**1. Introduction**

The IPM motor is increasingly being utilized throughout industry as a primary source of propulsion due to its good efficiency, torque and power density. Examples include the development of battery electric vehicle traction motors [1,2], plugin hybrid electric vehicles [3], and hybrid electric vehicles [4]. Ideally, the traction machine provides an average torque produced from a sinusoidal distribution of the airgap flux density. In reality, embedding the magne<sup>t</sup> within the salient structure of the rotor lamination and distributing windings in discrete locations result in airgap flux density harmonics. These harmonics result in increased torque ripple, radial forces, losses, and other unwanted phenomena.

In this paper, an approach to minimize torque ripple with rotor features is presented, based upon analytically modeling the machine features. The analytical modeling approach enables efficient use of computational resources, without the sacrifice of harmonic content, prior to the use of more expensive finite element methods. The calculation of the IPM machines' spatially-dependent torque harmonics is performed through the extension of the winding function method. New to the winding function framework is a method to model the equipotential nature of the rotor's salient features and rotor surface modifications. The non-homogeneous airgap of rotor surface modifications is included in the model through an additional MMF term. Unique to this analytical method, both the constituents and aggregates of the torque harmonics are found. A detailed investigation into the rotor geometry design space to minimize torque harmonics while managing average torque design trade-off is presented.

Inherent to the design of IPM machines, torque ripple is a persistent problem. Design choices to increase torque density or decrease manufacturing cost are often at odds with minimizing the torque ripple [5]. Rotor features, including surface modifications or sculpt

**Citation:** Hayslett, S.; Strangas, E. Analytical Design of Sculpted Rotor Interior Permanent Magnet Machines. *Energies* **2021**, *14*, 5109. https:// doi.org/10.3390/en14165109

Academic Editor: Athanasios Karlis

Received: 30 June 2021 Accepted: 17 August 2021 Published: 19 August 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

features, are utilized to minimize the torque harmonics. The ability to reduce ripple by design must consider the speed, current, and control angle ranges of the IPM machine.

Analytical expressions for the airgap and torque harmonics are developed for the IPM in [6,7]. The synchronous reluctance of torque harmonics presented in [8] is extended to the IPM machine in [7]. The expressions are useful in setting the stator slot and rotor barrier counts but do not model the machine.

Analytical models better relate the physical geometry of the machine to its airgap and flux density harmonics. Directly solving the Laplacian–Poisson is difficult [9,10]. Subdomain models break the model into pieces in which the Laplacian–Poisson can be more readily solved [11,12]. Magnetically Equivalent Circuits (MEC) divide the geometry into smaller manageable pieces [13]. Methods depending on winding functions allow for the geometry and harmonics to be described, but the second reluctance path can be difficult to model. The airgap harmonics of the salient pole permanent magne<sup>t</sup> synchronous machine are presented in [14] but does not address torque ripple or the secondary reluctance path of the IPM machine. The rotor permeance path is approximated in [15] to determine the torque ripple of the machine under study, but does not fully describe an IPM machine. The double V shaped IPM is presented in [16], in which flux densities are calculated through an MEC model and described with a Fourier series. The single V IPM presented in [17] considers the pole cap effect but does not consider torque ripple harmonics. The single V, delta and double V IPM rotor configurations are shown in Figure 1. Moreover, the airgap harmonics in permanent magne<sup>t</sup> synchronous machines were calculated in [18,19], but the effect of the second reluctance path on the airgap harmonics was not included in the calculations.

Recently, researchers have investigated rotor modifications to alter the airgap, modify airgap flux, and improve torque harmonics. The first feature type is pole shaping, which creates a small airgap near the *d*-axis and an increased airgap in the region of the *q*-axis. The torque ripple was reduced for the single magne<sup>t</sup> flat magne<sup>t</sup> IPM and optimized with a differential evolution algorithm and finite elements [20]. A surface-mount PM pole-shaped machine was studied with an analytical solution to the field in [21]. The 2D solution was confirmed both by finite element and testing. The pole-shaped single flat magne<sup>t</sup> IPM was optimized with a response surface method within FE [22]. This included the use of rotor core modifications as well; both FE and experimental results were presented. The flat magne<sup>t</sup> IPM pole shape was optimized, along with the creation of design rules for the ratio of *q*-axis and d-axis airgap length in [23]. The single V magnet-shape IPM was improved with pole shaping using finite elements in [24]. Cogging torque and back emf were measured. A third harmonic was added to the pole shape in [25], which studied the machine in finite elements. A second feature type is in the rotor core, which creates a small hole in the rotor core near the airgap in order to redirect flux. Holes in the rotor core's second reluctance path of the single magne<sup>t</sup> IPM were shown to decrease torque ripple using finite elements in [26]. The double V magne<sup>t</sup> IPM machine with improved torque ripple, due to holes in rotor iron core and rotor surface sculpt features, was shown to improve torque ripple but lower average torque in [27]. The delta magne<sup>t</sup> IPM shape included modified internal rotor features to improve for average torque and decrease iron loss in [28].

