Clarification to published paper.

force_torque_clarification.md

Clarification on DPNV Force/Torque Model in Combined Windings for Bearingless Motors-An Overview

Objective

Clarify force $k_f$ and torque $k_t$ constants in this paper:

A. Khamitov, N. P. Petersen and E. L. Severson, "Combined Windings for Bearingless Motors—An Overview," 2023 IEEE Energy Conversion Congress and Exposition (ECCE), Nashville, TN, USA, 2023, pp. 4628-4635, doi: 10.1109/ECCE53617.2023.10362696.

Context

There has been some confusion recently on Section IV-E of this paper as to exactly how force and torque constants are defined and related to coil current and current space vectors.

Clarification

This confusion is best resolved by understanding $k_t$ and $k_f$ to be the fundamental quantities that map motor coil currents into torque and force. That is, when

$i_k = \hat{I}_t \cos \left(\phi_t - [k-1]\alpha_t\right) + \hat{I}_s \cos \left(\phi_s - [k-1]\alpha_s\right)$ (Equation 10 of the paper)

current space vectors are calculated as:

$\vec{i}_t = \hat{I}_t e^{j\phi_t}$ and $\vec{i}_s = \hat{I}_s e^{j\phi_s}$

and these current space vectors are transformed to the synchronous frame:

$\vec{i}_t^T = (i_d + ji_q) = \vec{i}_t e^{-j\theta_t}$

$\vec{i}_s^S = (i_x + ji_y) = \vec{i}_s e^{-j\theta_s}$

they result in shaft torque and force calculated as:

$\vec{T} = [T_d + j\tau] = k_t \vec{i}_t^T$ and $\vec{F} = [F_x + jF_y] = k_f \vec{i}_s^S$.

The end of Section IV-E then provides relations to how $\hat{k}_t$ and $\hat{k}_f$ are calculated from $k_t$ and $k_f$ so that equations (24) and (25) work for each winding type.

Revision to the Paper

While the paper is factually correct, I find the explanation could be made simpler with the following revision. If we were still drafting this paper, I would change this text:

to be:

" $\vec{T} = k_t \vec{i}_t^T$ and $\vec{F} = k_f \vec{i}_s^S$ "