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Experimental adaptation of model parameters for microelectromechanical systems (MEMS)
NtlINR5 A ELSEVI ER
Sensors and Actuators A 62 (1997) 760-764
PHYSICAL
Experimental adaptation of model parameters for microelectromechanical systems (MEMS) S. Kurth *, W. Drtzel Department of Electrical Engitzeering and Informatiott Technology, Technical Universi~ of Chemnitz-Zwickau, D-09107 Chemnitz, Germany
Abstract With the use of simulation tools it is possible to predict the mechanical behaviour of micromechanical devices. However, there often are some differences between theoretical and experimental results characterizing the behaviour. Some problems of experimental analysis of capacitively operated coupled mass-spring oscillators, of setting up a suitable model and experimental identification of unknown or erroneous model parameters are discussed in this paper. The measurement of transfer functions and an adaptation method for model parameters are explained for cells of a capacitive micromechanical array. A more accurate simulation of the behaviour is possible using such an improved model. © 1997 Elsevier Science S.A. Keywords: Dynamic testing; Micromirror arrays; System identification
1. Introduction Complicated mechanical structures are permitted in the field of micromechanics, Systems with coupled microresonators or with springs of a complicated shape occur in various cases (e.g., [1,2] ). KOH etching of { 100} silicon in a twostep etch process results in wails which correspond with crystal faces of higher index out o f the { 111 } or { 100} plane by removing mask structures, for instance. The modelling of more complex resonant structures (e.g., sensor arrays, gyroscopes) causes several difficulties. Thermal stress by anodic bonding often influences the stiffness of silicon springs [3]. Calculations of the mass moments of inertia o f complex structures osciIlating at higher modes of vibration and their interaction with the surrounding air are involved. The air acts as a viscous damping and fluid coupling element as well [4]. Several model parameters are influenced by technological tolerances, or time-consuming calculations are necessary to get the parameter values. Experimental investigations are required to verify simulation models and they are appropriate to accommodate erroneous parameters of these modets. It is possible to modify such an improved simulation model to predict the behaviour of these or of similar micromechanical devices more accurately. This paper deals with improving the commonly used technique for treading experirnental results back into theoretical models. * Corresponding author. Tel.: +49 37 l 531 32 64. Fax: -I-49 371 531 32 59. 0924-4247/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved PllS0924-4247 ( 9 7 ) 0 1578- I
A micromechanical array of 25 silicon plates elastically suspended by silicon beams, shown in Fig. 1, acts as an exemplary test object [5]. These resonators are fixed on a glass substrate opposite aluminium electrodes by a mesh grating of thicker silicon. The electrodes and the silicon plates f o r m capacitors to drive and to read out the oscillation. The outer frame of the silicon part and the glass are joined by anodic bonding. The mirror array can be utilized to deflect rapidly a high-intensity laser beam.
2. Modelling Some kinds of models describing the mechanical behaviour are imaginable: models superposing oscillating quantities weighted by the transfer functions including crosstalk between several array cells, finite-element models or models using concentrated parameter elements. For straightforward simulation of the behaviour all these models are suitable. Nevertheless, adequate models for simulation and experimental investigation are required for adapting parameters. A model with a good conformity to the mechanical structure is proficient. Almost up to three degrees of freedom of each structural element have to be taken into account; it simplifies modelling, simulation and experimental investigation. A model with concentrated parameter elements is used to describe the array or a part of it (see Fig. 2). The array is split into different beams which are similar to the beam ele-
761
S. Kurth, W. DOtzel / Sensors and Actuators A 62 (1997) 760-764
~rt
des
~,a)
(b) Glass top
Fig. l. Schematic view of the micrornirrorarray (a) and a photographof the array chip (b). ments used in FEM analysis with the bending and torsional moments and normal force described by the following equations: - 6El m b = lZ ( 1 -- v~) ( wl -
4El
W2) ÷ ~ (
¢~Dbl-- f~Ol'2)
GJ M, = - 7 - ( cp,,- ~Pt2) 12EI
F,
13( 1 - u) '(wl - w a )
(1)
(2) 6EI q 12( 1 -- v---~)( q~bl- %2)
( - coZM + j coC ÷ K) cp= (3)
where the Youngs modulus is E, the shear modulus is G, the torsion moment is J, the area moment of inertia is L the beam length is 1, the Poisson ratio is v and the deflections are wl and Cpb,- or q~t~, respectively. Velocity proportional reaction loads created by fluid mechanic processes cause damping and cross coupling of ceils. A simulation result of the fluid flow in the electrode gap is shown in Fig. 3. Analytic or FEM calculations lead to initial model parameters for elasticity, inertia, driving electrostatic moment and damping moment. Parameter values for fluid and electrostatic coupling are unknown.
