Comparison of Electromagnetic Performance in Interior Permanent Magnet Motors with Different Central and Bilateral Bridges

: Studies on the central and bilateral bridges of interior permanent magnet (IPM) motors often focus on individual mechanical strength or electromagnetic performance, lacking comparative studies on the electromagnetic performance of motors with different central and bilateral bridges under the same mechanical strength. This paper designs three rotors with different central and bilateral bridges and compares the electromagnetic performance of the three motors. First, to ensure the safe operation of the three rotors at high speeds, the mechanical stress of each rotor has been analyzed using the finite-element method (FEM). Subsequently, the major electromagnetic performances of the three motors are analyzed and compared, including the air-gap flux density, back electromotive force (back-EMF), inductance, salience, torque, power, loss, efficiency, and demagnetization. The results indicate that the rotor without central bridges has the largest leakage flux and the lowest torque but exhibits minimal torque ripple. The rotor with narrower bilateral bridges has the highest torque and maximum torque ripple. The torque performance of the rotor with wider bilateral bridges lies between the two aforementioned motors, and it possesses the highest efficiency. In the end, by adjusting the dimensions of the permanent magnets, the torque of all three models increases, but the motor with narrower bilateral bridges still has the largest torque. These findings provide valuable references for rotor design.


Introduction
In recent years, the issue of environmental pollution has prompted a heightened global concern for environmental protection.As a crucial response to this increasingly serious problem, the automotive industry has undergone a significant shift from internal combustion engine vehicles to electric vehicles (EVs) [1][2][3].The power source of EVs can be obtained from renewable energy systems, such as solar or wind power.This feature holds enormous potential for reducing greenhouse gas emissions and the release of toxic wastes [4].Permanent Magnet Synchronous Motors (PMSMs), with advantages such as small size, high efficiency, and high power density, have become a key component of EVs [5][6][7][8].
PMSMs can be further categorized into surface permanent magnet (SPM) motors and interior permanent magnet (IPM) motors [9].SPM motors require the use of a protective sleeve for the permanent magnets (PMs), leading to a substantial increase in manufacturing costs [10].In contrast, IPM motors, with the PMs located inside the rotor, provide better protection, eliminating the need for additional sheathing for safe operation.Furthermore, the significant difference between the q-axis and d-axis inductance in IPM motors results in a substantial increase in reluctance torque, enhancing overall motor performance.The unique characteristics of IPM motors have made them particularly attractive for integration into EVs [11].Meanwhile, Fractional slot concentrated windings have been widely utilized various rotor topologies, such as spoke-type PMs, tangential-type PMs, U-shape PMs, and V-shape PMs.The findings suggest that the IPM motor with V-shape PMs exhibits superior performance.The above studies either solely investigate electromagnetic performance or focus on rotor strength, with scarce literature examining the impact of different combinations of central and bilateral bridge widths on motor performance while maintaining consistent rotor strength.
The specific arrangement of this article is as follows.First, three rotors with differing central and bilateral bridge widths will be designed, and a mechanical strength verification will be conducted to ensure that these three rotors possess equivalent mechanical strength under maximum rotational speed.Following this, an analysis and comparison of the no-load performance and load performance of the three motors will be carried out.There are various combinations of central and bilateral bridge widths that can meet the mechanical strength requirements of the rotor.However, to highlight the differences in these designs, three models with significantly divergent central and bilateral bridge widths have been selected.The specific widths of the central and bilateral bridges are determined by the mechanical stress analysis in Section 2.2, and the exact values are marked in Figure 1.In terms of other parameters, such as PM dimensions and stator specifications, these three motors remain consistent.The cost of three motors is roughly the same.To decrease the axial length of the motors, the stators of all motors have adopted the same concentrated winding scheme.The main motor parameters are detailed in Table 1.The rated current is 20 A, and the rated speed is 3000 rpm.

Motor Specification and Mechanical Stress Analysis 2.1. Motor Specification
Machines 2024, 12, x FOR PEER REVIEW 3 of 14 motors featuring various rotor topologies, such as spoke-type PMs, tangential-type PMs, U-shape PMs, and V-shape PMs.The findings suggest that the IPM motor with V-shape PMs exhibits superior performance.The above studies either solely investigate electromagnetic performance or focus on rotor strength, with scarce literature examining the impact of different combinations of central and bilateral bridge widths on motor performance while maintaining consistent rotor strength.
The specific arrangement of this article is as follows.First, three rotors with differing central and bilateral bridge widths will be designed, and a mechanical strength verification will be conducted to ensure that these three rotors possess equivalent mechanical strength under maximum rotational speed.Following this, an analysis and comparison of the no-load performance and load performance of the three motors will be carried out.There are various combinations of central and bilateral bridge widths that can meet the mechanical strength requirements of the rotor.However, to highlight the differences in these designs, three models with significantly divergent central and bilateral bridge widths have been selected.The specific widths of the central and bilateral bridges are determined by the mechanical stress analysis in Section 2.2, and the exact values are marked in Figure 1.In terms of other parameters, such as PM dimensions and stator specifications, these three motors remain consistent.The cost of three motors is roughly the same.To decrease the axial length of the motors, the stators of all motors have adopted the same concentrated winding scheme.The main motor parameters are detailed in Table 1.The rated current is 20 A, and the rated speed is 3000 rpm.

