a-1-4s-b-a. Enclosed area 1-2s-3-4s-1 represents the net heat added per unit of mass flowing. For any power cycle, the net heat added equals the net work done. Expressions for the principal energy transfers shown on the schematics of Table 12.6 are provided by Eqs. (1) to (4) of the table. They are obtained by reducing Eq. (12.10a) with the assumptions of negligible heat loss and negligible changes in kinetic and potential energy from the inlet to the exit of each component. All quantities are positive in the directions of the arrows on the figure. The thermal efficiency of a power cycle is defined as the ratio of the net work developed to the total energy added by heat transfer. Using expressions (1)–(3) of Table 12.6, the thermal efficiency is (12.27) To obtain the thermal efficiency of the ideal cycle, h 2s replaces h 2 and h 4s replaces h 4 in Eq. (12.27). TABLE 12.6 Rankine and Brayton Cycles Rankine Cycle Brayton Cycle (>0) (1) (>0) (2) (>0) (3) (>0) (4) W p ˙ W c ˙    m ˙ h 2 h 1 –()= Q ˙ in m ˙ h 3 h 2 –()= W ˙ t m ˙ h 3 h 4 –()= Q ˙ out m ˙ h 1 h 4 –()= h h 3 h 4 –()h 2 h 1 –()– h 3 h 2 – = 1 h 4 h 1 – h 3 h 2 – –= 0066_frame_C12 Page 26 Wednesday, January 9, 2002 4:22 PM ©2002 CRC Press LLC a-1-4s-b-a. Enclosed area 1-2s-3-4s-1 represents the net heat added per unit of mass flowing. For any power cycle, the net heat added equals the net work done. Expressions for the principal energy transfers shown on the schematics of Table 12.6 are provided by Eqs. (1) to (4) of the table. They are obtained by reducing Eq. (12.10a) with the assumptions of negligible heat loss and negligible changes in kinetic and potential energy from the inlet to the exit of each component. All quantities are positive in the directions of the arrows on the figure. The thermal efficiency of a power cycle is defined as the ratio of the net work developed to the total energy added by heat transfer. Using expressions (1)–(3) of Table 12.6, the thermal efficiency is (12.27) To obtain the thermal efficiency of the ideal cycle, h 2s replaces h 2 and h 4s replaces h 4 in Eq. (12.27). TABLE 12.6 Rankine and Brayton Cycles Rankine Cycle Brayton Cycle (>0) (1) (>0) (2) (>0) (3) (>0) (4) W p ˙ W c ˙    m ˙ h 2 h 1 –()= Q ˙ in m ˙ h 3 h 2 –()= W ˙ t m ˙ h 3 h 4 –()= Q ˙ out m ˙ h 1 h 4 –()= h h 3 h 4 –()h 2 h 1 –()– h 3 h 2 – = 1 h 4 h 1 – h 3 h 2 – –= 0066_frame_C12 Page 26 Wednesday, January 9, 2002 4:22 PM ©2002 CRC Press LLC 13 Modeling and Simulation for MEMS 13.1 Introduction 13.2 The Digital Circuit Development Process: Modeling and Simulating Systems with Micro- (or Nano-) Scale Feature Sizes 13.3 Analog and Mixed-Signal Circuit Development: Modeling and Simulating Systems with Micro- (or Nano-) Scale Feature Sizes and Mixed Digital (Discrete) and Analog (Continuous) Input, Output, and Signals 13.4 Basic Techniques and Available Tools for MEMS Modeling and Simulation Basic Modeling and Simulation Techniques • A Catalog of Resources for MEMS Modeling and Simulation 13.5 Modeling and Simulating MEMS, i.e., Systems with Micro- (or Nano-) Scale Feature Sizes, Mixed Digital (Discrete) and Analog (Continuous) Input, Output, and Signals, Two- and Three-Dimensional Phenomena, and Inclusion and Interaction of Multiple Domains and Technologies 13.6 A “Recipe” for Successful MEMS Simulation 13.7 Conclusion: Continuing Progress