# Analytical Models for Fast and Accurate Calculation of Electromagnetic Performances of Segmented Permanent Magnet Synchronous Machines with Large Angular Gaps

^{*}

*Applied Sciences*: Invited Papers in Electrical, Electronics and Communications Engineering Section)

## Abstract

**:**

^{TM}2D) for the specifications of a large diameter, low speed tidal high power current turbine generator. The presented method allows fast and accurate evaluation of the performances of this kind of particular machine and can be used in a systematic design process.

## 1. Introduction

## 2. Context, Hypotheses and Segmentation Protocol

## 3. EM Analytical Model

#### 3.1. EM Models for Non-Segmented PMS Machines

#### 3.1.1. Magnetic Flux Density Due to PM

#### 3.1.2. Cogging Torque

#### 3.1.3. Electromotive Forces and Electromagnetic Torque

#### 3.2. New Models for Stator-Segmented Structures

#### 3.2.1. Magnetic Flux Density Due to PM

^{TM}2D.

#### 3.2.2. Cogging Torque

- Superposition principleThe estimation of the cogging torque for a segmented structure is based on the superposition principle [11]. It is assumed that the contributions of slots and angular gaps to the cogging torque are independent and can be added. The cogging torque is evaluated for two basic structures: a non-segmented slotted machine with the same geometry as the studied segmented machine, and a structure with the same dimensions with no slot and with only one gap on the stator (named mono-gap structure). An example of this theoretical mono-gap configuration is presented in Figure 9b. The contribution of slots and angular gaps are estimated separately and then added. The classical analytical model for calculating the cogging torque (as in [10]) is used to estimate the first contribution (due to slots).For the proposed mono-gap model, it is supposed that angular gaps contain entire number of pole pairs. In fact, to keep a regular winding distribution, the gap angular width must be a multiple of the winding period which corresponds to a number of slots defined by Equation (7) and called reduced number of slots.$${Q}_{s0-red}={S}_{pp}\times 2\times \frac{{p}_{0}}{ppcm({Q}_{s0},{p}_{0})}\times m$$In this equation, ${S}_{pp}$ is the number of slots per pole and per phase.2D FEM (with the software Flux
^{TM}2D) is used in a preliminary step to validate the mono-gap model. This numerical study shows that the gap width does not influence the cogging torque peak value nor its frequency for a mono-gap structure, if the gaps contain entire number of pole pair. Figure 10 presents the cogging torque calculated by 2D FEM for two single-gap structures with the same geometry (as depicted in Figure 9b) and two different gap widths, in function of the rotor position ${\theta}_{r}$. It can then be seen that the contribution of one gap to the cogging torque is the same for similar structures with several gap widths.For a stator-segmented structure, the cogging torque is estimated for the whole slotted machine ${T}_{c,slotted}$ (Figure 9a) (with the Equation (3)). This contribution is multiplied by the active part proportion $(1-{p}_{gap})$. In addition, the mono-gap contribution to the cogging torque ${T}_{c,MonoGap}$ is evaluated and multiplied by the number of gaps ${n}_{gap}$ included in the structure, as shown by (8). This calculation method is valid because all the gaps are in the same relative position relatively to the rotor poles.$${T}_{c,StatorGap}\left({\theta}_{r}\right)={T}_{c,slotted}\left({\theta}_{r}\right)\times (1-{p}_{gap})+{T}_{c,MonoGap}\left({\theta}_{r}\right)\times {n}_{gap}$$ - Mono-gap cogging torque estimationThe proposed mono-gap model to evaluate the cogging torque is based on the classical analytical calculation for a non-segmented structure presented in [10]. The estimation of the mono-gap cogging torque depends on the magnetic pressure in each active sector/gap interface. The calculation is done on the red shaded area of the Figure 6. This calculation area corresponds to half a pole pitch in each gap extremity. The method is extended to the stator-segmented machine by using the flux density created by permanent magnets into gaps and the extended permeance function adapted for gaps in the stator.The proposed mono-gap model is validated for the mono-gap structure used for the previous numerical study (Figure 10). Figure 11 presents the mono-gap cogging torque obtained with the numerical (2D FEM) and proposed analytical methods for a mono-gap machine with a gap width of ten pole pitches.For the test case of a large diameter structure, the proposed analytical mono-gap model gives good evaluations of the cogging torque related to the mono-gap effect.

#### 3.2.3. Electromotive Forces

#### 3.3. New Models for Rotor-Segmented Structures

#### 3.3.1. Magnetic Flux Density Due to PM

#### 3.3.2. Cogging Torque

#### 3.3.3. Electromotive Forces

## 4. Validation

#### 4.1. Methods and Objectives

#### 4.1.1. Non-Segmented Reference Machines

#### 4.1.2. Segmented Machines Test Cases

#### 4.1.3. Methodology

^{TM}2D software it takes around 15 min, and around 30 min for the SS-machine (simulation of 1/7th of the whole machine) in a Intel(R) Core(TM) i5-6500 CPU with 8 Go of RAM. This time, that it is rather long, attests that a fast calculation procedure can be useful for systematic design processes.

