# Five-Terminal Dual-Polarity MEMS Electrostatic Transducer For Near-Limits Kinetic Energy Harvesting From Irregular Vibrations

Moein Rahmani, Armine Karami, Francisco Ambia, Alexis Brenes, Dimitri

Galayko, Elie Lefeuvre, Philippe Basset

► To cite this version:

Moein Rahmani, Armine Karami, Francisco Ambia, Alexis Brenes, Dimitri Galayko, et al. Five-Terminal Dual-Polarity MEMS Electrostatic Transducer For Near-Limits Kinetic Energy Harvesting From Irregular Vibrations. 2024 IEEE 23rd International Conference on Micro and Miniature Power Systems, Self-Powered Sensors and Energy Autonomous Devices (PowerMEMS), Nov 2024, Tonsberg, Norway. pp.143-146, 10.1109/PowerMEMS63147.2024.10814232. [hal-04943842](https://hal.science/hal-04943842v1)

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# FIVE-TERMINAL DUAL-POLARITY MEMS ELECTROSTATIC TRANSDUCER FOR NEAR-LIMITS KINETIC ENERGY HARVESTING FROM IRREGULAR VIBRATIONS

Moein Rahmani¹, Armine Karami¹, Francisco Ambia²,

Alexis Brenes², Dimitri Galayko³, Elie Lefeuvre², Philippe Basset¹

¹Univ Gustave Eiffel, CNRS, ESYCOM, F-77454 Marne-la-Vallée, France

²University Paris Saclay, CNRS, C2N, F-91120 Palaiseau, France

³Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

## ABSTRACT

This paper presents the design, fabrication, and testing of the first silicon electrostatic MEMS device specifically engineered to implement the near-limits kinetic energy harvesting (NL-KEH) principle. A novel five-terminal dual-polarity MEMS transducer is fabricated and experimentally evaluated, demonstrating its potential for improved vibration energy harvesting performance.

## KEYWORDS

Kinetic energy harvesting, capacitive transducers, MEMS

## INTRODUCTION

It is notoriously difficult to design resonant inertial kinetic energy harvesters (KEHs) of both small sizes ($\leq 1$ cm³) and good power conversion capabilities from low frequency vibrations ($\leq 10$ Hz). This is fundamentally due to the typical density and Young modulus of popular materials for KEH fabrication.

This issue has led researchers in the field to consider clever designs based on nonlinearities introduced from the mechanics [1] or the conditioning electronics [2], with various degrees of adaptability. Since the superposition principle does not hold in general for these designs, they require specific studies to address their performances under non-monochromatic excitation, both deterministic or stochastic, Even for devices that are linear by design, some nonlinearity will eventually manifest, for instance due to the limited size of small-scale KEHs. This limited-size constraint is sufficient to cause a decrease of the maximum power output reachable with resonant-based designs, relatively to the architecture-independent power output limit. This decrease is even more drastic as excitations with richer harmonic content are considered [3].

To address both low-frequency or irregular vibrations, the concept of near-limits kinetic energy harvesting (NL-KEH) proposes to implement a fully active control of the KEH electromechanical dynamics [4]. The unique constraints posed by the implementation of an NL-KEH (closed-loop control of the electromechanical dynamics) require specific transducer designs. In this paper, we present such a transducer designed in MEMS technology,

and relying on electrostatic transduction. The transducer features two terminals for bi-directional electrostatic control and two terminals for differential dynamic capacitance measurement, which is needed to estimate the mechanical state of the system.

The paper starts by quantitatively recalling the concept of NL-KEH. Then, the design process of the MEMS transducer is described. The paper ends with some experimental results, among which initial tests showing the possibility of electromechanical control.

## NEAR-LIMITS KINETIC ENERGY HARVESTING

A linear kinetic energy harvester (KEH) along some axis (Ox) can be effectively modeled as a mass-spring system influenced by an external mechanical force $\Xi(t)$ and by a transducer force $F_t(t)$. Here, mechanical damping effects are ignored. The equations of motion for this system, constrained by a displacement limit, are given by:

$$ \begin{cases} m\ddot{x}(t) + K_x x(t) + F_t(t) = \Xi(t) \\ |x(t)| \leq X_0 \end{cases} \quad (1) $$

Here, m is the mass of the movable part, x(t) is the mass displacement in (Ox) relatively to rest position, $K_x$ is the spring stiffness in (Ox), and $X_0$ is the displacement limit.

