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),,,( =⋅∇ tzyxH , 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 Eσ (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 HEP ×= . 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 ),,,( tzyxJ 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 HE 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 HE ωµj−=×∇ , which is rewritten as BE ωj−=×∇ , and using JDH +=×∇ ωj , 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)( =+×∇ AE ωj , and thus ËAE −∇=+ ωj , where Ë is the scalar potential. To guarantee that JDH +=×∇ ωj 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 ËA ωµεj−=⋅∇ , 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 Taking note of the spherical symmetry, we conclude that the magnetic vector potential A z is not a function of the polar and azimuth angles φθ and . In particular, the following equation results 0 1 22 2 =+ ∂ ∂ ∂ ∂ zv z Ak r A r r r . It is well-known that the solution of equation 0 2 2 2 =+ ψ ψ ψ v k d d has two components. In particular, rjk v e (outward propagation) and rjk v e − (inward propagation). The inward propagation is not a part of solution for the filament located in the origin. Thus, we have rjktj v aert − = ω ψ ),( (outward propagating spherical wave). In free space, we have )/( ),( crtj aert − = ω ψ . Substituting r A z ψ = , one obtains c rj z e r a rA ω − =)( . To find the constant a, we use the volume integral ∫∫∫∫ −−=⋅∇=∇ v z v z s drz v z dvJdvA c ddrAdvA 0 2 2 22 sin µ ω φθθ a , where the differential spherical volume is drddrdv d φθθsin 2 = ; r d is the differential radius. Making use of r c j z rz e r a r c j r A A ω ω −       +−= ∂ ∂ =⋅∇ 2 1a , we have idladdaer c j d d r c j d r 0 2 0 0 0 4sin1lim µπφθθ ω ππ ω −=−=       +− ∫∫ − → , one has π µ 4 0 idl a = . Thus, the following expression results c rj z e r idl rA ω π µ − = 4 )( 0 . Therefore, the final equation for the magnetic vector potential (outward propagating spherical wave) is z c rj e r idl r aA ω π µ − = 4 )( 0 . © 2001 by CRC Press LLC From θθ θ sincos aaa −= rz , we have )sincos( 4 )( 0 θθ π µ θ ω aaA −= − r c rj e r idl r The magnetic and electric field intensities are found using AB ×∇= and ωµε ω j j A AE ⋅∇∇ +−= . Then, one finds φ ω ω π θ µ aAH c rj e r cr j r idl rr −       +=×∇= 2 0 1 4 sin )( 1 )( , . 1 sin 4 1 cos 4 )( 322 2 0 0 32 0 0 θ ω ω ωω θ πω ε µ ω θ πω ε µ a aE c rj r c rj e rcr j rc cidlj e rcr j cidlj r − −         ++−−       += The intrinsic impedance is given as 0 0 0 ε µ =Z , and 0 0 0 0 1 µ ε == Z Y . 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 φ ω π θω µ aAH c rj e cr idl jrr − =×∇= 2 0 4 sin )( 1 )( and .sin 4 )( 2 0 0 θ θ π ω ε µ aE rc cidl jr = The complex Pointing vector can be found as )()( * 2 1 rr HE × . The following expression for the complex power flowing out of a sphere of radius r results ( ) π ωµ π εµµω 1212 )()( 22 0 22 000 2 * 2 1 dlik dli drr v s ==⋅× ∫ sHE . The real quality is found, and the power dissipated in the sense that it travels away from source and cannot be recovered. © 2001 by CRC Press LLC Example 4.1.2. Derive the expressions for the magnetic vector potential and electromagnetic field intensities for a magnetic dipole (small current loop) which is shown in Figure 4.1.2. Figure 4.1.2. Current loop in the xy plane Solution. The magnetic dipole moment is equal to the current loop are times current. That is, zz Mir aaM == 2 0 π . For the short current filament, it was derived in Example 4.1.1 that z c rj e r idl r aA ω π µ − = 4 )( 0 . In contrast, we have ∫ = l dl r i ' 1 4 0 π µ A . The distance between the source element dl and point ),,( 2 π θrO is denoted as r’. It should be emphasized that the current filament is lies in the xy plane, and φφφφ φ drdrdl yx 00 )cossin( aaa +−== . Thus, due to the symmetry ∫ − = 2/ 2/ 00 ' sin 2 π π φ φ φ π µ d r ir aA , where using the trigonometric identities one finds φθ sinsin2' 0 2 0 22 rrrrr −+= . Assuming that 2 0 2 rr >> , we have       +≈ φθ sinsin1 1 ' 1 0 r r rr . x y z θ r 'r 0 r φdir 0 φ ),,( 2 π θrO dl © 2001 by CRC Press LLC Therefore .sin 4 sinsinsin1 2 ' sin 2 2 2 00 2/ 2/ 000 2/ 2/ 00 θ µ φφφθ π µ φ φ π µ φ π π φ π π φ r ir d r r r ir d r ir aa aA =       += = ∫ ∫ − − Having obtained the explicit expression for the vector potential, the magnetic flux density is found. In particular, )sincos2( 4 sin 4 3 2 00 2 2 00 θθ µ θ µ θφ aaaAB +=×∇=×∇= r r ir r ir . Taking note of the expression for the magnetic dipole moment z ir aM 2 0 π= , one has r rr ir aMaA ×== 2 0 2 2 00 4 sin 4 π µ θ µ φ . It was shown that using ∫ = l dl r i ' 1 4 0 π µ A , the desired results are obtained. Let us apply ∫ − = l r c j dl r e i '4 ' 0 ω π µ A . From r c jr c j err c je ωω ω −−       −−≈ )'(1 ' , we have ( ) θ π µ π µ ω ω φ ω ω sin1 4 ' )]'(1[ 4 2 00 r c j c l r c j c erj r M dl r errj i − − += −− = ∫ aA . Therefore, one finds θ π ωµ ω ω ω φ sin 11 4 2 2 3 0 2 2 r c