**Figure 1.** IPM rotor types: single V (**left**), delta (**center**), double V (**right**).

The third and final feature type is sculpting the rotor surface at the airgap to redirect flux. The single flat magne<sup>t</sup> IPM machine cogging torque was reduced in [29] and

experimentally verified. A grid on/off optimization of the rotor surface was conducted on the single flat magne<sup>t</sup> IPM using finite elements in [30,31], resulting in an asymmetric rotor surface with reduction in torque ripple and maintaining average torque. The double V-magnet IPM torque ripple was minimized with both rotor core and surface sculpted features in [27]. The delta-magnet IPM machine torque ripple was minimized with rotor surface sculpt features in [32]. Then, a general analytical expression for torque harmonics was developed and utilized to optimize the solution with finite elements.

This paper presents a detailed analytical model of the sculpted rotor IPM machine. The model allows for a break down of flux and torque into magnet, primary reluctance, secondary reluctance, and sculpt features. Multiple sculpt features configurations are demonstrated to achieve similar torque harmonic reductions. Results are validated with finite elements and utilized to improve the torque harmonic characteristics of an existing industrialized machine. This is because finite elements accurately predict experimental results across a broad range of machines [33–36] and have been utilized to evaluate and compare machine types [37–39] and validate analytical solutions [40–50]. Section 2 introduces the topics relevant to the design of IPM motor construction and control. Section 3 provides details on how to model an IPM motor magneto motive force (MMF), permeance, and linear current density in order to model the machine geometry, flux, and torque harmonics. The model developed in Section 3 is applied to that of a well-known industrial IPM machine in Section 4. Design features are explored in Section 5. Contributions of this paper include a novel analytical winding function-based IPMSM model, the analytical description of rotor sculpt features, and modeling of magne<sup>t</sup> and reluctance torque component alignment due to asymmetric sculpt features. In addition, this paper demonstrates the torque effects of reluctance path pole arc, sculpt feature type (symmetrical/asymmetrical), sculpt feature location, and sculpt feature depth and sculpt feature width.

#### **2. Flux Distribution and Control of IPM Machines**

Performance, harmonics, and control are all dependent upon the distribution of flux density within the machine. Permanent magnets provide a constant source of flux density, which enable efficient torque production but can limit high speed operation. Reluctance features provide a source of torque dependent upon armature current at high current angles, useful for extending operation at high speeds. The ratio of magne<sup>t</sup> and reluctance torque is balanced to enable the machine to stay within its operation constraints while efficiently using the voltage and current available at the terminals. This section provides a brief overview of the machine's construction, design features, and flux paths. In addition, the necessary framework for control is introduced.

### *2.1. Flux Distribution*

Figure 2 shows a two-dimensional illustration of a four pole IPM machine. The IPM motor is fundamentally constructed of a stator and a rotor. The stator is the mechanically grounded part of the machine. It is constructed of slots, teeth, a yoke, and the three phase windings. The stator teeth and yoke are constructed of a magnetically permeable iron alloy. The teeth and yoke allow for easy flow of magnetic flux to and from the airgap of the machine. The slots allow space for the copper windings. The windings are distributed within the slots to produce a current dependant magneto motive force (MMF), which in turn creates the radial magnetic flux density. The placement of the windings also creates a current density along the bore of the stator, resulting in a tangential component of flux density. When arranged and controlled properly, the currents in the windings produce a rotating set of fields to produce torque.

**Figure 2.** IPM flux paths: first reluctance path (solid blue), second reluctance path (dashed blue), and magne<sup>t</sup> path (solid red).

