The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

Liitepartikkelien vokaalisointusuhteita suomen murteissa

Summary: Non-harmonic enclitic particles in Finnish dialects

Performance Characteristics of an Axial-Flux Solid-Rotor-Core Induction Motor

The integration of electric motors and industrial appliances such as pumps, fans, and compressors is rapidly increasing. For instance, the integration of an electric motor and a centrifugal pump provides cost savings and improved performance characteristics. Material cost savings are achieved when an electric motor is integrated into the shaft of a centrifugal pump, and the motor utilizes the bearings of the pump. This arrangement leads to a smaller configuration that occupies less floor space. The performance characteristics of a pump drive can be improved by using the variable-speed technology. This enables the full speed control of the drive and the absence of a mechanical gearbox and couplers. When using rotational speeds higher than those that can be directly achieved by the network frequency the structure of the rotor has to be mechanically durable. In this thesis the performance characteristics of an axial-flux solid-rotor-core induction motor are determined. The motor studied is a one-rotor-one-stator axial-flux induction motor, and thus, there is only one air-gap between the rotor and the stator. The motor was designed for higher rotational speeds, and therefore a good mechanical strength of the solid-rotor-core rotor is required to withstand the mechanical stresses. The construction of the rotor and the high rotational speeds together produce a feature, which is not typical of traditional induction motors: the dominating loss component of the motor is the rotor eddy current loss. In the case of a typical industrial induction motor instead the dominating loss component is the stator copper loss. In this thesis, several methods to decrease the rotor eddy current losses in the case of axial-flux induction motors are presented. A prototype motor with 45 kW output power at 6000 min-1 was designed and constructed for ascertaining the results obtained from the numerical FEM calculations. In general, this thesis concentrates on the methods for improving the electromagnetic properties of an axial-flux solid-rotor-core induction motor and examines the methods for decreasing the harmonic eddy currents of the rotor. The target is to improve the efficiency of the motor and to reach the efficiency standard of the present-day industrial induction motors equipped with laminated rotors.

Power quality improving with virtual flux-based voltage source line converter

Line converters have become an attractive AC/DC power conversion solution in industrial applications. Line converters are based on controllable semiconductor switches, typically insulated gate bipolar transistors. Compared to the traditional diode bridge-based power converters line converters have many advantageous characteristics, including bidirectional power flow, controllable de-link voltage and power factor and sinusoidal line current. This thesis considers the control of the lineconverter and its application to power quality improving. The line converter control system studied is based on the virtual flux linkage orientation and the direct torque control (DTC) principle. A new DTC-based current control scheme is introduced and analyzed. The overmodulation characteristics of the DTC converter are considered and an analytical equation for the maximum modulation index is derived. The integration of the active filtering features to the line converter isconsidered. Three different active filtering methods are implemented. A frequency-domain method, which is based on selective harmonic sequence elimination, anda time-domain method, which is effective in a wider frequency band, are used inharmonic current compensation. Also, a voltage feedback active filtering method, which mitigates harmonic sequences of the grid voltage, is implemented. The frequency-domain and the voltage feedback active filtering control systems are analyzed and controllers are designed. The designs are verified with practical measurements. The performance and the characteristics of the implemented active filtering methods are compared and the effect of the L- and the LCL-type line filteris discussed. The importance of the correct grid impedance estimate in the voltage feedback active filter control system is discussed and a new measurement-based method to obtain it is proposed. Also, a power conditioning system (PCS) application of the line converter is considered. A new method for correcting the voltage unbalance of the PCS-fed island network is proposed and experimentally validated.

Taajuusmuuttajiakoskevat EMC-standardit

Taajuusmuuttajat aiheuttavat toimintansa seurauksena runsaasti laajakaistaisia sähkömagneettisia häiriöitä. Häiriöt etenevät sekä johtumalla että säteilemällä ja ne ovat seurausta pääasiassa taajuusmuuttajien toimintaan perustuvista nopeista kytkentäilmiöistä sekä puolijohdekomponenttien epälineaarisuuksista. Sähkömagneettisille häiriöille määritellään suurimmat sallitut tasot useissa EMC-standardeissa. Taajuusmuuttajia koskevia EMC-standardia on neljä; tuotestandardi EN 61800-3, harmonisia virtoja pienjänniteverkoissa käsittelevä EN 61000-3-12 sekä yleiset standardit EN 61000-6-1, -6-2 ja EN 61000-6-3, -6-4. Tämän diplomityön tarkoituksena on tutkia ja koota yhteen edellä mainittujen standardien sisältämät vaatimukset. Työn pääpaino on kohdistettu häiriöpäästöille asetettuihin vaatimuksiin, sillä häiriöiden sietoon liittyvien vaatimusten täyttäminen ei taajuusmuuttajien kohdalla yleensä tuota ongelmia.

