# Low Frequency Wireless Power Transfer Using Magnetically Coupled Mechanical Resonators

Séverine Gauthier, Adrien Morel, Adrien Badel

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Séverine Gauthier, Adrien Morel, Adrien Badel. Low Frequency Wireless Power Transfer Using Magnetically Coupled Mechanical Resonators. Journées Nationales sur la Récupération et le Stockage de l'Energie, Jun 2025, Bourget-Du-Lac, France. hal-05479416

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Submitted on 27 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Low Frequency Wireless Power Transfer Using
# Magnetically Coupled Mechanical Resonators

Séverine GAUTHIER¹,*, Adrien MOREL¹ and Adrien BADEL¹

¹ SYMME, Université Savoie Mont-Blanc, F-74000, Annecy, France

Corresponding author e-mail: severine.gauthier@univ-smb.fr

**Abstract**—This paper presents an electrodynamic system for low-frequency wireless power transmission, in which both the transmitter and the receiver are mechanically resonant beams coupled by magnets. The transmitted power of the current experimental system is around 0.4mW for a transmission distance of 5cm and a frequency of 36.1Hz.

**Keywords**—WPT, electrodynamic, low frequency, resonator

## I. INTRODUCTION

Wireless Power Transfer (WPT) has gained significant attention in recent years. Different technologies can be used to transfer power from a transmitter to a receiver. For example, inductive coupling WPT is often chosen for charging electric vehicles [1] or smartphones. This method requires a high-frequency magnetic field which limits power transfer through conductive media. To overcome this limitation, electrodynamic WPT systems have been proposed as a low-frequency (<1 kHz) alternative to inductive wireless power transfer [2]. Electrod dynamic WPT systems are typically based on a transmitter coil that generates a low frequency magnetic field, and a magneto-mechanical receiver that converts this magnetic field into mechanical motion. The mechanical energy in the receiver is then converted to electrical energy, for example by piezoelectric materials.

However, EWPT presents several limitations. The weak coupling between the transmitter and receiver reduces the energy transfer efficiency (typically in the range of 0.1–1%) and makes information transmission difficult. Additionally, the energy transfer in conventional systems is unidirectional

The system presented here consists of two identical electromechanical resonators, allowing the roles of transmitter and receiver to be interchanged. Symmetrizing the wireless power transfer system enhances both coupling and efficiency, providing a high-efficiency, low-frequency alternative to conventional inductive wireless power transfer. In the first section, we describe the system as well as its model, followed by a second section detailing the parameter identification process. Finally, we present some initial experimental results that validate the proposed approach.

## II. SYSTEM

### A. Principle

The proposed wireless power transfer system is shown in Fig. 1.a. Each unit is a resonator consisting of a beam (modeled as a spring-damper system) with a bonded piezoelectric patch (PZT-J) and a magnetic mass. Each mass consists of two 1 cm³ NdFeB magnets. The model of this system is shown in Fig. 1.b. A voltage applied to the first piezoelectric element causes its mass to move, which drives the second beam via magnetic interaction. The resulting motion is converted into electricity by the second piezoelectric

a)

<figure><img src="image_2.png" /></figure>

b)

<figure><img src="image_3.png" /><figcaption>Fig. 1 System a) experimental setup and b) corresponding model</figcaption></figure>

element. The natural frequency of each resonator are 35 Hz and 36 Hz respectively.

### B. Equations of motion

If we consider each magnet as a point dipole, the magnetic coupling force can be expressed as in (1), where $\mu_0$ is the vacuum permeability, $m_i$ is the magnetic moment and $d$ is the distance between the two magnets. The equation can be linearized to give the term $K_{mag}$, which represents the equivalent stiffness resulting from the magnetic interaction between the magnets.

$$F_{mag} = \frac{3*\mu_0*m_1*m_2}{2*\pi*(d-x_1+x_2)^4} \approx K_{mag}(x_1 - x_2) \quad (1)$$

Using Newton's laws, the system of equations describing the motion can be obtained (2). It consists of two coupled mechanical equations and two electrical equations.

$$ \left\{ \begin{array}{l} M_1 \ddot{x}_1 + D_1 \dot{x}_1 + K_1 x_1 - K x_2 - \alpha_1 V_1 - F_{mag} = 0 \\ M_2 \ddot{x}_2 + D_2 \dot{x}_2 + K_2 x_2 - K x_1 + \alpha_2 V_2 + F_{mag} = 0 \\ I_1 = \alpha_1 \dot{x}_1 + C p_1 \dot{V}_1 \\ \frac{V_2}{R} = \alpha_2 \dot{x}_2 - C p_2 \dot{V}_2 \end{array} \right. \quad (2) $$

Where $V_i$ is the voltage associated with the $i^{th}$ beam, $D_i$ is the damping coefficient, $K_i$ is the stiffness, $M_i$ is the mass, $\alpha_i$ is the piezoelectric coefficient, $Cp_i$ is the capacitance of the piezoelectric material, $I_i$ is the electric current and $x_i$ is the displacement of the $i^{th}$ beam around its equilibrium position.
### III. PARAMETER IDENTIFICATION

In order to identify the parameters of the WPT system shown in Fig.1.a, a method was established using an impedance analyzer.

