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Design, optimization and simulation on microelectromagnetic pump
Sensors and Actuators 83 Ž2000. 200–207 www.elsevier.nlrlocatersna
Design, optimization and simulation on microelectromagnetic pump Qiulian Gong ) , Zhaoying Zhou, Yihua Yang, Xiaohao Wang Micromachine DiÕision, Department of Precision Instruments, Tsinghua UniÕersity, Beijing 100084, People’s Republic of China Accepted 17 December 1999
Abstract A four-layer electromagnetic micropump was designed, and its static and dynamic characters were studied from elements to full system. The mechanical model of the electromagnetic actuator was established, and some its geometric structural parameters were then optimized. The deflection of the pump membrane caused by the varying local magnetic driving forces was analyzed in particular by ANSYS FEM. Fluid theory and experiment data were used to analyze the microvalve. Full system model of the micropump and its differential equations were set up on the basis of physical liquid transmission procedure and system function block. Pulse and periodic pulse driving current responses were evaluated. The flow-frequency characteristic shows that the most effective driving frequency is 125 Hz. The effect of driving signal duty cycle and microvalve flow resistance was studied. The transfer function of this full system of micropump was also discussed. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Micropump; FEM; Electromagnetic; Model and simulation
1. Introduction Micropump, as the actuator for microfluid transmission and control, has been actively researched in MEMS field w1x. As a microsystem, how to produce effective mechanical driving effect and precision flow control at microscale is the key topic that has been investigated. Electromagnetic actuation has stronger driving force over long distance w2x compared with electrostatic actuation. It has been widely used in micropump, micromotor and microrelay w1x. Electromagnetic micropump is also an attractive topic, especially low-voltage driving pump w3x. Its actuating procedure has a close relationship with its microstructure, solid–liquid coupling and individual elements inside the system. To achieve optimal microfluidic actuation with limited driving energy and to save development time and cost as well as explain the experimental results, it is necessary to do microstructure modeling, theoretical calculation, optimization and system dynamic numerical simulation. The micropump’s modeling involves elastic mechanics, fluid transmission, energy transformation among electricity, magnetism, heat and mechanism as well as energy coupling effects. Industrial macroscale hydraulic system modeling and dynamic simulation have had fruitful )
Corresponding author.
achievements in recent decades. But at microscale, limited by measurement or test conditions, few experimental data may be referred as system parameter identification. Previous researches were mainly focused on system scale or functional element scale such as microvalve and membrane. Finite element method ŽFEM. is applied to analyze and optimize microvalve w4x. Non-linear membrane deformation under electrostatic actuation was analyzed by differential equations w5x and FEM w6x. Regarding full system modeling and simulation, one solution is to create differential equations to describe the micropump’s working procedure after all individual blocks have been functionally construed. Another method applies mechanical–electrical equivalent network representation w7,8x to build system transfer functions, and then figure out micropump’s state or process parameters by computer-aided electrical circuit analysis software, such as SPICE, AMS, etc. This paper designed a prototype micropump and makes detailed investigations, from element to system, on its static characteristics and microsystem dynamics.
2. Micropump structure design and parameter optimization One four-layer structure micropump was designed as shown in Fig. 1. This electromagnetic micropump consists
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 3 8 4 - 2
Q. Gong et al.r Sensors and Actuators 83 (2000) 200–207
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Fig. 1. Structure diagram of four-layer micropump.
of electromagnetic actuator, pump chamber, passive microvalve and accessory interfaces. The top-located microelectromagnetic actuator was made up of planar coils shown in Fig. 2. Its base material is Fe–Nickel alloy. The cross-section of magnetic core is shaped as an ‘‘E’’ form; thus, the central air gap was formed during fabrication and provided free space to produce vertical displacement under electromagnetic driving force. The magnetic resistance of the magnetic core may be ignored in traditional structures. But at microscale, the magnetic resistance of ‘‘E’’ shape core cannot be ignored w9x due to its minor size and fabrication process Žusually by electroplating.. Its relative permeability is very low without heat treatment. This reduces electromagnetic force. The mechanical model of the actuator is shown in Fig. 3. To get quantitative design data, suppose that: Ø There is minor energy leakage around magnetic loop; Ø The magnetic core is not fully magnetized; and
Fig. 2. Top view of microelectromagnetic actuator.
Ø Any cross-section along the integral loop, l 0 y l 1 y l 2 y l 3 y l 4 y l 0 , has average magnetic induction intensity. Thus, the electromagnetic force Ž Fm . at initial position may be concluded w10x by Gauss Magnetic Law and Ample Law as follows: 2 a 2m 0 N 2 I 2
Fm s y
ž
l0 q
= lq
a2
mr4 h 4
2 l4 ln aq a
/
l4 a 2 ln aq a a q l4 mr4 h 4 l 0 a2 ln a
ž
/0
.
Ž 1.
Here, mr i and h i are relative permeability and magnetic core thickness, respectively. Parameter a is central pole dimension. The optimum goal was to obtain the strongest driving force under limited conditions by changing the structure
Fig. 3. Mechanical model of electromagnetic actuator.
