# Analytical Model for the Design of Axial Flux Induction Motors with Maximum Torque Density

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Thought Experiment

#### 2.2. Model Expression Generation

#### 2.2.1. Stator Resistance

#### 2.2.2. Rotor Cage Resistance

#### 2.2.3. Slot Cross-Sectional Area

#### 2.2.4. General Model Expression

#### 2.2.5. Area of Rectangular Slots

#### 2.2.6. Relative Dimensions

## 3. Results

#### 3.1. Calculation Example

#### 3.1.1. Design Setup

#### 3.1.2. Applications of the Analytical Model

#### 3.2. Validation of Calculated Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AFIM | Axial flux induction motor |

FEM | Finite element method |

p | the number of pole pairs |

${T}_{em}$ | electromagnetic toque [$\mathrm{Nm}$] |

${m}_{1}$ | number of stator phases |

${\mathrm{U}}_{1}$ | the phase voltage of the stator [$\mathrm{V}$], RMS |

${\omega}_{s}$ | electrical angular velocity |

${R}_{1}$ | stator resistance [Ohm] |

${R}_{2}^{\prime}$ | the resistance of the rotor referred to the stator [Ohm] |

${c}_{1}$ | Coefficient taking into account voltage drop at full resistance of stator winding |

${X}_{1}$ | stator reactance [Ohm] |

${X}_{2}^{\prime}$ | the rotor reactance referred to the stator [Ohm] |

${E}_{10}$ | electromotive force (emf) [$\mathrm{V}$], [RMS] |

${W}_{1}$ | the number of series-connected turns of the stator winding |

${\Phi}_{m}$ | magnetic flux [$\mathrm{Wb}$] |

${D}_{i}$ | inner diameter [$\mathrm{m}$] |

${D}_{o}$ | outer diameter [$\mathrm{m}$] |

${\tau}_{p}$ | pole pitch [$\mathrm{m}$] |

${l}_{1}$ | the stator thickness [$\mathrm{m}$] |

${l}_{2}$ | the rotor thickness [$\mathrm{m}$] |

$\delta $ | thickness of air gap [$\mathrm{m}$] |

L | the total thickness of the machine [$\mathrm{m}$] |

${B}_{\delta}$ | magnetic flux density in air gap [$\mathrm{T}$] |

${B}_{t1}$ | magnetic flux density in stator teeth [$\mathrm{T}$] |

${B}_{t2}$ | magnetic flux density in rotor teeth [$\mathrm{T}$] |

${B}_{c1}$ | magnetic flux density in stator core [$\mathrm{T}$] |

${B}_{c2}$ | magnetic flux density in rotor core [$\mathrm{T}$] |

${k}_{\vartheta 1}$ | the thermal factor of the stator |

${k}_{\vartheta 2}$ | the thermal factor of the rotor |

${\gamma}_{1}$ | conductivity of stator conductors [$\mathrm{S}/\mathrm{m}$] |

${\gamma}_{2}$ | conductivity of rotor conductors [$\mathrm{S}/\mathrm{m}$] |

${q}_{a1}$ | area of the stator conductor [${\mathrm{m}}^{2}$] |

${q}_{bar}$ | area of the rotor rods [${\mathrm{m}}^{2}$] |

${q}_{ring}$ | area of short-circuited rings [${\mathrm{m}}^{2}$] |

${J}_{bar}$ | current density of rotor rods [$\mathrm{A}/{\mathrm{m}}^{2}$] |

${J}_{ring}$ | current density of short-circuited rings [$\mathrm{A}/{\mathrm{m}}^{2}$] |

