# Comparative Analysis of the Steady-State Model Including Non-Linear Flux Linkage Surfaces and the Simplified Linearized Model when Applied to a Highly-Saturated Permanent Magnet Synchronous Machine—Evaluation Based on the Example of the BMW i3 Traction Motor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Machine Geometry and Winding Configuration

#### 2.2. Definition of the Used Materials

#### 2.3. Model Summary

## 3. Results

#### 3.1. FEM Simulation Results

#### 3.2. Calculations Based on the Non-Linear Model

#### 3.3. Calculations Based on the Linearized Model

- ‘Non-linear model’—the space-vector model in steady-state defined with Equation (1), which includes voltage drop across stator resistance and non-linear flux linkage surfaces.
- ‘Linearized model’—the simplified model defined with Equation (3), which neglects voltage drop across stator resistance and assumes constant machine parameters (hence the term ‘linearized’).

#### 3.4. Comparison of the Results—Maximal Torque

#### 3.5. Comparison of the Results—Partial Load Operation

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FEM | Finite Element Method |

IPMSM | Interior Permanent Magnet Synchronous Machine |

MTPC | Maximal Torque Per Current |

MTPV | Maximal Torque Per Voltage |

ORNL | Oak Ridge National Laboratory |

PMSM | Permanent Magnet Synchronous Machine |

## Appendix A. BMW i3 Traction Machine Data and FEM Simulation Results

Parameter Name | Value | Unit |
---|---|---|

overall motor assembly mass | 42 | kg |

outer housing mass (incl. one bearing) | 6.9 | kg |

stator mass | 20.8 | kg |

rotor mass | 14.2 | kg |

stator outer diameter | 242.1 | mm |

stator inner diameter | 180.0 | mm |

rotor outer diameter | 178.6 | mm |

rotor inner diameter | 130.3 | mm |

shaft diameter | 60.0 | mm |

stack length | 132.3 | mm |

tooth width | 5.0 | mm |

slot opening | 1.7 | mm |

slot depth | 22.0 | mm |

Parameter Name | Value | Unit |
---|---|---|

number of slots | 72 | - |

number of pole pairs | 6 | - |

stator turns per coil | 9 | - |

number of wires “in hand” | 12 | strands/turn |

wire size | 21 | AWG |

slot fill factor | 54.43 | % |

parallel circuits per phase | 6 one neutral/phase | |

coils in series per phase | 1 per leg | |

winding notes | full pitch, concentrically wound | |

6 independent neutrals |

Machine Part | Material |
---|---|

stator core | magnetic steel: M-19 29 Ga |

stator coil | copper: 100% IACS |

stator liner | epoxy resin |

rotor core | magnetic steel: M-19 29 Ga |

rotor magnet | neodymium iron boron: 38/23 |

rotor sleeve | stainless steel: 304 |

shaft | cold rolled steel: CR10 |

hub | non-magnetic |

**Table A4.**Flux linkage in the d-axis calculated with the FEM model (mWb) (values averaged over one electrical period).

q-axis Current (A) | ||||||||
---|---|---|---|---|---|---|---|---|

0 | 100 | 200 | 300 | 400 | 500 | 600 | ||

ine d-axis current (A) | −600 | 1.0 | 1.3 | 2.0 | 2.7 | 3.2 | 3.4 | 3.6 |

−500 | 7.7 | 8.0 | 8.6 | 9.0 | 9.1 | 9.0 | 8.8 | |

−400 | 14.5 | 14.8 | 15.3 | 15.4 | 15.1 | 14.6 | 14.1 | |

−300 | 21.5 | 21.8 | 22.3 | 22.2 | 21.3 | 20.3 | 19.4 | |

−200 | 28.6 | 29.0 | 29.5 | 29.0 | 27.6 | 26.0 | 24.6 | |

−100 | 35.9 | 36.5 | 37.0 | 35.6 | 33.6 | 31.5 | 29.8 | |

0 | 43.6 | 44.4 | 44.0 | 41.5 | 39.0 | 36.6 | 34.5 |

**Table A5.**Flux linkage in the q-axis calculated with the FEM model (mWb) (values averaged over one electrical period).

