# Characterization of the response maps of alternating-current networks

## TL;DR

This paper characterizes the response maps of AC networks, proving that known conditions are sufficient and providing a method to construct networks for given response maps.

## Contribution

It establishes the sufficiency of known conditions for response maps and offers a constructive approach for designing AC networks based on these maps.

## Key findings

- Necessary conditions for response maps are also sufficient.
- A method to construct AC networks from response maps.
- Theoretical validation of response map characterization.

## Abstract

In an alternating-current network, each edge has a complex "conductance" with positive real part. The response map is the linear map from the vector of voltages at a subset of "boundary nodes" to the vector of currents flowing into the network through these nodes. We prove that the known necessary conditions for these response maps are sufficient, and we construct an appropriate alternating-current network for a given response map.