# Design, Modelling and Optimization of a High Power Density Axial Flux SRM with Reduced Torque Ripple for Electric Vehicles

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Hybrid Design Procedure for the DSAFSRM

#### 2.1. DSAFSRM’s Design Equations

_{dc}is the dc link voltage, and R

_{ph}and λ

_{ph}are the wire resistance and flux linkage per phase, respectively. It should be noted that (1) applies to the period of time when the rotor moves from the unaligned to aligned position and i

_{(t)}reaches to its peak value (I), which is assumed constant. By neglecting the phase resistance, it can be re-written as in (2) [42]:

^{sat}

_{a}and L

_{u}are saturated aligned and unaligned phase inductances, respectively. T is the period of time when the rotor moves from the unaligned to aligned position and can be calculated as in (3).

_{m}is the rotor angular velocity and β

_{s}is the stator pole shoe angle. The variable K

_{L}is defined as the ratio of unaligned inductance to saturated aligned inductance as in (4).

_{ph}is the coil turn per phase, and B

_{pole}and A

_{pole}are the average magnetic flux density and area, respectively, of one stator pole. The stator pole area (A

_{pole}) can be calculated as in (7).

_{o}and D

_{i}are the outer and inner diameters, respectively, of the stator disk, and K

_{s}is the ratio of the inner diameter to the outer diameter. Thus, (6) can be re-written as in (8).

_{d}is the ratio of the actual duty cycle of the phase current to its maximum allowable duty cycle, which is calculated as in (11).

_{r}is the number of rotor poles, and α is the actual conduction angle in each stroke. Finally, the combination of (5), (8), (9) and (10) yields the output power of the double-stator single-rotor AFSRM.

_{m}is the mechanical speed of the rotor in rpm. The above equation can be rearranged to calculate the outer diameter of the DSAFSRM:

_{rms}):

_{w}and F

_{f}are the number of phases, wire cross-sectional area and slot fill factor, respectively. By choosing rectangular stator slots, the slot opening (W

_{so}) and slot height (d

_{sh}) can be determined by the following formulations.

_{w}, α and θ are the conductor resistance per meter at 20 °C, resistance coefficient, and working temperature, respectively.

_{L}in (4)). This signifies the need for a complementary approach to increase the accuracy of the design procedure without imposing considerable complexity and computational effort. To do so, a non-linear MEC is developed in the next section and then combined with the classical equations, leading to a hybrid design procedure.

#### 2.2. Non-Linear MEC

_{L}is very sensitive to iron saturation level and the machine’s detailed geometry, it is not possible to calculate it using classical formulations. In this section, a non-linear MEC is developed to calculate the value of K

_{L}, leading to a more accurate analytical design procedure. The MEC is only established at unaligned and aligned positions of the rotor since the value of K

_{L}is solely dependent on these two positions.

#### 2.2.1. Permeance Network

_{i}, i = 1, 2, 3, …) at all nodes is determined by solving the network. Then, the flux generated by the active phase can be calculated as in (22).

_{L}is simply calculated as in (23).

#### 2.2.2. Air Gap Permeances

_{0}) is associated with the front surfaces of rotor and stator whose corresponding permeance can be calculated as in (24).

_{1}is associated with the rotor’s front surface and stator’s side surface. This flux tube consists of a straight line and a curved line, which for simplicity, is assumed to be a circular curve. The permeance associated with this flux tube is calculated as in (25).

_{2}) is related to the side surfaces of the stator and rotor. As can be seen from Figure 3a, FT

_{2}is composed of two curved paths and one straight path whose corresponding permeances are calculated as in (26).

_{3}and FT

_{4}are comprised of two straight lines and a circular curve which relate a side surface to a front surface. The corresponding permeances of FT

_{3}and FT

_{4}are written as in (28) and (29).

_{3}and FT

_{4}, the air gap permeance between the rotor and stator poles at the unaligned position is calculated as in (30).

_{ss}in Figure 2b, is calculated as in (31).

