Wavelet Analysis in Turbulence

The objective of this thesis is to study wavelets and their role in turbulence applications. Under scrutiny in the thesis is the intermittency in turbulence models. Wavelets are used as a mathematical tool to study the intermittent activities that turbulence models produce. The first section generally introduces wavelets and wavelet transforms as a mathematical tool. Moreover, the basic properties of turbulence are discussed and classical methods for modeling turbulent flows are explained. Wavelets are implemented to model the turbulence as well as to analyze turbulent signals. The model studied here is the GOY (Gledzer 1973, Ohkitani & Yamada 1989) shell model of turbulence, which is a popular model for explaining intermittency based on the cascade of kinetic energy. The goal is to introduce better quantification method for intermittency obtained in a shell model. Wavelets are localized in both space (time) and scale, therefore, they are suitable candidates for the study of singular bursts, that interrupt the calm periods of an energy flow through various scales. The study concerns two questions, namely the frequency of the occurrence as well as the intensity of the singular bursts at various Reynolds numbers. The results gave an insight that singularities become more local as Reynolds number increases. The singularities become more local also when the shell number is increased at certain Reynolds number. The study revealed that the singular bursts are more frequent at Re ~ 107 than other cases with lower Re. The intermittency of bursts for the cases with Re ~ 106 and Re ~ 105 was similar, but for the case with Re ~ 104 bursts occured after long waiting time in a different fashion so that it could not be scaled with higher Re.

Preservation and Management of Added Value in Networked R&D environment

Tämä tutkielma käsittelee lisäarvon syntymistä, ylläpitämistä ja hallintaa verkostoi-tuneessa tuotekehitysympäristössä. Teemahaastattelu-menetelmää käyttäen, tavoitteena on tunnistaa ja kuvata ne prosessit, käytännöt ja toimintatavat, joissa kohdeyritys on onnistunut ja joissa lisäarvoa on syntynyt. Toinen keskeinen tavoite on löytää ongelmalliset alueet lisäarvon tuottamisessa ja analysoida, miksi nämä alueet ovat ongelmallisia. Käsitteiden arvo, arvoketju ja arvoverkosto, sekä viitekirjallisuuden esimerkkien perusteella muodostetaan teoreettinen viitekehys ja kuvataan niitä hyödyllisiä toimintatapoja ja käytäntöjä, joihin panostamalla lisäarvoa syntyy. Erityisesti informaatioteknologian alalla verkostoituminen ja arvoverkosto ovat yhä merkittävämpiä tuotekehityksen toimintatapoja, mihin horisontaalisen yhteistyön kehittyminen, globalisoituminen ja informaatioteknologian nopea kehitys on johtanut. Keskeisiä tuloksia ovat tarve yhtenäisempään, prosessinomaisempaan toimintatapaan ja liiketoimintaprosessien muokkaamiseen verkostoituneen T&K ympäristön vaatimusten mukaisesti. Myös tarve paremman näkyvyyden luomiseen sekä aktiviteettien hallintaan uudentyyppisen arvoverkoston vaatimusten mukaisesti korostui tuloksissa.

Clustering and International Competitiveness of Information Technology Industry in the Saint Petersburg Area

The main objective of this study is to assess the potential of the information technology industry in the Saint Petersburg area to become one of the new key industries in the Russian economy. To achieve this objective, the study analyzes especially the international competitiveness of the industry and the conditions for clustering. Russia is currently heavily dependent on its natural resources, which are the main source of its recent economic growth. In order to achieve good long-term economic performance, Russia needs diversification in its well-performing industries in addition to the ones operating in the field of natural resources. The Russian government has acknowledged this and started special initiatives to promote such other industries as information technology and nanotechnology. An interesting industry that is basically less than 20 years old and fast growing in Russia, is information technology. Information technology activities and markets are mainly concentrated in Russia’s two biggest cities, Moscow and Saint Petersburg, and areas around them. The information technology industry in the Saint Petersburg area, although smaller than Moscow, is especially dynamic and is gaining increasing foreign company presence. However, the industry is not yet internationally competitive as it lacks substantial and sustainable competitive advantages. The industry is also merely a potential global information technology cluster, as it lacks the competitive edge and a wide supplier and manufacturing base and other related parts of the whole information technology value system. Alone, the industry will not become a key industry in Russia, but it will, on the other hand, have an important supporting role for the development of other industries. The information technology market in the Saint Petersburg area is already large and if more tightly integrated to Moscow, they will together form a huge and still growing market sufficient for most companies operating in Russia currently and in the future. Therefore, the potential of information technology inside Russia is immense.

