Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics

Bharat Khushalani

doi:doi:10.11648/j.epes.20251403.12

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Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics

Bharat Khushalani^*

Published in American Journal of Electrical Power and Energy Systems (Volume 14, Issue 3)

Received: 12 May 2025 Accepted: 27 May 2025 Published: 13 June 2025

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Abstract

This paper presents a comprehensive symbolic and numerical analysis of the dynamic behavior of Line-Start Permanent Magnet Synchronous Motors (LSPMSMs). Unlike conventional modeling approaches that rely heavily on numerical simulation, we develop a symbolic model that captures the electrical and mechanical dynamics of the motor with enhanced analytical clarity. Using the dq0 transformation, the motor differential equations are formulated in the synchronous reference frame. These equations are then linearized around a steady-state operating point to derive a small-signal model, facilitating deeper insights into local stability and transient response characteristics. A significant contribution of this work lies in the derivation of the Jacobian matrix and its eigenvalue analysis, which allows for direct observation of system stability under different operating conditions. By symbolically computing the Jacobian, we avoid numerical approximation errors and preserve parameter dependencies, making the model highly adaptable for control design and optimization studies. We further demonstrate how the linearized model can predict small perturbations in system behavior, offering a practical tool for early-stage control system development. Simulation results validate the accuracy of the symbolic and linearized models by comparing them against the nonlinear system response under typical startup and load conditions. The results show that the linearized system closely approximates the behavior of the full nonlinear model within a defined operating region, confirming the reliability of the approach.

Published in	American Journal of Electrical Power and Energy Systems (Volume 14, Issue 3)
DOI	10.11648/j.epes.20251403.12
Page(s)	64-71
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

LSPMSM, Small Signal Model, Linearization, Local Stability

References

[1]	A. H. Isfahani and S. Vaez-Zadeh, “Line Start Permanent Magnet Synchronous Motors: Challenges and Opportunities,” Energy, vol. 34, no. 11, pp. 1755–1763, 2009. https://doi.org/10.1016/j.energy.2009.04.022
[2]	Y. Yang, B. Yan, X. Wang, “Quantitative assessment on synchronisation capability of a line-start permanent magnet synchronous motor with hybrid rotor,” IET Electric Power Applications, vol. 18, no. 1, pp. 141–152, 2024. https://doi.org/10.1049/elp2.12373
[3]	P. Gnacinski, M. Peplinski, A. Muc, and D. Hallmann, “Line-Start Permanent Magnet Synchronous Motor Supplied with Voltage Containing Negative-Sequence Subharmonics,” Energies, vol. 17, no. 1, p. 91, 2024. https://doi.org/10.3390/en17010091
[4]	A. Chama, A. J. Sorgdrager, and R.-J. Wang, “Analytical Synchronization Analysis of Line-Start Permanent Magnet Synchronous Motors,” Progress In Electromagnetics Research M, vol. 48, pp. 183–193, 2016. https://doi.org/10.2528/PIERM16050311
[5]	M. Gwozdziewicz and K. Jankowska, “Analysis of a new concept of Line Start Permanent Magnet Synchronous Motor,” in Przeglad Elektrotechniczny, vol. 98, pp. 168– 173, 2022. https://doi.org/10.15199/48.2022.08.31
[6]	S. Tasoujian, J. Lee, K. Grigoriadis, and M. Franchek, “Robust Linear Parameter Varying Output Feedback Control of Permanent Magnet Synchronous Motors,” Systems Science and Control Engineering, vol. 9, pp. 612–622, 2021. https://doi.org/10.1080/21642583.2021.1974600
[7]	X. Liu, Y. Pan, L. Wang, X. Xu, Y. Zhu and Z. Li, “Model Predictive Current Control of PMSM Based on Parameter Identification and Dead Time Compensation,” Progress in Electromagnetics Research C, vol. 120, pp. 253–263, 2022. https://doi.org/10.2528/PIERC22040103
[8]	D. Li, G. Feng et al., “Irreversible Demagnetization of a Large Capacity Line-Start Permanent Magnet Synchronous Motors considering Influence of Permanent Magnet Temperature,” International Transactions on Electrical Energy Systems, 2023, e6798493. https://doi.org/10.1155/2023/6798493
[9]	W. Szelag, C. Jedryczka and M. Baranski, “A New Method of Reducing the Inrush Current and Improving the Starting Performance of a Line-Start Permanent- Magnet Synchronous Motor,” Energies, 17(5), 1040. https://doi.org/10.3390/en17051040
[10]	T. Anh, T. Bien et al., “Analysis of Permanent Magnet Demagnetization during the Starting Process of a Line-start Permanent Magnet Synchronous Motor,” Engineering, Technology and Applied Science Research, 14(6), 2024. https://doi.org/10.48084/etasr.8576
[11]	J. Barta, M. Toman et al., “Line-Start Permanent Magnet Machines for Low-Power Applications,” International ConferenceonElectricalMachines(ICEM),Torino, Italy, 2024, pp. 1-7, https://doi.org/10.1109/ICEM60801.2024.10700149
[12]	K. Jankowska, M. Gwozdziewicz, M. Dybkowski, “Application of Line-Start Permanent-Magnet Synchronous Motor in Converter Drive System with Increased Safety Level,” Electronics, 14(9), 1787, 2025. https://doi.org/10.3390/electronics14091787

