2D-finite-element simulations for long-range capacitive position sensor
INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF MICROMECHANICS AND MICROENGINEERING
J. Micromech. Microeng. 13 (2003) S183–S189
PII: S0960-1317(03)60542-5
2D-finite-element simulations for long-range capacitive position sensor A A Kuijpers, G J M Krijnen, R J Wiegerink, T S J Lammerink and M Elwenspoek Transducer Science and Technology Group, MESA + Research Institute, University of Twente, PO Box 217, 7500 AE, Enschede, The Netherlands E-mail: [email protected]
Received 7 May 2003, in final form 22 May 2003 Published 13 June 2003 Online at stacks.iop.org/JMM/13/S183 Abstract This paper presents the results of 2D-finite-element simulations of the periodic change in capacitance between two periodic structures, one sliding and one fixed (i.e. sense-structure). These structures are potentially interesting for long-range and high-accuracy position detection of microactuators. The use of periodic geometries and the combination of a discrete (incremental) and analog measurement relieves the demands for accuracy. The discrete measurement involves counting the number of periods. The (capacitance) analog measurement determines the position within one period, preferably with nm-accuracy. Two concepts are presented: (i) open-loop measurement of capacitance change versus slider displacement, (ii) closed-loop control of capacitance change versus slider displacement (i.e. sense-structure is actuated in orthogonal direction to the slider motion). These concepts are independent of their application in micro-scale devices but the realization raises particular challenges involving capacitance measurement and micromachining techniques. The periodic patterns examined in these simulations contain rectangular, triangular and sinusoidal shapes and several combinations. Simulation results for both concepts show measurable periodic changes in capacitance. For two arrays of 50 rectangular finger pairs along the sides of the slider the capacitance Cs ≈ 15 fF and Cmax ≈ 7 fF are predicted. Application of this concept in a micromachined device is advantageous because sensor and actuator are integrated in the same structural layer and allows fabrication using one lithographic mask only.
1. Introduction 1.1. Sensing principle for a periodic capacitive position sensor In this paper we present the modeling and results of finiteelement simulations for a micromachined capacitive position sensor, whose concept is given in figure 1. With this sensor we aim at nm-range accuracy over ten’s of µm-range displacement. We want to combine a discrete or incremental measurement with an analog measurement to relieve the demands on accuracy. In a discrete measurement the number of periods is counted over the total range. In the analog 0960-1317/03/040183+07$30.00
measurement the position is measured with nm-accuracy within one period. This combination makes it easier to measure with nm-accuracy over the total range. Both the sliderbeam and sense-structure have a periodic pattern on their sides (e.g. rectangular-shaped fingers or a sinusoidal pattern). As the sliderbeam is moving (x-direction) the capacitance will change periodically. Counting the number of periods and measuring the change in capacitance will give a long-range position measurement with potentially high accuracy. Advantages of this novel concept are: • Accurate long-range measurement is possible due to periodic geometry and combination of discrete and analog measurement.
© 2003 IOP Publishing Ltd Printed in the UK
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Figure 1. (Left) The concept of a periodic capacitive position sensor alongside a slider beam, which is driven by two drive-actuators. (Right) Example of a micromachined polysilicon structure that forms a periodic capacitive position sensor.
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Figure 2. (Left) Photograph of a realization of a capacitive position sensor for an electrostatic comb-drive actuator. (Top-right) Part of the mask-design of the sense-structure with a periodic capacitive position sensor alongside the sliderbeam. (Bottom-right) Cross-view of the structure showing its layer profile.
• Electrostatic forces are balanced in a symmetric and inplane design. • Fabrication is relatively easy because, sensor and actuator are in the same layer and made with the same technology. The large-range capacitive position sensor presented by Pedrocchi et al [1] and tested on a 10× scale PC board, contains arrays of long strip electrodes on both stator and movable plates. These arrays form a capacitance with out-ofplane orientation (i.e. normal to the substrate), which raises the problem of balancing the electrostatic force [2]. It is also more difficult to make with micromachining technology. 1.2. Choice of drive-actuator We use an electrostatic comb-drive actuator as a test vehicle for our designed position sensor. As with all electrostatic microactuators, comb-drive actuators have good scaling properties as the ratio of electrostatic force to volume scales with λ−1 increasing with downscaling (λ = dimensionless scale parameter) [3]. Typical characteristics of our comb-drive actuators for large displacements are a continuous actuation with deflections of about 30 µm at driving voltages around 20 V [4]. Furthermore, with two coupled comb-actuators in a so-called push–pull configuration [5], the displacement-tovoltage relation can be made linear and the range of motion S184
is increased. The fabrication of comb-drives is relatively easy using a one-mask surface micromachining technology. Naturally, a comb-structure can also be used as a capacitive position sensor. A good example is given in [6] where two comb-structures are used, one for driving, one as a position sensor in a Kalman filtering scheme for a statevariable feedback control loop. A drawback of a comb-sensor is that for large displacements the sensitivity (i.e. C/C) decreases. 1.3. Fabrication The micromachined capacitive position sensor in figure 2 (left) consists of three layers as depicted by the cross-view in figure 2 (right). The first layer is a boron-doped silicon substrate. The second layer is a 3 µm thick sacrificial siliconoxide (SiO2) layer with a 5 µm thick poly-silicon structural layer on top. This thickness is limited to 5 µm by the standard available poly-silicon deposition process (PECVD) of the MESA + laboratory. The structure is made by lithographic patterning and reactive ion etching (RIE) of the poly layer. The sacrificial SiO2 layer under the structure is etched selectively by isotropic BHF-etching. The structural poly-silicon parts become movable and suspended by the flexure beams. All large movable parts have a truss-like shape to let the etchant uderetch these parts well before the solid anchor points are
2D-finite-element simulations for long-range capacitive position sensor
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Figure 3. (Left) Basic periodic geometry as used in the FE model to simulate the change in capacitance between two periodic patterns where one is moving in relation to the other. (Right) This study includes simulations with different shapes of ‘fingers’.
