## 3 SHG Mathematical Model  
The SHG magnetic core has two SRSCs linked to each other in the axial direction by the stator housing and the rotor sleeve made of non-laminated low-carbon steel. The excitation winding fixed on the stator induces the excitation flux, which passes through the SRSCs, the stator sleeve on the rotor shaft, and the stator housing [8; 30] (see Figure 3). The modulation of the magnetic field by the teeth of the rotor allows this magnetic flux to interact with the poles of the armature winding on the stator. In the absence of magnets, only half of the poles would be involved in the torque production. The addition of permanent magnets between the teeth of the rotor makes all poles of the armature winding to usable [31; 32]. The developed mathematical model consists of a set of \(g\) boundary 2D magnetostatic problems for a magnetic field in the cross section of an SRCS and a magnetic circuit equation with lumped parameters in the axial direction. The range of considered rotor position angles is chosen taking into account the symmetry with respect to a shift by a third of the electric period and cyclic phase permutation and is equal to a third of the electric period. One third of the electric period is divided into \(g\) sections and boundary value problems are considered for the beginning of each section. In this study, \(g=24\). As in conventional electric machines, the magnetic field in a good approximation can be assumed to be uniform along the axis and lying in the transverse plane. Therefore, the equations traditional for 2D problems of magnetostatics are solved [17]:  
\[ \begin{array}{l}\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{ \partial y}=0;\\ \frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}=J_{z}, \end{array}  \tag{1}\]  
where \(J_{z}\) is the z-component of the current density, which is not equal to zero only in the stator slots filled with a winding: \(B_{x}\) and \(B_{y}\) are the components of the magnetic flux density; \(H_{x}\) and \(H_{y}\) are the components of the magnetic field. A distinctive feature of SHM is the presence of a magnetic monopole, i.e., excitation flux flowing in on the inner boundary of the rotor and flowing out of the outer boundary of the stator (or vice versa). Then, the general solution of the Gauss law for magnetism (1) can be expressed as follows:  
\[ \begin{array}{l}B_{x}=\frac{\partial A_{x}}{\partial y}+\frac{\phi\ x}{2\pi( x^{2}+y^{2})};\\ B_{y}=-\frac{\partial A_{z}}{\partial x}+\frac{\phi\ y}{2\pi(x^{2}+y^{2})}. \end{array}  \tag{2}\]  
where \(A_{z}\) is the \(z\)-component of the vector magnetic potential; \(\phi\) is the linear density of the magnetic charge. Only the component of the vector magnetic potential \(A\) along the \(z\)-axis is different from zero. In these equations, the first terms on the right side of these  
Figure 3: Excitation winding flux path through the stator housing, the SRCSs, and the rotor sleeve. equations are common for a magnetostatic problem. The second terms determine the linear flux density of the magnetic monopole, which is set based on the following relation:  
\[\Phi=\phi\cdot L_{stator}/2. \tag{3}\]  
where \(L_{stator}\) is the total stator laminated stack length. The calculation area is one sector corresponding to the pole pitch of the generator, with periodic boundary conditions \(P_{I}\) and \(P_{II}\) on the boundaries of this sector, as shown in Figure 4. The change in the rotor angular position is modeled not by changing the geometry of the calculation area, but by changing the connecting boundary condition on a circular line lying in the middle of the air gap between the rotor and stator. The equations of magnetostatics are complemented by constitutive equations. For the rotor and stator stacks, the magnetization curve \(H(B)\) is set. For magnets, the residual magnetic flux density and the magnetic permeability are set.