Abstract | U ovoj disertaciji predstavljena je iterativna metoda za proračun magnetskog polja u zračnom rasporu zasićenih sinkronih strojeva s površinskim trajnim magnetima bazirana na analitičko-numeričkom modelu. Visoke razine zasićenja željezne jezgre stroja utječu na harmonički sastav i amplitudu valnog oblika indukcije u zračnom rasporu, što u krajnjoj liniji ima utjecaj i na ostale proračunate veličine kao što su napon, moment, gubici itd. Analitički modeli koji pretpostavljaju beskonačno permeabilno željezo, bez obzira na svoju matematičku točnost, ne mogu opisati efekte koji nastaju zbog magnetskog zasićenja željezne jezgre te je dobar matematički opis nelinearnih magnetskih krugova nužan za precizan elektromagnetski proračun električnih strojeva. Metoda opisana u ovom radu predlaže novi pristup u kombiniranju konformnih preslikavanja i magnetskih ekvivalentnih krugova te je implementirana za proračun zasićenih strojeva s trajnim površinskim magnetima s razlomljenim i cjelobrojnim namotom. Metoda uzima u obzir nazubljenost statora i rotora, zasićenje željezne jezgre i promjenu zasićenja s položajem rotora. Uz poznatu prostornu raspodjelu vektorskog magnetskog potencijala u glatkom zračnom rasporu, konformno preslikavanje omogućava proračun raspodjele vektorskog magnetskog potencijala u mnogo kompleksnijoj geometrijskoj poddomeni kao što je nazubljeni zračni raspor. Razvijena metoda koristi računalno učinkovitu transformaciju zračnog raspora kojom se domena zračnog raspora sastavlja od niza manjih poddomena širine pola utorskoga koraka te u usporedbi s do sada razvijenim metodama za proračun rotacijskih strojeva koje koriste konformna preslikavanja i koje mogu biti računalno zahtjevne nudi veliku uštedu vremena pri proračunu. Magnetsko zasićenje željeza u modelu opisano je dodatnim strujnicama u domeni zračnog raspora, utorima i međupolnom prostoru između magneta. Te strujnice stvaraju magnetomotornu silu koja ima jednak iznos kao skalarni magnetski pad napona na željeznim dijelovima stroja. Magnetski padovi napona proračunati su koristeći metodu magnetskih ekvivalentnih krugova. Model također u obzir uzima promjenu magnetizacije magneta zbog utjecaja vanjskog magnetskog polja. Predložena metoda implementirana je na primjerima dvaju strojeva s 36 utora i 6 polova s različitim razinama zasićenja u jarmu statora i zubima. Metoda je također testirana na zasićenom stroju s 12 utora i 10 polova s razlomljenim koncentriranim namotom te stroju s 33 utora i 8 polova s razlomljenim distribuiranim namotom. Svi dobiveni rezultati uspoređeni su s rezultatima tranzijentnih simulacija metodom konačnih elemenata. |

Abstract (english) | Precise mathematical models of electrical machines are essential for their reliable design. In the design process it is required to determine the motor dimensions and relate them using the model to motor characteristic quantities such as inductances, winding resistance, torque characteristic, power losses etc. In the case of permanent magnet motors in no-load and on-load conditions, mathematical models are commonly used in the design stage to predict the induced voltage and torque waveforms. For calculation of the induced voltage and torque as well as other motor characteristic quantities, finite element method is commonly used. In order to obtain the best possible characteristics while achieving the best utilization of materials, electric machine design requires the use of mathematical optimization which is often computationally expensive and time-consuming if the model is complex. Hence for modelling of electrical machines, analytical and combined analytical-numerical models are being developed due to their better performance compared to finite element method with regard to time, while still providing satisfying accuracy. Such models can be used as a good tool for better understanding of laws that are applicable in the design of electrical machines and can usually be classified into one of the three main categories: - subdomain models that directly calculate the air gap field by solving partial differential equations, - models based on magnetic equivalent circuits, - models that utilize the properties of conformal mapping. In addition to the above methods, for modelling of electrical machines, the methods that are based on conformal mapping and magnetic equivalent circuits are developed and they combine good properties of both methods. Although they offer a number of advantages, to maintain simplicity they often neglect certain effects that affect the magnetic field distribution in the machine, and thus all characteristic values such as voltage, torque, etc. This thesis presents an iterative method for calculation of magnetic field in the air gap of a saturated surface permanent magnet machine in on-load and no-load conditions based on combined analytical-numerical model. In the case of high saturation in a motor, the air gap flux density waveform is affected in terms of magnitude and harmonic content, which then has the influence on all the other calculated quantities. The analytical models that assume infinitely permeable iron, no matter how mathematically correct they are, cannot capture correctly the effect of saturated iron in the motor. The presence of iron in magnetic circuits of electrical machines with