekä kokeellisin mittauksin. Tehdyt mittaukset osoittavat, että suunniteltu suodin vaimentaa yhteismuotoista jännitettä noin 20 dB verkkosillan kytkentätaajuudella ja tämän jälkeen yli 20 dB/dekadi taajuuteen 100 kHz asti. Lisäksi yhteismuotoisen virran suuruus syöttökaapelin kautta pieneni ehdotetun suotimen vaikutuksesta.

Modeling and Parameter Estimation of Double-Star Permanent Magnet Synchronous Machines

The power rating of wind turbines is constantly increasing; however, keeping the voltage rating at the low-voltage level results in high kilo-ampere currents. An alternative for increasing the power levels without raising the voltage level is provided by multiphase machines. Multiphase machines are used for instance in ship propulsion systems, aerospace applications, electric vehicles, and in other high-power applications including wind energy conversion systems. A machine model in an appropriate reference frame is required in order to design an efficient control for the electric drive. Modeling of multiphase machines poses a challenge because of the mutual couplings between the phases. Mutual couplings degrade the drive performance unless they are properly considered. In certain multiphase machines there is also a problem of high current harmonics, which are easily generated because of the small current path impedance of the harmonic components. However, multiphase machines provide special characteristics compared with the three-phase counterparts: Multiphase machines have a better fault tolerance, and are thus more robust. In addition, the controlled power can be divided among more inverter legs by increasing the number of phases. Moreover, the torque pulsation can be decreased and the harmonic frequency of the torque ripple increased by an appropriate multiphase configuration. By increasing the number of phases it is also possible to obtain more torque per RMS ampere for the same volume, and thus, increase the power density. In this doctoral thesis, a decoupled d–q model of double-star permanent-magnet (PM) synchronous machines is derived based on the inductance matrix diagonalization. The double-star machine is a special type of multiphase machines. Its armature consists of two three-phase winding sets, which are commonly displaced by 30 electrical degrees. In this study, the displacement angle between the sets is considered a parameter. The diagonalization of the inductance matrix results in a simplified model structure, in which the mutual couplings between the reference frames are eliminated. Moreover, the current harmonics are mapped into a reference frame, in which they can be easily controlled. The work also presents methods to determine the machine inductances by a finite-element analysis and by voltage-source inverters on-site. The derived model is validated by experimental results obtained with an example double-star interior PM (IPM) synchronous machine having the sets displaced by 30 electrical degrees. The derived transformation, and consequently, the decoupled d–q machine model, are shown to model the behavior of an actual machine with an acceptable accuracy. Thus, the proposed model is suitable to be used for the model-based control design of electric drives consisting of double-star IPM synchronous machines.

LVDC power distribution system: computational modelling

In the doctoral dissertation, low-voltage direct current (LVDC) distribution system stability, supply security and power quality are evaluated by computational modelling and measurements on an LVDC research platform. Computational models for the LVDC network analysis are developed. Time-domain simulation models are implemented in the time-domain simulation environment PSCAD/EMTDC. The PSCAD/EMTDC models of the LVDC network are applied to the transient behaviour and power quality studies. The LVDC network power loss model is developed in a MATLAB environment and is capable of fast estimation of the network and component power losses. The model integrates analytical equations that describe the power loss mechanism of the network components with power flow calculations. For an LVDC network research platform, a monitoring and control software solution is developed. The solution is used to deliver measurement data for verification of the developed models and analysis of the modelling results. In the work, the power loss mechanism of the LVDC network components and its main dependencies are described. Energy loss distribution of the LVDC network components is presented. Power quality measurements and current spectra are provided and harmonic pollution on the DC network is analysed. The transient behaviour of the network is verified through time-domain simulations. DC capacitor guidelines for an LVDC power distribution network are introduced. The power loss analysis results show that one of the main optimisation targets for an LVDC power distribution network should be reduction of the no-load losses and efficiency improvement of converters at partial loads. Low-frequency spectra of the network voltages and currents are shown, and harmonic propagation is analysed. Power quality in the LVDC network point of common coupling (PCC) is discussed. Power quality standard requirements are shown to be met by the LVDC network. The network behaviour during transients is analysed by time-domain simulations. The network is shown to be transient stable during large-scale disturbances. Measurement results on the LVDC research platform proving this are presented in the work.

CFD modeling of dispersion water feed in wastewater cleaning application

Fluid particle breakup and coalescence are important phenomena in a number of industrial flow systems. This study deals with a gas-liquid bubbly flow in one wastewater cleaning application. Three-dimensional geometric model of a dispersion water system was created in ANSYS CFD meshing software. Then, numerical study of the system was carried out by means of unsteady simulations performed in ANSYS FLUENT CFD software. Single-phase water flow case was setup to calculate the entire flow field using the RNG k-epsilon turbulence model based on the Reynolds-averaged Navier-Stokes (RANS) equations. Bubbly flow case was based on a computational fluid dynamics - population balance model (CFD-PBM) coupled approach. Bubble breakup and coalescence were considered to determine the evolution of the bubble size distribution. Obtained results are considered as steps toward optimization of the cleaning process and will be analyzed in order to make the process more efficient.

