Frequency domain models for power electronic circuits are either based on iterative techniques such as Newton's method or linearised around an operating point. Iterative frequency domain models provide great accuracy as they are capable of calculating the exact switching instants of the device. On the other hand, the accuracy of a linearised frequency domain model relies on the magnitude of input waveform to be small so that the circuit's operating point does not vary or varies very little. However, an important advantage of a linearised model is its ability to provide insight into waveform distortion interaction, more specifically, the frequency cross-coupling around a power electronic circuit. In general, a linearised model for harmonic analysis would not normally include the description of feedback control. Likewise a linearised model for control analysis would usually disregard frequency interactions above the fundamental (or the most significant component); that is assuming the cross-coupling between harmonic frequencies does not affect the dynamics of control. However, this thesis proposes that a linearised model for control analysis shall also include the complete description of frequency cross-coupling between harmonics to produce the correct dynamic response. This thesis presents a harmonic state-space (HSS) model of an HVdc converter that incorporates the full effect of varying switching instants, both through control and commutation period dynamics, while remaining within the constraints of a linear time-invariant (LTI) system. An example is given using the HSS model to explain how a close to fifth harmonic resonance contributes to the dominant system response through the frequency cross-coupling of the converter and the controller feedback loop. The response of the system is validated against a time domain model built in PSCAD/EMTDC, and more importantly, the correct response cannot be produced without including the harmonic interactions beyond the fundamental frequency component.

abstract: In this work, we focused on the stability and reducibility of quasi-periodic systems. We examined the quasi-periodic linear Mathieu equation of the form x ̈+(ä+ϵ[cost+cosùt])x=0 The stability of solutions of Mathieu's equation as a function of parameter values (ä,ϵ) had been analyzed in this work. We used the Floquet type theory to generate stability diagrams which were used to determine the bounded regions of stability in the ä-ù plane for fixed ϵ. In the case of reducibility, we first applied the Lyapunov- Floquet (LF) transformation and modal transformation, which converted the linear part of the system into the Jordan form. Very importantly, quasi-periodic near-identity transformation was applied to reduce the system equations to a constant coefficient system by solving homological equations via harmonic balance. In this process we obtained the reducibility/resonance conditions that needed to be satisfied to convert a quasi-periodic system to a constant one. / Dissertation/Thesis / M.S.Tech Engineering 2012

On s’intéresse dans cette thèse à des systèmes couplés de type champ moyen en étudiant l’existence de l’état de synchronisation qui se caractérise par une distance uniformément bornée dans le temps entre chaque paire de composantes d’une solution. L’étude se base sur une méthode perturbative. Néanmoins les résultats obtenus ne sont pas évidents dans le cas non-perturbé. En outre dans le cas où le système couplé est périodique et grâce au Théorème du point fixe on montre l’existence d’une solution périodique sur le tore. L’étude de stabilité et de stabilité exponentielle est établie dans le cas linéaire et appliquée à ce type de systèmes couplés / We study in this thesis a class of a perturbed interconnected mean-field system, also known as a coupled systems. Under some assumptions we prove the existence of an invariant open set by the flow of the perturbed system ; in other word, we prove that the distance between the components of an orbit is uniformly bounded, this property is also called synchronization. We use the perturbation method to obtain the result. However the result is not trivial for the not perturbed system. We use the fixed point theorem to prove the existence of a periodic orbit in the torus. We study in addition the stability and the exponential stability of such systems by studying the stability of a linear systems.

Dans un contexte de densification des villes et de leurs réseaux de transport, les gens sont de plus en plus exposés au bruit. Ainsi, le résultat de l'étude d'impact vibro-acoustique joue un rôle primordial dans l'expansion du réseau ferroviaire. L'une des principales sources est le bruit de roulement : La rugosité de la surface de la roue et du rail produit un déplacement imposé sur ces derniers. Ce déplacement entraine une réponse vibratoire des roues et de la voie ferrée et leurs rayonnements acoustiques. Cette thèse propose une amélioration de la modélisation vibro-acoustique de la voie ferrée.Pour la réponse vibratoire, le coté infini de la voie et sa déformation dans les 3 dimensions rendent les modèles analytiques et les éléments finis non-optimales dans la gamme de fréquence de l’audible. La méthode élément fini semi-périodique (SAFEM) est utilisée dans cette thèse pour modéliser une voie à support continue. Elle est ensuite couplée au théorème de Floquet pour modéliser une voie à support périodique. Cependant, cette technique génère des problèmes numériques qui ont imposé un algorithme adapté. La méthode d'Arnoldi du second ordre (SOAR) est utilisée avant de résoudre l'équation SAFEM permet de résoudre ces problèmes ainsi qu’apporter la stabilité requise. Des comparaisons avec d’autres modèles et des données expérimentales permettent de valider la méthode.Pour le rayonnement acoustique, la simulation de grand domaine en haute fréquence rendent inadapté l'utilisation de techniques conventionnelles (FEM, BEM, ...). La méthode proposée ici : la théorie variationnelle du rayon complexe est particulièrement bien adaptée à ce cas. Les principales caractéristiques de l'approche VTCR sont l'utilisation d'une formulation faible du problème acoustique, qui permet de considérer automatiquement les conditions limites entre sous-domaines. Ensuite, l'utilisation d'une répartition intégrale des ondes planes dans toutes les directions permet de simuler le champ acoustique. Les inconnues du problème sont leurs amplitudes. Cette méthode qui a déjà montré son efficacité pour les domaines fermés a été étendue au domaine ouvert et couplée à la réponse vibratoire. Des comparaisons avec des solutions analytiques et des simulations FEM à basse fréquence permettent de valider la méthode. / In a context of urban and transport network densification, people are increasingly exposed to noise. Consequently, the result of vibro-acoustic impact assessment has a pivotal role in rail network expansion. One of the main sources is the rolling noise: Roughness on the wheel and rail surface produce an imposed displacement one the both. This last, generates vibrational response of wheels and the railway track and their acoustic radiation. This PhD thesis presents some improvements of the vibro-acoustic railway track modelling.Concerning vibrational response, the infinite dimension in the longitudinal direction of the track and its deformation in the 3 dimensions, make the analytical models and finite elements non-optimal. The Semi-analytical finite element method (SAFEM), used in this thesis, is particularly well adapted in this case. Firstly, it is used to model railway track on a continuous support. Then, it is coupled with Floquet theorem to model tracks with a periodic support. However, this technique suffers from numerical problems that imposed an adapted algorithm. The second-order Arnoldi method (SOAR) is used to tackle them. This reduction allows to eliminate critical values improving the robustness of the method. Comparison with existing techniques and experimental results validate this model.Concerning acoustic radiation, big domains simulations at high frequency are almost unfeasible when using conventional techniques (FEM, BEM,…). The method used in this thesis, the Variational theory of complex ray (VTCR) is particularly well adapted to these cases. The principal features of VTCR approach are the use of a weak formulation of the acoustic problem, which allows to consider automatically boundary conditions between sub-domains. Then, the use of an integral repartition of plane waves in all the direction allow to simulate the acoustic field. The unknowns of the problem are their amplitudes. This method well assessed for closed domain, has been extended to open domain and coupled to vibrational response of the rail. Comparison with analytic solution and FEM simulation at low frequency allow to validate the method.Coupling these both methods allowed to simulate complex real life vibro-acoustic scenarios. Result of different railway tracks are presented and validated