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Fig. 4. Electromagnetic force and central pole area under different relative permeabilities.
parameters of the actuator. From Eq. Ž1., the driving force Fm will become stronger with the increase of coil numbers N and central pole scale a. But pump membrane area limits the increases of N and a at the same time. For given actuator area Ž6 = 6 mm2 ., parameter a can be chosen from Fig. 4 to get the strongest driving force. Here, coil width is 20 mm and coil gap is 10 mm. Initial air gap is 30 mm. Pump area is 3 = 3 mm2 . After fixing favorite a with mr4 s 5000, the relationship between Fm , the membrane central deflection d 0 , and driving current I can be expressed as formula Ž2. via curve fitting: Fm s Ž 62,752 d 03 y 722.3d 02 q 40.55d 0 q 2.78 . l 2 .
Ž 2.
3. FEM analysis on membrane deformation Pump membrane vibration process, if it is driven by period square wave signal, is more complex because the
electromagnetic force keeps changing due to the varying air gap lengths, which is an important factor to the electromagnetic force wcf. formula Ž1.x. The varying electromagnetic force on local membrane region deflects pump membrane furthermore. It is difficult to create analytic equations between membrane deflection and external driving force. FEM was applied to analyze the deformation procedure. As pump membrane and valve have minor mass, their resonant frequencies should be much higher than that of the micropump’s. Therefore, their dynamic characteristics can be skipped and their static deformation was emphasized. A quarter of the pump membrane was modeled due to its symmetry characters. The permalloy membrane was assumed to behave isotropic elastic. Shell element type and four-nodal quadrilateral element was used in FEM grid making. All degrees of the boundary were fixed to zero. In addition, large deflection analysis option was turned on to get the non-linear static effect of the membrane. The meshed quarter membrane and the local magnetic pressure applied on the membrane are shown in Fig. 5. FEM software ANSYSrED5.2 was applied to analyze the procedure. Firstly, in FEM analysis, iteration method was used to get the final position of membrane under electrical current I s 0.1 A. The curve 1 in Fig. 6 illustrates the electromagnetic driving pressure pm and different central deflection d 0 . Curve 2 presents further central deflection d 0 under variable electromagnetic driving pressures. Pump membrane will eventually approach its balance location because it has larger stiffness and the variation of electromagnetic force in this range can only cause limited deflection. According to FEM analysis, the offset between balance
Fig. 5. Boundary restriction and local load applied on membrane.
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Fig. 6. Electromagnetic driving pressure and membrane deformation.
Fig. 8. Membrane deformation under average distribution load.
position and the first deflection due to the initial electromagnetic pressure is very small compared with the first deflection. To simplify analysis later, we neglected this offset and derived the equation below: Pm s K 1 I 2 ,
Pump membrane stiffness under the uniform pressure was also analyzed by FEM as Žcf. Fig. 8.: Pe s aV 3 q bV 2 q cV .
Ž 4.
Ž 3. 4. Microvalve and micropipe
where K 1 is a constant figured out from formula Ž2.. Note that the electromagnetic force is sensitive to the initial air gap length. If the initial air gap is too narrow, the electromagnetic force will have rapid change. Narrow gap might cause pump membrane to be clamped onto central magnetic pole of the actuator. Furthermore, the membrane displacement will become larger and cannot be ignored. The reasonable and optimized initial air gap can be determined according to this analysis. Next, the FEM analysis was applied to study the relationship between electromagnetic driving pressure and its equivalent pressure distributed on pump membrane, because hydraulic pressure acts toward the membrane from cavity side at the same time when electromagnetic force is applied from planar coils. Our FEM analysis data showed this relationship within the pump pressure range Žcf. Fig. 7.. Average distributing pressure Pe can be linearly expressed by local pressure Pm equivalently as: Pe s K 2 Pm .
Based on laminar flow hypothesis under slow fluidic speed, the microvalve fluidics characteristics may be formulated as: Pv s Pi y Po s Pl13 q P35 . Pl13 is the pressure difference from parts 1 to 3 of Fig. 9. It can be expressed as: P l 1 3 s Ž 128 m LQr p D 4 . q Ž j 1 2 r Q 2 r2 d 24 . q Ž j 23 Q 2r2 d 34 .. P35 is the pressure difference from chamber 3 to outside Žpart 5. and can be expressed as: Q s CA q P35 where Q is volume flow, r is water density, m , C r are constants and j i is structure-related constant. The fluidic area Žacross. from port 3 through cantilever beam Ž A 4 . is a function of valve deflection that is also analyzed by ANSYS FEM. It is a linear function. The simulation curve 1 of the cantilever microvalve and its experiment data curve 2 are shown in Fig. 10. Curve 3 is the experiment data of another microvalve made by our
Fig. 7. Equivalent local and average distributing load.
Fig. 9. Cantilever valve model.