Fig. 2. Part of a model to describe tile mirror array.
A part of the exemplary resonator array will be used to explain some algorithms of adapting model parameters (Fig. 4). The most interesting degrees of freedom of the resonator motion are a tilting concerning the x- and y-axes and the translation in the z-direction. Simplifying the problem, the discussed model contains the angular displacement concerning a line parallel to the x-axis of two neighbouring array cells. The differential equation
(4)
upFu
with the matrices ,l~x2 0
M =
Crxl2 - Crxt2
C r x I -t-
C~
CrxI2 Cr:cl2
0
K=
k~l
0
crxg
0 - k,..,~ / kr~.2 krx2 l/ -- krx2 krxg+ kr.~.l+ krxa) -
-- krxI
0O 1
Crx 2 ÷
-
i'
..............
¢ s ¢1 I '
", "l . * . " / ' J ' F
A'v'"' ~ I
~
o v)~',O~
L I
Fig. 3. Velocity of the fluid in the electrode gap of a single tilting mirror, FEM calculation.
S. Kurth, W. Drtzel / Sensors and Actuators A 62 (1997) 760-764
762
Cross section A- k: Silicon pad with elastic suspended silicon plates
Glass support wilh evaporated aluminum electrodes Fig. 4. Part of a micromechanical array of silicon plates. Fl(u, up)
: i i
•
Fluid coupling C rx12
: : !
F2{u, up) 1 .
U Suspending grid
Mirror 1
Mirror 2
Fig. 5. Sire ~lified model for two adjacent array cells.
and bC,
bC~2 O! /
(P,:rI
~ ~Orx2
aC2~ aC2 F=0.5 8q~.~ 8~p~.~2 0
0
describes the model (Fig. 5). It is assumed that the stiffness of the frame suspending the silicon resonators is infinite and the spring-mass system of the mesh grating is concentrated to the stiffness coefficient k~.~g,the inertia J~g and the damping C~xg.The elasticity krxg and inertia J,g of the mesh grating and the damping coefficients c~.~and c~,,2 are unknown. Appropriate estimated values are available for the other parameters of the model.
3. Experimental Microelectromechanical systems are strongly non-linear. The stiffness of the spring, the damping force and the reaction load of magnetic and electric fields depend on displacement. The assumption of linear and time-invariant systems under the condition of small-signal operation makes it possible to estimate the frequency response function by exciting the structure and detecting the oscillation. The measurement of the mechanical reaction in a range of several nanometres up to 1 ixm is necessary. It is essential to superpose an additional d.c. voltage electrostatically exciting the structures. The voltage Up + u with u = Uosin~otleads to the electrostatic force Fe, = ~ ~
Uo. U~ + 2 UpUosin wt + -2-sin 209
(5)
with C the capacitance, w the displacement and w the angular frequency. The part with double frequency is negligible under the condition Uo << Up. The electric field causes a displacement depending force and decreases the elasticity k of springmass oscillators to k* = k - C( Up2/ d2) for plane electrodes with the electrode gap d. The natural frequency is decreased as well. Although some well-established test signals, e.g., multi-frequency sinous, random noise, harmonic sweep, contain an equal amplitude density about the interesting frequency range [ 6], a negative slope to zero is to be preferred because it prevents the field influencing the mechanical behaviour. A Doppler interferometer is used to separate the rotation concerning the x- or y-axis and the out-of-plane motion (Fig. 6). The use of an optical position-sensitive detector to measure the angular displacement, as shown for instance in Ref. [7], is a simpler method. However, the displacement in the z-direction produces a distortion signal in our case. Low-frequency noise signals (vibration of the building, man-made noise, etc.) are diminished by measuring the velocity instead of the displacement signal. Furthermore, statistical methods of signal processing decrease the noise. The estimated frequency response function H1 (jto)
c/)~,(jw) - ¢,,,,(j~)
~,..~(jo))H(jog) - q~,.,.(j o)) + ¢,,,,(jw)
_ Y( q~.w(t) + qo~,,(t) + p,,,y(t) + ~o,,,,,(t) )
(6)
J - (¢p.~,.(t) + {p.~,,,(t)+ q~,,,x(t) + %,,,(t) ) with the Fourier-transformed cross-correlated signals at signal analyser inputs a and b, q~,b(jw), and the Fourier-transformed auto-correlated signal qS,,~,(jw) leads with decreasing power density of noise @,,,,,@o) to the frequency response function H(jw) without distortion. The cross-correlated noise signals are neglected assuming random noise and sufficiently long measuring time.
