Despite continuous effort to minimize torque ripple, the reduction of vibration has not been yet achieved due to the lack of link between these two performances. The difficulty of investigating the relationship stems from the fact that torque is solely influenced by tangential force whereas vibration is directly tied to radial force. In this paper, the correlation between torque ripple and vibration in an IPMSM (Interior Permanent Magnet Synchronous Motor) is established by examining the air-gap flux density in both radial and tangential directions. Its temporal and spatial harmonics are analyzed. A six-pole nine-slot IPMSM is used as a base configuration, and the air-gap flux density is varied by changing the height of a stator tooth tip. In the previous study, the same approach has been done to observe how to reduce torque ripple effectively. In this study, however, it is found that the trend of torque ripple is not the same as that of vibration with respect to the height of a pole tip, and the relation between the two is thoroughly observed and explained.

## I. INTRODUCTION

Interior permanent magnet synchronous motors (IPMSMs) are widely used in various applications due to their key features such as high efficiency, high power density, and good controllability.^{1,2} Inevitably, the placement of permanent magnets (PMs) inside the rotor creates electromagnetic saliency as a source of torque ripple and vibration, and hence, this saliency has a possibility to decrease the overall performance of the motor.^{3,4} There has been a plethora of research on the vibration of IPMSMs relating to torque ripple, cogging torque, and radial force.^{5,6} In the previous study of,^{7} the effect of air-gap flux on torque ripple has been already examined. In particular, the height of stator tooth tips is varied as a design parameter to change radial and tangential flux densities simultaneously. According to many Refs. 8–10, the reduction of torque ripple leads directly to the improvement of vibration. However, torque and vibration are affected by the two flux densities in a different way, and in other words, torque and vibration are determined by each different governing equation.^{11} The objective of this paper is to observe whether a trade-off between torque ripple and vibration exists or not under the same condition of design modification.^{7}

## II. PREVIOUS STUDY OF TORQUE RIPPLE

A six-pole nine-slot IPMSM having concentrated windings is a base model in this paper since this configuration is widely adopted in industries due to the ease of manufacturing. The two tips of a stator pole are selected as a design parameter, and as noted in 1, these two positions are critical in determining torque ripple. In Fig. 1, their thickness is depicted as SH1 and SH2 on the left and right side of a stator pole, respectively. The thickness of a pole tip is reduced from 0.7 mm up to 0.4 mm with an interval of 0.1 mm. In the base model, the thickness of SH1 and SH2 is 0.8 mm in common, and there are three major cases regarding the variation of thickness in a pole tip. In the first case, both SH1 and SH2 are changed at the same time. In Case 2, there is the decrease of SH1 with the fixed value of 0.8 mm in SH2, and the variation of Case 3 in SH1 and SH2 is exactly opposite to that of Case 2.

*F*

_{t}) along air gap.

*A*and

*r*

_{g}are the cross-sectional area of air gap and an average radius from the center of a rotor to air gap. The tangential force density (

*F*

_{t}) is determined by two magnetic flux densities in air gap as expressed in Eq. (2).

*B*

_{r},

*B*

_{t},

*μ*

_{0},

*θ*

_{m}, and

*ω*are radial magnetic flux density in air gap, tangential magnetic flux density in air gap, permeability in air, mechanical angle, and the angular frequency of rotation, respectively.