Remote sensing for three-dimensional modelling of hydromorphology

Successful management of rivers requires an understanding of the fluvial processes that govern them. This, in turn cannot be achieved without a means of quantifying their geomorphology and hydrology and the spatio-temporal interactions between them, that is, their hydromorphology. For a long time, it has been laborious and time-consuming to measure river topography, especially in the submerged part of the channel. The measurement of the flow field has been challenging as well, and hence, such measurements have long been sparse in natural environments. Technological advancements in the field of remote sensing in the recent years have opened up new possibilities for capturing synoptic information on river environments. This thesis presents new developments in fluvial remote sensing of both topography and water flow. A set of close-range remote sensing methods is employed to eventually construct a high-resolution unified empirical hydromorphological model, that is, river channel and floodplain topography and three-dimensional areal flow field. Empirical as well as hydraulic theory-based optical remote sensing methods are tested and evaluated using normal colour aerial photographs and sonar calibration and reference measurements on a rocky-bed sub-Arctic river. The empirical optical bathymetry model is developed further by the introduction of a deep-water radiance parameter estimation algorithm that extends the field of application of the model to shallow streams. The effect of this parameter on the model is also assessed in a study of a sandy-bed sub-Arctic river using close-range high-resolution aerial photography, presenting one of the first examples of fluvial bathymetry modelling from unmanned aerial vehicles (UAV). Further close-range remote sensing methods are added to complete the topography integrating the river bed with the floodplain to create a seamless high-resolution topography. Boat- cart- and backpack-based mobile laser scanning (MLS) are used to measure the topography of the dry part of the channel at a high resolution and accuracy. Multitemporal MLS is evaluated along with UAV-based photogrammetry against terrestrial laser scanning reference data and merged with UAV-based bathymetry to create a two-year series of seamless digital terrain models. These allow the evaluation of the methodology for conducting high-resolution change analysis of the entire channel. The remote sensing based model of hydromorphology is completed by a new methodology for mapping the flow field in 3D. An acoustic Doppler current profiler (ADCP) is deployed on a remote-controlled boat with a survey-grade global navigation satellite system (GNSS) receiver, allowing the positioning of the areally sampled 3D flow vectors in 3D space as a point cloud and its interpolation into a 3D matrix allows a quantitative volumetric flow analysis. Multitemporal areal 3D flow field data show the evolution of the flow field during a snow-melt flood event. The combination of the underwater and dry topography with the flow field yields a compete model of river hydromorphology at the reach scale.

Influence of multi-phase phenomena on semibatch crystallization processes of aqueous solutions

Crystal properties, product quality and particle size are determined by the operating conditions in the crystallization process. Thus, in order to obtain desired end-products, the crystallization process should be effectively controlled based on reliable kinetic information, which can be provided by powerful analytical tools such as Raman spectrometry and thermal analysis. The present research work studied various crystallization processes such as reactive crystallization, precipitation with anti-solvent and evaporation crystallization. The goal of the work was to understand more comprehensively the fundamentals, phenomena and utilizations of crystallization, and establish proper methods to control particle size distribution, especially for three phase gas-liquid-solid crystallization systems. As a part of the solid-liquid equilibrium studies in this work, prediction of KCl solubility in a MgCl2-KCl-H2O system was studied theoretically. Additionally, a solubility prediction model by Pitzer thermodynamic model was investigated based on solubility measurements of potassium dihydrogen phosphate with the presence of non-electronic organic substances in aqueous solutions. The prediction model helps to extend literature data and offers an easy and economical way to choose solvent for anti-solvent precipitation. Using experimental and modern analytical methods, precipitation kinetics and mass transfer in reactive crystallization of magnesium carbonate hydrates with magnesium hydroxide slurry and CO2 gas were systematically investigated. The obtained results gave deeper insight into gas-liquid-solid interactions and the mechanisms of this heterogeneous crystallization process. The research approach developed can provide theoretical guidance and act as a useful reference to promote development of gas-liquid reactive crystallization. Gas-liquid mass transfer of absorption in the presence of solid particles in a stirred tank was investigated in order to gain understanding of how different-sized particles interact with gas bubbles. Based on obtained volumetric mass transfer coefficient values, it was found that the influence of the presence of small particles on gas-liquid mass transfer cannot be ignored since there are interactions between bubbles and particles. Raman spectrometry was successfully applied for liquid and solids analysis in semi-batch anti-solvent precipitation and evaporation crystallization. Real-time information such as supersaturation, formation of precipitates and identification of crystal polymorphs could be obtained by Raman spectrometry. The solubility prediction models, monitoring methods for precipitation and empirical model for absorption developed in this study together with the methodologies used gives valuable information for aspects of industrial crystallization. Furthermore, Raman analysis was seen to be a potential controlling method for various crystallization processes.