The rotor, the mechanically rotating part, is constructed of iron ribs, air-filled barriers, iron bridges, and permanent magnets. The iron core is constructed of ribs and bridges. The ribs control the distribution of flux density while the bridges mechanically couple all parts together. Barriers provide air pockets, which assist the ribs in directing flux, and also contain embedded magnets. The permanent magnets are embedded within the rotor and produce an MMF, which is independent of current.

From the perspective of the rotor, a direct axis (*d*-) and a quadrature axis (*q*-) of the machine are electromagnetically aligned to the rotor characteristics. The *d*- axis is the primary axis of which the permanent magne<sup>t</sup> flux density flows. The magne<sup>t</sup> flux density flows through the magne<sup>t</sup> into the central rib of the magne<sup>t</sup> pole, into the airgap, through the stator teeth and yoke and returns into the adjacent opposite magne<sup>t</sup> pole. This permanent magne<sup>t</sup> flux density path is shown in Figure 2 as a red ellipse. The *q*- axis is the axis in which the armature-induced flux flows through the rotor. This flux is produced from the armature MMF and the reaction of the rotor/stator permeance. Two armature-induced paths result; one through the primary reluctance path, and a second through the secondary reluctance path. The primary reluctance path flux is shown as the solid blue ellipse, and the second reluctance path flux is shown as the dashed blue ellipse in Figure 2.

### *2.2. Control of IPM Machine*

Control must be considered in the design of the IPM motor. The steady state torque and voltage equations are shown in Equations (1)–(3). These are the fundamental starting points to develop the necessary analysis for the control of electric machines. The equations are based upon *d*- and *q*- axis voltages, *vd* and *vq*, currents, *id* and *iq*, inductance *λd* and *<sup>λ</sup>q*, magne<sup>t</sup> flux linkage, *λ<sup>m</sup>*, and phase resistance *Rs*. Magnet offset, *δ*, as shown in Figure 3, is included to account for magne<sup>t</sup> alignment relative to the reluctance path, which may be caused by rotor sculpting features [51,52]. Traditional IPM alignment would feature *δ* = 0◦, with the *d*-axis aligned to the maximum of magne<sup>t</sup> flux linkage. For purposes of this paper, the *q*-axis remains aligned to the minimum reluctance of the rotors first reluctance path. By inspection, the resistance or loss terms do not have an effect on the torque and only affect the voltage and electrical power. It may be useful to assume the phase resistance is negligible.

$$\tau = \frac{1}{2} \frac{3}{2} \left( (l\_d - l\_q) i\_d i\_q + \lambda\_m \left( \cos(\delta) i\_q - \sin(\delta) i\_d \right) \right) \tag{1}$$

$$
\omega v\_d = R\_s i\_d + -\omega l\_q i\_q - \omega \lambda\_m \sin(\delta) \tag{2}
$$

$$
\omega\_q = R\_s i\_q + \omega l\_d i\_d + \omega \lambda\_m \cos(\delta) \tag{3}
$$

Performance assessments require including an analysis of the phase current constraint, *Imax*, and phase voltage *Vmax*. For a wye connected machine, *Imax* is equal to the phase current, *Iss*, and is limited by the power devices of the inverter and the electric machines thermal capability. The voltage limit is the maximum phase voltage that the inverter can apply, limited by the specific pulse width modulation (PWM) technique used. The voltage limit for space vector PWM is *Vmax* = *Vdc* ·*MI* √<sup>2</sup>· √3 and six step PWM is *Vmax* = *Vdc* ·*MI*· √<sup>2</sup>·*<sup>π</sup>* . The maximum modulation index is set to *MI* = 0.95 to account for cable and device voltage drops.

**Figure 3.** Vector diagram of IPM rotor with unaligned magnet, with variables: *φ* rotor position, *θ* rotor spatial coordinate, *δ* magne<sup>t</sup> alignment, *ω* rotor speed.

#### **3. Analytical Model: MMF, Permeance, Flux, and Torque**

This section develops the necessary analytical winding function model for the IPM machine idealized to focus on the effects of the rotor geometry. Assumptions include closed stator slots and no saturation leading to infinite permeability, leaving the permeability of the airgap assumed to be that of free space *μ*◦ = 4*π*10−<sup>7</sup> H m . Stator conductors are modeled by discrete current sheets along the stator bore inner diameter and phases are assumed to be wye connected.

