## 3 EM Analytical Model
### New Models for Stator-Segmented Structures
#### 3.2.2 Cogging Torque  
In a stator-segmented structure with a magnetic gap, the cogging torque is due to PM electromagnetic interaction with the sectors/gaps in addition to the interaction with slots/teeth. So it is necessary to determine the contribution of gaps to the cogging torque. 1. Superposition principle The estimation of the cogging torque for a segmented structure is based on the superposition principle [11]. It is assumed that the contributions of slots and angular gaps to the cogging torque are independent and can be added. The cogging torque is evaluated for two basic structures: a non-segmented slotted machine with the same geometry as the studied segmented machine, and a structure with the same dimensions with no slot and with only one gap on the stator (named mono-gap structure). An example of this theoretical mono-gap configuration is presented in Figure 9b. The contribution of slots and angular gaps are estimated separately and then added. The classical analytical model for calculating the cogging torque (as in [10]) is used to estimate the first contribution (due to slots). For the proposed mono-gap model, it is supposed that angular gaps contain entire number of pole pairs. In fact, to keep a regular winding distribution, the gap angular width must be a multiple of the winding period which corresponds to a number of slots defined by Equation (7) and called reduced number of slots. \[Q_{s0-real}=S_{pp}\times 2\times\frac{p_{0}}{ppcm(Q_{s0},p_{0})}\times m\] (7) In this equation, \(S_{pp}\) is the number of slots per pole and per phase. 2D FEM (with the software FluxTM 2D) is used in a preliminary step to validate the mono-gap model. This numerical study shows that the gap width does not influence the cogging torque peak value nor its frequency for a mono-gap structure, if the gaps contain entire number of pole pair. Figure 10 presents the cogging torque calculated by 2D FEM for two single-gap structures with the same geometry (as depicted in Figure 9b) and two different gap widths, in function of the rotor position \(\theta_{r}\). It can then be seen that the contribution of one gap to the cogging torque is the same for similar structures with several gap widths. For a stator-segmented structure, the cogging torque is estimated for the whole slotted machine \(T_{c,slotted}\) (Figure 9a) (with the Equation (3)). This contribution is multiplied by the active part proportion \((1-p_{gap})\). In addition, the mono-gap contribution to the cogging torque \(T_{c,MonoGap}\) is evaluated and multiplied by the number of gaps  
Figure 8: Analytical evaluation of the ratio between the magnetic flux density of a machine with an infinite air gap and of a slotless machine. included in the structure, as shown by (8). This calculation method is valid because all the gaps are in the same relative position relatively to the rotor poles. \[T_{c,StatorGap}(\theta_{r})=T_{c,slotted}(\theta_{r})\times(1-p_{gap})+T_{c, MonoGap}(\theta_{r})\times n_{gap}\] (8)
2. Mono-gap cogging torque estimation The proposed mono-gap model to evaluate the cogging torque is based on the classical analytical calculation for a non-segmented structure presented in [10]. The estimation of the mono-gap cogging torque depends on the magnetic pressure in each active sector/gap interface. The calculation is done on the red shaded area of the Figure 6. This calculation area corresponds to half a pole pitch in each gap extremity. The method is extended to the stator-segmented machine by using the flux density created by permanent magnets into gaps and the extended permeance function adapted for gaps in the stator. The proposed mono-gap model is validated for the mono-gap structure used for the previous numerical study (Figure 10). Figure 11 presents the mono-gap cogging torque obtained with the numerical (2D FEM) and proposed analytical methods for a mono-gap machine with a gap width of ten pole pitches.