Pintaemittoivan VCSEL-diodin käyttö aallonpituusmodulaatiospektroskopiaan perustuvissa hapen konsentraation mittauksissa

Teollisuuden sovelluksissa kaasun mittauslaitteilta vaaditaan suurta luotettavuutta sekä niiden huollontarve tulee olla minimaalista. Jatkuva-aikaisissa mittauksissa nämä vaatimuksetlunastaa viritettävän diodilaser-spektroskopian käyttö. Pienen kokonsa, vähäisten tehohäviöiden ja hyvien säätömahdollisuuksien ansiosta, VCSEL:t (vertical cavity surface emitting laser) ovat sopivia vaihtoehtoja valonlähteeksi mitattaessapäästökaasuja, tässä tapauksessa happea. Työssä rakennettiin mittauslaitteisto, jota käytettiin hapen konsentraation määrittämiseen. Mittauslaitteistolla suoritettiin hapen pitoisuusmittauksia ja saatuja tuloksia verrattiin teoreettisen mallin antamien tulosten kesken. Mittaukset perustuvat toisen harmonisen signaalin havainnointiin. Lisäksi tarkasteltiin paineen ja lämpötilanvaikutusta mittaustuloksiin.

Fractional slot permanent magnet synchronous motors for low speed applications

This study compares different rotor structures of permanent magnet motors with fractional slot windings. The surface mounted magnet and the embedded magnet rotor structures are studied. This thesis analyses the characteristics of a concentrated two-layer winding, each coil of which is wound around one tooth and which has a number of slots per pole and per phase less than one (q < 1). Compared to the integer slot winding, the fractional winding (q < 1) has shorter end windings and this, thereby, makes space as well as manufacturing cost saving possible. Several possible ways of winding a fractional slot machine with slots per pole and per phase lessthan one are examined. The winding factor and the winding harmonic components are calculated. The benefits attainable from a machine with concentrated windingsare considered. Rotor structures with surface magnets, radially embedded magnets and embedded magnets in V-position are discussed. The finite element method isused to solve the main values of the motors. The waveform of the induced electro motive force, the no-load and rated load torque ripple as well as the dynamic behavior of the current driven and voltage driven motor are solved. The results obtained from different finite element analyses are given. A simple analytic method to calculate fractional slot machines is introduced and the values are compared to the values obtained with the finite element analysis. Several different fractional slot machines are first designed by using the simple analytical methodand then computed by using the finite element method. All the motors are of thesame 225-frame size, and have an approximately same amount of magnet material, a same rated torque demand and a 400 - 420 rpm speed. An analysis of the computation results gives new information on the character of fractional slot machines.A fractional slot prototype machine with number 0.4 for the slots per pole and per phase, 45 kW output power and 420 rpm speed is constructed to verify the calculations. The measurement and the finite element method results are found to beequal.

Analysis of the voltage source inverter with small DC-link capacitor

IIn electric drives, frequency converters are used to generatefor the electric motor the AC voltage with variable frequency and amplitude. When considering the annual sale of drives in values of money and units sold, the use of low-performance drives appears to be in predominant. These drives have tobe very cost effective to manufacture and use, while they are also expected to fulfill the harmonic distortion standards. One of the objectives has also been to extend the lifetime of the frequency converter. In a traditional frequency converter, a relatively large electrolytic DC-link capacitor is used. Electrolytic capacitors are large, heavy and rather expensive components. In many cases, the lifetime of the electrolytic capacitor is the main factor limiting the lifetime of the frequency converter. To overcome the problem, the electrolytic capacitor is replaced with a metallized polypropylene film capacitor (MPPF). The MPPF has improved properties when compared to the electrolytic capacitor. By replacing the electrolytic capacitor with a film capacitor the energy storage of the DC-linkwill be decreased. Thus, the instantaneous power supplied to the motor correlates with the instantaneous power taken from the network. This yields a continuousDC-link current fed by the diode rectifier bridge. As a consequence, the line current harmonics clearly decrease. Because of the decreased energy storage, the DC-link voltage fluctuates. This sets additional conditions to the controllers of the frequency converter to compensate the fluctuation from the supplied motor phase voltages. In this work three-phase and single-phase frequency converters with small DC-link capacitor are analyzed. The evaluation is obtained with simulations and laboratory measurements.

