# Analysis and Preliminary Design of Variable Flux Reluctance Machines: A Perspective from Working Field Harmonics

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## Abstract

**:**

## 1. Introduction

## 2. Structure and Operation Principle

#### 2.1. Topology and Drive System

_{a}) of the VFRM is fed by a standard 3-phase inverter with sinusoidal currents (I

_{a}). The field current (I

_{f}) of the field winding (W

_{f}) is regulated by an H-bridge converter.

#### 2.2. Operation Principle

_{e}of the VFRM can be calculated as

_{r}is the number of rotor slots, and Ω

_{r}is the mechanical angular speed of the rotor.

_{e}is expressed as

_{r}. For example, a VFRM with 10 rotor slots has the same electrical frequency as an RESM with 20 rotor poles.

_{e}is defined as the fundamental electrical frequency, and the corresponding waveform is defined as the fundamental wave. The armature windings are fed by the 3-phase fundamental armature currents, written as

## 3. Analysis of No-Load Airgap Field Harmonics

#### 3.1. Definition of Working Field Harmonics

_{k}is the magnitudes of kth-order flux density component, P

_{spat}is the spatial order, P

_{temp}is the temporal order, and θ

_{initial}is the initial position.

_{spat}characterizes the pole pairs of the flux density harmonic, while the temporal order P

_{temp}characterizes the rotation speed. The flux linkage of a single stator coil A1 is given by the integral of the flux density over the airgap surface corresponding to the coil, as

_{a}is the turn number of a stator coil, R

_{r}is the rotor outer radius, L is the core length, and τ

_{s}is the coil span angle. Equation (5) highlights that the value of the flux linkage depends on the rotor position.

_{temp}of the working harmonics is fixed, while their spatial order P

_{spat}is variable. For the VFRM, the working harmonics should satisfy

#### 3.2. Spatial and Temporary Order of Working Harmonics

_{0}is the vacuum permeability, g

_{0}is the airgap length, N

_{f}is the turn number of the field coil on one tooth, Λ

_{sfn}is the amplitude of the nth-order component of the stator permeance polarized by the field winding, and Λ

_{rk}is the amplitude of the kth-order component of the rotor permeance.

#### 3.3. Winding Theory

_{s}and rotor pole pairs p, that is

_{p}is determined by the stator/rotor slots combination of the VFRM, but its sign is influenced by the spatial order of the working harmonic.

_{spat}= 4 and P

_{spat}= 10, respectively. The pitch factor is the ratio of the shaded area to a half-cycle sine wave area. In Figure 5a, the coil pitch is lower than the pole pitch, and k

_{p}is positive. On the contrary, in Figure 5b, the coil pitch is higher than 2 pole pitches, and k

_{p}is negative.

## 4. Preliminary Parameter Design

#### 4.1. Torque Model

_{s}= 12 and varying rotor slots N

_{r}. Each FEA model has the same geometry parameters and winding configuration setting as the corresponding analytical torque model. The relative permeability of cores in these FEA models is set to infinity to simulate the linear case. The good agreement at different rotor slot numbers verifies the accuracy of the torque model. Therefore, the torque model can be used to guide the preliminary design of the VFRM.

#### 4.2. Distribution Rule of the Armature and Field Magnetomotive Force

_{a}I

_{a}and the field winding MMF N

_{f}I

_{f}. Under the constraint of constant copper loss P

_{cu}, the maximum average torque can be achieved when

_{cu}is the slot fill factor, l

_{coil}is the length of the one-turn coil, ρ

_{cu}is the electrical conductivity of copper, and S

_{slot}is the area of one stator slot. This distribution rule applies to VFRMs with different stator/rotor slot combinations.

#### 4.3. Approximate Torque Model and Stator/Rotor Slot Combination

_{avgn}, generated by the working harmonics with spatial orders of (N

_{r}± nN

_{s}/2). Then, the torque can be reformulated as

_{avgn}of a 12s/10r VFRM. It can be found that the majority of the average torque is due to the low-order harmonics. Thus, the high-order harmonic components can be neglected to simplify the torque model used for the preliminary design.