Mechanical Stress Analysis
To ensure the safe operation of the rotors, finite-element method (FEM) analysis is conducted on the mechanical stress of three rotors at 1.2 times the maximum speed (21,600 rpm).Figure 2 illustrates the mechanical stress distribution of the three rotors.To show the deformation clearly, the deformation has been magnified 200 times.Clearly, Model A exhibits the maximum deformation.After repeatedly adjusting the central and bilateral bridge widths, the maximum mechanical stresses for the three models are 387.5 MPa, 373.6 Mpa, and 379.1 Mpa, respectively.The maximum mechanical stresses of the three models are comparable, meeting the design requirements.The maximum mechanical stress in Model A occurs on the bilateral bridges, whereas in Model B and Model C, it occurs on the central bridges.

Mechanical Stress Analysis
To ensure the safe operation of the rotors, finite-element method (FEM) analysis is conducted on the mechanical stress of three rotors at 1.2 times the maximum speed (21,600 rpm).Figure 2 illustrates the mechanical stress distribution of the three rotors.To show the deformation clearly, the deformation has been magnified 200 times.Clearly, Model A exhibits the maximum deformation.After repeatedly adjusting the central and bilateral bridge widths, the maximum mechanical stresses for the three models are 387.5 MPa, 373.6 MPa, and 379.1 MPa, respectively.The maximum mechanical stresses of the three models are comparable, meeting the design requirements.The maximum mechanical stress in Model A occurs on the bilateral bridges, whereas in Model B and Model C, it occurs on the central bridges.The material parameters for the rotor are listed in Table 2.The absence of central bridges in Model A results in increased bilateral bridge widths.Model B and Model C, with central bridges, experience reduced rotor deformation, consequently lowering the mechanical stress on the rotor.Consequently, this leads to a reduction in the combined central and bilateral bridge widths.The sum of the central and bilateral bridge widths at each rotor pole is 6.0 mm, 2.8 mm, and 2.5 mm, respectively.Therefore, among these three models, Model C exhibits the smallest leakage flux, while Model A has the highest leakage flux.

Comparison of No-Load Performance
The no-load performance mainly includes the air-gap flux density, back electromotive force (back-EMF), and cogging torque.Figure 3a illustrates the air-gap flux density for three models.Figure 3b depicts the Fourier decomposition of the air-gap flux density.It can be seen that the air-gap flux distribution of Model A is closer to the sinusoidal waveform.Model C, due to its minimal leakage flux, exhibits the highest fundamental amplitude (fifth).However, its harmonic amplitudes also increase.The fundamental amplitudes of the air-gap flux for the three models are 0.67 T, 0.91 T, and 0.93 T, respectively.A larger fundamental amplitude can increase torque density.The total harmonic distortion (THD) of the air-gap flux for the three models is 9.13%, 15.43%, and 18.94%, respectively.High THD can lead to significant torque ripple and losses.The material parameters for the rotor are listed in Table 2.The absence of central bridges in Model A results in increased bilateral bridge widths.Model B and Model C, with central bridges, experience reduced rotor deformation, consequently lowering the mechanical stress on the rotor.Consequently, this leads to a reduction in the combined central and bilateral bridge widths.The sum of the central and bilateral bridge widths at each rotor pole is 6.0 mm, 2.8 mm, and 2.5 mm, respectively.Therefore, among these three models, Model C exhibits the smallest leakage flux, while Model A has the highest leakage flux.

