# Sizing of the Motor Geometry for an Electric Aircraft Propulsion Switched Reluctance Machine Using a Reluctance Mesh-Based Magnetic Equivalent Circuit

^{*}

## Abstract

**:**

## 1. Introduction

## 2. NASA Maxwell X-57 Electric Aircraft

#### 2.1. NASA Maxwell X-57 Aircraft

#### 2.2. Design Specifications of the HLM

^{2}. The maximum wire fill factor and current density requirements, as well as the achieved fill factor and current density for the PMSM design, are shown in Table 2.

## 3. Reluctance Mesh-Based MEC Model

#### 3.1. Developing the Reluctance Mesh-Based MEC Model

#### 3.2. Solution of the Magnetic Field

^{th}iteration, ${I}_{ref}$, ${\theta}_{ON}$, and ${\theta}_{OFF}$, phase voltages ${\mathit{V}}_{p,k}$ are applied to the MEC model. Hence, phase flux linkages at the next iteration are determined by solving Equation (3). Then, ${\mathit{I}}_{s}$ and ${\mathit{\varphi}}_{l}$ can be calculated by Equation (4) using the iterative procedure in Figure 4 [18]. Initially, zero vectors are assigned to both ${\mathit{I}}_{s}$ and ${\mathit{\varphi}}_{l}$. The relative reluctivity, ${\nu}_{r}$, of nonlinear materials is considered to be unity at first. The matrices $\mathit{N}$ and $\mathit{R}$ are then built, and the Jacobian and Hessian matrices are calculated. Next, the values of ${\mathit{I}}_{s}$ and ${\mathit{\varphi}}_{l}$ at the (k+1)

^{th}are calculated using the Gauss Newton method [18]. Then, the relative reluctivity of nonlinear mesh elements is recalculated for the ${\mathit{\varphi}}_{l}$ at the (k+1)

^{th}. The convergence criteria are checked. If ${\mathit{r}}_{e}$ is less than the tolerance $\u03f5$, the iterative procedure is stopped. Otherwise, the relative reluctivity of nonlinear elements is obtained again using the BH characteristics of the utilized steel material in the core and matrix $\mathit{R}$ is rebuilt. The iterative procedure continues until it satisfies the convergence criteria. The phase currents, self, and mutual flux linkage during the phase commutation are considered in Equation (3). Then, the self and mutual flux linkage calculated in Equation (3) are used in Equation (4) to account for the phase interaction effects while solving for the phase currents.

## 4. Sizing of the SRM Geometry Using the Proposed Reluctance Mesh-Based MEC Method

#### 4.1. Selection of the Pole Configuration

#### 4.2. Selection of Geometry Parameters

#### 4.3. Material for the Magnetic Cores

#### 4.4. Selection of Winding Configuration and Stack Length

^{2}. With 32 turns and three strands per coil wound with 17 AWG heavy-build magnet wire, the 12/16 SRM achieves 24 N·m at 5450 r/min with an RMS current of 34.3 A at the minimum DC link voltage of 385 V. The SRM achieves this requirement at an RMS current density of 10.96 A/mm

^{2}with a fill factor of 58.4%, which are lower values than the design constraints defined in Table 2. The parameters of the selected winding configuration and stack length are provided in Table 6. The mass of the stator core, rotor core, and coils is 1.53 kg, 2.33 kg, and 1.77 kg, respectively. The total mass of the electromagnetic components is 5.63 kg. The rotor mass can be further reduced by adding cut outs in the rotor back-iron. The same IEEE 95-2002 standard used in the PMSM design is considered for selecting the thickness of slot and wire insulations in the 12/16 SRM [27]. According to IEEE 95-2002, the insulation should be capable of handling twice the maximum DC link voltage plus 1000 V. The 0.13 mm Nomex-410 insulation has breakdown voltage of 3575 V. Therefore, the 0.13 mm-thick Nomex-410 insulation layer used in the PMSM is considered for the stator slots. Furthermore, the Nomex-410 can withstand up to 250 ${}^{\circ}\mathrm{C}$ [28].

#### 4.5. Thermal Analysis

## 5. Static Characteristics of the Proposed SRM Geometry

#### 5.1. Magnetic Flux Density

^{th}mesh element can be determined by [18]

_{1}, P

_{2}, and P

_{3}in Figure 15 and their percentage errors relative to the FEM are shown in Table 9. The maximum percentage error that occurred in the flux density calculation is less than 10%.

