A CAPACITIVE ACCELEROMETER MODEL

Transcription

1 Électronique et transmission de l information A CAPACITIVE ACCELEROMETER MODEL FLORIN CONSTANTINESCU, ALEXANDRU GABRIEL GHEORGHE, MIRUNA NIŢESCU *1 Key words: MEMS models, Transient analysis, Capacitive accelerometer. A new capacitive accelerometer model able to be used in the transient analysis of a circuit measuring acceleration is proposed in this paper. This model ensures the convergence of the transient analysis made by a commercial simulator (SPICE or SPECTRE) by avoiding the use of large values for the capacities, unlike the known models. 1. INTRODUCTION In the last years the microelectromechanical systems (MEMS) like actuators, pressure sensors, accelerometers, gyroscopes and others know an unprecedented development. MEMS are parts of some complex electrical/electronic systems working at relatively high frequencies. In order to design these complex systems the MEMS circuit models are incorporated in intricate equivalent schematics of these systems [1, 2]. There are two kinds of MEMS circuit models: behavioral models and physical models. The behavioral models are circuits with a minimum number of elements which have the same behavior as the modeled device, but their parameters don t depend directly on the properties of the materials used for device manufacturing. For example in [3] are proposed some behavioral models of the power BAW resonators built with AlN which are able to reproduce the device small signal behavior, and two large signal nonlinear effects: the amplitudefrequency effect and the frequency characteristic of the second harmonic of the reflected power. In [4] are proposed some physical models for the same power BAW resonators built with AlN. These models are more intricate than the behavioral ones, but their parameters depend directly on the materials properties. Both behavioral and physical models are valid for a certain frequency range. The circuits used actually in various areas as mobile communications, automation and control and automotive applications are excited by modulated digital or analog signals which have a broad frequency spectrum. The periodic 1 * Politehnica University, Department of Electrical Engineering, Bucharest, Romania, florin.constantinescu@lce.pub.ro Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 58, 2, p , Bucarest, 2013

2 164 Florin Constantinescu, Alexandru Gabriel Gheorghe, Miruna Niţescu 2 steady state of these circuits is often determined using methods which involve transient analysis. This is because the MEMS circuit models must support this kind of analysis. To this end the commercial circuit simulators are using the companion model circuits which are solved using the Newton-Raphson iterations. Each iteration method involves the solving of a linear system of algebraic equations using a direct method as the LU factorization [5]. Due to unusual parameter values in the MEMS models this method of transient analysis fails sometimes to converge. In [7] a new equivalent circuit of a time-variable capacitor is used to build a capacitive accelerometer model. As this equivalent circuit creates convergence problems in transient analysis made by SPICE or SPECTRE, it has been modified in the same paper in order to support this kind of analysis. This paper deals with a new capacitive accelerometer model able to be used in the transient analysis of a circuit measuring acceleration. Section 2 deals with the accelerometer model, the transient analysis results are presented in Section 3 and Section 4 contains a discussion on this kind of models and on transient analysis algorithms. 2. A CAPACITIVE ACCELEROMETER AND ITS CIRCUIT MODEL The acceleration sensor structure is given in Fig. 1 [1, 8]. A movable microstructure is attached by means of two springs to the substrate, which remains fixed. This microstructure can move only up and down. All fixed plates are connected together as well as the mobile plates. A lot of capacitors connected in parallel are created by this way between the movable and the fixed plates. For a pair of plates two capacities can be identified: εa C 1 = = C C d + x 0 εa and C 2 = = C C d x 0 +, (1) where A is the capacitor area, d is the distance between the plates corresponding to a null acceleration of the movable microstructure, ε is the dielectric permittivity, and x is the displacement between the fixed and the movable plates. It follows that: x C 2 C 1 = 2 C = 2ε A or Cx 2 + εax Cd 2 = 0. d 2 x 2 (2) For x << 1 it results x d C / C 0. (3) The equilibrium equation between the elastic and inertial forces gives a = kx / m, where k is the elastic constant of the spring, and m is the mass of the

3 3 A capacitive accelerometer model 165 movable structure (Fig. 1). It follows that a signal proportional to C is a measure of the instantaneous acceleration: a = K C. (4) Fig. 1 Capacitive accelerometer sensor. The current through a linear time-variable capacitor with a capacity C(t) is: dq d dc( t) du( t) i = = ( C( t) u( t)) = u( t) + C( t), (5) where u(t) is the capacitor voltage. In order to obtain a signal proportional with C(t) the first term in (5) must be negligible with respect to the second one [1]. The schematic for the acceleration measurement is given in Fig. 2 [1]. The output potential V o can be computed as: C a V 0 = V S sin ωt = V S sin ωt. (6) C KC