fully underetched too. The width of beams and gaps in this design is 2 µm. It is close to the achievable resolution of the standard photolithography process in the MESA+ clean-room laboratory and it makes the (under) etch time to be uniform for the whole design. The next section gives the 2D finite-element modeling and simulation results for a periodic capacitive position sensor.
2. FEM simulations for capacitance sensor 2.1. Calculation of capacitance change versus displacement: open-loop operation According to Feynman, the only general methods of solution of Laplace’s equation are numerical. For analytical solutions of field distributions the complex variable technique is often powerful but limited to 2D problems, and it also is an indirect method [7]. Therefore, the approach has been to perform finite-element model (FEM) simulations to study the concept presented in this paper. Figure 2 presents the basic periodic geometry used in the program FEMLAB to run the FEM simulations. Initially, the parameters used in the 2D-simulations for the FE-model in figure 3 with rectangular fingers, assuming infinite structure height, are (µm): width of fingers w = 2, length fingers lf = 6, gap distance g = 1, (lateral) period Px = 12, bias-gap-distance ly = 2.lf + g. The width w of 2 µm comes from practical considerations as explained in section 1.3. We have also simulated different shapes and combinations of dissimilar shapes of ‘fingers’ as well as for different period Px, length lf and gap distance g. The capacitance (per meter of height) is calculated by calculating the electrostatic energy in the volume between the fingers with a given voltage difference U, i.e. 1 1 ε0 · E 2 · dv We = · C · U 2 = 2 2 1 → C = 2 ε0 · E 2 · dv. (1) U After meshing the geometry into finite elements the electrostatic E-vector is calculated by solving the Laplace’s
equation ∇ 2 V = 0 for all (area) elements. Dirichlet boundary conditions are used on the real geometric boundaries, by stating the potential. The Neumann boundary conditions on the vertical ‘artificial’ segment boundaries are implemented as periodic boundary conditions so that the calculated values for the E-field on the left side of the geometry in figure 2 are identical to values on the right side. This insures that the calculated field distribution corresponds to infinite repetition of a geometry segment. The simulated capacitance per meter of height is multiplied with the structure height of h = 5 µm (i.e. polysilicon layer thickness). This results in the (approximate) capacitance per geometry-segment, containing one pair of fingers. Due to the scaling properties of the 2D-Laplace equation, the units of parameters w, Px , Lf , g are not important but only their relative proportions. Only the units of the dielectric permitivity ε0 (F/m) and the structure height h (m) for calculation of the energy or capacitance (equation (1)) are important. It is preferred to have a unitfree normalized displacement (xn = x/Px), to better compare between geometries with different period (Px). Figure 4 gives the simulated capacitances against normalized displacement (x/Px) for three differently shaped finger pairs with a lateral period of Px = 12 (µm). As the displacement in x-direction between the pair of fingers is zero, the fingers are ‘in-phase’ and the capacitance is maximal. For a displacement of 12 Px the fingers are ‘outof-phase’ and the capacitance is minimum and rising again for larger displacements. For each shape of finger pair the change in capacitance for every displacement is calculated i.e. C(x) = C(x) − Cmin. From figure 4 it is clear that the combination of two arrays of rectangular-shaped fingers is showing the largest absolute capacitance as well as the largest change in capacitance for each displacement of the arrays with respect to each other. We also examined the influence of combinations of dissimilar ‘finger’ shapes, as depicted in figure 5. Figure 6 gives the results of the FE-calculations for these combinations, with the same figures for width w, period Px, finger length lf , gap g and with amplitudes As, Atr = lf . S185
A A Kuijpers et al Difference in capacitance for different finger-shapes (1)
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measurement signal. Increasing the structure height h is more advantageous because then C(x) is increased linearly. However, the height h is limited for the poly-silicon structural layer as explained in section 1.3. Figure 8 gives the simulated difference in capacitance for rectangular-shaped fingers and different period Px. Lowering Px does increase the sensor-density, and thus the discrete (counting) accuracy, but at the expense of a decreasing C at equal gap distances. However, a larger period Px shows around x = Px/2 (figure 8 right) a non-linear and smaller dC/dx over a larger range. This means C(x) will change less in this range, thus making it harder to detect this change in capacitance.