Comparison of No-Load Performance
The no-load performance mainly includes the air-gap flux density, back electromotive force (back-EMF), and cogging torque.Figure 3a illustrates the air-gap flux density for three models.Figure 3b depicts the Fourier decomposition of the air-gap flux density.It can be seen that the air-gap flux distribution of Model A is closer to the sinusoidal waveform.Model C, due to its minimal leakage flux, exhibits the highest fundamental amplitude (fifth).However, its harmonic amplitudes also increase.The fundamental amplitudes of the air-gap flux for the three models are 0.67 T, 0.91 T, and 0.93 T, respectively.A larger fundamental amplitude can increase torque density.The total harmonic distortion (THD) of the air-gap flux for the three models is 9.13%, 15.43%, and 18.94%, respectively.High THD can lead to significant torque ripple and losses.Figure 5 presents the line voltages of these three models.Due to the Y-type connection of the stator windings, these waveforms do not include the third harmonic and its multiples.The fundamental amplitudes of the line voltages for the three models are 234.5 V, 309.1 V, and 316.0 V, respectively.The phenomenon of cogging torque, arising from the interaction between the stator teeth and the rotor PMs, poses a significant challenge in motor design.Its presence frequently results in undesirable consequences, notably the elevation in torque ripple.Techniques such as skewed slots or skewed poles are frequently utilized to address stringent design criteria and alleviate cogging torque.
The cogging torque waveforms of the three models are shown in The phenomenon of cogging torque, arising from the interaction between the stator teeth and the rotor PMs, poses a significant challenge in motor design.Its presence frequently results in undesirable consequences, notably the elevation in torque ripple.Techniques such as skewed slots or skewed poles are frequently utilized to address stringent design criteria and alleviate cogging torque.
The cogging torque waveforms of the three models are shown in

Inductance and Salience
Salience is defined as the discrepancy between the d-axis inductance (Ld) and the qaxis inductance (Lq).A high level of salience is essential for achieving elevated torque output and extending the constant power region.Ld and Lq are defined separately.
where ψd and ψq represent the d-axis and q-axis magnetic flux, respectively.ψPM is defined as the magnetic flux generated by PMs.id and iq represent the d-axis and q-axis currents, respectively.
Figure 7a shows the variation of Ld with current.As the current increases, the motor gradually tends to saturate, leading to a decrease in Ld for all three models.Model A demonstrates the highest Ld, which can be attributed to its widest bilateral bridges.Conversely, Model C, characterized by the smallest total widths of central and bilateral bridges, exhibits the minimum Ld.At the rated current of 20 A, the Ld values for the three models are 4.09 mH, 3.53 mH, and 3.43 mH, respectively.

Comparison of Load Performance 4.1. Inductance and Salience
Salience is defined as the discrepancy between the d-axis inductance (L d ) and the q-axis inductance (L q ).A high level of salience is essential for achieving elevated torque output and extending the constant power region.L d and L q are defined separately.
where ψ d and ψ q represent the d-axis and q-axis magnetic flux, respectively.ψ PM is defined as the magnetic flux generated by PMs.i d and i q represent the d-axis and q-axis currents, respectively.
Figure 7a shows the variation of L d with current.As the current increases, the motor gradually tends to saturate, leading to a decrease in L d for all three models.Model A demonstrates the highest L d , which can be attributed to its widest bilateral bridges.Conversely, Model C, characterized by the smallest total widths of central and bilateral bridges, exhibits the minimum L d .At the rated current of 20 A, the L d values for the three models are 4.09 mH, 3.53 mH, and 3.43 mH, respectively.
Figure 7b illustrates the variation of L q with current.Similarly, as the current increases, the L q values decrease for all three models.Model A has the largest L q , while the L q values for Model B and Model C are approximately the same.At the rated current of 20 A, the L q values of the three models are 4.49 mH, 4.318 mH, and 4.319 mH, respectively.When the current is 40 A, the L q values of the three models tend to be identical.Figure 7c shows the variation of salience with current.Due to Model C having the minimum Ld, it results in Model C having the maximum salience.At the rated current of 20 A, the salience values of the three models are, respectively, 0.40 mH, 0.788 mH, and 0.889 mH.As the current increases due to the saturation of the motor, the salience of Model A first becomes less than zero, and then, nearing 40 A, the salience of Model B turns less than zero.The salience of Model C remains consistently greater than zero.The salience of less than zero results in the reluctance torque becoming negative, leading to a degradation in motor performance.Figure 7c shows the variation of salience with current.Due to Model C having the minimum L d , it results in Model C having the maximum salience.At the rated current of 20 A, the salience values of the three models are, respectively, 0.40 mH, 0.788 mH, and 0.889 mH.As the current increases due to the saturation of the motor, the salience of Model A first becomes less than zero, and then, nearing 40 A, the salience of Model B turns less than zero.The salience of Model C remains consistently greater than zero.The salience of less than zero results in the reluctance torque becoming negative, leading to a degradation in motor performance.Figure 7c shows the variation of salience with current.Due to Model C having the minimum Ld, it results in Model C having the maximum salience.At the rated current of 20 A, the salience values of the three models are, respectively, 0.40 mH, 0.788 mH, and 0.889 mH.As the current increases due to the saturation of the motor, the salience of Model A first becomes less than zero, and then, nearing 40 A, the salience of Model B turns less than zero.The salience of Model C remains consistently greater than zero.The salience of less than zero results in the reluctance torque becoming negative, leading to a degradation in motor performance.The waveform of torque at MTPA control when the current is 40 A is shown in Figure 8b.It shows that the Model C has the highest average torque and the torque ripples.Table 4 shows the average torque, torque component, and torque ripple of three IPM motors when the current is 40 A. To present the magnitude relationship between average torque and harmonic torque more clearly, Figure 9 shows the Decibel graph of torque at different currents.The 0th order represents the average torque.The waveform of torque at MTPA control when the current is 40 A is shown in Figure 8b.It shows that the Model C has the highest average torque and the torque ripples.Table 4 shows the average torque, torque component, and torque ripple of three IPM motors when the current is 40 A. To present the magnitude relationship between average torque and harmonic torque more clearly, Figure 9 shows the Decibel graph of torque at different currents.The 0th order represents the average torque.As is well known, the composition of electromagnetic torque for motors can be formulated with Equation (2).The first term represents the PM torque, and the second term represents the reluctance torque.