4 166 Florin Constantinescu, Alexandru Gabriel Gheorghe, Miruna Niţescu 4 Fig. 2 Acceleration measurement. For the transient analysis we consider a sinusoidal acceleration with f a = 1 khz and a sinusoidal supply V s = 1V with f = 1MHz, so that the capacity is: C + C = sin(2πf a t) [ff]. (7) A time-variable capacitor model has two-ports: the main one at which it behaves as the modeled device, and the control port at which the function representing the time variation of the capacity is applied. A model of this kind, containing a 1F capacity, is given in [6]. Because this model doesn t allow C(t) to take a zero value, we developed a new one starting from the equation (5) [7]. This model, shown in Fig. 3, exhibits a linear time-variable capacitor behavior between the terminals 1 and 2. The control voltage is applied between the node CTRL and ground. This model has been designed using C = 1F and C c = 1F for the sake of simplicity. The transient analysis of this model has been performed with commercial simulators PSPICE, SPICEP, and SPECTRE (CADENCE). In all cases simulation has been aborted. In order to reach a solution, the command V(CTRL) has been multiplied with a factor K = , accompanied by suitable modifications in control factors aimed to preserve the model characteristic between the terminals 1 and 2 [7]. Computation of the condition number of the Jacobian for various SPICEP solutions show that it increases as the time step decreases. This suggests that the time step is reduced due to the erroneous solutions obtained in solving the companion model circuit. If the trapezoidal rule is used, the companion model of a linear 1F capacitor contains a resistor having a h/2 resistance where h is the time step. If the signal period of 10 6 s is divided in 10 3 intervals this resistance will be Ω which can be too small to lead to an acceptable value for the condition number. At the same time a 1 pf capacitor is associated to a resistance of 0.5 kω which may be seen as an acceptable value [7].

5 5 A capacitive accelerometer model 167 Fig. 3 Linear time-variable capacitor model. After a simple algebraic manipulation the relation between the current i 12 and the voltage u 12 at the time variable capacitor terminals in Fig. 3 is: d u12 i12 = C + ( V( CTRL) 1) V(7) + u12 V(8) = d t du12 du12 du12 d V( CTRL) = C + V( CTRL) C R7 C R7 + u12c R8. C For R 7 =1 Ω and R 8 =1 Ω (8) becomes: du d V( CTRL) = +. (9) 12 i12 V( CTRL) C u12 C C (8) it follows: Let be C = C C = C 0. If C(t) = C 0 V(CTRL), (10)

6 168 Florin Constantinescu, Alexandru Gabriel Gheorghe, Miruna Niţescu 6 d u12 ( t) d C( t) i12 = C() t + u12() t, (11) which is identical to (5). Now we can use a small value for C 0, avoiding the convergence problems in transient analysis pointed out in [7]. It has been proved that the 1F capacities used in [6] and [7] make the program to abort the transient analysis: the program stops before tstop giving the error message time step too small. 3. TRANSIENT ANALYSIS RESULTS The transient analysis of the measuring circuit in Fig. 2 has been performed for various values of C 0 in order to determine the range in which this parameter can be chosen to get valid results from the transient analysis. As C(t) is given by (7) and (10) the product C 0 V(CTRL) is the same for all variants and can be written as CK 0 ( sin(2 π fat)) [F]. The errors have been set to reltol = 10 4, iabstol = = A, vabstol = V. Other simulation parameters are final time tstop = 2ms method = gear, and order = 2. The results are given in Table 1. Table 1 Transient analysis results for f = 1MHz, fa = 1kHz C 0 K Accepted steps Rejected steps 1mF analysis aborted 1µF nF pF fF 1 erroneous result For C 0 = 1mF with the specified parameters analysis is aborted. Changing reltol from 10 4 to 10 3 all transient analysis interval is swept, but the high frequency detail is not computed accurately (erroneous result). A solution computed correctly (for C 0 = 1µF, or C 0 = 1nF, or C 0 = 1pF) is given in Fig. 4. In this case the ratio between the 1kHz component and the 1MHz component of V 0 is , which can be considered a good approximation of the ratio between the amplitude of the variable component and the average component of the capacity, which is as is pointed out in (7).