in MEMS Modeling and Simulation 13.1 Introduction Accurate modeling and efficient simulation, in support of greatly reduced development cycle time and cost, are well established techniques in the miniaturized world of integrated circuits (ICs). Simulation accuracies of 5% or less for parameters of interest are achieved fairly regularly [1], although even much less accurate simulations (25–30%, e.g.) can still be used to obtain valuable information [2]. In the IC world, simulation can be used to predict the performance of a design, to analyze an already existing component, or to support automated synthesis of a design. Eventually, MEMS simulation environments should also be capable of these three modes of operation. The MEMS developer is, of course, most interested in quick access to particular techniques and tools to support the system currently under development. In the long run, however, consistently achieving acceptably accurate MEMS simulations will depend both on the ability of the CAD (computer-aided design) community to develop robust, efficient, user-friendly tools which will be widely available both to cutting-edge researchers and to production engineers and on the existence of readily accessible standardized processes. In this chapter we focus on fundamental approaches which will eventually lead to successful MEMS simulations becoming routine. Carla Purdy University of Cincinnati ©2002 CRC Press LLC 14 Rotational and Translational Microelectromechanical Systems: MEMS Synthesis, Microfabrication, Analysis, and Optimization 14.1 Introduction 14.2 MEMS Motion Microdevice Classifier and Structural Synthesis 14.3 MEMS Fabrication Bulk Micromachining • Surface Micromachining • LIGA and LIGA-Like Technologies 14.4 MEMS Electromagnetic Fundamentals and Modeling 14.5 MEMS Mathematical Models Example 14.5.1: Mathematical Model of the Translational Microtransducer • Example 14.5.2: Mathematical Model of an Elementary Synchronous Reluctance Micromotor • Example 14.5.3: Mathematical Model of Two-Phase Permanent-Magnet Stepper Micromotors • Example 14.5.4: Mathematical Model of Two-Phase Permanent-Magnet Synchronous Micromotors 14.6 Control of MEMS Proportional-Integral-Derivative Control • Tracking Control • Time-Optimal Control • Sliding Mode Control • Constrained Control of Nonlinear MEMS: Hamilton–Jacobi Method • Constrained Control of Nonlinear Uncertain MEMS: Lyapunov Method • Example 14.6.1: Control of Two-Phase Permanent-Magnet Stepper Micromotors 14.7 Conclusions Sergey Edward Lyshevski Purdue University Indianapolis ©2002 CRC Press LLC . (4) W p ˙ W c ˙    m ˙ h 2 h 1 –()= Q ˙ in m ˙ h 3 h 2 –()= W ˙ t m ˙ h 3 h 4 –()= Q ˙ out m ˙ h 1 h 4 –()= h h 3 h 4 –()h 2 h 1 –()– h 3 h 2 – = 1 h 4 h 1 – h 3 h 2 – –= 0066_frame_C12 Page 26 Wednesday, January 9, 2002 4:22 PM 2002 CRC Press LLC a -1- 4s-b-a. Enclosed area 1- 2s-3-4s -1. (4) W p ˙ W c ˙    m ˙ h 2 h 1 –()= Q ˙ in m ˙ h 3 h 2 –()= W ˙ t m ˙ h 3 h 4 –()= Q ˙ out m ˙ h 1 h 4 –()= h h 3 h 4 –()h 2 h 1 –()– h 3 h 2 – = 1 h 4 h 1 – h 3 h 2 – –= 0066_frame_C12 Page 26 Wednesday, January 9, 2002 4:22 PM 2002 CRC Press LLC 13 Modeling and . of Table 12 .6 are provided by Eqs. (1) to (4) of the table. They are obtained by reducing Eq. (12 .10 a) with the assumptions of negligible heat loss and negligible changes in kinetic and potential