In this thesis I present new insights into aspects of Peter Guthrie Tait's life and work, derived principally from largely-unexplored primary source material: Tait's scrapbook, the Tait–Maxwell school-book and Tait's pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen's College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin. Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait's life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait's surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey's 1828 work, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey.

Les équations de Maxwell en régime harmonique comportent plusieurs difficultés lorsque la fréquence est élevée. On peut notamment citer le fait que leur formulation variationnelle n’est pas définie positive et l’effet de pollution qui oblige à utiliser des maillages très fins, ce qui rend problématique la construction de solveurs itératifs. Nous proposons une stratégie de solution précise et rapide, qui associe une discrétisation par des éléments finis d’ordre élevé à des préconditionneurs de type décomposition de domaine. La conception, l’implémentation et l’analyse des deux méthodes sont assez difficiles pour les équations de Maxwell. Les éléments finis adaptés à l’approximation du champ électrique sont les éléments finis H(rot)-conformes ou d’arête. Ici nous revisitons les degrés de liberté classiques définis par Nédélec, afin d’obtenir une expression plus pratique par rapport aux fonctions de base d’ordre élevé choisies. De plus, nous proposons une technique pour restaurer la dualité entre les fonctions de base et les degrés de liberté. Nous décrivons explicitement une stratégie d’implémentation qui a été appliquée dans le langage open source FreeFem++. Ensuite, nous nous concentrons sur les techniques de préconditionnement du système linéaire résultant de la discrétisation par éléments finis. Nous commençons par la validation numérique d’un préconditionneur à un niveau, de type Schwarz avec recouvrement, avec des conditions de transmission d’impédance entre les sous-domaines. Enfin, nous étudions comment des préconditionneurs à deux niveaux, analysés récemment pour l’équation de Helmholtz, se comportent pour les équations de Maxwell, des points de vue théorique et numérique. Nous appliquons ces méthodes à un problème à grande échelle qui découle de la modélisation d’un système d’imagerie micro-onde, pour la détection et le suivi des accidents vasculaires cérébraux. La précision et la vitesse de calcul sont essentielles dans cette application. / The time-harmonic Maxwell’s equations present several difficulties when the frequency is large, such as the sign-indefiniteness of the variational formulation, the pollution effect and the problematic construction of iterative solvers. We propose a precise and efficient solution strategy that couples high order finite element (FE) discretizations with domain decomposition (DD) preconditioners. High order FE methods make it possible for a given precision to reduce significantly the number of unknowns of the linear system to be solved. DD methods are then used as preconditioners for the iterative solver: the problem defined on the global domain is decomposed into smaller problems on subdomains, which can be solved concurrently and using robust direct solvers. The design, implementation and analysis of both these methods are particularly challenging for Maxwell’s equations. FEs suited for the approximation of the electric field are the curl-conforming or edge finite elements. Here, we revisit the classical degrees of freedom (dofs) defined by Nédélec to obtain a new more friendly expression in terms of the chosen high order basis functions. Moreover, we propose a general technique to restore duality between dofs and basis functions. We explicitly describe an implementation strategy, which we embedded in the open source language FreeFem++. Then we focus on the preconditioning of the linear system, starting with a numerical validation of a one-level overlapping Schwarz preconditioner, with impedance transmission conditions between subdomains. Finally, we investigate how two-level preconditioners recently analyzed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view. We apply these methods to the large scale problem arising from the modeling of a microwave imaging system, for the detection and monitoring of brain strokes. In this application accuracy and computing speed are indeed of paramount importance.

In the first part of the master’s thesis dealing with the increasing efficiency of induction motors there are briefly presented basic information about induction motors, followed by an overview of the losses of induction motors. The next part deals with the ways to increase efficiency of induction motors without increasing tooling costs. The practical part consists of four measurements of four induction motors, with their various mechanical adjustments to make comparing benefits of these modifications possible. The measured results are compared by a finite element method in Maxwell 2D Design program, in which the same motors are simulated as measured. Theoretical knowledge about the increase of efficiency is practically applied while being implemented in the simulations.

The master’s thesis Improvement power parameter of small induction motors deals with issues of lowering the losses of small induction motors. The first part introduces with design and principles of operation of induction motors. Also introduces to theoretical problematic of losses, their lowering and measuring. In the practical part there are results of the measuring the losses in the induction motor ATAS Elektromotory Náchod a.s. T22VT512 (71-0512). There are proposed methods of increasing the efficiency of induction motor due to measuring and their verification in the Maxwell software. The last part is dedicated to measuring the losses of prototype motor from ATAS and comparison of results with previous motor.

<p dir="ltr">The short lifespan of capacitors in power electronics converters is a significant challenge. These capacitors, often electrolytic, are vital for voltage smoothing and frequency filtering. However, their susceptibility to heat, ripple current, and aging can lead to premature faults. This can cause issues like output voltage instability and short circuits, ultimately resulting in catastrophic failure and system shutdown. Capacitors are responsible for 30% of power electronics failures.</p><p dir="ltr">To tackle this challenge, scientists, researchers, and engineers are exploring various approaches detailed in technical literature. These include exploring alternative capacitor technologies, implementing active and passive cooling solutions, and developing advanced monitoring techniques to predict and prevent failures. However, these solutions often come with drawbacks such as increased complexity, reduced efficiency, or higher upfront costs. Additionally, research in material science is ongoing to develop corrosion-resistant capacitors, but such devices are not readily available.</p><p dir="ltr">This dissertation presents a capacitorless solution for dc-dc and dc-ac converters. The proposed solution involves harnessing parasitic elements and integrating them as intrinsic components in power converter technology. This approach holds the promise of enhancing power electronics reliability ratings, thereby facilitating breakthroughs in electric vehicles, compact power processing units, and renewable energy systems. The central scientific premise of this proposal is that the capacitance requirement in a power converter can be met by deliberately augmenting parasitic components.</p><p dir="ltr">Our research hypothesis that incorporating high dielectric material-based thin-films, fabricated using nanotechnology, into planar magnetics will enable the development of a family of capacitorless electronic converters that do not rely on discrete capacitors. This innovative approach represents a departure from the traditional power converter schemes employed in industry.</p><p dir="ltr">The first family of converters introduces a novel capacitorless solid-state power filter (SSPF) for single-phase dc-ac converters. The proposed configuration, comprising a planar transformer and an H-bridge converter operating at high frequency, generates sinusoidal ac voltage without relying on capacitors. Another innovative dc-ac inverter design is the twelve step six-level inverter, which does not incorporate capacitors in its structure.</p><p dir="ltr">The second family of capacitorless topologies consists of non-isolated dc-dc converters, namely the buck converter and the buck-boost converter. These converters utilize alternative materials with high dielectric constants, such as calcium copper titanate (CCTO), to intentionally enhance specific parasitic components, notably inter capacitance. This innovative approach reduces reliance on external discrete capacitors and facilitates the development of highly reliable converters.</p><p dir="ltr">The study also includes detailed discussions on the necessary design specifications for these parasitic capacitors. Furthermore, comprehensive finite element analysis solutions and detailed circuit models are provided. A design example is presented to demonstrate the practical application of the proposed concept in electric vehicle (EV) low voltage side dc-dc power converters used to supply EVs low voltage loads.</p>

Induction machine; outer rotor; water pump; thermal network; thermal analysis; optimization; electromagnetic model; finite element method; thermal insulation class