# Design and Characterization of an Electrostatic Constant-Force Actuator Based on a Non-Linear Spring System

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Electrostatic Comb Drive Actuator

#### 2.2. Non-Linear Spring System

#### 2.3. Design of the Constant-Force Generator

#### 2.4. Fabrication Process

#### 2.5. Electronical Voltage Control

**1**). In its conducting state, triggered by a base-emitter voltage ${U}_{\mathrm{BE}}$ of 5 V, the supply voltage is attenuated by the voltage divider consisting of the resistors ${R}_{3}$ and ${R}_{4}$. A base-emitter voltage ${U}_{\mathrm{BE}}$ of 0 V leads to the isolation of the transistor and, thus, to a transfer of the entire supply voltage to the next circuit stage. After lowpass filtering, the resulting output voltage ${U}_{\mathrm{o}}$ (

**2**) is given by

**3**). The high lowpass resistance leads to a very low collector current into this second transistor, which is in the range of 10 $\mathsf{\mu}$A to 50 $\mathsf{\mu}$A. Parasitic effects start to affect the collector-emitter voltage ${u}_{\mathrm{CE}}$ whose usual linear dependency to the base-emitter voltage ${u}_{\mathrm{BE}}$ is disrupted [21]. For a known collector-emitter capacitance ${C}_{\mathrm{CE}}$ in combination with the base-collector resistance ${r}_{\mathrm{BC}}$ and the base-collector capacitance ${C}_{\mathrm{BC}}$, the relation between those two voltages can be described by

**4**) as the limited lowpass output current is unable to deliver a sufficiently large current to the electrodes in time. The current amplification is realized in a buffer amplifier circuit, that provides a maximum current of 1.4 mA, a sufficient amount to charge the capacitor electrodes within nanoseconds. An electromechanical relay provides the required steep voltage rise that leads to the abrupt force generation within the microsystem. In the Appendix A, Table A1 displays the values and types of the electrical components used within the described circuit.

#### 2.6. Experimental Setup for the Force Measurement

## 3. Experiments and Characterization

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PWM | Pulse width modulation |

SOI | Silicon-on-insulator |

PCB | printed circuit board |

## Appendix A. Electrical Components

Component | Value | Type |
---|---|---|

${R}_{1}$ | 47 k$\Omega $ | - |

${R}_{2}$ | 1 k$\Omega $ | - |

${R}_{3}$ | 4.7 k$\Omega $ | - |

${R}_{4}$ | 1 k$\Omega $ | - |

NPN transistor | - | ON 2N5551 160 V bipolar junction transistor |

${R}_{\mathrm{Lowpass}1}$ | 1 M$\Omega $ | - |

${C}_{\mathrm{Lowpass}1}$ | 10 nF | - |

${R}_{\mathrm{Lowpass}2}$ | 1 M$\Omega $ | - |

${C}_{\mathrm{Lowpass}2}$ | 10 nF | - |

Relay | - | KY-019 5V magnetic relay |

${R}_{6}$ | 100 k$\Omega $ | - |

${R}_{7}$ | 470 $\Omega $ | - |

${R}_{8}$ | 1 M$\Omega $ | - |

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**Figure 1.**(

**a**) Schematic view of a single-sided electrostatic comb drive actuator in y-direction with the finger overlap ${l}_{0}$ and the spring stiffness in x- and y-direction ${k}_{\mathrm{x}}$ and ${k}_{\mathrm{y}}$. (

**b**) A to A cross-section of the finger electrodes with finger gap g, width b and depth t.

**Figure 2.**(

**a**) Schematic of the combined spring system featuring a linear serpentine spring and a curved cosine-shaped buckling beam. (

**b**) The individual and combined force-displacement characteristics of the serpentine and the curved spring.

**Figure 3.**Simulation result of the constant force spring system consisting of one linear serpentine spring and two curved cosine-shaped buckling beams (shown above the graphs). The simulated maximal displaced spring system is pictured under the graphs. The following parameters were used for the analytical model: ${E}_{\left[110\right]}=169$ GPa, ${k}_{\mathrm{s},\mathrm{y}}\approx 3\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{N}}{\mathrm{m}}$, ${L}_{\mathrm{c}}=860$ $\mathsf{\mu}$m, ${b}_{\mathrm{c}}$$=3.5$ $\mathsf{\mu}$m, $h\approx $ 35 $\mathsf{\mu}$m, $t=20$ $\mathsf{\mu}$m, ${I}_{\mathrm{z},\mathrm{c}}\approx 71.5$ $\mathsf{\mu}$${\mathrm{m}}^{4}$. The simulated constant-force is ${F}_{\mathrm{guid}}$$\approx 137\pm 2$ $\mathsf{\mu}$N between ${\omega}_{\mathrm{y}}=13$ $\mathsf{\mu}$m and 53 $\mathsf{\mu}$m, and the analytical force ${F}_{\mathrm{g}}\approx 123$ $\mathsf{\mu}$N starts at ${\omega}_{1}=5$ $\mathsf{\mu}$m.

