# Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Loss Prediction Model

_{Fe}consists of hysteresis loss, eddy current loss and excess loss, and the core loss yields:

_{e}is the excess loss in W, k

_{h}, k

_{c}, and k

_{e}are the corresponding coefficient of the above losses, respectively, f is the fundamental frequency of a magnetizing flux in Hz, and B

_{m}is the maximum flux density in core in T.

_{gr}/B

_{gt}) are predicted, as shown in Figure 1b. Clearly, for the stator points 1 and 2, the surrounded areas by the B

_{gr}/B

_{gt}loci are to be almost zero, which means the averaged B

_{gr}/B

_{gt}values are nearly zero. However, for points 3 and 4, the corresponding areas (the blue one and the pink one) are not centrosymmetric, which means a DC-biased component exists. For the points 5~8 in the rotor, the B

_{gr}/B

_{gt}loci are all centrosymmetric and the averaged values are close to zero. A typically DC-biased component and a minor hysteresis loop are shown in Figure 1c,d, respectively.

_{elem}is the finite elements number, ΔA

_{i}is the ith finite element area in m

^{2}, N

_{pr}

^{i}and N

_{pt}

^{i}are the radial and tangential minor loops numbers of the ith element during one period, respectively, N

_{step}is the calculation steps number, B

_{rm}

^{ij}and B

_{tm}

^{ij}are the maximum radial and tangential flux-densities of the jth hysteresis loop in the ith element in T, respectively, B

_{rmi}

^{k}and B

_{tmi}

^{k}are the maximum radial and tangential flux-density of the ith element in the kth calculation step in T, and Δt is the time step in s.

_{gr}/B

_{gt}results of each meshed iron element can be acquired, and based on MATLAB the core loss versus rotating speeds under different conditions can be assessed by a series of data processing calculations according to Equations (2)–(5). The no-load core loss density distribution of the stator and rotor is shown in Figure 2. Consequently, the predicted core losses versus rotor speed are compared with those obtained by commercial software, e.g., by JMAG and ANSYS EM as shown in Figure 3. It can be seen that the core losses obtained by the improved method are slightly higher than those from software, which validates the influence of the DC-biased component and minor hysteresis loop, and also validates the feasibility of the improved core loss prediction method.

_{pmc}and housing P

_{hc}versus rotating speeds are shown in Figure 5. It can be found that with the increase of the speed, the eddy current losses in PMs and housing increase gradually, which is caused by the air-gap harmonic fields and can be calculated by Equation (6) [20],

_{eddy}is the eddy current loss in PM and housing in W, J

_{e}is the current density in each element in A/m

^{2}, Δ

_{Ai}is the ith element area in m

^{2}, σ

_{r}is the conductivity of the eddy current zone in S/m, and t

_{c}is the time corresponding to a period in each element in s.

_{fri}of the FSPM machine yields [21]:

_{r}is the rotor peripheral speed in m/s. It is found that P

_{fri}= 10.7 W under the rated speed of 1000 r/min.

## 3. Two Thermal Models

_{rf}yields

_{f}is the specific heat capacity of fluid in J/(kg·°C), η

_{f}is fluid dynamic viscosity in N·s/m

^{2}, and λ

_{f}is the fluid thermal conductivity in W/(m·°C).

_{e}yields

_{f}is the fluid kinetic viscosity in m

^{2}/s, and d

_{e}is the hydraulic radius in m by Equation (10).

_{cs}is a cross-section area of a single cooling duct in m

^{2}. s, b, and h are the wetted perimeter, width, and height of cooling duct in m, respectively.

_{uf}

_{l}yields [22],

_{uft}yields

_{w}is the dynamic viscosity of housing in N·s/m

^{2}.

_{f}

_{0}yields

_{f}of the cooling jacket is affected by the geometric parameters and the velocity of the fluid, and can be given by

#### 3.1. Lumped Parameter Thermal Network Model

- Symmetrical temperature distribution and the same cooling conditions along the circumference;
- Uniformly distributed thermal capacity and heat generation;
- Independent heat flow in radial and axial directions

_{conv}

_{i}(i = 1, 2, 3) representing the heat dissipation by cooling medium convection between the housing external surface and ambient can be calculated by Equation (16) [25],

_{convi}is the thermal resistance due to convection heat transfer in °C/W, h

_{convi}is the convection heat transfer coefficient in W/(m

^{2}·°C), and A

_{convi}is the convective area in m

^{2}. Here, the area of the end-part winding is considered.

