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Euler–Lagrange method for low-frequency metamaterials

Zhi Gong (Research Center for Intelligent Chips and Devices, Zhejiang Lab, Hangzhou, China)
Shiyou Yang (College of Electrical Engineering, Zhejiang University, Hangzhou, China)
Chongxi Guan (Jiazhou Power Supply Center of Leshan Power Supply Company of State Grid, Leshan, China)

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering

ISSN: 0332-1649

Article publication date: 20 June 2022

Issue publication date: 12 January 2023




The purpose of this paper is to develop a modeling method for the analysis of low-frequency metamaterials (MTMs) and their near-field applications.


The Euler–Lagrange method is introduced. An MTM is modeled as a multi-degree-of-freedom system without homogenization. The properties and the responses of the MTM in a near-field device are readily and rigorously studied through the motion equation derived from the Lagrange equations. The resonance frequencies and the corresponding resonance modes are solved from the characteristic equation.


The numerical results of the proposed method show good agreement with the experimental ones. A measurement of MTM-core coil resistance and inductance shows high accuracy of the proposed method.


The proposed Euler–Lagrange method provides a new study perspective and enables more flexible, rigorous and straightforward analysis of low-frequency MTMs in near-field applications. Consequently, the presented work greatly facilitates further explorations and studies on various novel MTM-based low-frequency near-field devices and systems.



The authors acknowledge the support from the National Natural Science Foundation of China (NNSFC) under Grant No. 52077192, the National Key Research and Development Project under Grant 2020AAA0109001, and the Zhejiang Province Key R & D programs under Grant 2021C05004.


Gong, Z., Yang, S. and Guan, C. (2023), "Euler–Lagrange method for low-frequency metamaterials", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 42 No. 1, pp. 121-131.



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