Torque and Power
where p is the number of pole pairs.Ia is the stator current.θ is the current angle lagging the q-axis.Therefore, keeping the stator current constant, increasing ψpm can increase the PM torque, and increasing salience can enhance the reluctance torque.Because the ψpm and salience of Model C are the highest, the torque of Model C is always the largest.As is well known, the composition of electromagnetic torque for motors can be formulated with Equation (2).The first term represents the PM torque, and the second term represents the reluctance torque.
where p is the number of pole pairs.I a is the stator current.θ is the current angle lagging the q-axis.Therefore, keeping the stator current constant, increasing ψ pm can increase the PM torque, and increasing salience can enhance the reluctance torque.Because the ψ pm and salience of Model C are the highest, the torque of Model C is always the largest.The torque-angle characteristics of the three motors are shown in Figure 10.It can be seen that when the current is 20 A, the current angles for maximum torque of the three models are 20, 20, and 21 degrees, respectively.When the current is 40 A, the current angles for maximum torque of the three models are 26, 27, and 28 degrees, respectively.Due to the maximum salience of Model C, the current angle θ of Model C remains the largest.
Figure 11 illustrates the proportion of PM torque and reluctance torque for three models under varying currents.At 20 A, the order of PM torque and reluctance torque for all three models is as follows: Model C > Model B > Model A. Additionally, the reluctance torque for all models is positive.At 40 A, due to saturation effects, both Model A and Model B exhibit negative salience, resulting in negative values for their respective reluctance torque.Since the salience of Model B approaches zero, its reluctance torque tends toward zero as well.Conversely, at this point, Model C demonstrates positive saliency; hence, its reluctance torque retains a positive value.
The torque-angle characteristics of the three motors are shown in Figure 10.It can be seen that when the current is 20 A, the current angles for maximum torque of the three models are 20, 20, and 21 degrees, respectively.When the current is 40 A, the current angles for maximum torque of the three models are 26, 27, and 28 degrees, respectively.Due to the maximum salience of Model C, the current angle θ of Model C remains the largest.Figure 12 illustrates the power characteristic curves of three models under the 40 A current limit, with maximum power values of 6.0 kW, 9.2 kW, and 9.7 kW, respectively.Model C, due to its lower leakage flux and maximum salience, exhibits the highest power output across the entire speed range.Figure 11 illustrates the proportion of PM torque and reluctance torque for three models under varying currents.At 20 A, the order of PM torque and reluctance torque for all three models is as follows: Model C > Model B > Model A. Additionally, the reluctance torque for all models is positive.At 40 A, due to saturation effects, both Model A and Model B exhibit negative salience, resulting in negative values for their respective reluctance torque.Since the salience of Model B approaches zero, its reluctance torque tends toward zero as well.Conversely, at this point, Model C demonstrates positive saliency; hence, its reluctance torque retains a positive value.Figure 12 illustrates the power characteristic curves of three models under the 40 A current limit, with maximum power values of 6.0 kW, 9.2 kW, and 9.7 kW, respectively.Model C, due to its lower leakage flux and maximum salience, exhibits the highest power output across the entire speed range.Figure 12 illustrates the power characteristic curves of three models under the 40 A current limit, with maximum power values of 6.0 kW, 9.2 kW, and 9.7 kW, respectively.Model C, due to its lower leakage flux and maximum salience, exhibits the highest power output across the entire speed range.Figure 11 illustrates the proportion of PM torque and reluctance torque for three models under varying currents.At 20 A, the order of PM torque and reluctance torque for all three models is as follows: Model C > Model B > Model A. Additionally, the reluctance torque for all models is positive.At 40 A, due to saturation effects, both Model A and Model B exhibit negative salience, resulting in negative values for their respective reluctance torque.Since the salience of Model B approaches zero, its reluctance torque tends toward zero as well.Conversely, at this point, Model C demonstrates positive saliency; hence, its reluctance torque retains a positive value.