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APA Style

Khushalani, B. (2025). Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics. American Journal of Electrical Power and Energy Systems, 14(3), 64-71. https://doi.org/10.11648/j.epes.20251403.12

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ACS Style

Khushalani, B. Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics. Am. J. Electr. Power Energy Syst. 2025, 14(3), 64-71. doi: 10.11648/j.epes.20251403.12

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AMA Style

Khushalani B. Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics. Am J Electr Power Energy Syst. 2025;14(3):64-71. doi: 10.11648/j.epes.20251403.12

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@article{10.11648/j.epes.20251403.12,
  author = {Bharat Khushalani},
  title = {Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics
},
  journal = {American Journal of Electrical Power and Energy Systems},
  volume = {14},
  number = {3},
  pages = {64-71},
  doi = {10.11648/j.epes.20251403.12},
  url = {https://doi.org/10.11648/j.epes.20251403.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.epes.20251403.12},
  abstract = {This paper presents a comprehensive symbolic and numerical analysis of the dynamic behavior of Line-Start Permanent Magnet Synchronous Motors (LSPMSMs). Unlike conventional modeling approaches that rely heavily on numerical simulation, we develop a symbolic model that captures the electrical and mechanical dynamics of the motor with enhanced analytical clarity. Using the dq0 transformation, the motor differential equations are formulated in the synchronous reference frame. These equations are then linearized around a steady-state operating point to derive a small-signal model, facilitating deeper insights into local stability and transient response characteristics. A significant contribution of this work lies in the derivation of the Jacobian matrix and its eigenvalue analysis, which allows for direct observation of system stability under different operating conditions. By symbolically computing the Jacobian, we avoid numerical approximation errors and preserve parameter dependencies, making the model highly adaptable for control design and optimization studies. We further demonstrate how the linearized model can predict small perturbations in system behavior, offering a practical tool for early-stage control system development. Simulation results validate the accuracy of the symbolic and linearized models by comparing them against the nonlinear system response under typical startup and load conditions. The results show that the linearized system closely approximates the behavior of the full nonlinear model within a defined operating region, confirming the reliability of the approach.
},
 year = {2025}
}

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TY  - JOUR
T1  - Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics

AU  - Bharat Khushalani
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.epes.20251403.12
DO  - 10.11648/j.epes.20251403.12
T2  - American Journal of Electrical Power and Energy Systems
JF  - American Journal of Electrical Power and Energy Systems
JO  - American Journal of Electrical Power and Energy Systems
SP  - 64
EP  - 71
PB  - Science Publishing Group
SN  - 2326-9200
UR  - https://doi.org/10.11648/j.epes.20251403.12
AB  - This paper presents a comprehensive symbolic and numerical analysis of the dynamic behavior of Line-Start Permanent Magnet Synchronous Motors (LSPMSMs). Unlike conventional modeling approaches that rely heavily on numerical simulation, we develop a symbolic model that captures the electrical and mechanical dynamics of the motor with enhanced analytical clarity. Using the dq0 transformation, the motor differential equations are formulated in the synchronous reference frame. These equations are then linearized around a steady-state operating point to derive a small-signal model, facilitating deeper insights into local stability and transient response characteristics. A significant contribution of this work lies in the derivation of the Jacobian matrix and its eigenvalue analysis, which allows for direct observation of system stability under different operating conditions. By symbolically computing the Jacobian, we avoid numerical approximation errors and preserve parameter dependencies, making the model highly adaptable for control design and optimization studies. We further demonstrate how the linearized model can predict small perturbations in system behavior, offering a practical tool for early-stage control system development. Simulation results validate the accuracy of the symbolic and linearized models by comparing them against the nonlinear system response under typical startup and load conditions. The results show that the linearized system closely approximates the behavior of the full nonlinear model within a defined operating region, confirming the reliability of the approach.