(
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Fig. 10. Simulation and experiment data of microvalve behavior. 3r2 . group, PSPI microvalve. In low flow range, Pl35 Ž Q A P35 is the main fact of Q ; Pv curve because Pl35 is too small. 1r2 . Pl13 ŽQ A Pl13 becomes larger and P35 gets smaller comparatively when the flow increases. When the differential pressure Pv ranges from zero to several decade kiloPascals Žthe driving pressure is just in this range., the simulation curve Žflow. Q via Pv behaves almost linearly. This can meet the experimental data of the cantilever microvalve, which also has linear flow resistance of about 2 kParŽmlrs. in lower pressure range. Thus, the valve may be expressed by a linear flow resistance R v as:
One pumping cycle, if we ignore the check valve’s backward leakage, can be divided into two phases. The liquid-in pumping process has the flowchart:
The liquid-out process is:
Pv s R v Q Turbulence and non-linear deformation of the cantilever beam under large pressure are not considered in curve 1. The actual Q in large pressure should be lower. The mechanical capacitor effect of the pipe caused by fluidic compression can also be neglected. The mechanical inductive effect due to water quantity effect is very small and may be ignored.
Thus, the model of the pump is shown in Fig. 11. A set of constant coefficient differential equations was established to express the micropump’s liquid-in and liquid-out behavior. Liquid-in behaves as w11,12x: Pm s K 1 I 2 ,
5. Micropump modeling In order to research the micropump’s dynamics, it is necessary to create mathematical models according to the fluid transmission procedure and the element characteristics. Integrated parameter modeling was applied in this paper w11,12x.
dÕ dt
s Qi ,
Pv s R v Q i ,
Pm ™ e y Pchamber s Pm rK 2 y Pv s aV 3 q bV 2 q cV . and liquid-out characters as: dV dt
s Qo ,
Pv s R v Q o ,
Pelastic s Pchamber s Pv s aV 3 q bV 2 q cV , where V is chamber volume variation. Parameter a, b, c, K 1 and K 2 are constants from previous analysis.
6. Numerical simulation results
Fig. 11. Micropump fluid transmission model.
The Runga–Kutta method w12x was applied to solve the above equations, supposing that the stimulating electrical
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Fig. 14. Volume variation curves under different signal duty cycles. Fig. 12. Micropump’s pulse responses and stimulating pulse width.
current was 0.1 A and the valve’s flow resistance was 2kParŽmlrs..
6.1. Pulse response To illustrate the micropump’s dynamic characteristics, system pulse response was first figured out. Fig. 12 shows V under different pulse widths. From the above numerical simulation, the following may be concluded. Ø When in-valve, chamber and out-valve behave as a serial system, positive step forcing responses are quicker than negative step forcing response. Positive step is an active pumping procedure caused by electromagnetic force.
Fig. 13. Volume variation with different impulse periods.
Fig. 15. Fluidic resistance and its effect on micropump dynamic response.
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Negative response is a passive pumping caused by membrane elastic effects. Ø To ensure sufficient pumping volume, input pulse should have sufficient amplitude and width Žstimulating duration.. The membrane, in outlet phase, needs some time to completely restore back to its initial position. 6.2. Periodic pulse response Periodic pulse responses were studied. Fig. 13 shows the examples of the micropump’s volume variation when the frequency of driving current is 100 Hz and 25 Hz. 6.3. Duty cycle Some optimizations were studied on impulse signal’s duty cycle. Fig. 14 shows the micropump’s volume variation with different duty cycles Ž1:1 and 1:2.. These simulation curves illustrated that the micropump’s pumping volume can restore to its initial position completely if the driving electric current has a reasonable duty cycle. This is especially important to improve the micropump’s working effect for energy-saving design. 6.4. The effect of ÕalÕe’s flow resistance Fig. 15 shows pumping volume rate under different valve resistance Žpulse width 0.04 s.. Curves 1, 2 and 3 correspond to valve flow resistance valued at 2, 3 and 4 kParŽmlrs., respectively. When flow resistance decreases, the micropump’s dynamic responses improved.
6.5. Volume flow at different driÕing frequency Flow volume–frequency simulation data were illustrated in Fig. 16. From simulation of curve 1, it may be seen that pumping volume rapidly increases when stimulated pulse signals have lower working frequencies. There is a maximum point at 125 Hz, and then it goes down slowly with the increase of pumping frequency. The actual flow curve should be curve 2, which took account of microvalve’s opening pressure and the dynamic characteristics of the membrane and valve. In summary, the maximum pumping flow and pulse frequency should be carefully determined. The flow will decrease at high driving frequency. Optimal working mode exists if you have a reasonable operation.
7. Conclusion This paper designed an electromagnetic micropump, and studied the micropump’s static and dynamic characters from element to system. An iteration method was used in FEM to get membrane deflection under varying electromagnetic driving forces. The final central deflection of the membrane is determined by the initial air gap and driving electrical current. Reasonable choice of initial air gap can avoid membrane clamped onto central magnetic pole and favor fabrication and assembling later. The micropump’s dynamic response has a close relationship with its chamber structure, compliance, fluid resistance along micropipe and through check valves. Chamber liquid capacitor Žcompliance. is the main factor which limits pump dynamics. Based on the paper’s simulation results, the maximum flow occurred when driving frequency was around 125 Hz for the paper’s model system. Elastic pump membrane and microvalve have much less mass than chamber liquid quantity. They have higher resonant frequencies and quick dynamic responses. The paper’s numerical simulation results — ignoring membrane and valve dynamics — have fluidic curves similar to experimental data. Future research may focus on their static characteristics.