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The combination of rectangular-versus sinusoidal-shaped ‘finger’ is showing the largest difference in capacitance owing to the smallest ‘average gap distance’. The combination of rectangular fingers in figure 4 is showing a larger change, because of charge accumulation at the edges. Also the minimum value of the capacitance for this combination is lower, because the maximum distance ly (figure 3) is larger. Figure 7 shows the results of the difference in capacitance versus displacement for rectangular fingers when the initial gap is changed from 2 to 0.5 µm (i.e. g = w, 12 w, 14 w). Halving the gap increases the maximum difference by a factor larger than 2. This indicates that decreasing the gap by open-loop control of the sense-actuator provides a way of increasing the change in capacitance C(x) and therefore the position
2.2. Closed-loop operation for constant capacitance In the next part we will give the results of simulations for a slightly different concept given in figure 9. Here, the capacitance between sense-actuator and slider is held constant as the sliderbeam is moved by the drive-actuators. Each senseactuator is driven using a closed-loop control scheme and can only move in the y-direction. In general, this technique is used for larger dynamic range and minimization of the influence of parasitic capacitances, e.g., accelerometers [3, 8]. The control signal for the senseactuator becomes a measure for the x-displacement of the slider.
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Figure 10 gives the required motion in the y-direction of the sense-actuator to keep the capacitance at the initial value C0 for x = 0. If the initial gap for x = 0 is smaller, the initial capacitance C0 is larger. For a displacement of the slider the fingers of the sense-actuator will have to follow the pattern on the slider more closely in order to keep the capacitance at the larger value of C0. This increases the amplitude of the motion in the y-direction. Figure 10 also shows that the y-displacement of the sense-actuator increases when the length of its fingers is increased from 6 to 8 µm. (The fingers on the slider remain 6 (µm).) Figure 11 gives the simulated y-displacement of the senseactuator for four combinations of finger shapes: (a) triangular– triangular, (b) rectangular–triangular (c) rounded–rounded, (d) rectangular-rectangular.
Combination (d) in figure 11 shows the largest y-displacement but the differential change in y-motion (dy/dx) around x = 12 Px is slightly less than for combination (b). This means combination (d) at this position has a lower sensitivity to detect a displacement of the slider. Figure 12 depicts the calculated differential change dy/dx versus displacement in the x-direction for different finger shapes. Although close to zero, the difference in dy/dx around x = 12 Px between combination (d) and the others appears to be not so bad. More interesting is the large value of dy/dx around xn = 1/4 and xn = 3/4. If rectangular fingers are not realistic because of the achievable precision of the micromachining techniques, figure 12 shows that also rounded fingers can give a reasonable y-displacement and a differential change around these points. One can try to find a different combination of finger shapes that combines the amplitude in y-displacement of the rectangular fingers (combination (d)) with the almost linear behavior of dy/dx for rounded fingers (combination (b)) around x = 0 and x = 12 Px. However, as a better alternative, one can use two geometrically shifted arrays as in figure 9 (right), so that a minimum in sensitivity for one array is balanced by a maximum in sensitivity of the other (quadrature detection [13]). Implementation of this principle would require more space (area/volume), though.
3. Discussion Although the FE-simulations are only 2D and the influence of parasitic capacitances is neglected, the results indicate, the periodic sensor-capacitance can be Cs ≈ 15 fF and Differential change in capacitance for different period Px
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Figure 8. (Left) For rectangular-shaped fingers the difference in capacitance simulated for different period Px and initial gap distance g = 0.5 µm. (Right) The differential change in capacitance dC/dx.
Figure 9. (Left) Extension of the concept in figure 1. For each displacement of the slider in x-direction the sense-structure is now actuated in y-direction to keep the sense-capacitance constant using closed-loop control. (Right) Photograph of micromachined periodic capacitance sense-structure.
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characterization. Legtenberg et al [12] have studied this before and performed quasi-statical capacitance versus dcbias voltage measurements using a gain-phase analyzer (HP4194A). Over a deflection range of 30 µm he measured a linear change C/x ≈ 2.67 (fF µm−1) versus squared voltage (V 2 ), with initial capacitance Ccomb ≈ 300 fF (Cmax ≈ 80 fF). For each comb-structure in a push–pull configuration of two comb-drives, the capacitance C(y) and change in capacitance C(y) between rotor and stator can be expressed as [4]
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Figure 10. Simulated displacement in the y-direction, required to keep the capacitance equal to the initial capacitance C0 (x = 0).
