# Effects of the Magnetic Model of Interior Permanent Magnet Machine on MTPA, Flux Weakening and MTPV Evaluation

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## Abstract

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## 1. Introduction

## 2. Operating Strategies in the Linear Model

- R: phase resistance;
- ${\overline{v}}_{dq}$: voltage vector;
- ${\overline{i}}_{dq}$: current vector;
- ${\omega}_{m}$: mechanical speed;
- ${p}_{p}$: pole pairs number;
- ${\overline{\lambda}}_{dq}={\overline{L}}_{dq}{\overline{i}}_{dq}+{\overline{\lambda}}_{pm,dq}$: flux linkage vector;
- ${\overline{\lambda}}_{pm,dq}=\left[\begin{array}{c}{\lambda}_{pm}\\ 0\end{array}\right]$: flux linkage due to PM;
- ${\overline{L}}_{dq}=\left[\begin{array}{cc}{L}_{d}& 0\\ 0& {L}_{q}\end{array}\right]$: inductance matrix.

- Pure sinusoidal currents and air gap magnetic field distribution;
- No cross-coupling between the equivalent magnetic circuit and the d-q axes. The two axes are magnetically decoupled because they are at 90 electrical degrees;
- No saturation effects, i.e., no variation of the inductances versus currents.

## 3. Finite Element Analysis for Magnetic Model Mapping

#### 3.1. Standard Method

#### 3.2. Frozen Permeability Method

## 4. Nonlinear d-q Model Computation

- Define a starting value of ${L}_{d}$, ${L}_{q}$ considering a known point belonging to the desired control curve;
- Re-compute ${L}_{d}$, ${L}_{q}$ with the current values of ${I}_{d}$, ${I}_{q}$ and the inductances map;
- Use the new value of ${L}_{d}$, ${L}_{q}$ computed in the previous bullet for the next iteration.

- ${L}_{d}$ and ${L}_{q}$ are computed in the point $({I}_{ch};0)$ with the inductances map;
- ${I}_{ch}$ is re-computed with Equation (8);
- The difference $\Delta {I}_{ch}$ between the current ${I}_{ch}$ value and the previous one is computed and compared with a desired threshold.

#### 4.1. Maximum Torque Per Ampere (MTPA)

- A space vector of current amplitude $\left[I\right]$ is defined ranging from 0 to current limit ${I}_{lim}$ value;
- For each value of $\left[I\right]$, a vector of current phase $\left[\gamma \right]]$ is defined ranging from 90° to 180°;
- Each couple of the matrix $[I;\gamma ]$ identifies an operating point in the second quadrant of the d-q plane;
- For each couple, ${L}_{d}$ and ${L}_{q}$ are interpolated from inductances maps at disposal. Torque is computed by Equation (4) and stored in a matrix;
- For each value of $\left[I\right]$, the maximum torque is computed comparing the matrix elements corresponding to each value of $\left[\gamma \right]$. Also the corresponding values of ${L}_{d}$, ${L}_{q}$, ${I}_{d}$ and ${I}_{q}$ are computed and stored.

#### 4.2. Flux Weakening (FW)

- The last element of the base speed vector $\left[{\omega}_{base}\right]$ of MTPA is selected as the starting element for the FW speed (${\omega}_{FW,start}$);
- The limit speed in FW (${\omega}_{FW,end}$) is computed according to Equation (15);
- The FW speed vector ranges from ${\omega}_{start}$ to ${\omega}_{lim}$;
- Starting values of ${L}_{d}$ and ${L}_{q}$ are computed in the end point of MTPA trajectory: ${L}_{d}={L}_{d}({I}_{{d}_{end,MTPA}},{I}_{{q}_{end,MTPA}})$ and ${L}_{q}={L}_{q}({I}_{{d}_{end,MTPA}},{I}_{{q}_{end,MTPA}})$;
- For each element of the speed vector, the following parameters are computed and stored:

#### 4.3. Maximum Torque Per Volt (MTPV)

- Starting values of ${L}_{d}$ and ${L}_{q}$ are computed in the starting point $({I}_{ch};0)$ with the stabilizing loop described in Section 4;
- For each element of the speed vector with ${\gamma}^{\prime}=180\xb0$, the ${L}_{d}$ and ${L}_{q}$ values computed at the previous iteration are taken as starting values and the stabilizing loop is performed. After their values are fixed with high accuracy, the following parameters are computed:
- -
- -
- -
- Torque is computed by Equation (4) and stored in a matrix;
- -
- The new values of ${L}_{d}$ and ${L}_{q}$ to be used for the next speed value are interpolated from the map with the current ${I}_{d}$ and ${I}_{q}$ values.