8. Discussions
Fig. 16. The volume variation and stimulated pulse frequencies.
Micropump flow is determined by driving signal frequency and elastic membrane vibration amplitude. The maximum flow Žoccurring at 125 Hz. does not mean that the membrane approaches resonant status, or maximum deflection. This only implies its effective working frequency. For those solo-direction-driven pump Žwith electrostatic or electromagnetic microactuator., pump membrane can
Q. Gong et al.r Sensors and Actuators 83 (2000) 200–207
only passively recover by its elastic force. No external action contributes or accelerates the procedure. It has worse dynamic features than bi-directional-driven pump via the same valve’s flow resistant effect. When the electrical current I varies, formula Ž3. should be expressed as: Pm s K 1Ž I . I 2 , and K Ž I . can be determined by formulas Ž2. and Ž4.. Micropump’s transfer function in inlet and outlet procedures can be expressed, respectively, as: K 1Ž I . I 2 s R v
dV dt
w8x
w9x
w10x
w11x
q aV 3 q bV 2 q cV ,
and Rv
dV dt
s aV 3 q bV 2 q cV .
w12x
207
of a flap valve micropump based on Kirchhoffian network theory, Sensors and Actuators A 66 Ž1998. 9–14. T. Bourouina, J.P. Grandchamp, Modeling micropumps with electrical equivalent networks, Journal of Micromechanics and Microengineering 6 Ž1996. 398–404. C.H. Ann, Micromachined components as integrated inductors and magnetic micro actuators, PhD Dissertation of Geogia Institute of Technology, 1993. Q. Gong, Z. Zhou, Y. Yang, in: Micro Electromagnetic Actuator for Micropump, Proceedings of 3rd International Symposium on Test and Measurement, Xian, China, June 2–4, 1999, pp. 15–18. Y. Yang et al., in: Ischemic Disease Mechanical Auxiliary Counterpulsating System Based Upon Electro-Pneumatic Servo Control And Hemodynamic Simulation, Proceedings the 1996 18th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Part 3 Žof 5., Amsterdam, Netherlands, Oct. 31– Nov. 3 Vol. 31996. Z. Liu, Kindkessel modeling and its application in cardiovascular system analysis, China Science Press Ž1987..
Acknowledgements
Biographies
The authors are grateful to Prof. Xiongying, Ye, Shenshu Xiong and other colleagues in microfluid system division who have shared their knowledge with us and provided the valuable material used in this paper. We are also thankful to Prof. Zongxia Jiao of BUAA for his valued suggestions on modeling and numeric simulation of microfluid transmission.
Qiulian Gong received her MS and BS degrees in 1999 and 1987 respectively. She is now an assistant professor in the micromachine division of the Department of Precision Instruments, Tsinghua University. Since 1996, she started to research electromagnetic actuator, sensor and Quasi-LIGA fabrication processing. Her research interest is in microfluidic system modeling, numerical simulation, MEMS fabrication for sensors and actuators. She has published two technical books and eight academic papers on computer science, micromachine and dynamic measurement.
References w1x Z.Y. Zhou, X.Y. Ye, Y. Li, Z. You, T.H. Cui, Y. Yang, X.N. Jiang, M. Hu, S.S. Xiong, W.D. Zhang, H.N. Cai, J. Zou, L. Zhang, X.H. Wang, X.Q. Lu, Research on microfluid handling systems, Journal of the Institution of Engineers ŽSingapore. 38 Ž4. Ž1998. 7–14. w2 x Q. Gong, Z. Zhou, in: Micromachined Electromagnetic Actuator,Proceedings of 3rd International Symposium on Test and Measurement, Xian, China, June 2–4, 1999, pp. 23–26. w3x W. Zhang, C.H. Ahn, A bi-directional magnetic micropump on a silicon wafer, in: Tech. Dig. Solid-State Sens. Actuators Workshop, Hilton Head, South Carolina, June 3–6, 1996, pp. 94–97. w4x A. Hzhofer, B. Ritter, Ch. Tsakmakis, Development of passive microvalves by the finite element method, Journal of Micromechanics and Microengineering 5 Ž3. Ž1995. 226–230, September. w5x O. Francais, I. Dufour, E. Sarraute, Analytical static modeling and optimization of electrostatic micropumps, Journal of Micromechanics and Microengineering 7 Ž3. Ž1997. 183–185, September. w6x A. Cozma, R. Puers, Electrostatic actuation as a self-testing method for silicon pressure sensors, Sensors and Actuators A 60 Ž1997. 32–36. w7x P. Voigt, G. Schrag, G. Wachutka, Electrofluid full-system modeling
Zhaoying Zhou graduated from the Department of Precision Instruments, Tsinghua University, Beijing, China, in 1961. He is a professor and chairman of academic committee, the MicrorNano Technology Research Center, Tsinghua University. He is the President of the Beijing Instrument Society, a fellow of the Chinese Instrument Society, and the Chinese Mechanical Engineering Society, and deputy chief editor of the Journal of Chinese Instrumentation. Prof. Zhou has published five technical books, and more than 200 scientific and technical papers. His research interest is in the fields of measurement, control and MEMS, and he is involved in three state key MEMS projects. Yihua Yang earned his PhD in 1991, major in Fluid Power Transmission and Control from Zhejiang University, Hangzhou, China. His current research interest is in microsensor and microsystem integration, fluidic modeling and simulation. He is a senior member of the China Mechanical Engineering Society and has published four technical books and 25 journal or conference papers. Xiaohao Wang received his PhD degree in engineering science from the Department of Precision Instruments and Mechanology of Tsinghua University where he was engaged in research in the field of microfluidics and microfluidic MEMS devices. At present, he is working in the Micro and Nano Technology Research Center, and studying microactuators and their application.
