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System Level Simulation of Mixed-signal Multi-domain Microsystems with Piecewise Linear Behavioral Models

José A. Martínez, Timothy P. Kurzweg, Steven P. Levitan, Abhijit J. Davare, Mark Kahrs, Donald M. Chiarulli
University of Pittsburgh, US

Keywords: behavioral modeling, piecewise linear simulation, MEMs simulation, modified nodal analysis, mixed-signal multi-domain simulation, system simulation of microsystems

Abstract:
The creation of simulation tools for mixed-signal, multi-domain (MSMD) systems is challenging because these systems span the physical domains of electronics, photonics, fluidics, and mechanics, as well as multiple orders of magnitude in both time and length scales. The difficulties are compounded by the fact that computational performance and accuracy of the simulations are directly related to the level of detail in the underlying models. In this paper, we present a component-based multi-level mixed-signal design and simulation environment that provides a solution to the problem of accurate modeling and simulation of multi-domain devices at the system level. This is achieved by partitioning the system into components that are modeled by analytic expressions at the behavioral level. These expressions are reduced via linearization into regions of operation for each element of the component and solved with Modified Nodal Analysis (MNA) in the frequency domain, which guarantees convergence. At the system level, a discrete event simulator sends composite signals between components and manages multiple timescales and feedback among components. For electrical and mechanical components, interaction is via physical connectivity [1] while optical signals are modeled using complex scalar wavefronts [2], providing the accuracy necessary to model micro-optical components. Simulation speed vs. simulation accuracy can be tuned by controlling the granularity of the regions of operation of the devices, sample density of the optical wavefronts, and/or the time steps of the discrete event simulator. The methodology is specifically optimized for loosely coupled systems of complex components such as are found in multi-domain microsystems. Fig. 1(a) and (b) show the linearization of a single CMOS transistor. Each non-linear element in a component can be linearized into regions of operation by an automatic hyperplane generator procedure [3]. These models, together with linear models can be used in a SPICE compatible netlist (Fig. 1(b)) for use in our general MNA piecewise linear behavioral solver. The use of a general PWL solver for both electrical and mechanical simulation decreases the computational task and allows for a trade-off between accuracy and speed. The additional advantage of using the same technique to characterize electrical and mechanical models allows us to easily merge both technologies in complex devices that interact in mixed domains. We use the above techniques to perform system-level simulations of multi-domain microsystems. Fig. 2 shows one pixel of a Grating Light Valve (GLV) display system based on a square well diffraction MEM component [4]. Each pixel is composed of a set of anchored micro-mechanical ribbons, suspended above the silicon substrate. A pixel is operated by electro-static attraction pulling alternating ribbons toward the substrate a distance of * of the wavelength of the incident light. This is seen in Fig. 2(c). When this occurs, the optical power is diffracted and steered through the optical system for projection. Fig. 3(a) and (b) show the CMOS driver which was modeled with the lineaization technique and the piecewise linear input and output waveforms. This driver is used to electro-statically drive the GLV beams. Fig. 3(c) shows the dynamic response of the GLV ribbon driven at a high switching frequency. Note, the high stiffness of the structure gives it a fast response time. However, under this stimulus, resonant effects are observed in the displacement of the nodes. The visible pattern of underdamped oscillations shows that the stiffness and the degree of damping affect the maximum operating speed of this device. Fig. 4 shows the multi-domain GLV system in operation. The graphs show voltage vs. ribbon displacement and ribbon displacement vs. optical power captured in a detector centered on the +1st diffractive mode. The contour plots show the optical diffraction pattern at different points in time. Table 1 shows system simulation time as a function of both the optical mesh resolution and the number of PWL segments in the ribbons. The mechanical subsystem time includes the initialization of the MNA as well as the solution times for the entire movement for the 2.4ms stimulus. The optical subsystem time includes both the scalar propagation time and the detector power integration time. The system time includes the electrical simulation of the CMOS driver, as well as initialization overhead. What is interesting to note is the range of the simulation times: 3 seconds for the 5-element, 128x128 case to 168 seconds for the 41-element, 512x512 case and the commensurate increase in fidelity of the resulting optical waveforms. What this illustrates is that we can use the same behavioral descriptions, in the same system-level simulation environment, to perform both interactive “what if” design exploration as well as more detailed investigations of higher order effects by simply changing the simulation parameters (e.g., optical mesh size, number of mechanical nodes, number of regions of operation for non-linear elements, and minimum timestep) without recourse to lower level simulation tools.

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