靜電懸浮磁盤的水平系統(tǒng)仿真外文翻譯/中英文翻譯/外文文獻翻譯

System Level Simulation of an Electrostatically Levitated Disk Michael Kraft and Alan Evans University of Southampton, Highfield, Southampton, SO17 1BJ ABSTRACT This paper describes the derivation of a system-level model for a micromachined disk which is levitated by electrostatic forces. Such a system has many potential applications for inertial sensors, micro-motors, microfluidic and micro-optical devices. As a simulation tool Matlab/Simulink was used since it can easily handle different domains such as electrical and mechanical without special arrangements. Multi-axis electro-mechanical sigmadelta modulators were chosen to control the various degrees of freedom of the levitated disk. Of special interest is the power-up phase at the end of which the control system has to ensure that the disk is centred between the top andbottom electrodes. Keywords: Electrostatic levitation, inertial sensors, systemlevel simulation. 1 INTRODUCTION Electrostatic forces are commonly used for actuation of micromachined devices e.g. force-balanced inertial sensors, microfluidic valves and micro-optical mirrors. However, in all these devices a mechanical connection exists between the actuated part and the substrate, the properties of which are subject to considerable process tolerances and cannot be changed easily once the device has been manufactured. In this paper a micromachined disk is suggested which is levitated by electrostatic forces. This has many potential applications such as accelerometers with online dynamic characteristics tuning, gyroscopes if the disk is spun and the precession of the disk induced by the Coriolis force is measured, microfluidic mixers and pumps, frictionless bearings for micro-motors and micro-optical light choppers. Surprisingly little work has been undertaken in this direction . The system, as shown in fig.1, consists of a nickel disk manufactured by electroplating which is encaged by sets of electrodes on top and bottom and electroplated pillars at the sides. The manufacturing process will be described elsewhere in detail. Each set of electrodes forms five capacitors which are used for both sensing the position of the disk as well as actuating it in such a way that it is maintained at or close to the centre position. In this work the system level simulation of such a levitated disk is presented. Simulink was chosen as a simulation tool since it is ideal for the simulation of a micromachined system because it easily can handle different domains such as electrical and mechanical without special arrangements. The system is unstable in the openloop configuration, consequently, the control strategy has to be considered with great care. Here, multi-axis electromechanical sigma-delta modulators are used 3 to control the various degrees of freedom of the disk. This prevents the possibility of electrostatic latch-up 4 and also results in an inherently digital system. 2 DERIVATION OF THE MODEL In order to derive a model capturing the governing features of the system the following building blocks have to be considered: - The dynamic behaviour of the disk, - conversion from the mechanical to the electrical domain, i.e. the position measurement interface, - the multi-axis electromechanical sigma-delta modulator, - the compensator, - the reset mechanism, i.e. conversion from the feedback voltage to the electrostatic forces and moments including the effects of the current position of the disk. 2.1 Disk Dynamics The dynamic behaviour of the levitated disk is characterized by three second order differential equations: where m is the mass of the disk, z the displacement from the mid-position between the electrodes,and the angular deflections about the x and y axis of the disk, bz, b and b the damping coefficient in z,and direction respectively, Ix and Iy the moments of inertia of