3.2.2. Magnet

The permanent magne<sup>t</sup> is represented as an equivalent current *IPM*, which is related to the remnant flux *Br*, permeability of free space *u*◦, relative permeability *μ<sup>r</sup>*, and magne<sup>t</sup> thickness. Magnet MMF for the IPM machine interacts with the magne<sup>t</sup> path permeance, where *τm* is the magnet's salient iron pole pitch, and *wm* is the combined width of the magnets for a single pole.

$$I\_{PM} = \frac{B\_r}{\mu\_0 \mu\_r} l\_m \tag{11}$$

$$F\_{PM}(\theta,\phi) = N\_{PM}(\theta)I\_{PM}\frac{\tau\_{\text{ff}}}{w\_{\text{ff}}} \tag{12}$$

### 3.2.3. Second Reluctance Path Modification

The equipotential nature of the rotor's second reluctance path reacts only to the regional harmonics of the stator MMF. In this case, all looping flux that enters the second reluctance path pole arc through the airgap must exit through the same airgap. Modification to the armature MMF by removing its mean satisfies this condition and is made possible through Equation (13). The symbols *F*<*abc*> and < *Fabc*(*<sup>θ</sup>*, *φ*)*NPM*(*θ*) > represent the modified MMF, which interacts with the second reluctance path and the mean of the MMF across this same boundary. As a matter of convenience, the permanent magne<sup>t</sup> winding function *NPM* is also used to consider the stator MMF in the region of the second reluctance path, invert it, remove the mean, and revert to the original polarity.

$$F\_{\text{}>}(\theta,\phi) = \left(F\_{\text{abc}}(\theta,\phi)N\_{\text{PM}}(\theta) - \lhd F\_{\text{abc}}(\theta,\phi)N\_{\text{PM}}(\theta) > \right)N\_{\text{PM}}(\theta) \tag{13}$$

$$<\langle F\_{\rm abc}(\theta,\phi)N\_{PM}(\theta)\rangle \Rightarrow = \frac{\frac{1}{2\pi}\int\_0^{2\pi} F\_{\rm abc}(\theta,\phi)N\_{PM}(\theta)d\theta}{\frac{1}{2\pi}\int\_0^{2\pi} |\ N\_{PM}(\theta)\rangle\,|\,d\theta} \tag{14}$$

### *3.3. Sculpt Feature Description: Equivalent Magnetizing Dipole Current*

MMF-permeance methods allow for flux density for the IPM smooth rotor homogeneous airgap to be calculated. Rotor sculpting affects both magne<sup>t</sup> flux distribution and reluctance flux distribution. This smooth rotor MMF-permeance theory does not adequately describe stator slots and rotor sculpt features as it has been developed with a constant airgap dimension [53]. In this section, an extension of winding function theory is developed, which can be used in the description of both slots and sculpts based upon equivalent magnetic currents (EMC) [54] and the equivalent magnetic dipole [55]. These non-homogeneous airgap features are represented by additional MMF terms utilizing the description of a magnetic dipole and its equivalent magnetic currents. The redistribution of flux density and MMF is possible with the use of the equivalent dipole concept.

The magnetic dipole in free space is formed by a loop of radius *b* and current of *I*. The solution at far fields, when *R* >> *b*, solved in spherical coordinates, using the magnetic vector potential, *A*, is shown in Equation (15), where the magnetic dipole moment *m* is written as *m* = *az Iπb*<sup>2</sup> [55].

$$A = \frac{\mu\_0 m \times a\_R}{4\pi R^2} \tag{15}$$

This dipole in free space can be used to explain the magnetism at the atomistic level, where small circulating currents are formed by the process of magnetization. This magnetization aligns the individual atomic dipoles and modifies the orbital spin of the electrons for each atom.

The macroscopic volume density of magnetization, *M*, with units of A/m, is computed through a sum of the individual microscopic dipoles. Shown in [55], the magnetization vector *M* is equivalent to both a volume current density, *Jm* with units of Am2 , and a surface current density *Jms* with units of Am.

$$J\_{\mathfrak{m}} = \nabla \times \mathbf{M} \tag{16}$$

$$J\_{ms} = M \times a\_n \tag{17}$$

Given *M*, the flux density *B* can be found by computing both *Jm* and *Jms*. These values are used to determine the magnetic vector potential *A*. Uniform *M* within a magnetic material will result in no volume current density and only a surface current density *Jms* on its borders. If space variations of *M* exist within a material, a net volume current density will exist. Hence, a magnetic dipole inside a material with constant magnetization *M* can be represented by a current loop in the air, formed at the exterior boundary material.

Figure 5 illustrates the process of analyzing the rotor sculpt feature effects. Rather than account for the changing flux density over the sculpt feature, the assumption of homogeneous flux density holds when breaking the geometry into smaller discrete dipoles (*i*) of fixed width. For purposes of this analysis, it is assumed the sculpt features have a constant depth *lms*. A sufficient number of points (*i*) must be defined in order to hold the assumption of homogeneous flux density. For each point (*i*) contained within the sculpt feature, the first current *Ims*1(*i*) is applied based upon prior analysis of the magnetic dipole.

$$I\_{\rm ms}(i) = \frac{l\_{\rm ms}(i) \* B\_{\rm ms}(i)}{\mu\_{\odot}} \tag{18}$$

A second dipole counter current in the adjacent point *Ims*2(*<sup>i</sup>* + 1) = <sup>−</sup>*Ims*1(*i*). The net effective dipole current for each point, *Ims*3(*i*) is formed through summation dipole currents *Ims*3(*i*) = *Ims*1(*i*) + *Ims*2(*i*). A third dipole current *Ims*3, or summed current, becomes the current-turns function for the sculpted feature, in which an equivalent MMF for the rotor surface features can be determined through the use of a winding function. It should be noted that the sculpt features analyzed with this process do not create flux but only distribute flux away from the sculpt feature.

**Figure 5.** Sculpted Rotor Reluctance Counter Dipole Current.

### *3.4. Flux Density*

Total airgap flux density, *Btot*, is a result of the flux creating sources and flux distributing features. Primary reluctance path, secondary reluctance path, and the permanent magnets create and distribute flux. The sculpt and slotting features serve to redistribute flux, with the assumption of small features. Flux densities can be computed for each individual component or in summation.

$$B\_{\rm tot} = 2\Lambda\_{\rm R1} F\_{\rm abc} + 2\Lambda\_{\rm R2} F\_{\rm } + 2\Lambda\_{\rm PM} F\_{\rm PM} + 2\Lambda\_{\rm sculpt} F\_{\rm sculpt} + 2\Lambda\_{\rm slot} F\_{\rm slot} \tag{19}$$