Design, optimization and simulation on microelectromagnetic pump Qiulian Gong ) , Zhaoying Zhou, Yihua Yang, Xiaohao Wang Micromachine DiÕision, Department of Precision Instruments, Tsinghua UniÕersity, Beijing 100084, People’s Republic of China Accepted 17 December 1999
Abstract A four-layer electromagnetic micropump was designed, and its static and dynamic characters were studied from elements to full system. The mechanical model of the electromagnetic actuator was established, and some its geometric structural parameters were then optimized. The deflection of the pump membrane caused by the varying local magnetic driving forces was analyzed in particular by ANSYS FEM. Fluid theory and experiment data were used to analyze the microvalve. Full system model of the micropump and its differential equations were set up on the basis of physical liquid transmission procedure and system function block. Pulse and periodic pulse driving current responses were evaluated. The flow-frequency characteristic shows that the most effective driving frequency is 125 Hz. The effect of driving signal duty cycle and microvalve flow resistance was studied. The transfer function of this full system of micropump was also discussed. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Micropump; FEM; Electromagnetic; Model and simulation
1. Introduction Micropump, as the actuator for microfluid transmission and control, has been actively researched in MEMS field w1x. As a microsystem, how to produce effective mechanical driving effect and precision flow control at microscale is the key topic that has been investigated. Electromagnetic actuation has stronger driving force over long distance w2x compared with electrostatic actuation. It has been widely used in micropump, micromotor and microrelay w1x. Electromagnetic micropump is also an attractive topic, especially low-voltage driving pump w3x. Its actuating procedure has a close relationship with its microstructure, solid–liquid coupling and individual elements inside the system. To achieve optimal microfluidic actuation with limited driving energy and to save development time and cost as well as explain the experimental results, it is necessary to do microstructure modeling, theoretical calculation, optimization and system dynamic numerical simulation. The micropump’s modeling involves elastic mechanics, fluid transmission, energy transformation among electricity, magnetism, heat and mechanism as well as energy coupling effects. Industrial macroscale hydraulic system modeling and dynamic simulation have had fruitful )
Corresponding author.
achievements in recent decades. But at microscale, limited by measurement or test conditions, few experimental data may be referred as system parameter identification. Previous researches were mainly focused on system scale or functional element scale such as microvalve and membrane. Finite element method ŽFEM. is applied to analyze and optimize microvalve w4x. Non-linear membrane deformation under electrostatic actuation was analyzed by differential equations w5x and FEM w6x. Regarding full system modeling and simulation, one solution is to create differential equations to describe the micropump’s working procedure after all individual blocks have been functionally construed. Another method applies mechanical–electrical equivalent network representation w7,8x to build system transfer functions, and then figure out micropump’s state or process parameters by computer-aided electrical circuit analysis software, such as SPICE, AMS, etc. This paper designed a prototype micropump and makes detailed investigations, from element to system, on its static characteristics and microsystem dynamics.
2. Micropump structure design and parameter optimization One four-layer structure micropump was designed as shown in Fig. 1. This electromagnetic micropump consists
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 3 8 4 - 2
Q. Gong et al.r Sensors and Actuators 83 (2000) 200–207
201
Fig. 1. Structure diagram of four-layer micropump.
of electromagnetic actuator, pump chamber, passive microvalve and accessory interfaces. The top-located microelectromagnetic actuator was made up of planar coils shown in Fig. 2. Its base material is Fe–Nickel alloy. The cross-section of magnetic core is shaped as an ‘‘E’’ form; thus, the central air gap was formed during fabrication and provided free space to produce vertical displacement under electromagnetic driving force. The magnetic resistance of the magnetic core may be ignored in traditional structures. But at microscale, the magnetic resistance of ‘‘E’’ shape core cannot be ignored w9x due to its minor size and fabrication process Žusually by electroplating.. Its relative permeability is very low without heat treatment. This reduces electromagnetic force. The mechanical model of the actuator is shown in Fig. 3. To get quantitative design data, suppose that: Ø There is minor energy leakage around magnetic loop; Ø The magnetic core is not fully magnetized; and
Fig. 2. Top view of microelectromagnetic actuator.
Ø Any cross-section along the integral loop, l 0 y l 1 y l 2 y l 3 y l 4 y l 0 , has average magnetic induction intensity. Thus, the electromagnetic force Ž Fm . at initial position may be concluded w10x by Gauss Magnetic Law and Ample Law as follows: 2 a 2m 0 N 2 I 2
Fm s y
ž
l0 q
= lq
a2
mr4 h 4
2 l4 ln aq a
/
l4 a 2 ln aq a a q l4 mr4 h 4 l 0 a2 ln a
ž
/0
.
