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.

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.

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.

Multilevel converter modulation: implementation and analysis

Multilevel converters provide an attractive solution to bring the benefits of speed-controlled rotational movement to high-power applications. Therefore, multilevel inverters have attracted wide interest in both the academic community and in the industry for the past two decades. In this doctoral thesis, modulation methods suitable especially for series connected H-bridge multilevel inverters are discussed. A concept of duty cycle modulation is presented and its modification is proposed. These methods are compared with other well-known modulation schemes, such as space-vector pulse width modulation and carrier-based modulation schemes. The advantage of the modified duty-cycle modulation is its algorithmic simplicity. A similar mathematical formulation for the original duty cycle modulation is proposed. The modified duty cycle modulation is shown to produce well-formed phase-to-neutral voltages that have lower total harmonic distortion than the space-vector pulse width modulation and the duty cycle modulation. The space-vector-based solution and the duty cycle modulation, on the other hand, result in a better-quality line-to-line voltage and current waveform. The voltage of the DC links in the modules of the series-connected H-bridge inverter are shown to fluctuate while they are under load. The fluctuation causes inaccuracies in the voltage production, which may result in a failure of the flux estimator in the controller. An extension for upper-level modulation schemes, which changes the switching instants of the inverter so that the output voltage meets the reference voltage accurately regardless of the DC link voltages, is proposed. The method is shown to reduce the error to a very low level when a sufficient switching frequency is used. An appropriate way to organize the switching instants of the multilevel inverter is to make only one-level steps at a time. This causes restrictions on the dynamical features of the modulation schemes. The produced voltage vector cannot be rotated several tens of degrees in a single switching period without violating the above-mentioned one-level-step rule. The dynamical capabilities of multilevel inverters are analyzed in this doctoral thesis, and it is shown that the multilevel inverters are capable of operating even in dynamically demanding metal industry applications. In addition to the discussion on modulation schemes, an overvoltage in multilevel converter drives caused by cable reflection is addressed. The voltage reflection phenomenon in drives with long feeder cables causes premature insulation deterioration and also affects the commonmode voltage, which is one of the main reasons for bearing currents. Bearing currents, on the other hand, cause fluting in the bearings, which results in premature bearing failure. The reflection phenomenon is traditionally prevented by filtering, but in this thesis, a modulationbased filterless method to mitigate the overvoltage in multilevel drives is proposed. Moreover, the mitigation method can be implemented as an extension for upper-level modulation schemes. The method exploits the oscillations caused by two consecutive voltage edges so that the sum of the oscillations results in a mitigated peak of the overvoltage. The applicability of the method is verified by simulations together with experiments with a full-scale prototype.

St. Petersburg Court Chant and the Tradition of Eastern Slavic Church Singing

The present study examines the repertory of liturgical chant known as St. Petersburg Court Chant which emerged within the Imperial Court of St. Petersburg, Russia, and appeared in print in a number of revisions during the course of the 19th century, eventually to spread throughout the Russian Empire and even abroad. The study seeks answers to questions on the essence and composition of Court Chant, its history and liturgical background, and most importantly, its musical relationship to other repertories of Eastern Slavic chant. The research questions emerge from previous literary accounts of Court Chant (summarized in the Introduction), which have tended to be inaccurate and generally not based on critical research. The study is divided into eight main chapters. Chapter 1 provides a survey of the history of Eastern Slavic chant and the Imperial Court Chapel of St. Petersburg until 1917, with special emphasis on the history of singing traditional chant in polyphony, the status of the Court Chapel as a government authority, and its endeavours in publishing church music. Chapter 2 deals with the liturgical background of Eastern chant, the chant genres, and main repertories of Eastern Slavic chant. Chapter 3 concentrates on chant sources: it introduces the musical notations utilised, after which a typology of chant books is presented. The discussion continues with a survey of the sources of Court Chant and their content, the specimens selected for closer analysis, the comparative materials from other repertories, and ends with a commentary on some chant sources that have been excluded. The comparative sources include a specimen from around the beginning of the 12th century, a few manuscripts from the 17th century, and printed and manuscript chant books from the early 18th to early 20th century, covering the geographical area that delimits to the western Ukraine, Astrakhan, Nizhny Novgorod, and the Solovetsky Monastery. Chapter 4 presents the approach and methods used in the subsequent analytical comparisons. After a survey of the pitch organization of Eastern Slavic chant, the customary harmonization strategy of traditional chant polyphony is examined, according to which a method for meaningful analysis of the harmony is proposed. The method is based on the observation that the harmonic framework of chant polyphony derives from the standard pitch collection of monodic chant known as the Church Gamut, specific pitches of which form eight harmonic regions that behave like the usual tonalities of major and harmonic minor. Because of the considerable quantity of comparative chant forms, computer-assisted statistical methods are applied to the analysis of chant melodies. The primary chant forms and their respective comparative forms have been pre-processed into reduced chant prototypes and divided into redactions. The analyses are carried out by measuring the formal dissimilarities of the primary chant forms of the Court Chant repertory against each comparative form, and also by measuring the reciprocal dissimilarities of all chant versions in a redaction, the results of which are subjected to agglomerative hierarchical clustering in order to find out how the chant forms relate to each other. The dissimilarities are determined by applying a metric dissimilarity function that is based on the Levenshtein Distance. Chapter 5 provides the melodic and harmonic analyses of generic chants (chants used for multiple texts of different lengths), i.e., chants for stichera samoglasny and troparia, Chapter 6 of pseudo-generic chants (chants that are used for multiple texts but with certain restrictions), i.e., chants for heirmoi, prokeimena, and three other hymns, and Chapter 7 of non-generic chants, covering nine chants that in the Court repertory are not shared by multiple texts. The results are summarized and evaluated in Chapter 8. Accordingly, it can be established that, contrary to previous conceptions, melodically, Court Chant is in effect a full part of the wider Eastern Slavic chant tradition. Even if it is somewhat detached from the chant versions of the Synodal square-note chant books and the local tradition of Moscow, it is particularly close to chant forms of East Ukraine and some vernacular repertories from Russia. Respectively, the harmonization strategies of Court Chant do not show significant individuality in comparison with those of the available polyphonic comparative sources, the main difference being the part-writing, which generally conforms to western common practice standard, whereas the deviations from this tend to be more significant in other analysed repertories of polyphonic chant. Thus, insofar as the subsequent prevalence of Court Chant is not based on its forceful dissemination by authorities (as suggested in previous literature but for which little tangible evidence could be found in Chapter 1), in the present author’s interpretation, Court Chant attained its dominance principally because musically it was considered sufficiently traditional, and as a chant body supported by the government, was conveniently available in print in serviceable harmonizations.