4. Results
The method of least squares is used to adapt the parameter values (fundamental discussion in Ref. [8]). The experimental investigation determines very preciselj_~, inherent frequencies of the structure to be (ooi= oor,-/1/1- D,.a with the 9 2 "9 natural damping coefficients Di = ( eObF-- mai) / (4c%#). The frequency values of w,,; and o)b,. are indicated by d(Re(H(jo2)))/dto=0, The use of the condition d2(Re (/-/(j¢o)) )/do) 2= 0 indicates the natural frequencies more
S. Kurth, W. D6tzel / Sensors and Actuators A 62 (1997) 760-764
763
Pulser._~~eneratort.l 2 6 10 14 F~quency [kHz]
1(
2
18
Vibrometer 6
10
14
aser beam
x ~
. z-direction Displacement in -Vibratingmicro- ] mechanicaldevice// displacemont Fig. 6, Experimental
IH (jco) I
/,
. . . . . . . . Re(H(j~)
Magnitude
+
do.,-Fle(H(J :))
'
with characteristic
peaks
i
0~1
s e t - u p to m e a s u r e
.
O)a2 too2 (%2
function of a two-degree-of-freedom
(o resonator
vaIues.
accurately than the peaks of magnitude IH(jw) I (Fig, 7), It is required to assign measured frequencies to different oscillation mode shapes. Vibration nodes are detected by observing the amplitude and phase shift of the vibration measured at several locations of the structure, harmonic exciting them at resonance frequencies. Square values of the inherent frequencies correspond to the eigenvalues of the homogeneous part o f Eq. ( 1) without damping. The matrices M and K are separated into a part with well-known parameters and some other parts with more erroneous parameters. The measured frequency response functions exciting one resonator at very low frequency give a weighted stiffness matrix K: = (H(jw)[.,~o)-1/-]1"/7'
functions,
(9)
with H ( s ) = (s 21~'I+ s ( E . = 1_3anOn) -t-R) -I and to the correction coefficients a. of the erroneous damping sub-matrices C., Searching for the minimum of the residual vector v = p - H ( s ) -~0, the method of least-square sum of the residue is used once again to get appropriate values for a.. Finally the damping matrix is adapted by
c= Z a°c.
~lo)
n ~ 1-3
Fig. 8 shows a simulation result of the frequency-response function with adapted model parameters compared with experimentally determined values. The cross-coupling atten0.001
0.00001
, []
(7)
The method of least-square sum of the eigenvalues leads iteratively to more accurate values ~rx~, ~,.~z, /~rx~,)~g. The inertia matrix I~ and the stiffness matrix I~ are now corrected assuming ),.xl: =Jrxt and 3~.2: = J,-x2. A second procedure is used to adapt the damping matrix C. The numerical values of electrostatic exciting moments are given by p = u r , Fu
the frequency-response
~o(s) =uraq( s)u
"" " -
('%1
analyzer
with the transposed voltage vector u r = (urn,0,0) and the tranposed vector of additional d.c. voltages u~ = (up,0,0). The matrix of measured frequency-response functions leads to the Laplace-transformed angular displacements
~
.~.
0
Fig. 7. Frequency-response
~
Dual channel signal
Support with vibration damping
.
(8)
10-8 i
'10
1
t the mesh grating
. .
, . . . .
250
500
, , .
F i g . 8. S i m u l a t i o n o f f r e q u e n c y - r e s p o n s e and measured
values (squares),
, . i
.
750 "1000 Frequency [Hz] functions
, i ....
~[250
with adapted
i , .
1500 parameters
764
S. Kurttl. W. DOtzel / Sensors and Actuators A 62 (1997) 760-764
uation to the adjacent mirror cell is more then 26 dB in this case.