^{12}

. | Case1_1 . | Case1_2 . | Case1_3 . | Case1_4 . |
---|---|---|---|---|

SH1 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

SH2 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

Torque ripple (%) | 21.55 | 19.62 | 19.46 | 20.09 |

Case2_1 | Case2_2 | Case2_3 | Case2_4 | |

SH1 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

SH2 (mm) | 0.80 | 0.80 | 0.80 | 0.80 |

Torque ripple (%) | 25.02 | 25.61 | 26.18 | 28.23 |

Case3_1 | Case3_2 | Case3_3 | Case3_4 | |

SH1 (mm) | 0.80 | 0.80 | 0.80 | 0.80 |

SH2 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

Torque ripple (%) | 20.40 | 17.09 | 13.90 | 13.33 |

. | Case1_1 . | Case1_2 . | Case1_3 . | Case1_4 . |
---|---|---|---|---|

SH1 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

SH2 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

Torque ripple (%) | 21.55 | 19.62 | 19.46 | 20.09 |

Case2_1 | Case2_2 | Case2_3 | Case2_4 | |

SH1 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

SH2 (mm) | 0.80 | 0.80 | 0.80 | 0.80 |

Torque ripple (%) | 25.02 | 25.61 | 26.18 | 28.23 |

Case3_1 | Case3_2 | Case3_3 | Case3_4 | |

SH1 (mm) | 0.80 | 0.80 | 0.80 | 0.80 |

SH2 (mm) | 0.70 | 0.60 | 0.50 | 0.40 |

Torque ripple (%) | 20.40 | 17.09 | 13.90 | 13.33 |

Compared to the base, three big design modifications, Case1_4, Case2_4, and Case3_4, are given in Fig. 2 in terms of their instantaneous torque. It is noted that their pole height as a design variable is dominant on torque with respect to electrical angle, but there is almost no change of their average. Therefore, the analysis of torque ripple is performed at the same average torque of 2.1 Nm as a rated condition, and the results are summarized in Table I. In case of the base having the pole height of 0.8 mm on both SH1 and SH2, torque ripple is 24.25%. The biggest improvement of torque ripple is obtained in Case 3 where SH2 is reduced from 0.7 mm to 0.4 mm with the fixed height of 0.8 mm in case of SH1. On the other hand, torque ripple deteriorates in Case 2 by decreasing only the height of SH1 up to 0.4 mm. It is noted that regardless of SH1, the variation of SH2 is critical in the performance of torque ripple. The reason for this result is explained in Table II as following.

. | Time . | Case2_1 . | Case2_2 . | Case2_3 . | Case2_4 . |
---|---|---|---|---|---|

Sum of F_{t} SH1 | Maximum torque | −193 | −193 | −195 | −182 |

Minimum torque | 186 | 181 | 177 | 167 | |

Subtraction | −379 | −374 | −372 | −349 |

. | Time . | Case2_1 . | Case2_2 . | Case2_3 . | Case2_4 . |
---|---|---|---|---|---|

Sum of F_{t} SH1 | Maximum torque | −193 | −193 | −195 | −182 |

Minimum torque | 186 | 181 | 177 | 167 | |

Subtraction | −379 | −374 | −372 | −349 |

. | Time . | Case3_1 . | Case3_2 . | Case3_3 . | Case3_4 . |
---|---|---|---|---|---|

Sum of F_{t} SH2 | Maximum torque | 1278 | 1265 | 1239 | 1226 |

Minimum torque | 650 | 693 | 722 | 752 | |

Subtraction | 628 | 572 | 517 | 474 |

. | Time . | Case3_1 . | Case3_2 . | Case3_3 . | Case3_4 . |
---|---|---|---|---|---|

Sum of F_{t} SH2 | Maximum torque | 1278 | 1265 | 1239 | 1226 |

Minimum torque | 650 | 693 | 722 | 752 | |

Subtraction | 628 | 572 | 517 | 474 |

In Table II, the sum of *F*_{t} is given at the maximum and minimum torque positions, and their subtraction is compared in Cases 2 and 3. In the result of Case 2, the gap of *F*_{t} at the maximum and minimum torque increases as SH1 length becomes shorter, and it means that the torque ripple increases as described in Table I. Torque ripple in Case 3 decreases as SH2 becomes shorter due to the decreased gap of Ft at the maximum and minimum torque. As a result, the length of SH2 is dominant on the reduction of torque ripple.

## III. TRADE-OFF BETWEEN TORQUE RIPPLE AND VIBRATION

### A. Stator vibration

*F*

_{0}denotes the DC component of radial force.

*F*

_{nm}and

*θ*

_{nm}are radial force density and phase angle with respect to spatial (m) and temporal (n) harmonics, respectively. The vibration displacement of a stator core is calculated by Eq. (5) without the DC component of radial force since vibration displacement varies only when time changes.

*R*

_{si},

*L*

_{stk},

*M*,

*ω*

_{m},

*ω*

_{n}, and

*ζ*

_{m}denote inner radius, stack length, weight, natural frequency, angular frequency, and damping ratio per mode, respectively, only on the stator side.

