PMSM Modeling Strategies

• PMSM models are comprehensive representations of coupled electromagnetic, thermal, and mechanical behaviors, essential for control and design in applications like automotive and wind energy.
• Analytical and state-space formulations enable precise system-level analysis, while energy-based models capture nonlinear effects such as saturation and spatial harmonics.
• Geometric, data-driven, and thermal models facilitate robust fault diagnosis and real-time parameter adaptation through efficient simulation and optimization techniques.

A permanent magnet synchronous machine (PMSM) model mathematically encapsulates the coupled electromagnetic, electrical, thermal, and structural behavior of PMSMs by describing dynamic and steady-state relationships among stator/rotor currents, magnetic fields, voltages, torque, losses, and—in advanced variants—effects such as saturation, spatial harmonics, temperature, and parameter imbalance. Precise modeling underpins analysis, control, design optimization, and fault diagnosis, and is essential to contemporary electric drives, wind energy systems, and automotive applications.

1. Fundamental Electromagnetic Formulations

The governing equation for PMSM modeling in the electromagnetic domain is typically derived from the magnetostatic Maxwell–Ampère law, augmented to include the magnetization of the permanent magnets. In three-dimensional form:

$$\nabla \times (\nu\,\nabla \times \mathbf{A}) = \mathbf{J}_s + \mathbf{J}_m$$

where $\mathbf{A}$ is the magnetic vector potential, $\nu$ is the local reluctivity, $\mathbf{J}_s$ encodes source (coil) currents, and $\mathbf{J}_m = \nabla \times \mathbf{M}$ characterizes permanent magnet excitation with $\mathbf{M}$ the magnetization. Boundary conditions are application-specific, frequently homogeneous Dirichlet ($\mathbf{A}\times n=0$ on $\partial\mathcal{D}$). For design and field computations, the machine’s two-dimensional cross-section is usually considered (with $A_z$ the principal degree of freedom), reducing the PDE to a scalar Poisson problem:

$$-\nabla \cdot (\nu\,\nabla u) = J_{src,z} + (\nabla \times \mathbf{M}) \cdot e_z$$

This forms the basis for finite element, isogeometric, or surrogate modeling strategies (Merkel et al., 2019, Bontinck et al., 2017, Backmeyer et al., 25 Jul 2025).

2. Analytical and State-Space Models

The classical analytical PMSM model in the $abc$ or $dq$ reference frame enables system-level studies, control syntheses, and observer design. For a three-phase, star-connected PMSM, the $abc$-frame continuous-time model is:

$$\mathbf{v}_{abc}(t) = R_s\,\mathbf{i}_{abc}(t) + \frac{d}{dt}\, \psi_{s,abc}(t)$$

$$\psi_{s,abc} = \mathbf{L}_s\,\mathbf{i}_{abc} + \psi_{pm,abc}(t)$$

where $R_s$ is the phase resistance, $\mathbf{L}_s$ is the inductance matrix (isotropic or anisotropic), and $\psi_{pm,abc}$ the spatially rotating PM flux (Hackl et al., 2018). Via Park transformation, the equations are recast in the $dq$ frame:

$$\begin{aligned} v_d &= R_s i_d + L_d\frac{di_d}{dt} - \omega_e L_q i_q \ v_q &= R_s i_q + L_q\frac{di_q}{dt} + \omega_e (L_d i_d + \psi_{pm}) \ \tau_e &= \frac{3}{2} n_p [ \psi_{pm} i_q + (L_d-L_q) i_d i_q ] \end{aligned}$$

where $L_d$, $L_q$ are the direct and quadrature axis inductances, $\psi_{pm}$ is the PM flux (d-axis aligned), and $n_p$ the pole pair count. This model forms the canonical core for inner current control, field-oriented control, and model-based estimation.

Reduced-order state-space models are obtained via time-scale separation, neglecting fast electrical dynamics under adequate time-constant disparity (i.e., $\tau_{elec} \ll \tau_{mech}$). This is essential in system-level simulations and high-level controllers (Hackl et al., 2018).

