Chapter three: Structural design, modeling, and simulation 191 Figure 18. Timing Diagram © 2001 by CRC Press LLC 192 Chapter three: Structural design, modeling, and simulation © 2001 by CRC Press LLC Chapter three: Structural design, modeling, and simulation 193 © 2001 by CRC Press LLC 194 Chapter three: Structural design, modeling, and simulation © 2001 by CRC Press LLC Chapter three: Structural design, modeling, and simulation 195 © 2001 by CRC Press LLC CHAPTER 4 CONTROL OF NANO- AND MICROELECTROMECHANICAL SYSTEMS 4.1. FUNDAMENTALS OF ELECTROMAGNETIC RADIATION AND ANTENNAS IN NANO- AND MICROSCALE ELECTROMECHANICAL SYSTEMS The electromagnetic power is generated and radiated by antennas. Time- varying current radiates electromagnetic waves (radiated electromagnetic fields). Radiation pattern, beam width, directivity, and other major characteristics can be studied using Maxwell’s equations, see Section 2.2. We use the vectors of the electric field intensity E , electric flux density D, magnetic field intensity H, and magnetic flux density B. The constitutive equations are ED ε = and HB µ = where ε is the permittivity; µ is the permiability. It was shown in Section 2.2 that in the static (time-invariant) fields, electric and magnetic field vectors form separate and independent pairs. That is, E and D are not related to H and B, and vice versa. However, for time- varying electric and magnetic fields, we have the following fundamental electromagnetic equations t tzyx tzyx ∂ ∂ µ ),,,( ),,,( H E −=×∇ , ),,,( ),,,( ),,,(),,,( tzyx t tzyx tzyxtzyx J E EH ++=×∇ ∂ ∂ εσ , ε ρ ),,,( ),,,( tzyx tzyx v =⋅∇ E , 0),,,( = ⋅ ∇ tzyx H , where J is the current density, and using the conductivity σ , we have EJ σ= ; v ρ is the volume charge density. The total current density is the sum of the source current J S and the conduction current density E σ (due to the field created by the source J S ). Thus, EJJ σ+= Σ S . The equation of conservation of charge (continuity equation) is ∫∫ −=⋅ v v s dv dt d d ρ sJ , and in the point form one obtains t tzyx tzyx v ∂ ∂ −=⋅∇ ),,,( ),,,( ρ J . © 2001 by CRC Press LLC Therefore, the net outflow of current from a closed surface results in decrease of the charge enclosed by the surface. The electromagnetic waves transfer the electromagnetic power. That is, the energy is delivered by means of electromagnetic waves. Using equations t ∂ ∂ µ H E −=×∇ and J E H +=×∇ t ∂ ∂ ε , we have +⋅−⋅−=×∇⋅−×∇⋅=×⋅∇ J E E H HHEEHHE tt ∂ ∂ ε ∂ ∂ µ)()()( . In a media, where the constitute parameters are constant (time-invariant), we have the so-called point-function relationship ( ) 22 2 1 2 2 1 )( EHE t σµε ∂ ∂ −+−=×⋅∇ HE . In integral form one obtains ( ) EHE sHE of presence in the dissipatedpower ohmic 2 field magnetic and field electric the in storedenergy of change of ratetime 2 2 1 2 2 1 .)( ∫∫∫ −+−=⋅× − vvs dvEdvHE t d σµε ∂ ∂ The right side of the equation derived gives the rate of decrease of the electric and magnetic energies stored minus the ohmic power dissipated as heat in the volume v. The pointing vector, which is a power density vector, represents the power flows per unit area, and H EP × = . Furthermore, ( ) ∫∫∫∫ ++=⋅=⋅× vv HE ss dvdvww t dd σ ρ ∂ ∂ volumeenclosed theleavingpower )( sPsHE , where 2 2 1 Ew E ε= and 2 2 1 Hw H µ= are the electric and magnetic energy densities; 22 1 JE σ σρ σ == is the ohmic power density. The important conclusion is that the total power transferred into a closed surface s at any instant equals the sum of the rate of increase of the stored electric and magnetic energies and the ohmic power dissipated within the enclosed volume v. If the source charge density ),,,( tzyx v ρ and the source current density ),,,( tzyx J vary sinusoidally, the electromagnetic field also vary sinusoidally. Hence, we have deal with the so-called time-harmonic electromagnetic fields. The sinusoidal time-varying electromagnetic fields will be studied. Hence, the phasor analysis is applied. For example, zzyyxx EEE arararrE )()()()( + + = . © 2001 by CRC Press LLC The electric field