As I delve into the intricate field of inverter-fed Permanent Magnet Synchronous Machines (PMSMs), I find that electromagnetic vibration and noise (vibroacoustic) issues are paramount in modern high-performance applications, including aerospace, electric vehicles, and renewable energy systems. The presence of current harmonics introduced by inverters—especially those associated with different types of solar inverter systems—significantly exacerbates the electromagnetic force harmonics, leading to undesirable acoustic noise and structural vibrations. In this article, I present a comprehensive first-person review of the state-of-the-art modeling and computational methods for predicting and analyzing these phenomena. I systematically cover key aspects: multi-category current harmonic modeling, electromagnetic force computation under non-sinusoidal excitation, structural modal analysis, vibration response, and radiated noise. Throughout, I incorporate numerous tables and formulas to consolidate knowledge and highlight the critical role of various types of solar inverter in shaping the harmonic spectrum.
Before proceeding, I must emphasize that the accurate prediction of electromagnetic vibration and noise in PMSMs is a multi-physics challenge that couples electromagnetic fields, structural dynamics, and acoustics. Traditional finite element analysis (FEA) offers high fidelity but suffers from high computational cost, especially when considering the rich harmonic content produced by pulse-width modulation (PWM) inverters. To overcome this, researchers have developed a variety of analytical, semi-analytical, and hybrid methods. The following sections summarize these approaches while I repeatedly draw attention to the influence of different types of solar inverter on current harmonic characteristics.
Multi-Category Current Harmonic Modeling and Computation
Inverter-fed PMSMs experience current harmonics that can be classified into four main categories: (i) high-frequency sideband harmonics around the switching frequency and its multiples, (ii) low- to medium-frequency harmonics arising from back electromotive force (EMF) distortion or inverter nonlinearities (e.g., dead time), (iii) low-frequency sideband harmonics due to position sensor errors, and (iv) harmonics intentionally injected for control purposes such as harmonic injection for torque ripple reduction. The types of solar inverter used in photovoltaic applications often employ similar PWM strategies, leading to analogous harmonic patterns. In my analysis, I focus on the analytical and numerical methods for predicting these harmonics.
Low- and Medium-Frequency Current Harmonics
Low- and medium-frequency current harmonics (e.g., 5th, 7th, 11th, 13th orders) are primarily caused by non-ideal back EMF waveforms and inverter nonlinearities like dead time and device voltage drops. These harmonics are difficult to model analytically due to magnetic saturation and cross-coupling effects. In the reviewed literature, a common approach is to combine electromagnetic finite element models (FEM) with control system simulations to capture the full inverter-machine interaction. For instance, one study proposed a differential evolution algorithm to predict current harmonics by fitting the relationship between inverter output voltage harmonics and PMSM terminal voltage harmonics. The optimization minimizes the error between measured and calculated phase currents.
Another important source is the DC-link voltage ripple, which is particularly relevant in types of solar inverter where the input from solar panels can be unsteady. The ripple can be modeled by generating inverter switching functions in the time domain and performing convolution in the frequency domain, combined with a DC-link equivalent transfer function. This method accounts for different modulation strategies and motor control states.
High-Frequency Sideband Current Harmonics
High-frequency sideband current harmonics, located around the switching frequency \(f_c\) and its integer multiples, are a direct consequence of PWM. Using the double Fourier series decomposition, I can derive analytical expressions for these harmonics. For a symmetrical regular-sampled space-vector PWM (SVPWM) inverter, the sideband current harmonics at frequencies \(m f_c \pm n f_1\) (where \(m\) and \(n\) are integers) can be accurately predicted. The amplitude of these harmonics depends on the modulation index, DC-link voltage, stator impedance, and the motor’s operating point. In the literature, analytical models have been extended to account for magnetic saturation and cross-coupling inductance, especially in interior permanent magnet (IPM) machines. The resulting dq-axis current harmonics at the \(k\)-th sideband frequency \(\omega_k\) are given by:
$$
\begin{cases}
i_{d\_\omega_k} = \frac{L_q u_{d\_\omega_k} – \sigma_M \sqrt{L_d L_q} u_{q\_\omega_k}}{j\omega_k L_d L_q (1 – \sigma_M^2)} \\
i_{q\_\omega_k} = \frac{L_d u_{q\_\omega_k} – \sigma_M \sqrt{L_d L_q} u_{d\_\omega_k}}{j\omega_k L_d L_q (1 – \sigma_M^2)}
\end{cases}
$$
where \(L_d, L_q\) are d- and q-axis inductances, \(\sigma_M\) is the cross-coupling coefficient, and \(u_{d\_\omega_k}, u_{q\_\omega_k}\) are the corresponding voltage harmonics. I note that different types of solar inverter (e.g., single-phase vs. three-phase string inverters) may have different carrier frequencies and modulation schemes, leading to distinct sideband harmonic distributions.