_{avg}calculated by the torque model (19) and the torque error δ

_{T}caused by the approximate torque model (25).

_{r}/N

_{s}is close to 1 for various stator slot numbers ranging from 6 to 24. If N

_{r}/N

_{s}is outside the range from 0.5 to 1.5, the torque decreases significantly. Combinations of stator/rotor slots outside this range are filtered out and not considered in the subsequent analysis.

_{r}/N

_{s}ranging from 0.5 to 1.5, the torque error is within ±5%. The error is most pronounced for low N

_{s}and decreases significantly with increasing N

_{s}. Therefore, the feasibility and accuracy of the approximate torque model can be guaranteed for the VFRMs with constrained stator/rotor slot combinations. Also, it can be concluded that the spatial order of the working harmonics that significantly contributes to the average torque is N

_{r}± N

_{s}/2 and N

_{r}± 3N

_{s}/2.

#### 4.4. Rotor Geometric Design

_{r}

_{1}. The average torque is proportional to Λ

_{r}

_{1}.

_{r}

_{1}is determined by the geometric parameters, including the airgap length g

_{0}, the rotor outer radius R

_{r}, and the rotor slot opening to slot pitch ratio β

_{r}(abbreviated as the rotor slot opening ratio). For the rotor geometric design, the key geometric parameter to be optimized is the rotor slot opening ratio. The increase in the rotor outer radius leads to higher Λ

_{r}

_{1}, but it is constrained by the stator slot area, which will discussed in the following section.

_{r}

_{1}by scanning the rotor slot opening ratio β

_{r}. The same curve can be observed at different rotor slot numbers N

_{r}. The maximum Λ

_{r}

_{1}is achieved at an optimal rotor slot opening ratio of approximately 0.5 to 0.55. The increase in the airgap length leads to a slight increase in the optimal rotor slot opening ratio. This optimal value can be used in the preliminary rotor design of different VFRMs. Taking β

_{r}= 0.5 for instance, the rotor slot pitch angle of 12s/8r, 12s/10r, 12s/11r, and 12s/13r is 45, 36, 32.7, and 27.7 degrees, respectively, and thus, the corresponding rotor slot opening angle is 22.5, 18, 16.35, and 13.85 degrees.

#### 4.5. Stator Geometric Design

_{sf}

_{1}and Λ

_{sf}

_{3}, and the area of the stator slots S

_{slot}. For the stator geometric design, the key geometric parameters to be optimized are the stator slot opening to slot pitch ratio β

_{s}(abbreviated as the stator slot opening ratio) and the stator inner radius R

_{s}. Since the core saturation is not considered, the stator yoke thickness is kept at a constant value.

_{y}, the stator inner radius is normalized and named the split ratio

_{sfn}= 0 and T

_{avgn}= 0 when n is even.

_{sfn}is mainly determined by the stator inner radius R

_{s}and the stator slot opening ratio β

_{s}. Figure 12 illustrates Λ

_{sf}

_{1}and Λ

_{sf}

_{3}at different split ratio and stator slot opening ratio combinations. Λ

_{sf}

_{1}increases with the decrease in the stator slot opening ratio. In comparison, Λ

_{sf}

_{3}is relatively lower, and the curve has a local maximum point at about the stator slot opening ratio of 0.7. On the contrary, Λ

_{sf}

_{1}and Λ

_{sf}

_{3}are little affected by the changes in the split ratio.

_{s}and split ratio d

_{s}on the average torque at constant copper loss can be expressed by Function F

_{1}and a

_{3}are decided by the stator/rotor slots combination N

_{r}/N

_{s}. Figure 13 plots a

_{1}and a

_{3}versus N

_{r}/N

_{s}. As analyzed before, both T

_{avg}

_{1}and T

_{avg}

_{3}contribute significantly to the total torque. Thus, the average torque output can be guaranteed when both of the two components are positive. That means N

_{r}/N

_{s}should be restricted from 0.5 to 1.5, which is consistent with the finding in the earlier analysis.