intensity components are the complex functions. In particular, ImRe )( xxx jEEE +=r , ImRe )( yyy jEEE + = r , ImRe )( zzz jEEE +=r . For the real electromagnetic field, we have tEtEtE xxx ωω sin)(cos)(),( ImRe rrr − = . One obtains the time-harmonic electromagnetic field equations. In particular, • Faraday’s law H E ωµ j − = × ∇ , • generalized Amphere’s law JEJEEH + +=++=×∇ ε ω σ ωωεσ j jj , • Gauss’s law ε ω σ ρ + =⋅∇ j v E , • Continuity of magnetic flux 0 = ⋅ ∇ H , • Continuity law v j ωρ−=⋅∇ J , (4.1.1) where +ε ω σ j is the complex permittivity. However, for simplicity we will use ε keeping in mind that the expression for the complex permittivity ε ω σ + j must be applied. T he electric field intensity E, electric flux density D, magnetic field intensity H, magnetic flux density B, and current density J are complex-valued functions of spatial coordinates. From the equation (4.1.1) taking the curl of H E ωµ j − = × ∇ , which is rewritten as B E ω j − = × ∇ , and using JDH + = × ∇ ω j , one obtains JEEE ωµµεω jk v −===×∇×∇ 22 , where k v is the wave constant µεω= v k , and in free space c k v ω εµω == 000 because the speed of light is 00 1 εµ =c , sec m 8 103×=c . The wavelength is found as µεω ππ λ 22 == v v k , and in free space ω ππ λ c k v v 22 0 0 == . © 2001 by CRC Press LLC Using the magnetic vector potential A, we have AB × ∇ = . Hence, 0)( = + × ∇ A E ω j , and thus Ë A E −∇ = + ω j , where Ë is the scalar potential. To guarantee that JDH + = × ∇ ω j holds, it is required that JEAAAH µωµεµ +=∇−⋅∇∇=×∇×∇=×∇ j 2 . Therefore, one finally finds the equation needed to be solved JËAAA µωµε −+⋅∇∇=+∇ )( 22 jk v . Taking note of the Lorentz condition Ë A ωµε j − = ⋅ ∇ , one obtains. JAA µ−=+∇ 22 v k . Thus, the equation for Ë is found. In particular, ε ρ v v k −=+∇ ËË 22 . The equation for the magnetic vector potential is found solving the following inhomogeneous Helmholtz equation JAA µ−=+∇ 22 v k . The expression for the electromagnetic field intensity, in terms of the vector potential, is ωµε ω j j A AE ⋅∇∇ +−= . To derive E, one must have A. The Laplacian for A in different coordinate systems can be found. For example, we have xxvx JAkA µ−=+∇ 22 , yyvy JAkA µ−=+∇ 22 , zzvz JAkA µ−=+∇ 22 . It was shown that the magnetic vector potential and the scalar potential obey the time-dependent inhomogeneous wave equation ),(),( 2 2 2 tFt t k rr −=Ω ∂ ∂ −∇ . The solution of this equation is found using Green’s function as ∫∫∫∫ −−−=Ω '')';'()','(),( τddtttGtFt rrrr , where ( ) '' '4 1 )';'( rr rr rr −−− − −=−− kttttG δ π . The so-called retarded solution is © 2001 by CRC Press LLC ∫∫∫ − −− −=Ω ' ' )'','( ),( τd ktF t rr rrr r . For sinusoidal electromagnetic fields, we apply the Fourier analysis to obtain ∫∫∫ − −=Ω −− ')'( '4 1 )( ' τ π dF e v jk r rr r rr . Thus, we have the expressions for the phasor retarded potentials dv e v jk v ∫ − = −− )'( '4 )( ' rJ rr A rr r π µ , dv e v jk v ∫ − = −− )'( '4 1 )( ' r rr Ë rr r ρ πε . Example 4.1.1. Consider a short (dl) thin filament of current located in the origin, see Figure 4.1.1. Derive the expressions for magnetic vector potential and electromagnetic field intensities. Figure 4.1.1. Current filament in the spherical coordinate system Solution. The magnetic vector potential has only a z component, and thus, from JAA µ−=+∇ 22 v k , we have ds i JAkA zzvz µµ −=−=+∇ 22 , where ds is the cross-sectional area of the filament. x y z idl θ φ r φ a r a θ a © 2001 by CRC Press LLC [...]