The near-limits strategy focuses on maximizing the energy input into the system by the external vibration.

$$ \int_{-\tau}^{\tau} \Xi(\tau) \dot{x}(\tau) d\tau = - \int_{-\tau}^{\tau} F_t(\tau) \dot{x}(\tau) d\tau + o(1) \quad (2) $$

It shows that the energy input from the external mechanical excitation $\Xi(t)$ equals the energy converted by the transducer force $F_t(t)$, up to a bounded term as time t increases

The displacement constraint (1) stems from the necessarily limited size of any small-scale KEH system. The KEH must maximize the energy input from the external force under this constraint, for given $\Xi(t)$. To do so, the KEH must implement the trajectory

$$ x^* = -\mathrm{sign}[\Xi(t)]X_0 \quad (3) $$

thanks to the proper transducer force $F_t(t)$.
<figure><img src="image_2.png" /><figcaption>Figure 1: NL-KEH trajectory example (red) in response to an input force Ξ(t) (blue). The right figure shows the time-magnification of an example of toggling event that must be implemented at extrema of the input force.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Initial schematic of an AO transducer illustrating the definition of its design parameters.</figcaption></figure>

Practically, the optimal trajectory keeps the mass at ±X₀ and toggles it from -X₀ to +X₀ and vice versa, in response to local maxima and minima in the external excitation Ξ(t). Fig. 1 illustrates this, with the red line showing the mass trajectory x(t) and the dashed blue curve representing the mechanical input Ξ(t).

The trajectory in (3) describes instantaneous togglings from ±X₀ to ∓X₀ as soon as extrema of Ξ(t) are detected, an unphysical idealization. However, approximating this trajectory by dynamically controlling the system allows us to approach the power bound, hence the term "near-limits."

To approach x*, the toggling process must be fast enough so that the mechanical force remains close to its extrema value throughout the displacement.

A system that implements the trajectory x* thanks to the transducer force F_t will maximize the power converted from the mechanical to the electrical domain. As it is based on the detection of the external force extrema, this scheme can be implemented indifferently for low-frequency (compared to the mechanical system's resonant frequency), or irregular vibrations. Evidently, to maximize the total net harvested energy available for usage by a useful load, the efficiency of the electrical interface used to implement F_t and collect the converted energy should also be maximized.

## MEMS DEVICE DESIGN

In this section, we discuss the design of a MEMS device tailored for implementing near-limits control. Among various configurations of electrostatic micro-scale energy harvesters, area-overlapping (AO) configurations are selected. Since electrostatic transducers can only apply forces in one direction, two such transducers are needed. Two additional transducers are necessary to perform differential measurements of the device capacitance. This capacitance can be mapped to displacement to implement the near-limits control with feedback on the mechanical state. In all, a sketch of the device can be proposed, as depicted in Fig. 2.

We further require that, at any position, the overlap between fingers of any terminal and fingers of the central terminal is sufficient to neglect any fringing field effects. Under this additional hypothesis, following the schematic in Fig. 2, the net force along (Ox) implemented by the blue (+) and green (-) transducer reads

$$F_t(t) = \frac{1}{2} \frac{\varepsilon L_Y L_Z}{D(D + \Delta)} (V_+^2(t) - V_-^2(t)) \quad (4)$$

where ε is the permittivity of air, L_Z is the device dimension onto the page. The other symbols are defined in Fig. 2.

This justifies our choice of an AO geometry a posteriori: the straightforward relationship between the electrostatic force and the applied voltage.

The rest of the design process can be viewed as a constrained optimization problem starting from the above pre-defined choices. The objective function to maximize is the energy E converted by the NL-KEH per extrema of the input force. Since the input force is related to the KEH's acceleration A(t) by Ξ(t) = -mA(t), then for a local extrema of A(t) of value A₀, E reads [4]:

$$E = 2mA_0X_0 \quad (5)$$

The mass m is linked to the system dimensions by

$$m = \rho L_X L_Y L_Z \quad (6)$$

where ρ represents the density of the material used in the MEMS fabrication (silicon), L_X and L_Y which are shown in Fig. 2 are the total device dimensions. We choose to design the system to operate under harmonic excitation with a frequency ≤ 5 Hz and amplitude of A₀ = 10 m s⁻². For the toggling to finish while the input is still close to its sensed extremum, the system's resonance frequency (f_res) must be significantly higher than the excitation frequency. We require that Ξ(t) must vary less than 5% from its extremum during each toggling. Therefore, we require

$$\sqrt{K_x/m} \ge 100\pi \quad (7)$$

To keep the mass at the end-stops, the force should exceed mA₀ + KₓX₀. This sets a minimum value on the voltage to be applied (from (4)). On the other hand, the voltage across the transducer terminals is limited by the electronics. In our case, that is at least the maximum voltage of the high-voltage amplifier, so V_max = 400 V. Hence, the maximum voltage constraint is formulated as:

$$V_{\text{max}} > \sqrt{\frac{2(mA_0 + K_x X_0)D(D + \Delta)}{\varepsilon L_Y L_Z}} \quad (8)$$