Clustering and international competitiveness of information technology industry in the Saint Petersburg area

The main objective of this study is to assess the potential of the information technology industry in the Saint Petersburg area to become one of the new key industries in the Russian economy. To achieve this objective, the study analyzes especially the international competitiveness of the industry and the conditions for clustering. Russia is currently heavily dependent on its natural resources, which are the main source of its recent economic growth. In order to achieve good long-term economic performance, Russia needs diversification in its well-performing industries in addition to the ones operating in the field of natural resources. The Russian government has acknowledged this and started special initiatives to promote such other industries as information technology and nanotechnology. An interesting industry that is basically less than 20 years old and fast growing in Russia, is information technology. Information technology activities and markets are mainly concentrated in Russia’s two biggest cities, Moscow and Saint Petersburg, and areas around them. The information technology industry in the Saint Petersburg area, although smaller than Moscow, is especially dynamic and is gaining increasing foreign company presence. However, the industry is not yet internationally competitive as it lacks substantial and sustainable competitive advantages. The industry is also merely a potential global information technology cluster, as it lacks the competitive edge and a wide supplier and manufacturing base and other related parts of the whole information technology value system. Alone, the industry will not become a key industry in Russia, but it will, on the other hand, have an important supporting role for the development of other industries. The information technology market in the Saint Petersburg area is already large and if more tightly integrated to Moscow, they will together form a huge and still growing market sufficient for most companies operating in Russia currently and in the future. Therefore, the potential of information technology inside Russia is immense.

Reconstruction and analysis of surface variation using photometric stereo

In many industrial applications, accurate and fast surface reconstruction is essential for quality control. Variation in surface finishing parameters, such as surface roughness, can reflect defects in a manufacturing process, non-optimal product operational efficiency, and reduced life expectancy of the product. This thesis considers reconstruction and analysis of high-frequency variation, that is roughness, on planar surfaces. Standard roughness measures in industry are calculated from surface topography. A fast and non-contact method to obtain surface topography is to apply photometric stereo in the estimation of surface gradients and to reconstruct the surface by integrating the gradient fields. Alternatively, visual methods, such as statistical measures, fractal dimension and distance transforms, can be used to characterize surface roughness directly from gray-scale images. In this thesis, the accuracy of distance transforms, statistical measures, and fractal dimension are evaluated in the estimation of surface roughness from gray-scale images and topographies. The results are contrasted to standard industry roughness measures. In distance transforms, the key idea is that distance values calculated along a highly varying surface are greater than distances calculated along a smoother surface. Statistical measures and fractal dimension are common surface roughness measures. In the experiments, skewness and variance of brightness distribution, fractal dimension, and distance transforms exhibited strong linear correlations to standard industry roughness measures. One of the key strengths of photometric stereo method is the acquisition of higher frequency variation of surfaces. In this thesis, the reconstruction of planar high-frequency varying surfaces is studied in the presence of imaging noise and blur. Two Wiener filterbased methods are proposed of which one is optimal in the sense of surface power spectral density given the spectral properties of the imaging noise and blur. Experiments show that the proposed methods preserve the inherent high-frequency variation in the reconstructed surfaces, whereas traditional reconstruction methods typically handle incorrect measurements by smoothing, which dampens the high-frequency variation.

Inactive endosialidase-based detection of bacterial and oncofetal polysialic acid

Polysialic acid is a carbohydrate polymer which consist of N-acetylneuraminic acid units joined by alpha2,8-linkages. It is developmentally regulated and has an important role during normal neuronal development. In adults, it participates in complex neurological processes, such as memory, neural plasticity, tumor cell growth and metastasis. Polysialic acid also constitutes the capsule of some meningitis and sepsis-causing bacteria, such as Escherichia coli K1, group B meningococci, Mannheimia haemolytica A2 and Moraxella nonliquefaciens. Polysialic acid is poorly immunogenic; therefore high affinity antibodies against it are difficult to prepare, thus specific and fast detection methods are needed. Endosialidase is an enzyme derived from the E. coli K1 bacteriophage, which specifically recognizes and degrades polysialic acid. In this study, a novel detection method for polysialic acid was developed based on a fusion protein of inactive endosialidase and the green fluorescent protein. It utilizes the ability of the mutant, inactive endosialidase to bind but not cleave polysialic acid. Sequencing of the endosialidase gene revealed that amino acid substitutions near the active site of the enzyme differentiate the active and inactive forms of the enzyme. The fusion protein was applied for the detection of polysialic acid in bacteria and neuroblastoma. The results indicate that the fusion protein is a fast, sensitive and specific reagent for the detection of polysialic acid. The use of an inactive enzyme as a specific molecular tool for the detection of its substrate represents an approach which could potentially find wide applicability in the specific detection of diverse macromolecules.