Evaluation of the Efficiency of Line-Start Permanent-Magnet Machines as a Function of the Operating Temperature

The standard squirrel-cage induction machine has nearly reached its maximum efficiency. In order to further increase the energy efficiency of electrical machines, the use of permanent magnets in combination with the robust design and the line start capability of the induction machine is extensively investigated. Many experimental designs have been suggested in literature, but recently, these line-start permanent-magnet machines (LSPMMs) have become off-the-shelf products available in a power range up to 7.5 kW. The permanent magnet flux density is a function of the operating temperature. Consequently, the temperature will affect almost every electrical quantity of the machine, including current, torque, and efficiency. In this paper, the efficiency of an off-the-shelf 4-kW three-phase LSPMM is evaluated as a function of the temperature by both finite-element modeling and by practical measurements. In order to obtain stator, rotor, and permanent magnet temperatures, lumped thermal modeling is used.

Effect of antifoams on oxygen mass transfer in pulp and paper mill effluents

Vaahdonestoaineiden haitallinen vaikutus hapen liukenemisnopeuteen biologisen puhdistamon jätevesissä on yleisesti tunnettua. Aineiden eri vaikutusmekanismien takia on silti vaikea etukäteen arvioida, miten ja kuinka paljon aineensiirto muuttuu. Työn tavoitteena oli saada tietoa vaahdonestoaineiden ja muiden pinta-aktiivisten aineiden vaikutuksesta kuplakokoon, kaasun tilavuusosuuteen ja kaasu-neste aineensiirtoon. Työn teoriaosassa on kuvailtu vaahdon muodostumiseen vaikuttavia tekijöitä sekä eri vaahdonestoaineiden vaikutusmekanismeja sellu- ja paperitehtaan jätevedessä. Edelleen on esitetty useita hapen siirtoa estäviä ja parantavia aineita. Työn kokeellisessa osassa tutkittiin kahdenkymmenenviiden eri pinta-aktiivisen aineen vaikutusta hapen liukenemisnopeuteen yksivaiheisessa kuplakolonnissa. Kokeet tehtiin kahdella pitoisuudella, kahdella eri kaasunjakolaitteella ja kolmella eri kaasun tyhjäputkinopeudella. Aineensiirtokokeiden rinnalla tutkittiin jätevesien laatu- ja fysikaalisiaominaisuuksia, niiden vaikutusta hapen liukenemisnopeuteen sekä testattavien koeaineiden vaikutusta fysikaalisin ominaisuuksiin. Kokeet osoittavat että pinta-aktiivisten aineiden vaikutus hapen aineensiirtoon vaihtelee riippuen kaasunjakolaitteesta ja aineen pitoisuuksista. Testatuista vaahdonestonaineista pienin negatiivinen vaikutus oli aineella AT 35 ja positiivinen vaikutustodettiin olevan vaahdonestoaineiden komponenteilla: P2, S1, F4 ja T9.

Converter-flux-based current control of voltage source PWM rectifiers - analysis and implementation

Pulsewidth-modulated (PWM) rectifier technology is increasingly used in industrial applications like variable-speed motor drives, since it offers several desired features such as sinusoidal input currents, controllable power factor, bidirectional power flow and high quality DC output voltage. To achieve these features,however, an effective control system with fast and accurate current and DC voltage responses is required. From various control strategies proposed to meet these control objectives, in most cases the commonly known principle of the synchronous-frame current vector control along with some space-vector PWM scheme have been applied. Recently, however, new control approaches analogous to the well-established direct torque control (DTC) method for electrical machines have also emerged to implement a high-performance PWM rectifier. In this thesis the concepts of classical synchronous-frame current control and DTC-based PWM rectifier control are combined and a new converter-flux-based current control (CFCC) scheme is introduced. To achieve sufficient dynamic performance and to ensure a stable operation, the proposed control system is thoroughly analysed and simple rules for the controller design are suggested. Special attention is paid to the estimationof the converter flux, which is the key element of converter-flux-based control. Discrete-time implementation is also discussed. Line-voltage-sensorless reactive reactive power control methods for the L- and LCL-type line filters are presented. For the L-filter an open-loop control law for the d-axis current referenceis proposed. In the case of the LCL-filter the combined open-loop control and feedback control is proposed. The influence of the erroneous filter parameter estimates on the accuracy of the developed control schemes is also discussed. A newzero vector selection rule for suppressing the zero-sequence current in parallel-connected PWM rectifiers is proposed. With this method a truly standalone and independent control of the converter units is allowed and traditional transformer isolation and synchronised-control-based solutions are avoided. The implementation requires only one additional current sensor. The proposed schemes are evaluated by the simulations and laboratory experiments. A satisfactory performance and good agreement between the theory and practice are demonstrated.

