Modeling and Input Optimization under Uncertainty for a Collection of Rf Mems Devices

The dynamic response of an RF MEMS device to a timevarying electrostatic force is optimized to enhance robustness to variations in material properties and geometry. The device functions as an electrical switch, where an applied voltage is used to close a circuit. The objective is to minimize the severity of the mechanical impact that occurs each time the switch closes, because severe impacts have been found to significantly decrease the design life of these switches. The switch is modeled as a classical vibro-impact system: a single degree-of-freedom oscillator subject to mechanical impact with a single rigid barrier. Certain model parameters are described as random variables to represent the significant unit-to-unit variability observed during fabrication and testing of the collection of nominally-identical switches; these models for unit-to-unit variability are calibrated to available experimental data. Our objective is to design the shape and duration of the voltage waveform so that impact velocity at switch closure for the collection of nominally-identical switches is minimized subject to design constraints. The methodology is also applied to search for design changes that reduce the impact velocity and to predict the effect of fabrication process improvements. INTRODUCTION Radio Frequency Micro Electro Mechanical System (RF MEMS) switches have been the subject of study for a number of applications because they can potentially provide very low power consumption, high isolation, and greater linearity at low cost and ∗Address all correspondence to this author. Email: [email protected]. 1Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. in a compact package [1] [2] [3]. Unfortunately, current designs for RF switches fail to achieve the high reliability demanded for many applications. The high velocity with which the switches can impact the electrical contacts is one contributing factor. Recent research at Sandia has revealed that the actuating voltage pulse can be shaped to limit the velocity with which the plate impacts the electrical contacts, increasing a switch’s life by orders of magnitude. Unfortunately, there is considerable unit-to-unit variability in the dimensions and the properties of these switches, so a waveform designed to minimize the contact velocity, or provide a soft landing, for the nominal switch is not effective for a batch of switches manufactured using current processes. This work demonstrates that the actuating voltage waveform can be optimized for a collection of RF switches with random physical parameters in order to minimize the contact velocity experienced by the ensemble. This can be cast as a problem of optimization under uncertainty or Reliability-Based Design Optimization (RBDO) [4]. The procedure is also used to optimize the design of the RF switches to reduce the contact velocity of the ensemble and to study the effect of reducing the degree of variation due to the manufacturing process. Reliability-Based Design Optimization has received considerable attention in recent years. A few researchers have applied RBDO to MEMS systems in order to improve the robustness of device designs, in part because MEMS systems tend to suffer from significant manufacturing variation and exhibit complex or uncertain physical phenomena. Optimization under uncertainty methodologies can be classified as Robust Design Optimization (RDO) methods or as ReliabilityBased Design Optimization (RBDO) methods [4]. RDO methods use deterministic analysis to attempt to maximize deterministic performance while minimizing the sensitivity of the optimum design to uncertain or random parameters. For example, Han and Kwak [5] used this ap1 Copyright c © 2006 by ASME proach to optimize the design of a MEMS accelerometer and a resonant-type micro probe by augmenting their objective function with the gradient of the objective function with respect to the random parameters. Some limitations of RDO methods are that they cannot provide information about the probability that a device will fail and, because they are gradient based, they may not perform properly if the objective function is noisy or highly nonlinear over the range spanned by the uncertain parameters. RBDO methods are based on stochastic analysis and are therefore preferred in many applications because they provide an estimate of the reliability of a design and can more accurately account for variability in uncertain parameters. Heo, Yoon and Kim used what may be classified as an RBDO method to optimize the design of a MEMS thermal actuator [6]. Allen et al [7], presented an application of RBDO to a variable capacitance MEMS capacitor. They validated the First-Order Reliability Method (FORM) for their application by comparing it to Monte Carlo Simulation (MCS) and then used FORM to optimize the design of the capacitor. FORM can be significantly less computationally expensive than MCS, yet FORM may be inaccurate if the response is non-Gaussian or if the failure boundary is not well approximated by a linear function. Allen et al observed that the FORM algorithm worked well even though their system was nonlinear, yet all of the uncertain variables in their system were assumed to be Gaussian with relatively small coefficients of variation. This assumption is often inappropriate for MEMS applications. Maute and Frangopol also used the FORM algorithm as part of an optimization strategy for a MEMS device [4]. One potential limitation of the FORM algorithm is that it includes an iterative search cast as an optimization problem to find the most probable point of failure. As a result, the FORM algorithm becomes much more difficult to apply as the number of uncertain parameters, and hence the dimension of this iterative optimization problem, increases. Also, because FORM is based on optimization, one may encounter situations in which slow or local convergence is obtained, greatly increasing the complexity of implementing FORM and diminishing its computational efficiency. Neither Allen et al for Maute and Frangopol mentioned any difficulties obtaining convergence with the FORM algorithm, although their problems were limited to a small number of uncertain variables. The work discussed here involves an objective function that is highly nonlinear due to mechanical impact and whose uncertainties are large and highly non-Gaussian. For these reasons, Monte Carlo Simulation (MCS) was used to evaluate unit-to-unit variability in these RF MEMS switches. Also, a low order mathematical model exists that captures the physics of the device remarkably well, so the computational burden is low enough that the problem is tractable with MCS. This paper is organized as follows. We first present a derivation of a reduced order model that provides a good representation of the dynamics of the RF Switch to an actuating voltage. The objective function and optimization procedure are then discussed and some results presented. Finally, the effects of design and process improvement is illustrated followed by some conclusions. Figure 1. Schematic of RF MEMS switch. MODEL DEFINITION The RF MEMS switch design of interest is shown in Fig. 1. The switch consists of a stiff plate supported above a rigid substrate by four flexible supports. A 100 nm thick electrostatic pad is adhered to the substrate below the switch plate to provide electrostatic actuation. When voltage is applied to the pad, the plate deflects downward and the contact tabs make mechanical contact with the waveguide to close the circuit. Dyck et al described the design and characterization of this switch in [1]. A single degree-of-freedom model for the RF switch is used for analysis. Previous works have demonstrated the accuracy and utility of this model for these systems, especially when the input is shaped to limit excitation to higher frequency modes [8] [9] [10]. Let X(t) denote the displacement of the contact tabs; the equations of motion are