q-axis Current (A) | ||||||||
---|---|---|---|---|---|---|---|---|

0 | 100 | 200 | 300 | 400 | 500 | 600 | ||

ine d-axis current (A) | −600 | 0.0 | 18.8 | 35.2 | 47.4 | 55.4 | 60.0 | 62.7 |

−500 | 0.0 | 19.8 | 36.8 | 48.9 | 56.3 | 60.3 | 62.8 | |

−400 | 0.0 | 20.8 | 38.2 | 49.9 | 56.6 | 60.2 | 62.5 | |

−300 | 0.0 | 21.8 | 39.4 | 50.2 | 56.3 | 59.6 | 61.8 | |

−200 | 0.0 | 22.8 | 40.3 | 49.9 | 55.2 | 58.5 | 60.8 | |

−100 | 0.0 | 23.9 | 40.8 | 48.9 | 53.4 | 56.9 | 59.3 | |

0 | 0.0 | 24.9 | 40.1 | 46.7 | 51.3 | 54.8 | 57.6 |

**Table A6.**Machine torque calculated with the FEM model (Nm) (values averaged over one electrical period).

q-axis Current (A) | ||||||||
---|---|---|---|---|---|---|---|---|

0 | 100 | 200 | 300 | 400 | 500 | 600 | ||

ine d-axis current (A) | −600 | −1.4 | 100.6 | 190.2 | 259.5 | 307.8 | 337.7 | 357.1 |

−500 | −1.1 | 94.2 | 177.7 | 241.0 | 283.8 | 310.5 | 329.5 | |

−400 | −0.9 | 86.3 | 162.1 | 218.6 | 256.3 | 281.3 | 300.3 | |

−300 | −0.6 | 76.8 | 144.1 | 193.3 | 227.0 | 251.2 | 270.9 | |

−200 | −0.3 | 65.9 | 123.9 | 166.6 | 197.3 | 221.4 | 241.8 | |

−100 | −0.1 | 53.7 | 101.9 | 138.8 | 167.9 | 192.3 | 213.4 | |

0 | 0.0 | 39.7 | 78.1 | 110.9 | 139.4 | 164.0 | 185.7 |

q-axis Current (A) | ||||||||
---|---|---|---|---|---|---|---|---|

0 | 100 | 200 | 300 | 400 | 500 | 600 | ||

ine d-axis current (A) | −600 | 21.06 | 21.62 | 21.49 | 22.51 | 21.69 | 18.63 | 19.99 |

−500 | 18.53 | 18.94 | 16.86 | 16.89 | 16.93 | 14.06 | 16.55 | |

−400 | 15.90 | 16.15 | 13.83 | 12.84 | 12.50 | 10.27 | 15.28 | |

−300 | 12.30 | 12.71 | 12.56 | 12.01 | 11.99 | 11.36 | 16.45 | |

−200 | 8.56 | 9.17 | 10.22 | 11.61 | 10.86 | 10.19 | 13.83 | |

−100 | 5.02 | 5.74 | 8.03 | 10.37 | 10.35 | 11.08 | 12.12 | |

0 | 2.16 | 3.66 | 6.77 | 9.39 | 11.18 | 12.23 | 13.79 |

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**Figure 5.**Magnetic circuit saturation of the BMW i3 traction machine: (

**a**) flux density plot at no current state, (

**b**) flux density plot at the working point corresponding to the peak torque (${i}_{d\_\mathrm{Tpk}}=-401\phantom{\rule{3.33333pt}{0ex}}\mathrm{A},{i}_{q\_\mathrm{Tpk}}=399\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$). These plots were obtained with the “Ansys Maxwell” software.

**Figure 6.**Surface plots of the FEM simulation results for the BMW i3 traction machine: (

**a**) d-axis flux linkage averaged over one electrical period, (

**b**) q-axis flux linkage averaged over one electrical period, (

**c**) torque averaged over one electrical period, (

**d**) peak to peak torque ripple.

**Figure 7.**Comparison of the experimental locked rotor test results obtained by the Oak Ridge National Laboratory (ORNL) [19] (dotted lines with cross markers) with corresponding values obtained with the FEM model (solid lines with circle markers). Simulation results are averaged over one electrical period and the shaded areas indicate the amount of torque ripple.

