**Figure 1.**
Structures of permanent magnet synchronous machines with (**a**) stator and (**b**) rotor segmentations.

**Figure 1.**
Structures of permanent magnet synchronous machines with (**a**) stator and (**b**) rotor segmentations.

**Figure 2.**
Stator-segmented structure.

**Figure 2.**
Stator-segmented structure.

**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 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 4.**
Flux lines in a slot opening for a radial magnetization.

**Figure 4.**
Flux lines in a slot opening for a radial magnetization.

**Figure 5.**
Permeance function taking into account the slot openings for a non-segmented machine.

**Figure 5.**
Permeance function taking into account the slot openings for a non-segmented machine.

**Figure 6.**
Areas used for the calculation of the cogging torque.

**Figure 6.**
Areas used for the calculation of the cogging torque.

**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 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 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 9.**
Magnetic circuit of the two basics structures. (**a**) Slotted machine. (**b**) Mono-gap machine.

**Figure 9.**
Magnetic circuit of the two basics structures. (**a**) Slotted machine. (**b**) Mono-gap machine.

**Figure 10.**
Cogging torque for two mono-gap widths estimated by numerical method (FEM).

**Figure 10.**
Cogging torque for two mono-gap widths estimated by numerical method (FEM).

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

**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 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 13.**
Representation of the geometric parameters.

**Figure 13.**
Representation of the geometric parameters.

**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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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.

**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]).

**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 |

**Table 2.**
Main geometric parameters of reference machine designed to the rotor segmentation.

**Table 2.**
Main geometric parameters of reference machine designed to the rotor segmentation.

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 |

**Table 3.**
Main geometric parameters of reference machine designed to the stator segmentation.

**Table 3.**
Main geometric parameters of reference machine designed to the stator segmentation.

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 |

**Table 4.**
Test case topology for the parametric variations.

**Table 4.**
Test case topology for the parametric variations.

${\mathit{Q}}_{\mathit{s}}$ | 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.

**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.

**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 |

**Table 7.**
Ratio between analytical and numerical value calculated for the studied EM quantities.

**Table 7.**
Ratio between analytical and numerical value calculated for the studied EM quantities.

${\mathit{Q}}_{\mathit{s}}$ | 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 |