the disk about the corresponding axis and Fext,z, Mext,x and Mext,y the external force and moments in the corresponding directions. The main difference to the usual dynamic equations describing a micromachined proof mass is that a mechanical spring force term is missing. The effective spring constant of the disk is entirely controlled by the external electrostatic forces and moments. As a drawback it obvious that the system cannot operate in an open loop mode, as usually possible for inertial sensors. A further building block is required to simulate the physical constraint of the disk, this is achieved by saturation blocks. 2.2 Position Measuring Interface The position of the disk, characterized by its z,and coordinates, has to be measured. This is achieved by measuring the differential capacitance of the four outer, pieshaped electrodes. The system level model does not simulate an electronic implementation, however, it is envisaged to apply an excitation voltage to the round,middle electrodes and sense the currents from the outer electrodes from which it possible to measure the four differential capacitances between the corresponding top and bottom electrodes. The information about z,and deflections is encoded in the differential capacitance values. Two possible control approaches are therefore possible. Firstly, one could extract the position information at this point of the control loop which would result in a three-axis sigma-delta modulator control scheme. Secondly, a sigma-delta loop with four paths is possible, one for each differential capacitance. The information about disk position and external inertial forces and moments is then encoded in the digital bitstream providing the output signals of the system. This approach was chosen here since the decoding of the position information can then be undertaken in the digital domain with obvious advantages. For the system level model an analytical expression for the capacitance between the pie-shaped electrodes and the disk has to be derived. The expression for this geometry is very lengthy and it was decided to use an approximation for a square electrode. As an example the expression for the top-right segment is given: where0 is the dielectric constant of vacuum (assumed to be the same as for air), R is the disk radius, z0 the nominal distance between disk and electrodes when the disk is at mid-position.Furthermore, small angular displacements are assumed so that the usual approximation of parallel-plate capacitors is made. The expressions for the other electrodes can be derived by symmetry considerations. The use of the equation is illustrated in fig. 2a. This square plate approximation is justified by the fact that in the forward path of the sigma-delta modulator control system an ideal comparator is located whose output is not determined by a numerical exact solution but rather governed by the sign of its input signal.It should be noted that the expression for the capacitance has a singularity at= 0 or= 0, which causes a simulation error at zero deflection and incorrect numerical results for very small angles due to numerical noise. This problem can be overcome by using a linearized version of eq. (2) around = 0 and= 0, both expressions are then implemented in the model, depending on the magnitude of the angular displacements either the linerized or full expression is used. 2.3 Sigma-delta Modulator and Compensator The building blocks for the sigma-delta modulator can be easily simulated by a relay and a sample and hold. The preceding compensator is required to stabilize the loop and ensure a high-frequency limit cycle in the unforced condition by adding a zero at a frequency at approximately 1/10 of the sampling frequency. 