VL  - 14
IS  - 3
ER  -

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Author Information

Bharat Khushalani

Department of Artificial Intelligence, Shri Vishnu Engineering College for Women, Bhimavaram, India

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APA Style

Khushalani, B. (2025). Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics. American Journal of Electrical Power and Energy Systems, 14(3), 64-71. https://doi.org/10.11648/j.epes.20251403.12

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ACS Style

Khushalani, B. Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics. Am. J. Electr. Power Energy Syst. 2025, 14(3), 64-71. doi: 10.11648/j.epes.20251403.12

Copy | Download

AMA Style

Khushalani B. Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics. Am J Electr Power Energy Syst. 2025;14(3):64-71. doi: 10.11648/j.epes.20251403.12

Copy | Download

@article{10.11648/j.epes.20251403.12,
  author = {Bharat Khushalani},
  title = {Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics
},
  journal = {American Journal of Electrical Power and Energy Systems},
  volume = {14},
  number = {3},
  pages = {64-71},
  doi = {10.11648/j.epes.20251403.12},
  url = {https://doi.org/10.11648/j.epes.20251403.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.epes.20251403.12},
  abstract = {This paper presents a comprehensive symbolic and numerical analysis of the dynamic behavior of Line-Start Permanent Magnet Synchronous Motors (LSPMSMs). Unlike conventional modeling approaches that rely heavily on numerical simulation, we develop a symbolic model that captures the electrical and mechanical dynamics of the motor with enhanced analytical clarity. Using the dq0 transformation, the motor differential equations are formulated in the synchronous reference frame. These equations are then linearized around a steady-state operating point to derive a small-signal model, facilitating deeper insights into local stability and transient response characteristics. A significant contribution of this work lies in the derivation of the Jacobian matrix and its eigenvalue analysis, which allows for direct observation of system stability under different operating conditions. By symbolically computing the Jacobian, we avoid numerical approximation errors and preserve parameter dependencies, making the model highly adaptable for control design and optimization studies. We further demonstrate how the linearized model can predict small perturbations in system behavior, offering a practical tool for early-stage control system development. Simulation results validate the accuracy of the symbolic and linearized models by comparing them against the nonlinear system response under typical startup and load conditions. The results show that the linearized system closely approximates the behavior of the full nonlinear model within a defined operating region, confirming the reliability of the approach.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics

AU  - Bharat Khushalani
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.epes.20251403.12
DO  - 10.11648/j.epes.20251403.12
T2  - American Journal of Electrical Power and Energy Systems
JF  - American Journal of Electrical Power and Energy Systems
JO  - American Journal of Electrical Power and Energy Systems
SP  - 64
EP  - 71
PB  - Science Publishing Group
SN  - 2326-9200
UR  - https://doi.org/10.11648/j.epes.20251403.12
AB  - This paper presents a comprehensive symbolic and numerical analysis of the dynamic behavior of Line-Start Permanent Magnet Synchronous Motors (LSPMSMs). Unlike conventional modeling approaches that rely heavily on numerical simulation, we develop a symbolic model that captures the electrical and mechanical dynamics of the motor with enhanced analytical clarity. Using the dq0 transformation, the motor differential equations are formulated in the synchronous reference frame. These equations are then linearized around a steady-state operating point to derive a small-signal model, facilitating deeper insights into local stability and transient response characteristics. A significant contribution of this work lies in the derivation of the Jacobian matrix and its eigenvalue analysis, which allows for direct observation of system stability under different operating conditions. By symbolically computing the Jacobian, we avoid numerical approximation errors and preserve parameter dependencies, making the model highly adaptable for control design and optimization studies. We further demonstrate how the linearized model can predict small perturbations in system behavior, offering a practical tool for early-stage control system development. Simulation results validate the accuracy of the symbolic and linearized models by comparing them against the nonlinear system response under typical startup and load conditions. The results show that the linearized system closely approximates the behavior of the full nonlinear model within a defined operating region, confirming the reliability of the approach.

VL  - 14
IS  - 3
ER  -

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