A platform for anti-biofilm assays combining viability, biomass and matrix quantifications in susceptibility assessments of antimicrobials against Staphylococcus aureus biofilms

Bacteria can exist as planktonic, the lifestyle in which single cells exist in suspension, and as biofilms, which are surface-attached bacterial communities embedded in a selfproduced matrix. Most of the antibiotics and the methods for antimicrobial work have been developed for planktonic bacteria. However, the majority of the bacteria in natural habitats live as biofilms. Biofilms develop dauntingly fast high resistance towards conventional antibacterial treatments and thus, there is a great need to meet the demands of effective anti-biofilm therapy. In this thesis project it was attempted to fill the void of anti-biofilm screening methods by developing a platform of assays that evaluate the effect that screened compounds have on the total biomass, viability and the extracellular polysaccharide (EPS) layer of the biofilms. Additionally, a new method for studying biofilms and their interactions with compounds in a continuous flow system was developed using capillary electrochromatography (CEC). The screening platform was utilized with a screening campaign using a small library of cinchona alkaloids. The assays were optimized to be statistically robust enough for screening. The first assay, based on crystal violet staining, measures total biofilm biomass, and it was automated using a liquid handling workstation to decrease the manual workload and signal variation. The second assay, based on resazurin staining, measures viability of the biofilm, and it was thoroughly optimized for the strain used, but was then a very simple and fast method to be used for primary screening. The fluorescent resazurin probe is not toxic to the biofilms. In fact, it was also shown in this project that staining the biofilms with resazurin prior to staining with crystal violet had no effect on the latter and they can be used in sequence on the same screening plate. This sequential addition step was indeed a major improvement on the use of reagents and consumables and also shortened the work time. As a third assay in the platform a wheat germ agglutinin based assay was added to evaluate the effect a compound has on the EPS layer. Using this assay it was found that even if compounds might have clear effect on both biomass and viability, the EPS layer can be left untouched or even be increased. This is a clear implication of the importance of using several assays to be able to find “true hits” in a screening setting. In the pilot study of screening for antimicrobial and anti-biofilm effects using a cinchona alkaloid library, one compound was found to have antimicrobial effect against planktonic bacteria and prevent biofilm formation at low micromolar concentration. To eradicate biofilms, a higher concentration was needed. It was also shown that the chemical space occupied by the active compound was slightly different than the rest of the cinchona alkaloids as well as the rest of the compounds used for validatory screening during the optimization processes of the separate assays.

Theoretical Analysis and Numerical Simulation of Spectral Radiative Properties of Combustion Gases in Oxy/Air-Fired Combustion Systems

Energy efficiency is one of the major objectives which should be achieved in order to implement the limited energy resources of the world in a sustainable way. Since radiative heat transfer is the dominant heat transfer mechanism in most of fossil fuel combustion systems, more accurate insight and models may cause improvement in the energy efficiency of the new designed combustion systems. The radiative properties of combustion gases are highly wavelength dependent. Better models for calculating the radiative properties of combustion gases are highly required in the modeling of large scale industrial combustion systems. With detailed knowledge of spectral radiative properties of gases, the modeling of combustion processes in the different applications can be more accurate. In order to propose a new method for effective non gray modeling of radiative heat transfer in combustion systems, different models for the spectral properties of gases including SNBM, EWBM, and WSGGM have been studied in this research. Using this detailed analysis of different approaches, the thesis presents new methods for gray and non gray radiative heat transfer modeling in homogeneous and inhomogeneous H2O–CO2 mixtures at atmospheric pressure. The proposed method is able to support the modeling of a wide range of combustion systems including the oxy-fired combustion scenario. The new methods are based on implementing some pre-obtained correlations for the total emissivity and band absorption coefficient of H2O–CO2 mixtures in different temperatures, gas compositions, and optical path lengths. They can be easily used within any commercial CFD software for radiative heat transfer modeling resulting in more accurate, simple, and fast calculations. The new methods were successfully used in CFD modeling by applying them to industrial scale backpass channel under oxy-fired conditions. The developed approaches are more accurate compared with other methods; moreover, they can provide complete explanation and detailed analysis of the radiation heat transfer in different systems under different combustion conditions. The methods were verified by applying them to some benchmarks, and they showed a good level of accuracy and computational speed compared to other methods. Furthermore, the implementation of the suggested banded approach in CFD software is very easy and straightforward.