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Figure 11. Y-displacement of closed-loop controlled sense-actuator for four combinations of dissimilar finger shapes. Constant Capacitance: Differential y-displacement (dy/dx) vs. Normalized x-displacement 40 30 20
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Cmax ≈ 7 fF, if we apply two arrays of 50 finger pairs along the sides of the slider. We think it should be possible to measure the change in capacitance versus displacement, using a charge amplifier and synchronous detection [3]. Zwijze [9] and Toth [10] report that a capacitance of about 2 pF can be measured with an accuracy of 0.01% using a modified Martin oscillator. This corresponds to an accuracy of 200 aF with a resolution of about 50 aF. Kung and Lee [11] have reported an integrated air-gap-capacitor pressure sensor where a 100 fF air-gap-capacitor could be measured with a resolution of less than 30 aF. We consider using the changing capacitance of the comb-actuators as a reference displacement sensor for S188
2 · n · ε0 · h · (y + y0 ) g
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At larger displacements the change in capacitance per change in displacement has to be detected from an increasing capacitance C(y). In other words the ratio ∂C C(y) = 1/y ∂y (i.e. sensitivity) is decreasing for increasing displacement y. When two comb-sensors are read-out differentially with C1 = C + C(y) and C2 = C − C(y) we can measure (C1 − C2)/(C1 + C2) = C(y)/C ∝ y/y0 and the sensitivity has become constant. The total range of displacement is limited of course by the length of the comb-fingers. This may be a drawback, but with the combination of a comb-structure, functioning as a coarse position sensor, and the periodic capacitive position sensor presented in this paper, a highaccuracy, long-range position measurement for microactuator systems can be gained. The given figures of measured resolution of capacitances also prove that measurement of the capacitance changes of the periodic capacitive position sensor is feasible. Thus, our future plans are to perform measurements, initially open-loop operation, and compare these with the simulation results. Also, the simulations will be extended to reach a more thorough conclusion about the best geometry and parameter set (period Px, gap g, length lf ) to choose in our design. Preferably, we would have a periodic geometry that produces a capacitance function C(x) with a capacitance change C(x) that is linear and as large as possible, e.g. triangular-shaped. Because of the periodicity of C(x) it will be necessary to implement a quadrature detection technique to measure the position unambiguously [13] (figure 9 (right)).
4. Conclusions We have performed 2D-finite-element simulations for a microsized in-plane periodical capacitive position sensor. The basic idea is to have a periodic geometry or structure (slider) moving in relation to a fixed periodic sense-structure, giving a periodic capacitive read-out. Two concepts have been presented: (i) open-loop measurement of capacitance-change versus displacement (i.e. sense-structure is fixed), (ii) closed-loop control of capacitance-change versus displacement (i.e. sense-structure is actuated).
2D-finite-element simulations for long-range capacitive position sensor
Simulation results for both concepts show that the periodic change in capacitance is large enough to measure, providing an incremental position-measurement over ten’s of µm range with potentially high accuracy. These concepts will be tested using two comb-drives in push–pull mode, allowing relative easy fabrication through surface micromachining technology.
Acknowledgments This research is financed by STW. The author likes to thank M de Boer and E Berenschot for the micromachining advice and assistance, H van Wolferen and R Sanders for technical assistance. Student I vanUitert is acknowledged for work on the simulations. Finally, the people of Comsol (www.comsol.com) are acknowledged for the assistance with the finite-element program ‘FEMLAB’.
References [1] Pedrocchi A et al 2000 Perspectives on MEMS in bioengineering: a novel capacitive position micro-sensor Trans. Biomed. Eng. 47 8–11 [2] Hoen S et al 1997 Electrostatic surface drives: theoretical considerations and fabrication Proc. IEEE Int. Conf. Solid-State Sensors and Actuators pp 41–4