#### 4.4. Voltage Limit Ellipses

- The corresponding ${I}_{d}$ on the MTPA trajectory is identified;
- The starting values of ${L}_{d}$ and ${L}_{q}$ are interpolated from the inductance map;
- The corresponding speed value is computed by Equation (7);
- A vector ranging from the identified ${I}_{d}$ to the current limit $-{I}_{lim}$ is defined. For each value of this vector:
- -
- -
- The new ${L}_{d}$ and ${L}_{q}$ with the current values of ${I}_{d}$ and ${I}_{q}$ are interpolated from the maps for the next iteration.

#### 4.5. Constant Torque Hyperbolas

- The corresponding ${I}_{d}$ on the MTPA trajectory is identified;
- The starting values of ${L}_{d}$ and ${L}_{q}$ are interpolated from the inductance map;
- The corresponding torque value is computed by Equation (4);
- A vector ranging from the identified ${I}_{d}$ to the current limit $-{I}_{lim}$ is defined. For each value of this vector:
- -
- -
- The new ${L}_{d}$ and ${L}_{q}$ with the current values of ${I}_{d}$ and ${I}_{q}$ are interpolated from the map for the next iteration.

## 5. Results and Comparisons

#### 5.1. Generated Control Trajectories

#### 5.2. Comparison of the Two Different Nonlinear Methods

- A range of current density $\left[J\right]$ is defined;
- A range of current phase angle $\left[\gamma \right]$ is defined;
- For each couple of $[J;\gamma ]$ matrix, five different rotor positions are defined (6 mechanical degrees apart one from the other in order to cover 360 electrical degrees);
- The solution of the magnetic model is found for all the different rotor positions and torque is evaluated by the Maxwell’s stress tensor;
- The mean value of the torque on all position is computed;

## 6. Conclusions and Future Development

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 12.**Comparison of MTPA and MTPV curves, obtained with Standard Method (blue) and FP (yellow).

**Figure 13.**Comparison of Voltage Limit Ellipse (at base speed) and Constant Torque Hyperbola (at rated torque), obtained with Standard Method (blue) and FP (yellow).

**Figure 14.**Comparison of MTPA curve obtained by the conventional linear model, nonlinear model based on Standard Method, nonlinear model based on FP and direct computation via 2D FEA.

**Figure 15.**Comparison of torque–speed curves obtained by the conventional linear model, nonlinear model based on Standard Method and nonlinear model based on FP.

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

n° of Pole Pairs | ${p}_{p}$ | 2 | - |

n° of Stator Slots | Q | 24 | - |

n° of Conductors per Slot | ${n}_{c}$ | 2 | - |

Stack Length | ${L}_{stk}$ | 100 | mm |

Phase Resistance | R | 0.0032 | $\Omega $ |

Flux Linkage of PM | ${\lambda}_{pm}$ | 0.0128 | Wb |

Rated Current | ${I}_{lim}$ | 400 | A |

Overload Current | ${I}_{OL}$ | 720 | A |

DC BUS Voltage | ${V}_{lim}$ | 48 | V |

Maximum Speed | ${\omega}_{max}$ | 30 | krpm |

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## Share and Cite

**MDPI and ACS Style**

Bianchini, C.; Bisceglie, G.; Torreggiani, A.; Davoli, M.; Macrelli, E.; Bellini, A.; Frigieri, M. Effects of the Magnetic Model of Interior Permanent Magnet Machine on MTPA, Flux Weakening and MTPV Evaluation. *Machines* **2023**, *11*, 77.
https://doi.org/10.3390/machines11010077

**AMA Style**

Bianchini C, Bisceglie G, Torreggiani A, Davoli M, Macrelli E, Bellini A, Frigieri M. Effects of the Magnetic Model of Interior Permanent Magnet Machine on MTPA, Flux Weakening and MTPV Evaluation. *Machines*. 2023; 11(1):77.
https://doi.org/10.3390/machines11010077

**Chicago/Turabian Style**

Bianchini, Claudio, Giorgio Bisceglie, Ambra Torreggiani, Matteo Davoli, Elena Macrelli, Alberto Bellini, and Matteo Frigieri. 2023. "Effects of the Magnetic Model of Interior Permanent Magnet Machine on MTPA, Flux Weakening and MTPV Evaluation" *Machines* 11, no. 1: 77.
https://doi.org/10.3390/machines11010077