INF5490 RF MEMS. L3: Modeling, design and analysis. S2008, Oddvar Søråsen Department of Informatics, UiO

Transcription

1 INF5490 RF MEMS L3: Modeling, design and analysis S2008, Oddvar Søråsen Department of Informatics, UiO 1

2 Today s lecture MEMS - function Transducer principles Sensor principles Methods for RF MEMS modeling 1. Simple mathematical models 2. Convert to electrical equivalents (3. Analyzing using Finite Element Methods) L4 2

3 Transducers for (RF) MEMS Electromechanical transducers Transforming electrical energy mechanical energy Transducer principles Electrostatic Electromagnetic Electro thermal Piezoelectric 3

4 Transducer principles Electrostatic transducers Principle: Forces between electric charges Coulombs law Stored energy when mechanical or electrical work is performed on the unit can be converted to the other form of energy The most used form of electromechanical energy conversion Fabrication is simple Often implemented using a capacitor with movable plates Vertical movement: parallel plates Horizontal movement: Comb structures 4

5 Electrostatic transducers + Beneficial due to simplicity + Actuation controlled by voltage voltage charge attractive force movement + Movement gives current movement variable capacitor current when voltage is constant Need environmental protection (dust) Packaging required (vacuum) Transduction mechanism is non-linear Gives distortions (force is not proportional to voltage) Solution: small signal variations around a DC voltage 5

6 Transducer principles, contd. Electromagnetic transducers Magnetic winding pulls the element Electro thermal actuator Different thermal expansion on different locations due to temperature gradients Large deflections can be obtained Slow 6

7 Transducer principles, contd. Piezoelectric transducers In some anisotropic crystalline materials the charges will be displaced when stressed electric field stress = mechanical stress Similarly, strain results when an electric field is applied strain = mechanical strain Ex. PZT (lead zirconate titanates) ceramic materials (Electrostrictive transducers Mechanical deformation by electric field Magnetostrictive transducers Deformation by magnetic field) 7

8 Comparing different principles 8

9 Sensor principles Piezoresistive detection Capacitive detection Piezoelectric detection Resonance detection 9

10 Sensor principles Piezoresistive detection Resistance varies due to external pressure/stress Used in pressure sensors deflection of membrane Piezo-resistors placed on membrane where strain is maximum Resistor value is proportional to strain Performance of piezoresistive micro sensors are temperature dependent 10

11 Pressure sensor Senturia 11

12 Sensor principles, contd. Capacitive detection Exploiting capacitance variations Pressure electric signal Detected by change in oscillation frequency, charge, voltage (V) Potentially higher performance than piezoresistive detection + Better sensitivity + Can detect small pressure variations + High stability 12

13 Sensor principles, contd. Piezoelectric detection Electric charge distribution changed due to external force electric field current Resonance detection Analogy: stress variation on a string gives strain and is changing the natural resonance frequency 13

14 Methods for modeling RF MEMS 1. Simple mathematical models Ex. parallel plate capacitor 2. Converting to electrical equivalents 3. Analysis using Finite Element Methods 14

15 1. Simple mathematical models Equations, formulas describing physical phenomena Simplification, approximations Explicit solutions for simple problems linearization around a bias point Numerical solution of the set of equations Typical differential equations + Gives the designer insight/ understanding How the performance changes by parameter variations May be used for initial estimates 15

16 Ex. On mathematical modeling Important equatins for many RF MEMS components: Parallel plate capacitor! Electrostatic actuation of the capacitor with one spring-suspended plate Calculating pull-in Formulas and figures 16

17 Electrostatics + q - q F Electric force between charges: Coulombs law 1 q q F = 4 πε r r Electric field = force pr. unit charge E = Work done by a force = change in potential energy F q 0 W a b b = F a dl = U a U b Potential, V = potential energy pr. unit charge V = U q 0 Voltage = potential difference V a V b b = E dl a 17

18 Capacitance a b E d + Q V ab Definition of capacitance C = Q V ab A -Q Surface charge density = σ σ Q 1 E = = ε A ε 0 0 C = Q V ab = ε 0 A d Voltage V ab = E d = Q A ε 0 d Energy stored in a capacitor, C, that is charged to a voltage V 0 at a current dt i = Q & = C dv U = dv v i dt = v C dt = C dt V0 1 2 ε0 v dv = CV0 = A V d

19 Parallel plate capacitor Attractive force between plates F U = d = d εa ( V 2d 2 ) = εav 2d

20 Movable capacitor plate Assumptions for calculations: Suppose air between plates Spring attached to upper plate Spring constant: k Voltage is turned on Electrostatic attraction At equilibrium Forces up and forces down are in balance 20

21 Force balance k = spring constant d0 = gap at 0V and zero spring strain d = d0 z z=d0 d deflection from start position Force on upper plate at V and d: = 0 at equilibrium 21