[3] Elwenspoek M and Wiegerink R J 2000 Mechanical Microsensors, Microtechnology and MEMS (Berlin: Springer) pp 13–21 [4] Legtenberg R et al 1996 Comb-drive actuators for large displacements J. Micromech. Microeng. 6 320–9 [5] Bao M H et al 2000 Handbook of Sensors and Actuators vol 8 (Amsterdam: Elsevier) pp 171–7 [6] Cheung P and Horowitz R 1996 Design, fabrication, position sensing and control of an electrostatically-driven polysilicon microactuator IEEE Trans. Magn. 32 122–8 [7] Feynman R P, Leighton R B and Sands M 1977 The Feynman Lectures on Physics vol 2 (Reading, MA: Addison-Wesley) 7-1/7-5 [8] Boser B E and Howe R T 1996 Surface micromachined accelero-meters IEEE J. Solid-State Circuits 31 366–75 [9] Zwijze A F 2000 Micro-machined high capacity siliconload cells PhD Thesis University of Twente, Enschede pp 89–122 [10] Toth F N 1997 A design methodology for low-cost, high performance capacitive sensors PhD Thesis University of Twente, Enschede [11] Kung J T and Lee H S 1992 An integrated air-gap-capacitor pressure sensor and digital readout with sub-100 attofarad resolution J. Microelectromech. Syst. 1 121–8 [12] Legtenberg R et al 1995 Towards position control of electrostatic comb drives Proc. 6th Workshop on Micromachining Micromechanics and Microsystems (MME’95) (Copenhagen, Denmark, 3–5 Sept.) [13] Baxter L K et al 1997 Capacitive Sensors: Design and Applications (New York: IEEE)
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JOURNAL OF MICROMECHANICS AND MICROENGINEERING
J. Micromech. Microeng. 13 (2003) S183–S189
PII: S0960-1317(03)60542-5
2D-finite-element simulations for long-range capacitive position sensor A A Kuijpers, G J M Krijnen, R J Wiegerink, T S J Lammerink and M Elwenspoek Transducer Science and Technology Group, MESA + Research Institute, University of Twente, PO Box 217, 7500 AE, Enschede, The Netherlands E-mail: [email protected]
Received 7 May 2003, in final form 22 May 2003 Published 13 June 2003 Online at stacks.iop.org/JMM/13/S183 Abstract This paper presents the results of 2D-finite-element simulations of the periodic change in capacitance between two periodic structures, one sliding and one fixed (i.e. sense-structure). These structures are potentially interesting for long-range and high-accuracy position detection of microactuators. The use of periodic geometries and the combination of a discrete (incremental) and analog measurement relieves the demands for accuracy. The discrete measurement involves counting the number of periods. The (capacitance) analog measurement determines the position within one period, preferably with nm-accuracy. Two concepts are presented: (i) open-loop measurement of capacitance change versus slider displacement, (ii) closed-loop control of capacitance change versus slider displacement (i.e. sense-structure is actuated in orthogonal direction to the slider motion). These concepts are independent of their application in micro-scale devices but the realization raises particular challenges involving capacitance measurement and micromachining techniques. The periodic patterns examined in these simulations contain rectangular, triangular and sinusoidal shapes and several combinations. Simulation results for both concepts show measurable periodic changes in capacitance. For two arrays of 50 rectangular finger pairs along the sides of the slider the capacitance Cs ≈ 15 fF and Cmax ≈ 7 fF are predicted. Application of this concept in a micromachined device is advantageous because sensor and actuator are integrated in the same structural layer and allows fabrication using one lithographic mask only.
1. Introduction 1.1. Sensing principle for a periodic capacitive position sensor In this paper we present the modeling and results of finiteelement simulations for a micromachined capacitive position sensor, whose concept is given in figure 1. With this sensor we aim at nm-range accuracy over ten’s of µm-range displacement. We want to combine a discrete or incremental measurement with an analog measurement to relieve the demands on accuracy. In a discrete measurement the number of periods is counted over the total range. In the analog 0960-1317/03/040183+07$30.00
measurement the position is measured with nm-accuracy within one period. This combination makes it easier to measure with nm-accuracy over the total range. Both the sliderbeam and sense-structure have a periodic pattern on their sides (e.g. rectangular-shaped fingers or a sinusoidal pattern). As the sliderbeam is moving (x-direction) the capacitance will change periodically. Counting the number of periods and measuring the change in capacitance will give a long-range position measurement with potentially high accuracy. Advantages of this novel concept are: • Accurate long-range measurement is possible due to periodic geometry and combination of discrete and analog measurement.
© 2003 IOP Publishing Ltd Printed in the UK
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Figure 1. (Left) The concept of a periodic capacitive position sensor alongside a slider beam, which is driven by two drive-actuators. (Right) Example of a micromachined polysilicon structure that forms a periodic capacitive position sensor.
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Figure 2. (Left) Photograph of a realization of a capacitive position sensor for an electrostatic comb-drive actuator. (Top-right) Part of the mask-design of the sense-structure with a periodic capacitive position sensor alongside the sliderbeam. (Bottom-right) Cross-view of the structure showing its layer profile.