its inherent non-linearity is an unavoidable technical fact that in most cases should not be neglected. The method described in this thesis proposes a novel approach in combining conformal mapping and magnetic equivalent circuits and is implemented for saturated slotted surface permanent magnet machines with both fractional slot and integer slot windings. The model takes into account slotting, saturation of the iron core and its variation due to rotor movement. With the known field distribution in the circular slotless air gap, conformal mapping enables calculation of the field distribution in the slotted air gap, which is a more complex geometrical subdomain. Commercial software tools for numerical conformal transformation have enabled more accurate mapping of analytical magnetic field solutions from the initially developed analytical transformation, but at same time they have prolonged the calculation time due to the increased complexity of geometric configurations which are transformed. Along with a detailed description of the application of conformal mappings for magnetic field calculation, this dissertation presents a method for computationally efficient conformal mapping of magnetic field solutions that makes this method more appropriate for calculation of electrical machines compared to the most developed numerical models based on a similar principle. Simplified conformal mapping of the air gap using transformation of only one half of a slot pitch is developed, which significantly reduces the calculation time in comparison with conventional conformal mapping models because the geometric shape for transformation has less polygon edges which are then used for reconstruction of the transformation of the complete machine air gap. Magnetic saturation is taken into account by using additional point wires in the slots and inter-polar region between magnets that will create a magnetomotive force equal to the scalar magnetic voltage drop across the iron. The magnetic voltage drop across the iron is calculated using magnetic equivalent circuit. The model also takes into account variation of permanent magnet magnetization due to external magnetic field. The proposed method has been implemented and evaluated on selected examples of 36 slot 6 pole motors with radially magnetized magnets and different levels of saturation in teeth and stator yoke. The method has been also tested on a saturated 12 slot, 10 pole surface permanent magnet machine with radially magnetized magnets and non-overlapping concentrated winding and 33 slot, 8 pole machine with bread-loaf magnets and fractional slot distributed winding. Comparisons are made with results obtained using time-stepping finite element analysis with rotor motion for flux density, back-EMF and torque waveforms, magnet and iron losses as well as d and q axis inductances, cross-saturation inductance and permanent magnet flux linkage in both axes. The scientific contributions of this thesis can have a wide application for modelling of electrical machines. Some suggestions for further research and development of the method are: - Development of models based on conformal mappings and magnetic equivalent circuits for other topologies of electrical machines such as synchronous reluctance machines or synchronous machines with internal permanent magnets. - Application of the saturation model using magnetic voltage drops on the analytical subdomain models, which would enable faster execution of calculations due the absence of conformal mappings. The methodology for calculation of electrical machines developed as a part of this thesis is using a new approach compared to the previously described methods in the literature. Listed below are the original scientific contributions of the thesis: 1. Model of a saturated surface permanent magnet synchronous machine in which magnetic voltage drops across the stator and rotor iron are replaced with additional point wires in the slots and inter-polar region between magnets This scientific contribution includes modelling of saturation effect in the machine. The equation for analytical solution of magnetic vector potential is derived under assumption of infinitely permeable iron core, while in reality the iron has finite permeability. In order to model iron with finite permeability, it is necessary to set the equivalent magnetization currents which correspond to the difference between magnetization currents of the finite and infinitely permeable iron core. Magnetization of the iron caused by saturation is modelled by using point current sources placed in the air gap and the slots of the machine. Added point currents have the same effect on the magnetic field in the air gap as the magnetic voltage drops across the machine iron with finite permeability, which is placed outside the limits of the air gap. Since values of magnetic voltage drops across iron flux tubes depend on the air gap flux, it is clear that the calculation must be conducted iteratively. An iterative method for calculation of equivalent magnetization point currents varies predefined magnetic vector potentials until the relative difference between two consecutive iterations is smaller than a predefined tolerance. The currents of point wires which are used for modelling magnetic saturation have the same values as magnetic voltage drops across iron core reluctances and can be obtained from the magnetic equivalent circuits. Each flux tube in the machine is presented with a separate magnetic circuit and links between all magnetic equivalent circuits are established by means of analytical field solution which provides the values of flux for the flux sources. 