*3.5. Torque*

The Maxwell stress tensor, Equation (20), allows for torque computation given the radial flux density and the conductor's linear current density. Torque due to individual components of flux density can be separated or combined for the net effects.

$$T = \int\_{\theta=0}^{2\pi} r^2 l \cdot B\_{\text{hot}}(\theta) K(\theta) d\theta \tag{20}$$

### *3.6. Rotation and Convolution*

In this section, a computationally efficient method to evaluate the phenomenon of rotation and the ensuing interaction of the time and space domain harmonics is reviewed. To simplify analysis of the magnetic fields, the rotor is left frozen while the stator rotates in the counter direction. The Fourier spatial coefficients of the permeance and winding functions are determined at the initial position of the rotor through use of the Fast Fourier Transform (FFT). For each rotor position, *φ*(*i*), a complex rotation, Equation (21), is used to form a rotation vector, Equation (22). The complex rotation vector is used to transform the FFT coefficients of permeance and winding functions at each position.

$$\epsilon = \left(\cos(\phi) + i\sin(\phi)\right) \tag{21}$$

$$
\Phi = \begin{bmatrix}
\epsilon^0 & \epsilon^1 & \epsilon^2 & \dots & \epsilon^{\frac{\mu}{2}-1} & \epsilon^0 & \overline{\epsilon^1} & \overline{\epsilon^2} & \dots & \overline{\epsilon^{\frac{\mu}{2}-1}}
\end{bmatrix}^\top \tag{22}
$$

In place of the convolution of the Fourier coefficients to determine flux density, the permeance and winding function is reconstructed with the inverse fast Fourier transform.

### *3.7. Comparative Analysis to Recent Analytical Methods*

This section will compare the analytical methods developed in this paper to three recent methods [45,49,50].

The analytical MEC model is used to design a reduced magne<sup>t</sup> cost single V consequent pole (CP) machine with the same average torque as single V IPM in [45]. The models are developed based on zones and regions, allowing for an assumption of the open circuit flux density distribution. Two flux sources (magnets) and six reluctance network paths are used to create the open circuit single V IPM model. Open circuit flux density is assumed to take a trapezoidal waveform where the flux density is determined spatially from the fluxes in the MEC reluctance network. Similarly, the single V CP IPM network consists of one magne<sup>t</sup> flux source and five reluctance paths. In both cases, the rotors reluctance path reaction to the armature loading is not computed. The open circuit flux densities are used to quickly determine an equivalent single V CP IPM fundamental to that of the single V IPM fundamental. In order to guide design, the method is used to find an equivalent consequent pole open circuit flux density fundamental to the traditional IPM flux density. Finite elements are relied on to complete the study of the torque performance.

Multi-barrier synchronous reluctance and Permanent Magnet Assisted Synchronous Reluctance Machines (PMSynRM) are modeled using conformal mapping and magnetic equivalent circuits in [50]. Hyperbolic shaped flux barriers are assumed. Conformal transformations are employed to the rotors flux barrier geometry to compute the magnetic reluctance. The reluctance values calculated from conformal mapping are subsequently used in the reluctance network values of the MEC model. The MEC model considers MMF sources of both the armature and magnet. Loaded and open circuit flux densities, average torque, and torque ripple are compared to finite elements with reasonable accuracy.

The slotless U-type IPM machine open circuit flux density is analytically modeled with a subdomain method solving Laplace's and Poisson's equations in [49]. Analytical equations are derived and presented for each subdomain. Results are validated against finite elements. The model is divided into four regions, which consist of the airgap and magnets. The governing system of partial differential equations is developed, along with simplifications, interface and boundary conditions. A separation of variables is used to develop the general solutions of the PDEs, and they are written as a Fourier series. The system of equations is solved and compared to FE. Strong agreemen<sup>t</sup> of the radial and tangential flux density is shown between the FE and the subdomain methods.

The analytical models discussed were developed for multiple purposes. The MEC method is used in [45] to quickly estimate the open circuit flux density fundamental of the single-V IPM and single-V CP IPM machines. The armature reaction of the reluctance features is not considered by the model, and finite elements are used to finish the designs. Conformal mapping is used in [50] to determine the reluctances of a multi-barrier PMSynRM and further evaluated using a MEC network. Both open circuit and loaded conditions are evaluated for airgap flux densities and torque performance and compared to finite elements. The analysis is not extended to the design. The subdomain methodology is employed in [49] for the analysis of the U-shape IPM machine open circuit conditions. Both tangential and radial flux densities are shown to match finite element results. The model requires further extension to consider the torque performance due to a loaded armature. The single-V sculpted rotor IPM winding function model developed in this paper considers both open circuit and loaded conditions. The model is constructed such that computational efficiency is possible without the sacrifice of spatially dependent harmonic content to drive the design. The choice of which analytical model to develop is dependent upon the application, computational resources, and intended purpose. In this case, the new winding factor IPM model was developed.