Ž 1.
Here, mr i and h i are relative permeability and magnetic core thickness, respectively. Parameter a is central pole dimension. The optimum goal was to obtain the strongest driving force under limited conditions by changing the structure
Fig. 3. Mechanical model of electromagnetic actuator.
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Fig. 4. Electromagnetic force and central pole area under different relative permeabilities.
parameters of the actuator. From Eq. Ž1., the driving force Fm will become stronger with the increase of coil numbers N and central pole scale a. But pump membrane area limits the increases of N and a at the same time. For given actuator area Ž6 = 6 mm2 ., parameter a can be chosen from Fig. 4 to get the strongest driving force. Here, coil width is 20 mm and coil gap is 10 mm. Initial air gap is 30 mm. Pump area is 3 = 3 mm2 . After fixing favorite a with mr4 s 5000, the relationship between Fm , the membrane central deflection d 0 , and driving current I can be expressed as formula Ž2. via curve fitting: Fm s Ž 62,752 d 03 y 722.3d 02 q 40.55d 0 q 2.78 . l 2 .
Ž 2.
3. FEM analysis on membrane deformation Pump membrane vibration process, if it is driven by period square wave signal, is more complex because the
electromagnetic force keeps changing due to the varying air gap lengths, which is an important factor to the electromagnetic force wcf. formula Ž1.x. The varying electromagnetic force on local membrane region deflects pump membrane furthermore. It is difficult to create analytic equations between membrane deflection and external driving force. FEM was applied to analyze the deformation procedure. As pump membrane and valve have minor mass, their resonant frequencies should be much higher than that of the micropump’s. Therefore, their dynamic characteristics can be skipped and their static deformation was emphasized. A quarter of the pump membrane was modeled due to its symmetry characters. The permalloy membrane was assumed to behave isotropic elastic. Shell element type and four-nodal quadrilateral element was used in FEM grid making. All degrees of the boundary were fixed to zero. In addition, large deflection analysis option was turned on to get the non-linear static effect of the membrane. The meshed quarter membrane and the local magnetic pressure applied on the membrane are shown in Fig. 5. FEM software ANSYSrED5.2 was applied to analyze the procedure. Firstly, in FEM analysis, iteration method was used to get the final position of membrane under electrical current I s 0.1 A. The curve 1 in Fig. 6 illustrates the electromagnetic driving pressure pm and different central deflection d 0 . Curve 2 presents further central deflection d 0 under variable electromagnetic driving pressures. Pump membrane will eventually approach its balance location because it has larger stiffness and the variation of electromagnetic force in this range can only cause limited deflection. According to FEM analysis, the offset between balance
Fig. 5. Boundary restriction and local load applied on membrane.
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Fig. 6. Electromagnetic driving pressure and membrane deformation.
Fig. 8. Membrane deformation under average distribution load.
position and the first deflection due to the initial electromagnetic pressure is very small compared with the first deflection. To simplify analysis later, we neglected this offset and derived the equation below: Pm s K 1 I 2 ,
Pump membrane stiffness under the uniform pressure was also analyzed by FEM as Žcf. Fig. 8.: Pe s aV 3 q bV 2 q cV .
Ž 4.
Ž 3. 4. Microvalve and micropipe
where K 1 is a constant figured out from formula Ž2.. Note that the electromagnetic force is sensitive to the initial air gap length. If the initial air gap is too narrow, the electromagnetic force will have rapid change. Narrow gap might cause pump membrane to be clamped onto central magnetic pole of the actuator. Furthermore, the membrane displacement will become larger and cannot be ignored. The reasonable and optimized initial air gap can be determined according to this analysis. Next, the FEM analysis was applied to study the relationship between electromagnetic driving pressure and its equivalent pressure distributed on pump membrane, because hydraulic pressure acts toward the membrane from cavity side at the same time when electromagnetic force is applied from planar coils. Our FEM analysis data showed this relationship within the pump pressure range Žcf. Fig. 7.. Average distributing pressure Pe can be linearly expressed by local pressure Pm equivalently as: Pe s K 2 Pm .
Based on laminar flow hypothesis under slow fluidic speed, the microvalve fluidics characteristics may be formulated as: Pv s Pi y Po s Pl13 q P35 . Pl13 is the pressure difference from parts 1 to 3 of Fig. 9. It can be expressed as: P l 1 3 s Ž 128 m LQr p D 4 . q Ž j 1 2 r Q 2 r2 d 24 . q Ž j 23 Q 2r2 d 34 .. P35 is the pressure difference from chamber 3 to outside Žpart 5. and can be expressed as: Q s CA q P35 where Q is volume flow, r is water density, m , C r are constants and j i is structure-related constant. The fluidic area Žacross. from port 3 through cantilever beam Ž A 4 . is a function of valve deflection that is also analyzed by ANSYS FEM. It is a linear function. The simulation curve 1 of the cantilever microvalve and its experiment data curve 2 are shown in Fig. 10. Curve 3 is the experiment data of another microvalve made by our
Fig. 7. Equivalent local and average distributing load.