Non-Markovian Dynamics and the Quantum-To-Classical Transition for Quantum Brownian Motion

In this Thesis I discuss the dynamics of the quantum Brownian motion model in harmonic potential. This paradigmatic model has an exact solution, making it possible to consider also analytically the non-Markovian dynamics. The issues covered in this Thesis are themed around decoherence. First, I consider decoherence as the mediator of quantum-to-classical transition. I examine five different definitions for nonclassicality of quantum states, and show how each definition gives qualitatively different times for the onset of classicality. In particular I have found that all characterizations of nonclassicality, apart from one based on the interference term in the Wigner function, result in a finite, rather than asymptotic, time for the emergence of classicality. Second, I examine the diverse effects which coupling to a non-Markovian, structured reservoir, has on our system. By comparing different types of Ohmic reservoirs, I derive some general conclusions on the role of the reservoir spectrum in both the short-time and the thermalization dynamics. Finally, I apply these results to two schemes for decoherence control. Both of the methods are based on the non-Markovian properties of the dynamics.

Modeling and Control of the Power Conversion Unit in a Solid Oxide Fuel Cell Environment

In this doctoral thesis, a power conversion unit for a 10 kWsolid oxide fuel cell is modeled, and a suitable control system is designed. The need for research was identified based on an observation that there was no information available about the characteristics of the solid oxide fuel cell from the perspective of power electronics and the control system, and suitable control methods had not previously been studied in the literature. In addition, because of the digital implementation of the control system, the inherent characteristics of the digital system had to be taken into account in the characteristics of the solid oxide fuel cell (SOFC). The characteristics of the solid oxide fuel cell as well the methods for the modeling and control of the DC/DC converter and the grid converter are studied by a literature survey. Based on the survey, the characteristics of the SOFC as an electrical power source are identified, and a solution to the interfacing of the SOFC in distributed generation is proposed. A mathematical model of the power conversion unit is provided, and the control design for the DC/DC converter and the grid converter is made based on the proposed interfacing solution. The limit cycling phenomenon is identified as a source of low-frequency current ripple, which is found to be insignificant when connected to a grid-tied converter. A method to mitigate a second harmonic originating from the grid interface is proposed, and practical considerations of the operation with the solid oxide fuel cell plant are presented. At the theoretical level, the thesis discusses and summarizes the methods to successfully derive a model for a DC/DC converter, a grid converter, and a power conversion unit. The results of this doctoral thesis can also be used in other applications, and the models and methods can be adopted to similar applications such as photovoltaic systems. When comparing the results with the objectives of the doctoral thesis, we may conclude that the objectives set for the work are met. In this doctoral thesis, theoretical and practical guidelines are presented for the successful control design to connect a SOFC-based distributed generation plant to the utility grid.

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.

Modal testing and numerical modeling of the dynamic properties of layered sheet-steel structure

Recently, due to the increasing total construction and transportation cost and difficulties associated with handling massive structural components or assemblies, there has been increasing financial pressure to reduce structural weight. Furthermore, advances in material technology coupled with continuing advances in design tools and techniques have encouraged engineers to vary and combine materials, offering new opportunities to reduce the weight of mechanical structures. These new lower mass systems, however, are more susceptible to inherent imbalances, a weakness that can result in higher shock and harmonic resonances which leads to poor structural dynamic performances. The objective of this thesis is the modeling of layered sheet steel elements, to accurately predict dynamic performance. During the development of the layered sheet steel model, the numerical modeling approach, the Finite Element Analysis and the Experimental Modal Analysis are applied in building a modal model of the layered sheet steel elements. Furthermore, in view of getting a better understanding of the dynamic behavior of layered sheet steel, several binding methods have been studied to understand and demonstrate how a binding method affects the dynamic behavior of layered sheet steel elements when compared to single homogeneous steel plate. Based on the developed layered sheet steel model, the dynamic behavior of a lightweight wheel structure to be used as the structure for the stator of an outer rotor Direct-Drive Permanent Magnet Synchronous Generator designed for high-power wind turbines is studied.