#### **4. Application of Analytical Model to Example Machine**

The method developed in Section 3 is validated with a well-known industrialized IPM machine. Details of the 2004 Toyota Prius traction motor are included in Table 2, and the geometry is modeled analytically within Matlab and finite elements within Ansys Maxwell.


**Table 2.** Example motor parameters.

Focusing on the effects of rotor, the stator geometry has been idealized with no slots. Both sinusoidally distributed stator windings and the production configuration of discretely placed windings are modeled. Only the stator winding harmonics interactions with rotor geometry harmonics are considered. With the assumption of infinite permeability, the bridge features are omitted. Airgap fringing in the region of the magne<sup>t</sup> barrier is not considered.

Design parameters are studied within this section using the analytical winding function model previously validated. Rotor sculpt features are included along with their additional MMF term developed in Section 3.

### *4.1. Sculpting Geometry*

The sculpted rotor IPM machine geometry design space to be explored is shown on a single pole of the example machine in Figure 6. Rotor primary and secondary reluctance paths are shown in green with no bridge features. The stator, shown in gray, continues to have omitted its slot features, and distributed windings, orange, are placed within the airgap.

**Figure 6.** Rotor sculpt features.

The primary design parameters are centered on the rotor effects, which include the ratio of primary and secondary reluctance path and sculpt features. The magne<sup>t</sup> pole arc width is varied. Up to two sets of symmetrically placed sculpt features are placed on the second reluctance path. The symmetrical feature span locations *τ*1 and *τ*2 define the symmetrical location of the feature in terms of its percentage of the magne<sup>t</sup> pole arc span. Single asymmetrical features are described with a similar parameter but with only one sculpt feature on the pole. In this single asymmetrical case, the feature location, *τ*, is set to

be positive for right hand side placement and negative for the left hand side placement. The depths *D*1 and *D*2 are measured from the outer surface of the rotor to the root of the sculpt feature. The widths *W*1 and *W*2 are measured in terms of a single feature percentage of the pole span.

A Fourier analysis of design feature effects on torque harmonic analysis is presented. The phase of the torque harmonics is set with respect to the negative zero crossing of phase A back EMF, i.e., the north pole of the machine. Torque components are separated in terms of total, primary reluctance, secondary reluctance and magne<sup>t</sup> torque with both the torque amplitude and its corresponding phase.

### *4.2. Model Implementation*

Implemented in Matlab, the analytical model in Section 3 has been utilized to explore the design space. Rotor permeances of primary reluctance, secondary reluctance, and magne<sup>t</sup> and the permanent magne<sup>t</sup> winding function are modeled spatially as the rectangular waves. Stator phase winding functions and linear turn densities are spatially modeled, and the spatial harmonic coefficients are determined with the FFT. For each rotor position, the complex rotation vector is created and applied to the stator winding and linear turn functions. With the current applied, the modified secondary reluctance path is determined from Equation (13). Radial flux density is now determined for the primary reluctance, second reluctance, and magne<sup>t</sup> paths. A broad range of phase current, control angle, and rotor sculpting features are studied. Maxwell's stress tensor in conjunction with Equation (19) provides the ability to separate components of torque.

The analytical model is implemented utilizing both the FFT and the inverse FFT algorithms to efficiently move between frequency and spatial domains. Rotation is best performed within the frequency domain, Equation (22), and convolution is performed within the spatial domain. This provides the most efficient use of resources by a factor of five times. Results are computed over an entire electrical cycle with a sufficient number of points to provide a smooth torque waveform. The analytical model is executed within Matlab in 6.7 s, and the finite element results are executed in 20 min. Ansys Maxwell was also used to perform the finite element analysis.

### *4.3. Model Validation: Radial Flux Density*

This section compares the radial flux density results of the analytical model and finite elements while varying: (1) winding type, (2) current, (3) control angle, and (4) rotor sculpt features. Figures 7–10 plot the flux density along the rotors spatial coordinate, *θ*, over a single pole pair. Both sinusoidal and distributed windings are compared. The *q*-axis, which is aligned to the rotor minimum reluctance, occurs at *θelec* = 90◦ and *θelec* = 270◦. The *d*-axis is aligned, which is aligned to the smooth rotors permanent magne<sup>t</sup> maximum flux linkage, occurs at *θelec* = 0◦, *θelec* = 180◦, and *θelec* = 360◦. In all results, the finite element and the analytical model result in comparable flux densities.