• Electrostatic forces are balanced in a symmetric and inplane design. • Fabrication is relatively easy because, sensor and actuator are in the same layer and made with the same technology. The large-range capacitive position sensor presented by Pedrocchi et al [1] and tested on a 10× scale PC board, contains arrays of long strip electrodes on both stator and movable plates. These arrays form a capacitance with out-ofplane orientation (i.e. normal to the substrate), which raises the problem of balancing the electrostatic force [2]. It is also more difficult to make with micromachining technology. 1.2. Choice of drive-actuator We use an electrostatic comb-drive actuator as a test vehicle for our designed position sensor. As with all electrostatic microactuators, comb-drive actuators have good scaling properties as the ratio of electrostatic force to volume scales with λ−1 increasing with downscaling (λ = dimensionless scale parameter) [3]. Typical characteristics of our comb-drive actuators for large displacements are a continuous actuation with deflections of about 30 µm at driving voltages around 20 V [4]. Furthermore, with two coupled comb-actuators in a so-called push–pull configuration [5], the displacement-tovoltage relation can be made linear and the range of motion S184
is increased. The fabrication of comb-drives is relatively easy using a one-mask surface micromachining technology. Naturally, a comb-structure can also be used as a capacitive position sensor. A good example is given in [6] where two comb-structures are used, one for driving, one as a position sensor in a Kalman filtering scheme for a statevariable feedback control loop. A drawback of a comb-sensor is that for large displacements the sensitivity (i.e. C/C) decreases. 1.3. Fabrication The micromachined capacitive position sensor in figure 2 (left) consists of three layers as depicted by the cross-view in figure 2 (right). The first layer is a boron-doped silicon substrate. The second layer is a 3 µm thick sacrificial siliconoxide (SiO2) layer with a 5 µm thick poly-silicon structural layer on top. This thickness is limited to 5 µm by the standard available poly-silicon deposition process (PECVD) of the MESA + laboratory. The structure is made by lithographic patterning and reactive ion etching (RIE) of the poly layer. The sacrificial SiO2 layer under the structure is etched selectively by isotropic BHF-etching. The structural poly-silicon parts become movable and suspended by the flexure beams. All large movable parts have a truss-like shape to let the etchant uderetch these parts well before the solid anchor points are
2D-finite-element simulations for long-range capacitive position sensor
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fully underetched too. The width of beams and gaps in this design is 2 µm. It is close to the achievable resolution of the standard photolithography process in the MESA+ clean-room laboratory and it makes the (under) etch time to be uniform for the whole design. The next section gives the 2D finite-element modeling and simulation results for a periodic capacitive position sensor.
2. FEM simulations for capacitance sensor 2.1. Calculation of capacitance change versus displacement: open-loop operation According to Feynman, the only general methods of solution of Laplace’s equation are numerical. For analytical solutions of field distributions the complex variable technique is often powerful but limited to 2D problems, and it also is an indirect method [7]. Therefore, the approach has been to perform finite-element model (FEM) simulations to study the concept presented in this paper. Figure 2 presents the basic periodic geometry used in the program FEMLAB to run the FEM simulations. Initially, the parameters used in the 2D-simulations for the FE-model in figure 3 with rectangular fingers, assuming infinite structure height, are (µm): width of fingers w = 2, length fingers lf = 6, gap distance g = 1, (lateral) period Px = 12, bias-gap-distance ly = 2.lf + g. The width w of 2 µm comes from practical considerations as explained in section 1.3. We have also simulated different shapes and combinations of dissimilar shapes of ‘fingers’ as well as for different period Px, length lf and gap distance g. The capacitance (per meter of height) is calculated by calculating the electrostatic energy in the volume between the fingers with a given voltage difference U, i.e. 1 1 ε0 · E 2 · dv We = · C · U 2 = 2 2 1 → C = 2 ε0 · E 2 · dv. (1) U After meshing the geometry into finite elements the electrostatic E-vector is calculated by solving the Laplace’s
equation ∇ 2 V = 0 for all (area) elements. Dirichlet boundary conditions are used on the real geometric boundaries, by stating the potential. The Neumann boundary conditions on the vertical ‘artificial’ segment boundaries are implemented as periodic boundary conditions so that the calculated values for the E-field on the left side of the geometry in figure 2 are identical to values on the right side. This insures that the calculated field distribution corresponds to infinite repetition of a geometry segment. The simulated capacitance per meter of height is multiplied with the structure height of h = 5 µm (i.e. polysilicon layer thickness). This results in the (approximate) capacitance per geometry-segment, containing one pair of fingers. Due to the scaling properties of the 2D-Laplace equation, the units of parameters w, Px , Lf , g are not important but only their relative proportions. Only the units of the dielectric permitivity ε0 (F/m) and the structure height h (m) for calculation of the energy or capacitance (equation (1)) are important. It is preferred to have a unitfree normalized displacement (xn = x/Px), to better compare between geometries with different period (Px). Figure 4 gives the simulated capacitances against normalized displacement (x/Px) for three differently shaped finger pairs with a lateral period of Px = 12 (µm). As the displacement in x-direction between the pair of fingers is zero, the fingers are ‘in-phase’ and the capacitance is maximal. For a displacement of 12 Px the fingers are ‘outof-phase’ and the capacitance is minimum and rising again for larger displacements. For each shape of finger pair the change in capacitance for every displacement is calculated i.e. C(x) = C(x) − Cmin. From figure 4 it is clear that the combination of two arrays of rectangular-shaped fingers is showing the largest absolute capacitance as well as the largest change in capacitance for each displacement of the arrays with respect to each other. We also examined the influence of combinations of dissimilar ‘finger’ shapes, as depicted in figure 5. Figure 6 gives the results of the FE-calculations for these combinations, with the same figures for width w, period Px, finger length lf , gap g and with amplitudes As, Atr = lf . S185
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measurement signal. Increasing the structure height h is more advantageous because then C(x) is increased linearly. However, the height h is limited for the poly-silicon structural layer as explained in section 1.3. Figure 8 gives the simulated difference in capacitance for rectangular-shaped fingers and different period Px. Lowering Px does increase the sensor-density, and thus the discrete (counting) accuracy, but at the expense of a decreasing C at equal gap distances. However, a larger period Px shows around x = Px/2 (figure 8 right) a non-linear and smaller dC/dx over a larger range. This means C(x) will change less in this range, thus making it harder to detect this change in capacitance.
