Hypothesizing a bipolar reference signal, it can be assumed that the range V ref > 0 corresponds to the supply–user connection, while the field V ref < 0 corresponds to the user–discharge connection. The appropriate equation of flow above must be rewritten in the same way. Calculation of the conductance of the flow proportional valve is made considering the static and dynamic link between the reference voltage V ref and the opening of the passage aperture A V in accordance with modeling of the second order of the type: (20.65) where ζ is the damping factor, σ n is the valve’s natural frequency, and K s is its area static gain. Assuming a static relation of the linear type between the opening A V and the conductance C, as an initial approximation, we get (20.66) where K V is the flow static gain of the valve, function of the maximum conductance C max , and of the maximum value of the reference voltage V ref max : (20.67) The complete dynamic relation between reference voltage and conductance is, therefore, (20.68) The nonlinear model of the pneumatic servosystem with the position reference x set and the force disturbance F e as inputs, is made up of a nonlinear system of nine equations, of order eight overall, of the type: a) C 1 = C 1 (V ref 1 , t) order 2 conductance of valve V 1 (see (20.68)) b) C 2 = C 2 (V ref 2 , t) order 2 flow rate of valve V 1 (see (20.68)) c) G 1 = G 1 (C 1 , P 1 ) order 0 flow rate of valve V 1 (see (20.61)–(20.64)) d) G 2 = G 2 (C 2 , P 2 ) order 0 flow rate of valve V 2 (see (20.61)–(20.64)) e) G 1 = G 1 (P 1 , 1 , x, ) order 1 continuity chamber 1 (see (20.58)) f) G 2 = G 2 (P 2 , 2 , x, ) order 1 continuity chamber 2 (see (20.59)) g) = (F e , P 1 , P 2 , ) order 2 piston equilibrium (see (20.60)) h) V ref 1 = V ref 1 (x set , x ret ) order 0 V 1 valve control i) V ref 2 = V ref 2 (x set , x ret ) order 0 V 2 valve control If we want to carry out a linear analysis, it can be assumed that the equations a), b), g), h), i) are already written in linear form. As far as the flow rates of valves c) and d) are concerned, it is hypothesized that the flow rate for each of them is subsonic in feed, with V ref > 0, and sonic in discharge, with V ref < 0. This means that for valve V 1 , for example, the pressure P 1 must be within the range bP s < P 1 ≤ P s in feed and in the range P 1 ≥ P amb /b d 2 A V dt 2 2zs n dA V dt s n 2 A V ++ K s s n 2 V ref = CK c A V K c K s V ref K V V ref == = K V C max V ref max = d 2 C dt 2 2zs n dC dt s n 2 C++ K c s n 2 V ref = P ˙ x ˙ P ˙ x ˙ x ˙˙ x ˙˙ x ˙ 0066_Frame_C20 Page 92 Wednesday, January 9, 2002 5:49 PM ©2002 CRC Press LLC Hypothesizing a bipolar reference signal, it can be assumed that the range V ref > 0 corresponds to the supply–user connection, while the field V ref < 0 corresponds to the user–discharge connection. The appropriate equation of flow above must be rewritten in the same way. Calculation of the conductance of the flow proportional valve is made considering the static and dynamic link between the reference voltage V ref and the opening of the passage aperture A V in accordance with modeling of the second order of the type: (20.65) where ζ is the damping factor, σ n is the valve’s natural frequency, and K s is its area static gain. Assuming a static relation of the linear type between the opening A V and the conductance C, as an initial approximation, we get (20.66) where K V is the flow static gain of the valve, function of the maximum conductance C max , and of the maximum value of the reference voltage V ref max : (20.67) The complete dynamic relation between reference voltage and conductance is, therefore, (20.68) The nonlinear model of the pneumatic servosystem with the position reference x set and the force disturbance F e as inputs, is made up of a nonlinear system of nine equations, of order eight overall, of the type: a) C 1 = C 1 (V ref 1 , t) order 2 conductance of valve V 1 (see (20.68)) b) C 2 = C 2 (V ref 2 , t) order 2 flow rate of valve V 1 (see (20.68)) c) G 1 = G 1 (C 1 , P 1 ) order 0 flow rate of valve V 1 (see (20.61)–(20.64)) d) G 2 = G 2 (C 2 , P 2 ) order 0 flow rate of valve V 2 (see (20.61)–(20.64)) e) G 1 = G 1 (P 1 , 1 , x, ) order 1 continuity chamber 1 (see (20.58)) f) G 2 = G 2 (P 2 , 2 , x, ) order 1 continuity chamber 2 (see (20.59)) g) = (F e , P 1 , P 2 , ) order 2 piston equilibrium (see (20.60)) h) V ref 1 = V ref 1 (x set , x ret ) order 0 V 1 valve control i) V ref 2 = V ref 2 (x set , x ret ) order 0 V 2 valve control If we want to carry out a linear analysis, it can be assumed that the equations a), b), g), h), i) are already written in linear form. As far as the flow rates of valves c) and d) are concerned, it