**Figure 8.**The results of the BMW i3 traction drive maximal torque analysis, based on a numerical search algorithm and the non-linear flux linkage surfaces obtained with FEM simulation: (

**a**) current space-vector locus, (

**b**) torque vs. speed curve. The filled dots are the operation points for different speeds.

**Figure 9.**Comparison of the maximal BMW i3 traction drive torque calculated with two different methods, i.e., the numerical search algorithm based on the non-linear machine model (black) and analytical equations based on the linearized machine model (gray): (

**a**) torque surfaces, (

**b**) torque vs. speed curves. The filled dots are the operation points for different speeds.

**Figure 10.**Comparison of the partial load torque vs. speed calculations for the BMW i3 traction drive calculated with two different methods, i.e., the numerical search algorithm based on the non-linear machine model (black) and analytical equations based on the linearized machine model (gray): (

**a**) current space-vector loci, (

**b**) torque vs. speed curves. The consecutive curves were calculated for many different values of the current limitation.

**Figure 11.**Comparison of the partial load torque vs. current magnitude for the BMW i3 traction drive calculated with two different methods, i.e., a numerical search algorithm based on the non-linear machine model (black) and analytical equations based on the linearized machine model (gray): (

**a**) operation points locus at torque surfaces, (

**b**) torque vs. phase current magnitude. The filled dots are the operation points calculated for different phase current limitations. The points corresponding to the actual peak phase current of the drive (i.e., 565.7 A) are marked with blue.

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

max. current | ${i}_{\mathrm{max}}$ | 565.7 | A |

max. voltage | ${u}_{\mathrm{max}}$ | 159.2 | V |

no. of pole pairs | p | 6 | - |

max. speed | ${n}_{\mathrm{max}}$ | 11,400 | rpm |

phase resistance | R | 5.3 | m$\Omega $ |

winding configuration | - | Y | - |

d-axis flux linkage | ${\psi}_{d}$ | see Table A4 | Wb |

q-axis flux linkage | ${\psi}_{q}$ | see Table A5 | Wb |

**Table 2.**The BMW i3 traction drive parameters obtained for the linearized model (3) for the current and voltage limitations listed in Table 1.

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

motor saliency factor | $\epsilon $ | 1.98 | - |

drive characteristic factor | ${k}_{\mathrm{ch}}$ | 0.92 | - |

motor characteristic current | ${i}_{\mathrm{ch}}$ | 612.4 | A |

permanent magnet flux linkage | ${\psi}_{\mathrm{PM}\_\mathrm{lin}}$ | 0.0436 | Wb |

d-axis inductance | ${L}_{d\_\mathrm{lin}}$ | 71.2 | $\mu $H |

q-axis inductance | ${L}_{q\_\mathrm{lin}}$ | 141.3 | $\mu $H |

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**MDPI and ACS Style**

Gierczynski, M.; Grzesiak, L.M.
Comparative Analysis of the Steady-State Model Including Non-Linear Flux Linkage Surfaces and the Simplified Linearized Model when Applied to a Highly-Saturated Permanent Magnet Synchronous Machine—Evaluation Based on the Example of the BMW i3 Traction Motor. *Energies* **2021**, *14*, 2343.
https://doi.org/10.3390/en14092343

**AMA Style**

Gierczynski M, Grzesiak LM.
Comparative Analysis of the Steady-State Model Including Non-Linear Flux Linkage Surfaces and the Simplified Linearized Model when Applied to a Highly-Saturated Permanent Magnet Synchronous Machine—Evaluation Based on the Example of the BMW i3 Traction Motor. *Energies*. 2021; 14(9):2343.
https://doi.org/10.3390/en14092343

**Chicago/Turabian Style**

Gierczynski, Michal, and Lech M. Grzesiak.
2021. "Comparative Analysis of the Steady-State Model Including Non-Linear Flux Linkage Surfaces and the Simplified Linearized Model when Applied to a Highly-Saturated Permanent Magnet Synchronous Machine—Evaluation Based on the Example of the BMW i3 Traction Motor" *Energies* 14, no. 9: 2343.
https://doi.org/10.3390/en14092343