Another critical constraint is to avoid lateral pull-in between comb fingers. The lateral pull-in voltage is checked with the inequality [5]:

$$V_{\text{max}} < \sqrt{1.83} \sqrt{\frac{ED^3\Delta^3}{6\varepsilon W^4}} \quad (9)$$

where E is the Young's modulus of silicon, and W the length of fingers. Several other constraints are considered in the design problem: ensuring sufficient spring stiffness
<figure><img src="image_4.png" /><figcaption>Figure 3: Schematic view of the designed device. (a) Mechanical layout: moving parts (pink), fixed parts (grey). (b) Electrical layout: ground (pink), electrical terminal for upward electrostatic force (blue), electrical terminal for downward electrostatic force (green), electrical terminals for measurement purposes (red and purple). (c) The final device fabricated at C2N clean-room.</figcaption></figure>

<table><caption>Table 1: The device parameters resulted from the optimization problem.</caption><thead><tr><td>Parameter</td><td>Value</td><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>K<sub>x</sub> (N/m)</td><td>20</td><td>D, Δ (μm)</td><td>16</td></tr><tr><td>K<sub>y</sub> (N/m)</td><td>3500</td><td>X<sub>0</sub> (μm)</td><td>300</td></tr><tr><td>m (g)</td><td>0.1</td><td>W (μm)</td><td>780</td></tr></tbody></table>

in undesired directions, or checking frequencies of higher modes and mechanical deflections to avoid contact between different electrical terminals.

After solving the optimization problem using a grid search, the selected design for fabrication is presented. The design targets a resonant mechanical structure with side beams to prevent rotational motion [6]. Fig. 3 shows schematics of the mechanical and electrical parts, along with a view of the final fabricated device. The transducers used for application of the bidirectional electromechanical forces (control and harvesting) have $N = 950$ fingers. The transducers for the reading of the mechanical state through capacitance measurement counts $N_m = 160$ fingers. The device parameters are summarized in Table 1.

To verify the characteristics of the designed device, finite-element simulations were conducted using the COMSOL Multiphysics software. The first natural mode is in the desired direction of motion ($Ox$) and resonates at 65 Hz, which is within the acceptable range for our design. The second mode occurs in the perpendicular out-of-plane direction, and is at 510 Hz, significantly higher than the first mode. A pull-in simulation for a pair of isolated fingers was performed. The obtained results are consistent with (9), which predicts a pull-in voltage of approximately 500 V.

The device was fabricated in silicon-on-glass technology with 200 µm thickness of the structural layer. The silicon is etched by DRIE and released by liquid HF etching at the Centre de Nanosciences et Nanotechnologies (C2N) clean room. A picture of the device is shown in Fig. 3.d.

## EXPERIMENTAL RESULTS

To validate the models used in the device design and the possibility for control implementation, experimental tests were conducted. The experimental setup is depicted in Fig. 4. The device is mounted on a custom PCB and attached to a shaker for experiments requiring mechanical vibrations. Due to the absence of connections for the two measurement transducers (red and purple in Fig. 3.c), capacitance variation was measured from one of the harvesting transducers (blue transducer in Fig. 3.c) using an HF2LI lock-in amplifier from Zurich Instruments. The am-

<figure><img src="image_5.png" /><figcaption>Figure 4: Experimental setup schematic</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Frequency response of the mechanical resonator for different vibration amplitudes in units of $A_0$.</figcaption></figure>

plifier applies a high frequency carrier of 10 kHz to the selected transducer, thereby inducing a high frequency current through it. This current is amplitude-modulated by the transducer's capacitance value. This current is then read and demodulated (by demodulator1 in Fig. 4), yielding an output signal proportional to the transducer's capacitance. This signal can, in turn, be correlated with the displacements of the mobile mass.

This setup is used to evaluate the dynamic response of the MEMS energy harvester to a frequency-swept mechanical input, which is applied by a mechanical shaker. For this experiment, the output of this demodulator1 is further demodulated using the signal driving the shaker (demodulator2 in Fig. 4), to extract the first harmonic of capacitance variation as the driving frequency varies. The result from demodulator 2 is used to generate Fig. 5.

The nonlinear mechanical behavior of the device, evident in Fig. 5, contrasts with the initial simulation predictions. This discrepancy may be attributed to fabrication imperfections, such as spring over-etching observed under the microscope. However, the resonance frequency of the device ($f_{res} = 56$ Hz) remains within the expected range.

For control input tests, a function generator produces the required signals. The control signals are amplified by a *Trek 2205* high-voltage amplifier, whose output is in turn applied across the transducers (blue and green in Fig. 4). When both harvesting transducers are employed for control input experiments, a laser station (Keyence LK-H050) is used to monitor the system displacements.

