This paper presents a physics-informed neural network approach for dynamic modeling of saturable synchronous machines, including cases with spatial harmonics. We introduce an architecture that incorporates gradient networks directly into the fundamental machine equations, enabling accurate modeling of the nonlinear and coupled electromagnetic constitutive relationship. By learning the gradient of the magnetic field energy, the model inherently satisfies energy balance (reciprocity conditions). The proposed architecture can universally approximate any physically feasible magnetic behavior and offers several advantages over lookup tables and standard machine learning models: it requires less training data, ensures monotonicity and reliable extrapolation, and produces smooth outputs. These properties further enable robust model inversion and optimal trajectory generation, often needed in control applications. We validate the proposed approach using measured and finite-element method (FEM) datasets from a 5.6-kW permanent-magnet (PM) synchronous reluctance machine. Results demonstrate accurate and physically consistent models, even with limited training data.
D YNAMIC models of electric machines are essential for control, estimation, monitoring, and for the design and optimization of drives. The most challenging aspect of machine modeling is the magnetic model, which describes the relationship between flux linkages, currents, rotor angle, and electromagnetic torque, assuming magnetostatic conditions [1], [2]. In modern power-dense electric machines, magnetic saturation effects are significant and must be incorporated into the models. High-fidelity models that include spatial harmonics are also needed, e.g., in time-domain simulations during the design and optimization stage.
The nonlinear magnetics are modeled using analytical functions [3]- [7], lookup tables [8]- [13], or neural networks [14]- [17]. These models can be characterized based on finiteelement method (FEM) data [18], laboratory measurements [19], or automatic identification routines [20]. The accuracy of the analytical models is limited, and they are difficult to extend to higher dimensions (e.g., for spatial harmonics or multi-phase machines). Lookup tables work well in two dimensions but suffer from the curse of dimensionality, high memory requirements, and non-smooth output when using linear interpolation. Black-box neural networks can approximate high-dimensional complex maps and require less memory than lookup tables, but their training still demands large datasets, their extrapolation capability is limited, and energy balance is not guaranteed.
To address the limitations of black-box neural networks, physics-informed neural networks combine data with known physical principles [21]- [24]. Hamiltonian neural networks [23], [24] are particularly relevant in this context, as electric machines are port-Hamiltonian systems [25] with magnetic field energy serving as the Hamiltonian. According to fundamental physical principles [1], [2], the current vector and electromagnetic torque are the gradients of the field energy with respect to the flux-linkage vector and rotor angle, respectively.
The Hamiltonian neural network architectures [23], [24] model the Hamiltonian as a neural network and obtain the gradients via numerical differentiation. This approach improves data efficiency and physical consistency compared to blackbox networks. However, it still faces challenges related to numerical differentiation of the scalar neural network to obtain gradients [26].
Recent gradient networks [26] directly model conservative vector fields, enabling universal approximation of any gradient field without the need for numerical differentiation of a scalar neural network. This approach allows the model to inherently satisfy physical laws such as energy balance and reciprocity, while also improving data efficiency and generalization.
In this paper, we propose a physics-informed magnetic modeling framework for synchronous machines that combines fundamental electromechanical dynamics [1], [2] with gradient networks [26]. The stator current and electromagnetic torque are directly modeled as gradients of a scalar field energy function, which guarantees physical consistency by design. We employ monotone gradient networks to ensure the field energy is convex. This property enforces a unique, invertible relationship between flux linkages and currents, enabling the formulation of both current and flux-linkage (dual) maps. Additionally, Fourier features [27] are incorporated to capture spatial harmonics while preserving the lossless (conservative) field structure. The resulting model offers a universal approximation of complex magnetic behavior, including cross- saturation and angle dependency. We also compare activation functions and propose a computationally efficient 𝑝-norm gradient activation as an alternative to the commonly used softmax. This paper is organized as follows. Section II reviews the physics-based machine model. Section III details the proposed gradient network architecture. In Section IV, the model is validated using measured and FEM datasets from a 5.6-kW permanent-magnet (PM) synchronous reluctance machine. Results demonstrate accurate and physically consistent models, even with limited training data. Section V concludes the paper.
Scalars are denoted by italic letters (e.g., 𝑥), column vectors by bold lowercase letters (e.g., x), and matrices by bold uppercase letters (e.g., A). All quantities are in per-unit unless otherwise specified. where 𝛙 s s = [𝜓 α , 𝜓 β ] ⊤ is the stator flux-linkage vector, i s s = [𝑖 α , 𝑖 β ] ⊤ is the stator current vector, u s s = [𝑢 α , 𝑢 β ] ⊤ is the stator voltage vector, and 𝑅 s is the stator resistance. Furthermore, 𝜗 m is the electrical angle of the rotor d-axis with respect to the stator coordinates, and 𝜔 m is the electrical angular speed of the rotor.
Assuming a lossless (conservative) magnetic field system, the magnetic behavior is fully described by its field energy function 𝑊 s ( 𝛙 s s , 𝜗 m ). Hence, the stator current and electromagnetic torque are given
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