Interdisciplinary Applied Mathematics

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much less compared to general purpose simulators like SABER29 (Mantooth and Vlach, 1992) or MATLAB that rely on user-provided HDL models.


Nodal Analysis


Nodal analysis has been widely used for formulating system equations in circuit analysis tools such as SPICE. The circuit is decomposed into N -terminal devices, and each device is modeled by ordinary differential equations (ODEs) with coefficients parameterized by device geometry and material properties (Zhou et al., 1998). The devices are linked together at their terminals or nodes, and the resulting coupled differential equations can be solved as a system of nonlinear ODEs. This approach is fast, reasonably accurate, flexible, and can be used for a higher-level simulation of



FIGURE 17.9. (a) A cantilever-beam-based microdevice. (b) Nodal representation of the microdevice.


microsystems.


In nodal analysis, the microdevice is represented using atomic elements like anchors, gaps, and beams (Zhou et al., 1998). Figure 17.9 shows a microdevice and its nodal representation. The nodal representation contains three anchor elements, one beam element, and an electrostatic gap element. Each atomic element has a lumped behavioral model with geometric parameters that can be specified individually. This simplifies the evaluation of changes in size on the device performance in each design iteration. The system matrices formed are much smaller than those in finite element analysis, and the models are implemented in analog HDLs supporting mixed physical domain simulations. The total system is formulated by formulating each individual element first. For the beam element defined between nodes 1 and 2, we have

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