# Iron Loss Modelling of Electrical Traction Motors for Improved Prediction of Higher Harmonic Losses

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## Abstract

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## 1. Introduction

^{®}, a specific product line of electrical steels which are optimised for automotive traction electrical machines. In order to further reduce the magnetic core losses of these electrical steels for wide frequency operation, research is ongoing on the modelling of various mechanisms of core loss dissipation. A robust prediction of these losses under realistic operating conditions of the machine remains challenging, as core losses are strongly influenced by a number of factors which are not yet completely understood. Losses measured on an electrical machine can be considerably higher than those expected from laboratory measurements on steel sheets, due to the presence of stresses, material degradation from production techniques, operation at high temperature, etc. Additionally, the presence of time- and space-harmonics, for example due to inverter-fed operation, will result in additional losses, possibly in combination with minor loops and the occurrence of skin effect in the laminations. It is the aim of this paper to present a loss modelling approach that takes into account skin effect and minor loops, whilst relying on methodologies that can be combined with commercial Finite Element (FE) software packages without the need to access FE formulations [1].

## 2. Theoretical Analysis of Loss Models which do not Account for Skin Effect

#### 2.1. Hysteresis Loss Component

_{p}, and the fundamental frequency, f

_{0}[3]:

_{hyst}, α and β are fitting parameters that depend on the electrical steel grade and are determined via Epstein frame measurements. The hysteresis energy loss per cycle, W

_{hyst}, can actually be directly measured under quasi-static conditions, as it is the only loss component that does not depend on the dynamics of the waveform. Hysteresis losses can only be calculated a posteriori, after the complete hysteresis loop has been closed, and thus a direct description as a function of time cannot be given. Further, it is possible to account for the effect of rotational magnetisation by adapting the above equation, which was derived for alternating magnetization [3]:

_{min}/J

_{p}) and r(J

_{p}) is an empirical rotational loss factor, which must be determined via experiments.

#### 2.2. Excess loss component

_{0}is a parameter which depends on J

_{p}.

#### 2.3. Classical Loss Component

_{class}(t), which is due to a homogeneous distribution of eddy currents flowing in the lamination, is given by [10,12]:

#### 2.4. Overall Core Loss Model

_{p}, can be obtained as follows:

## 3. Loss Modelling Taking Skin Effect into Account

#### 3.1. Prediction of Eddy Current Loss Using Analytical Models

_{p}is the peak value of the average flux density through the material. The resulting eddy current loss density can then be calculated as follows [10]:

#### 3.2. Non-Linear Post-Processing Calculation Using a Finite Difference Modelling

- B
_{1}, B_{2}, ..., B_{N}are the flux densities in each of the N layers, - H
_{1}, H_{2}, …, H_{N}are the magnetic fields in each layer, - h is the thickness of each layer,
- u(t) is the applied voltage over the winding with w
_{1}turns, - and S is the cross-sectional area of the core.

#### 3.3. FE Modelling of Eddy Current Loss

#### 3.4. Increase in Hysteresis and Excess Loss Due to Skin Effect

_{hyst}given in Equation 3 and derived from low-frequency measurement results, where skin effect can be ignored. As hysteresis loss density generally increases with increasing polarisation (i.e., normally α+βJ

_{p}> 1), skin effect is expected to result in an increased hysteresis loss. This may lead to only a small increase in total losses, however, as dynamic losses are normally dominant at high frequencies.

_{0}, that is used to calculate the excess loss (Equation (5)) depends on the incremental permeability at the point where the minor loop and major loops are connected. For small amplitudes of the minor loop, the following equation can then be used [10]:

_{m}and J

_{b}are the peak amplitude and DC bias polarisation of the minor loop, respectively.

## 4. Global Modelling Approach

## 5. Experimental Verification

#### 5.1. Sinusoidal Waveforms Without DC Bias

_{hyst_skin}refers to the hysteresis losses that have been corrected for the skin effect, and P

_{exc}refers to the excess loss based on the average flux density in the lamination. Figure 6 shows a comparison between the eddy current loss constant, k

_{eddy}, that was obtained from measurement and various calculation methodologies, where k

_{eddy}was defined as the volumetric loss divided by the square of the frequency. It is clear that the downward slope of the eddy current losses with frequency is predicted by both the FE and analytical methods. The classical analysis is given by Equation (6) as it appears in the method from Bertotti, however, this overestimates the eddy current losses at high frequencies because the skin-effect is ignored.