^{13}

### B. Vibration analysis regarding radial force

As given in Eq. (5), the vibration displacement of an electric motor is influenced by interaction between the natural frequency (*ω*_{m}) of its stator and the radial force density of *F*_{nm}. Using Eq. (4), force density is decomposed with respect to its temporal and spatial components to end up with *F*_{nm}. As shown in Fig. 3, the spectrum of *F*_{nm} is illustrated in terms of its temporal and spatial harmonic orders in case of the base model. It is noted that Eq. (3) is the same as Eq. (4) in a different format, and the raw data of *F*_{nm} is *F*_{r} directly from air-gap flux densities in the radial and tangential direction. From modal analysis, it is also found that the natural frequency of the third mode is 4077 Hz in Fig. 3.

In terms of temporal harmonic (n) in Fig. 3, the magnitude of the sixth components is strongest compared to other spectra except the 0*th* order as shaded in yellow along the axis of m. Among the temporal sixth spectra, two dominant numbers regarding spatial harmonic (m) are the third and sixth orders. The smaller order (m = 3) is the greatest common factor of pole and slot numbers, and the next one (m = 6) is determined by the number of rotor poles.

As mentioned in Fig. 3, it is seen that the spectrum of radial force density is distinguished in terms of temporal (n) and spatial (m) order, and two dominant spectra are *F*_{(n=6, m=3)} and *F*_{(n=6, m=6)}. In Table III, two spectra in radial force density are given in case of the base model. As circled twice in red, *F*_{(n=6, m=3)} and *F*_{(n=6, m=6)} are 67 243 *N*/*m*^{2} and 140 701 *N*/*m*^{2}, respectively. Also, using the result of *F*_{(n=6, m=3)} and *F*_{(n=6, m=6)}, vibration displacement is calculated by Eq. (5) on the right side of Table III. Just like the procedure of the base model, that of Case 1, 2, and 3 is conducted in the same manner, and their radial force density and vibration displacement are given in Table III.

Base . | F_{nm} (N/m^{2})
. | A_{nm} (mm)
. | ||||
---|---|---|---|---|---|---|

n = 6, m = all . | n = 6, m = 3 . | n = 6, m = 6 . | n = 6, m = all . | n = 6, m = 3 . | n = 6, m = 6 . | |

344 235 (100%) . | 67 243 (19.5%) . | 140 701 (40.9%) . | 16.653 (100%) . | 16.406 (98.5%) . | 2.852 (17.1%) . | |

Case1_1 | 322 492 | 67 886 | 140 797 | 16.809 | 16.563 | 2.854 |

Case1_2 | 317 199 | 69 235 | 141 485 | 17.136 | 16.890 | 2.869 |

Case1_3 | 321 381 | 70 602 | 142 332 | 17.467 | 17.226 | 2.887 |

Case1_4 | 327 091 | 71 634 | 142 956 | 17.718 | 17.476 | 2.901 |

Case2_1 | 341 313 | 67 548 | 139 927 | 16.724 | 16.481 | 2.836 |

Case2_2 | 344 140 | 68 150 | 139 201 | 16.867 | 16.628 | 2.822 |

Case2_3 | 345 425 | 68 550 | 138 635 | 16.961 | 16.726 | 2.811 |

Case2_4 | 344 603 | 69 814 | 137 912 | 17.263 | 17.032 | 2.797 |

Case3_1 | 325 783 | 66 987 | 141 668 | 16.595 | 16.344 | 2.871 |

Case3_2 | 322 613 | 67 521 | 143 232 | 16.720 | 16.470 | 2.903 |

Case3_3 | 324 419 | 67 816 | 144 174 | 16.804 | 16.542 | 2.923 |

Case3_4 | 330 301 | 68 299 | 145 431 | 16.924 | 16.664 | 2.949 |

Base . | F_{nm} (N/m^{2})
. | A_{nm} (mm)
. | ||||
---|---|---|---|---|---|---|

n = 6, m = all . | n = 6, m = 3 . | n = 6, m = 6 . | n = 6, m = all . | n = 6, m = 3 . | n = 6, m = 6 . | |

344 235 (100%) . | 67 243 (19.5%) . | 140 701 (40.9%) . | 16.653 (100%) . | 16.406 (98.5%) . | 2.852 (17.1%) . | |