3. Physics-Based Nonlinear Extensions: Saturation, Cross-Saturation, and Harmonics

Accurate PMSM modeling for high torque, field-weakening, or low-speed sensorless control necessitates inclusion of nonlinearities such as saturation and cross-saturation. The energy-based methodology expresses flux-current relations as gradients of an augmented magnetic energy function:

$$\mathcal{H}(\phi_d, \phi_q) = \frac{1}{2L_d} \phi_d^2 + \frac{1}{2L_q} \phi_q^2 + \alpha_{3,0}\phi_d^3 + \alpha_{1,2}\phi_d\phi_q^2 + \alpha_{4,0}\phi_d^4 + \alpha_{2,2}\phi_d^2\phi_q^2 + \alpha_{0,4}\phi_q^4$$

with corresponding currents given by:

$$i_d = \frac{\partial \mathcal{H}}{\partial \phi_d}, \quad i_q = \frac{\partial \mathcal{H}}{\partial \phi_q}$$

This formulation enforces symmetry, reciprocity, and admits cross-saturation and higher-order nonlinearities in the flux-current mapping. The resulting parametric model (seven parameters: $L_d$, $L_q$, five $\alpha$ coefficients) is efficiently identified through HF voltage injection and measurement of current ripple amplitudes, enabling real-time capable nonlinear flux estimators (Jebai et al., 2011, Jebai et al., 2012, Jebai et al., 2014).

Harmonic effects due to slotting and non-sinusoidal windings are introduced by explicit angular dependence in the energy function, leading to periodic torque and current ripples; e.g., a periodic $6$-th order dependency in $H_{dq}(\theta)$. These phenomena are experimentally validated in both surface- and interior-mounted PMSMs (Jebai et al., 2014).

Parameter imbalance (resistance, inductance, magnet flux) is incorporated analytically by perturbing each phase's parameter and transforming via Clarke and Park. This introduces $2\omega_e$ voltage and current harmonics, directly linked to torque pulsations and amenable to both analytical prediction and real-time compensation schemes (Pramod, 2023).

4. Geometric and Surrogate Field Models: Isogeometric, Finite Element, and Data-Driven Approaches

Contemporary PMSM simulation and optimization make extensive use of geometric models that capture the exact CAD geometry, permit shape optimization, and manage arbitrarily complex boundary/interface conditions. Isogeometric Analysis (IGA) employs Non-Uniform Rational B-Splines (NURBS) as the shape and solution basis, enabling single-mesh or multipatch representations:

$x(\xi,\eta) = \sum_{i} N_i(\xi,\eta)P_i$

Discrete magnetostatic equations are solved patchwise; stator and rotor meshes are coupled either by Dirichlet-to-Neumann or harmonic mortar-type methods, sidestepping remeshing under rotation (Merkel et al., 2019, Bontinck et al., 2017, Backmeyer et al., 25 Jul 2025). The use of parametric control points for geometry allows for efficient and robust shape optimization, e.g., minimizing electromotive force (EMF) total harmonic distortion (THD) under manufacturability and minimum air-gap constraints.

Parametric or surrogate models can further accelerate field prediction and optimization by combining IGA with Proper Orthogonal Decomposition (POD) and machine learning. POD reduces the solution space to a low-dimensional manifold; deep neural networks map parameter tuples to POD coefficients, reconstructing full fields in under 1 ms with errors $\sim$1% and torque error $\sim$1%—facilitating real-time optimization and digital-twin implementations (Backmeyer et al., 25 Jul 2025).

5. Thermal and Temperature Effects

Accurate PMSM modeling must account for temperature-dependent phenomena, especially the irreversible demagnetization and flux reduction of rotor permanent magnets and copper/iron losses. Two modeling paradigms dominate:

• Lumped Parameter Thermal Networks (LPTN): Representing thermal masses and resistances as a network of RC elements, integrating nodal energy balances. Real-time capable, typically with $\sim$50 parameters, and fit empirically or via CFD (Kirchgässner et al., 2020).
• Data-driven regression models: Learning the mapping from electrical, environmental, and operating-state features to PM temperature using regression, ensemble, or deep learning methods. Empirical studies show feed-forward neural networks and OLS regressors achieve RMSE $<2^\circ\text{C}$, matching LPTN accuracy with low parameter counts when trained on large datasets (Kirchgässner et al., 2020).