_{r}/N

_{s}since the function F includes the variable N

_{r}/N

_{s}. Figure 15 collects the optimal split ratio and stator slot opening ratio of the VFRMs with various stator/rotor slot combinations. The trend is similar for the VFRMs with different stator slot numbers. There is a slight increase in both the optimal stator slot ratio and the optimal split ratio with increasing N

_{r}/N

_{s}. For the VFRMs with N

_{r}/N

_{s}near 1, the initial value of the stator split ratio and stator opening ratio can be set as 0.54 and 0.6, respectively. In the recommended range of N

_{r}/N

_{s}from 0.5 to 1.5, the same initial values can also be used, to unify and simplify the preliminary design process.

## 5. Conclusions

_{r}but different spatial order of N

_{r}± nN

_{s}/2 (n = 1, 3, 5…). It is found that a unified star of slots can consider all these working harmonics since the differences between their slot electrical angles are equal to a multiple of 2π. It is also deduced that the winding factors of these harmonics are the same, except for the sign, which is affected by the spatial order. Then, an average torque model of the VFRM is developed and simplified. This model takes into account the 1st-order rotor permeance and the 1st- and 3rd-order polarized stator permeance. Permeance harmonics of other orders are eliminated. By analyzing the torque model, the guideline for the preliminary design of the VFRM is obtained.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

α_{s}^{e} | slot electrical angle | μ_{0} | vacuum permeability |

β_{r} | rotor slot opening to slot pitch ratio | N_{a} | turn number of a stator coil |

β_{s} | stator slot opening to slot pitch ratio | N_{f} | turn number of a field coil |

B_{g} | no-load airgap flux density | N_{r} | rotor slot number |

B_{k} | magnitudes of kth-order flux density | N_{s} | stator slot number |

d_{s} | split ratio | P_{cu} | copper loss |

δ_{T} | torque error | P_{spat} | spatial order |

f_{e} | electrical frequency | P_{temp} | temporal order |

F_{f} | MMF of the field winding | ρ_{cu} | electrical conductivity of copper |

f_{i} | ith-order unit field winding MMF | R | stator outer radius |

g_{0} | airgap length | R_{s} | stator inner radius |

h_{y} | stator yoke thickness | R_{r} | rotor outer radius |

I_{f} | field current | S_{slot} | area of one stator slot |

I_{a} | armature current amplitude | τ_{s} | coil span angle |

k_{d} | distribution factor | T_{avg} | average torque |

k_{p} | pitch factor | T_{e} | electrical period |

k_{cu} | slot fill factor | W_{a} | armature winding |

k_{w} | winding factor | W_{f} | field winding |

Λ | radial airgap permeance | Ω_{r} | mechanical angular speed |

Λ_{r} | radial airgap permeance of single-side salient-rotor model | Ψ_{a} | phase flux linkage amplitude |

Λ_{rk} | kth-order rotor permeance | θ_{intial} | initial position |

Λ_{s} | radial airgap permeance of single- side salient-stator model | L | core length |

Λ_{sf} | radial stator permeance polarized by the field winding | l_{coil} | length of the one-turn coil |

## Appendix A

_{f}(θ) is the MMF of the field winding, and Λ(θ,t) is the airgap permeance.

_{0}is the vacuum permeability, g

_{0}is the airgap length, Λ

_{s}(θ) is the radial airgap permeance of the single-side salient-stator model, and Λ

_{r}(θ,t) is the airgap permeance of the single-side salient-rotor model.

_{sf}(θ) is stator permeance polarized by the field winding (introduced in Appendix C), N

_{f}is the turn number of the field winding on each stator tooth, and I

_{f}is the field current.

_{r}(θ,t) is written as

_{rk}is the amplitude of the kth-order component of rotor permeance, N

_{r}is the rotor slot number, and Ω

_{r}is the mechanical angular speed of the rotor.

_{sf}(θ) is written as

_{sfn}is the amplitude of the nth-order component of the polarized stator permeance, and N

_{s}is the stator slot number.

## Appendix B

_{a}is the turn number of armature winding on each stator tooth, R

_{r}is the rotor radius, and L is the core length. Finally, Ψ

_{a}is the amplitude of the flux linkage.