#### 4.2. No-Load Flux Densities

#### 4.2.1. Stator-Segmented Case

#### 4.2.2. Rotor-Segmented Case

#### 4.3. Cogging Torque

#### 4.3.1. Stator-Segmented Case

#### 4.3.2. Rotor-Segmented Case

#### 4.4. Electromotive Force

#### 4.4.1. Stator-Segmented Case

#### 4.4.2. Rotor-Segmented Case

#### 4.5. Electromagnetic Torque

#### 4.5.1. Stator-Segmented Case

#### 4.5.2. Rotor-Segmented Case

^{TM}2D as shown in Table 6.

#### 4.6. Summary

## 5. Conclusions

^{TM}2D) results in several typical test cases exhibits the accuracy of the proposed methods.

## Author Contributions

## Funding

## Acknowledgments

^{TM}.

## Conflicts of Interest

## Abbreviations

PMSM | Permanent Magnet Synchronous Machine |

PM | permanent magnet |

FEM | Finite Element Method |

EM | electromagnetic |

EMF | electromotive force |

SS-machine | stator-segmented machine |

RS-machine | rotor-segmented machine |

## References

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**Figure 1.**Structures of permanent magnet synchronous machines with (

**a**) stator and (

**b**) rotor segmentations.

**Figure 3.**Magnetic circuits of the studied machines: (

**a**) non-segmented machine, (

**b**) stator-segmented machine with amagnetic gaps, (

**c**) rotor-segmented machine with amagnetic gaps.

**Figure 7.**Permeance function (relative value) for the machine under consideration with stator segmentation by an amagnetic material (336 slots, 140 pole pairs, 7 gaps, and 25% of gap).

**Figure 8.**Analytical evaluation of the ratio between the magnetic flux density of a machine with an infinite air gap and of a slotless machine.

**Figure 11.**Cogging torques obtained by numerical and analytical models for a machine with only one gap of ten-pole pitches.

**Figure 12.**Modulation function to take into account the parts without magnet for a structure with ${p}_{gap}$ around 20% and ${n}_{gap}=6$ gaps.

**Figure 14.**Axial length depending on the proportion of segmentation, with and without considering the 3D correction, for the machines (

**a**) with ${S}_{pp}=2/5$ and (

**b**) with ${S}_{pp}=1/2$.

**Figure 15.**Studied machines. (

**a**) Non-segmented machine with ${S}_{pp}$ = 2/5. (

**b**) Stator-segmented machine with 25% of amagnetic gap. (

**c**) Non-segmented machine with ${S}_{pp}$ = 1/2. (

**d**) Rotor-segmented machine with around 20% of amagnetic gap.

**Figure 16.**Permeance function for the SS-machine with ${p}_{gap}$ = 0.50. (

**a**) ${n}_{gap}$ = 7 gaps. (

**b**) ${n}_{gap}$ = 14 gaps.

**Figure 17.**Waveforms and spectral analysis of magnetic flux density for the SS-machine with different p

_{gap}and n

_{gap}(

**a1**,

**a2**) Reference machine. (

**b1**,

**b2**) SS-machine with p

_{gap}= 0.25 and n

_{gap}= 7 gaps. (

**c1**,

**c2**) SS-machine with p

_{gap}= 0.50 and n

_{gap}= 7 gaps. (

**d1**,

**d2**) SS-machine with p

_{gap}= 0.50 and n

_{gap}= 14 gaps.

**Figure 18.**Comparison between numerical and analytical calculation of magnetic flux densities due to PM for the SS-machine with ${n}_{gap}$ = 7 gaps and with different ${p}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 19.**Comparison between numerical and analytical calculation of magnetic flux densities due to PM for the SS-machine with ${p}_{gap}$ = 0.50 and with different ${n}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 20.**Waveforms and spectral analysis of magnetic flux density for the RS-machine with different p

_{gap}and n

_{gap}= 6 gaps. (

**a1**,

**a2**) Reference machine. (

**b1**,

**b2**) RS-machine with p

_{gap}= 0.217. (

**c1**,

**c2**) RS-machine with p

_{gap}= 0.391.

**Figure 21.**Comparison between numerical and analytical calculation of magnetic flux densities due to PM for the RS-machine with ${n}_{gap}$ = 6 gaps and with different ${p}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 22.**Waveforms and spectral analysis of cogging torques for the SS-machine with different p

_{gap}and n

_{gap}. (

**a1**–

**a3**) Reference machine. (

**b1**–

**b3**) SS-machine with p

_{gap}= 0.25 and n

_{gap}= 7 gaps. (

**c1**–

**c3**) SS-machine with p

_{gap}= 0.50 and n

_{gap}= 7 gaps. (

**d1**–

**d3**) SS-machine with p

_{gap}= 0.50 and n

_{gap}= 14 gaps.

