Circuit theory via algebraic topology

Abstract

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.