# Efficiency Maps of Shifted Inductances Axes Permanent Magnet Synchronous Motors

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## Abstract

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## 1. Introduction

## 2. Shifted Inductances Axes Permanent Magnet Synchronous Motors (SIAPMSMs)

#### 2.1. SIAPMSM Machines Principle

#### 2.2. SIAPMSM Structures

## 3. Modeling of SIAPMSMs

#### 3.1. Decomposition into Two Machines

#### 3.2. Lossless Model

- Symbols ${i}_{d},{i}_{q}$: d and q axes components of armature current;
- Symbols ${v}_{d},{v}_{q}$: d and q axes components of terminal voltage;
- Symbols ${\Phi}_{d},{\Phi}_{q}$: d and q axes components of magnetic flux.

#### 3.2.1. Representation in the ${(d}_{PM},{q}_{PM})$ Referential Frame

#### 3.2.2. Representation in the ${(d}_{Reluc.},{q}_{Reluc.})$ Referential Frame

#### 3.3. Model including the Electromagnetic Losses

- Symbols ${i}_{fd},{i}_{fq}$: d and q axes components of iron loss current;
- Symbols ${R}_{a}$: armature winding resistance per phase;
- Symbols ${R}_{f}$: iron loss resistance.

#### 3.3.1. Representation in the ${(d}_{PM},{q}_{PM})$ Referential Frame

#### 3.3.2. Representation in the ${(d}_{Reluc.},{q}_{Reluc.})$ Referential Frame

#### 3.4. Per-Unit System

## 4. Efficiency Maps of SIAPMSMs

#### 4.1. Presentation of Developed Tool

Algorithm 1. MATLAB script for the determination of ${V}_{nmax}$. | |||

$\mathbf{S}\mathbf{t}\mathbf{a}\mathbf{r}\mathbf{t}$ | |||

$\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s}\text{}\mathrm{o}\mathrm{f}\text{}{L}_{dn},\rho ,{R}_{an}\text{}\mathrm{a}\mathrm{n}\mathrm{d}\text{}{R}_{fn}\text{}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{d}\text{}\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}\text{}\mathrm{b}\mathrm{e}\mathrm{e}\mathrm{n}\text{}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d};$ | |||

${I}_{n}=[0:\u2206{I}_{n}:1];$ | |||

$\psi =[-\pi :\u2206\psi :\pi ];$ | |||

$\beta =[0:\u2206\beta :\pi ];$ | |||

$For\text{}i=1:length\left({I}_{n}\right);$ | |||

$For\text{}j=1:length\left(\psi \right)$ | |||

$For\text{}k=1:length\left(\beta \right)$ | |||

${\Gamma}_{n}^{\prime}\left(k\right)=h\left({I}_{n}\left(i\right),\psi \left(j\right),\beta \left(k\right)\right);\left[\mathrm{s}\mathrm{e}\mathrm{e}\text{}\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{}\left(36\right)\right]$ | |||

$End$ | |||

$\left[Y1\left(j\right),X1\left(j\right)\right]=max\left({\Gamma}_{n}^{\prime}\right);$ | |||

${\Gamma}_{n1}^{\prime}\left(j\right)={\Gamma}_{n}^{\prime}\left(X1\left(j\right)\right);$ | |||

${\beta}_{1}\left(j\right)=\beta \left(X1\left(j\right)\right);$ | |||

$End$ | |||

$\left[Y2\left(i\right),X2\left(i\right)\right]=max\left({\Gamma}_{n1}^{\prime}\right);$ | |||

${\Gamma}_{n2}^{\prime}\left(i\right)={\Gamma}_{n1}^{\prime}\left(X2\left(i\right)\right);$ | |||