#### 5.2. Quasi-Static Measurements at DC Bias

#### 5.3. Verification on a Waveform with High-Frequency Harmonics

## 6. Application to Loss Modelling of Automotive Traction Motor

^{®}range that was selected for its low losses at elevated frequencies. As shown in Figure 9, the symmetry of the geometry is used to model only one pole-pair of the machine. For this machine, a previous study showed that the calculation of the skin effect did not significantly change the total core loss behaviour when the motor is supplied by sinusoidal waveforms. For this analysis, the motor was simulated at 4800 rpm, and supplied with a three-phase PWM waveform with a carrier frequency of 16 kHz, by using the standard library function that is implemented in the JMAG software. A timestep of 4.1 µs was used for the simulation.

## 7. Conclusions

^{®}range, which is a portfolio of electrical steels specifically developed to satisfy the demanding requirements of automotive traction applications. Laboratory magnetic measurements at high frequencies showed good agreement with the theoretical behaviour of the material at high frequencies. Further, FE calculations were carried out to predict losses in an automotive traction motor. This showed that a time-domain model may lead to somewhat different results compared to a frequency-domain model, which is based on linear superposition of the contribution of the individual harmonics. It could further be concluded that the explicit calculation of skin-effect and minor hysteresis loops did not affect the loss prediction considerably for this specific machine geometry and control. For the analysed case, it could be shown that the harmonics that are generated by the of PWM control add an additional 15% of core losses.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Flux density profile in half a lamination of 0.3 mm when an average sinusoidal flux density is applied with an amplitude of 0.15 T at 10 kHz, (

**b**) corresponding current density distribution in lamination and (

**c**) 1D model for the calculation of skin effect, where all parameters only depend on the z-dimension.

**Figure 2.**Flux density distribution within a lamination, when the average flux density is enforced to be a sinewave with (

**a**) average Bpk = 0.5 T at a frequency of 2 kHz and (

**b**) average B

_{pk}= 1.5 T at a frequency of 2 kHz.

**Figure 3.**Comparison of eddy-current loss constant, calculated using three different methodologies, for two average sinusoidal waveforms of small amplitude.

**Figure 4.**Polarisation waveforms in the material (near centre, near skin and average) for an average waveform of 0.025 T at 10 kHz, superposed on a DC bias polarisation of (

**a**) 0.5 T and (

**b**) 1 T.

**Figure 6.**Predicted and measured eddy current loss density as function of frequency, for sinusoidal polarisation waveforms with peak amplitudes of 0.05 T, 0.1 T and 0.15 T.

**Figure 7.**(

**a**) Quasi-static waveform which includes a quasi-static minor loop and (

**b**) shape of quasi-static minor loops for different DC bias fields.

**Figure 8.**(

**a**) Measured hysteresis loop for a 50 Hz, 0.9 T sinusoidal polarisation that was generated via a PWM scheme and (

**b**) corresponding measured polarisation waveform. The minor loops that were separately calculated are marked in red.

**Table 1.**Comparison of loss predictions for the data shown in Figure 8, for which a total loss of 0.967 W/kg was measured.

Methodology | Total Loss |
---|---|

Frequency-domain | 1 W/kg from which 0.97 W/kg at the first harmonic |

Time-domain without skin effect | 0.97 W/kg |

Time-domain with skin effect and minor loops | 0.95 W/kg from which 0.15 W/kg at all minor loops |

Specification | Value |
---|---|

Outer diameter | 195 mm |

Axial length | 171 mm |

Nominal power | 50 kW |

Maximum torque | 150 Nm |

Maximum speed | 12000 rpm |

Methodology | Total Core Loss |
---|---|

Frequency-domain | 546 W |

Time-domain without skin effect | 665 W |

Time domain with skin effect and separate minor loop calculation | 678 W |

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**MDPI and ACS Style**

Rens, J.; Vandenbossche, L.; Dorez, O. Iron Loss Modelling of Electrical Traction Motors for Improved Prediction of Higher Harmonic Losses. *World Electr. Veh. J.* **2020**, *11*, 24.
https://doi.org/10.3390/wevj11010024

**AMA Style**

Rens J, Vandenbossche L, Dorez O. Iron Loss Modelling of Electrical Traction Motors for Improved Prediction of Higher Harmonic Losses. *World Electric Vehicle Journal*. 2020; 11(1):24.
https://doi.org/10.3390/wevj11010024

**Chicago/Turabian Style**

Rens, Jan, Lode Vandenbossche, and Ophélie Dorez. 2020. "Iron Loss Modelling of Electrical Traction Motors for Improved Prediction of Higher Harmonic Losses" *World Electric Vehicle Journal* 11, no. 1: 24.
https://doi.org/10.3390/wevj11010024