Fig. 9. Cantilever valve model.
(
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Fig. 10. Simulation and experiment data of microvalve behavior. 3r2 . group, PSPI microvalve. In low flow range, Pl35 Ž Q A P35 is the main fact of Q ; Pv curve because Pl35 is too small. 1r2 . Pl13 ŽQ A Pl13 becomes larger and P35 gets smaller comparatively when the flow increases. When the differential pressure Pv ranges from zero to several decade kiloPascals Žthe driving pressure is just in this range., the simulation curve Žflow. Q via Pv behaves almost linearly. This can meet the experimental data of the cantilever microvalve, which also has linear flow resistance of about 2 kParŽmlrs. in lower pressure range. Thus, the valve may be expressed by a linear flow resistance R v as:
One pumping cycle, if we ignore the check valve’s backward leakage, can be divided into two phases. The liquid-in pumping process has the flowchart:
The liquid-out process is:
Pv s R v Q Turbulence and non-linear deformation of the cantilever beam under large pressure are not considered in curve 1. The actual Q in large pressure should be lower. The mechanical capacitor effect of the pipe caused by fluidic compression can also be neglected. The mechanical inductive effect due to water quantity effect is very small and may be ignored.
Thus, the model of the pump is shown in Fig. 11. A set of constant coefficient differential equations was established to express the micropump’s liquid-in and liquid-out behavior. Liquid-in behaves as w11,12x: Pm s K 1 I 2 ,
5. Micropump modeling In order to research the micropump’s dynamics, it is necessary to create mathematical models according to the fluid transmission procedure and the element characteristics. Integrated parameter modeling was applied in this paper w11,12x.
dÕ dt
s Qi ,
Pv s R v Q i ,
Pm ™ e y Pchamber s Pm rK 2 y Pv s aV 3 q bV 2 q cV . and liquid-out characters as: dV dt
s Qo ,
Pv s R v Q o ,
Pelastic s Pchamber s Pv s aV 3 q bV 2 q cV , where V is chamber volume variation. Parameter a, b, c, K 1 and K 2 are constants from previous analysis.
6. Numerical simulation results
Fig. 11. Micropump fluid transmission model.
The Runga–Kutta method w12x was applied to solve the above equations, supposing that the stimulating electrical
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Fig. 14. Volume variation curves under different signal duty cycles. Fig. 12. Micropump’s pulse responses and stimulating pulse width.
current was 0.1 A and the valve’s flow resistance was 2kParŽmlrs..
6.1. Pulse response To illustrate the micropump’s dynamic characteristics, system pulse response was first figured out. Fig. 12 shows V under different pulse widths. From the above numerical simulation, the following may be concluded. Ø When in-valve, chamber and out-valve behave as a serial system, positive step forcing responses are quicker than negative step forcing response. Positive step is an active pumping procedure caused by electromagnetic force.
Fig. 13. Volume variation with different impulse periods.
Fig. 15. Fluidic resistance and its effect on micropump dynamic response.
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Negative response is a passive pumping caused by membrane elastic effects. Ø To ensure sufficient pumping volume, input pulse should have sufficient amplitude and width Žstimulating duration.. The membrane, in outlet phase, needs some time to completely restore back to its initial position. 6.2. Periodic pulse response Periodic pulse responses were studied. Fig. 13 shows the examples of the micropump’s volume variation when the frequency of driving current is 100 Hz and 25 Hz. 6.3. Duty cycle Some optimizations were studied on impulse signal’s duty cycle. Fig. 14 shows the micropump’s volume variation with different duty cycles Ž1:1 and 1:2.. These simulation curves illustrated that the micropump’s pumping volume can restore to its initial position completely if the driving electric current has a reasonable duty cycle. This is especially important to improve the micropump’s working effect for energy-saving design. 6.4. The effect of ÕalÕe’s flow resistance Fig. 15 shows pumping volume rate under different valve resistance Žpulse width 0.04 s.. Curves 1, 2 and 3 correspond to valve flow resistance valued at 2, 3 and 4 kParŽmlrs., respectively. When flow resistance decreases, the micropump’s dynamic responses improved.
6.5. Volume flow at different driÕing frequency Flow volume–frequency simulation data were illustrated in Fig. 16. From simulation of curve 1, it may be seen that pumping volume rapidly increases when stimulated pulse signals have lower working frequencies. There is a maximum point at 125 Hz, and then it goes down slowly with the increase of pumping frequency. The actual flow curve should be curve 2, which took account of microvalve’s opening pressure and the dynamic characteristics of the membrane and valve. In summary, the maximum pumping flow and pulse frequency should be carefully determined. The flow will decrease at high driving frequency. Optimal working mode exists if you have a reasonable operation.