#### 5.2. Static Flux Linkage, Torque and Voltage

## 6. Dynamic Characteristics of the Proposed SRM Geometry

#### 6.1. Dynamic Current and Electromagnetic Torque

#### 6.2. Magnetic Flux Density

_{1}, P

_{2}, P

_{3}, P

_{4}, P

_{5}, and P

_{6}in Figure 26a are given in Table 11.

#### 6.3. Torque–Speed Characteristics of the 12/16 SRM

## 7. Conclusions

^{2}. This can result in a higher operating temperature, but SRM can operate at a higher temperature than the PMSM due to the lack of permanent magnets. The obtained motor geometry of the 12/16 SRM using the MEC model should be further improved with more detailed FEM simulations. The losses and efficiency of the motor should be calculated in FEM. Thermal analysis is required to determine the operating temperature of the 12/16 SRM and validate that the stack length can be further reduced by enabling a higher current density operation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Top view of the NASA Maxwell X-57 distributed propulsion aircraft [19].

**Figure 2.**An example reluctance mesh for 12/16 SRM [18].

**Figure 4.**The procedure for implementing the dynamic model, including the magnetic saturation effects and mutual coupling.

**Figure 5.**Static electromagnetic torque at a 60 A excitation current for 6/16, 12/16, and 24/16 SRMs.

**Figure 6.**Static electromagnetic torque at a 60 A excitation current for 12/8, 12/16, and 12/20 SRMs.

**Figure 11.**Characteristics of Hiperco-50 0.15 mm laminations: (

**a**) BH characteristics and (

**b**) specific core losses.

**Figure 14.**Airgap flux density waveforms at 60 A static excitation current: (

**a**) radial flux density at the unaligned position, (

**b**) tangential flux density at the unaligned position, (

**c**) radial flux density at the aligned position, and (

**d**) tangential flux density at the aligned position.

**Figure 15.**Magnetic flux density contours at 60 A static excitation current: (

**a**) unaligned position, MEC, (

**b**) unaligned position, FEM, (

**c**) aligned position, MEC, and (

**d**) aligned position, FEM.

**Figure 19.**Dynamic characteristics at 2000 r/min at 385 V DC: (

**a**) phase currents ${I}_{ref}=52$ A, ${\theta}_{ON}=-{5}^{\circ}$, and ${\theta}_{OFF}={165}^{\circ}$ and (

**b**) developed electromagnetic torque.

**Figure 20.**Dynamic characteristics at 4000 r/min at 385 V DC: (

**a**) phase currents ${I}_{ref}=50.5$ A, ${\theta}_{ON}=-{50}^{\circ}$, and ${\theta}_{OFF}={149}^{\circ}$ and (

**b**) developed electromagnetic torque.

**Figure 21.**Dynamic characteristics at 5450 r/min at 385 V DC: (

**a**) phase currents ${I}_{ref}=55$ A, ${\theta}_{ON}=-{52.5}^{\circ}$, and ${\theta}_{OFF}={125}^{\circ}$ and (

**b**) developed electromagnetic torque.

**Figure 22.**Dynamic characteristics at 5460 r/min at 385 V DC: (

**a**) phase currents ${I}_{ref}=55$ A, ${\theta}_{ON}=-{52.5}^{\circ}$, and ${\theta}_{OFF}={125}^{\circ}$ and (

**b**) developed electromagnetic torque.

**Figure 23.**Dynamic characteristics at 7000 r/min at 385 V DC: (

**a**) phase currents ${I}_{ref}=55$ A, ${\theta}_{ON}=-{79}^{\circ}$, and ${\theta}_{OFF}={100}^{\circ}$ and (

**b**) developed electromagnetic torque.

**Figure 24.**Dynamic characteristics at 8000 r/min at 385 V DC: (

**a**) phase currents ${I}_{ref}=55$ A, ${\theta}_{ON}=-{85}^{\circ}$, and ${\theta}_{OFF}={95}^{\circ}$ and (

**b**) developed electromagnetic torque.

**Figure 25.**Airgap flux density at ${\theta}_{elec,phA}={240}^{\circ}$ for phase currents in Figure 21: (a) radial component and (b) tangential component.

**Figure 26.**Magnetic flux density contours at ${\theta}_{elec,phA}={240}^{\circ}$ for the current waveform in Figure 21.