Analysis of torque and speed ripple producing non-idealities of frequency converters in electric drives

Electric motors driven by adjustable-frequency converters may produce periodic excitation forces that can cause torque and speed ripple. Interaction with the driven mechanical system may cause undesirable vibrations that affect the system performance and lifetime. Direct drives in sensitive applications, such as elevators or paper machines, emphasize the importance of smooth torque production. This thesis analyses the non-idealities of frequencyconverters that produce speed and torque ripple in electric drives. The origin of low order harmonics in speed and torque is examined. It is shown how different current measurement error types affect the torque. As the application environment, direct torque control (DTC) method is applied to permanent magnet synchronous machines (PMSM). A simulation model to analyse the effect of the frequency converter non-idealities on the performance of the electric drives is created. Themodel enables to identify potential problems causing torque vibrations and possibly damaging oscillations in electrically driven machine systems. The model is capable of coupling with separate simulation software of complex mechanical loads. Furthermore, the simulation model of the frequency converter's control algorithm can be applied to control a real frequency converter. A commercial frequencyconverter with standard software, a permanent magnet axial flux synchronous motor and a DC motor as the load are used to detect the effect of current measurement errors on load torque. A method to reduce the speed and torque ripple by compensating the current measurement errors is introduced. The method is based on analysing the amplitude of a selected harmonic component of speed as a function oftime and selecting a suitable compensation alternative for the current error. The speed can be either measured or estimated, so the compensation method is applicable also for speed sensorless drives. The proposed compensation method is tested with a laboratory drive, which consists of commercial frequency converter hardware with self-made software and a prototype PMSM. The speed and torque rippleof the test drive are reduced by applying the compensation method. In addition to the direct torque controlled PMSM drives, the compensation method can also beapplied to other motor types and control methods.

Virtuaaliprototyypin käyttö roottorin dynamiikan analysoinnissa

Työn tavoitteena oli selvittää kaupallisen dynamiikansimulointiohjelmiston soveltuvuus roottoridynamiikan analysointiin. Työssä keskityttiin erityisesti roottorin dynamiikkaan vaikuttavien epäideaalisuuksien mallintamiseen. Simulointitulosten tarkkuutta selvitettiin mittauksilla. Lisäksi vertailtiin yleiskäyttöisen dynamiikan simulointiohjelmiston ja roottoridynamiikan erikoisohjelmiston teoriaa. Tutkittava roottori oli paperikoneen putkitela. Telan joustavuus kuvattiin elementtimenetelmällä ratkaistujen moodien avulla. Elementtimallissa huomioitiin telan vaipan seinämänpaksuusvaihtelu, joka vaikuttaa telan massa- ja jäykkyysjakaumaan. Dynamiikkaohjelmistossa mallinnettiin telan tuennasta tulevat herätteet. Dynamiikkaohjelmistona käytettiin ADAMS:ia ja FEM-ohjelmana ANSYS:stä. Tuloksista havaittiin käytetyn menetelmän soveltuvan roottoridynamiikan analysointiin ja roottorin epäideaalisuuksien mallintamiseen. Simulointimallilla saatiin esille murtolukukriittiset pyörimisnopeudet ja telan kriittinen pyörimisnopeus vastasi hyvin mittaustuloksia.