**Figure 23.**Comparison between numerical and analytical calculation of cogging torque for the SS-machine with ${n}_{gap}$ = 7 gaps and with different ${p}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 24.**Comparison between numerical and analytical calculation of cogging torque for the SS-machine with ${p}_{gap}$ = 0.50 and with different ${n}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 25.**Waveforms and spectral analysis of cogging tprques for the RS-machine with different p

_{gap}and n

_{gap}= 6 gaps. (

**a1**–

**a3**) Reference machine. (

**b1**–

**b3**) p

_{gap}= 0.217. (

**c1**–

**c3**) p

_{gap}= 0.391.

**Figure 26.**Comparison between numerical and analytical calculation of cogging torque for the RS-machine with ${n}_{gap}$ = 6 gaps and with different ${p}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 27.**Waveforms of electromotive forces in one phase (neutral phase) at 1rd/s for the SS-machine with different ${p}_{gap}$. (

**a**) Reference machine. (

**b**) SS-machine with ${p}_{gap}=0.25$ and ${n}_{gap}$ = 7 gaps. (

**c**) SS-machine with ${p}_{gap}=0.50$ and ${n}_{gap}$ = 7 gaps.

**Figure 28.**Comparison between electromotive forces obtained by numerical and analytical methods in one phase at 1 rd/s for the SS-machine with ${n}_{gap}$ = 7 gaps and with different ${p}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 29.**Calculated waveforms of electromotive forces at 1 rd/s for the RS-machine with different ${p}_{gap}$. (

**a**) Reference machine. (

**b**) RS-machine with ${p}_{gap}=0.217$ and ${n}_{gap}$ = 7 gaps. (

**c**) RS-machine with ${p}_{gap}=0.391$ and ${n}_{gap}$ = 6 gaps.

**Figure 30.**Comparison between EMF obtained by numerical and analytical methods at 1 rd/s for the RS-machine with ${n}_{gap}$ = 6 gaps and with different ${p}_{gap}$. (

**a**) Peak values. (

**b**) Correlation.

**Figure 31.**Waveforms of EM torques for the SS-machine with different ${p}_{gap}$. (

**a**) Reference machine. (

**b**) SS-machine with ${p}_{gap}=0.25$ and ${n}_{gap}$ = 7 gaps. (

**c**) SS-machine with ${p}_{gap}=0.50$ and ${n}_{gap}$ = 7 gaps.

**Figure 32.**Comparison between EM torques obtained by numerical and analytical methods for the SS-machine with ${n}_{gap}$ = 7 gaps and with different ${p}_{gap}$. (

**a**) Amplitude of the ripples. (

**b**) Correlation.

**Figure 33.**Waveforms of EM torques for the RS-machine with different ${p}_{gap}$. (

**a**) Reference machine. (

**b**) RS-machine with ${p}_{gap}=0.217$ and ${n}_{gap}$ = 7 gaps. (

**c**) RS-machine with ${p}_{gap}=0.391$ and ${n}_{gap}$ = 6 gaps.

**Figure 34.**Comparison between EM torques obtained by numerical and analytical methods for the RS-machine with ${n}_{gap}$=6 and with different ${p}_{gap}$. (

**a**) Amplitude of the ripples. (

**b**) Correlation.

**Table 1.**Main parameters of Rim-Driven machine (derived from [7]).

Power P (kW) | 300 |

Turbine speed N (rpm) | 15 |

Air gap length ${h}_{g}$ (m) | 0.02 |

Linear electric loading ${A}_{L}$ (kA×m${}^{-1}$) | 60 |

Current density J (A×mm${}^{-2}$) | 4 |

Magnet width (${\tau}_{m}$) to pole pitch (${\tau}_{p}$) ratio ${\beta}_{m}$ | 0.7 |