2.4 Feedback Arrangement In the feedback path, each of the eight top and bottom electrodes is either energized by the feedback voltage or held at zero potential, depending on the output state of the four comparators. The control system has to ensure that the electrode further away from the disk is energized,consequently forces and moments are generated to move the disk back to the mid-position parallel to the top and bottom electrodes.Provided the disk is held at zero potential (which is assumed here) the magnitude of the electrostatic force inthe z-direction can be calculated and is given by: for the top right electrode, where Vfb is the feedback voltage which is either a fixed voltage or zero depending on the state of the comparator output. The result of using the equation is illustrated in fig. 2b.As a minimum value the electrostatic force must at least be able to counterbalance the gravitational force on the disk, however, any larger inertial forces would make the disk touch the electrodes, hence the system momentarily inoperative. Simple force calculation reveal that a feedback voltage of 40 V can counterbalance approximately 100 g inertial force for a disk with a thickness of 200m and a radius of 0.5 mm, which is believed to be sufficient.The magnitude of the moment generated by the feedback voltage can be calculated and is given by: for the moment about the y-axis for the top right electrode. The result of using the equation is illustrated infig.2c.Both the expressions for the force and moments suffer from the same singularity as the capacitance equation causing numerical problems. The same method as described for the capacitance was chosen to circumvent the problem.Since the expression for the moments contain a squared 2.5 Overall Model Fig. 3 shows the overall model implemented in Simulink. The nonlinear expressions have been directly implemented as function blocks. One sampling period is divided in a position measuring phase and a actuation phase, the delay between and the length are important parameters determining the loop stability since they add further lag. 3 RESULTS Simulations with initial conditions assuming the disk already at the mid-position show the expected behaviour；the four output signals from the comparators are repetitive sequences of two high periods and two low periods, i.e. a stable (2,2) mode which is the lowest possible mode for a second order sigma-delta modulator . Of special interest is the power-up phase, at the beginning of which the disk lies flat on one set of electrodes, i.e. initial conditions z = z0,= 0 and = 0. For a system with R = 500   the simulation result is shown in Fig. 4. It is obvious that the disk is forced to the centre position between the electrodes and the tilt angles are zeroed as well. This simulation does not take into account any surface forces which might cause stiction of the disk to the electrodes. Such a behaviour is extremely difficult to describe analytically and will be investigated in the hardware implementation. Since the disk will be fabricated from electroplated nickel the surface roughness will be considerable hence will reduce the contact area between disk and electrode which it is believed to alleviate the problem. 4 CONCLUSIONS For any control system comprising a sigma-delta modulator system level simulation is often the only way to predict its behaviour since a complete analytical description is extremely difficult if not impossible. Consequently, the model for the levitated disk has proved to be a valuable aid for the design of the proposed system. Although the model presented here does not consider all effects which could be included, it still describes the governing features of the system. Especially the parameters that determine the geometry, capacitance, sigma-delta modulator dynamics and stability etc. can be assessed and optimised. A trade-off between the degree of realism and the simulation time is also important. A typical simulation run of the presented model takes about 10-20 minutes on a modern PC.Further work will concentrate on the inclusion of the behaviour of lateral deflections and simulation of spinning the disk about its main axis. REFERENCES 1 Fukatsu,K,et.al.,“Electrostatically levitated micro motor for inertia measurement system,” Transducer 99,3P2.16, 1999. 