Loss and Efficiency
Figure 13 illustrates the distribution of iron loss density for three models at the same output (20 Nm and 3000 rpm).At this moment, the specific parameters of the three motors are as shown in Table 5: It can be observed that under these conditions, Model A performs the worst.Although it has smaller stator iron loss, its higher stator current leads to the lowest efficiency.Model C, despite requiring the smallest stator current, incurs total losses approximately greater than those of Model B, resulting in Model B achieving the highest efficiency.The mechanical losses of the three models are determined by empirical formulas from Ref. [32], all approximately equal to 27.5 W. The reluctance torque of the three models remains highest in Model C; although the value is smaller, it still contributes to increased motor output power.
the worst.Although it has smaller stator iron loss, its higher stator current leads to the lowest efficiency.Model C, despite requiring the smallest stator current, incurs total losses approximately greater than those of Model B, resulting in Model B achieving the highest efficiency.The mechanical losses of the three models are determined by empirical formulas from Ref. [32], all approximately equal to 27.5 W. The reluctance torque of the three models remains highest in Model C; although the value is smaller, it still contributes to increased motor output power.Demagnetization analysis is an important aspect of motor design.Demagnetization of the PMs can affect the performance of the motor.The higher the temperature, the more susceptible the PMs are to demagnetization.Therefore, this article conducts a demagnetization analysis of three motors under extreme conditions (temperature of 180 °C, stator current of 40 A, current angle of 90 deg).
Figure 14 shows the magnetic flux density of the three motors under extreme conditions.The minimum magnetic flux densities for Model A, Model B, and Model C are 0.59 T, 0.52 T, and 0.55 T, respectively, all located near the central bridges.All three models do not undergo demagnetization.Model A has the maximum magnetic flux density and the lowest demagnetization risk due to the wide bilateral bridges, allowing a large amount of demagnetizing magnetic field to pass through the bilateral bridges.This results in a higher magnetic flux density within the PMs of Model A. Model B has a smaller minimum  Demagnetization analysis is an important aspect of motor design.Demagnetization of the PMs can affect the performance of the motor.The higher the temperature, the more susceptible the PMs are to demagnetization.Therefore, this article conducts a demagnetization analysis of three motors under extreme conditions (temperature of 180 • C, stator current of 40 A, current angle of 90 deg).
Figure 14 shows the magnetic flux density of the three motors under extreme conditions.The minimum magnetic flux densities for Model A, Model B, and Model C are 0.59 T, 0.52 T, and 0.55 T, respectively, all located near the central bridges.All three models do not undergo demagnetization.Model A has the maximum magnetic flux density and the lowest demagnetization risk due to the wide bilateral bridges, allowing a large amount of demagnetizing magnetic field to pass through the bilateral bridges.This results in a higher magnetic flux density within the PMs of Model A. Model B has a smaller minimum magnetic flux density compared to Model C, as the narrow central bridges of Model B cause some demagnetizing magnetic flux to enter the PMs.

Analysis of the Impact of PM Dimensions on Demagnetization and Torque
This section will use PMs with the same volume to obtain the maximum torque of three models by optimizing the length and thickness of the PMs.Due to the smaller size change, the impact on rotor strength is being neglected.The minimum magnetic flux density of the PMs changes with the increase of the thickness of the PMs, as shown in Figure 15a.The motor torque varies with the increase in the thickness of the PMs, as shown in Figure 15b.As the thickness of the PMs increases, the motor's ability to resist demagnetization improves.However, due to the increased thickness of the PMs, the length of the PMs decreases, resulting in a decrease in motor torque.This section will use PMs with the same volume to obtain the maximum torque of three models by optimizing the length and thickness of the PMs.Due to the smaller size change, the impact on rotor strength is being neglected.The minimum magnetic flux density of the PMs changes with the increase of the thickness of the PMs, as shown in Figure 15a.The motor torque varies with the increase in the thickness of the PMs, as shown in Figure 15b.As the thickness of the PMs increases, the motor's ability to resist demagnetization improves.However, due to the increased thickness of the PMs, the length of the PMs decreases, resulting in a decrease in motor torque.To ensure uniform demagnetization conditions across all three models, torque comparisons are made with the minimum magnetic flux density of the PMs set at 0.5 T. At this point, the thicknesses of the PMs for Model A, Model B, and Model C are 2.6 mm, 2.9 mm, and 2.7 mm, respectively.The torque values for these models stand at 19.7 Nm, 24.0 Nm, and 25.6 Nm, respectively.Remarkably, after optimization, all three models exhibit increased torque performance, with Model C maintaining the highest torque output among them.

