# Design and Analysis of Impedance Pumps Utilizing Electromagnetic Actuation

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## Abstract

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## 1. Introduction

_{3}layer for high-precision positioning applications. Alternatively, electromagnetic actuators represent an ideal solution for many modern MEMS-based applications with their simple driving mode, low actuation frequencies, large displacements and planar structures. Liu et al. [4] developed an active MEMS-based fluid control system incorporating surface micromachined magnetic actuators, and showed that the actuators were capable of achieving a large deflection (100 μm) under the application of a magnetic force with a magnetic flux density of 1.76 kGauss at 2.5 A current input. Lagorce et al. [5] presented a micro actuator based on a polymer magnet, and demonstrated that a good agreement existed between its theoretical and experimental response. However, the use of the device was limited with its maximum deflection of just 20 μm as a pumping component for practical microfluidic systems. In 2005, Hickerson et al. [6] proposed a valveless impedance pump in which a net flow was induced by periodically pinching a flexible section asymmetrically from its ends. In their design, the optimized lengths of elastic and inelastic sections are 1.91 and 15.2 cm, respectively. Their experimental results showed the flow rates are sensitive to duty cycle and pinching frequency. In their study, the pump was also simulated and showed the wave speed traveling on the tube did not necessarily have the same velocity, nor must be in phase with the flow rate. The flow exiting the impedance pump is typically pulsatile and the net flow rate (−10.9∼9.0 mL/min) has a non-linear relationship to the frequency of activation with characteristic peaks and flow reversals. The same authors also constructed a one-dimensional wave model which predicted many of the characteristics exhibited by the experiments of impedance pumping [7]. Yeo et al. [8] presented an impedance pump utilizing a PZT cantilever beam with a high frequency actuation. Their microchannel had the dimensions of 15 × 3 × 0.4 mm

^{3}, and a flow rate of 36 μL/s. Recently, Chang et al. [9] designed and analyzed a valveless impedance pump in which the actuation mechanism comprised a permanent magnet mounted on a flexible PDMS diaphragm positioned above a copper plated micro-coil at a height of 630 μm, corresponding to the position of the maximum electromagnetic force on the magnet. The valveless impedance pumping effect (Liebau Phenomenon) was first reported by Gerhart Liebau in 1954 and numerically examined by Bozi and Propst [14]. Based on partially elastic rigid walls, the impedance pump was operated by the interaction between traveling waves emitted from the compression and reflected waves at the impedance-mismatched positions, it exhibits a non-linear response to the actuating compression frequency and flow reversal with actuating frequencies at certain ranges. In their study, the theoretical results showed that a diaphragm deflection of 15 μm could be obtained by passing a current of 0.6–0.7 A through the micro-coil in order to produce a compression force of 11 μN. The design of the micropump was easily fabricated and was readily integrated with existing microfluidic chips due to its planar structure.

## 2. Designs

## 3. Analysis

#### 3.1. Magnetic Analysis

_{z}is the vertical magnetic field produced by the coil, B

_{r}is the retentivity of the magnet, and S

_{m}, h

_{m}and V

_{m}are the surface area, thickness and volume of the magnet, respectively. From Figure 1, it can be inferred that H

_{z}is symmetrical about both the x- and the y-axes. Equation (1) indicates that the magnitude of the magnetic force varies with the rate of change of the magnetic field in the vertical direction. To maximize the diaphragm deflection, the magnetic diaphragm should be positioned such that it coincides with the point in the magnetic field at which the gradient attains its maximum value. In the current study, this position is identified by evaluating the magnetic field, H

_{z}, and the gradient of the magnetic field, ∂H

_{z}/ ∂z, numerically using the Ansoft/Maxwell 3D FEA software [10].

## 3.2. Actuator Displacement Analysis

_{y}= 20 kPa) [16]. Therefore, the diaphragm ensures a safe yet efficient pumping operation even under resonance conditions. In Figure 1, the maximum deflection takes place in the center of the diaphragm. Since the load imposed by the magnet is uniformly distributed, the diaphragm experiences a deflection over the circular area corresponding to the position of the magnet (Figure 2). The analysis commenced by considering the general case in which the load is uniformly distributed over a circle of radius b (0 < b < a). From Figure 2, the displacement field of the diaphragm is given respectively by (2) [9, 10, 12, 17]:

^{3}/ 12(1 − ν

^{2}) is the flexural rigidity of the diaphragm, in which E, ν and h are the elastic modulus, Poisson’s ratio and thickness of the diaphragm, respectively.

#### 3.3. Pumping Analysis

_{0}, was calculated in equation (4) at various applied electrical current. Non-uniform meshes are generated over the pumping chamber, the buffers and channels for the simulation, which are illustrated in Figure 3. In order to capture the detailed flow fields, the meshes are refined in the junctions between the chamber and the channels, and the junctions between the buffers and the channel ends. A mesh independent analysis was conducted to ensure reasonable results. The simulations were carried out by using the FLUENT 6.3 CFD software. The pressure at the pump inlet and outlet were set to be equal, and the pumping flow rates were calculated by averaging the periodic flow rates over one period.

## 4. Results and Discussion

## 5. Conclusions

## Acknowledgments

## References and Notes

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**Figure 1.**(a) Schematic illustration of valveless impedance pump. (b) Details of (a) [13].

**Figure 4.**Variation of flux density with distance as function of input current for micro-coils with different widths: (a) 125 μm, (b) 100 μm and (c) 75 μm.

**Figure 5.**Variations of magnetic field gradient with distance as function of input current for micro-coils with different widths: (a) 125 μm, (b) 100 μm and (c) 75 μm.

**Figure 6.**Variation of maximum diaphragm deflection with input current as function of coil width without driving liquid.

**Figure 7.**Variation of maximum diaphragm deflection with input current as function of PDMS diaphragm thickness without driving liquid.

**Figure 8.**Variation of maximum diaphragm deflection with input current as function of magnetic layer thickness without driving liquid.

**Figure 9.**Variation of flow rate with input current as function of magnetic layer thickness at constant actuation frequency of 240 Hz.

**Figure 10.**Variation of maximum diaphragm deflection with input current as function of PDMS diaphragm thickness at constant actuation frequency of 240 Hz.

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**MDPI and ACS Style**

Wang, Y.-H.; Tsai, Y.-W.; Tsai, C.-H.; Lee, C.-Y.; Fu, L.-M.
Design and Analysis of Impedance Pumps Utilizing Electromagnetic Actuation. *Sensors* **2010**, *10*, 4040-4052.
https://doi.org/10.3390/s100404040

**AMA Style**

Wang Y-H, Tsai Y-W, Tsai C-H, Lee C-Y, Fu L-M.
Design and Analysis of Impedance Pumps Utilizing Electromagnetic Actuation. *Sensors*. 2010; 10(4):4040-4052.
https://doi.org/10.3390/s100404040

**Chicago/Turabian Style**

Wang, Yu-Hisang, Yao-Wen Tsai, Chien-Hsiung Tsai, Chia-Yen Lee, and Lung-Ming Fu.
2010. "Design and Analysis of Impedance Pumps Utilizing Electromagnetic Actuation" *Sensors* 10, no. 4: 4040-4052.
https://doi.org/10.3390/s100404040