M\"obius Transformations and the Analytic--Geometric Reconstruction of the Induction--Machine Circle Diagram

TL;DR

This paper formalizes the classical Heyland circle diagram for induction machines using M"obius transformations, providing an analytic reconstruction that aligns with the steady-state current locus from the equivalent circuit.

Contribution

It introduces a complete Euclidean and complex-analytic framework for reconstructing the Heyland circle diagram from measured data, linking geometric and analytic perspectives.

Findings

The reconstructed circle matches the steady-state current locus.

A M"obius transformation explains the diagram's circularity.

The method uses only two measured phasors and elementary geometric operations.

Abstract

The Heyland circle diagram is a classical graphical method for representing the steady--state behavior of induction machines using no--load and blocked--rotor test data. Despite its long pedagogical history, the traditional geometric construction has not been formalized within a closed analytic framework. This note develops a complete Euclidean reconstruction of the diagram using only the two measured phasors and elementary geometric operations, yielding a unique circle, a torque chord, a slip scale, and a maximum--torque point. We prove that this constructed circle coincides precisely with the analytic steady--state current locus obtained from the per--phase equivalent circuit. A M\"obius transformation interpretation reveals the complex--analytic origin of the diagram's circularity and offers a compact explanation of its geometric structure.