An efficient and accurate MEMS accelerometer model with sense finger dynamics for applications in mixed-technology control loops

Transcription

1 BMAS 2007, San Jose, September 2007 An efficient and accurate MEMS accelerometer model with sense finger dynamics for applications in mixed-technology control loops Chenxu Zhao, Leran Wang and Tom J Kazmierski {cz05r,lw04r,tjk}@ecs.soton.ac.uk 1

2 Outline Introduction MEMS capacitive accelerometer Accurate VHDL-AMS accelerometer model with sense finger dynamics Simulation results Conclusion 2

3 Introduction Effects of sense finger dynamics in MEMS capacitive accelerometer studied and modeled accurately Distributed mechanical sensing element model Finite Difference Approximation (FPA) approach Well known failure of Sigma-Delta accelerometers when fingers resonate is modeled correctly 3

4 MEMS capacitive accelerometer Deflection of proof mass is caused by input acceleration Differential change in capacitance, which is proportional to deflection, is measured 4

5 2nd-order electromechanical Sigma-Delta modulation accelerometer MEMS sensing element is an integral part of the system Advantages Better bandwidth, dynamic range, linearity Direct digital output 5

6 Conventional sensing element model Mass-Damper-Spring system is modeled by a 2nd order differential equation: 2 F(t)=M d x 2 dt + D dx + Kx dt Too simple and inaccurate Does not reflect finger dynamics 6

7 Mechanical Sensing Element Higher resonant modes of sense finger cannot be reflected in conventional model Distributed model for sense fingers is needed Non-collocated system 7

8 ACCURATE MODEL WITH SENSE FINGER DYNAMICS Motion of fingers could be modeled by following partial differential equation (PDE): ρ yxt (, ) yxt (, ) yxt (, ) S + C (, ) 2 DI + EI = F 4 4 e x t t x t x E, I, CD, ρ, S are all physical properties of the beam. Fe(x, t) - distributed electrostatic force along the beam: 1 V V Fe ( x, t) = ε A[ ] 2 ( d y ( x, t )) ( d y ( x, t ))

9 Finite Difference Approximation (FDA) approach VHDL-AMS cannot support PDEs Convert PDE to a series of ODEs Finger is divided into N segments (5 segments in this design) Partial derivatives wrt position can be replaced with: yn() t yn() t yn 1() t = x Δx n= 0,1,2... N 9

10 Distributed model for sense finger Partial differential equation (PDE) for the motion of the sense fingers: ρ yxt (, ) yxt (, ) yxt (, ) S + C (, ) 2 DI + EI = F 4 4 e x t t x t x Series of ODE (Ordinary Differential Equations) converted from the PDE by FDA: fe () ( ) ( ) n t S yn+ yn+ + yn yn + yn = dt ( Δx) dt dt dt dt dt ( Δx) Δx 2 d yn CDI dyn+ 2 dyn+ 1 dyn dyn 1 dyn 2 EI ρ n = 0,1, 2... N 10

11 Boundary conditions Equations for the border segments are a little different At root (x=0) y(0, t) = z( t) y (0, t) x = 0 At free end (x=l) 2 y(,) l t 2 x = 0 3 y(,) l t 3 x = 0 11

12 Simulation results of conventional model Frequency response of conventional model Lowest resonant mode caused by the dynamics of the structure mass ω = K M 0 / 12

13 Simulation results of conventional model Output bitstream Pulse density is inversely proportional to the input signal 13

14 Higher resonant mode The higher resonant mode is related to the sense finger resonance Fingers bend significantly while the lumped mass has a small deflection. Resonant frequency ω i W L E 12ρ 2 = αi α1 α2 = 1.875, =

15 Simulation results of the distributed model The finger dimensions in this design are: L = 140μm, W =2μm, T = 1μm Higher order resonant modes are exhibited in the accurate distributed model 15

16 Simulation results ---- movement of the sense fingers Deflection of the sense fingers in the conventional model 16

17 Simulation results ---- movement of the sense fingers The finger dimensions in this design are: L = 140μm, W =2μm, T = 1μm Displacement of the lumped-mass (Blue) and average position of the sense finger (Red) 17

18 Simulation results - Distributed model Finger dimensions are: L = 200μm, W =2μm, T = 1μm 18

19 Movement of the sense fingers Displacement of the lumped-mass (Blue) and average position of the sense finger (Red) Sense fingers bend significantly at their resonant frequency 19

20 Simulation results of the distributed model Sense finger resonance can affect the performance of the Sigma-Delta control loop Here Sigma-Delta control breaks down - proposed model reflects this behavior correctly 20

21 Simulation results of conventional model Output bitstream of the conventional model. The Sigma-Delta control loop still appears to work but in reality it fails 21

22 CPU time comparison Model Conventional Model CPU time (SystemVision) 9s 766ms Distributed Model 21s 812ms SystemVision v.4 from Mentor Graphics was used to carry out simulation experiments 22

23 Conclusion An accurate distributed MEMS accelerometer model with sense finger dynamics has been developed and implemented in VHDL-AMS. The proposed model correctly reflects the way in which finger dynamics affect the performance of the control loop. 23