# A PINN-Based Nonlinear PMSM Electromagnetic Model Using Differential Inductance Theory
## 1 Introduction  
Permanent magnet synchronous motors (PMSMs) offer advantages over traditional asynchronous motors, such as high efficiency, high power density, fast response, and low noise. These features enable electric vehicles (EVs) to achieve better acceleration performance, longer driving range, and improved control accuracy, making PMSMs widely used in EV drivetrain systems [1]. Additionally, PMSMs can function as energy storage components in lithium-ion battery AC heating systems [2, 3]. The construction of traditional PMSM models relies on parameters including stator equivalent resistance \(R_{s}\), dq-axis equivalent inducances \(L_{d}\), \(L_{q}\), and permanent magnet flux linkage \(\psi_{f}\). By adopting the synchronous rotating coordinate system, the control complexity of alternating quantities in the motor is effectively simplified; meanwhile, the double-reaction theory addresses inductance fluctuations caused by uneven air gaps in salient-pole motors, forming a simplified constant-parameter PMSM model with concise formulations. However, to facilitate model development, this approach simplifies and linearizes the real physical model of PMSMs under four key assumptions:  
* Magnetic saturation effects in the core magnetic circuit are neglected. * The magnetic potential in the air-gap field and the back-EMF generated by permanent magnets contain no harmonics. * The rotor has no damper windings, and permanent magnets exhibit no damping effects. * Hysteresis and eddy current losses in the motor, as well as magnetic circuit nonlinearity, are not considered. These assumptions deviate from the motor's actual operating conditions, especially in vehicle drivetrain systems where stator and rotor cores often operate in a high magneticsaturation state to achieve high power density [4]. Under load, the air-gap magnetic field of the motor rarely maintains an ideal sinusoidal distribution [5]. Therefore, to accurately simulate the voltage-current excitation response characteristics at specific operating points, it is necessary to comprehensively consider physical phenomena affecting parameter characteristics, such as magnetic saturation [6; 7], cross-coupling effects [8; 9], rotor magnetic field harmonics, and loss-induced temperature rise. The relative permeability of motor ferromagnetic materials exhibits a nonlinear relationship with magnetic field strength \(H\): when \(H\) is low, the relative permeability of the core is high; once \(H\) exceeds a critical value, the relative permeability decreases rapidly, making core reluctance effects non-negligible. As the degree of magnetic saturation increases, nonlinear reluctance differences at different spatial positions in the motor significantly exacerbate the non-sinusoidal distribution of the air-gap field, causing obvious changes in the dq-axis inductance parameters under different stator current excitations \(i_{d}\), \(i_{q}\)[10; 11]. Furthermore, core magnetic saturation induces cross-coupling phenomena, generating mutual inductances \(M_{dq}\), \(M_{qd}\) between orthogonal dq-axis windings, and the magnitude of these mutual inductances can approach that of the dq-axis self-inductances under deep saturation [5; 12]. Many scholars have studied the nonlinear relationships of physical quantities in PMSMs. [PERSON] et al. [13] abandon the explicit calculation of absolute inductances in PMSM modeling and instead directly construct a stator flux linkage model. This method comprehensively considers magnetic saturation, cross-coupling, and rotor magnetic field harmonic effects, defining the stator flux linkage as a three-dimensional function of dq-axis currents and rotor angle. Flux linkage MAP tables are generated through finite element analysis (FEA) combined with radial basis function (RBF) interpolation, and a nonlinear model is then constructed using look-up table modules. This modeling approach replaces complex functional operations with discretization processing, significantly reducing computational load and improving real-time performance. [PERSON] et al.