# On the notion of effective impedance

## TL;DR

This paper extends the concept of effective impedance to complex electrical networks with inductors and capacitors, providing two mathematical models to analyze their properties.

## Contribution

It introduces two new mathematical models for electrical networks with coils and capacitors, broadening the understanding of effective impedance beyond resistor networks.

## Key findings

- Defined effective impedance in complex-weighted graph models
- Established models using rational functions for network analysis
- Provided a framework for analyzing general electrical networks

## Abstract

It is known that electrical networks with resistors are related to the Laplace operator and random walk on weighted graphs. In this paper we consider more general electrical networks with coils, capacitors, and resistors. We give two mathematical models of such networks: complex-weighted graphs and graphs with weight from the ordered field of rational functions. The notion of effective impedance in both approaches is defined.