## 2 Analytical Approach  
The current density distribution and the Joule losses can be analytically evaluated as a function of the operating frequency for a single rectangular slot without the polar shoes. Alternative hairpin winding layouts are taken into account in the model developed in [7]. The current density distribution can be analytically found following the Maxwell's equations, considering the reference frame of Figure 3 and assuming the magnetic field in quasi-static  
Figure 2: Manufacturing process of hairpin winding. approximation, see Equation (1). This approximation is reasonable, since the displacement current \(\frac{\partial\vec{D}}{\partial t}\) for good conductors is negligible compared to the conduction (drift) current \(\vec{J}\)  
\[ \begin{cases}\
abla\times\vec{H}=\vec{J_{z}}\\ \
abla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}\end{cases}  \tag{1}\]  
Merging the equations of system in Equation (1), the second-order partial differential equation for \(J_{z}\) is obtained as in Equation (2), where \(a\) is defined as in Equation (3). \[\frac{\partial J_{z}^{2}}{\partial x^{2}}+\frac{\partial J_{z}^{2}}{\partial y ^{2}}=a^{2}J_{z} \tag{2}\]  
\[a^{2}=j\omega\mu\sigma \tag{3}\]  
Considering the edges of the slot, due to the high difference between air and iron magnetic permeabilities, it becomes possible to consider only the \(y\) components for the \(\vec{B}\) and \(\vec{H}\) fields. Therefore, the partial derivative of \(J\) along the y-axis could be considered equal to zero, thus simplifying the expression for the \(J_{z}\), as in Equtaion (4). The general solution of Equation (4) could be expressed as in Equation (5). \[\frac{\partial J_{z}^{2}}{\partial x^{2}}=a^{2}J_{z} \tag{4}\]  
\[J_{z}\left(x\right)=c_{1}e^{ax}+c_{2}e^{-ax} \tag{5}\]  
Applying the boundary conditions provided in Equation (6) to this equation, the mathematical expression for \(J_{z}\) is obtained as shown in Equation (7). \[ \begin{cases}H_{y}\left(0\right)=0\\ I=\int\int\vec{J_{z}}\cdot d\vec{S}\end{cases}  \tag{6}\]  
\[J_{z}\left(x\right)=\frac{aI}{w_{s}}\frac{\cosh\left(ax\right)}{\sinh\left(ah \right)} \tag{7}\]  
Figure 3: Single slot fully filled by a single conductor. Substituting the expression for \(J_{z}\) in the relation that defines the Joule losses, it is possible to obtain Equation (8), which represents the ratio between AC resistance and DC resistance, both per unit-length, where \(\xi\) is defined as in Equation (9). \[\varphi\left(\xi\right)=\frac{r_{AC}}{r_{DC}}=\xi\cdot\frac{\sinh\left(2\xi \right)+\sin\left(2\xi\right)}{\cosh\left(2\xi\right)+\cos\left(2\xi\right)} \tag{8}\]  
\[\xi=\frac{h}{\delta} \tag{9}\]  
Figure 4 shows the ratio between AC resistance and DC resistance as a function of \(\xi\): the smaller the _skin depth_\(\delta\), the greater the operating frequency and the AC resistance will be. In particular, \(\varphi(\xi)\) grows linearly if \(\xi\geq 2\) because the hyperbolic functions of Equation (8) become much greater than the trigonometric ones. This behavior is due to both skin and proximity effects, since the current density tends to distribute itself in a smaller surface than over the whole conductor, hence increasing the resistance. Since the current densities due to skin and proximity effects are orthogonal, as given in Equation (10), they can be treated separately, according to Chapter 6 and Appendix A of [8; 9], respectively. Hence, the plots of the current density module and phase and the plot of the power losses can be found considering only one of the two effects. Chapters 3 and 4 of [10] treat both cases accurately. \[\int\int\left(J_{z,s}\cdot J_{z,p}\right)\cdot dS=0 \tag{10}\]  
An analytical expression for the current density can be found, always starting from [PERSON]'s equation, Equation (1), when multiple conductors are placed inside the slot. Figure 4: Ratio between \(r_{AC}\) and \(r_{DC}\). In this case, different boundary conditions have to be applied to the second-order partial differential Equation (2).