# Multi-orbit jump for electrodynamic wireless power transfer systems with non-linear dynamics

Adrien Ameye, Adrien Morel, Nicolas Garraud, Adrien Badel

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## To cite this version:

Adrien Ameye, Adrien Morel, Nicolas Garraud, Adrien Badel. Multi-orbit jump for electrodynamic wireless power transfer systems with non-linear dynamics. JNRSE 2025 - Journées nationales de la récupération et du stockage de l'énergie, SYMME USMB, Jun 2025, Annecy, France. hal-05111735

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https://hal.science/hal-05111735v1

Submitted on 13 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Multi-orbit jump for electrodynamic wireless power transfer systems with non-linear dynamics

Adrien AMEYE¹*, Adrien MOREL¹, Nicolas GARRAUD², and Adrien BADEL¹

¹ SYMME, Université Savoie Mont Blanc, France

² CEA-Leti, Univ. Grenoble Alpes, France

Corresponding author e-mail: adrien.ameye@univ-smb.fr

**Abstract**— This paper presents a method to optimize the orbital transition time of nonlinear wireless power transfer (WPT) receivers by exploiting stable intermediate orbits. The proposed strategy reduces the energy accumulation time required by the energy management circuit when the receiver is in low-energy orbits, thereby accelerating the transition to high-energy orbits. The approach is shown numerically on an electrodynamic receiver operating in both continuous rotation and resonant modes. The system and its analytical model are detailed, and a 52 % reduction in the transition time to the high-energy orbit is demonstrated.

**Keywords**—Electrodynamic wireless power transfer, nonlinear mechanical system, orbit jump, sub-harmonic

## I. INTRODUCTION

Electrodynamic power transfer systems based on permanent magnets operate at very low frequencies (typically below 1 kHz), enabling transmission through conductive media or biological tissues, and minimizing electromagnetic interference with surrounding systems.

In the literature, various rotating systems have been explored for energy transmission and harvesting due to their potential for high power density. However, these systems can exhibit nonlinear behavior, leading to multiple coexisting orbits for the same excitation parameters (frequency and amplitude). Since the output power depends on the orbit, it is desirable to control the receiver towards the high-energy orbit in order to maximize power transfer [1].

During startup or after a disturbance, such nonlinear systems may settle into low-energy orbits, where output power is reduced. This suboptimal state can be corrected using orbit-jumping strategies [2], which involve injecting energy at the right moment to shift the system from a low- to a high-energy orbit.

In this work, we use a previously developed electrodynamic receiver [3] which exhibits multiple orbits under specific excitation frequencies and amplitudes. Using an analytical model, we numerically assess the feasibility of orbit jumps and evaluate the time required to accumulate the necessary energy when the system resides in a low-energy orbit. We propose a "multi-jump" strategy, involving successive transitions between intermediate orbits to minimize the total energy accumulation time for reaching the high-energy state.

## II. CONSIDERED DEVICE

The studied receiver presented in Figure 1 consists of a diametrically magnetized permanent magnet, free to rotate about its axis. A second fixed permanent magnet is placed nearby, acting as magnetic stiffness.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Studied device: (A) schematic, (B) photograph.</figcaption></figure>

The receiver is placed at the center of a Helmholtz coil, allowing control of the excitation magnetic field, which, in this study, is set at 1 mT with a sinusoidal waveform, and the receiver is connected to an Energy Management System. A preliminary phase of mechanical and electrical measurements was carried out to identify the parameters listed in Table 1.

<table><caption>Table 1. Identified device parameters. The amplitude of the excitation torque is determined for the excitation field set at 1 mT.</caption><thead><tr><th>Symbol</th><th>Description</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>J</td><td>Rotation inertia</td><td>8.1 10<sup>-9</sup></td><td>[kg. m<sup>2</sup>]</td></tr><tr><td>D</td><td>Damping coefficient</td><td>5.2</td><td>[nN.m.s]</td></tr><tr><td>R<sub>c</sub></td><td>Coil resistance</td><td>114</td><td>[Ω]</td></tr><tr><td>R<sub>l</sub></td><td>Load resistance</td><td>2000</td><td>[Ω]</td></tr><tr><td>B<sub>0</sub></td><td>Transduction coefficient</td><td>10.2</td><td>[mN.m/A]</td></tr><tr><td>K<sub>0</sub></td><td>Magnetic stiffness</td><td>12</td><td>[mN.m]</td></tr><tr><td>τ<sub>0</sub></td><td>Excitation torque amplitude</td><td>1.2</td><td>[mN.m]</td></tr></tbody></table>

## III. MODEL AND DYNAMIC BEHAVIOUR

The notations are listed in Table 1, and the complete model of the receiver can be found in [3]. The magnetic field with angular frequency $\omega$, generated by the transmitter, exerts a varying torque $\tau$ on the moving magnet. This torque depends on the angle $\theta$ between the magnet's magnetic moment and the transmitter's axis:

$$ \tau = \tau_0 \sin(\omega t) \sin(\theta) \quad (1) $$

Three additional torques act on the moving magnet: a magnetic stiffness torque $\tau_K = -K_0 \cos(\theta)$, a mechanical damping torque $\tau_D = -D\dot{\theta}$, and an electrodynamic transduction torque $\tau_B = -B_0 \sin(\theta) i$, where $i$ is the current in the receiving coil. Newton's second law yields the following mechanical equation:

$$ \tau = j\ddot{\theta} + K \cos(\theta) + D\dot{\theta} + B_0 \sin(\theta) i \quad (2) $$