_{airi}.

_{o}and r

_{i}are the outer and inner diameter of the cylinder in m, λ is the thermal conductivity of the material in W/m/°C, and L is the cylinder length in m.

_{cond}is the area for the conduction in m

^{2}.

_{sh}

_{1}-R

_{sh}

_{8}, R

_{sy}

_{1}-R

_{sy}

_{6}, R

_{air}

_{1}-R

_{air}

_{9}, R

_{st}

_{1}-R

_{st}

_{3}, R

_{coil}

_{1}-R

_{coil}

_{3}, and R

_{pm}

_{1}-R

_{pm}

_{5}represent the thermal resistances of the housing, stator yoke, air-gap, stator tooth, stator winding coils, and PMs. R

_{rt}

_{1}, R

_{ry}

_{1,}and R

_{shaft}

_{1}represent the thermal resistances of rotor tooth, rotor yoke, and shaft, respectively. C

_{sh}

_{1}-C

_{sh}

_{3}, C

_{sy}

_{1}-C

_{sy}

_{2}, C

_{st}

_{1}, C

_{coil}

_{1}, C

_{pm}

_{1}-C

_{pm}

_{2}, and C

_{rt}represent the thermal capacitances of the housing, stator yoke, stator tooth, winding coils, PMs, and rotor tooth. Here, since the heat dissipated by forced convection is much larger than that by radiation, the radiation heat dissipation is ignored.

#### 3.2. One-Dimensional Steady Heat Conduction Model

_{avg}yields [26,27]:

_{s}, A

_{wind}, and A

_{pm}are the cross-section area of the stator, windings, and PMs in m

^{2}, respectively, λ

_{s}, λ

_{wind}, and λ

_{pm}are the thermal conductivity of stator, windings, and PMs in W/(m·°C).

_{I}is the thickness of the ith layer in m, λ

_{i}is the thermal conductivity of the ith layer in W/(m·°C).

_{Vs}and rotor q

_{Vr}can be obtained by

_{eave}is equivalent average loss in W, P

_{s}, P

_{cu}, P

_{pm}, and P

_{r}are the stator core loss, winding joule loss, eddy current loss in PMs, and rotor core loss in W, respectively, V

_{s}, V

_{cu}, V

_{pm}, V

_{equ}, and V

_{r}are the volume of stator, winding, PM, equivalent stator, and rotor in m

^{3}, respectively.

_{s}/q

_{r}) yields

_{s}/S

_{r}is the cross-section area of stator/rotor lamination in m

^{2}.

_{f}equals

_{sh}can be derived by

^{2}·K), Rsho is the housing outer radius in m, Rshao is the shaft outer radius in m, and Rsi is the stator inner radius in m.

_{sy}yields

_{so}is the stator outer radius in m, and h

_{sh}is the housing thermal conductivity in W/(m·°C).

_{r}is the rotor thermal conductivity in W/(m·°C).

## 4. CFD-Based 3D Temperature Field Verification

## 5. Experiment Verification

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The flux-density loci of key points in the FSPM machine. (

**a**) The key stator and/rotor core points in the FSPM machine. (

**b**) The B

_{gr}/B

_{gt}loci of key stator and rotor core points. (

**c**) The DC-biased components of B

_{gr}/B

_{gt}. (

**d**) The local minor hysteresis loop.

**Figure 7.**The cooling system of the FSPM machine. (

**a**) FSPM machine structure, (

**b**) housing, (

**c**) coolant flow path, and (

**d**) machine assembly. (

**e**) Cross-section of the cooling jacket, and (

**f**) schematic diagram of cooling duct.

**Figure 8.**The LPTN model of the FSPM machine: (

**a**) 3D module structure; (

**b**) stator module; (

**c**) the LPTN model of the 1/24 FSPM machine.

**Figure 9.**The predicted temperature rises by the LPTN model under water-cooling conditions @ nN = 1000 r/min and Iph = 30.7 A and 60 A (RMS).