Sampling delay effects, particularly under low carrier-frequency-ratio operation, are modeled using Lagrangian expansion and complex coordinate transformations. Hysteresis and eddy-current reaction in the magnetic core are captured via small-signal time-harmonic FEM, which I find improves the accuracy of sideband harmonic predictions. Rotor position errors from resolvers introduce additional sideband harmonics at \(f_c \pm n f_1 \pm P_r f_1\) and \(f_c \pm n f_1 \pm 2P_r f_1\), where \(P_r\) is the resolver pole-pair ratio.
| Harmonic Category | Main Causes | Analytical Methods | Numerical/Hybrid Methods | Comments on Solar Inverter Types |
|---|---|---|---|---|
| Low/Medium frequency (5th, 7th, etc.) | Back EMF distortion, dead time, voltage drop | Differential evolution, double Fourier series with dead time | FEM + control co-simulation | Single-phase solar inverters exhibit strong 3rd harmonic; three-phase string inverters have characteristic 5th/7th |
| High-frequency sideband (around \(f_c\)) | PWM switching | Double Fourier series (SVPWM, SPWM) | Time-harmonic FEM considering saturation | Different \(f_c\) choices in micro vs. central solar inverters |
| Sampling-delay induced | Digital control delay | Lagrangian expansion + complex coordinate | FEM with small-step transient | Low carrier ratio common in high-speed solar inverters |
| Rotor position error induced | Resolver inaccuracy | Analytical derivation with error harmonics | Finite element validation | Affects both grid-tied and standalone solar inverters |
Electromagnetic Force Modeling Under Non-Sinusoidal Excitation
The electromagnetic force acting on the stator core is the primary source of vibration and noise. I compute the radial and tangential force densities using Maxwell stress tensor (MST) formulation:
$$
P_{\text{rad}}(\theta,t) = \frac{1}{2\mu_0} \left[ b_{\text{rad}}^2(\theta,t) – b_{\text{tan}}^2(\theta,t) \right]
$$
$$
P_{\text{tan}}(\theta,t) = \frac{b_{\text{rad}}(\theta,t) b_{\text{tan}}(\theta,t)}{\mu_0}
$$
where \(b_{\text{rad}}, b_{\text{tan}}\) are radial and tangential air-gap flux densities, \(\mu_0\) is vacuum permeability. For axial flux PMSMs, the axial force density is approximately \(b_{\text{axi}}^2/(2\mu_0)\). The MST method is computationally efficient and allows clear identification of the harmonic sources of each spatial and temporal order. In contrast, the virtual work principle gives nodal forces via integration over element Jacobians, which is more accurate but requires detailed mesh data.
When considering high-frequency sideband harmonics from any types of solar inverter, the required number of time steps per electrical cycle becomes:
$$
N_{\text{cyc}} = S_{\text{cyc}} \frac{f_{\text{max}}}{f_1}
$$
where \(S_{\text{cyc}} \ge 2\) (sampling factor) and \(f_{\text{max}}\) is the highest frequency of interest. I recommend \(S_{\text{cyc}} = 3 \sim 5\) to limit error to \(\pm 10\%\).
Meshless Methods
Meshless methods (e.g., meshless finite difference) have been applied to PMSM field computation. They discretize the rotor with regular point clouds and solve for air-gap flux density and stress. This approach eliminates the need for mesh regeneration and can handle large deformations in structural coupling. In the reviewed study, the method shows higher accuracy and speed than FEM for steady-state and transient magnetic and stress fields, especially when combined with small-signal excitation for high-frequency harmonics.
Subdomain Model (Hybrid with Magnetic Circuit)
The subdomain (SD) method divides the motor geometry into simple regions (air gap, slots, magnets) and solves Laplace/Poisson equations analytically. To consider saturation, a hybrid subdomain–magnetic circuit model iterates the iron permeability. The stator slots are represented by equivalent current sheets. This method provides explicit expressions for air-gap flux density components, enabling fast computation of electromagnetic forces even with high-frequency current harmonics from any types of solar inverter. Recent extensions handle U-shaped interior permanent magnets and V-shaped rotors with flux barriers, using conformal mapping for shaped surfaces.