7 7 A capacitive accelerometer model 169 a) b) Fig. 4 Output voltage V 0 (correct solution): a) a modulation signal period; b) detail. The erroneous solution for C 0 = 1fF is given in Fig. 5.

8 170 Florin Constantinescu, Alexandru Gabriel Gheorghe, Miruna Niţescu 8 a) b) Fig. 5 Output voltage V 0 (erroneous solution): a) a modulation signal period; b) detail.

9 9 A capacitive accelerometer model 171 It can be observed that even though the signal envelope seems to be correctly computed, some periods in the high frequency detail are not. In order to verify the validity range of the proposed model the source frequency has been modified to f = 1GHz. A maximum time step of 1ns must be forced in order to obtain a correct solution. The transient analysis parameters are the same as in the previous case, except tstop = 1 ms. The results obtained with SPECTRE are given in Table 2. Table 2 Transient analysis results for f = 1GHz, fa = 1 khz C 0 K Accepted steps Rejected steps 1mF erroneous result 1µF nF pF fF All solutions, except that for C 0 = 1mF, are correct. 4. DISCUSSION As the electrical/electronic systems containing MEMS are in tremendous progress on the actual industrial market, there is a need for MEMS models including those for the transient analysis. This problem is not a trivial one if the signals in these systems have a broadband spectrum. A new model of a capacitive accelerometer, able to support the transient analysis made by commercial simulators as SPICE and SPECTRE, is proposed in this paper. Unlike the known models, this model contains small value capacities which avoid the failure of the transient analysis due to the ill-conditioned matrix associated with the resistive circuit containing companion models. The proposed model has been tested successfully with two couples of sinusoidal excitations. The iterative method of conjugate gradients has better performances to solve linear systems with ill-conditioned matrices than the direct methods as the LU decomposition used by SPICE or SPECTRE. A disadvantage of this method is that it works only for symmetric and positive definite matrices, so that a preconditioning of the circuit matrix is necessary prior to use it. There are some circuit simulators which use the conjugate gradients method for solving the resistive circuits containing companion models of linear dynamic elements [9]. The model presented in this paper is significant for these circuit simulators also. The

10 172 Florin Constantinescu, Alexandru Gabriel Gheorghe, Miruna Niţescu 10 criteria used in this paper can be employed for building some circuit models of phemomena which are described by broad spectrum signals as knock in internal combustion machines [10, 11, 12]. Received on January 8, 2013 REFERENCES 1. S. Senturia, Microsystem Design, Boston, Kluwer Academic Publishers, M. Nitescu, Microelectromechanical Systems, Course draft (in Romanian), /studenti, F. Constantinescu, M. Nitescu, A. G. Gheorghe, A. Florea, O. Llopis, Behavioral circuit models of power BAW resonators and filters, Analog Integrated Circuits and Signal Processing (Kluwer), F. Constantinescu, A.G. Gheorghe, M. Nitescu, A. Florea, Nonlinear artificial line models for power BAW resonators, AFRICON 2011, September 13 15, 2011, Livingstone, Zambia. 5. A. Vladimirescu, SPICE Book, John Wiley & Sons, C. Basso, SPICE analog behavioral modelling of variable passives, Power Electronics Technology, April 2005, pp. 58, 7. F. Constantinescu, A. G. Gheorghe, M. Nitescu, Convergence problems in transient analysis of circuits containing MEMS, Simpozionul Naţional de Electrotehnică Teoretică (SNET 12), 14 Decembrie 2010, Bucureşti. 8. M. Andrejašic, MEMS Accelerometers Seminar, University of Ljubljana, Faculty for mathematics and physics, Department of Physics, March Z. Li, C. J.R. Shi, A coupled iterative/direct method for efficient time-domain simulation of nonlinear circuits with power/ground networks, International Symposium on Circuits and Systems (ISCAS) 2004, pp. V-165 V Radoi A., Lazarescu V., Florescu A, Wavelet analysis to detect the knock on internal combustion engines, Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 54, 3, pp , C. Oros, C. Radoi, A. Florescu, Comparison among Computational Intelligence Methods for Engine Knock Detection. Part 1, Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 56, 4, pp , A. Florescu, C. Oros, A. Radoi, Comparison among Computational Intelligence Methods for Engine Knock Detection. Part 2, Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 57, 1, pp , 2012.