2. Model of the entire air gap and slot contour of a surface synchronous permanent magnet machine based on conformal mapping of a reduced geometry consisting of one half of the slot pitch This scientific contribution includes a computationally efficient conformal mapping of the geometry of the entire air gap. Considering the layout of slots, it is sufficient to transform the air gap and slot contour within only one half of the slot pitch, which significantly reduces calculation time in comparison with calculation when an entire pole pitch of the air gap is transformed. The transformation transforms only geometry and since the geometry of the slot pitches repeats with every half slot pitch, it is sufficient to perform this geometric transformation within local coordinates of the half slot pitch. The air gap should be subdivided into half-slot sectors in the physical domain, which are transformed into rectangular sectors in the canonical domain. Using these fundamental transformed elements (now rectangular in the canonical domain), a smooth air gap within the angular span that corresponds to the number of poles in one repetitive winding pattern is assembled. After forming the complete rectangular canonical domain, exponential conformal mapping is conducted in order to obtain the final circular canonical domain for magnetic field calculation. The complete canonical domain consists of even number of poles after which the winding pattern repeats. 3. Application of the developed model of a synchronous surface permanent magnet machine on the examples of machines with integer slot distributed windings and fractional slot concentrated and distributed windings and verification of the model by comparing the results with results obtained by numerical finite element method This scientific contribution includes calculation of the air gap flux density waveforms, induced voltage and torque waveforms as well as calculation of losses, inductances and magnet flux linkages using the proposed model of a saturated machine. The obtained results show very good agreement with the results of finite element method with included saturation. The comparison with results obtained by using unsaturated finite element model and deviation of those results from the ones obtained with inclusion of saturation confirm the importance of modelling the iron saturation effect. The thesis is divided into seven chapters. Chapter 1.- Introduction describes the motivation for development of the presented method. Original scientific contributions are listed and described as well as the structure of the thesis. Chapter 2.- Literature Review gives an overview of existing literature in the field. Conventional methods for modelling of machines with permanent magnets are listed. Special attention is given to the methods based on conformal mappings and magnetic equivalent circuits. Chapter 3.- Air Gap Magnetic Field Analysis Using Conformal Mapping provides a general description of conformal mappings and the principle of their application to the magnetic field calculation in the slotted air gap. The second scientific contribution of this thesis is featured, which includes the computationally efficient forming of the domain for mapping the analytical magnetic field solution. The models of radial, parallel and bread-loaf magnets are presented as well as the models of single-layered, non-overlapping concentrated winding and double-layered distributed winding. Chapter 4.- Modelling of Iron Magnetic Saturation Effect describes the first scientific contribution of the thesis. The expressions for equivalent magnetization currents and variation of permanent magnet magnetization are derived. The iterative equivalent magnetic circuit method which is used for calculation of magnetic voltage drop across the machine iron core is described. Chapter 5.- No-Load and On-Load Analysis explains the third scientific contribution of this thesis. Detailed description of the flux density, back-EMF, cogging torque and total torque calculations using the proposed method is given. The computation of magnet flux linkages and inductances using conformal mapping in combination with the frozen permeability method is described. This chapter also contains the detailed description of copper and iron loss calculation as well as the derivation of Helmholtz differential equation solution for calculation of magnet losses . Chapter 6.- Results and Comparison With Finite Element Analysis comparisons have been made with 2D nonlinear time-stepping finite element calculations for four different surface permanent magnet machines. In order to emphasize the importance of modelling the magnetic saturation, comparisons have also been made with FEM calculations of machines with infinitely permeable iron core. Chapter 7.- Conclusion is the last chapter and it provides short analysis of the presented method and summarizes the main contributions. Proposals for further research are given. |