Equilibrium Structure of a 1-dof Electrostatic Mems Model with Parasitics

Feedback control of electrostatic microelectromechanical systems (MEMS) is significantly complicated by the presence of parasitic surfaces. This note considers the stabilization of a onedegree-of-freedom (1-DOF) piston actuator with capacitivelycoupled parasitics. Previous work by the authors has shown how, in the absence of parasitics, any feasible equilibrium point of this system may be made globally asymptotically stable using passivity-based control. However if parasitics are present this nominal closed-loop system may be destabilized by capacitive coupling, through a phenomenon called charge pull-in. This note shows how the nominal controller formulation may be modified to eliminate multiple equilibria. If the movable electrode is completely screened from the parasitic electrode by the control electrode, the unique equilibrium is globally asymptotically stable. Otherwise, though the desired equilibrium is still unique, its region of attraction may be finite and the equilibrium may lose stability through a Hopf bifurcation. ∗To whom correspondence should be addressed. INTRODUCTION Electrostatic actuation of microelectromechanical systems (MEMS) makes use of the attractive coulomb forces that develop between capacitively-coupled conductors differing in voltage. Electrostatic actuation is nonlinear, making open-loop control over a large operating range difficult. Furthermore, the nonlinearity gives rise to a saddle-node bifurcation known as voltage pull-in that necessitates operational limitations. Eliminating this effect would allow for enhanced functionality in a number of applications by increasing the operational range of the movable electrode, reducing the need for motion limiters and anti-stiction measures, and preventing disturbances from causing the movable electrode to depart from its stable operating region. A number of controls approaches have been presented in the literature to address pull-in. In the context of the current paper the most relevant are those that—implicitly or explicitly—make use of the substantial improvements in stability associated with control of electrode charge versus control of electrode voltage. Some notable examples include [10, 14, 16]. However, an additional challenge is to implement these controllers in the presence of dynamics arising from resistive and capacitive coupling both 1 Copyright c © 2007 by ASME between device components, and between these components and the surroundings. These interactions are commonly referred to as parasitics. It is known that the effects of parasitic capacitance can cause loss of stability of charge-controlled electrostatic MEMS through a saddle-node bifurcation known as charge pullin [4,5,12,14]. Explicit compensation of parasitic effects is only now beginning to attract attention from the controls community, for example [17]. In [2, 7–9] we present a series of results on passivity-based global and semi-global stabilization of electrostatically actuated MEMS. The one-degree-of-freedom case is considered in [7]. The system input and output are the control voltage applied to the fixed electrode and the charge on the movable electrode, respectively. Two controllers are derived that eliminate the pull-in bifurcation and stabilize any point in the capacitive gap. These are the only points for which a feasible equilibrium exists. One controller, based on the energy-shaping version of passivity [13], results in a charge feedback controller. The other, based on feedback passivation techniques [3], requires an additional velocity feedback term. Unlike the energy-shaping controller, the feedback-passivation controller may be used to inject damping into the mechanical subsystem, improving transient performance. The generalized model and controller presented in [8] extends the 1-DOF results to a broad class of electrostatically forced mechanical systems, including a variety of interesting MEMS devices. That extension requires measurement of the voltage and charge associated with all electrodes, including parasitic surfaces. Typically this is not feasible. When these measurements are not available the controllers of [7, 8] may fail to stabilize equilibrium points low in the gap, with loss of stability due to charge pull-in. The main object of the present note is to examine whether the global 1-DOF result of [7] may be recovered in the presence of parasitics without requiring charge or voltage measurements on the parasitic electrodes. The analysis presented here considers only stabilization and not improved transient behavior. The energy-shaping controller and the feedbackpassivation controller are equivalent with respect to the location and stability of closed-loop equilibrium points. However so far only the behavior of the energy-shaping controller has been thoroughly studied. The feedback-passivation controller may provide a larger domain of attraction, and may influence the loss of stability through dynamic mechanisms such as Hopf bifurcation. This question is a topic of current research, and further discussion is beyond the scope of the present note. The model considered is the standard 1-DOF model used in [7]—a grounded movable electrode suspended by a spring and damper above a fixed control electrode—with an additional fixed parasitic electrode added below the control electrode (see Fig. 1). As in [1,7,8], the voltage across the movable and fixed electrode is assumed to be measured, along with either the charge on the fixed electrode or the mutual capacitance between the movable and fixed electrodes. In the present note the parasitic electrode is connected to ground through a resistor. Therefore both the parasitic voltage and the charge can vary dynamically. At equilibrium however the parasitic voltage will be zero. This condition may be relaxed by including a voltage source in series [2], but such an extension is beyond the scope of the