Lightweight, liquid-cooled, direct-drive generator for highpower wind turbines: motivation, concept, and performance

Thesis: A liquid-cooled, direct-drive, permanent-magnet, synchronous generator with helical, double-layer, non-overlapping windings formed from a copper conductor with a coaxial internal coolant conduit offers an excellent combination of attributes to reliably provide economic wind power for the coming generation of wind turbines with power ratings between 5 and 20MW. A generator based on the liquid-cooled architecture proposed here will be reliable and cost effective. Its smaller size and mass will reduce build, transport, and installation costs. Summary: Converting wind energy into electricity and transmitting it to an electrical power grid to supply consumers is a relatively new and rapidly developing method of electricity generation. In the most recent decade, the increase in wind energy’s share of overall energy production has been remarkable. Thousands of land-based and offshore wind turbines have been commissioned around the globe, and thousands more are being planned. The technologies have evolved rapidly and are continuing to evolve, and wind turbine sizes and power ratings are continually increasing. Many of the newer wind turbine designs feature drivetrains based on Direct-Drive, Permanent-Magnet, Synchronous Generators (DD-PMSGs). Being low-speed high-torque machines, the diameters of air-cooled DD-PMSGs become very large to generate higher levels of power. The largest direct-drive wind turbine generator in operation today, rated just below 8MW, is 12m in diameter and approximately 220 tonne. To generate higher powers, traditional DD-PMSGs would need to become extraordinarily large. A 15MW air-cooled direct-drive generator would be of colossal size and tremendous mass and no longer economically viable. One alternative to increasing diameter is instead to increase torque density. In a permanent magnet machine, this is best done by increasing the linear current density of the stator windings. However, greater linear current density results in more Joule heating, and the additional heat cannot be removed practically using a traditional air-cooling approach. Direct liquid cooling is more effective, and when applied directly to the stator windings, higher linear current densities can be sustained leading to substantial increases in torque density. The higher torque density, in turn, makes possible significant reductions in DD-PMSG size. Over the past five years, a multidisciplinary team of researchers has applied a holistic approach to explore the application of liquid cooling to permanent-magnet wind turbine generator design. The approach has considered wind energy markets and the economics of wind power, system reliability, electromagnetic behaviors and design, thermal design and performance, mechanical architecture and behaviors, and the performance modeling of installed wind turbines. This dissertation is based on seven publications that chronicle the work. The primary outcomes are the proposal of a novel generator architecture, a multidisciplinary set of analyses to predict the behaviors, and experimentation to demonstrate some of the key principles and validate the analyses. The proposed generator concept is a direct-drive, surface-magnet, synchronous generator with fractional-slot, duplex-helical, double-layer, non-overlapping windings formed from a copper conductor with a coaxial internal coolant conduit to accommodate liquid coolant flow. The novel liquid-cooling architecture is referred to as LC DD-PMSG. The first of the seven publications summarized in this dissertation discusses the technological and economic benefits and limitations of DD-PMSGs as applied to wind energy. The second publication addresses the long-term reliability of the proposed LC DD-PMSG design. Publication 3 examines the machine’s electromagnetic design, and Publication 4 introduces an optimization tool developed to quickly define basic machine parameters. The static and harmonic behaviors of the stator and rotor wheel structures are the subject of Publication 5. And finally, Publications 6 and 7 examine steady-state and transient thermal behaviors. There have been a number of ancillary concrete outcomes associated with the work including the following. X Intellectual Property (IP) for direct liquid cooling of stator windings via an embedded coaxial coolant conduit, IP for a lightweight wheel structure for lowspeed, high-torque electrical machinery, and IP for numerous other details of the LC DD-PMSG design X Analytical demonstrations of the equivalent reliability of the LC DD-PMSG; validated electromagnetic, thermal, structural, and dynamic prediction models; and an analytical demonstration of the superior partial load efficiency and annual energy output of an LC DD-PMSG design X A set of LC DD-PMSG design guidelines and an analytical tool to establish optimal geometries quickly and early on X Proposed 8 MW LC DD-PMSG concepts for both inner and outer rotor configurations Furthermore, three technologies introduced could be relevant across a broader spectrum of applications. 1) The cost optimization methodology developed as part of this work could be further improved to produce a simple tool to establish base geometries for various electromagnetic machine types. 2) The layered sheet-steel element construction technology used for the LC DD-PMSG stator and rotor wheel structures has potential for a wide range of applications. And finally, 3) the direct liquid-cooling technology could be beneficial in higher speed electromotive applications such as vehicular electric drives.