The rotation of the permanent magnet induces a voltage $U_e = B_0 \sin(\theta) \dot{\theta}$ at the coil terminals. This equivalent voltage source is connected in series to the coil internal resistance $R_c$ and the load, such as:

$$ B_0 \sin(\theta) \dot{\theta} - iR_c = u \quad (3) $$

with $u$ the output voltage applied to the load. The effect of the coil's inductance is negligible at the operating frequencies of
our system. When the receiver is connected to a resistive load $R_L$, the output power $P_m$ is defined as:

$$P_m = R_L(i^2) \qquad (4)$$

Using numerical simulations, it is possible to plot the device output power of the receiver as a function of the excitation frequency $\omega/2\pi$. This power depends on the initial conditions of the magnet $(\theta, \dot{\theta})$ due to the existence of several stable orbits in specific frequency bands (Figure 2). Specifically, five types of orbits coexist in the 100-200 Hz range: two rotational orbits (Rot1:1 and the subharmonic Rot1:3) and three oscillatory orbits (Osc1:1, and the subharmonics Osc1:3 and Osc1:5).

<figure><img src="image_4.png" /><figcaption>Fig. 2. Receiver output power as a function of frequency for 1 mT excitation. The red arrows represent the multiple jump strategy presented in part 4.</figcaption></figure>

## IV. MULTIPLE JUMP STRATEGY

To transition from a low- to a higher-energy orbit, it is necessary to add energy to the moving magnet $E_c$

$$E_c(t) = \int_{ti}^{t} R_L i(\tau)^2 d\tau = \int_{ti}^{t} R_L \left( \frac{B_0 \sin(\theta) \dot{\theta}}{R_L + R_c} \right)^2 d\tau \quad (5)$$

When sufficient energy has been accumulated, it can be discharged into the magnet by applying a voltage $U_d$ across the electrodynamic transducer (Fig. 3), resulting in a significant mechanical torque $\tau_B$:

$$\tau_B = \frac{B_0^2 \sin^2(\theta) \dot{\theta} - B_0 \sin(\theta) U_d}{R_c} \qquad (6)$$

<figure><img src="image_5.png" /><figcaption>Fig. 3. Flowchart of the orbit-jump strategy.</figcaption></figure>

For example, if the receiver is operated arbitrarily at 127 Hz, a minimum energy input of $E_{O1R1} = 3.96 mJ$ has been numerically determined to transition from the low-energy orbit Osc1:1 to its high-energy orbit Rot1:1. At this frequency, if the receiver remains in the Osc1:1 orbit, accumulating this energy via the load takes approximately 240 s.

To shorten this time, it is possible to use the intermediate orbit Osc1:3, as illustrated with red arrows in Figure 2.

The minimum energy required for a jump (J1) from the low-energy stable orbit Osc1:1 to the intermediate stable orbit Osc3:1 is $E_{O1R3} = 1.82 mJ$, corresponding to a reduced accumulation time of 112 seconds.

<figure><img src="image_6.png" /><figcaption>Fig. 4. Numerical simulation of the multi-orbit jump strategy enabling fast transition to the high-energy orbit. (A) Magnet's rotational speed. (B) Accumulated energy in the load. Discharges are indicated with red arrow. (C) Phase space.</figcaption></figure>

Since the intermediate orbit Osc1:3 provides higher output power, accumulating the energy for the second jump (J2) is much faster: once this orbit is reached, the power delivered from this orbit is high enough to enable a second jump to the Rot1:1 orbit, requiring only 0.9 seconds of energy accumulation. This strategy is illustrated in Figure 4. In the presented case, the total transition time reduced to approximately 113 seconds, compared to 240 seconds for a direct transition. This result represents a 52% improvement in terms of the system's settling speed.

## V. CONCLUSION

In this paper, we presented a multi-orbit jump strategy to optimize the orbital transition time of nonlinear electrodynamic WPT receivers. By numerically exploiting intermediate stable orbits, such as the Osc1:3 subharmonic orbit, as gateways, we demonstrated that it is possible to significantly reduce the required energy accumulation time to reach the highest-performing operating state. For the system under study, this approach reduced by more than 52% the time needed to transition from the low-energy orbit Osc1:1 to the high-energy orbit Rot1:1.

## REFERENCES

[1] S. Zhou et al., 'Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting', Communications in Nonlinear Science and Numerical Simulation, vol. 61, pp. 271-284, Aug. 2018, doi: 10.1016/j.cnsns.2018.02.017.

[2] C. Saint-Martin et al., 'Optimized and robust orbit jump for nonlinear vibration energy harvesters', Nonlinear Dyn, vol. 112, no. 5, pp. 3081-3105, Mar. 2024, doi: 10.1007/s11071-023-09188-x.

[3] A. Ameye et al., 'Wireless Electrodynamic Power Transfer: Modeling And Discussion Of a Dual-Mode Receiver', in 2024 PowerMEMS, Nov. 2024, pp. 2-5. doi: 10.1109/PowerMEMS63147.2024.10814423.