Thermal Analysis
Under the same output conditions (20 Nm, 3000 rpm), thermal analysis is conducted on three motors utilizing natural cooling.As shown in Figure 16, the axial temperature distributions of the three motors are illustrated.Since this operating condition is the most commonly encountered for the motors, the temperatures of all three motors meet the To ensure uniform demagnetization conditions across all three models, torque comparisons are made with the minimum magnetic flux density of the PMs set at 0.5 T. At this point, the thicknesses of the PMs for Model A, Model B, and Model C are 2.6 mm, 2.9 mm, and 2.7 mm, respectively.The torque values for these models stand at 19.7 Nm, 24.0 Nm, and 25.6 Nm, respectively.Remarkably, after optimization, all three models exhibit increased torque performance, with Model C maintaining the highest torque output among them.

Thermal Analysis
Under the same output conditions (20 Nm, 3000 rpm), thermal analysis is conducted on three motors utilizing natural cooling.As shown in Figure 16, the axial temperature distributions of the three motors are illustrated.Since this operating condition is the most commonly encountered for the motors, the temperatures of all three motors meet the requirements, allowing for long-term stable operation.At this juncture, Model A exhibits a higher overall temperature due to its higher stator current compared to Model B and Model C. Model B maintains the lowest temperature among the three models, owing to its lowest losses.All three motors meet the requirements of maximum rotor temperature below 150 • C and maximum winding temperature below 130 • C, providing a significant safety margin for the motors under extreme operating conditions (short-term overspeed and overload).a higher overall temperature due to its higher stator current compared to Model B and Model C. Model B maintains the lowest temperature among the three models, owing to its lowest losses.All three motors meet the requirements of maximum rotor temperature below 150 °C and maximum winding temperature below 130 °C, providing a significant safety margin for the motors under extreme operating conditions (short-term overspeed and overload).

Conclusions
This article designs three rotors with different widths of central and bilateral bridges and conducts a detailed comparison, reaching the following conclusions:

Figure 1
Figure 1 illustrates three IPM motors with different rotor designs, denoted as Model A, Model B, and Model C. Model A distinguishes itself by the absence of central bridges, whereas both Model B and Model C integrate central and bilateral bridges.However, in comparison to Model B, Model C features narrower bilateral bridges and wider central bridges.There are various combinations of central and bilateral bridge widths that can meet the mechanical strength requirements of the rotor.However, to highlight the differences in these designs, three models with significantly divergent central and bilateral bridge widths have been selected.The specific widths of the central and bilateral bridges are determined by the mechanical stress analysis in Section 2.2, and the exact values are marked in Figure1.In terms of other parameters, such as PM dimensions and stator specifications, these three motors remain consistent.The cost of three motors is roughly the same.To decrease the axial length of the motors, the stators of all motors have adopted the same concentrated winding scheme.The main motor parameters are detailed in Table1.The rated current is 20 A, and the rated speed is 3000 rpm.

Figure 1
Figure 1 illustrates three IPM motors with different rotor designs, denoted as Model A, Model B, and Model C. Model A distinguishes itself by the absence of central bridges, whereas both Model B and Model C integrate central and bilateral bridges.However, in comparison to Model B, Model C features narrower bilateral bridges and wider central bridges.There are various combinations of central and bilateral bridge widths that can meet the mechanical strength requirements of the rotor.However, to highlight the differences in these designs, three models with significantly divergent central and bilateral bridge widths have been selected.The specific widths of the central and bilateral bridges are determined by the mechanical stress analysis in Section 2.2, and the exact values are marked in Figure1.In terms of other parameters, such as PM dimensions and stator specifications, these three motors remain consistent.The cost of three motors is roughly the same.To decrease the axial length of the motors, the stators of all motors have adopted the same concentrated winding scheme.The main motor parameters are detailed in Table1.The rated current is 20 A, and the rated speed is 3000 rpm.

Figure 2 .
Figure 2. Distribution of rotor mechanical stress.(a) Model A. (b) Model B. (c) Model C.

Figure 2 .
Figure 2. Distribution of rotor mechanical stress.(a) Model A. (b) Model B. (c) Model C.

Figure 4 Figure 4 .
Figure 4 illustrates the back-EMF and harmonic decomposition of three models at 3000 rpm.The waveform of Model A closely resembles a sine wave, and Model C exhibits the highest fundamental and harmonic amplitudes.The back-EMF results of the three models are consistent with the air-gap flux density waveforms.The fundamental amplitudes of the back-EMF for the three models are 135.6V, 178.4 V, and 183.7 V, respectively.

Figure 5 Figure 5 .
Figure5presents the line voltages of these three models.Due to the Y-type connection of the stator windings, these waveforms do not include the third harmonic and its multiples.The fundamental amplitudes of the line voltages for the three models are 234.5 V, 309.1 V, and 316.0 V, respectively.