Case1_1 | 322 492 | 67 886 | 140 797 | 16.809 | 16.563 | 2.854 |

Case1_2 | 317 199 | 69 235 | 141 485 | 17.136 | 16.890 | 2.869 |

Case1_3 | 321 381 | 70 602 | 142 332 | 17.467 | 17.226 | 2.887 |

Case1_4 | 327 091 | 71 634 | 142 956 | 17.718 | 17.476 | 2.901 |

Case2_1 | 341 313 | 67 548 | 139 927 | 16.724 | 16.481 | 2.836 |

Case2_2 | 344 140 | 68 150 | 139 201 | 16.867 | 16.628 | 2.822 |

Case2_3 | 345 425 | 68 550 | 138 635 | 16.961 | 16.726 | 2.811 |

Case2_4 | 344 603 | 69 814 | 137 912 | 17.263 | 17.032 | 2.797 |

Case3_1 | 325 783 | 66 987 | 141 668 | 16.595 | 16.344 | 2.871 |

Case3_2 | 322 613 | 67 521 | 143 232 | 16.720 | 16.470 | 2.903 |

Case3_3 | 324 419 | 67 816 | 144 174 | 16.804 | 16.542 | 2.923 |

Case3_4 | 330 301 | 68 299 | 145 431 | 16.924 | 16.664 | 2.949 |

In each temporal and spatial spectrum, the magnitude of vibration displacement is proportional to that of radial force density. However, this result is not coherent at all in case of changing spatial harmonic order (m). More specifically, *F*_{(n=6, m=3)} is lower than *F*_{(n=6, m=6)} by the gap of 21.4%p, but in case of vibration displacement, *A*_{(n=6, m=3)} is higher than *A*_{(n=6, m=6)} by the gap of 81.4%p in an opposite way. The reason for this result is that spatial harmonic order (m) is proportional to natural frequency (*ω*_{m}), and as shown in Eq. (5), natural frequency is inversely proportional to vibration displacement.^{13} The key point until this part is that the magnitude of vibration displacement is dominated mainly by the smaller harmonic in space which is the third order in this paper.

### C. Investigation of torque ripple and vibration

Figure 4 shows the vibration displacement and torque ripple of Case 1, 2, and 3 compared to the base model. The comparison of torque ripple regarding the height of a stator pole tip is given in Table I, and the result is given as a line plot in Fig. 4 including Case 1, 2, and 3 compared to the base model. The data of vibration displacement is obtained from Table III and depicted as a bar plot in Fig. 4 as well. In terms of torque ripple, the variation of SH2 is critical regardless of SH1, and hence, the pattern of torque ripple is different in Case 1, 2, and 3. However, that of vibration displacement is similar in all three cases, and this means that the smaller height of SH1 and SH2 leads to the bigger vibration displacement. In summary, torque and vibration are determined by each different governing equation. More specifically, torque is fundamentally dependent on Eq. (2), and on the other hand, vibration is basically affected by Eq. (3). There are radial and tangential magnetic flux densities in the two equations. The product of the two flux densities is dominant in torque, but vibration is affected by difference between their squares. As shown in Fig. 4, Case 2 has a rising trend simultaneously in both torque ripple and vibration, but this result is not the same in the other two cases. In particular, the variation of vibration is more dominant in Case 1, but this is totally opposite in Case 3.

Case 3 is the best fit due to its torque ripple performance. From the trade-off point of view between torque ripple and vibration, the four cases (No. 3_1 ∼ No. 3_4) have to be compared each other under the condition of a given specification of vibration.

## IV. CONCLUSION

In this paper, the height of a stator pole tip is employed as a design parameter to see how it affects vibration performance along with the variation of torque ripple. In terms of torque ripple, the variation of SH2 is critical regardless of SH1, and hence, the pattern of torque ripple is different in Case 1, 2, and 3. However, that of vibration displacement is similar in all three cases, and this means that the smaller height of SH1 and SH2 leads to the bigger vibration displacement. This key conclusion comes from the fact that torque and vibration are determined by each different governing equation. More specifically, there are radial and tangential magnetic flux densities in air gap. The product of the two flux densities and difference between their squares are dominant in torque and vibration, respectively. Vibration is dominated mainly by radial force density which is decomposed into its temporal and spatial components by using Fourier transform.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Sangjin Lee**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). **Gyeonghwan Yun**: Conceptualization (lead); Formal analysis (lead); Methodology (equal); Writing – original draft (supporting); Writing – review & editing (equal). **Grace Firsta Lukman**: Conceptualization (equal); Data curation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Cheewoo Lee**: Conceptualization (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Permanent Magnet Motor Technology: Design and Applications*

*Noise of Polyphase Electric Motors*