[4] T. Veijola, T. Ryhfinen, H. Kuisma and J. Lahdenperfi, Circuit simulation model of gas damping in microstructures with nontrivial geometries, Teeh. Digest, 8th Int. Conf. Solid-State Sensors and Actuators (Transduceers '96/Eurosensors IX), Stockholm, Sweden, 2529June, 1996, Vol. 2, pp. 36-39.
5. Conclusions Methods of experimental characterization of capacitive micromechanical resonators are presented. Algorithms to adapt appropriate erroneous model parameters are explained using a micromechanical resonator array. Experimentally determined natural frequencies and the frequency-response functions at very low frequencies and at frequencies near the magnitude peaks are the input data of calculations. The model describes the behaviour of the electromechanical system more accurately then the model with predicted parameters. Experimental improvement of models including time-consuming parameter calculations (e.g., terms to describe crosscoupling of array cells) seems to be an appropriate way o f getting a base for further simulations of s i m i l ~ mechanical devices.
Acknowledgements The authors would like to acknowledge the support by the DFG (Deutsche Forschungsgemeinschaft) under Sonderforschungsbereich 379 and the Federal Ministry for Education and Research (con tract 16SV397 / 8 ).
References [I] S. Marco, J. Samitier, J.R, Morante, A. Grtz and L Esteve, Novel structures for miniature pressure transducers obtained by electrochemical etch-stop on diffused membranes,Sensors Mater., 7 (t995) 331-345. [2] P. Hsiang, A. Garcia-Valenzula,M.A. Neifeld and M. Tabib-Azar, Micromachined 50 /.zmX250 /zm silicon torsional mirror arrays for optical signal processing,SPIE Proc., Vol. 1793, huegrated Optics"and Microstructures, 1992, pp. 190-198. [3] A, Cozma and B. Puers, Characterizationof the electrostaticbonding of silicon and Pyrex glass. J. Micromech. Microeng., 5 (1995) 190198.
[5] T. Gessner, W. Drtzel, D. Billep, R. Hahn, C. Kaufmann and S. Kurth, Design and technology of a mirror-array in silicon-micromechanics, Proc. 3rd hzt. Col~ Micro Electro. Opto, Mechanic Systems and Component3; MICRO SYSTEM Technologies '96, Potsdam, Germany, 17-19 Sept., 1996, pp. 657-662. [6] H.G. Natke and N. Cottin, in H.G. Natke (ed.), Application of System Identification in Engineering, Springer,Vienna~ 1988, Ch. l, pp. 3-110.
[7] N. Asada, H. Matsuki and E. Esashi, Two-dimensional silicon micro optical scanner, Proc. Int. Symp. Microsystems, h~telligent Materials and Robots, Sendal. Japan, 27-29 Sept., 1995, pp. 626-629. 18] H.G. Natke, Einfiihrang in Theorie and Praxis der Zeitreihen- und Modalanalyse: ldentifikation sehwingungsfOhiger elastomechanischer Systeme, Vieweg, Braunschweig, 1992, Ch. 6, pp. 395-47l.
Biographies Steffen Kurth was born in Stollberg, Germany, in 1965. He received the Dipl.-Ing. degree in 1991 and the Ph.D. degree in electronic engineering from the Technical University Chemnitz-Zwickau, Germany, in 1995. He is currently w o r k ing as a research assistant in the Department of Electronic Engineering and Information Technology at the same university, where he is engaged in research on micromechanical sensors and light-deflecting actuators. W o l f r a m DOtzel was born in Erfurt, Germany, in 1941. He received the Dipl.-Ing. degree in electrical and precision engineering from Technical University Dresden in 1966 and the Dr.-Ing, degree from the Technical University K a r l - M a r x Stadt (Chemnitz) in 1971. In 1973 he worked at the Energetic Institute in M o s c o w on the reliability o f electromechanical systems. From 1974 to 1986 he worked in the field o f peripheral computer equipment. Since 1987 he has been involved with the research and development of m i c r o m e c h a n i c a l c o m ponents. Since 1993 he has been a professor of m i c r o s y s t e m and device technology at the Technical University ChemnitzZwickau. His current work is focused on design and simulation of micromechanical structures and their application, especially in precision engineering.
