# Classification Method to Define Synchronization Capability Limits of Line-Start Permanent-Magnet Motor Using Mesh-Based Magnetic Equivalent Circuit Computation Results

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## Abstract

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## 1. Introduction

^{®}, has been developed [16]. However, some parasitic effects, such as magnetic saturation, cross-magnetization and differences in rotor bar resistances, are neglected in the transient simulation models [5] and could affect the correct modeling output. To get more reliable and accurate results, numerical techniques like magnetic equivalent circuit (MEC) modeling [7,8,11] and finite elements analysis (FEA) have to be involved. It is also possible to use FEA complementarily with SPEED

^{®}, in order to correct the equivalent circuit parameters or to calculate iron loss [17]. The magnetic flux in an LS-PMSM depends on the rotor position and the saturation of the ferromagnetic material. The magnetic saturation will cause variations in phase inductances and back-electromotive force (EMF). Such variations of the inductances for interior permanent magnet synchronous machines (IPMSM) can be computed as functions of the line current by means of an FEA model done in preprocessing before simulation in the time domain [18]. On the one hand, FEA will give the most accurate results; on the other hand, changing the design parameters requires a reconstruction of the FEA model [7]. The process takes a long time and is very computationally intensive. A method of interest to meet the compromise between accuracy and speed requirements, MEC can be used, especially in design optimization software [7,8,11,17,19,20]. The computation of an MEC model can be done by two methods [21]. The first method is the node-based analysis that is derived from Kirchhoff’s current law (KCL) and used in [7,11], while the other one, namely the mesh-based analysis, is based on Kirchhoff’s voltage law (KVL) [19,22]. However, in [8], stator slotting is neglected, and position dependency is not taken into account. In [17,23], a more complex air gap reluctance modeling for IPMSM taking slotting into account is presented. This modeling method is also used to compute the start-up torque in LS-PMSM [7]. To increase the accuracy of the MEC start-up model, the number of flux paths or branches can be increased, subsequently causing a computation time and complexity increase, such that the advantage relative to FEA will be lost [17].

^{®}environment is presented in order to be able to study the dynamic behavior of LS-PMSM. The energy conversion is calculated with MEC modeling, such as in [7,19,20]. In contrast to [7], wherein node-based MEC is used to model the LS-PMSM dynamics, here a mesh-based analysis is applied. As in this paper, nonlinear parameters will be used, this method facilitates convergence compared to the node-based analysis [24,25]. In [23], an analytical air gap permeance function is presented to be used in the node-based analysis. In mesh-based analysis, an algorithm that excludes the zero permeances from the air gap permeance/reluctance matrix is essential [22,26], which is called the dynamic management of air gap reluctances in [22]. The contribution in this paper is an algorithm to define the dynamic management of the air gap reluctances based on the aforementioned analytical air gap permeance function.

^{®}FEA. The simulation results obtained with the MEC model are compared to experimental results.

## 2. Magnetic Network Algorithm

#### 2.1. Constant Reluctances

#### 2.2. Position-Dependent Reluctances

#### Air Gap Reluctances

#### 2.3. Nonlinear Reluctances

#### 2.4. Active Elements

#### 2.4.1. Magnet MMF Source

#### 2.4.2. Stator and Rotor MMF

#### 2.5. MEC Network

## 3. Analysis of an LS-PMSM by Using MEC-Based Time Simulations

#### 3.1. Dynamical Model of the LS-PMSM

^{®}Simulink R2015b (The MathWorks, Inc., Natick, MA, USA) is used. To solve the differential Equations (27) and (28) in a discrete time step, the Euler method is used. Each time iteration, the nonlinear magnetic field equation is solved with the NR-method. In this method, the rotor position ${\theta}_{m}$ and rotor/stator current are input. To reduce the number of NR iterations, the branch flux vector from the last time step has been set as the initial state for the NR algorithm in the next time step. The block diagram to model and simulate the dynamical behavior of the LS-PMSM is shown in Figure 10.

#### 3.2. Synchronization Capability

#### 3.3. Results

#### 3.3.1. Simulation Results

^{®}(Version 10.4, CEDRAT S.A., Meylan Cedex, France).The input parameters for both methods can be found in Table 1, and the solving process options can be found in Table 2. The computation time has been compared, and the results are given in Table 3. Considering that the number of elements in the FEA method is reduced, but not minimized, the computation time of the MEC computation is $0.38\%$ of the computation time of the FEA computation, which is about 260-times less. In this paper, the simulation program and the MEC model are evaluated step by step. In Figure 14a, one period of the waveform of the electromotive force (EMF) is given at 3000 rpm. In Figure 14b, a fourth of the period has been enlarged. In this figure, the differences between the MEC results and the FEA results are clearer. The harmonic distortion amplitudes of the EMF are higher in the case of FEA. Due to a transitional phenomenon caused by the mesh algorithm in the air gap, there is more numeric noise in the waveforms of the MEC. The harmonic content of the EMF waveform, obtained after a fast Fourier transform (FFT) analysis, is shown in Figure 15. In this figure, the values are given. In Figure 15a, the harmonic content of the FEA simulation is given, while Figure 15b shows the harmonics of MEC. The orders of the harmonics correspond; the magnitude of the fundamental harmonic is almost the same for MEC and FEM. For the higher harmonics, the magnitude is less in most cases. The cogging torque is given in Figure 16. In this figure, the effect of the slotting is much more pronounced in the waveform calculated by FEA simulations compared to the waveform calculated by MEC simulations. In the MEC model, a smooth air gap permeance function is used. This function is only position dependent. In contrast with the FEA model, the saturation in the tooth tips is not implemented in the MEC model.