Torque vibration model of axial-flux surface-mounted permanent magnet synchronous machine

In order that the radius and thus ununiform structure of the teeth and otherelectrical and magnetic parts of the machine may be taken into consideration the calculation of an axial flux permanent magnet machine is, conventionally, doneby means of 3D FEM-methods. This calculation procedure, however, requires a lotof time and computer recourses. This study proves that also analytical methods can be applied to perform the calculation successfully. The procedure of the analytical calculation can be summarized into following steps: first the magnet is divided into slices, which makes the calculation for each section individually, and then the parts are submitted to calculation of the final results. It is obvious that using this method can save a lot of designing and calculating time. Thecalculation program is designed to model the magnetic and electrical circuits of surface mounted axial flux permanent magnet synchronous machines in such a way, that it takes into account possible magnetic saturation of the iron parts. Theresult of the calculation is the torque of the motor including the vibrations. The motor geometry and the materials and either the torque or pole angle are defined and the motor can be fed with an arbitrary shape and amplitude of three-phase currents. There are no limits for the size and number of the pole pairs nor for many other factors. The calculation steps and the number of different sections of the magnet are selectable, but the calculation time is strongly depending on this. The results are compared to the measurements of real prototypes. The permanent magnet creates part of the flux in the magnetic circuit. The form and amplitude of the flux density in the air-gap depends on the geometry and material of the magnetic circuit, on the length of the air-gap and remanence flux density of the magnet. Slotting is taken into account by using the Carter factor in the slot opening area. The calculation is simple and fast if the shape of the magnetis a square and has no skew in relation to the stator slots. With a more complicated magnet shape the calculation has to be done in several sections. It is clear that according to the increasing number of sections also the result will become more accurate. In a radial flux motor all sections of the magnets create force with a same radius. In the case of an axial flux motor, each radial section creates force with a different radius and the torque is the sum of these. The magnetic circuit of the motor, consisting of the stator iron, rotor iron, air-gap, magnet and the slot, is modelled with a reluctance net, which considers the saturation of the iron. This means, that several iterations, in which the permeability is updated, has to be done in order to get final results. The motor torque is calculated using the instantaneous linkage flux and stator currents. Flux linkage is called the part of the flux that is created by the permanent magnets and the stator currents passing through the coils in stator teeth. The angle between this flux and the phase currents define the torque created by the magnetic circuit. Due to the winding structure of the stator and in order to limit the leakage flux the slot openings of the stator are normally not made of ferromagnetic material even though, in some cases, semimagnetic slot wedges are used. In the slot opening faces the flux enters the iron almost normally (tangentially with respect to the rotor flux) creating tangential forces in the rotor. This phenomenon iscalled cogging. The flux in the slot opening area on the different sides of theopening and in the different slot openings is not equal and so these forces do not compensate each other. In the calculation it is assumed that the flux entering the left side of the opening is the component left from the geometrical centre of the slot. This torque component together with the torque component calculated using the Lorenz force make the total torque of the motor. It is easy to assume that when all the magnet edges, where the derivative component of the magnet flux density is at its highest, enter the slot openings at the same time, this will have as a result a considerable cogging torque. To reduce the cogging torquethe magnet edges can be shaped so that they are not parallel to the stator slots, which is the common way to solve the problem. In doing so, the edge may be spread along the whole slot pitch and thus also the high derivative component willbe spread to occur equally along the rotation. Besides forming the magnets theymay also be placed somewhat asymmetric on the rotor surface. The asymmetric distribution can be made in many different ways. All the magnets may have a different deflection of the symmetrical centre point or they can be for example shiftedin pairs. There are some factors that limit the deflection. The first is that the magnets cannot overlap. The magnet shape and the relative width compared to the pole define the deflection in this case. The other factor is that a shifting of the poles limits the maximum torque of the motor. If the edges of adjacent magnets are very close to each other the leakage flux from one pole to the other increases reducing thus the air-gap magnetization. The asymmetric model needs some assumptions and simplifications in order to limit the size of the model and calculation time. The reluctance net is made for symmetric distribution. If the magnets are distributed asymmetrically the flux in the different pole pairs will not be exactly the same. Therefore, the assumption that the flux flows from the edges of the model to the next pole pairs, in the calculation model from one edgeto the other, is not correct. If it were wished for that this fact should be considered in multi-pole pair machines, this would mean that all the poles, in other words the whole machine, should be modelled in reluctance net. The error resulting from this wrong assumption is, nevertheless, irrelevant.