Sensors and Actuators A 62 (1997) 760-764
PHYSICAL
Experimental adaptation of model parameters for microelectromechanical systems (MEMS) S. Kurth *, W. Drtzel Department of Electrical Engitzeering and Informatiott Technology, Technical Universi~ of Chemnitz-Zwickau, D-09107 Chemnitz, Germany
Abstract With the use of simulation tools it is possible to predict the mechanical behaviour of micromechanical devices. However, there often are some differences between theoretical and experimental results characterizing the behaviour. Some problems of experimental analysis of capacitively operated coupled mass-spring oscillators, of setting up a suitable model and experimental identification of unknown or erroneous model parameters are discussed in this paper. The measurement of transfer functions and an adaptation method for model parameters are explained for cells of a capacitive micromechanical array. A more accurate simulation of the behaviour is possible using such an improved model. © 1997 Elsevier Science S.A. Keywords: Dynamic testing; Micromirror arrays; System identification
1. Introduction Complicated mechanical structures are permitted in the field of micromechanics, Systems with coupled microresonators or with springs of a complicated shape occur in various cases (e.g., [1,2] ). KOH etching of { 100} silicon in a twostep etch process results in wails which correspond with crystal faces of higher index out o f the { 111 } or { 100} plane by removing mask structures, for instance. The modelling of more complex resonant structures (e.g., sensor arrays, gyroscopes) causes several difficulties. Thermal stress by anodic bonding often influences the stiffness of silicon springs [3]. Calculations of the mass moments of inertia o f complex structures osciIlating at higher modes of vibration and their interaction with the surrounding air are involved. The air acts as a viscous damping and fluid coupling element as well [4]. Several model parameters are influenced by technological tolerances, or time-consuming calculations are necessary to get the parameter values. Experimental investigations are required to verify simulation models and they are appropriate to accommodate erroneous parameters of these modets. It is possible to modify such an improved simulation model to predict the behaviour of these or of similar micromechanical devices more accurately. This paper deals with improving the commonly used technique for treading experirnental results back into theoretical models. * Corresponding author. Tel.: +49 37 l 531 32 64. Fax: -I-49 371 531 32 59. 0924-4247/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved PllS0924-4247 ( 9 7 ) 0 1578- I
A micromechanical array of 25 silicon plates elastically suspended by silicon beams, shown in Fig. 1, acts as an exemplary test object [5]. These resonators are fixed on a glass substrate opposite aluminium electrodes by a mesh grating of thicker silicon. The electrodes and the silicon plates f o r m capacitors to drive and to read out the oscillation. The outer frame of the silicon part and the glass are joined by anodic bonding. The mirror array can be utilized to deflect rapidly a high-intensity laser beam.
2. Modelling Some kinds of models describing the mechanical behaviour are imaginable: models superposing oscillating quantities weighted by the transfer functions including crosstalk between several array cells, finite-element models or models using concentrated parameter elements. For straightforward simulation of the behaviour all these models are suitable. Nevertheless, adequate models for simulation and experimental investigation are required for adapting parameters. A model with a good conformity to the mechanical structure is proficient. Almost up to three degrees of freedom of each structural element have to be taken into account; it simplifies modelling, simulation and experimental investigation. A model with concentrated parameter elements is used to describe the array or a part of it (see Fig. 2). The array is split into different beams which are similar to the beam ele-
761
S. Kurth, W. DOtzel / Sensors and Actuators A 62 (1997) 760-764
~rt
des
~,a)
(b) Glass top
Fig. l. Schematic view of the micrornirrorarray (a) and a photographof the array chip (b). ments used in FEM analysis with the bending and torsional moments and normal force described by the following equations: - 6El m b = lZ ( 1 -- v~) ( wl -
4El
W2) ÷ ~ (
¢~Dbl-- f~Ol'2)
GJ M, = - 7 - ( cp,,- ~Pt2) 12EI
F,
13( 1 - u) '(wl - w a )
(1)
(2) 6EI q 12( 1 -- v---~)( q~bl- %2)
( - coZM + j coC ÷ K) cp= (3)
where the Youngs modulus is E, the shear modulus is G, the torsion moment is J, the area moment of inertia is L the beam length is 1, the Poisson ratio is v and the deflections are wl and Cpb,- or q~t~, respectively. Velocity proportional reaction loads created by fluid mechanic processes cause damping and cross coupling of ceils. A simulation result of the fluid flow in the electrode gap is shown in Fig. 3. Analytic or FEM calculations lead to initial model parameters for elasticity, inertia, driving electrostatic moment and damping moment. Parameter values for fluid and electrostatic coupling are unknown.