_{sc}and G

_{s}, which are air gap permeances associated with the current-carrying and non-current-carrying slots, respectively, are calculated as follows.

#### 2.2.3. Non-Linear Hybrid Algorithm

_{L}requires taking the iron saturation effect into account. In this regard, permeances positioned on the main flux path (highlighted in red in Figure 2) are considered to be non-linear. They are subjected to a considerable amount of magnetic saturation, especially at the aligned position. It should be mentioned that the stator teeth without a magneto motive force (MMF) source are not subjected to saturation. Therefore, their corresponding permeances are considered linear (i.e., G

_{t2}, G

_{t3}). The equations associated with the iron part permeances are given in (33).

_{t1}and μ

_{y1}are the permeabilities of the aligned stator tooth and its adjacent yoke, respectively.

_{L}and efficiency (η) are initialized, which is followed by obtaining the machine’s dimensions and electrical parameters using the classical formulations (10)–(20). At this stage, the algorithm moves to the MEC section by developing DSAFSRM’s permeance network. Next, the permeances associated with air gap and linear iron parts are calculated. In order to determine the permeability of the saturated parts, a function is fitted on the B-H curve of the iron. Then, B and H values are calculated using an iterative algorithm. Thus, the non-linear permeances are calculated and the permeance network is solved using (21, Appendix A). Next, K

_{L}is calculated and compared with its initial value. If the assumed values for K

_{L}and η match their initial values, the algorithm finishes; otherwise, it returns to the step in which the two parameters are initialized.

## 3. Electromagnetic Performance Analysis

_{r}. In the case of the 18/12 DSAFSRM, the conduction period should not exceed 15°. However, due to the machine’s large inductance, the current would not immediately fall to zero after reaching the aligned position, which results in negative torque production. Consequently, the phase switches should be turned off at some point before reaching the aligned position. In this regard, it is convenient to take the conduction angle as 360°/mN

_{r}, where m is the number of phases. Thus, the conduction angle would be 10° in the DSAFSRM under study.

## 4. Torque Ripple Minimization

#### 4.1. Conventional Pole Coverage Optimization

_{s}and B

_{r}are the stator pole and rotor pole coverage, respectively, which are defined as the ratio of the pole shoe to their corresponding pole pitch. In this regard, the results of the sensitivity analysis are presented in Figure 8. As can be seen, the DSAFSRM’s torque ripple is influenced by these two parameters fluctuating between 115% and 130%. How ever, this range of torque ripple cannot be tolerated in many applications, including in EV powertrains. Consequently, alternative methods should be adopted to effectively reduce the torque ripple of the DSAFSRM.

#### 4.2. Proposed Two-Step Optimization

_{s}and B

_{r}) are evaluated when a and b are fixed at their optimum values. In this regard, the torque ripple versus B

_{s}and B

_{r}is shown in Figure 10. It can be seen that the torque ripple is very sensitive to variations in B

_{s}and B

_{r}, fluctuating between 35% and 75%. Selecting (B

_{s},B

_{r}) = (0.525,0.5) and/or (B

_{s},B

_{r}) = (0.5,0.525) would yield the minimum torque ripple (35%). Examining the DSAFSRM’s efficiency versus B

_{s}and B

_{r}in Figure 11 can determine the most suitable combination. It can be seen that the efficiency is greatly affected by the rotor pole coverage (B

_{r}) rather than B

_{s}. It is also understood that the relationship between efficiency and B

_{r}is inverse linear, where higher B

_{r}results in lower efficiency. Consequently, (B

_{s},B

_{r}) = (0.525,0.5) would be a better choice since it yields the minimum torque ripple at a higher efficiency value.

## 5. Comparative Analysis

^{3}), while the double-sided radial flux SRM has the lowest power density (6.66 N m/cm

^{3}).

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{m}has different values in the aligned and non-aligned positions as in (A2) and (A3), respectively.

