Page 297 - The Mechatronics Handbook

0066_frame_C14.fm Page 11 Wednesday, January 9, 2002 1:49 PM

For the Cartesian, cylindrical, and spherical coordinate systems, which can be used to develop the
mathematical model, we have

E x = E 1 , E y = E 2 , E z = E 3 , D x = D 1 , D y = D 2 , D z = D 3 ,
H x = H 1 , H y = H 2 , H z = H 3 , B x = B 1 , B y = B 2 , B z = B 3
E r = E 1 , E θ = E 2 , E z = E 3 , D r = D 1 , D θ = D 2 , D z = D 3 ,
H r = H 1 , H θ = H 2 , H z = H 3 , B r = B 1 , B θ = B 2 , B z = B 3
E ρ = E 1 , E θ = E 2 , E φ = E 3 , D ρ = D 1 , D θ = D 2 , D φ = D 3 ,
H ρ = H 1 , H θ = H 2 , H φ = H 3 , B ρ = B 1 , B θ = B 2 , B φ = B 3

Maxwell’s equations can be solved using the MATLAB environment.
In motion microdevices, the designer analyzes the torque or force production mechanisms.
Newton’s second law for rotational and translational motions is

--------- = 1 ∑ T Σ, -------- = ω r
dω r
dθ r
--
dt J dt
dv ---- ∑ F Σ, dx v
1
----- =
------ =
dt m dt
where ω r and θ r are the angular velocity and displacement, v and x are the linear velocity and displacement,
∑T Σ is the net torque, ∑F Σ is the net force, J is the equivalent moment of inertia, and m is the mass.
14.5 MEMS Mathematical Models

The problems of modeling and control of MEMS are very important in many applications. A mathe-
matical model is a mathematical description (in the form of functions or equations) of MEMS, which
integrate motion microdevices (microscale actuators and sensors), radiating energy microdevices, micro-
scale driving/sensing circuitry, and controlling/signal processing ICs. The purpose of the model devel-
opment is to understand and comprehend the phenomena, as well as to analyze the end-to-end behavior.
To model MEMS, advanced analysis methods are required to accurately cope with the involved highly
complex physical phenomena, effects, and processes. The need for high-fidelity analysis, computationally-
efficient algorithms, and simulation time reduction increases significantly for complex microdevices, restrict-
ing the application of Maxwell’s equations to problems possible to solve. As was illustrated in the previous
section, nonlinear electromagnetic and energy conversion phenomena are described by the partial differ-
ential equations. The application of Maxwell’s equations fulfills the need for data-intensive analysis capa-
bilities with outcome prediction within overall modeling domains as particularly necessary for simulation
and analysis of high-performance MEMS. In addition, other modeling and analysis methods are applied.
The lumped mathematical models, described by ordinary differential equations, can be used. The process
of mathematical modeling and model development is given below.
The first step is to formulate the modeling problem:
• examine and analyze MEMS using a multilevel hierarchy concept, develop multivariable input-
output subsystem pairs, e.g., motion microstructures (microscale actuators and sensors), radiating
energy microdevices, microscale circuitry, ICs, controller, input/output devices;
• understand and comprehend the MEMS structure and system configuration;
• gather the data and information;
• develop input-output variable pairs, identify the independent and dependent control, disturbance,
output, reference (command), state and performance variables, as well as events;

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