7. Conclusion This paper designed an electromagnetic micropump, and studied the micropump’s static and dynamic characters from element to system. An iteration method was used in FEM to get membrane deflection under varying electromagnetic driving forces. The final central deflection of the membrane is determined by the initial air gap and driving electrical current. Reasonable choice of initial air gap can avoid membrane clamped onto central magnetic pole and favor fabrication and assembling later. The micropump’s dynamic response has a close relationship with its chamber structure, compliance, fluid resistance along micropipe and through check valves. Chamber liquid capacitor Žcompliance. is the main factor which limits pump dynamics. Based on the paper’s simulation results, the maximum flow occurred when driving frequency was around 125 Hz for the paper’s model system. Elastic pump membrane and microvalve have much less mass than chamber liquid quantity. They have higher resonant frequencies and quick dynamic responses. The paper’s numerical simulation results — ignoring membrane and valve dynamics — have fluidic curves similar to experimental data. Future research may focus on their static characteristics.
8. Discussions
Fig. 16. The volume variation and stimulated pulse frequencies.
Micropump flow is determined by driving signal frequency and elastic membrane vibration amplitude. The maximum flow Žoccurring at 125 Hz. does not mean that the membrane approaches resonant status, or maximum deflection. This only implies its effective working frequency. For those solo-direction-driven pump Žwith electrostatic or electromagnetic microactuator., pump membrane can
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only passively recover by its elastic force. No external action contributes or accelerates the procedure. It has worse dynamic features than bi-directional-driven pump via the same valve’s flow resistant effect. When the electrical current I varies, formula Ž3. should be expressed as: Pm s K 1Ž I . I 2 , and K Ž I . can be determined by formulas Ž2. and Ž4.. Micropump’s transfer function in inlet and outlet procedures can be expressed, respectively, as: K 1Ž I . I 2 s R v
dV dt
w8x
w9x
w10x
w11x
q aV 3 q bV 2 q cV ,
and Rv
dV dt
s aV 3 q bV 2 q cV .
w12x
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Acknowledgements
Biographies
The authors are grateful to Prof. Xiongying, Ye, Shenshu Xiong and other colleagues in microfluid system division who have shared their knowledge with us and provided the valuable material used in this paper. We are also thankful to Prof. Zongxia Jiao of BUAA for his valued suggestions on modeling and numeric simulation of microfluid transmission.
Qiulian Gong received her MS and BS degrees in 1999 and 1987 respectively. She is now an assistant professor in the micromachine division of the Department of Precision Instruments, Tsinghua University. Since 1996, she started to research electromagnetic actuator, sensor and Quasi-LIGA fabrication processing. Her research interest is in microfluidic system modeling, numerical simulation, MEMS fabrication for sensors and actuators. She has published two technical books and eight academic papers on computer science, micromachine and dynamic measurement.
References w1x Z.Y. Zhou, X.Y. Ye, Y. Li, Z. You, T.H. Cui, Y. Yang, X.N. Jiang, M. Hu, S.S. Xiong, W.D. Zhang, H.N. Cai, J. Zou, L. Zhang, X.H. Wang, X.Q. Lu, Research on microfluid handling systems, Journal of the Institution of Engineers ŽSingapore. 38 Ž4. Ž1998. 7–14. w2 x Q. Gong, Z. Zhou, in: Micromachined Electromagnetic Actuator,Proceedings of 3rd International Symposium on Test and Measurement, Xian, China, June 2–4, 1999, pp. 23–26. w3x W. Zhang, C.H. Ahn, A bi-directional magnetic micropump on a silicon wafer, in: Tech. Dig. Solid-State Sens. Actuators Workshop, Hilton Head, South Carolina, June 3–6, 1996, pp. 94–97. w4x A. Hzhofer, B. Ritter, Ch. Tsakmakis, Development of passive microvalves by the finite element method, Journal of Micromechanics and Microengineering 5 Ž3. Ž1995. 226–230, September. w5x O. Francais, I. Dufour, E. Sarraute, Analytical static modeling and optimization of electrostatic micropumps, Journal of Micromechanics and Microengineering 7 Ž3. Ž1997. 183–185, September. w6x A. Cozma, R. Puers, Electrostatic actuation as a self-testing method for silicon pressure sensors, Sensors and Actuators A 60 Ž1997. 32–36. w7x P. Voigt, G. Schrag, G. Wachutka, Electrofluid full-system modeling
Zhaoying Zhou graduated from the Department of Precision Instruments, Tsinghua University, Beijing, China, in 1961. He is a professor and chairman of academic committee, the MicrorNano Technology Research Center, Tsinghua University. He is the President of the Beijing Instrument Society, a fellow of the Chinese Instrument Society, and the Chinese Mechanical Engineering Society, and deputy chief editor of the Journal of Chinese Instrumentation. Prof. Zhou has published five technical books, and more than 200 scientific and technical papers. His research interest is in the fields of measurement, control and MEMS, and he is involved in three state key MEMS projects. Yihua Yang earned his PhD in 1991, major in Fluid Power Transmission and Control from Zhejiang University, Hangzhou, China. His current research interest is in microsensor and microsystem integration, fluidic modeling and simulation. He is a senior member of the China Mechanical Engineering Society and has published four technical books and 25 journal or conference papers. Xiaohao Wang received his PhD degree in engineering science from the Department of Precision Instruments and Mechanology of Tsinghua University where he was engaged in research in the field of microfluidics and microfluidic MEMS devices. At present, he is working in the Micro and Nano Technology Research Center, and studying microactuators and their application.