Equivalent Magnetic Network (EMN)
EMN models divide the magnetic circuit into many small reluctance elements (e.g., diamond, pentagonal shapes). The air-gap region is discretized, and radial/axial flux densities are obtained directly. For rotating motion, the connection between rotor and stator reluctance elements is updated at each step. To efficiently handle high-frequency harmonics, a discrete EMN model for surface-mounted PMSMs varies the magnetomotive force and reluctance values inside rotor elements without changing mesh connections. This reduces computational burden while preserving accuracy.
Magnetomotive Force–Permeance Method
The MMF–permeance method computes radial and tangential air-gap flux densities as products of MMF and permeance functions:
$$
\begin{cases}
b_{\text{rad}} = (f_{\text{rad\_mag}} + f_{\text{arm}})\lambda_{\text{rad}} + f_{\text{rad\_mag}}\lambda_{\text{tan}} \\
b_{\text{tan}} = f_{\text{tan\_mag}}\lambda_{\text{rad}} – (f_{\text{rad\_mag}} + f_{\text{arm}})\lambda_{\text{tan}}
\end{cases}
$$
where \(f_{\text{rad\_mag}}, f_{\text{tan\_mag}}\) are radial and tangential MMF from permanent magnets, \(f_{\text{arm}}\) is armature MMF, and \(\lambda_{\text{rad}}, \lambda_{\text{tan}}\) are air-gap radial/tangential permeances. This method is fast for parametric design. Complex permeance models use FEM to precompute the modulation functions from stator and rotor sources, including the effects of high-frequency harmonics.
| Method | Advantages | Disadvantages | Suitability for High-Freq Harmonics from Solar Inverters |
|---|---|---|---|
| Meshless (finite difference) | No mesh generation; handles large displacement; good for multi-physics coupling | Complex boundary treatment; requires dense points for fine motion steps | Moderate – dense points needed; can use small-signal linearization |
| Hybrid subdomain–magnetic circuit | Closed-form expressions; fast; accounts for saturation | Complex derivation for irregular geometries; many equations | Excellent – no mesh increase for high k; explicit harmonics |
| Equivalent magnetic network (discrete EMN) | Fast; direct force extraction; handles various topologies | Motion requires many reluctance elements for small steps; accuracy depends on element size | Good with motion simplification; extra elements needed |
| MMF–permeance (including complex permeance) | Very fast; clear physical insight; suitable for optimization | Relies on numerical precomputation of permeance; accuracy limited if saturation severe | Excellent – time-domain method works directly with harmonic currents |
Structural Modal Modeling Methods
Modal analysis of the PMSM structure is essential to avoid resonance with electromagnetic force harmonics. I focus on the stator system (core, windings, housing) and rotor system (shaft, magnets, bearings). The stator radial modes (0th order breathing, 2nd, 4th, etc.) are most critical. Material properties of laminated cores and windings must be homogenized.
Homogenized Material Properties
The stator core is a stack of electrical steel laminations and air gaps (due to insulation). Using Voigt–Reuss rules, the Young’s modulus in the \(x, y, z\) directions is:
$$
E_{x,y,z} = \frac{\phi_{\text{air}} E_{\text{air}}^{x,y,z}(q) + E_{\text{steel}}^{x,y,z} + \phi_{\text{steel}} E_{\text{steel}}^{x,y,z}(q) + E_{\text{air}}^{x,y,z}}{\phi_{\text{air}}(q) + E_{\text{steel}}^{x,y,z} + \phi_{\text{steel}}(q) + E_{\text{air}}^{x,y,z}}
$$
with
$$
q = \frac{\phi_{\text{air}} E_{\text{air}}^{x,y,z}}{2(1+\nu_{\text{air}})} + \frac{\phi_{\text{steel}} E_{\text{steel}}^{x,y,z}}{2(1+\nu_{\text{steel}})}
$$
where \(\phi_{\text{air}}, \phi_{\text{steel}}\) are volume fractions (stacking factor), and \(\nu\) are Poisson’s ratios. Shear moduli are computed similarly. Impregnated windings are treated as an orthotropic composite with stiffness and mass contributions. I note that the types of solar inverter housing materials (e.g., aluminum vs. steel) also affect overall modal frequencies, though the PMSM structure is the dominant factor.
