Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Möbius Transformations and the Analytic--Geometric Reconstruction of the Induction--Machine Circle Diagram

Anubhav Gupta

TL;DR

Problem: the Heyland circle diagram has a long geometric heritage but lacked a closed analytic framework. Approach: a complete Euclidean reconstruction using two phasors is developed, proving existence, uniqueness, and equivalence with the per-phase analytic current locus; a Möbius-transformation view explains the circle’s origin. Contributions: unique circle, torque chord, slip scale, and maximum-torque point are derived analytically; the circle diagram is shown to be exact for balanced sinusoidal operation and its Möbius interpretation unifies motoring and generating branches. Impact: provides a rigorous foundation for computation and interpretation of circle diagrams and motivates generalizations to other machines.

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 Möbius transformation interpretation reveals the complex--analytic origin of the diagram's circularity and offers a compact explanation of its geometric structure.

Möbius Transformations and the Analytic--Geometric Reconstruction of the Induction--Machine Circle Diagram

TL;DR

Problem: the Heyland circle diagram has a long geometric heritage but lacked a closed analytic framework. Approach: a complete Euclidean reconstruction using two phasors is developed, proving existence, uniqueness, and equivalence with the per-phase analytic current locus; a Möbius-transformation view explains the circle’s origin. Contributions: unique circle, torque chord, slip scale, and maximum-torque point are derived analytically; the circle diagram is shown to be exact for balanced sinusoidal operation and its Möbius interpretation unifies motoring and generating branches. Impact: provides a rigorous foundation for computation and interpretation of circle diagrams and motivates generalizations to other machines.

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 Möbius transformation interpretation reveals the complex--analytic origin of the diagram's circularity and offers a compact explanation of its geometric structure.

Paper Structure

This paper contains 11 sections, 6 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric Setup and Classical Reconstruction
  3. Equivalence to the Analytic Current Locus
  4. Möbius Transformations and the Current Locus
  5. Rotor Current as a Möbius Map
  6. Geometric interpretation.
  7. Circularity of the Rotor--Current Locus
  8. Geometric Meaning of the Maximum--Torque Condition
  9. Example: Möbius and Analytic Reconstruction of the Circle Diagram
  10. Conclusion
  11. Analytic Derivation of the Current Locus

Key Result

Proposition 1

Let $p_0, p_A \in \mathbb{R}^2$ with $p_0 \neq p_A$ and fix the horizontal line $y = y_0$ through $p_0$. Then Construction constr:recon yields a unique circle that passes through both $p_0$ and $p_A$ whose center lies on $y = y_0$.

Figures (1)

  • Figure 1: Image of the slip axis under the Möbius transformation $I_r(z)=V_{\mathrm{th}}/(\gamma z+\delta)$. Positive real values of $z$ ($z>0$) correspond to motoring operation ($s>0$), while negative values ($z<0$) correspond to generating operation ($s<0$). Both branches map to the same circular current locus (blue arc). The no–load point $O'$ and blocked–rotor point $A$ lie on this locus and uniquely determine its center $C$ once the reference horizontal through $O'$ is fixed. Also shown are the classical geometric constructions: the output line (dashed), torque chord (dash–dot), maximum–output line (solid), maximum–torque line (red dotted), and the efficiency line (grey).

Theorems & Definitions (16)

  • Definition 1: Test points
  • Definition 2: Output line
  • Proposition 1: Uniqueness given a reference horizontal
  • proof
  • Lemma 1: Orthogonality and maximum torque
  • proof
  • Remark 1
  • Theorem 1: Equivalence to the equivalent--circuit locus
  • proof
  • Remark 2
  • ...and 6 more