**Figure 10.**One-dimensional steady heat conduction model of the FSPM machine. (

**a**) Equivalent heat conduction model; (

**b**) equivalent heat flow path.

**Figure 13.**Three-dimensional CFD thermal model of the FSPM machine and steady-state temperatures @30.7 A and 60 A; (

**a**) 3D-CFD thermal model; (

**b**) water cooling.

**Figure 19.**Steady-state temperature distribution of the FSPM machine under forced water-cooling conditions.

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

DC-link voltage | U_{DC} | 144 | V |

Phase number | m | 3 | - |

Stator slots | N_{s} | 12 | - |

Rotor pole pairs | N_{r} | 10 | - |

PM pole pairs | N_{PM} | 6 | - |

Rated power | P_{N} | 10 | kW |

Rated speed | n_{N} | 1000 | r/min |

Rated torque | T_{N} | 95.5 | Nm |

Stator outer diameter | D_{so} | 260 | mm |

Rotor inner diameter | D_{ri} | 50 | mm |

Air-gap length | g_{0} | 0.9 | mm |

Stack length | L_{a} | 55 | mm |

Materials | Thermal Conductivity (W/m/°C) | Specific Heat Capacity (J/kg/°C) | Density (kg/m^{3}) |
---|---|---|---|

Steel silicon | 23 | 460 | 7650 |

Copper | 380 | 385 | 8978 |

PM | 9 | 504 | 7500 |

Aluminum | 237 | 833 | 2688 |

Air | 0.02624 | 1005 | 1.205 |

Thermal Resistances | Value (°C/W) | Thermal Resistances | Value (°C/W) |
---|---|---|---|

R_{sh}_{1}, R_{sh}_{2}, R_{sh}_{3} | 0.0004307 | R_{st}_{1}, R_{st}_{2} | 0.1479 |

R_{sh}_{4} | 0.1094 | R_{st}_{3} | 0.01449 |

R_{sh}_{5} | 0.07664 | R_{coil}_{1} | 0.6297 |

R_{sh}_{6}, R_{sh}_{7}, R_{sh}_{8} | 0.0004469 | R_{coil}_{2} | 4.895 |

R_{sy}_{1}, R_{sy}_{2} | 0.00359 | R_{coil}_{3} | 0.7244 |

R_{sy}_{3} | 0.2107 | R_{pm}_{1} | 0.01197 |

R_{sy}_{4} | 0.1128 | R_{pm}_{2} | 0.04699 |

R_{sy}_{5} | 0.003729 | R_{pm}_{3} | 0.05441 |

R_{sy}_{6} | 0.01632 | R_{pm}_{4} | 0.2054 |

R_{air}_{1}, R_{air}_{2}, R_{air}_{3} | 1.096 | R_{pm}_{5} | 0.04829 |

R_{air}_{4}, R_{air}_{5} | 596 | R_{rt}_{1} | 0.02435 |

R_{air}_{6}, R_{air}_{7} | 27.84 | R_{ry}_{1} | 0.03246 |

R_{air}_{8} | 30.76 | R_{shaft}_{1} | 0.2613 |

R_{air}_{9} | 56.14 | - | - |

Thermal Capacitance | Value (J/°C) | Thermal Capacitance | Value (J/°C) |
---|---|---|---|

C_{sh}_{1} | 30.82 | C_{st}_{1} | 71.37 |

C_{sh}_{2} | 21.86 | C_{coil}_{1} | 0.0036 |

C_{sh}_{3} | 10.75 | C_{pm}_{1} | 10.67 |

C_{sy}_{1} | 31.63 | C_{pm}_{2} | 33.14 |

C_{sy}_{2} | 22.98 | C_{r} | 497.2 |

Temperature Prediction Methods | Steady State | Transient |
---|---|---|

LPTN method | 2 s | 7 s |

1D-SHC method | 0.6 s | 1.4 s |

CFD method | 11 min | 100 min |

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## Share and Cite

**MDPI and ACS Style**

Yu, W.; Wu, Z.; Hua, W.
Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles. *World Electr. Veh. J.* **2023**, *14*, 130.
https://doi.org/10.3390/wevj14050130

**AMA Style**

Yu W, Wu Z, Hua W.
Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles. *World Electric Vehicle Journal*. 2023; 14(5):130.
https://doi.org/10.3390/wevj14050130

**Chicago/Turabian Style**

Yu, Wenfei, Zhongze Wu, and Wei Hua.
2023. "Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles" *World Electric Vehicle Journal* 14, no. 5: 130.
https://doi.org/10.3390/wevj14050130