_{d}is the distribution factor of the VFRM, and I

_{a}is the amplitude of the sinusoidal armature current.

## Appendix C

_{s}(θ) and Λ

_{r}(θ,t) can be carried out assuming an infinitely deep slot model, given in [30]. Figure A1 shows the equivalent airgap length in the infinitely deep slot model. The equivalent airgap length at the slot-facing position is the sum of the airgap length g

_{0}and the parallel length of two quarter circles.

**Figure A1.**Equivalent airgap length in the infinitely deep slot model. (

**a**) Single-slide salient-stator model; (

**b**) single-slide salient-rotor model.

_{s}(θ) and Λ

_{r}(θ,t) are expressed as

_{r}is the rotor slot opening to slot pitch ratio, β

_{s}is the stator slot opening to slot pitch ratio, R

_{r}is the rotor outer radius, and R

_{s}is the stator inner radius.

_{s}(θ) and Λ

_{r}(θ,t) can be given as (A4) and (A15).

_{i}is the amplitude of the ith-order component of the unit field winding MMF.

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**Figure 1.**Cross-section view of 12-stator-slots VFRMs with only A-phase armature coils. (

**a**) Eight rotor slots. (

**b**) Ten rotor slots. (

**c**) Eleven rotor slots. (

**d**) Thirteen rotor slots.

**Figure 4.**Star of slots; (

**a**) 12-stator-slots/20-rotor-poles RESM, ${\alpha}_{\mathrm{s}}^{\mathrm{e}}=300\xb0$; (

**b**) 12-stator-slots/10-rotor-slots VFRM, ${\alpha}_{\mathrm{s}}^{\mathrm{e}}=120\xb0\pm \left(\mathrm{n}-1\right)\pi $.

**Figure 5.**Pitch factors of working harmonics in a 12s/10r VFRM. (

**a**) P

_{spat}= 4; (

**b**) P

_{spat}= 16.

**Figure 6.**Analytical and FEA calculated average torque in linear condition, with N

_{s}= 12 and varying N

_{r}.

**Figure 8.**Average torque and error of approximate torque model. (

**a**) N

_{s}= 6; (

**b**) N

_{s}= 12; (

**c**) N

_{s}= 18; (

**d**) N

_{s}= 24.

**Figure 11.**Stator permeance. (

**a**) Distribution of the stator permeance; (

**b**) distribution of normalized field winding MMF; (

**c**) distribution of the polarized stator permeance; (

**d**) spectra of the polarized stator permeance.

**Figure 15.**Optimal stator slot opening ratio and split ratio versus N

_{r}/N

_{s}. (

**a**) Stator slot opening ratio; (

**b**) split ratio.

RESM | VFRM | |
---|---|---|

Stator slot number | 12 | 12 |

Rotor pole number | 20 | 10 |

Electrical frequency | $\frac{10{\mathsf{\Omega}}_{r}}{2\pi}$ | $\frac{10{\mathsf{\Omega}}_{r}}{2\pi}$ |

Temporal order | 10 | 10 |

Spatial order | 10 | 10 ± 6n (n = 1, 3, 5…) |

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## Share and Cite

**MDPI and ACS Style**

Gu, X.; Bianchi, N.; Zhang, Z.
Analysis and Preliminary Design of Variable Flux Reluctance Machines: A Perspective from Working Field Harmonics. *Vehicles* **2024**, *6*, 571-589.
https://doi.org/10.3390/vehicles6010026

**AMA Style**

Gu X, Bianchi N, Zhang Z.
Analysis and Preliminary Design of Variable Flux Reluctance Machines: A Perspective from Working Field Harmonics. *Vehicles*. 2024; 6(1):571-589.
https://doi.org/10.3390/vehicles6010026

**Chicago/Turabian Style**

Gu, Xiangpei, Nicola Bianchi, and Zhuoran Zhang.
2024. "Analysis and Preliminary Design of Variable Flux Reluctance Machines: A Perspective from Working Field Harmonics" *Vehicles* 6, no. 1: 571-589.
https://doi.org/10.3390/vehicles6010026