present note. Physically the parasitic surface might represent the wafer substrate upon which the MEMS device is fabricated. It is shown that feedback control using the total charge on the movable and parasitic electrodes as system output can prevent charge pull-in, and provide global asymptotic stability of the desired equilibrium. If the control electrode is bigger than either the movable electrode or the parasitic electrode then, neglecting fringing, the parasitic electrode is completely screened from the movable electrode. In this case the total charge feedback law can be implemented using the specified measurements. Otherwise the equilibrium is only locally asymptotically stable, and in fact the equilibrium may itself lose stability through a Hopf bifurcation. Furthermore, the feedback law can not be implemented using the specified measurements and will have to be approximated. This approximation does not affect the location of the equilibrium, but it may further reduce the region of attraction. These effects become more pronounced as the parasitic surface becomes larger with respect to the control surface, or as the parasitic surface moves closer to the control surface. This note is organized as follows: First we present a 1-DOF model of an electrostatic MEMS with a parasitic capacitance. We then revisit the nominal energy-shaping controller derived in [7,8] and show how parasitics may cause charge pull-in, which we interpret in terms of bifurcation of the system zero dynamics. We next demonstrate the use of total charge feedback to eliminates charge pull-in and recover global asymptotic stability, and examines how measured quantities may be used to implement this feedback. The controller is demonstrated using Matlab and ANSYS simulations. Finally, we summarize the result and discuss possible extensions. 1-DOF MODEL WITH PARASITIC CAPACITANCE We consider a MEMS device modeled by three parallel plates. On top is the movable electrode, suspended by spring and damping elements and constrained to translate in the vertical direction, in the middle is a fixed plate that we refer to as the drive or control electrode, and at bottom is a fixed parasitic electrode. The plates have area A0, Ac, and Ap, respectively, and are assumed to be centered on a common axis. The zero voltage gap between the moving and drive electrode is d and the distance between the parasitic and drive electrode is δ. Resistive crosscoupling is neglected here, and the movable electrode is assumed to be grounded, that is, connected to ground through a zero voltage bias, with negligible series resistance. Figure 1 shows a schematic of the model. The configuration of the parasitic electrode is motivated by MEMS designs in which a parallel-plate 2 Copyright c © 2007 by ASME Figure 1. 1-DOF MODEL OF A ELECTROSTATIC MICROACTUATOR. THE TOP PLATE OF THE MEMS IS FREE TO MOVE AND THE TWO BOTTOM PLATES ARE HELD FIXED. THE MIDDLE PLATE IS THE CONTROL ELECTRODE AND THE BOTTOM PLATE IS THE PARASITIC ELECTRODE. THIS PARTICULAR CONFIGURATION IS AN EXAMPLE OF CASE II. THE AREA OF THE BOTTOM PLATE MAY BE REDUCED TO THAT OF THE TOP PLATE WITHOUT AFFECTING THE SIMPLE CAPACITANCE MODELS USED FOR CONTROL DESIGN. THE PARASITIC BIAS VOLTAGE up IS ASSUMED TO BE ZERO IN THE CONTROLLER DESIGN AND ANALYSIS. device is surface micromachined on an insulating silicon dioxide or silicon nitride layer, which is in turn deposited on a (relatively conductive) silicon wafer. In such cases large parasitic capacitances may exist between the device and the underlying silicon, especially if the insulating layer is thin. In what follows, the parasitic bias voltage up shown in Fig. 1 is assumed to be zero. For the purpose of control design the capacitive coupling and electrostatic forces are derived using a simple infinite parallel plate model that neglects fringing. Thus the mutual capacitance between any two surfaces will be Ci j = εAi j/li j, where li j is the distance between the surfaces and Ai j is the area of the overlap between the surfaces with no intervening conductor. If the parasitic effects are due to interactions with a conductive substrate, this model allows us to consider only that portion of the substrate directly beneath the device. ε is the permittivity of the material in the gap between the plates, sometimes written as a dielectric constant times the permittivity of free space. In what follows, ε is simply a constant. Even using the simplified 1-DOF model of Fig. 1 several distinct electrode configurations are possible. The following observation facilitates the analysis: Under the simplified capacitance calculation, the largest electrode area may, without loss of generality, be reduced to that of the next largest, resulting in three possible situations, in which either the parasitic electrode is the smallest, the control electrode is the smallest, or the movable electrode is the smallest. The first and third are qualitatively very similar and the third may be easily treated by analogy to the first. Therefore, to avoid unnecessarily complex notation, we will consider only the first and second configurations, which we subsequently refer to as Case I and Case II. We define the parameter ρ = Ap/Ac. For Case I, ρ ≤ 1 and for Case II ρ ≥ 1. Note that in Case I the movable electrode is completely screened from the parasitic electrode by the control electrode. In Case II the movable electrode is directly affected by the parasitics. The configuration space of the movable electrode is G = R . Let {e1} be a coordinate frame for R , with e1 fixed at the center of mass of the moving electrode in zero voltage equilibrium and pointing away from the fixed control electrode, and with x denoting the displacement of the movable electrode along e1. Then application of the modeling procedure presented in [2, 8], gives the following equations of motion: Q̇c=− 1 rcc Vc + 1 rcc uc, (1) =− 1 rcc C(x)(α(x)Qc +Qp)+ 1 rcc uc, (2) Q̇p=− 1 rpp C(x)(Qc +α(x)Qp) , (3)