Focused on the MMF interaction with the second reluctance path, the analytical model describes MMF and permeance functions. These winding function-based MMFs, *Fx*(*<sup>θ</sup>*, *φ*), and permeance functions, <sup>Λ</sup>(*<sup>θ</sup>*, *φ*), express the harmonic content of stator and rotor features as Fourier series. Relationships between the rotor spatial coordinate *θ* and rotor position *φ* and current angle *β* are included. Flux densities are computed using Equation (4), and contributions of the stator and rotor harmonic interactions to the torque ripple are determined.

$$B\_r(\theta,\phi) = 2\Lambda\_r(\theta,\phi)F\_r(\theta,\phi) \tag{4}$$

### *3.1. Permeance Functions*

**Table 1.** Permeance functions and

The general form of the permeance functions Λ*<sup>x</sup>*, shown in Figure 4, can be written as Equation (5), where the amplitude *Y* and phase *γ* define the location of features relative to the *d*-axis of the machine.

$$\Lambda\_x(\theta) = \sum\_{n=0,2,4,6,\dots}^{\infty} \Upsilon\_x(n) \cos(n\theta + \gamma\_x(n))\tag{5}$$

Permeance functions, along with rotor and stator construction, are shown in Figure 4. The salient features of the rotor begin with the definition of the primary reluctance path, which assumes a small airgap *lg*, aligned with the minimum reluctance of the *q*-axis, and a large airgap considering barrier and magne<sup>t</sup> dimensions, *lm*. Above the magnet, a secondary reluctance path exists, which reacts as equipotential salient iron to the armatures MMF. The permanent magne<sup>t</sup> permeance path describes the total amount of air the magne<sup>t</sup> must push its flux through, including its thickness in the same region of the second reluctance path. Coefficients of Equation (5) are derived from the local definition of the permeance Equation (6).

$$
\Lambda = \frac{\mu\_{\circ}}{\mathcal{g}} \tag{6}
$$

The defining airgaps of the permeance functions are listed in Table 1. related

 minimum and maximum airgaps.


**Figure 4.** Conductor locations and permeability functions, (red) phase A conductor, (blue) phase B conductor locations, (green) phase C conductor location, (gold) first reluctance path permeability function, (lavender) second reluctance path permeability functions, (purple) magne<sup>t</sup> path permeability function, (gray) equivalent permanent magne<sup>t</sup> conductor locations.

### *3.2. Magneto Motive Forces (MMF)*

Equation (7) gives the general form of the winding function composed of the turns function *<sup>n</sup>*(*<sup>θ</sup>*, *φ*) and its mean < *<sup>n</sup>*(*φ*, *θ*) >, where the turns function is a result of the closed path integral of the conductor [53]. MMF functions, *<sup>F</sup>*(*<sup>θ</sup>*, *φ*), are comprised of the turns function multiplied by a current.

$$N(\theta, \phi) = n(\theta, \phi) - < n(\phi, \theta) > \tag{7}$$

$$ = \frac{1}{2\pi} \int\_0^{2\pi} n(\theta,\phi)d\theta\tag{8}$$

Expanded into a Fourier series, the winding takes the form as follows (9):

$$N\_{\mathbf{x}}(\theta,\phi) = \sum\_{n=1,3,5,7...}^{\infty} \Upsilon\_{wf\_{\mathbf{x}}}(n) \cos(n\theta + \gamma\_{wf\_{\mathbf{x}}}(n) + n\phi\_{i} + n\phi) \tag{9}$$

where *Yw fx* is the coefficient, *γw fx* (*n*) is the phase, mechanical order *n*, rotor position *φ*, and initial rotor position *φ<sup>i</sup>*.

### 3.2.1. Stator

Stator MMF is formed from the interaction of the phase currents (*Ia*, *Ib*, *Ic*) and the phase winding functions (*Na*, *Nb*, *Nc*). The summation of the three phases creates a rotating MMF, *Fabc*, which directly acts upon the primary reluctance path.

$$F\_{abc}(\theta,\phi) = N\_a(\theta,\phi)I\_a(\phi,\beta) + N\_b(\theta,\phi)I\_b(\phi,\beta) + N\_c(\theta,\phi)I\_c(\phi,\beta) \tag{10}$$
