# Low-Cost High-Speed In-Plane Stroboscopic Micro-Motion Analyzer

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## Abstract

**:**

## 1. Introduction

## 2. Low-Cost Stroboscopy Imaging System

#### 2.1. Timer Generator Circuit

_{L}< M. High-precision periodic digital strobe signals can be generated with high-speed finite-state machines using programmable gate arrays or high-speed transistor-transistor logic (TTL) or CMOS logic chips. In this paper, we utilize a TTL implementation due to its simplicity and low cost.

_{c}/M. The second module is a digital one-shot circuit that produces a single delayed pulse at a specified count N

_{i}starting from the rising edge of the START input. For example, the phase delay for such a digital pulse is thus ϕ

_{i}= 2π N

_{i}/M radians.

_{i}is loaded into the period and one-shot modules using an Arduino Mega 2560 microcontroller (Digi-Key Electronics, Thief River Falls, MN, USA). These values are set by the Arduino programming code and the digitized voltage at the Arduino input A

_{0}that is provided by a phase control potentiometer. Figure 4 shows the timing diagrams of the resulting digital timing signals.

_{D}) < M that is triggered by a high logic level at the START input. The one-shot consists of a 74HCT4059 counter [15] (NXP Semiconductors, Eindhoven, Netherlands), 2D flip-flops and 1 RS flip-flop. The down counter is configured such that the counter initial value N

_{D}is first loaded while the k

_{b}control pin is at logic low. The counter remains in the loading state until the state of FF4 is changed. When the START signal goes to logic low, it presets FF4 and the k

_{b}control pin goes to logic high and the counter starts counting down. When the count reaches zero its Q output produces a DD pulse, which is subsequently delayed by FF2 and FF3 and used to clear flip-flop FF4 setting the counter loading mode until a new START signal again goes to logic low, restarting the single-shot cycle.

#### 2.2. High Voltage MEMS and LED Driver Circuits

#### 2.3. Imaging Optics

## 3. Test Device Fabrication

## 4. Comb Drive Motion

_{eff}is the effective mass of the system under motion, b is the damping coefficient, k is the net stiffness constant for the folded beam system, F

_{ext}is the electrostatic excitation force which is a function of time (t) and frequency (ω), and x is the displacement in the x direction. The excitation force is the result of fluctuating electrostatic voltage from the high voltage comb drive pulse train and it can be derived from the rate of change of energy stored in the capacitor formed between a rotor finger and two stator fingers and is expressed as ${F}_{ext}=({n}_{f}{\epsilon}_{o}h){V}_{pulse}^{2}/{d}_{gap}$, where n

_{f}is the total number of fingers, ε

_{o}is the vacuum permittivity, h is the height of the stator and rotor fingers, d

_{gap}is the gap between the stator and rotor fingers, and V

_{pulse}is the voltage pulse train output from the high voltage driver circuit. To preserve the linearity of the equation of motion, we neglect the sideways movement. The periodic voltage pulse train can be expressed as a Fourier series, as given by Equation (2).

_{o}is the amplitude of voltage pulse, τ is the pulse width, and T is the time period which is equal to 2π/ω. The effective mass of the moving rotor can be estimated by applying the principle of conservation of energy on the resonating structure and assuming that all of beams deflect with mode shapes as if under static loads [16]. The energy conservation and beam theory together give the effective mass of the resonator as:

_{s}is the mass of the shuttle, m

_{f}that of moving fingers, m

_{b}represents mass of eight parallel beams, and m

_{t}is the mass of two trusses. Assuming that the trusses act as rigid support for the beams under deflection, the total stiffness for the folded beams under no residual stress can be calculated by using the series and parallel beam theory. Since each beam length is not identical the effective stiffness along x axis becomes:

_{x,b}

_{1}and k

_{x,b}

_{2}are the beam stiffness for beam elements b

_{1}and b

_{2}, as shown in the figure. From Euler beam theory the stiffness of each beam segment is $k=12{E}_{Si}{I}_{z,b}/{L}^{3}$, where E

_{Si}is the Young’s modulus of silicon, I

_{z,b}is the area moment of inertia along the z axis, and L is length of the beam segment. The area moment of inertia for beam cross section is given by ${I}_{z,b}=h{w}^{3}/12$, where h is the height, w is the width of the beam. The material properties that were used to calculate the variables in the equation of motion are given in Table 1 below. Using these values, the effective mass of the rotor and the total stiffness for the folded beam arrangement is calculated to be 1.23 µg and 210 N/m, respectively which give the analytical resonant frequency ${f}_{n}=\sqrt{{k}_{eff}/{m}_{eff}}/2\pi $ as 65.7 kHz.

## 5. Experimental Imaging Results

_{COMB}that was produced by the circuit of Figure 6a, and the device mechanical oscillation motion was captured by the microscope camera when illuminated by a stream of LED pulses that were strobed at a specific phase. The maximum comb drive voltage was kept below the side snap-in instability limit. The comb side-snap pull-in voltage [17] was calculated to be 300 V, but when corrected for dimensional errors in fabrication, it is close to 220 V.

#### 5.1. Frequency Analysis

_{adj}= 132 N/m. The Q factor from the experimental result was 113, which when compared to the calculated analytical value (~155) as given in [18,19], is off by ~27%. Adjusting the Q factor value with the experimentally found k

_{adj}value gives a Q factor of 118, which is in good agreement with the observed value. The images at peak resonance show that the maximum displacement in +x direction was ~30 μm, so the total stroke d

_{s}of the resonating comb drive was ~60 μm.