To associate capacitance variations of the transducer to its displacements, a high amplitude acceleration at the resonance natural frequency was applied on the shaker, ensuring that the moving part was doing the whole traveling range. Simultaneously, the capacitance variation was obtained from Demodulator1. Assuming steady-state oscilla-
<figure><img src="image_7.png" /><figcaption>Figure 6: Capacitance as a function of the overlapping area between the device's fingers for the harvesting transducer. On the top, the x-coordinate corresponding to displacement range of the mass are displayed.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Experimental results for the mass toggling control.</figcaption></figure>

tions of the mobile mass at full traveling range, one can plot the capacitance versus overlapping area shown in Fig. 6.

This capacitance versus displacement is estimated to be approximately half of the designed value. This discrepancy can be attributed to fabrication imperfections, such as broken comb fingers and over-etching. Nevertheless, the observed linear relationship between capacitance and displacement suggests that the transducer can still generate consistent forces under constant voltage conditions, which facilitates the control implementation.

The implementation of the control was the ultimate goal of the device design. However, deviations from the design predictions observed in the device's behavior present challenges for implementing full-range control. Nonetheless, reducing the displacement range could ensure a more linear response of the resonator, and lower the transducer forces required for control.

Both harvesting transducers were used to apply control voltage inputs. Simultaneously, a laser station was employed to measure relative displacement. At this stage, no mechanical excitation was applied to the harvester. The results are depicted in Fig. 7 where the blue and green lines represent the voltages applied to the transducers generating the electromechanical forces, respectively, while the red curve indicates the position of the mobile mass.

As illustrated in Fig. 7, the mobile mass is initially maintained in a positive position by applying a positive electrostatic force while keeping the negative force at zero. Approximately at $t = 500$ ms, the toggling process begins by eliminating the positive force and applying an appropriate amount of negative force at the precise moment to capture the mass in a negative position with zero velocity. A zoomed view of this process reveals the smooth capture at the end of the motion. A second toggling process commences around $t = 1300$ ms. However, during this phase, the control at the other transducer is deliberately not implemented accurately, to emphasize that improper (or no)

control leads to natural oscillations at the end of the mass movement. Here, the oscillations are free since the equilibrium position is far from the stopper.

Note that here, unlike in the final NL-KEH application, the mass travels under the sole effect of the spring restoring force. Still, this experiment is a step towards NL-KEH, demonstrating the required catch at the end of toggling.

## CONCLUSION

This work presented the first MEMS transducer specifically designed and fabricated to implement the near-limits kinetic energy harvesting principle. Experimental results show relatively good agreement with the design, and validate the possibility of fine electromechanical control of the device dynamics. Future work is warranted to control the device under external mechanical vibration to estimate the net power output and further validate the feasibility and relevance of the near-limits approach.

## ACKNOWLEDGEMENTS

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-22-CE05-0014 NearLimits. A CC-BY public copyright license has been applied by the authors to the present document, in accordance with the grant's open access conditions. https://creativecommons.org/licenses/by/4.0/

## References

[1] T. Huguet, A. Badel, and M. Lallart, “Exploiting bistable oscillator subharmonics for magnified broadband vibration energy harvesting,” *Applied Physics Letters*, vol. 111, no. 17, 2017.

[2] A. Morel, Q. Demouron, and A. Badel, “Enhancing bistable vibration energy harvesters with tunable circuits: A comparative analysis,” *arXiv preprint arXiv:2409.00469*, 2024.

[3] A. Karami, M. Rahmani, D. Galayko, and P. Basset, “Towards maximum power conversion from realistic vibrations: Limits for size-constrained inertial kinetic energy harvesters under bi-chromatic vibration inputs,” in *2023 IEEE 22nd International Conference on Micro and Nanotechnology for Power Generation and Energy Conversion (PowerMEMS)*. IEEE, 2023, pp. 167–170.

[4] A. Karami, J. Juillard, E. Blokhina, P. Basset, and D. Galayko, “Electrostatic near-limits kinetic energy harvesting from arbitrary input vibrations,” *arXiv preprint arXiv:2002.07086*, 2019.

[5] A. Ramezani, A. Alasty, and J. Akbari, “Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces,” *International Journal of Solids and Structures*, vol. 44, no. 14-15, pp. 4925–4941, 2007.

[6] F. Ambia, J. Chavez, M. Lallart, X. Leroux, and E. Lefeuvre, “Design and simulation of an electrostatic energy harvester for biomedical implants,” in *2021 Symposium on Design, Test, Integration & Packaging of MEMS and MOEMS (DTIP)*. IEEE, 2021, pp. 1–6.

## CONTACT

Armine Karami: armine.karami@univ-eiffel.fr