#### 3.3.2. Experimental Results

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Half cross-section and magnetic equivalent circuit (MEC) model of line start permanent magnet synchronous motors (LS-PMSM).

**Figure 2.**Tooth overlap at different positions: start/end of the overlap (

**a**) and completely aligned (

**b**).

**Figure 4.**(

**a**) Position-dependent air gap permeance function for one stator tooth to rotor tooth connection; (

**b**) derivative of the position-dependent air gap permeance function for one tooth-to-tooth connection; (

**c**) air gap permeance function in the stator tooth reference (dark gray) and rotor tooth reference (light gray); (

**d**) derivative of the air gap permeance function in the stator tooth reference (dark gray) and rotor tooth reference (light gray).

**Figure 6.**Air gap mesh number (left vertical axes) seen by a branch constantly linked to a rotor tooth $j=1$ with a pulsating air gap permeance (right vertical axes).

**Figure 7.**Air gap mesh number (left vertical axes) seen by a branch constantly linked to a stator tooth $i=1$ with a pulsating air gap permeance (right vertical axes).

**Figure 11.**Load angle of a successful synchronization and simulation stop from the moment the synchronization is nearly reached.

**Figure 12.**Synchronization capability at voltage variation (inertia of 0.0106 kg·m${}^{2}$ and nominal torque).

**Figure 14.**Electromotive force (EMF) at nominal speed (3000 rpm) for one period in (

**a**) and $\frac{1}{4}$ of the period in (

**b**); MEC is compared with finite element analysis (FEA).

**Figure 15.**Amplitudes of voltage harmonics of the EMF waveform, as a function of the harmonic order for FEA simulations (

**a**) and MEC simulations (

**b**).

**Figure 17.**Electromagnetic reluctance torque, at 7 A, in the case of (

**a**) linear and (

**b**) nonlinear magnetical material. There is no influence of permanent magnets. MEC is compared with FEA.

**Figure 18.**Simulation at nominal torque and five-times the inertia of the rotor, failing in synchronization at $85\%$ of the nominal voltage and succeeding in synchronization at $95\%$ of the nominal voltage, with (

**a**) FEA and (

**b**) MEC.

**Figure 20.**Effect of the voltage on the synchronization of the motor for a constant torque of 1.5-times the nominal torque, with measurements (MEAS) and MEC simulations.

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

${V}_{\mathrm{nom}}$ | nominal voltage | 380 V |

${P}_{\mathrm{nom}}$ | nominal power | 3 kW |

${f}_{\mathrm{nom}}$ | nominal frequency | 50 Hz |

${N}_{\mathrm{nom}}$ | nominal speed | 3000 rpm |

${T}_{\mathrm{nom}}$ | nominal torque | 9.55 Nm |

${J}_{\mathrm{m}}$ | motor inertia | 0.0053 kg·m${}^{2}$ |

${N}_{\mathrm{ph}}$ | number of phases | 3 |

${N}_{\mathrm{p}}$ | number of poles | 2 |

N${}_{s}$ | number of stator teeth | 36 |

${N}_{\mathrm{r}}$ | number of rotor teeth | 20 |

g | air gap length | 0.3 mm |

${L}_{\mathrm{stk}}$ | rotor stack length | 103 mm |

${r}_{\mathrm{rot}}$ | rotor radius | 42.2 mm |

${w}_{\mathrm{PM}}$ | permanent magnet width | 27 mm |

${l}_{\mathrm{PM}}$ | permanent magnet length | 103 mm |

${d}_{\mathrm{PM}}$ | permanent magnet thickness | 3 mm |

${B}_{\mathrm{r}}$ | permanent magnet remanent induction | 1.1 T |

Setting | Value |
---|---|

Relative precision (${K}_{\mathrm{r}}$) for NR | $1\times {10}^{-4}$ |

Absolute precision (${K}_{\mathrm{a}}$) for NR | $1\times {10}^{-6}$ |

Relaxation | no |

Material Properties | MEC | FEA |
---|---|---|

linear | 0.007 s | 1.6 s |

nonlinear | 0.014 s | 3.67 s |

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**MDPI and ACS Style**

Wymeersch, B.; De Belie, F.; Rasmussen, C.B.; Vandevelde, L.
Classification Method to Define Synchronization Capability Limits of Line-Start Permanent-Magnet Motor Using Mesh-Based Magnetic Equivalent Circuit Computation Results. *Energies* **2018**, *11*, 998.
https://doi.org/10.3390/en11040998

**AMA Style**

Wymeersch B, De Belie F, Rasmussen CB, Vandevelde L.
Classification Method to Define Synchronization Capability Limits of Line-Start Permanent-Magnet Motor Using Mesh-Based Magnetic Equivalent Circuit Computation Results. *Energies*. 2018; 11(4):998.
https://doi.org/10.3390/en11040998

**Chicago/Turabian Style**

Wymeersch, Bart, Frederik De Belie, Claus B. Rasmussen, and Lieven Vandevelde.
2018. "Classification Method to Define Synchronization Capability Limits of Line-Start Permanent-Magnet Motor Using Mesh-Based Magnetic Equivalent Circuit Computation Results" *Energies* 11, no. 4: 998.
https://doi.org/10.3390/en11040998