Circuit theory via algebraic topology
We are proposing a new formulation of circuit theory, taking in consideration its physical distribution in the space. For doing this we will use some concepts of the algebraic topology. Names as Hermann Weyl and Steve Smale did important contributions showing these connections between the theory of...
| Main Author: | ABATH, Leonardo Lopes |
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| Other Authors: | LEANDRO, Eduardo Shirlippe Goes |
| Format: | masterThesis |
| Language: | eng |
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Universidade Federal de Pernambuco
2019
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https://repositorio.ufpe.br/handle/123456789/31348 |
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ir-123456789-313482019-10-26T06:00:42Z Circuit theory via algebraic topology ABATH, Leonardo Lopes LEANDRO, Eduardo Shirlippe Goes http://lattes.cnpq.br/8245668136909160 http://lattes.cnpq.br/0559184209749319 Matemática Topologia algébrica We are proposing a new formulation of circuit theory, taking in consideration its physical distribution in the space. For doing this we will use some concepts of the algebraic topology. Names as Hermann Weyl and Steve Smale did important contributions showing these connections between the theory of circuits and the theory of algebraic topology. In this work, we will go to consider an electrical circuit as a graph or as a one-dimensional complex, where the domain of the boundary operator ∂ is the vector space C₁ generated by the branches (wires of the circuit) and its codomain is the vector space C₀ generated by the nodes. In chapter 3, the Kirchhoff ’s current law will be reformulate to the concise formula ∂I = 0 and the Kirchhoff ’s potential law will be reformulate to the concise formula V = −dΦ, where d : C₀ → C₁ is the coboundary map. The methods of mesh-current and node-potential are also discussed in this chapter, as well as a conclusive analysis of the existence and uniqueness of solutions for the electric circuit equations too is realized. In chapter 4 we will study some alternative methods for solving electric circuit equations. The Weyl’s method makes use of orthogonal projection operators and this method is summarized by the formula π = σ(sZσ)⁻¹sZ. The Kirchhoff’s method uses graph theory to find the values of voltages and electric currents and will be given by pλ = R⁻¹ΣᴛQᴛpᴛ. The Green’s reciprocity theorem exposes symmetries for some resistive circuits. In chapter 5, we will treat circuits where their branches have at most a battery in series with a capacitor. Here, the Gauss’ Law will be reformulated to ∂Q = −ρ, and the Poisson’s equation will be reformulated to −∂CdΦ = −ρ. In this chapter, we too study the Dirichlet problem, ending with the study of Green’s functions. CNPq Estamos propondo uma nova formulação da teoria dos circuitos, levando em consideração a sua distribuição física no espaço. Para fazer isto, usaremos alguns conceitos da topologia algébrica. Nomes como Hermann Weyl e Steve Smale fizeram importantes contribuições mostrando essas conexões entre a teoria dos circuitos e a da topologia algébrica. Neste trabalho, nós consideraremos um circuito elétrico como um grafo ou um complexo unidimensional, onde o domínio do operador fronteira ∂ é o espaço vetorial C₁ gerado pelos ramos (fios do circuito), e o seu codomínio é o espaço vetorial C₀ gerado pelos nós. No capítulo 3, a lei das correntes de Kirchhoff será reformulada para a fórmula concisa ∂I = 0 e a lei das voltagens de Kirchhoff será reformulada para a fórmula concisa V = −dΦ, onde d : C₀ → C₁ é a aplicação cofronteira. Os métodos da corrente na malha e do potencial nos nós são também discutidos neste capítulo, bem como uma análise conclusiva da existência e unicidade das soluções para as equações dos circuitos elétricos é também realizada. No capítulo 4, estudaremos alguns métodos alternativos para resolver equações de circuitos elétricos. O método de Weyl faz uso de operadores para projeção ortogonal e este método resume-se a fórmula π = σ(sZσ)⁻¹sZ. O método de Kirchhoff usa a teoria de grafos para encontrar os valores de tensões e correntes elétricas e será dado por pλ = R⁻¹ΣᴛQᴛpᴛ. O teorema da reciprocidade de Green expõe simetrias para alguns circuitos resistivos. No capítulo 5, vamos tratar circuitos onde seus ramos têm no máximo uma bateria em série com um capacitor. Aqui, a Lei de Gauss será reformulada para ∂Q = −ρ, e a equação de Poisson será reformulada para −∂CdΦ = −ρ. Neste capítulo, nós também estudamos o problema de Dirichlet, terminando com o estudo das funções de Green. 2019-07-04T19:56:30Z 2019-07-04T19:56:30Z 2014-07-28 masterThesis https://repositorio.ufpe.br/handle/123456789/31348 eng openAccess Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ application/pdf Universidade Federal de Pernambuco UFPE Brasil Programa de Pos Graduacao em Matematica |
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REPOSITORIO UFPE |
| language |
eng |
| topic |
Matemática Topologia algébrica |
| spellingShingle |
Matemática Topologia algébrica ABATH, Leonardo Lopes Circuit theory via algebraic topology |
| description |
We are proposing a new formulation of circuit theory, taking in consideration its physical distribution in the space. For doing this we will use some concepts of the algebraic topology. Names as Hermann Weyl and Steve Smale did important contributions showing these connections between the theory of circuits and the theory of algebraic topology. In this work, we will go to consider an electrical circuit as a graph or as a one-dimensional complex, where the domain of the boundary operator ∂ is the vector space C₁ generated by the branches (wires of the circuit) and its codomain is the vector space C₀ generated by the nodes. In chapter 3, the Kirchhoff ’s current law will be reformulate to the concise formula ∂I = 0 and the Kirchhoff ’s potential law will be reformulate to the concise formula V = −dΦ, where d : C₀ → C₁ is the coboundary map. The methods of mesh-current and node-potential are also discussed in this chapter, as well as a conclusive analysis of the existence and uniqueness of solutions for the electric circuit equations too is realized. In chapter 4 we will study some alternative methods for solving electric circuit equations. The Weyl’s method makes use of orthogonal projection operators and this method is summarized by the formula π = σ(sZσ)⁻¹sZ. The Kirchhoff’s method uses graph theory to find the values of voltages and electric currents and will be given by pλ = R⁻¹ΣᴛQᴛpᴛ. The Green’s reciprocity theorem exposes symmetries for some resistive circuits. In chapter 5, we will treat circuits where their branches have at most a battery in series with a capacitor. Here, the Gauss’ Law will be reformulated to ∂Q = −ρ, and the Poisson’s equation will be reformulated to −∂CdΦ = −ρ. In this chapter, we too study the Dirichlet problem, ending with the study of Green’s functions. |
| author2 |
LEANDRO, Eduardo Shirlippe Goes |
| format |
masterThesis |
| author |
ABATH, Leonardo Lopes |
| author_sort |
ABATH, Leonardo Lopes |
| title |
Circuit theory via algebraic topology |
| title_short |
Circuit theory via algebraic topology |
| title_full |
Circuit theory via algebraic topology |
| title_fullStr |
Circuit theory via algebraic topology |
| title_full_unstemmed |
Circuit theory via algebraic topology |
| title_sort |
circuit theory via algebraic topology |
| publisher |
Universidade Federal de Pernambuco |
| publishDate |
2019 |
| url |
https://repositorio.ufpe.br/handle/123456789/31348 |
| _version_ |
1648654534725074944 |
| score |
13.657419 |