Wind turbine indirect ice detection and operational analysis

Wind energy is one of the most promising and fast growing sector of energy production. Wind is ecologically friendly and relatively cheap energy resource available for development in practically all corners of the world (where only the wind blows). Today wind power gained broad development in the Scandinavian countries. Three important challenges concerning sustainable development, i.e. energy security, climate change and energy access make a compelling case for large-scale utilization of wind energy. In Finland, according to the climate and energy strategy, accepted in 2008, the total consumption of electricity generated by means of wind farms by 2020, should reach 6 - 7% of total consumption in the country [1]. The main challenges associated with wind energy production are harsh operational conditions that often accompany the turbine operation in the climatic conditions of the north and poor accessibility for maintenance and service. One of the major problems that require a solution is the icing of turbine structures. Icing reduces the performance of wind turbines, which in the conditions of a long cold period, can significantly affect the reliability of power supply. In order to predict and control power performance, the process of ice accretion has to be carefully tracked. There are two ways to detect icing – directly or indirectly. The first way applies to the special ice detection instruments. The second one is using indirect characteristics of turbine performance. One of such indirect methods for ice detection and power loss estimation has been proposed and used in this paper. The results were compared to the results directly gained from the ice sensors. The data used was measured in Muukko wind farm, southeast Finland during a project 'Wind power in cold climate and complex terrain'. The project was carried out in 9/2013 - 8/2015 with the partners Lappeenranta university of technology, Alstom renovables España S.L., TuuliMuukko, and TuuliSaimaa.

Release Process Effectiveness - Case of Operating Room Management Software

Työn päätavoitteena oli tuoda esiin tärkeimmät julkistamisprosessin tehokkuuteen vaikuttavat tekijät. Tutkimuksessa tarkasteltiin aihetta julkistamisprojektien vetäjän näkökulmasta. Kirjallinen selvitys kattaa keskeisimmät ohjelmistoprosessin, palvelun laadun sekä projektihallinnan teoriat. Kokeellisena aineistona käytettiin asiakkailta ja myynnin sekä käyttöönoton organisaatioilta tullutta palautetta ja asiantuntijahaastatteluita. Case-tuotteena tarkasteltiin suuren kansainvälisen yrityksen jälleenmyymää leikkaussalihallinnan ohjelmistoa. Tärkeimpiä julkistamisprosessin tehokkuuteen vaikuttavia tekijöitä ovat tiekartan ja julkistamispakettien sisällön hallinta, projektin aikataulujen pitäminen, rehellinen ja nopea kommunikaatio myyntikanavaan ja asiakkaille, sekä hyvin toteutettu testaus. Työssä käydään läpi esimerkkistrategioita kehittymiseen näillä alueilla.

Maastoajoneuvon käyttöliittymän reaaliaikasimulaattori

Työn tavoitteena oli kehittää ja toteuttaa reaaliaikasimulaattori maastoajoneuvon käyttöliittymälle, jota ajoneuvon kuljettajakäyttää ajon aikana. Simulaattori oli tarkoitettu ensisijaisesti käyttöliittymän testaukseen, mutta sen pitää olla helposti laajennettavissa esimerkiksi koulutuskäyttöön. Mallinnustyökaluina oli tarkoitus käyttääpääsääntöisesti markkinoilta saatavia valmiita ohjelmistoja. Simulaattorin toteutuksessa käytettiin myös manuaalista ohjelmointia, koska valituilla ohjelmistoilla ei suoraan voinut saavuttaa reaaliaikaista visualisointia. Käsin kirjoitetut koodit hoitavat valmiilla ohjelmistoilla tehtyjen osien välisen tiedonsiirron. Varsinainen mallintaminen oli valituilla ohjelmistoilla helppoa ja nopeaa. Työn tuloksena saatiintoteutettua simulaattori, jonka vaikutelma oli reaaliaikainen. Käytettävyystestit onnistuivat hyvin simulaattorin avulla. Simulointimallin modulaarisuuden ansiosta mallia on helppo päivittää. Simulaattorin jatkokehityksessä oleelliset seikat ovat visualisoinnin parantaminen ja todellista ajoneuvoa vastaavan dynamiikan lisääminen.