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The combination of rectangular-versus sinusoidal-shaped ‘finger’ is showing the largest difference in capacitance owing to the smallest ‘average gap distance’. The combination of rectangular fingers in figure 4 is showing a larger change, because of charge accumulation at the edges. Also the minimum value of the capacitance for this combination is lower, because the maximum distance ly (figure 3) is larger. Figure 7 shows the results of the difference in capacitance versus displacement for rectangular fingers when the initial gap is changed from 2 to 0.5 µm (i.e. g = w, 12 w, 14 w). Halving the gap increases the maximum difference by a factor larger than 2. This indicates that decreasing the gap by open-loop control of the sense-actuator provides a way of increasing the change in capacitance C(x) and therefore the position
2.2. Closed-loop operation for constant capacitance In the next part we will give the results of simulations for a slightly different concept given in figure 9. Here, the capacitance between sense-actuator and slider is held constant as the sliderbeam is moved by the drive-actuators. Each senseactuator is driven using a closed-loop control scheme and can only move in the y-direction. In general, this technique is used for larger dynamic range and minimization of the influence of parasitic capacitances, e.g., accelerometers [3, 8]. The control signal for the senseactuator becomes a measure for the x-displacement of the slider.
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Figure 10 gives the required motion in the y-direction of the sense-actuator to keep the capacitance at the initial value C0 for x = 0. If the initial gap for x = 0 is smaller, the initial capacitance C0 is larger. For a displacement of the slider the fingers of the sense-actuator will have to follow the pattern on the slider more closely in order to keep the capacitance at the larger value of C0. This increases the amplitude of the motion in the y-direction. Figure 10 also shows that the y-displacement of the sense-actuator increases when the length of its fingers is increased from 6 to 8 µm. (The fingers on the slider remain 6 (µm).) Figure 11 gives the simulated y-displacement of the senseactuator for four combinations of finger shapes: (a) triangular– triangular, (b) rectangular–triangular (c) rounded–rounded, (d) rectangular-rectangular.
Combination (d) in figure 11 shows the largest y-displacement but the differential change in y-motion (dy/dx) around x = 12 Px is slightly less than for combination (b). This means combination (d) at this position has a lower sensitivity to detect a displacement of the slider. Figure 12 depicts the calculated differential change dy/dx versus displacement in the x-direction for different finger shapes. Although close to zero, the difference in dy/dx around x = 12 Px between combination (d) and the others appears to be not so bad. More interesting is the large value of dy/dx around xn = 1/4 and xn = 3/4. If rectangular fingers are not realistic because of the achievable precision of the micromachining techniques, figure 12 shows that also rounded fingers can give a reasonable y-displacement and a differential change around these points. One can try to find a different combination of finger shapes that combines the amplitude in y-displacement of the rectangular fingers (combination (d)) with the almost linear behavior of dy/dx for rounded fingers (combination (b)) around x = 0 and x = 12 Px. However, as a better alternative, one can use two geometrically shifted arrays as in figure 9 (right), so that a minimum in sensitivity for one array is balanced by a maximum in sensitivity of the other (quadrature detection [13]). Implementation of this principle would require more space (area/volume), though.
3. Discussion Although the FE-simulations are only 2D and the influence of parasitic capacitances is neglected, the results indicate, the periodic sensor-capacitance can be Cs ≈ 15 fF and Differential change in capacitance for different period Px
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Figure 8. (Left) For rectangular-shaped fingers the difference in capacitance simulated for different period Px and initial gap distance g = 0.5 µm. (Right) The differential change in capacitance dC/dx.
Figure 9. (Left) Extension of the concept in figure 1. For each displacement of the slider in x-direction the sense-structure is now actuated in y-direction to keep the sense-capacitance constant using closed-loop control. (Right) Photograph of micromachined periodic capacitance sense-structure.
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characterization. Legtenberg et al [12] have studied this before and performed quasi-statical capacitance versus dcbias voltage measurements using a gain-phase analyzer (HP4194A). Over a deflection range of 30 µm he measured a linear change C/x ≈ 2.67 (fF µm−1) versus squared voltage (V 2 ), with initial capacitance Ccomb ≈ 300 fF (Cmax ≈ 80 fF). For each comb-structure in a push–pull configuration of two comb-drives, the capacitance C(y) and change in capacitance C(y) between rotor and stator can be expressed as [4]
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Figure 10. Simulated displacement in the y-direction, required to keep the capacitance equal to the initial capacitance C0 (x = 0).
