# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
## 3 Analytical Model
### Magnetic Flux Density Created by Coils in a Slotless Machine  
The magnetic flux density produced by the armature in the air gap is computed by (25), and that on the rotor surface by (26). These expressions are valid only when magnets are placed on the surface of the rotor. In case of inner magnet configurations, the length of the magnetic path seen by the coils is not constant along the air-gap perimeter, which depends on the rotor position. Furthermore, the problem becomes more nonlinear due to the fact that this length of the magnetic path depends strongly on saturations. Therefore, as the nonlinearities are not considered in this model, rotors with inner magnets placed were not analyzed. \[B_{g}^{a}(t,\theta)=\mu_{0}\cdot\frac{F_{MM}(t,\theta)}{g} \tag{25}\]  
\[B_{g}^{a}(t,\theta)=\mu_{0}\cdot\frac{F_{MM}(t,\theta)}{g+\frac{h_{m}}{\mu_{m }}} \tag{26}\]  
where \(F_{MM}\) is the magnetomotive force created by the coils in the air gap. To obtain the \(F_{MM}\), the distribution of the conductors must be known. This distribution can be defined using the widely known star of slots method. This method is valid for all types of windings: distributed or concentrated, single layer or double layer, and so on. In Figure 4, the stars of slots corresponding to the machines considered in this work are shown. As shown, the winding of the integer machine is defined by distributed coils in a single-layer  
Figure 3: (**a**) Spatial distribution of the field with \(a\) and without \(a\); (**b**) main mechanical dimensions of the machine. configuration, even though a double-layer configuration would also be possible. On the other hand, in the fractional machine the chosen winding consists of concentrated coils in a double-layer configuration, even though a single-layer configuration would also be possible. After the physical distribution of the conductors is obtained, the winding factor, \(Kw_{n}\), can be computed using (27). Therefore, the magnetomotive force per phase and per current unit can be calculated using (28) and (29). \[Kw_{n}=\frac{\sum_{k=0}^{k-k-1}e^{i\cdot n\cdot t_{p}\cdot\varphi_{k}}-e^{j \cdot n\cdot t_{p}\cdot(\varphi_{k}+\varphi)}}{\sum_{k=0}^{k-k-1}\left|e^{j \cdot n\cdot t_{p}\cdot\varphi_{k}}-e^{j\cdot n\cdot t_{p}\cdot(\varphi_{k}+ \varphi)}\right|}=\xi_{n}e^{j\cdot n\cdot t_{p}\cdot\varphi_{ph}} \tag{27}\]  
\[F_{an}=\frac{-j2N}{\pi n^{2}\pi t_{p}}\text{sin}\bigg{(}\frac{n\cdot t_{p} \cdot\alpha}{2}\bigg{)}\xi_{n}e^{i\cdot n\cdot t_{p}\cdot\varphi_{ph}} \tag{28}\]  
\[F_{a}(\theta)=\sum_{n=-\infty}^{n=+\infty}F_{an}e^{j\cdot n\cdot t_{p}\cdot( \varphi_{ph}-\theta)} \tag{29}\]  
where \(h\) is the number of coils per phase, \(\varphi_{ph}\) the position angle of the \(h\)th coil, \(\varphi\) the coil pitch angle, \(N\) the number of turns, and \(\alpha\) the angle at which the conductors are distributed (Figure 3b). The magnitude of the winding factor \(Kw_{n}\) for the n-order harmonic depends on the distribution of the conductors, so adjusting the winding configuration can modulate the winding factor value (in some cases, it is also possible to eliminate the winding factor for certain harmonics). Furthermore, for a given winding factor, adjusting the distribution angle can also minimize or eliminate certain harmonics, which is very useful in optimizing the vibroacoustic performance of electrical machines. Once \(F_{a}\) is calculated, it is multiplied by the current phase (30) and the magnetomotive forces are subsequently summed (31) in order to compute the overall \(F_{MM}\) in the air gap. \[F_{MMa}(t,\theta)=F_{a}(\theta)\cdot i_{A}(t) \tag{30}\]  
\[F_{MM}(t,\theta)=F_{MMa}(t,\theta)+F_{MMb}(t,\theta)+F_{MMc}(t,\theta) \tag{31}\]  
Finally, (25) is applied to calculate the magnetic flux density created by the armature in the air gap. It is important to note that in three-phase machines, the spatial shift between magnetomotive forces of different phases is \(2\pi/3\) radians, as for the currents. This is the reason why (32) can be deduced from (31).