${Q}_{\tau}$ | the area of the coil pith [${\mathrm{m}}^{2}$] |

${\tilde{Q}}_{s1}$ | the area of the stator slot [${\mathrm{m}}^{2}$] |

${Q}_{s1}$ | the total area of the all stator slots [${\mathrm{m}}^{2}$] |

${Q}_{t1}$ | the area of the stator tooth [${\mathrm{m}}^{2}$] |

${\tilde{Q}}_{s2}$ | the area of the rotor slot [${\mathrm{m}}^{2}$] |

${Q}_{s2}$ | the total area of the all rotor slots [${\mathrm{m}}^{2}$] |

${Q}_{t2}$ | the area of the rotor tooth [${\mathrm{m}}^{2}$] |

${k}_{p}$ | pitch factor |

${k}_{e\mathrm{w}}$ | radial length factor of the end winding |

${k}_{\mathrm{w}}$ | winding factor |

${k}_{Fe}$ | space factor for iron |

${k}_{Cu}$ | space factor for copper |

${k}_{Al}$ | space factor for aluminum |

${z}_{1}$ | the number of the stator slots |

${z}_{2}$ | the number of the rotor slots |

${b}_{s1}$ | the width of the stator slot [$\mathrm{m}$] |

${b}_{s2}$ | the width of the rotor slot [$\mathrm{m}$] |

${h}_{s1}$ | the height of the stator slot [$\mathrm{m}$] |

${h}_{c1}$ | the height of the stator core [$\mathrm{m}$] |

${h}_{s2}$ | the height of the rotor slot [$\mathrm{m}$] |

## References

- Dineva, A.; Mosavi, A.; Faizollahzadeh Ardabili, S.; Vajda, I.; Shamshirband, S.; Rabczuk, T.; Chau, K.W. Review of soft computing models in design and control of rotating electrical machines. Energies
**2019**, 12, 1049. [Google Scholar] [CrossRef] [Green Version] - Virtic, P.; Pisek, P.; Marcic, T.; Hadziselimovic, M.; Stumberger, B. Analytical analysis of magnetic field and back electromotive force calculation of an axial-flux permanent magnet synchronous generator with coreless stator. IEEE Trans. Magn.
**2008**, 44, 4333–4336. [Google Scholar] [CrossRef] - Krasopoulos, C.T.; Beniakar, M.E.; Kladas, A.G. Multicriteria PM motor design based on ANFIS evaluation of EV driving cycle efficiency. IEEE Trans. Transp. Electrif.
**2018**, 4, 525–535. [Google Scholar] [CrossRef] - Virtic, P.; Pisek, P.; Hadziselimovic, M.; Marcic, T.; Stumberger, B. Torque analysis of an axial flux permanent magnet synchronous machine by using analytical magnetic field calculation. IEEE Trans. Magn.
**2009**, 45, 1036–1039. [Google Scholar] [CrossRef] - Virtič, P.; Vražić, M.; Papa, G. Design of an axial flux permanent magnet synchronous machine using analytical method and evolutionary optimization. IEEE Trans. Energy Convers.
**2015**, 31, 150–158. [Google Scholar] [CrossRef] - Muselli, M.; Notton, G.; Louche, A. Design of hybrid-photovoltaic power generator, with optimization of energy management. Sol. Energy
**1999**, 65, 143–157. [Google Scholar] [CrossRef] - Bramerdorfer, G.; Zăvoianu, A.C. Surrogate-based multi-objective optimization of electrical machine designs facilitating tolerance analysis. IEEE Trans. Magn.
**2017**, 53, 1–11. [Google Scholar] [CrossRef] - Bramerdorfer, G.; Tapia, J.A.; Pyrhönen, J.J.; Cavagnino, A. Modern electrical machine design optimization: Techniques, trends, and best practices. IEEE Trans. Ind. Electron.
**2018**, 65, 7672–7684. [Google Scholar] [CrossRef] - Meo, S.; Zohoori, A.; Vahedi, A. Optimal design of permanent magnet flux switching generator for wind applications via artificial neural network and multi-objective particle swarm optimization hybrid approach. Energy Convers. Manag.
**2016**, 110, 230–239. [Google Scholar] [CrossRef] - Hejra, M.; Mansouri, A.; Trabeisi, H. Optimal design of a permanent magnet synchronous motor: Application of in-wheel motor. In Proceedings of the 2014 5th International Renewable Energy Congress (IREC), Hammamet, Tunisia, 25–27 March 2014; pp. 1–5. [Google Scholar]
- Raminosoa, T.; Blunier, B.; Fodorean, D.; Miraoui, A. Design and optimization of a switched reluctance motor driving a compressor for a PEM fuel-cell system for automotive applications. IEEE Trans. Ind. Electron.
**2010**, 57, 2988–2997. [Google Scholar] [CrossRef] - Knypiński, Ł.; Pawełoszek, K.; Le Menach, Y. Optimization of Low-Power Line-Start PM Motor Using Gray Wolf Metaheuristic Algorithm. Energies
**2020**, 13, 1186. [Google Scholar] - Frederico, L.S.B.F.C.; Min, G.G.J.A.R. Ant colony optimization for the topological design of interior permanent magnet (IPM) machines. Electron. Eng.
**2014**, 26, 1324–1345. [Google Scholar] - Mamede, A.C.F.; Camacho, J.R. Evolutionary algorithms for optimization of 4/4 single phase switched reluctance machine. IEEE Lat. Am. Trans.
**2018**, 16, 1684–1691. [Google Scholar] [CrossRef] - Benallal, M.; Vaganov, M.; Pantouhov, D.; Ailam, E.; Hamouda, K. Optimal value of air gap induction in an induction motor. In Proceedings of the XIX International Conference on Electrical Machines—ICEM 2010, Rome, Italy, 6–8 September 2010; pp. 1–4. [Google Scholar]
- Pyrhonen, J.; Jokinen, T.; Hrabovcova, V. Design of Rotating Electrical Machines; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Yang, Y.P.; Shih, G.Y. Optimal design of an axial-flux permanent-magnet motor for an electric vehicle based on driving scenarios. Energies
**2016**, 9, 285. [Google Scholar] [CrossRef] [Green Version] - Du-Bar, C. Design of an Axial Flux Machine for an In-Wheel Motor Application. Master’s Thesis, Chalmers Reproservice, Göteborg, Sweden, 2011. [Google Scholar]
- Gao, P.; Gu, Y.; Wang, X. The design of a permanent magnet in-wheel motor with dual-stator and dual-field-excitation used in electric vehicles. Energies
**2018**, 11, 424. [Google Scholar] [CrossRef] [Green Version] - Kreim, A.; Schäfer, U. An approach to an optimal design of permanent magnet synchronous machines for battery electric vehicles. World Electr. Veh. J.
**2013**, 6, 673–683. [Google Scholar] [CrossRef] [Green Version] - Nobahari, A.; Darabi, A.; Hassannia, A. Axial flux induction motor, design and evaluation of steady state modeling using equivalent circuit. In Proceedings of the 2017 8th Power Electronics, Drive Systems & Technologies Conference (PEDSTC), Mashhad, Iran, 14–16 February 2017; pp. 353–358. [Google Scholar]
- Valtonen, M.; Parviainen, A.; Pyrhonen, J. Influence of the air-gap length to the performance of an axial-flux induction motor. In Proceedings of the 2008 18th International Conference on Electrical Machines, Vilamoura, Portugal, 6–9 September 2008; pp. 1–5. [Google Scholar]
- Nasiri-Gheidari, Z.; Lesani, H. Design optimization of a single-phase axial flux induction motor with low torque ripple. Przegląd Elektrotechniczny
**2012**, 88, 168–172. [Google Scholar]