Stator internal diameter D (m) | 11.151 |

Magnet remanence ${B}_{r}$ (T) | 1.2 |

Magnet relative permeability ${\mu}_{r}$ | 1.05 |

Number of slots per pole and per phase ${S}_{pp}$ | 1/2 |

1st harmonic winding coefficient ${k}_{w}$ | 0.866 [14] |

Pole pair angular width (rad) | 2$\pi $/138 |

Magnet height ${h}_{m}$ (m) | 0.0208 |

Slot depth ${h}_{s}$ (m) | 0.0434 |

Tooth to slot pitch ratio ${\beta}_{t}$ | 0.54 |

Stator and rotor yoke height hy (m) | 0.0236 |

Iron axial length Lz0 (m) | 0.0564 |

Number of slots per pole and per phase ${S}_{pp}$ | 2/5 |

1st harmonic winding coefficient ${k}_{w}$ | 0.966 [14] |

Pole pair angular width (rad) | 2$\pi $/140 |

Magnet height ${h}_{m}$ (m) | 0.0210 |

Slot depth ${h}_{s}$ (m) | 0.0472 |

Tooth to slot pitch ratio ${\beta}_{t}$ | 0.58 |

Stator and rotor yoke height ${h}_{y}$ (m) | 0.0248 |

Iron axial length ${L}_{z0}$ (m) | 0.0518 |

${\mathit{Q}}_{\mathit{s}}$ | p | Segmented Part | ${\mathit{p}}_{\mathit{gap}}$ | ${\mathit{n}}_{\mathit{gap}}$ |
---|---|---|---|---|

336 | 140 | none | 0 | 0 |

336 | 140 | stator | 0.25 | 7 |

336 | 140 | stator | 0.50 | 7 |

336 | 140 | stator | 0.50 | 14 |

414 | 138 | none | 0 | 0 |

414 | 138 | rotor | 0.217 | 6 |

414 | 138 | rotor | 0.391 | 6 |

**Table 5.**Evolution of the mean electromagnetic torques for the SS-machine with ${n}_{gap}$ = 7 gaps.

Gap Proportion | 0% | 25% | 50% |
---|---|---|---|

Numerical $\u2329{T}_{em}\u232a\phantom{\rule{0.166667em}{0ex}}(\mathrm{kN}\times \mathrm{m})$ | 226 | 213 | 213 |

Analytical $\u2329{T}_{em}\u232a\phantom{\rule{0.166667em}{0ex}}(\mathrm{kN}\times \mathrm{m})$ | 222 | 209 | 209 |

Ratio (%) | 98.2 | 98.1 | 98.1 |

**Table 6.**Evolution of the mean electromagnetic torques for the RS-machine with ${n}_{gap}$ = 6 gaps.

Gap Proportion | 0% | ~20% | ~40% |
---|---|---|---|

Numerical $\u2329{T}_{em}\u232a\phantom{\rule{3.33333pt}{0ex}}(\mathrm{kN}\times \mathrm{m})$ | 215 | 205 | 198 |

Analytical $\u2329{T}_{em}\u232a\phantom{\rule{3.33333pt}{0ex}}(\mathrm{kN}\times \mathrm{m})$ | 217 | 208 | 201 |

Ratio (%) | 99.1 | 98.6 | 98.5 |

${\mathit{Q}}_{\mathit{s}}$ | p | Segmented Part | ${\mathit{p}}_{\mathit{gap}}$ | ${\mathit{n}}_{\mathit{gap}}$ | Maximum ${\mathit{B}}_{/\mathit{R}}$ | Maximum ${\mathit{C}}_{\mathit{d}}$ | Maximum EMF |
---|---|---|---|---|---|---|---|

336 | 140 | none | 0 | 0 | 0.93 | 0.47 | 1.00 |

336 | 140 | stator | 0.25 | 7 | 0.92 | 0.92 | 1.00 |

336 | 140 | stator | 0.50 | 7 | 0.92 | 0.93 | 1.00 |

336 | 140 | stator | 0.50 | 14 | 0.93 | 0.96 | 1.01 |

414 | 138 | none | 0 | 0 | 0.93 | 0.92 | 1.0 |

414 | 138 | rotor | 0.217 | 6 | 0.91 | 0.93 | 1.01 |

414 | 138 | rotor | 0.391 | 6 | 0.91 | 0.93 | 1.01 |

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**MDPI and ACS Style**

Fleurot, E.; Scuiller, F.; Charpentier, J.-F.
Analytical Models for Fast and Accurate Calculation of Electromagnetic Performances of Segmented Permanent Magnet Synchronous Machines with Large Angular Gaps. *Appl. Sci.* **2021**, *11*, 459.
https://doi.org/10.3390/app11010459

**AMA Style**

Fleurot E, Scuiller F, Charpentier J-F.
Analytical Models for Fast and Accurate Calculation of Electromagnetic Performances of Segmented Permanent Magnet Synchronous Machines with Large Angular Gaps. *Applied Sciences*. 2021; 11(1):459.
https://doi.org/10.3390/app11010459

**Chicago/Turabian Style**

Fleurot, Eulalie, Franck Scuiller, and Jean-Frédéric Charpentier.
2021. "Analytical Models for Fast and Accurate Calculation of Electromagnetic Performances of Segmented Permanent Magnet Synchronous Machines with Large Angular Gaps" *Applied Sciences* 11, no. 1: 459.
https://doi.org/10.3390/app11010459