is hypothesized that the flow rate for each of them is subsonic in feed, with V ref > 0, and sonic in discharge, with V ref < 0. This means that for valve V 1 , for example, the pressure P 1 must be within the range bP s < P 1 ≤ P s in feed and in the range P 1 ≥ P amb /b d 2 A V dt 2 2zs n dA V dt s n 2 A V ++ K s s n 2 V ref = CK c A V K c K s V ref K V V ref == = K V C max V ref max = d 2 C dt 2 2zs n dC dt s n 2 C++ K c s n 2 V ref = P ˙ x ˙ P ˙ x ˙ x ˙˙ x ˙˙ x ˙ 0066_Frame_C20 Page 92 Wednesday, January 9, 2002 5:49 PM ©2002 CRC Press LLC theory and mechanics comprise the fundamentals for analysis, modeling, simulation, design, and opti- mization, while fabrication is based on the micromachining and high-aspect-ratio techniques and pro- cesses, which are the extension of the CMOS technologies developed to fabricate ICs. For many years, the developments in microelectromechanical systems (MEMS) have been concentrated on the fabrica- tion of microstructures adopting, modifying, and redesigning silicon-based processes and technologies commonly used in integrated microelectronics. The reason for refining of conventional processes and technologies as well as application of new materials is simple: in general, microstructures are three- dimensional with high aspect ratios and large structural heights in contrast to two-dimensional planar microelectronic devices. Silicon structures can be formed from bulk silicon micromachining using wet or dry processes, or through surface micromachining. Metallic micromolding techniques, based upon photolithographic processes, are also widely used to fabricate microstructures. Molds are created in polymer films (usually photoresist) on planar surfaces, and then filled by electrodepositing metal (elec- trodeposition plays a key role in the fabrication of the microstructures and microdevices, which are the components of MEMS). High-aspect ratio technologies use optical, e-beam, and x-ray lithography to create trenches up to 1 mm deep in polymethylmethacrylate resist on the electroplating base (called seed layer). Electrodeposition of magnetic materials and conductors, electroplating, electroetching, and lift- off are extremely important processes to fabricate microscale structures and devices. Though it is recog- nized that the ability to use and refine existing microelectronics fabrication technologies and materials is very important, and the development of novel processes to fabricate MEMS is a key factor in the rapid growth of affordable MEMS, other emerging areas arise. In particular, devising, design, modeling, analysis, and optimization of novel MEMS are extremely important. Therefore, recently, the MEMS theory and microengineering fundamentals have been expanded to thoroughly study other critical prob- lems such as the system-level synthesis and integration, synergetic classification and analysis, modeling and design, as well as optimization. This chapter studies the fabrication, analysis, and design problems for electromagnetic microstructures and microdevices (microtransducers with ICs). The descriptions of the fabrication processes are given, modeling and analysis issues are emphasized, and the design is performed. Design and Fabrication In MEMS, the fabrication of thin film magnetic components and microstructures requires deposition of conductors, insulators, and magnetic materials. Some available bulk material constants (conductivity σ , resistivity ρ at 20 ° C, relative permeability µ r , thermal expansion t e , and dielectric constant—relative permittivity r ) in SI units are given in Table 20.12. TABLE 20.12 Material Constants Material σρ µ r t e × 10 − 6 r Silver 6.17 × 10 7 0.162 × 10 − 7 0.9999998 NA Copper 5.8 × 10 7 0.172 × 10 − 7 0.99999 16.7 Gold 4.1 × 10 7 0.244 × 10 − 7 0.99999 14 Aluminum 3.82 × 10 7 0.26 × 10 − 7 1.00000065 25 Tungsten 1.82 × 10 7 0.55 × 10 − 7 NA NA Zinc 1.67 × 10 7 0.6 × 10 − 7 NA NA Cobalt NA NA 250 NA Nickel 1.45 × 10 7 0.69 × 10 − 7 600 nonlinear NA Iron 1.03 × 10 7 1 × 10 − 7 4000 nonlinear NA Si 2.65 11.8 SiO 2 0.51 3.8 Si 3 N 4 2.7 7.6 SiC 3.0 6.5 GaAs 6.9 13 Ge 2.2 16.1 0066_Frame_C20.fm Page 97 Wednesday, January 9, 2002 1:44 PM ©2002 CRC Press LLC . nonlinear NA Iron 1. 03 × 10 7 1 × 10 − 7 4000 nonlinear NA Si 2.65 11 .8 SiO 2 0. 51 3. 8 Si 3 N 4 2.7 7.6 SiC 3. 0 6.5 GaAs 6.9 13 Ge 2.2 16 .1 0066_Frame_C20.fm. 7 0 .17 2 × 10 − 7 0.99999 16 .7 Gold 4 .1 × 10 7 0.244 × 10 − 7 0.99999 14 Aluminum 3. 82 × 10 7 0.26 × 10 − 7 1. 00000065 25 Tungsten 1. 82. of the type: a) C 1 = C 1 (V ref 1 , t) order 2 conductance of valve V 1 (see (20.68)) b) C 2 = C 2 (V ref 2 , t) order 2 flow rate of valve V 1 (see (20.68)) c) G 1 = G 1 (C 1 , P 1 ) order 0