Analytical Modal Frequency Prediction
The stator can be approximated as a finite-length cylindrical shell. For the \(m\)-th circumferential mode, the natural frequency \(f_m\) is:
$$
f_m =
\begin{cases}
\frac{1}{\pi D_c} \sqrt{\frac{E_c}{r_c k_i k_{\text{md}}}}, & m=0 \\[6pt]
\frac{f_0}{\sqrt{2}} \frac{\sqrt{1 + k^2 k_{\text{mrot}}/k_{\text{md}}}}{1}, & m=1 \\[6pt]
f_0 \frac{k_m (m^2 – 1)}{\sqrt{m^2 + 1}} \sqrt{k_a} F_m, & m \ge 2
\end{cases}
$$
where \(D_c\) is stator outer diameter, \(E_c\) is Young’s modulus, \(k_{\text{mrot}}, k_{\text{md}}\) are mass addition factors for rotation and displacement, and \(F_m\) includes inertia effects. More advanced analytical methods consider orthotropic properties and three-dimensional coupling with teeth. The beam element model (Euler–Bernoulli or Timoshenko) replaces the stator yoke and teeth with assembled beam elements, yielding accurate mode shapes and frequencies via the free-vibration equation:
$$
\det(-\omega_0^2 M + K) = 0
$$
| Method | Strengths | Weaknesses | Consideration for Inverter-Fed Harmonics |
|---|---|---|---|
| Simplified cylinder shell analytic | Fast preliminary estimate | Low accuracy; ignores orthotropy, teeth | Suitable for initial design; must verify with experiments |
| Orthotropic shell analytic | Accounts for laminated core and windings | Complex derivation; coefficients need calibration | Can be tuned using modal test data from actual motor |
| Beam element (Euler–Bernoulli / Timoshenko) | Reasonable accuracy; captures mode shapes; easy to parameterize | 2D elements ignore 3D coupling; Timoshenko accounts for shear | Better for predicting resonance with high-order spatial forces |
Electromagnetic Vibration Modeling Methods
Vibration response is governed by the multi-degree-of-freedom system equation in the frequency domain:
$$
\textbf{u}(j\omega) = \sum_{v=1}^{N} \frac{\boldsymbol{\phi}_v^T \boldsymbol{\phi}_v \textbf{F}(j\omega)}{\omega_v^2 + 2j\omega \eta_v \omega_v – \omega^2}
$$
where \(\omega_v, \boldsymbol{\phi}_v, \eta_v\) are the natural frequency, mass-normalized mode shape, and damping ratio of the \(v\)-th mode. Damping is often estimated empirically (e.g., \(\eta_v = 0.5(2.76\times 10^5 f_v + 0.062)\)), or extracted from step-response tests or half-power bandwidth method.
Direct Analytical Vibration Method
Using a simplified cylindrical shell model, the radial vibration equation is derived from equilibrium and is given by:
$$
\frac{\partial^6 u}{\partial \theta^2} + 2\frac{\partial^4 u}{\partial \theta^4} + \frac{\partial^2 u}{\partial \theta^2} – \frac{R^4}{EJ}\left( \frac{\partial p_{\text{rad}}}{\partial \theta} – p_{\text{tan}} \right) + \frac{\rho A R^4}{EJ} \frac{\partial^2}{\partial t^2}\left( \frac{\partial^2 u}{\partial \theta^2} – u \right) = 0
$$
This method is suitable only for simplistic designs and cannot account for complex housing or end caps.
Transfer Function Method (Modal Superposition)
The transfer function approach constructs a mapping from electromagnetic forces to vibration displacement using precomputed static deformation shapes for each spatial order. For a radial concentrated force of order \(v\), the vibration displacement at the housing surface is:
$$
Y = \sum_{\alpha=1}^{3} \sum_{i=0}^{i_{\max}} \sum_{v=0,\pm1}^{v_{\max}} F_{\alpha,i,v} \cdot |H_{\alpha,v}(\omega)| \cdot \phi_{\alpha,v} e^{j(v’\theta + i \omega_b t + \varphi_{\alpha,i,v} – \psi_{\alpha,v})}
$$
where \(\alpha=1,2,3\) represents radial, tangential, and torque directions, \(\omega_b\) is fundamental electric frequency, \(H\) is frequency response function, and \(\phi, \psi\) are phase and amplitude of static deformation. This method is highly efficient for repeated evaluations under different operating conditions, such as varying load or speed scenarios common in many types of solar inverter applications.
| Method | Design Phase Suitability | Consideration of Complex Structure | Computational Efficiency | Accuracy for High-Freq Harmonics |
|---|---|---|---|---|
| Direct analytic shell | Preliminary design | Low | Very high | Low |
| Transfer function (modal superposition) | Detailed design; parametric studies | High (via FEM preprocessing) | High (once TF built) | High |
Electromagnetic Noise Modeling Methods
Noise is computed from the vibration of the stator outer surface and end caps. Sound pressure level \(L_P\) and sound power level \(L_W\) are defined as:
$$
L_P = 20 \log_{10}\left( \frac{P_s}{P_{s,\text{ref}}} \right), \quad L_W = 10 \log_{10}\left( \frac{W_s}{W_{s,\text{ref}}} \right)
$$
with reference values \(2\times10^{-5}\,\text{Pa}\) and \(10^{-12}\,\text{W}\).