${\beta}_{2}\left(i\right)={\beta}_{1}\left(X2\left(i\right)\right);$ | |||

${\psi}_{1}\left(i\right)=\psi \left(X2\left(i\right)\right);$ | |||

$End$ | |||

$\left[Y3,X3\right]=max\left({\Gamma}_{n2}^{\prime}\right);$ | |||

${I}_{nOpt}={I}_{n}\left(X3\right);$ | |||

${\psi}_{Opt}={\psi}_{1}\left(X3\right);$ | |||

${\beta}_{Opt}={\beta}_{2}\left(X3\right);$ | |||

${V}_{nmaxd}=f\left({I}_{nOpt},{\psi}_{Opt},{\beta}_{Opt}\right);\left[\mathrm{s}\mathrm{e}\mathrm{e}\text{}\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{}\left(17\right)\right]$ | |||

${V}_{nmaxq}=g({I}_{nOpt},{\psi}_{Opt},{\beta}_{Opt});[\mathrm{s}\mathrm{e}\mathrm{e}\text{}\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{}(17)]$ | |||

${V}_{nmax}=\mathrm{s}\mathrm{q}\mathrm{r}\mathrm{t}({V}_{nmaxd}^2+{V}_{nmaxq}^2);$ | |||

$\mathbf{E}\mathbf{n}\mathbf{d}$ |

#### 4.2. Validation of Developed Tool

#### 4.2.1. Validation of Developed Tool for the Lossless Case

#### 4.2.2. Validation of Developed Tool for the Case including the Electromagnetic Losses

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Improvement of torque density using SIAPMSM structures (current angle is the phase shift between phase armature currents and EMF). (

**a**) Classical PM Machine, (

**b**) SIAMPMSM $({\psi}_{Opt}=0\xb0)$.

**Figure 5.**Lossless SIAMPMSM equivalent circuits under motor mode operation in the ($d,q$) referential frame. (

**a**) d axis equivalent circuit, (

**b**) q axis equivalent circuit.

**Figure 6.**SIAMPMM equivalent circuits including the electromagnetic losses. (

**a**) d axis equivalent circuit, (

**b**) q axis equivalent circuit.

**Figure 9.**Comparison of maximum torque capabilities obtained using new and old codes $({L}_{dn}=2.5,\rho =0.6)$.

**Figure 10.**Torque increase in the (${L}_{dn},\rho $) plane. (

**a**) ${L}_{dn}\in (0,10]$ and $\rho \in (0,5]$ (

**b**) ${L}_{dn}\in (0,10]$ and $\rho \in \left[0.5,1.5\right].$

**Figure 12.**Comparison of maximum torque capabilities obtained using new and old codes $({L}_{dn}=2.5,\rho =0.6,{R}_{an}=0.1,$ and ${R}_{fn}=20).$

**Figure 13.**Comparison of efficiency maps obtained using new and old codes $({L}_{dn}=2.5,\rho =0.6,{R}_{an}=0.1$, and ${R}_{fn}=20)$. (

**a**) Efficiency maps drawn using the old tool. (

**b**) Efficiency maps drawn using the new tool.

Quantities and Parameter | Variation Interval |
---|---|

${\Omega}_{n}$ | [0, +∞) |

${\Gamma}_{n}$ | [0, 1] |

${I}_{n}$ | [0, 1] |

${L}_{dn}$ | (0, +∞) |

$\rho $ | (0, +∞) |

${R}_{an}$ | [0, 10] |

${R}_{fn}$ | (0, +∞) |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Nasser, H.; Amara, Y.; Chabour, F.; Ghandour, M.
Efficiency Maps of Shifted Inductances Axes Permanent Magnet Synchronous Motors. *World Electr. Veh. J.* **2023**, *14*, 174.
https://doi.org/10.3390/wevj14070174

**AMA Style**

Nasser H, Amara Y, Chabour F, Ghandour M.
Efficiency Maps of Shifted Inductances Axes Permanent Magnet Synchronous Motors. *World Electric Vehicle Journal*. 2023; 14(7):174.
https://doi.org/10.3390/wevj14070174

**Chicago/Turabian Style**

Nasser, Hussein, Yacine Amara, Ferhat Chabour, and Mazen Ghandour.
2023. "Efficiency Maps of Shifted Inductances Axes Permanent Magnet Synchronous Motors" *World Electric Vehicle Journal* 14, no. 7: 174.
https://doi.org/10.3390/wevj14070174