静电悬浮磁盘的水平系统仿真外文翻译/中英文翻译/外文文献翻译

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. 静电悬浮磁盘的水平系统仿真  Michael Kraft ， Alan Evans 南安普顿大学  摘要  本论文叙述了一个微机械磁盘的水平系统模型的来历，这个磁盘是受静电力而悬浮起来的。这种系统已经有很多潜在的应用，例如惯性传感器，微发动机，微射流控制的和微光学的装置。作为仿真工具的 Matlab/Simulink都已被广泛使用，因为它们对于不同领域的东西都很易于 操作，就象电类的、机械类的，除非有特殊要求。多轴机电调节器被用来控制悬浮磁盘的各种自由度。由于某种特殊意义，这个控制系统的上限能量相位必须能够确保这个磁盘保持在在上下电极的中间位置。  关键字 ：静电悬浮，惯性传感器，水平系统仿真  1  简介  静电力通常被用来作为微机械装置的冲击力，例如惯性传感器的平衡力，射流控制电子管和微光学镜。但是，在这些装置中驱动部分和基底部分的机械联接是不可避免的，它们的道具受相当大的处理公差所支配，并且一旦人做出了这种装置，这些问题是很难改变的。在这篇文章中建议用一个受静电力而悬浮起来的微机械磁盘。这已经有了很多潜在的应用，例如加速度计的联机动态特征的调整；如果这个磁盘被看作是陀螺仪，测量它受科里奥利力而引起的运动；微射流控制混频器和微泵；微电动机的无摩擦力轴承和微光学发光断路器。令人惊讶的是至今这方面的工作还没有人做。  如图 1所示，这个系统由电镀的人造镍磁盘组成，它被一个旁边有电镀支撑柱，上下有电极的装置封闭起来。这个制作的过程将在其他刊物上详细介绍。每个电极装置都由五个电容器组成，这些电容器用来感知磁盘和驱动部分的位置，在某种程度上它被保持在中间位置或接近于中间位置。  这篇文章介绍的是 悬浮磁盘的水平系统仿真。在这里选用 Simulink为仿真工具，因为它是微机械系统的理想的仿真工具，它对各种领域都易于操作，比如没有特殊用途的电和机械等方面。这个系统在开放的结构中是不稳定的，因此，控制策略不得不谨慎考虑。这里，多轴机电的三角架调节器用来控制磁盘各个方向的自由度。还要防止静电屏蔽的可能性和导致一种固有的数字系统。  1 这个模型的来历  为了获取和控制这个装置的系统属性，我们不得按下面的积木形式考虑：    磁盘的动态特征，    从机械的到电的转化，也就是位置测量的分界面，    多轴机电的三角架调节器，    补偿    重新 安排机械装置，也就是从电压反馈到静电力的转化和磁盘瞬间姿态的影响。  2.1 磁盘动力学  悬浮磁盘的动态特征由下面三个二次微分方程来描述：   其中 m是磁盘的介子的质量， z是偏移电极中间位置的位移量，和是 x和 y轴的偏转角， ,bz是 z轴方向的阻尼系数，磁盘各个相应轴的转动惯量分别为 Ix 和Iy， Fext,z, Mext,x和  Mext,y是各个相应轴的瞬间外力。  图 1 磁盘悬浮和控制系统  这个方程与通常描述微机械薄片的动态方程的主要不同点是机械弹簧力不需要什么条件。磁盘的有效弹簧系数完全是受外部和瞬间静电力控制的。 作为一个明显的缺点，这个系统上不能在开放的环境中操作，通常类似于惯性传感器。再下一步的积木就要求对磁盘的物理局限性进行仿真，这要通过浸透木块来完成。  2.2  分界面位置的测量  以 z和与其相对应的参量为特征的磁盘的位置必须测量。通过测量四个外部微分电容，形成电极。这个系统的水平模型不能仿真一个电子仪器，但是，它的周围或电极中间可以加一个激发电压，从相应的上下电极之间可能能测得第四个微分电容量，从而可以从外部电极感知电流。将 z和偏移量的信息译成微分电容值。有两种可能的控制方法。第一，在控制回路上可以获取这点的 位置信息，这将导致一个三轴三角架调节器的控制策略。第二，一个三角形回路可能有四中路线，其中一条是每个微分电容。磁盘位置和外部惯性力和力矩的信息被译成数字串来提供系统的输出信号。这里选用这种方法是因为位置信息译码可以在数字方面控制，具有明显的优势。对于这个系统的水平模型，一种分析表达来源于各种形状的电极和磁盘之间的电容。这种几何学的表达非常冗长，而且用的是一个正方形电极的近似值。下面给出一个顶部表达式的例子：   其中 0是真空介电常数（假设与空气的一样）， R是磁盘半径， Z0是当磁盘在电极中间位置时磁盘与电极 的距离，此外，小角位移假定与普通平行电容器接近。这个等式的运用如图 2a所示。这个正方形的金属板的近似值被证明是合理的，这个三角架调节器作为一个理想的比较仪的正向传输路径的定位和输出是不能用精确的数字解算的，也不能用它的输入信号标记控制。表示成电容不应该有 0偏移，这样会导致仿真错误和由于数字噪声而引起的小角度的数学错误。这个问题可以用等式（ 2）的一个线性化的译本来解决，在 0值或 0值周围，两种表达式都可以用在该模型中，根据角位移的大小来选择是用线性表达式还是完整的表达式。  2.3  三脚架调节器和补偿  三脚架调节器的 积木可以通过一个继电器的取样很容易的仿真出来。先前的补偿要求必须是稳定的回路，并且确保在自然条件下的高频率循环限制，这要通过在频率上增加一个接近于取样频率 1/10的零点值。  2.4   反馈队列  在反馈路径中，根据四个比较仪的输出情况判断，每个上下电极不是靠反馈电压加强作用，就是存在一个潜在的零点。这个控制系统必须确保电极远离磁盘是加强的，从而使力和力矩能够远离从平行于上下电极又回到中间位置的磁盘。倘若磁盘保持在潜在的零点位置（这里只是假设），那么静电力在 Z方向的大小就可以用下面的式子计算：   此式为上极板的电 极， Vfb是指反馈电压，这个电压根据比较仪的输出情况判断，不是固定电压就是零点。这个等式的结果如图 2b所示。作为静电力的最小值必须要至少平衡掉磁盘的重力，但是，任何大一点的惯性力都将使得磁盘接触到电极，因此此刻系统不起作用。简单的力学计算显示 40V的反馈电压可以近似平衡掉一个 100g， 200mm厚和