Analytical Noise Prediction
For a simplified infinite cylindrical shell model, the radiated sound pressure in the far field is:
$$
p(r,\theta,z) = -j v_0 \rho_0 c_0 \frac{k_0}{k_r} \frac{H_m^{(2)}(k_r a)}{dH_m^{(2)}(k_r a)/d(k_r a)} \cos(m\theta) H_m^{(2)}(k_r r) e^{-j k_z z}
$$
where \(v_0\) is radial velocity amplitude, \(\rho_0, c_0\) are air density and speed of sound, \(H_m^{(2)}\) is the Hankel function of the second kind. Finite-length correction uses spatial Fourier transform of axial vibration distribution. The sound power level can also be approximated by:
$$
L_W = 4 \sigma_{\text{rel}} \rho_0 c_0 \pi^3 f_{\text{exc}}^2 x^2 R_{\text{out}} L_{\text{stk}}
$$
with relative radiation efficiency \(\sigma_{\text{rel}} = k^2/(1+k^2)\) and \(k = 2\pi R_{\text{out}} f_{\text{exc}} / c_0\).
Numerical Methods (BEM and FEM)
Boundary element method (BEM) solves the Helmholtz equation on the surface only, using Green’s function:
$$
c(\mathbf{r}_0) p(\mathbf{r}_0) = \int_{\Gamma} \left( p(\mathbf{r}) \frac{\partial G(\mathbf{r},\mathbf{r}_0)}{\partial n} – \frac{\partial p(\mathbf{r})}{\partial n} G(\mathbf{r},\mathbf{r}_0) \right) d\Gamma
$$
with \(G(\mathbf{r},\mathbf{r}_0) = e^{ik|\mathbf{r}-\mathbf{r}_0|}/(4\pi|\mathbf{r}-\mathbf{r}_0|)\). BEM is preferred for external radiation problems because only surface mesh is needed, reducing dimensionality. However, for high-frequency harmonics from any types of solar inverter (e.g., switching frequency up to 20 kHz), the required mesh size must be finer than one-sixth of the acoustic wavelength, which increases computational resources. Finite element method (FEM) for acoustics requires volumetric meshing and is less efficient for unbounded domains. I note that BEM with fast multipole acceleration can mitigate high-frequency numerical instability.
| Method | Complexity | Accuracy | Computational Cost | High-Frequency Handling | Best Use Stage |
|---|---|---|---|---|---|
| Analytic (cylindrical shell) | Low | Moderate (low for complex structures) | Very low | Moderate | Initial design |
| FEM acoustic | High (volume mesh) | High | Very high | Poor (mesh size requirement) | Detailed validation |
| BEM | Moderate (surface mesh) | High | Moderate to high | Good with fast multipole | Detailed analysis and optimization |
Conclusions and Future Outlook
In this review, I have systematically presented the computational methods for electromagnetic vibration and noise in inverter-fed PMSMs, with a specific emphasis on the influence of current harmonics. The types of solar inverter—whether single-phase microinverters, three-phase string inverters, or central inverters for utility-scale plants—each introduce distinct harmonic patterns that must be accurately modeled. I have shown that multiple modeling approaches exist for current harmonics, electromagnetic forces, structural dynamics, and acoustics, each with trade-offs between accuracy and speed. Hybrid methods that combine analytical solutions (e.g., subdomain or MMF-permeance) with numerical calibration (e.g., FEM for saturation or modal parameters) offer the best balance for engineering practice.
Looking ahead, I identify several promising directions:
- Fast multi-operating-point force computation – developing reduced-order models (e.g., proper orthogonal decomposition) to handle the many speed-torque combinations encountered in real-world inverter-fed drives, including those from different types of solar inverter.
- Multi-physics coupling – incorporating thermal, stress, and eddy-current effects into electromagnetic force models for greater fidelity, particularly when high-frequency harmonics cause additional copper and iron losses.
- Advanced structural models – using beam elements or shell models with orthotropic properties to capture complex housing and winding effects without full 3D FEM.
- Digital twin technology – integrating real-time sensor data (current, vibration, temperature) with physics-based models to enable continuous prediction and optimization of vibroacoustic performance under varying solar inverter operating conditions.
- Combined BEM–analytical methods – employing isogeometric BEM or meshless BEM to overcome high-frequency numerical instability and reduce preprocessing effort.
I believe that the continued development of these techniques will significantly enhance our ability to design quieter and more reliable PMSMs for aerospace, electric vehicle, and renewable energy applications, where the types of solar inverter play a central role in the harmonic excitation spectrum.