j c c e r rj c M jE −           −= , θ π ε µ ωµ ω ωω cos 11 4 2 32 0 0 2 3 0 3 3 2 2 r c j cc r e rjr c M jH −           += , θ π ε µ ωµ ω ωω ω θ sin 111 4 32 0 0 2 3 0 3 3 2 2 r c j cc c e rjr rj c M jH −           −−−= . © 2001 by CRC Press LLC The electromagnetic fields in near- and far-fields can be straightforwardly derived, and thus, the corresponding approximations for the φ E , r H and θ H can be obtained. Let the current density distribution in the volume is given as )( 0 rJ , and for far-field from Figure 4.1.3 one has 0 ' rrr −≈ . Figure 4.1.3. Radiation from volume current distribution The formula to calculate far-field magnetic vector potential is dvee r v jkrjk vv ∫ −− = 0 )( 4 )( 0 r rJA r π µ , and the electric and magnetic field intensities are found using ωµε ω j j A AE ⋅∇∇ +−= and AB ×∇= . We have [ ] dvee r Zjk v jk rr rjk vv vv ∫ −− −⋅= 0 )()( 4 )( 00 r rJarJaE r π , )()( rEaH r ×= rv Y . 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, vv z λλ 4 1 4 1 ≤≤− . The current distribution is zkizi v cos)( 0 = . O btain the equations for electromagnetic field intensities and radiated power. x z 0 r r r a y 'r J Source © 2001 by CRC Press LLC [...]... INTRODUCTION TO INTELLIGENT CONTROL OF NANOAND MICROELECTROMECHANICAL SYSTEMS Hierarchical distributed closed-loop systems must be designed for largescale multi-node NEMS and MEMS in order to perform a number of complex functions and tasks in dynamic and uncertain environments In particular, the goal is the synthesis of control algorithms and architectures which maximize performance and efficiency minimizing system... evolution, and organization; • adaptive decision making, • coordination and autonomy of multi-node NEMS and MEMS through tasks and functions generation, organization and decomposition, • performance analysis with outcomes prediction and assessment, • real-time diagnostics, health monitoring, and estimation, • real-time adaptation and reconfiguration, • fault tolerance and robustness, • etc Control theory 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 augment... =0 assigned command and events ri(t) and system outputs yi(t), and the end-toend error vector is e( t ) = r (t ) − y (t ) ; j • ∑ x (t ) is the state, event, and decision variable vector; i i =0 j • ∑ s (t ) i is the sensed information (inputs, outputs, state and decision i =0 variables, events, disturbances, noise, parameters, et cetera) measured by k jth and lower level sensors, and, in general,... (NEMS/MEMS with subsystems – sensors, actuators, and ICs), one sensor and actuator were failed These types of failures must be identified in real-time (through diagnostics and health monitoring), and closedloop NEMS/MEMS must be reconfigurated through intelligence and adaptive decision making with performance analysis with outcome prediction and assessment Hierarchically distributed closed-loop systems must... specified requirements and priorities, monitoring (sensing) the external environment for entities of interest, recognizing those entities and then infer high-level attributes about those entities, etc The closed-loop systems use the data from different sensors, feedback commands (controls) are generated and executed, and intelligent updates and evolution are performed The feedback for sensor and control mechanisms... 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... coordination and autonomy through tasks generation/organization and decomposition, adaptive decision making with performance analysis and outcome prediction, diagnostics and estimation, © 2001 by CRC Press LLC adaptation and reconfiguration, fault tolerance and robustness, as well as other functions must be performed through sensing-actuation, learning, evolution, analysis, evaluation, behavioral (dynamic and. .. performance) optimization and adaptation, etc Architectures for hierarchically distributed complex closed-loop systems can be synthesized based upon the decomposition of tasks and functions The analysis of complexity, hierarchy, data flow (sensing and actuation), and controllers design, allows the designer to synthesize architectures starting from lowest structural level and then governing and augmenting lower... the overall analysis and high-level decision making It must be emphasized that high-, medium-, and low-level layers communicate with each other, and the high-level layer possesses a key role Decision-making theory must be applied to develop and integrate key enabling methods, algorithms, and tools for the use in intelligent large-scale multi-node NEMS and MEMS These intelligent systems must make optimal . CONTROL OF NANO- AND MICROELECTROMECHANICAL SYSTEMS 4.1. FUNDAMENTALS OF ELECTROMAGNETIC RADIATION AND ANTENNAS IN NANO- AND MICROSCALE ELECTROMECHANICAL SYSTEMS. illustrated that NEMS and MEMS must be controlled. Nano- and microelectromechanical systems augment a great number of subsystems, and to control microscale