Flux densities shown for sinusoidal windings, Figure 7, illustrate the changing airgap flux density harmonics with current and control angle. When the phase current is set to zero, only the permanent magne<sup>t</sup> field is present. As current and current angle increases, the flux density becomes more jagged, with the case of a fully negative *d*-axis current displaying the most harmonic content. It is clear that as the negative *d*-axis current becomes dominant, so do the reluctance path harmonics. Harmonic effects of the discretely distributed windings are shown in Figure 8. As current increases, so do the airgap reluctance harmonics. In all cases, the analytical model and finite element results agree with reasonable accuracy.

**Figure 7.** Radial flux densities with sinusoidally distributed windings (*N* = 200) at various currents and control angles.

**Figure 8.** Radial flux densities with distributed windings (2 SPP) at various currents and control angles.

The effects of rotor sculpt features on the second reluctance path are shown in Figures 9 and 10. A symmetrical pair of sculpt features are shown with distributed windings in Figure 9. The flux densities of the smooth rotor and sculpted rotor are plotted. Reduced flux density in the region of the sculpt features is observed, and sculpt features are located approximately at *θelec* = 30◦, 150◦, 210◦, 270◦. This flux density from the sculpt features is conserved and redistributed across the regions of the second reluctance path.

**Figure 9.** Radial flux densities with single symmetrical sculpt feature located at *τ*1 = 50%, *W*1 = 10%, and *D*1 = 1.2 mm.

The reluctance flux density single sculpt feature is plotted in Figure 10. Similar to the symmetrical sculpt features, the flux density is reduced in the region of the sculpt feature. In all cases, the analytical model and finite element results agree with reasonable accuracy.

**Figure 10.** Radial flux densities (reluctance only) with single asymmetrical sculpt feature located at *τ* = 50%, *W* = 20%, and *D* = 1.2 mm.

### *4.4. Model Validation: Torque Ripple*

Torque ripple of the smooth rotor IPM, Table 2, is compared between finite element and the analytical model in Figure 11. Good agreemen<sup>t</sup> between the finite element and analytical models is observed. The torque ripple effects of two symmetrical rotor sculpt features are demonstrated in Figure 12, directly calculated by the analytical model, whereas the finite element model requires two runs, once with and once without sculpt features, to determine the sculpt feature effects.

Good correlation between the model and finite elements is demonstrated and shown.

**Figure 11.** Smooth rotor IPM torque ripple.

**Figure 12.** Sculpt feature torque ripple.

### *4.5. Torque Ripple Components*

In this section, the torque ripple results of the analytical model are studied while varying current and control angle. Figure 13 plots the torque ripple for a complete electrical cycle of the example machine. The torque components for the first reluctance path, second reluctance path, total reluctance torque, magne<sup>t</sup> torque, and total machine torque are plotted. Magnet torque and its harmonics are dominant at lower currents, but the reluctance paths cannot be ignored. As current is increased, the reluctance torque increases relative to the magne<sup>t</sup> torque. The stronger field weakening currents cause the contribution of the reluctance features torque to increase. The dominant torque harmonic orders are the 6th and 12th electrical orders. In the design, both the torque harmonic amplitudes and phases of each of the components need to be considered as the sculpt feature design will provide the counter torque at the counter phase.

**Figure 13.** Analytical model torque ripple components at various control angles.

### **5. Investigation of Design Features**

In this section, the effects of design features are demonstrated to influence both torque harmonic amplitudes and phases. Carefully applied, these effects are used to design counter torque harmonics. Second reluctance path pole arc, sculpt feature type (symmetrical/asymmetrical), sculpt feature location, sculpt feature depth, and sculpt feature width can all be used to design an appropriate counter torque to reduce the machine's torque harmonics. While mildly affecting average torque, the second reluctance path pole arc, *<sup>τ</sup>p*, strongly affects the phase of the 12th electrical order torque harmonic. A single pair of symmetrical sculpt features placed upon the second reluctance path pole arc reduce the average torque. Feature position provides 12th electrical order torque harmonic phasing, and the feature width and depth directly affect the torque harmonics amplitude. The single asymmetrical feature is shown to increase average torque when placed on a specific side of the second reluctance path pole arc. The asymmetrical feature placement can also be used to modify the phase of both 6th and 12th electrical order torque harmonics. Finally, feature phasor summation is shown to be effective in combining the effects of multiple design features, further providing the ability to design both the

amplitude and phase of these minimizing torque harmonics. These relationships provide the necessary intuition to reduce computationally intensive design steps.