#### A. Experimental method

Impedance analysis to identify the mechanical and electrical parameters of a piezoelectric resonator is a well-known method [3]. In this case, we apply the same procedure to the two coupled beams, measuring the impedance of one beam and short-circuiting the other one to simplify the impedance expression. The impedance expression in frequency domain can be obtained from (2):

$$ \frac{I_1}{V_1} = \frac{j\omega\alpha_1^2(-\omega^2 M_2 + j\omega D_2 + K_2 + K_{mag})}{(-\omega^2 M_1 + j\omega D_1 + K_1 + K_{mag})(-\omega^2 M_2 + j\omega D_2 + K_2 + K_{mag}) - K_{mag}^2} + \frac{j\omega C_{p1}(-\omega^2 M_1 + j\omega D_1 + K_1 + K_{mag})(-\omega^2 M_2 + j\omega D_2 + K_2 + K_{mag}) - K_{mag}^2}{(-\omega^2 M_1 + j\omega D_1 + K_1 + K_{mag})(-\omega^2 M_2 + j\omega D_2 + K_2 + K_{mag}) - K_{mag}^2} + \frac{-j\omega C_{p1}K_{mag}^2}{(-\omega^2 M_1 + j\omega D_1 + K_1 + K_{mag})(-\omega^2 M_2 + j\omega D_2 + K_2 + K_{mag}) - K_{mag}^2} \quad (3) $$

#### B. Results

Two impedance measurements are taken, either from the electrodes of beam n°1 or from the electrodes of beam n°2. These two measurements are then fitted with (3) in order to identify all the parameters, except $\alpha_1$ and $\alpha_2$ which each require an additional measurement. Results of this identification for the resonators shows in Fig.1.a are given in Table 1.

Fig. 2 presents the measurements and fitted model of the impedance magnitude and phase of both beams.

<table><caption>Table 1 Model parameters</caption><thead><tr><th>Beams parameters</th><th>Beam n°1</th><th>Beam n°2</th></tr></thead><tbody><tr><td>Mass M<sub>i</sub> (g)</td><td>15.2</td><td>15.5</td></tr><tr><td>Stiffness K<sub>i</sub> (N.m)</td><td>720</td><td>790</td></tr><tr><td>Damping D<sub>i</sub> (g/s)</td><td>69.2</td><td>65.9</td></tr><tr><td>Capacitance C<sub>p</sub> (nF)</td><td>128</td><td>115</td></tr><tr><td>Coef. Piezoeletric α<sub>i</sub> (N/V)</td><td>1.4e-3</td><td>1.7e-3</td></tr><tr><td>Coupling K<sub>mag</sub> at d=5cm</td><td>33</td><td></td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 2 Comparison of an impedance analysis of beam n°1 (left) and beam n°2(right) with the same model</figcaption></figure>

### IV. POWER TRANSFER

A dedicated experimental test bench has been developed to apply a sinusoidal voltage to the transmitter and observe the dynamic response of the system.

#### A. Results

An example of the transmitted power under specific experimental conditions is shown in Fig 3. The input voltage amplitude is 50 V, the receiver load resistance is 20 kΩ, and the distance between the beams is 5 cm. The RMS voltage at the receiver was measured for frequencies in the range of 32 Hz to 40 Hz. The corresponding transmitted power is then computed as the square of the RMS voltage divided by the resistance.

<figure><img src="image_5.png" /><figcaption>Fig. 3 Transmitted power curve as a function of frequency</figcaption></figure>

#### B. Discussion

Two peaks in the transmitted power correspond to the resonant frequencies of the coupled system. At these frequencies, the displacements of each mass are maximized, resulting in higher output voltage and power. Outside this range, excitation is weak and energy harvesting is minimal. Operating at a resonant frequency is therefore essential.

### V. CONCLUSION

This work explores an alternative approach to low-frequency wireless energy transfer based on coupled magnetomechanical resonators. The system has been experimentally characterized and accurately modeled using analytical tools. It has been demonstrated that energy transfer between the resonators is indeed possible. The next step will be to optimize such a system and explore its advantages over a conventional coil-resonant receiver configuration.

### ACKNOWLEDGMENT

This work is supported by the French. National Agency for Research under grant ANR23CE510007.

### REFERENCES

[1] Sagar, A., Kashyap, A., Nasab, M. A., Padmanaban, S., Bertoluzzo, M., Kumar, A., & Blaabjerg, F. (2023). A Comprehensive Review of the Recent Development of Wireless Power Transfer Technologies for Electric Vehicle Charging Systems. *IEEE Access*, 11, 83703-83751. https://doi.org/10.1109/access.2023.3300475

[2] Adrien Ameye, Pierre Gasnier, Nicolas Decroix, Adrien Badel, Nicolas Garraud. Increasing the robustness of electrodynamic wireless power receivers with hybrid transduction. 2022 IEEE Wireless Power Week Conference (WPW 2022), Jul 2022, Bordeaux, France. (hal-03760301).

[3] Tsurumi, T., Ichihara, T., Asaga, K., & Daimon, M. (1990). Admittance-Curve-Fitting Method for Semiconductive Piezoelectric Ceramics. *Journal Of The American Ceramic Society*, 73(5), 1330-1333. https://doi.org/10.1111/j.1151-2916.1990.tb05200.