## 2 Modeling the Electric Drive Transmission System
### Equivalent Magnetic Network Model of DTP-PMSM
With the magnetic permeability and potential determined, an equivalent magnetic network model of the motor can be formulated using scalar magnetic potential and current as variables. Applying [PERSON]'s current law, node analysis is conducted across the entire equivalent magnetic network, resulting in a set of node algebraic equations, as illustrated in Equation (5) [12]. \[ \begin{bmatrix}G_{1,1}&G_{1,2}&\dots&G_{1,n}\\ G_{2,1}&G_{2,2}&\dots&G_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ G_{n,1}&G_{n,2}&\dots&G_{n,n}\end{bmatrix}\begin{bmatrix}F_{1}\\ F_{2}\\ \vdots\\ F_{3}\end{bmatrix} = \begin{bmatrix}\phi_{1}\\ \phi_{2}\\ \vdots\\ \phi_{3}\end{bmatrix}  \tag{5}\]  
In the equations, \([G]_{n\times n}\) is the magnetic permeability matrix, \([F]_{n\times 1}\) is the node scalar magnetic potential, and \([\phi]_{n\times 1}\) is the flux generated by the excitation source at the node location. Figure 2: Relative positions relationship between the equivalent teeth of the stator and rotor. The differential equations governing the relationships among magnetic flux linkage, voltage, and current in the stator windings of the non-phase shift DTP-PMSM are expressed as follows [12; 21],  
\[\frac{d\upl_{dq012}}{dt}=\mathbf{V}_{dq012}-R_{s}\mathbf{i}_{dq012} \tag{6}\]  
where \(R_{s}\) represents the stator winding resistance, and \(\upl_{dq012}\), \(\mathbf{V}_{dq012}\), and \(\mathbf{i}_{dq012}\) correspond to the stator flux linkages, terminal voltages, and phase currents in the \(dq0\) reference frame, respectively. For the electromagnetic torque \(T_{e}\) expression [21],  
\[T_{e}=\frac{3}{2}P_{n}(i_{q1}+i_{q2})\psi_{f} \tag{7}\]  
where \(P_{n}\) denotes the number of pole pairs, and \(\psi_{f}\) signifies the permanent magnet flux linkage amplitude. The mechanical motion equation of the electric machine is expressed as follows [12; 21]  
\[\frac{d\omega_{m}}{dt}=\frac{1}{J}(T_{e}-T_{L}-B\omega_{m}) \tag{8}\]  
where \(\omega_{m}\) indicates the mechanical angular velocity, \(J\) represents the total rotational inertia, \(T_{L}\) specifies the external load torque, and \(B\) characterizes the viscous damping coefficient. The magnetic saturation effect in electric machines can significantly influence transient torque and current during system disturbances. The motor model calculates the magnetic field intensity within permeance units based on node magnetomotive forces, subsequently solving for the magnetic flux density. Based on the computed magnetic flux density, it dynamically updates the permeability of iron core regions (primarily the stator yoke and rotor yoke, see Equation (2) for the rotor yoke calculation). The nonlinear permeability function \(\mu_{Fe}\) of the silicon steel material in the machine can be approximately represented by an equation with magnetic field intensity \(H\) as the independent variable [12; 19]:  
\[\mu_{Fe}(H)=\left\{ \begin{matrix}\frac{K_{2}\ln(K_{s}|H|+1)}{|H|}&H\
eq 0\\ K_{1}K_{2}&H=0\end{matrix} \right. \tag{9}\]  
In the equation, \(H=\Delta F/\hbar\) represents the magnetic potential difference across the steel material unit, where \(l\) denotes the unit length. Figure 3 illustrates the nonlinear permeability characteristics of the motor material. Figure 3: Nonlinear magnetic permeability of motor materials.