**Figure 27.**Motor characteristics for the peak operating point of the 12/16 SRM at 385 V DC link voltage: (

**a**) torque–speed characteristics and (

**b**) power–speed characteristics.

Electromagnetic Design Parameters | |
---|---|

Peak output power | 13.7 kW |

Peak torque | 24 N·m |

Minimum torque density | 18.8 N·m/L |

Base speed | 5450 r/min |

RMS phase current | 35 A |

DC link voltage | 385–538 V |

Number of phases | 3 |

Geometry Constraints | |

Maximum motor diameter (with casing) | 161.5 mm |

Maximum stator outer diameter (without casing) | 156.45 mm |

Maximum stator axial length including end turns | 66.4 mm |

Expected Performance | |

1. Provide 24 N·m of torque between 2000 and 5450 r/min and 22 N·m of torque at 5460 r/min. | |

2. Capable of producing 10.5 kW output power at 5460 r/min and at 460 V DC link voltage with a minimum efficiency of 93%. |

Characteristic | Maximum Allowable Value | Achieved by PMSM Design |
---|---|---|

Wire fill factor | 60% | 58.4% |

Current density | 11 A/mm^{2} | 10.7 A/mm^{2} |

**Table 3.**Average torque and torque ripple of the torque profiles in Figure 8.

Airgap Length | Avg. Torque | Pk–Pk Torque Ripple | Torque Density |
---|---|---|---|

0.4 mm | 23.6 N·m | 14 N·m | 18.5 N·m/L |

0.35 mm | 24.6 N·m | 14.59 N·m | 19.3 N·m/L |

0.3 mm | 25.5 N·m | 15.23 N·m | 20 N·m/L |

Parameter | Value |
---|---|

Outer diameter (${D}_{out}$) | 156.45 mm |

Bore diameter (${D}_{bore}$) | 115 mm |

Shaft diameter (${D}_{shaft}$) | 50 mm |

Airgap length (g) | 0.35 mm |

Stator pole arc angle (${\beta}_{s}$) | 8.2^{∘} |

Rotor pole arc angle (${\beta}_{r}$) | 8.5^{∘} |

Stator pole height (${h}_{s}$) | 15 mm |

Rotor pole height (${h}_{r}$) | 10 mm |

Stator back-iron thickness (${h}_{sb}$) | 5.725 mm |

Rotor back-iron thickness (${h}_{rb}$) | 22.15 mm |

Characteristic | Hiperco-50 | AFK-502 | Vacodur-49 | |
---|---|---|---|---|

Specific core loss | 400 Hz/2 T | 56.7 W/kg | 70 W/kg | 60 W/kg |

1000 Hz/2 T | 301 W/kg | 320 W/kg | 330 W/kg | |

Saturation flux density | 2.3 T | 2.32 T | 2.3 T | |

Resistivity | 0.4 $\mathsf{\mu}\mathsf{\Omega}\xb7\mathrm{m}$ | 0.4 $\mathsf{\mu}\mathsf{\Omega}\xb7\mathrm{m}$ | 0.42 $\mathsf{\mu}\mathsf{\Omega}\xb7\mathrm{m}$ | |

Yield strength | 331 MPa–414 Mpa | 300 MPa–420 Mpa | 210 MPa–390 Mpa | |

Thermal conductivity | 29.8 W/m·K | 32 W/m·K | 32 W/m·K |

Parameter | Value |
---|---|

Number of turns per coil (${N}_{t}$) | 32 |

Number of strands | 3 |

Wire fill factor | 58.4% |

Maximum current density | 10.96 A/mm^{2} |

Wire gauge | 17 AWG heavy-build |

Coil resistance (${R}_{coil}$) | 0.02768 $\mathsf{\Omega}$ |

Stator core stack length (${L}_{stk}$) | 47 mm |

Estimated stacking factor | 97% |

Estimated end turn length (${L}_{end}$) | 9.7 mm |

Total axial length (${L}_{ax}$) | 66.4 mm |

Design Parameters | SRM Design | PMSM Design |
---|---|---|

Outer diameter | 156.45 mm | 156.45 mm |

Axial length | 66.4 mm | 51.5 mm |

Torque density | 19.3 Nm/L | 24.2 Nm/L |

Current density | 10.96 A/mm^{2} (<11 A/mm^{2}) | 10.7 A/mm^{2} (<11 A/mm^{2}) |

Fill factor | 58.4% (<60%) | 58.4% (<60%) |

Efficiency | 95.46% | 96% |

Component | Material | Thermal Conductivity |
---|---|---|

Housing (heatsink) | Aluminum 2024-T3 | 120 W/m^{2} |

Stator and rotor | Hiperco-50 | 20 W/m^{2} |

Windings | Copper | 400 W/m^{2} |

Stator slot voids | Epoxy resin | 1 W/m^{2} |

Slot linear | Nomex 410 | 0.14 W/m^{2} |

**Table 9.**Flux density comparison at points P

_{1}, P

_{2}, and P

_{3}in Figure 15 in static simulations.