Taajuusmuuttajan verkkovirran yliaallot pienitehoisissa kolmivaihekäytöissä

Pienitehoisessa taajuusmuuttajassa verkkojännite tasasuunnataan yleisesti diodisillalla. Taajuusmuuttajaa kuormitettaessa diodisilta aiheuttaa sen, että verkosta otettava virta on epäsinimuotoista ja sisältää runsaasti eri kertalukujen yliaaltoja. Tällainen virta vääristää verkon jännitettä sekä aiheuttaa häviöitä, resonansseja ja toimintaepävarmuutta sähkönjakeluverkkoon liitetyissä laitteissa ja komponenteissa. Työssä on tarkasteltu pienitehoisen kolmivaiheisen taajuusmuuttajan tulovirrassa esiintyviä yliaaltoja ja keinoja niiden rajoittamiseksi. Laitevalmistajien osalta syynä aiheen tutkimiseen ja kiinnostavuuteen ovat lähinnä uudet yliaaltoja käsittelevät standardit. Työssä onkin tutustuttu tarvittavin osin standardoinnin nykytilaan sekä seurattu standardeja valmistelevien työryhmien työtä ja tätä kautta perehdytty alaa käsittelevien standardien tulevaisuudennäkymiin Euroopan Unionin alueella. Käsittely on kohdistettu kolmivaiheisiin laitteisiin. Standardi IEC 61000-3-2 asettaa yliaaltorajat ammattikäyttöön tarkoitetuille laitteille, kun laitteen tuloteho on korkeintaan 1 kW. Tällä tehoalueella riittää kolmivaihelaitteille kuristimilla suoritettu passiivinen suodatus täyttämään asetetut viranomaisvaatimukset. Tähän liittyen on suunniteltu kolmivaihekuristin, joka on tarkoitettu käytettäväksi taajuusmuuttajasarjan lisävarusteena tehoalueella 0.12…2.2 kW.

Päällystyskoneiden runkojen värähtelyiden FE-analysoinnin kehittäminen

Työssä kehitettiin päällystyskoneen runkojen FE-analysointia erityisesti vastelaskennan kannalta, joskin myös ominaistaajuuslaskennan tarkentamiseksi esitettiin parannuksia. Työssä rajoitutaan harmonisten telaherätteiden aiheuttamien vasteiden analysointiin. . Työssä käsitellään värähtelyjen teoriaa ja tarkastellaan Abaqus-ohjelmiston (versio 5.8) laskentamenetelmiä vakiotilan värähtelyvasteen laskemiseksi. FEM-mallin rakenteeseen liittyen käsitellään perustuksien ja maaperän, konepalkin ja telaherätteiden mallintamista ja mallinnuksen laajuutta. Telaherätteitä käsitellään yksittäisen telan ja telojen samanvaiheisuuden kannalta. Samanvaiheisuutta tutkitaan työssä kehitetyllä summafunktiolla. Värähtelyjen mittaukseen FEM-mallien verifioimiseksi esitetään parannuksia. Nykyinen mallinnustapa käsitellään lyhyesti. Parannusehdotuksia kokeiltiin mallintamalla Rauma 400-päällystyskone ja vertaamalla tuloksia mitattuihin. Laskennan tulokset vastasivat vaihtelevasti mittaustuloksia, mittaustulosten puutteellisuus vaikeutti vertailua. Tulosten perusteella herätetiedon parantaminen on perusteltua ja mallin laajennus lisää todenmukaisuutta. Maaperän huomiointi vaikuttaa ennen kaikkea ominaistaajuuksiin ja muotoihin ja suora ratkaisutapa on käyttökelpoinen vasteen laskentamenetelmä otettaessa maaperä huomioon.

High-Speed Solid-Rotor Induction Machine – Electromagnetic Calculation and Design

Within the latest decade high-speed motor technology has been increasingly commonly applied within the range of medium and large power. More particularly, applications like such involved with gas movement and compression seem to be the most important area in which high-speed machines are used. In manufacturing the induction motor rotor core of one single piece of steel it is possible to achieve an extremely rigid rotor construction for the high-speed motor. In a mechanical sense, the solid rotor may be the best possible rotor construction. Unfortunately, the electromagnetic properties of a solid rotor are poorer than the properties of the traditional laminated rotor of an induction motor. This thesis analyses methods for improving the electromagnetic properties of a solid-rotor induction machine. The slip of the solid rotor is reduced notably if the solid rotor is axially slitted. The slitting patterns of the solid rotor are examined. It is shown how the slitting parameters affect the produced torque. Methods for decreasing the harmonic eddy currents on the surface of the rotor are also examined. The motivation for this is to improve the efficiency of the motor to reach the efficiency standard of a laminated rotor induction motor. To carry out these research tasks the finite element analysis is used. An analytical calculation of solid rotors based on the multi-layer transfer-matrix method is developed especially for the calculation of axially slitted solid rotors equipped with wellconducting end rings. The calculation results are verified by using the finite element analysis and laboratory measurements. The prototype motors of 250 – 300 kW and 140 Hz were tested to verify the results. Utilization factor data are given for several other prototypes the largest of which delivers 1000 kW at 12000 min-1.