22 Two equilibrium positions ς = 1 d/d0 Senturia 22

23 Stability How the forces develop when d decreases Suppose a small perturbation in the gap at constant voltage Suppose the gap decreases If the upward force also deceases, the system is UNSTABLE! 23

24 Stability, contd. Stability condition: Pull-in when: 24

25 Pull-in Pull-in when: 25

26 Pull-in Senturia 26

27 2. Converting to electrical equivalents Mechanical behavior can be modeled using electrical circuit elements Mechanical structure simplifications equivalent electrical circuit ex. spring/mass R, C, L Possible to interconnect electrical and mechanical energy domains Simplified modeling and co-simulation of electronic and mechanical parts of the system Proper analysis-tools can be used Ex. SPICE 27

28 Converting to electrical equivalents, contd. We will discuss: Needed circuit theory Conversion principles effort - flow Example of conversion Mechanical resonator In a future lecture: Co-existence and coupling between various energy domains 28

29 Circuit theory Basic circuit elements: R, C, L Current and voltage equations for basic elements (low frequency) Ohms law, C and L-equations V = RI, I = C dv/dt, V = L di/dt Laplace transformation From differential equations to algebraic (s-polynomial) Complex impedances: R, 1/sC, sl Kirchhoffs equations Σ current into nodes = 0, Σ voltage in a loop = 0 29

30 Effort - flow Electrical circuits are described by a set of variables: conjugate power variables Voltage V: across or effort variable Current I: through or flow variable An effort variable drives a flow variable through an impedance, Z Circuit element is modeled as a 1-port with terminals Same current (f = flow) in and out and through the element Positive flow into a terminal defining a positive effort 30

31 Energy-domains, analogies Various energy domains exist Electric, elastic, thermal, for liquids etc. For every energy domain it is possible to define a set of conjugate power variables that may be used as basis for lumped component modeling using equivalent circuits elements Table 5.1 Senturia -> 31

32 Ex. of conjugate power variables 32

33 Conjugate power variables: e,f Assume conversion between energy domains were the energy is conserved! Properties e * f = power e / f = impedance Generalized displacement represents the state, f. ex. position or charge f ( t) = q& ( t) e * q = energy t q( t) = f ( t) dt + q( t0) t 0 33

34 Generalized momentum Mechanics: impulse F*dt= mv mv0 t p( t) = e( t) dt + p( t ) 0 t 0 General: p * f = energy 34

35 Ex.: Mechanical energy domain force velocity position momentum force x time work/time = power force*distance = work = energy energy 35

36 Ex.: Electrical energy domain voltage current charge power energy 36

37 e V - convention Senturia and Tilmans use the e V convention Ex. electrical and mechanical circuits e V (voltage) equivalent to F (force) f I (current) equivalent to v (velocity) q Q(charge) equivalent to x (position) e * f = power injected into the element H. Tilmans, Equivalent circuit representation of electromagnetical transducers: I. Lumped-parameter systems, J. Micromech. Microeng., Vol. 6, pp ,

38 Other conventions Different conventions exist for defining throughor across-variables * alternativt 38

39 Generalized circuit elements One-port circuit elements R, dissipating element C, L, energy-storing elements Elements can have a general function! Can be used in various energy domains 39

40 Generalized capacitance Compare with a simplified case: -a linear capacitor definisjon av C 40

41 Generalized capacitance, contd. Capacitance is associated with stored potential energy Co-energy: 41

42 Energy stored in parallel plate capacitor Energy: W ( Q) Q Q = e dq = 0 0 q C dq = 2 Q 2C Co-energy: W * ( V ) = V V q de = C v dv = 0 0 CV 2 2 * W ( V ) = W ( Q) for linear capacitance 42

43 Mechanical spring Hook s law: F = k x Stored energy Compare with capacitor 1 1 W ( Q) = Q 2 C 2 Q x1 displacement displacement 1/C equivalent to k 43

44 Compliance Compliance = inverse stiffness 1 C spring = k Stiff spring small capacitor Soft spring large capacitor 44

45 Generalized inductance Energy also defined as: Energy = stored kinetic energy 45

46 Ex.: Electrical inductor Co-energy: 46

47 Analogy between mass (mechanical inertance) and inductance L A mechanical system has linear momentum: p = mv Flow: Co-energy: 47

48 Analogy between m and L Compare with: L is equivalent to m m = L inertance Mechanical inertance = mass m is analog to inductance L 48

49 Interconnecting elements e V follows two basic principles Elements that share a common flow, and hence a common variation of displacement, are connected in series Elements that share a common effort are connected in parallel 49

50 Ex. of interconnection: Direct transformation 50

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54 System without damping (b=0, R=0) ω 1 ω LC 0 =, 0 = k m 54

55 System without damping, contd. 55

56 With damping 56

57 Damped system, contd. 57

58 58

59 What is the meaning of damping time? Power initial conditions 59

60 Q-factor and damping time General equation mechanical electrical 60

61 Amplitude at resonance for forced vibrations 61