Position | Point P_{1} | Point P_{2} | Point P_{3} | ||||||
---|---|---|---|---|---|---|---|---|---|

MEC | FEM | % Error | MEC | FEM | % Error | MEC | FEM | % Error | |

Unaligned | 0.98 T | 0.96 T | 2.1% | 1.43 T | 1.31 T | 9.1% | 0.55 T | 0.54 T | 1.9% |

Aligned | 1.89 T | 1.8 T | 5% | 2.41 T | 2.33 T | 3.4% | 2.31 T | 2.25 T | 2.7% |

**Table 10.**Comparison of RMS current, average torque, and peak–peak torque ripple from the MEC and FEM models.

Speed | RMS Current | Avg. Torque | Pk–Pk Torque Ripple | ||||||
---|---|---|---|---|---|---|---|---|---|

MEC | FEM | % Error | MEC | FEM | % Error | MEC | FEM | % Error | |

2000 r/min | 35.8 A | 34.9 A | 3.2% | 24.2 N·m | 25.3 N·m | 4.3% | 11.7 N·m | 8.7 N·m | 25.6% |

4000 r/min | 35.5 A | 34.8 A | 2% | 24.5 N·m | 24.4 N·m | 0.6% | 7.86 N·m | 10.37 N·m | 9.7% |

5450 r/min | 34.3 A | 34.2 A | 0.3% | 24.4 N·m | 24 N·m | 2.5% | 14.59 N·m | 13.97 N·m | 4.44% |

5460 r/min | 34 A | 34.1 A | 0.3% | 24.4 N·m | 24 N·m | 2.5% | 14.86 N·m | 14.18 N·m | 4.8% |

6000 r/min | 32 A | 33. 6A | 4.8% | 23.4 N·m | 23.5 N·m | 0.4% | 15.95 N·m | 14.89 N·m | 7.12% |

7000 r/min | 31.7 A | 32.5 A | 2.5% | 20.1 N·m | 19.9 N·m | 1% | 20.45 N·m | 20.13 N·m | 1.59% |

8000 r/min | 27.8 A | 28.6 A | 2.8% | 15.7 N·m | 15.6 N·m | 0.6% | 19.5 N·m | 18.72 N·m | 4.17% |

**Table 11.**Flux density comparison at points P

_{1}, P

_{2}, P

_{3}, P

_{4}, P

_{5}, and P

_{6}for the dynamic operation in Figure 26.

Location | MEC | FEM | % Error |
---|---|---|---|

Point P_{1} | 1.01 T | 0.89 T | 13.5% |

Point P_{2} | 0.39 T | 0.38 T | 2.6% |

Point P_{3} | 2.24 T | 2.26 T | 0.9% |

Point P_{4} | 2.03 T | 1.98 T | 2.5% |

Point P_{5} | 0.61 T | 0.52 T | 17.3% |

Point P_{6} | 0.63 T | 0.54 T | 16.7% |

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**MDPI and ACS Style**

Watthewaduge, G.; Bilgin, B.
Sizing of the Motor Geometry for an Electric Aircraft Propulsion Switched Reluctance Machine Using a Reluctance Mesh-Based Magnetic Equivalent Circuit. *Machines* **2023**, *11*, 59.
https://doi.org/10.3390/machines11010059

**AMA Style**

Watthewaduge G, Bilgin B.
Sizing of the Motor Geometry for an Electric Aircraft Propulsion Switched Reluctance Machine Using a Reluctance Mesh-Based Magnetic Equivalent Circuit. *Machines*. 2023; 11(1):59.
https://doi.org/10.3390/machines11010059

**Chicago/Turabian Style**

Watthewaduge, Gayan, and Berker Bilgin.
2023. "Sizing of the Motor Geometry for an Electric Aircraft Propulsion Switched Reluctance Machine Using a Reluctance Mesh-Based Magnetic Equivalent Circuit" *Machines* 11, no. 1: 59.
https://doi.org/10.3390/machines11010059