**Figure 4.**Schematic view of the constant-force actuator design that includes fixed combs for the electrostatic actuation and voltage supply (black), moveable combs on a frame with the free standing measurement tip that are suspended by the non-linear spring system on the four corners (grey) and the fixed combs for differential capacitance measurement (blue and green). The parameters $g,b,\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}n$ are described in Section 2.1.

**Figure 5.**Main fabrication steps of the constant-force chip with free moveable measurement tip: (

**a**) aluminium electrodes (

**b**) ${\mathrm{SiO}}_{2}$ hard mask (

**c**) DRIE handle layer (

**d**) patterning of the ${\mathrm{SiO}}_{2}$ hard mask (

**e**) DRIE device layer (

**f**) HF-vapor etching (

**g**) dicing free.

**Figure 6.**Excitation circuit for the reliable deflection of an electrostatic constant-force microsystem with PWM generation (

**1**), PWM filtering (

**2**), AC generation (

**3**), the charge control components (

**4**), and the constant-force microsystem (

**5**) displayed by its capacitance.

**Figure 7.**(

**a**) Final processed constant-force microchip with electronic connections. (

**b**) Experimental setup for the force measurement consists of a piezo linear stage to displace the system in y-direction and a high precision balance.

**Figure 8.**(

**a**) Force-displacement characteristic of the non-linear spring system without electrostatic actuation. (

**b**) Capacitance-displacement curve measured by the FDC1004Q chip and calculated by Formula (1) with ${l}_{0}=100$ $\mathsf{\mu}$m, ${n}_{{\mathrm{c}}^{+}}={n}_{{\mathrm{c}}^{-}}=62$, $g=4$ $\mathsf{\mu}$m, and $t=20$ $\mathsf{\mu}$m. (

**c**) Three constant-force measurements with an electrostatic operating voltage ${U}_{\mathrm{o}}=25$ V with optimized non-linear spring system parameters ${k}_{\mathrm{s},\mathrm{y}}\approx 3\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{N}}{\mathrm{m}}$ and ${b}_{\mathrm{c}}$$=3.5$ $\mathsf{\mu}$m comparable to Figure 3. (

**d**) Constant-force measurement with different operating voltages. For the analytical model, the following parameters were used: ${E}_{[}110]=169$ GPa, ${k}_{\mathrm{s},\mathrm{y}}\approx 2.47\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{N}}{\mathrm{m}}$, ${L}_{\mathrm{c}}=860$ $\mathsf{\mu}$m, ${b}_{\mathrm{c}}$$=3.3$ $\mathsf{\mu}$m, $h\approx $ 35 $\mathsf{\mu}$m, $t=20$ $\mathsf{\mu}$m, ${I}_{\mathrm{z},\mathrm{c}}\approx 59.9$ $\mathsf{\mu}$${\mathrm{m}}^{4}$.

Operating Voltage [V] | Constant-Force [$\mathsf{\mu}$N] | Displacement Range [$\mathsf{\mu}$m] |
---|---|---|

23 | $64\pm 2$ | 35 |

24 | $71\pm 3$ | 37 |

25 | $79\pm 2$ | 40 |

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**MDPI and ACS Style**

Thewes, A.C.; Schmitt, P.; Löhler, P.; Hoffmann, M.
Design and Characterization of an Electrostatic Constant-Force Actuator Based on a Non-Linear Spring System. *Actuators* **2021**, *10*, 192.
https://doi.org/10.3390/act10080192

**AMA Style**

Thewes AC, Schmitt P, Löhler P, Hoffmann M.
Design and Characterization of an Electrostatic Constant-Force Actuator Based on a Non-Linear Spring System. *Actuators*. 2021; 10(8):192.
https://doi.org/10.3390/act10080192

**Chicago/Turabian Style**

Thewes, Anna Christina, Philip Schmitt, Philipp Löhler, and Martin Hoffmann.
2021. "Design and Characterization of an Electrostatic Constant-Force Actuator Based on a Non-Linear Spring System" *Actuators* 10, no. 8: 192.
https://doi.org/10.3390/act10080192