... modeling, and optimization of NEMS and MEMS lead to the development of superior high-performance NEMS and MEMS In this section, we address introductory control issues Mathematical models of NEMS and MEMS were derived, and the application of the Lyapunov theory is studied as applied to solve the motion control problem It was illustrated that NEMS and MEMS must be controlled Nano- and microelectromechanical systems. .. of subsystems, and to control microscale electric motors, as discussed in previous chapters, power amplifiers (ICs) regulate the voltage or current fed to the motor windings These power amplifiers are controlled based upon the reference (command), output, decision making, and other variables Studying the end-to-end NEMS and MEMS behavior, usually the output is the nano- or microactuator linear and angular... References 1 2 3 Hayt W H., Engineering Electromagnetics, McGraw-Hill, New York, 1989 Collin R E., Antennas and Radiowave propagation,” McGraw-Hill, New York, 1985 Paul C R., Whites K W., and Nasar S A., Introduction to Electromagnetic Fields, McGraw-Hill, New York, 1998 © 2001 by CRC Press LLC 4.2 DESIGN OF CLOSED-LOOP NANO- AND MICROELECTROMECHANICAL SYSTEMS USING THE LYAPUNOV STABILITY THEORY The solution... disturbance attenuation, et cetera) Several methods have been developed to address and solve nonlinear design and motion control problems for multi-input/multi-output dynamic systems In particular, the Hamilton-Jacobi and Lyapunov theories are found to be the most straightforward in the design of control laws The NEMS and MEMS dynamics is described as & x(t ) = F ( x, r , d ) + B ( x)u , y = H (x) ,... fields in near- and far-fields can be straightforwardly derived, and thus, the corresponding approximations for the Eφ , H r and H θ can be obtained Let the current density distribution in the volume is given as J (r0 ) , and for far-field from Figure 4.1.3 one has r ≈ r '−r0 z ar J r0 y Source x r' r Figure 4.1.3 Radiation from volume current distribution The formula to calculate far-field magnetic... potential Az is not a function of the polar and azimuth angles θ and φ In particular, the following equation results 1 ∂ 2 ∂Az r + k v2 Az = 0 2 ∂r r ∂r It is well-known that the solution of equation d 2ψ + kv2ψ = 0 has two 2 dψ − jk r jk r components In particular, e v (outward propagation) and e v (inward propagation) The inward propagation is not a part of solution for the filament located in... vector; r∈R⊂b and y∈Y⊂b are the measured reference and output vectors; d∈D⊂s is the disturbance vector; F(⋅):c×b×s→c and B(⋅):c→ c×m are jointly continuous and Lipschitz; H(⋅):c→b is the smooth map defined in the neighborhood of the origin, H(0) = 0 Before engaged in the design of closed-loop systems, which will be based upon the Lyapunov stability theory, let us study stability of time-varying... (r0 )e − jk vr0 dv , 4πr v ∫ and the electric and magnetic field intensities are found using E = − jωA + ∇∇ ⋅ A jωµε and B = ∇ × A We have E(r ) = jkv Z v − jkv r [a r ⋅ J (r0 )a r − J (r0 )]e − jkvr0 dv , e 4πr v ∫ H (r ) = Yv a r × E(r ) Example 4.1.3 Consider the half-wave dipole antenna fed from a two-wire transmission line, as shown in Figure 4.1.4 The antenna is one-quarter wavelength; that is,... + 3 e c aθ c r cr r The intrinsic impedance is given as Z0 = ε0 µ0 µ0 1 = , and Y0 = Z0 ε0 Near-field and far-field electromagnetic radiation fields can be found, simplifying the expressions for H(r) and E(r) For near-field, we have idl sin θω − 1 H (r ) = ∇ × A(r ) = j e µ0 4πcr 2 and E(r ) = j µ0 cidlω ε0 4πc 2 r j ωr c a φ sin θaθ The complex Pointing vector can be found as 1 2 E( r... the exist the K ∞ -functions ρ 1 (⋅) and ρ 2 (⋅) , and K-function ρ 3 (⋅) such that ρ1 ( x ) ≤ V (t , x ) ≤ ρ 2 ( x ) and dV ( x ) ≤ − ρ3 ( x dt ) n Examples are studied to illustrate how this Theorem can be straightforwardly applied Example 4.2.1 Study stability of the following nonlinear system 2 & x1 (t ) = − x13 − x1 x 2 , 7 & x 2 (t ) = − x 2 , t ≥ 0 Solution A scalar positive-definite function . closed surface results in decrease of the charge enclosed by the surface. The electromagnetic waves transfer the electromagnetic power. That is, the energy is delivered by means of electromagnetic. and MEMS must be controlled. Nano- and microelectromechanical systems augment a great number of subsystems, and to control microscale electric motors, as discussed in previous chapters, power amplifiers. RADIATION AND ANTENNAS IN NANO- AND MICROSCALE ELECTROMECHANICAL SYSTEMS The electromagnetic power is generated and radiated by antennas. Time- varying current radiates electromagnetic waves (radiated