Fig. 2. Part of a model to describe tile mirror array.
A part of the exemplary resonator array will be used to explain some algorithms of adapting model parameters (Fig. 4). The most interesting degrees of freedom of the resonator motion are a tilting concerning the x- and y-axes and the translation in the z-direction. Simplifying the problem, the discussed model contains the angular displacement concerning a line parallel to the x-axis of two neighbouring array cells. The differential equation
(4)
upFu
with the matrices ,l~x2 0
M =
Crxl2 - Crxt2
C r x I -t-
C~
CrxI2 Cr:cl2
0
K=
k~l
0
crxg
0 - k,..,~ / kr~.2 krx2 l/ -- krx2 krxg+ kr.~.l+ krxa) -
-- krxI
0O 1
Crx 2 ÷
-
i'
..............
¢ s ¢1 I '
", "l . * . " / ' J ' F
A'v'"' ~ I
~
o v)~',O~
L I
Fig. 3. Velocity of the fluid in the electrode gap of a single tilting mirror, FEM calculation.
S. Kurth, W. Drtzel / Sensors and Actuators A 62 (1997) 760-764
762
Cross section A- k: Silicon pad with elastic suspended silicon plates
Glass support wilh evaporated aluminum electrodes Fig. 4. Part of a micromechanical array of silicon plates. Fl(u, up)
: i i
•
Fluid coupling C rx12
: : !
F2{u, up) 1 .
U Suspending grid
Mirror 1
Mirror 2
Fig. 5. Sire ~lified model for two adjacent array cells.
and bC,
bC~2 O! /
(P,:rI
~ ~Orx2
aC2~ aC2 F=0.5 8q~.~ 8~p~.~2 0
0
describes the model (Fig. 5). It is assumed that the stiffness of the frame suspending the silicon resonators is infinite and the spring-mass system of the mesh grating is concentrated to the stiffness coefficient k~.~g,the inertia J~g and the damping C~xg.The elasticity krxg and inertia J,g of the mesh grating and the damping coefficients c~.~and c~,,2 are unknown. Appropriate estimated values are available for the other parameters of the model.
3. Experimental Microelectromechanical systems are strongly non-linear. The stiffness of the spring, the damping force and the reaction load of magnetic and electric fields depend on displacement. The assumption of linear and time-invariant systems under the condition of small-signal operation makes it possible to estimate the frequency response function by exciting the structure and detecting the oscillation. The measurement of the mechanical reaction in a range of several nanometres up to 1 ixm is necessary. It is essential to superpose an additional d.c. voltage electrostatically exciting the structures. The voltage Up + u with u = Uosin~otleads to the electrostatic force Fe, = ~ ~
Uo. U~ + 2 UpUosin wt + -2-sin 209
(5)
with C the capacitance, w the displacement and w the angular frequency. The part with double frequency is negligible under the condition Uo << Up. The electric field causes a displacement depending force and decreases the elasticity k of springmass oscillators to k* = k - C( Up2/ d2) for plane electrodes with the electrode gap d. The natural frequency is decreased as well. Although some well-established test signals, e.g., multi-frequency sinous, random noise, harmonic sweep, contain an equal amplitude density about the interesting frequency range [ 6], a negative slope to zero is to be preferred because it prevents the field influencing the mechanical behaviour. A Doppler interferometer is used to separate the rotation concerning the x- or y-axis and the out-of-plane motion (Fig. 6). The use of an optical position-sensitive detector to measure the angular displacement, as shown for instance in Ref. [7], is a simpler method. However, the displacement in the z-direction produces a distortion signal in our case. Low-frequency noise signals (vibration of the building, man-made noise, etc.) are diminished by measuring the velocity instead of the displacement signal. Furthermore, statistical methods of signal processing decrease the noise. The estimated frequency response function H1 (jto)
c/)~,(jw) - ¢,,,,(j~)
~,..~(jo))H(jog) - q~,.,.(j o)) + ¢,,,,(jw)
_ Y( q~.w(t) + qo~,,(t) + p,,,y(t) + ~o,,,,,(t) )
(6)
J - (¢p.~,.(t) + {p.~,,,(t)+ q~,,,x(t) + %,,,(t) ) with the Fourier-transformed cross-correlated signals at signal analyser inputs a and b, q~,b(jw), and the Fourier-transformed auto-correlated signal qS,,~,(jw) leads with decreasing power density of noise @,,,,,@o) to the frequency response function H(jw) without distortion. The cross-correlated noise signals are neglected assuming random noise and sufficiently long measuring time.
