Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Perspective on the Algebra, Topology, and Logic of Electrical Networks

Marko Orešković, Ivana Kuzmanović Ivičić, Juraj Benić, Mario Essert

TL;DR

This work develops a unified $m$-theoretic framework for CRL one-port networks by encoding networks as $m$-words (jorbs) and organizing them within an $m$-topology that preserves impedance-equivalence. It introduces the $\lambda$–$\Delta$ metric and the valuation morphism $\Phi$ to compactly characterize impedance-degree through $(v(l_x),\lambda_s,\lambda_p,v(r_x))$, and embeds the algebra, topology, and logic into a constructive pipeline for generation, classification, and synthesis. The authors demonstrate equivalence classification, algorithmic generation of non-isomorphic SP topologies, and Cauer–Foster synthesis workflows, with validation against Ladenheim’s and Morelli’s catalogs. The approach enables symbolic-to-topological translation of impedance functions and supports automated CAD, model reduction, and formal verification in the emerging field of Jorbology, while outlining extensions to richer element alphabets. Overall, the paper provides a rigorous, computationally driven foundation for automatic network synthesis and verification within a novel algebra-topology-logic framework.

Abstract

This paper presents a unified algebraic, topological, and logical framework for electrical one-port networks based on Šare's $m$-theory. Within this formalism, networks are represented by $m$-words (jorbs) over an ordered alphabet, where series and parallel composition induce an $m$-topology on $m$-graphs with a theta mapping $\vartheta$ that preserves one-port equivalence. The study formalizes quasi-orders, shells, and cores, showing their structural correspondence to network boundary conditions and impedance behavior. The $λ--Δ$ metric, together with the valuation morphism $Φ$, provides a concise descriptor of the impedance-degree structure. In the computational domain, the framework is extended with algorithmic procedures for generating and classifying non-isomorphic series-parallel topologies, accompanied by programmatic Cauer/Foster synthesis workflows and validation against canonical examples from Ladenheim's catalogue. The resulting approach enables symbolic-to-topological translation of impedance functions, offering a constructive bridge between algebraic representation and electrical realization. Overall, the paper outlines a self-consistent theoretical and computational foundation for automated network synthesis, classification, and formal verification within the emerging field of Jorbology.

A Perspective on the Algebra, Topology, and Logic of Electrical Networks

TL;DR

This work develops a unified -theoretic framework for CRL one-port networks by encoding networks as -words (jorbs) and organizing them within an -topology that preserves impedance-equivalence. It introduces the – metric and the valuation morphism to compactly characterize impedance-degree through , and embeds the algebra, topology, and logic into a constructive pipeline for generation, classification, and synthesis. The authors demonstrate equivalence classification, algorithmic generation of non-isomorphic SP topologies, and Cauer–Foster synthesis workflows, with validation against Ladenheim’s and Morelli’s catalogs. The approach enables symbolic-to-topological translation of impedance functions and supports automated CAD, model reduction, and formal verification in the emerging field of Jorbology, while outlining extensions to richer element alphabets. Overall, the paper provides a rigorous, computationally driven foundation for automatic network synthesis and verification within a novel algebra-topology-logic framework.

Abstract

This paper presents a unified algebraic, topological, and logical framework for electrical one-port networks based on Šare's -theory. Within this formalism, networks are represented by -words (jorbs) over an ordered alphabet, where series and parallel composition induce an -topology on -graphs with a theta mapping that preserves one-port equivalence. The study formalizes quasi-orders, shells, and cores, showing their structural correspondence to network boundary conditions and impedance behavior. The metric, together with the valuation morphism , provides a concise descriptor of the impedance-degree structure. In the computational domain, the framework is extended with algorithmic procedures for generating and classifying non-isomorphic series-parallel topologies, accompanied by programmatic Cauer/Foster synthesis workflows and validation against canonical examples from Ladenheim's catalogue. The resulting approach enables symbolic-to-topological translation of impedance functions, offering a constructive bridge between algebraic representation and electrical realization. Overall, the paper outlines a self-consistent theoretical and computational foundation for automated network synthesis, classification, and formal verification within the emerging field of Jorbology.
Paper Structure (26 sections, 1 theorem, 62 equations, 9 figures, 5 tables)

This paper contains 26 sections, 1 theorem, 62 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Algebra of $m$‑Numbers
  3. The alphabet
  4. The concatenation
  5. The Set $M_\Gamma$
  6. $\lambda$-Length
  7. $m$-word Compression
  8. Unary Operators
  9. Relations $\leq_q$ and $\geq_q$
  10. Relation of Equivalence of $m$-space
  11. Jorb as an RLC Network Representation
  12. Series and Parallel Connections
  13. S-Core and P-Core
  14. S-Shell and P-Shell
  15. $m$-Topology
  16. ...and 11 more sections

Key Result

Proposition 1

Let $p, r \in \Gamma$ be arbitrary symbols. Then where $w_1,w_2$ are subwords (one of which can be empty) such that $w=w_1 \cdot p \cdot r \cdot p \cdot w_2\in M_\Gamma$.

Figures (9)

  • Figure 1: Permutation group
  • Figure 2: Action of group elements D, E, and F
  • Figure 3: Hasse diagram for $([M_\Gamma],\le_q)$ with $\Gamma=\{a,b,c,d,e\}$.
  • Figure 4: Electrical circuit with the corresponding $m$-graph
  • Figure 5: Hasse diagrams for $(\Omega_s,\le_q)$ i $(\Omega_p,\ge_q)$.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • ...and 4 more