## 2 Mathematical Modeling of VFRM  
In the VFRM design, the layout of the 6/4 VFRM incorporates both AC and DC components, as depicted in Figure 1a. Traditionally, the regulation of the DC segment of the VFRM was backed by an external H-bridge inverter connected to a separate DC bus voltage. In order to achieve maximum power output within the boundaries imposed by the DC bus voltage and to bolster system reliability, the DC and AC coils have been consolidated into a single winding. Furthermore, the open windings for both the positive and negative terminals in the VFRM are regulated through a dual inverter that employs a shared DC bus voltage, as demonstrated in Figure 1b. In this setup, the regulation of the DC component is accomplished through the zero-sequence current yielded by the dual inverter. As stated in [29], Equation (1) for the voltage in the VFRM includes an inductance element that varies with a third harmonic order. \[ \begin{split}\left[\begin{array}{c}u_{\text{d}}\\ u_{\text{q}}\\ u_{\text{0}}\end{array}\right]=R_{\text{s}}\left[\begin{array}{c}i_{\text{d}} \\ i_{\text{q}}\\ i_{\text{0}}\end{array}\right]+\left[\begin{array}{cc}\frac{L_{\text{g}}}{2} \cos 3\theta_{\text{e}}+L_{\text{s}}&-\frac{L_{\text{g}}}{2}\sin 3\theta_{\text{e}} &M_{\text{s}}\\ -\frac{L_{\text{g}}}{2}\sin 3\theta_{\text{e}}&\frac{L_{\text{g}}}{2}\cos 3 \theta_{\text{e}}-L_{\text{s}}&0\\ \frac{L_{\text{g}}}{2}\frac{L_{\text{g}}}{0}&0&M_{\text{s}}\end{array}\right] \left[\begin{array}{c}\frac{\text{d}i_{\text{d}}}{\text{d}t}\\ \frac{\text{d}i_{\text{q}}}{\text{d}t}\end{array}\right]\\ +\omega_{\text{e}}\left[\begin{array}{cc}-L_{\text{g}}\sin 3\theta_{\text{e} }&-L_{\text{s}}\cos 3\theta_{\text{e}}-L_{\text{s}}&0\\ -L_{\text{s}}\cos 3\theta_{\text{e}}+L_{\text{s}}&L_{\text{s}}\sin 3\theta_{\text{e} }&M_{\text{s}}\\ 0&0\end{array}\right]\left[\begin{array}{c}i_{\text{d}}\\ i_{\text{q}}\\ i_{\text{0}}\end{array}\right]\end{split}  \tag{1}\]  
Along the dq0-axis, the elements are labeled as \(i_{\text{d}}\), \(i_{\text{q}}\), and \(i_{\text{0}}\) for the respective current components, whereas \(U_{\text{d}}\), \(U_{\text{d}}\), and \(U_{\text{0}}\) signify the voltage components. Additionally, \(\omega_{\text{e}}\) and \(\theta_{\text{e}}\) correspond to the electrical angular velocity and phase angle, respectively, while \(R_{\text{s}}\) represents the winding resistance and \(M_{\text{s}}\) represents varying components of the mutual inductance between the field and armature windings. The primary frequency element is characterized as \(L_{\text{s}}\), while the invariant element of the self-inductance is denoted by \(L_{\text{s}}\). The waveform of the self-inductance and its harmonic properties, informed by the FEA modeling, are demonstrated in Figure 2.  
Figure 1: (**a**) Integrated armature and field winding VFRM topology, (**b**) Dual inverter with Shared DC Bus. Figure 2: (**a**) Waveforms of self-inductance (when \(i_{\text{0}}=1\) A and \(\omega_{\text{e}}=400\) rpm). (**b**) Harmonic content of self-inductance. In VFRM, the self-inductance is comprised of a constant component and a harmonic linked to the primary frequency. Although the intensity of self-inductance can vary with fluctuations in the DC component, higher-order harmonics are neglected in this model, based on their minimal contribution to system dynamics. As shown in Figure 2, the amplitudes of harmonics beyond the second order are significantly smaller. Analytical studies and FEA confirm that these higher-order components have negligible influence on the torque ripple and inter-axis coupling effects. Including them would add unnecessary complexity to the model without providing meaningful improvements in accuracy. Therefore, for computational efficiency and to focus on the most critical disturbances, only the dominant harmonic components are considered in this paper. It is also evident from Equation (1) that the self-inductance \(L\delta\) varies dynamically with \(\omega e\) and \(\theta e\).