Graphical electrical circuit theory

Thesis

Most physical systems are naturally compositional and relational, yet our standard mathematical tools rarely reflect that structure. Standard mathematical notation favors terms and functions over relations, making relational reasoning second-class. This changed with the recent development of string diagrams. This 2-dimensional syntax provides the extra flexibility needed for a practical calculus of relations. We build specifically on the development of Graphical Linear Algebra (GLA), a relational calculus for finite-dimensional linear algebra.

Based on this we introduce Graphical Electrical Circuit Theory, a string diagrammatic formalism for analyzing electrical circuits. In this framework, we show that we can reproduce standard results practically. We then extend Graphical Linear Algebra itself to go beyond linear relations to piecewise-linear relations. Using this more expressive semantic domain, we add diodes and transistors to our framework.

Authors: Boisseau, G

• DOI: 10.5287/ora-zb7dkd8w0
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: electrical circuits; string diagrams; PROP; GLA; relational mathematics; piecewise-linear
• Subjects: Mathematics--Charts, diagrams, etc.; Linear systems; Logic; Categories (Mathematics); Electric circuit analysis