2 Torti, R, et. al., “Electrostatically suspended and sensed micromechanical rate gyroscope,” SPIE, Vol 2200,pp 27-38, 1994. 3 Lemkin, M.A. and Boser, B.E., “A 3-axis micromachined accelerometer with a CMOS position-sense interface and digital offset-trim electrodes.” IEEE J. of Solid-State Circuits, Vol. 34, No. 4, pp. 456-468, 1999. 4 Kraft, M., “Closed loop accelerometer employing over sampling conversion.” Coventry Uni., Ph.D. dissertation, 1997. 5 Boser, B. E. and Howe, R. T., “Surface micromachined accelerometers,” IEEE J. of Solid-State Circuits, Vol. 31, No. 3, pp. 336-375, 1996. 靜電懸浮磁盤的水平系統(tǒng)仿真  Michael Kraft ， Alan Evans 南安普頓大學(xué)  摘要  本論文敘述了一個微機械磁盤的水平系統(tǒng)模型的來歷，這個磁盤是受靜電力而懸浮起來的。這種系統(tǒng)已經(jīng)有很多潛在的應(yīng)用，例如慣性傳感器，微發(fā)動機，微射流控制的和微光學(xué)的裝置。作為仿真工具的 Matlab/Simulink都已被廣泛使用，因為它們對于不同領(lǐng)域的東西都很易于 操作，就象電類的、機械類的，除非有特殊要求。多軸機電調(diào)節(jié)器被用來控制懸浮磁盤的各種自由度。由于某種特殊意義，這個控制系統(tǒng)的上限能量相位必須能夠確保這個磁盤保持在在上下電極的中間位置。  關(guān)鍵字 ：靜電懸浮，慣性傳感器，水平系統(tǒng)仿真  1  簡介  靜電力通常被用來作為微機械裝置的沖擊力，例如慣性傳感器的平衡力，射流控制電子管和微光學(xué)鏡。但是，在這些裝置中驅(qū)動部分和基底部分的機械聯(lián)接是不可避免的，它們的道具受相當大的處理公差所支配，并且一旦人做出了這種裝置，這些問題是很難改變的。在這篇文章中建議用一個受靜電力而懸浮起來的微機械磁盤。這已經(jīng)有了很多潛在的應(yīng)用，例如加速度計的聯(lián)機動態(tài)特征的調(diào)整；如果這個磁盤被看作是陀螺儀，測量它受科里奧利力而引起的運動；微射流控制混頻器和微泵；微電動機的無摩擦力軸承和微光學(xué)發(fā)光斷路器。令人驚訝的是至今這方面的工作還沒有人做。  如圖 1所示，這個系統(tǒng)由電鍍的人造鎳磁盤組成，它被一個旁邊有電鍍支撐柱，上下有電極的裝置封閉起來。這個制作的過程將在其他刊物上詳細介紹。每個電極裝置都由五個電容器組成，這些電容器用來感知磁盤和驅(qū)動部分的位置，在某種程度上它被保持在中間位置或接近于中間位置。  這篇文章介紹的是 懸浮磁盤的水平系統(tǒng)仿真。在這里選用 Simulink為仿真工具，因為它是微機械系統(tǒng)的理想的仿真工具，它對各種領(lǐng)域都易于操作，比如沒有特殊用途的電和機械等方面。這個系統(tǒng)在開放的結(jié)構(gòu)中是不穩(wěn)定的，因此，控制策略不得不謹慎考慮。這里，多軸機電的三角架調(diào)節(jié)器用來控制磁盤各個方向的自由度。還要防止靜電屏蔽的可能性和導(dǎo)致一種固有的數(shù)字系統(tǒng)。  1 這個模型的來歷  為了獲取和控制這個裝置的系統(tǒng)屬性，我們不得按下面的積木形式考慮：    磁盤的動態(tài)特征，    從機械的到電的轉(zhuǎn)化，也就是位置測量的分界面，    多軸機電的三角架調(diào)節(jié)器，    補償    重新 安排機械裝置，也就是從電壓反饋到靜電力的轉(zhuǎn)化和磁盤瞬間姿態(tài)的影響。  2.1 磁盤動力學(xué)  懸浮磁盤的動態(tài)特征由下面三個二次微分方程來描述：   其中 m是磁盤的介子的質(zhì)量， z是偏移電極中間位置的位移量，和是 x和 y軸的偏轉(zhuǎn)角， ,bz是 z軸方向的阻尼系數(shù)，磁盤各個相應(yīng)軸的轉(zhuǎn)動慣量分別為 Ix 和Iy， Fext,z, Mext,x和  Mext,y是各個相應(yīng)軸的瞬間外力。  圖 1 磁盤懸浮和控制系統(tǒng)  這個方程與通常描述微機械薄片的動態(tài)方程的主要不同點是機械彈簧力不需要什么條件。磁盤的有效彈簧系數(shù)完全是受外部和瞬間靜電力控制的。 作為一個明顯的缺點，這個系統(tǒng)上不能在開放的環(huán)境中操作，通常類似于慣性傳感器。再下一步的積木就要求對磁盤的物理局限性進行仿真，這要通過浸透木塊來完成。  2.2  分界面位置的測量  以 z和與其相對應(yīng)的參量為特征的磁盤的位置必須測量。通過測量四個外部微分電容，形成電極。這個系統(tǒng)的水平模型不能仿真一個電子儀器，但是，它的周圍或電極中間可以加一個激發(fā)電壓，從相應(yīng)的上下電極之間可能能測得第四個微分電容量，從而可以從外部電極感知電流。將 z和偏移量的信息譯成微分電容值。有兩種可能的控制方法。第一，在控制回路上可以獲取這點的 位置信息，這將導(dǎo)致一個三軸三角架調(diào)節(jié)器的控制策略。第二，一個三角形回路可能有四中路線，其中一條是每個微分電容。磁盤位置和外部慣性力和力矩的信息被譯成數(shù)字串來提供系統(tǒng)的輸出信號。這里選用這種方法是因為位置信息譯碼可以在數(shù)字方面控制，具有明顯的優(yōu)勢。對于這個系統(tǒng)的水平模型，一種分析表達來源于各種形狀的電極和磁盤之間的電容。這種幾何學(xué)的表達非常冗長，而且用的是一個正方形電極的近似值。下面給出一個頂部表達式的例子：   其中 0是真空介電常數(shù)（假設(shè)與空氣的一樣）， R是磁盤半徑， Z0是當磁盤在電極中間位置時磁盤與電極 的距離，此外，小角位移假定與普通平行電容器接近。這個等式的運用如圖 2a所示。這個正方形的金屬板的近似值被證明是合理的，這個三角架調(diào)節(jié)器作為一個理想的比較儀的正向傳輸路徑的定位和輸出是不能用精確的數(shù)字解算的，也不能用它的輸入信號標記控制。表示成電容不應(yīng)該有 0偏移，這樣會導(dǎo)致仿真錯誤和由于數(shù)字噪聲而引起的小角度的數(shù)學(xué)錯誤。這個問題可以用等式（ 2）的一個線性化的譯本來解決，在 0值或 0值周圍，兩種表達式都可以用在該模型中，根據(jù)角位移的大小來選擇是用線性表達式還是完整的表達式。  2.3  三腳架調(diào)節(jié)器和補償  三腳架調(diào)節(jié)器的 積木可以通過一個繼電器的取樣很容易的仿真出來。先前的補償要求必須是穩(wěn)定的回路，并且確保在自然條件下的高頻率循環(huán)限制，這要通過在頻率上增加一個接近于取樣頻率 1/10的零點值。  2.4   反饋隊列  在反饋路徑中，根據(jù)四個比較儀的輸出情況判斷，每個上下電極不是靠反饋電壓加強作用，就是存在一個潛在的零點。這個控制系統(tǒng)必須確保電極遠離磁盤是加強的，從而使力和力矩能夠遠離從平行于上下電極又回到中間位置的磁盤。倘若磁盤保持在潛在的零點位置（這里只是假設(shè)），那么靜電力在 Z方向的大小就可以用下面的式子計算：   此式為上極板的電 極， Vfb是指反饋電壓，這個電壓根據(jù)比較儀的輸出情況判斷，不是固定電壓就是零點。這個等式的結(jié)果如圖 2b所示。作為靜電力的最小值必須要至少平衡掉磁盤的重力，但是，任何大一點的慣性力都將使得磁盤接觸到電極，因此此刻系統(tǒng)不起作用。簡單的力學(xué)計算顯示 40V的反饋電壓可以近似平衡掉一個 100g， 200mm厚和