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**Figure 1.**Configurations of segment rotor double-sided axial flux SRMs. (

**a**) Single-stator and double-rotor topology. (

**b**) Double-stator and single-rotor topology. (

**c**) Pancake topology.

**Figure 2.**Non-linear permeance network of the DSAFSRM (non-linear permeances highlighted in red). (

**a**) Full model at aligned position. (

**b**) Reduced model at aligned position. (

**c**) Full model at unaligned position. (

**d**) Reduced model at unaligned position.

**Figure 3.**Parametric dimensions of the DSAFSRM and air gap flux tubes. (

**a**) Aligned position. (

**b**) Unaligned position.

**Figure 6.**Vector plot of the magnetic flux density distribution in the DSAFSRM at the rated current. (

**a**) Unaligned position. (

**b**) Aligned position.

**Figure 8.**Surface plot of torque ripple versus stator and rotor pole coverage using the conventional optimization method.

**Figure 10.**Surface plot of torque ripple versus stator and rotor pole coverage using the double-layer optimization method.

**Figure 12.**Magnetic flux density distribution in the double-sided radial flux topology at rated current. (

**a**) Aligned position. (

**b**) Unaligned position.

**Figure 13.**Flux linkage versus armature current of the DSAFSRM and radial flux SRM at aligned and unaligned position.

**Figure 14.**Full-load torque profile of the radial flux SRM and the proposed DSAFSRM before and after optimization.

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

Output power (kW) | P_{o} | 100 |

Nominal speed (rpm) | n_{m} | 3000 |

DC link voltage (V) | V_{dc} | 440 |

Outer diameter (mm) | D_{o} | 370 |

Inner diameter (mm) | D_{i} | 148 |

Active length (mm) | L_{m} | 111 |

Number of stator poles | N_{s} | 18 |

Number of rotor poles | N_{r} | 12 |

Air gap length per side (mm) | L_{g} | 0.5 |

Number of turns per coil | N_{c} | 5 |

Stator slot opening (mm) | W_{ss} | 20 |

Stator slot depth (mm) | T_{st} | 28 |

Stator yoke thickness | T_{sy} | 18 |

Rotor length | T_{rt} | 18 |

Slot fill factor | F_{f} | 50 |

Parameter | Optimized DSAFSRM | Non-Optimized DSAFSRM | Double-Sided Radial Flux SRM |
---|---|---|---|

Rated speed (rpm) | 3000 | ||

Outer diameter (mm) | 370 | ||

Active length (mm) | 111 | ||

Current density (A/mm^{2}) | 8 | ||

Air gap length (mm) | 0.5 | ||

Output power (kW) | 100 | 99.7 | 79.5 |

Copper loss (kW) | 2.4 | 2.3 | 2.1 |

Core loss (kW) | 1.5 | 1.8 | 1.7 |

Efficiency (%) | 96.2 | 96 | 95.4 |

Power density (W/cm^{3}) | 8.38 | 8.35 | 6.66 |

Torque ripple (%) | 35 | 120 | 92 |

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**MDPI and ACS Style**

Ajamloo, A.M.; Ibrahim, M.N.; Sergeant, P.
Design, Modelling and Optimization of a High Power Density Axial Flux SRM with Reduced Torque Ripple for Electric Vehicles. *Machines* **2023**, *11*, 759.
https://doi.org/10.3390/machines11070759

**AMA Style**

Ajamloo AM, Ibrahim MN, Sergeant P.
Design, Modelling and Optimization of a High Power Density Axial Flux SRM with Reduced Torque Ripple for Electric Vehicles. *Machines*. 2023; 11(7):759.
https://doi.org/10.3390/machines11070759

**Chicago/Turabian Style**

Ajamloo, Akbar Mohammadi, Mohamed N. Ibrahim, and Peter Sergeant.
2023. "Design, Modelling and Optimization of a High Power Density Axial Flux SRM with Reduced Torque Ripple for Electric Vehicles" *Machines* 11, no. 7: 759.
https://doi.org/10.3390/machines11070759