Suunnitelma Ethernet-pohjaisen TCP/IP-verkon toteuttamiseksi prosessitietoliikenteen tarpeisiin Helsingin Energiassa

Prosessitietoliikenneverkko toimii mm. energian tuotannon, siirron ja jakelun käytönvalvontajärjestelmien tiedonsiirron alustana. Tietoliikenneverkkojen rooli osana yrityksen operatiivisia prosesseja kasvaa järjestelmäuusintojen myötä ja järjestelmien tehokas hyödyntäminen vaatii luotettavat sekä nopeat tietoverkot. Uusittavien järjestelmien tietoliikenneratkaisut perustuvat pääsääntöisesti Ethernetpohjaiseen TCP/IP-verkkotekniikkaan. Tällä hetkellä prosessijärjestelmien tiedonsiirtoyhteydet kenttälaitteiden ja valvomojärjestelmien välillä on pääsääntöisesti toteutettu PCM-siirtojärjestelmällä tai kiinteän yhteyden modeemeilla. Käytössä oleva PCM-tekniikka on elinkaarensa loppupuolella ja tiedonsiirtonopeudet sekä liityntärajapinnat eivät täytä nykyaikaisten prosessijärjestelmien tiedonsiirto¬vaatimuksia. Työssä käytiin läpikeskeisimmät Ethernet-tekniikkaan ja TCP/IP-protokollaperheeseen liittyvät standardit ja verkon aktiivilaitteet. Vaatimusmäärittelyissä selvitettiin Helsingin Energian eri liiketoimintayksiköiden tarpeita sekä energia-alan lainsäädännön vaatimuksia prosessitietoliikenneverkolle. Tarpeiden ja vaatimusten pohjalta verrattiin kolmea eri vaihtoehtoista verkkotekniikkaa: kytkintekniikkaa, reititintekniikkaa ja MPLS-tekniikkaa. Vertailun perusteella valitulla verkkotekniikalla tehtiin suunnitelma avoimesta, valmistajasta riippumattomasta standardeihin perustuvasta yritystason Ethernet-pohjaisesta TCP/IP prosessiverkkoratkaisusta. Johtopäätöksissä pohditaan valitun ratkaisun käytettävyyttä, tietoturvaa sekä toimenpiteitä niiden kehittämiseksi.

Asiakaslähtöinentilaus-toimitus-prosessin kehittäminen räätälöivässä yrityksessä

Tapa, jolla yritys vastaa sen räätälöityvien tuotteiden kysyntään, määrittää yrityksen menestymisen. Sen on ymmärrettävä mikä arvo asiakkaalle syntyy, kun sen yksilökohtaiset tarpeet otetaan huomioon tuotetta räätälöitäessä. Toisaalta, yrityksen on ymmärrettävä kustannus, joka näiden tarpeiden huomioon ottamimisesta syntyy. Yhdistämällä nämä tiedot oikein tilaus-toimitus-prosessin suunnittelussa ja valinnassa voidaan kutakin valittua asiakasta tai asiakasryhmää palvella kannattavasti. Tämän diplomityöntavoite on kehittää tutkimuksen toimeksiantajan Vahterus Oy:n tilaus-toimitus-prosessia. Perusteena ovat kolmen tärkeimmän asiakasryhmän tarpeet sekä tieto nykyisestä prosessista. Tutkimuksen teoreettinen perusta luodaan tutustumalla kirjallisuuteen räätälöinnistä, asiakaslähtöisyydestä sekä toimitus-prosessien eriyttämisestä. Order-winner ja order-qualifier kriteerien selvittäminen ja niiden käyttö oikean prosessin valinnassa muodostaa tutkimuksen teoreettisen ytimen. Empiriaosassa analysoitiin case-yrityksen nykyisen tilaus-toimitus-prosessin toiminta. Prosessin kehittämistyön pohjaksi selvitetettiin case-yrityksen kolmen tärkeimmän asiakasryhmän erityispiirteet ja tarpeet haastattelututkimuksella. tilaus-toimitus-prosessin analyysissä havaittiin huomattavaa asiakasryhmäkohtaista vaihtelua läpimenoajoissa. Tämä johtui tilatun tuotteen tyypistä sekä asiakkaiden erityisvaatimuksista. Haastattelututkimuksessa vahvistuinäkemys asiakasryhmien tilaus-toimitus-prosessille asettamien vaatimuksien selkeistä eroista. Johtopäätöksenä esitetään erilaisia toimitusstrategioita eri asiakasryhmille. Kustannus-tehokkuutta ja nopeutta vaativalle asiakasryhmälle prosessia tehostetaan suunnitteluprosessin automatisoinnilla. Joustavuutta vaativaa, korkeampia kustannuksia sallivaa asiakasryhmää palvellaan paremmin kehittämällä projektihallinnan ja myynnin työkaluja sekä henkilökunnan osaamista. Tutkimuksen tärkein kontribuutio yritykselle oli asiakasryhmien erojen tiedostaminen ja niiden huomioon ottaminen toimitustrategioiden ja organisaation jatkuvassa kehittämisessä.

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.