Figure 4 Figure 3 .
Figure 4 illustrates the back-EMF and harmonic decomposition of three models at 3000 rpm.The waveform of Model A closely resembles a sine wave, and Model C exhibits the highest fundamental and harmonic amplitudes.The back-EMF results of the three models are consistent with the air-gap flux density waveforms.The fundamental amplitudes of the back-EMF for the three models are 135.6V, 178.4 V, and 183.7 V, respectively.

Figure 4 Figure 4 .
Figure 4 illustrates the back-EMF and harmonic decomposition of three models at 3000 rpm.The waveform of Model A closely resembles a sine wave, and Model C exhibits the highest fundamental and harmonic amplitudes.The back-EMF results of the three models are consistent with the air-gap flux density waveforms.The fundamental amplitudes of the back-EMF for the three models are 135.6V, 178.4 V, and 183.7 V, respectively.

Figure 5 Figure 5 .
Figure5presents the line voltages of these three models.Due to the Y-type connection of the stator windings, these waveforms do not include the third harmonic and its multiples.The fundamental amplitudes of the line voltages for the three models are 234.5 V, 309.1 V, and 316.0 V, respectively.

Figure 4
Figure 4 illustrates the back-EMF and harmonic decomposition of three models at 3000 rpm.The waveform of Model A closely resembles a sine wave, and Model C exhibits the highest fundamental and harmonic amplitudes.The back-EMF results of the three models are consistent with the air-gap flux density waveforms.The fundamental amplitudes of the back-EMF for the three models are 135.6V, 178.4 V, and 183.7 V, respectively.

Figure 5 Figure 5 .
Figure5presents the line voltages of these three models.Due to the Y-type connection of the stator windings, these waveforms do not include the third harmonic and its multiples.The fundamental amplitudes of the line voltages for the three models are 234.5 V, 309.1 V, and 316.0 V, respectively.

Figure 6 .
Due to the increased air-gap flux density, the cogging torque of Model B and Model C significantly increases compared to Model A. The peak-to-peak values of the cogging torque for the three models are 204.2mNm, 386.2 mNm, and 490.6 mNm, respectively.Machines 2024, 12, x FOR PEER REVIEW 6 of 14

Figure 6 .
Due to the increased air-gap flux density, the cogging torque of Model B and Model C significantly increases compared to Model A. The peak-to-peak values of the cogging torque for the three models are 204.2mNm, 386.2 mNm, and 490.6 mNm, respectively.

Figure 6 .
Figure 6.Cogging torque of the three models.

Figure 6 .
Figure 6.Cogging torque of the three models.

Figure 7b illustrates the
Figure 7b illustrates the variation of Lq with current.Similarly, as the current increases, the Lq values decrease for all three models.Model A has the largest Lq, while the Lq values for Model B and Model C are approximately the same.At the rated current of 20 A, the Lq values of the three models are 4.49 mH, 4.318 mH, and 4.319 mH, respectively.When the current is 40 A, the Lq values of the three models tend to be identical.Figure7cshows the variation of salience with current.Due to Model C having the minimum Ld, it results in Model C having the maximum salience.At the rated current of 20 A, the salience values of the three models are, respectively, 0.40 mH, 0.788 mH, and 0.889 mH.As the current increases due to the saturation of the motor, the salience of Model A first becomes less than zero, and then, nearing 40 A, the salience of Model B turns less than zero.The salience of Model C remains consistently greater than zero.The salience of less than zero results in the reluctance torque becoming negative, leading to a degradation in motor performance.

FigureFigure 8 .
Figure 8a depicts the waveform of torque under maximum torque per ampere (MTPA) control when the current is 20 A. Model C exhibits the highest average torque, attributed to its maximum air-gap flux density and salience.Due to the highest THD of the air-gap flux density in Model C, its torque ripple is maximal.Table3shows the average torque, torque component, and torque ripple of three IPM motors when the current is 20 A, where Tavg represents average torque, ΔTp-p represents peak-to-peak torque, Tripple represents torque ripple.

Figure 7 .
Figure 7.The curves of inductance and salience variation with current.(a) L d .(b) L q .(c) salience.

Figure
Figure 8a depicts the waveform of torque under maximum torque per ampere (MTPA) control when the current is 20 A. Model C exhibits the highest average torque, attributed to its maximum air-gap flux density and salience.Due to the highest THD of the airgap flux density in Model C, its torque ripple is maximal.Table3shows the average torque, torque component, and torque ripple of three IPM motors when the current is 20 A, where T avg represents average torque, ∆T p-p represents peak-to-peak torque, T ripple represents torque ripple.