### *5.1. Magnet Pole Arc*

Magnet pole arc span, *τm*, effects upon the torque harmonics, without rotor sculpt features, are explored. Figure 14 shows the average, 6th, and 12th harmonics of torque as a function of *τm*. For this case, magne<sup>t</sup> torque is dominant. Although not always the case, it is just as important to follow the trends of individual torque components. Total, first reluctance, and magne<sup>t</sup> average machine torque are reduced as the pole arc is increased, and only the second reluctance torque increases the average torque. The 12th order torque harmonic is dominant, with primary contributions from the magne<sup>t</sup> and the second reluctance path, whereas the 6th order torque harmonic is mostly contributed to by the primary reluctance and magnet. Rotor geometry has a strong influence on the phase of the 12th order torque harmonics, whereas the 6th harmonic is less affected by rotor geometry.

**Figure 14.** Effects of magne<sup>t</sup> pole arc *τp* loaded at *Iss* = 200 and *β* = 135◦.

### *5.2. Single Pair Symmetrical Rotor Sculpt Feature*

A single symmetrical rotor sculpt feature torque is studied in Figures 15–17. Only the effects of the sculpt feature torque are plotted. In Figures 15 and 16, a single sculpt feature position is varied, with fixed width, *W*1, and fixed depth, *D*1, along the magne<sup>t</sup> pole arc. Rotor sculpt features have a negative effect upon average torque, as the phase is 180◦ out of phase with the smooth rotor average torque. Figure 15 compares the analytical model to finite elements and shows precise agreemen<sup>t</sup> with the phase and matching trends for torque amplitude. Using a wider feature width, *D*1, Figure 16 translates amplitude and phase plots to a phasor representation. The 12th harmonic is clearly the dominant torque in both amplitudes and choice of phase.

**Figure 15.** Symmetrical sculpt feature effects compared to finite elements: *D*1 = 1.2 mm, *W*1 = 5%.

**Figure 16.** Symmetrical sculpt feature effects with phasor diagram: *D*1 = 1.2 mm, *W*1 = 9%.

Figure 17 includes the sculpt features width and depth effects. Increasing sculpt feature width, *W*1, and/or the sculpt features depth, *D*1, increases the amplitude of the sculpt features torque harmonic. The primary influence on the torque harmonic amplitude is the width of the sculpt feature. Sculpt feature width and depth have no effect on the torque harmonics phase.

### *5.3. Single Asymmetrical Rotor Sculpt Feature*

A single sculpt feature resulting in asymmetrical placement upon the rotor surface is studied in this section. Similar parameters *D*1, *W*1, and *τ*1 are used to describe the features width, depth, and location. In the asymmetrical case, the location, *τ*1, is described with the same location parameter, where in this case, a positive *τ*1 results on the right side of Figure 6 and a negative *τ*1 results in sculpt feature placement on the left hand side. Figure 18 compares the model to finite element and shows precise agreemen<sup>t</sup> with phase and matching trends for torque amplitude. In the asymmetric sculpt feature case, a torque improvement is possible due to the aligned axis effect from *τ*1 > 0. The placement of the sculpt feature allows for placement of the torque harmonic phase angle across a broad range of phases. Negative values of *τ*1 result in the largest amplitudes of the 12th electrical torque harmonic.

**Figure 17.** Sculpt feature effects *N* = 1.

**Figure 18.** Single asymmetrical sculpt feature effects with phasor diagram: *D*1 = 1.2 mm, *W*1 = 5%.

### *5.4. Two Symmetrical Rotor Sculpt Features*

More than one symmetrical rotor sculpt feature can be used. In this section, it is shown that the components of a first symmetrical feature can be combined with that of the second symmetrical rotor sculpt feature. The MMF-permeance model is validated by comparing to finite element results in Figure 12.

To illustrate this concept, the parameters of the two sculpt feature sets of Figure 6 are shown in Table 3. Through vector summation, the two vectors were used to create a 12th order counter torque with a phase of <sup>−</sup>116◦.

**Table 3.** Two sculpt feature parameters.


Figure 19 illustrates the first (red) and second 12th order electrical torque (green) phasors. The two phasors combine to create the effective total phasor (blue). This phasor summation is plotted along with the torque complex mapping of the previous single feature design sweep. These symmetrical rotor sculpt features are designed to mitigate the 12th order electrical torque harmonics to near zero. A single feature or multiple features can be designed to minimize the torque ripple. The sculpt features are not without consequence, as the average torque is negatively affected.

**Sculpt Feature Torque Phasor (12th Order Electrical), Iss=200 = 135<sup>o</sup>**

**Figure 19.** Two sculpt features of 12th order electrical torque phasor plot.