Recently, nonlinear magnetics models have explicitly separated PM demagnetization (modeled as a temperature-dependent current source, $I_{pm}(T_r)$) from iron saturation (embedding the saturation in the nonlinear magnetization mapping), decoupling temperature and field-strength effects, yielding 4–5% torque error under deep saturation versus $\sim$18% for conventional flux-linkage models (Srinivasan et al., 21 Oct 2024).

6. Advanced and Fault Modeling: Online Parameter Adaptation, Faults, and Control

Robust PMSM models for advanced control tasks (sensorless control, model-predictive current/torque control, diagnostics) increasingly deploy on-the-fly parameter identification, harmonic compensation, and explicit fault modeling. Examples include:

• Recursive Least Squares (RLS) parameter adaptation embedded in finite-control-set model predictive current controllers, dynamically identifying electrical parameters, dead-time compensation, and harmonic regressors for inverter-induced and winding harmonics. Experimental evidence shows a drop in steady-state current residuals from more than $1.45\,$A (LUT) to $0.05\,$A (RLS) at $2000\,$rpm (Brosch et al., 2019).
• Discrete-time fault models for interturn short circuits (ISC) in interior PMSMs, accommodating universal winding architectures and employing matrix exponential discretization for numerically stable simulation and model-based diagnostics. These models capture the interaction of ISC currents with main stator windings, connection resistance, and spatially-harmonic flux linkages (Zezula et al., 16 Apr 2025).
• Online detection and harmonic feedforward-compensation of parameter imbalance, leveraging analytic $2\omega_e$ signatures in the $dq$ voltages; online estimators extract and cancel these via bandpass filtering and feedforward subtraction, suppressing current and torque pulsations by up to 90% in practice (Pramod, 2023).
• Adaptive estimation of PM excitation for decoupling thermal and magnetic saturation behavior, realized by Lyapunov-stable adaptation laws on the PM current-source state to obviate explicit temperature sensing (Srinivasan et al., 21 Oct 2024).

7. Applications, Numerical Performance, and Model Selection Criteria

PMSM models are routinely benchmarked for fidelity, computational efficiency, and suitability for various tasks:

Modeling Approach	Typical Error	DoF/Param Count	Notable Application
$dq$-linear	$\mathcal{O}(2\%)$	6–10	Control, observer, analysis (Hackl et al., 2018)
Energy-based w/ saturation	$<2\%$ (static/step)	7–10	Sensorless control, field-weakening (Jebai et al., 2011)
Isogeometric/FEA	$1$–$2$% (EMF/THD)	$10^3$–$10^5$	Field computation, design (Bontinck et al., 2017, Merkel et al., 2019)
Data-driven surrogate (POD-DNN)	$1$–$1.5\%$ (field/torque)	$\sim$100	Real-time optimization (Backmeyer et al., 25 Jul 2025)
Thermal LPTN	$2$–$2.5^\circ$C	$40$–$50$	Onboard temperature estimation (Kirchgässner et al., 2020)
Adaptive harmonic compensation	$2$% ($dq$ voltage)	$8$–$12$	Online imbalance cancellation (Pramod, 2023)

Model choice is determined by the required fidelity, computational constraints, nature of task (control, optimization, FEA, diagnostics), and the importance of specific effects: nonlinearities (saturation, cross-saturation), parameter variations, harmonics, or faults.

---

A comprehensive PMSM modeling strategy integrates electromagnetic, electrical, and (when relevant) thermal/structural domains, employing the appropriate analytical rigor and numerical sophistication for the intended task, while supporting efficient simulation, analysis, and optimization across operating regimes and under real-world parameter deviations and faults (Merkel et al., 2019, Backmeyer et al., 25 Jul 2025, Srinivasan et al., 21 Oct 2024, Jebai et al., 2011, Pramod, 2023, Brosch et al., 2019, Zezula et al., 16 Apr 2025).