Figure 7b illustrates the
Figure 7b illustrates the variation of Lq with current.Similarly, as the current increases, the Lq values decrease for all three models.Model A has the largest Lq, while the Lq values for Model B and Model C are approximately the same.At the rated current of 20 A, the Lq values of the three models are 4.49 mH, 4.318 mH, and 4.319 mH, respectively.When the current is 40 A, the Lq values of the three models tend to be identical.Figure7cshows the variation of salience with current.Due to Model C having the minimum Ld, it results in Model C having the maximum salience.At the rated current of 20 A, the salience values of the three models are, respectively, 0.40 mH, 0.788 mH, and 0.889 mH.As the current increases due to the saturation of the motor, the salience of Model A first becomes less than zero, and then, nearing 40 A, the salience of Model B turns less than zero.The salience of Model C remains consistently greater than zero.The salience of less than zero results in the reluctance torque becoming negative, leading to a degradation in motor performance.

FigureFigure 8 .
Figure 8a depicts the waveform of torque under maximum torque per ampere (MTPA) control when the current is 20 A. Model C exhibits the highest average torque, attributed to its maximum air-gap flux density and salience.Due to the highest THD of the air-gap flux density in Model C, its torque ripple is maximal.Table3shows the average torque, torque component, and torque ripple of three IPM motors when the current is 20 A, where Tavg represents average torque, ΔTp-p represents peak-to-peak torque, Tripple represents torque ripple.

Figure 11
Figure 11  illustrates the proportion of PM torque and reluctance torque for three models under varying currents.At 20 A, the order of PM torque and reluctance torque for all three models is as follows: Model C > Model B > Model A. Additionally, the reluctance torque for all models is positive.At 40 A, due to saturation effects, both Model A and Model B exhibit negative salience, resulting in negative values for their respective reluctance torque.Since the salience of Model B approaches zero, its reluctance torque tends toward zero as well.Conversely, at this point, Model C demonstrates positive saliency; hence, its reluctance torque retains a positive value.

Figure 12 .
Figure 12.Power curves of three models.

Figure 12 .
Figure 12.Power curves of three models.

Figure 12
Figure12illustrates the power characteristic curves of three models under the 40 A current limit, with maximum power values of 6.0 kW, 9.2 kW, and 9.7 kW, respectively.Model C, due to its lower leakage flux and maximum salience, exhibits the highest power output across the entire speed range.

Figure 12 .
Figure 12.Power curves of three models.Figure 12. Power curves of three models.

Figure 12 .
Figure 12.Power curves of three models.Figure 12. Power curves of three models.

Figure 13 .
Figure 13.Distribution of iron loss density.(a) Model A. (b) Model B. (c) Model C.

Figure 13 .
Figure 13.Distribution of iron loss density.(a) Model A. (b) Model B. (c) Model C.

Machines 2024 ,Figure 14 .
Figure 14.Magnetic flux distribution of PMs under extreme conditions.(a) Model A. (b) Model B. (c) Model C. 4.4.2.Analysis of the Impact of PM Dimensions on Demagnetization and Torque This section will use PMs with the same volume to obtain the maximum torque of three models by optimizing the length and thickness of the PMs.Due to the smaller size change, the impact on rotor strength is being neglected.The minimum magnetic flux density of the PMs changes with the increase of the thickness of the PMs, as shown in Figure 15a.

Figure 14 .
Figure 14.Magnetic flux distribution of PMs under extreme conditions.(a) Model A. (b) Model B. (c) Model C.

Figure 14 .
Figure 14.Magnetic flux distribution of PMs under extreme conditions.(a) Model A. (b) Model B. (c) Model C.

Figure 15 .
Figure 15.Demagnetization and torque characteristics (a) Variation of minimum magnetic flux density with PMs thickness.(b) Variation of torque with PMs thickness.

Figure 15 .
Figure 15.Demagnetization and torque characteristics (a) Variation of minimum magnetic flux density with PMs thickness.(b) Variation of torque with PMs thickness.

( 1 )
Due to the absence of a central bridge, Model A requires a very thick bilateral bridge to ensure rotor operation, resulting in a significant increase in magnetic flux leakage and causing Model A to not meet torque requirements.(2) The total bridge width of Model B and Model C is significantly smaller than that of Model A, resulting in a reduction in inductance for both Model B and Model C.

Table 1 .
Main parameters of motor.

Table 1 .
Main parameters of motor.

Table 2 .
Material parameters of rotor.

Table 2 .
Material parameters of rotor.

Table 3 .
Torque components of current = 20 A.

Table 4 .
Torque components of current = 40 A.

Table 4 .
Torque components of current = 40 A.

Table 5 .
The main calculated parameters of the motor at the same output.

Table 5 .
The main calculated parameters of the motor at the same output.