[Paper Review] How to Model Brushless Electric Motors for the Design of Lightweight Robotic Systems

Ung Hee Lee, Tor Shepherd+6 more|arXiv (Cornell University)|Sep 29, 2023

Electric Motor Design and AnalysisEngineering3 citations

TL;DR

This paper presents a unified mathematical framework for modeling brushless DC (BLDC) motors in lightweight robotic systems, enabling accurate torque, voltage, and thermal analysis. It introduces a power-invariant d-q transformation to convert three-phase BLDC behavior into a simplified DC-equivalent model, correcting common errors in resistance, torque constant, and voltage estimation that can lead to 50–100% miscalculations in power loss and torque output.

ABSTRACT

A key step in the development of lightweight, high performance robotic systems is the modeling and selection of permanent magnet brushless direct current (BLDC) electric motors. Typical modeling analyses are completed a priori, and provide insight for properly sizing a motor for an application, specifying the required operating voltage and current, as well as assessing the thermal response and other design attributes (e.g.transmission ratio). However, to perform these modeling analyses, proper information about the motor's characteristics are needed, which are often obtained from manufacturer datasheets. Through our own experience and communications with manufacturers, we have noticed a lack of clarity and standardization in modeling BLDC motors, compounded by vague or inconsistent terminology used in motor datasheets. The purpose of this tutorial is to concisely describe the governing equations for BLDC motor analyses used in the design process, as well as highlight potential errors that can arise from incorrect usage. We present a power-invariant conversion from phase and line-to-line reference frames to a familiar q-axis DC motor representation, which provides a ``brushed'' analogue of a three phase BLDC motor that is convenient for analysis and design. We highlight potential errors including incorrect calculations of winding resistive heat loss, improper estimation of motor torque via the motor's torque constant, and incorrect estimation of the required bus voltage or resulting angular velocity limitations. A unified and condensed set of governing equations is available for designers in the Appendix. The intent of this work is to provide a consolidated mathematical foundation for modeling BLDC motors that addresses existing confusion and fosters high performance designs of future robotic systems.

Motivation & Objective

• Address widespread confusion and inconsistency in BLDC motor modeling due to non-standardized terminology and parameter reporting in manufacturer datasheets.
• Correct common errors in power loss and torque estimation that arise from mismatched reference frames (e.g., using line-to-line resistance with q-axis current).
• Provide a unified, mathematically consistent framework for converting three-phase BLDC motor characteristics into a single-phase DC-equivalent model for intuitive design and analysis.
• Enable precise thermal, voltage, and current predictions during early-stage robotic system design, especially for mass-sensitive applications like legged and wearable robots.

Proposed method

• Derive a power-invariant d-q transformation to map three-phase BLDC motor dynamics into a simplified q-axis DC motor representation.
• Use the d-q transformation to express voltage, current, and flux linkage in a reference frame compatible with field-oriented control (FOC) systems.
• Establish conversion rules between line-to-line (terminal) and phase quantities for resistance, voltage, and torque constants.
• Apply the transformation to derive accurate expressions for Joule heating: $ P = \frac{1}{2}I_q^2 R^{ll} $ for wye-wound and $ P = \frac{3}{2}I_q^2 R^{ll} $ for delta-wound motors.
• Correctly relate the back-EMF constant $ K_b^{ll} $, velocity constant $ K_v^{ll} $, and torque constant $ K_t^q $, showing $ K_t^q = \sqrt{\frac{3}{2}} K_b^{ll} $.
• Demonstrate that using $ K_v^{ll} $ directly with $ I^q $ leads to a $ \sqrt{\frac{3}{2}} $ underestimation of torque, a critical error in design.

Experimental results

Research questions

• RQ1 How can three-phase BLDC motor behavior be accurately and consistently modeled using a single-phase DC-equivalent representation?
• RQ2 What are the key errors introduced by mismatched reference frames when using manufacturer-provided parameters like terminal resistance and $ K_v $?
• RQ3 How do winding configurations (wye vs. delta) affect the conversion between line-to-line and phase resistance, and what impact does this have on power loss estimation?
• RQ4 What is the correct relationship between the torque constant $ K_t^q $, back-EMF constant $ K_b^{ll} $, and velocity constant $ K_v^{ll} $ in the context of q-axis current?
• RQ5 How can thermal and voltage requirements be accurately predicted during early robotic system design using a unified motor modeling framework?

Key findings

• Using line-to-line resistance $ R^{ll} $ directly with q-axis current $ I^q $ leads to a 100% overestimation of power loss in wye-wound motors and a 33% underestimation in delta-wound motors.
• The correct power loss for wye-wound motors is $ P = \frac{1}{2}I_q^2 R^{ll} $, and for delta-wound motors, it is $ P = \frac{3}{2}I_q^2 R^{ll} $, which corrects common miscalculations.
• Incorrectly using the velocity constant $ K_v^{ll} $ with q-axis current results in a $ \sqrt{\frac{3}{2}} \approx 1.22 $ times underestimation of torque output.
• The torque constant in the q-axis frame is $ K_t^q = \sqrt{\frac{3}{2}} K_b^{ll} $, and since $ K_b^{ll} = 1/K_v^{ll} $, the correct torque is $ \tau = \sqrt{\frac{3}{2}} \cdot \frac{1}{K_v^{ll}} \cdot I^q $.
• The d-q transformation ensures that the q-axis voltage equation reduces to a DC-like form, enabling accurate modeling of BLDC motors as equivalent brushed DC motors under ideal conditions.
• A unified mathematical framework is provided in Appendix A, enabling consistent and accurate motor modeling across different motor types and configurations.