#### 5.2. Phase Analysis

#### 5.3. Stroke vs. Applied Voltage

#### 5.4. Stroboscopic System Limitations

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Schematic representation of stroboscopic imaging by phase delayed imaging light-emitting diode (LED) pulse.

**Figure 3.**Schematic diagram of the timer circuit showing the generation of the output pulses using the digital delay modules.

**Figure 4.**Schematic logic diagram indicating generation of both strobe and drive signals from four narrow digital delay pulses at different digital phase ϕ

_{0}–ϕ

_{3}.

**Figure 5.**(

**a**) Schematic and timing diagram of the period module used as period pulse generator. The period module produces a repeating pulse every N

_{i}input pulses. (

**b**) Schematic and timing diagram of digital one-shot module. The digital one-shot produces a single delayed pulse after the START signal goes high.

**Figure 6.**(

**a**) Schematic of the half bridge gate driver configuration of high voltage MEMS driver ckt. and (

**b**) schematic of the high-speed LED switching ckt. using an ultrafast gate driver integrated circuit (IC) with an R*C differentiator ckt. for producing sharp LED pulses.

**Figure 8.**(

**a**) Two-dimensional (2D) schematic of the COMB drive actuator with critical dimensions labeled. (

**b**) 3D schematic of the COMB drive actuator with the rest of the critical dimensions labeled and (

**c**) SEM photographs of the released COMB drive microactuator used for testing.

**Figure 10.**Frequency response through stroboscopic freeze frame images of comb drive actuator at maximum displacements for different frequencies (

**a**) No actuation, (

**b**) 50 kHz, (

**c**) 52.1 kHz, (

**d**) 80 kHz and (

**e**) Frequency response graph for all of the frequencies tested giving observed Q factor ≈ 113 and resonance frequency of 52.1 kHz.

**Figure 11.**Phase resolved resonant motion of a test comb drive actuator imaged through stroboscopic imaging system at resonance at (

**a**) 7°, (

**b**) 90°, (

**c**) 184°, (

**d**) 330°, and (

**e**) graph of displacement vs. phase w.r.t. the drive pulse shows a cosine relation between displacement and time, which is consistent with the device theory in discussed in Section 4.

**Figure 12.**Effect of applied voltage to maximum displacement observed through stroboscopic imaging set-up at (

**a**) 90 V, (

**b**) 110 V, (

**c**) 130 V, (

**d**) 150 V, and (

**e**) stroke is plotted against the applied voltage, and it was observed to have a square dependence on applied voltage which agrees with the device theory.

**Figure 13.**Performance limits of the imaging system. (

**a**) The imaging system is shown to work well at srtobing frequencies up to 11 MHz frequency as shown by the pulsed LED output detected by a high-speed photodetector placed at the camera c-mount; (

**b**) The sharpening of the LED pulses is shown at the resonant frequency of 52.1 kHz with 50 ns sharp pulses from an R*C differentiator; Figure (

**c**) shows the motion blurring in the captured image at 52.1 kHz with rectangular 190 ns strobe pulse widths as achieved from a divide by 100 counter; (

**d**) shows a reduction in motion blurring when the LED signal is sharpened with a differentiation circuit producing 50 ns wide strobe pulses.

Name | Parameter | Value | Unit |
---|---|---|---|

Shuttle length | L_{s} | 294 | µm |

Truss length | L_{t} | 153 | µm |

Width shuttle | w_{s} | 20 | µm |

Width truss | w_{t} | 10 | µm |

1st beam length | L_{b}_{1} | 125 | µm |

2nd beam length | L_{b}_{2} | 90 | µm |

Beam width | w_{b} | 2.5 | µm |

Support length | L_{cs} | 293 | µm |

Support width | w_{cs} | 10 | µm |

Finger length | L_{f} | 50 | µm |

Finger width | w_{f} | 3 | µm |

Finger gap | d_{gap} | 2 | µm |

Finger overlap length | L_{o} | 22 | µm |

Number of rotor fingers | N_{fr} | 60 | - |

Number of stator fingers | N_{fs} | 58 | - |

Finger height | h | 30 | µm |

Young’s modulus for silicon | E_{Si} | 150 | GPa |

Density of silicon | ρ_{Si} | 2330 | Kg · m^{−3} |

Buffer oxide thickness | d_{SiO2} | 1 | µm |

Kinematic viscosity of air | ν_{air} | 1.57 × 10^{−5} | m^{2}/s |

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**MDPI and ACS Style**

Pandey, S.S.; Banerjee, A.; Karkhanis, M.U.; Mastrangelo, C.H.
Low-Cost High-Speed In-Plane Stroboscopic Micro-Motion Analyzer. *Micromachines* **2017**, *8*, 351.
https://doi.org/10.3390/mi8120351

**AMA Style**

Pandey SS, Banerjee A, Karkhanis MU, Mastrangelo CH.
Low-Cost High-Speed In-Plane Stroboscopic Micro-Motion Analyzer. *Micromachines*. 2017; 8(12):351.
https://doi.org/10.3390/mi8120351

**Chicago/Turabian Style**

Pandey, Shashank S., Aishwaryadev Banerjee, Mohit U. Karkhanis, and Carlos H. Mastrangelo.
2017. "Low-Cost High-Speed In-Plane Stroboscopic Micro-Motion Analyzer" *Micromachines* 8, no. 12: 351.
https://doi.org/10.3390/mi8120351