**Figure 1.**Illustration of a thought experiment. (

**a**) Illustration of the Axial Flux Induction Machine (AFIM) in which the stator occupies 80% of the axial length, (

**b**) the stator occupies 50% of the axial length, and (

**c**) the stator occupies 30% of the axial length.

Designation | Name | Value | Unit |
---|---|---|---|

U | Line voltage, RMS | 7.35 | $\mathrm{V}$ |

${U}_{max}$ | Maximum line voltage | 36 | $\mathrm{V}$ |

f | Rated frequency | 50 | $\mathrm{Hz}$ |

${f}_{max}$ | Maximum frequency | 100 | $\mathrm{Hz}$ |

${D}_{o}$ | Outer diameter | 0.11 | $\mathrm{m}$ |

${D}_{i}$ | Inner diameter | 0.05 | $\mathrm{m}$ |

L | Total thickness | 0.03 | $\mathrm{m}$ |

${m}_{1}$ | The number of phases | 3 | - |

p | The number of pole pairs | 4 | - |

q | The number of slots per pole and phase | 1 | - |

${y}_{1}$ | Step of span in slot pitches | 3 | - |

${z}_{2}$ | The number of rotor slots | 32 | - |

${k}_{Cu}$ | Space factor for copper | 0.5 | - |

${k}_{Fe}$ | Space factor for iron | 0.98 | - |

${k}_{Al}$ | Space factor for aluminum | 0.97 | - |

${k}_{\vartheta 1}$ | Thermal factor | 1.32 | - |

$\delta $ | Thickness of air gap | 1 | $\mathrm{mm}$ |

${k}_{m}$ | Overload capability | 2 | - |

${B}_{c1}$ | Inductance of stator core | 1.4 | $\mathrm{T}$ |

${B}_{c2}$ | Inductance of rotor core | 1.3 | $\mathrm{T}$ |

${B}_{t1}$ | Inductance in stator teeth | 1.8 | $\mathrm{T}$ |

${q}_{a1}$ | Cros-section of 1 conductor | $2.8\times {10}^{7}$ | ${\mathrm{m}}^{2}$ |

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**MDPI and ACS Style**

Baranov, G.; Zolotarev, A.; Ostrovskii, V.; Karimov, T.; Voznesensky, A.
Analytical Model for the Design of Axial Flux Induction Motors with Maximum Torque Density. *World Electr. Veh. J.* **2021**, *12*, 24.
https://doi.org/10.3390/wevj12010024

**AMA Style**

Baranov G, Zolotarev A, Ostrovskii V, Karimov T, Voznesensky A.
Analytical Model for the Design of Axial Flux Induction Motors with Maximum Torque Density. *World Electric Vehicle Journal*. 2021; 12(1):24.
https://doi.org/10.3390/wevj12010024

**Chicago/Turabian Style**

Baranov, Georgii, Alexander Zolotarev, Valerii Ostrovskii, Timur Karimov, and Alexander Voznesensky.
2021. "Analytical Model for the Design of Axial Flux Induction Motors with Maximum Torque Density" *World Electric Vehicle Journal* 12, no. 1: 24.
https://doi.org/10.3390/wevj12010024