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Cmax ≈ 7 fF, if we apply two arrays of 50 finger pairs along the sides of the slider. We think it should be possible to measure the change in capacitance versus displacement, using a charge amplifier and synchronous detection [3]. Zwijze [9] and Toth [10] report that a capacitance of about 2 pF can be measured with an accuracy of 0.01% using a modified Martin oscillator. This corresponds to an accuracy of 200 aF with a resolution of about 50 aF. Kung and Lee [11] have reported an integrated air-gap-capacitor pressure sensor where a 100 fF air-gap-capacitor could be measured with a resolution of less than 30 aF. We consider using the changing capacitance of the comb-actuators as a reference displacement sensor for S188
2 · n · ε0 · h · (y + y0 ) g
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At larger displacements the change in capacitance per change in displacement has to be detected from an increasing capacitance C(y). In other words the ratio ∂C C(y) = 1/y ∂y (i.e. sensitivity) is decreasing for increasing displacement y. When two comb-sensors are read-out differentially with C1 = C + C(y) and C2 = C − C(y) we can measure (C1 − C2)/(C1 + C2) = C(y)/C ∝ y/y0 and the sensitivity has become constant. The total range of displacement is limited of course by the length of the comb-fingers. This may be a drawback, but with the combination of a comb-structure, functioning as a coarse position sensor, and the periodic capacitive position sensor presented in this paper, a highaccuracy, long-range position measurement for microactuator systems can be gained. The given figures of measured resolution of capacitances also prove that measurement of the capacitance changes of the periodic capacitive position sensor is feasible. Thus, our future plans are to perform measurements, initially open-loop operation, and compare these with the simulation results. Also, the simulations will be extended to reach a more thorough conclusion about the best geometry and parameter set (period Px, gap g, length lf ) to choose in our design. Preferably, we would have a periodic geometry that produces a capacitance function C(x) with a capacitance change C(x) that is linear and as large as possible, e.g. triangular-shaped. Because of the periodicity of C(x) it will be necessary to implement a quadrature detection technique to measure the position unambiguously [13] (figure 9 (right)).
4. Conclusions We have performed 2D-finite-element simulations for a microsized in-plane periodical capacitive position sensor. The basic idea is to have a periodic geometry or structure (slider) moving in relation to a fixed periodic sense-structure, giving a periodic capacitive read-out. Two concepts have been presented: (i) open-loop measurement of capacitance-change versus displacement (i.e. sense-structure is fixed), (ii) closed-loop control of capacitance-change versus displacement (i.e. sense-structure is actuated).
2D-finite-element simulations for long-range capacitive position sensor
Simulation results for both concepts show that the periodic change in capacitance is large enough to measure, providing an incremental position-measurement over ten’s of µm range with potentially high accuracy. These concepts will be tested using two comb-drives in push–pull mode, allowing relative easy fabrication through surface micromachining technology.
Acknowledgments This research is financed by STW. The author likes to thank M de Boer and E Berenschot for the micromachining advice and assistance, H van Wolferen and R Sanders for technical assistance. Student I vanUitert is acknowledged for work on the simulations. Finally, the people of Comsol (www.comsol.com) are acknowledged for the assistance with the finite-element program ‘FEMLAB’.
References [1] Pedrocchi A et al 2000 Perspectives on MEMS in bioengineering: a novel capacitive position micro-sensor Trans. Biomed. Eng. 47 8–11 [2] Hoen S et al 1997 Electrostatic surface drives: theoretical considerations and fabrication Proc. IEEE Int. Conf. Solid-State Sensors and Actuators pp 41–4
[3] Elwenspoek M and Wiegerink R J 2000 Mechanical Microsensors, Microtechnology and MEMS (Berlin: Springer) pp 13–21 [4] Legtenberg R et al 1996 Comb-drive actuators for large displacements J. Micromech. Microeng. 6 320–9 [5] Bao M H et al 2000 Handbook of Sensors and Actuators vol 8 (Amsterdam: Elsevier) pp 171–7 [6] Cheung P and Horowitz R 1996 Design, fabrication, position sensing and control of an electrostatically-driven polysilicon microactuator IEEE Trans. Magn. 32 122–8 [7] Feynman R P, Leighton R B and Sands M 1977 The Feynman Lectures on Physics vol 2 (Reading, MA: Addison-Wesley) 7-1/7-5 [8] Boser B E and Howe R T 1996 Surface micromachined accelero-meters IEEE J. Solid-State Circuits 31 366–75 [9] Zwijze A F 2000 Micro-machined high capacity siliconload cells PhD Thesis University of Twente, Enschede pp 89–122 [10] Toth F N 1997 A design methodology for low-cost, high performance capacitive sensors PhD Thesis University of Twente, Enschede [11] Kung J T and Lee H S 1992 An integrated air-gap-capacitor pressure sensor and digital readout with sub-100 attofarad resolution J. Microelectromech. Syst. 1 121–8 [12] Legtenberg R et al 1995 Towards position control of electrostatic comb drives Proc. 6th Workshop on Micromachining Micromechanics and Microsystems (MME’95) (Copenhagen, Denmark, 3–5 Sept.) [13] Baxter L K et al 1997 Capacitive Sensors: Design and Applications (New York: IEEE)
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