4. Results
The method of least squares is used to adapt the parameter values (fundamental discussion in Ref. [8]). The experimental investigation determines very preciselj_~, inherent frequencies of the structure to be (ooi= oor,-/1/1- D,.a with the 9 2 "9 natural damping coefficients Di = ( eObF-- mai) / (4c%#). The frequency values of w,,; and o)b,. are indicated by d(Re(H(jo2)))/dto=0, The use of the condition d2(Re (/-/(j¢o)) )/do) 2= 0 indicates the natural frequencies more
S. Kurth, W. D6tzel / Sensors and Actuators A 62 (1997) 760-764
763
Pulser._~~eneratort.l 2 6 10 14 F~quency [kHz]
1(
2
18
Vibrometer 6
10
14
aser beam
x ~
. z-direction Displacement in -Vibratingmicro- ] mechanicaldevice// displacemont Fig. 6, Experimental
IH (jco) I
/,
. . . . . . . . Re(H(j~)
Magnitude
+
do.,-Fle(H(J :))
'
with characteristic
peaks
i
0~1
s e t - u p to m e a s u r e
.
O)a2 too2 (%2
function of a two-degree-of-freedom
(o resonator
vaIues.
accurately than the peaks of magnitude IH(jw) I (Fig, 7), It is required to assign measured frequencies to different oscillation mode shapes. Vibration nodes are detected by observing the amplitude and phase shift of the vibration measured at several locations of the structure, harmonic exciting them at resonance frequencies. Square values of the inherent frequencies correspond to the eigenvalues of the homogeneous part o f Eq. ( 1) without damping. The matrices M and K are separated into a part with well-known parameters and some other parts with more erroneous parameters. The measured frequency response functions exciting one resonator at very low frequency give a weighted stiffness matrix K: = (H(jw)[.,~o)-1/-]1"/7'
functions,
(9)
with H ( s ) = (s 21~'I+ s ( E . = 1_3anOn) -t-R) -I and to the correction coefficients a. of the erroneous damping sub-matrices C., Searching for the minimum of the residual vector v = p - H ( s ) -~0, the method of least-square sum of the residue is used once again to get appropriate values for a.. Finally the damping matrix is adapted by
c= Z a°c.
~lo)
n ~ 1-3
Fig. 8 shows a simulation result of the frequency-response function with adapted model parameters compared with experimentally determined values. The cross-coupling atten0.001
0.00001
, []
(7)
The method of least-square sum of the eigenvalues leads iteratively to more accurate values ~rx~, ~,.~z, /~rx~,)~g. The inertia matrix I~ and the stiffness matrix I~ are now corrected assuming ),.xl: =Jrxt and 3~.2: = J,-x2. A second procedure is used to adapt the damping matrix C. The numerical values of electrostatic exciting moments are given by p = u r , Fu
the frequency-response
~o(s) =uraq( s)u
"" " -
('%1
analyzer
with the transposed voltage vector u r = (urn,0,0) and the tranposed vector of additional d.c. voltages u~ = (up,0,0). The matrix of measured frequency-response functions leads to the Laplace-transformed angular displacements
~
.~.
0
Fig. 7. Frequency-response
~
Dual channel signal
Support with vibration damping
.
(8)
10-8 i
'10
1
t the mesh grating
. .
, . . . .
250
500
, , .
F i g . 8. S i m u l a t i o n o f f r e q u e n c y - r e s p o n s e and measured
values (squares),
, . i
.
750 "1000 Frequency [Hz] functions
, i ....
~[250
with adapted
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uation to the adjacent mirror cell is more then 26 dB in this case.
[4] T. Veijola, T. Ryhfinen, H. Kuisma and J. Lahdenperfi, Circuit simulation model of gas damping in microstructures with nontrivial geometries, Teeh. Digest, 8th Int. Conf. Solid-State Sensors and Actuators (Transduceers '96/Eurosensors IX), Stockholm, Sweden, 2529June, 1996, Vol. 2, pp. 36-39.
5. Conclusions Methods of experimental characterization of capacitive micromechanical resonators are presented. Algorithms to adapt appropriate erroneous model parameters are explained using a micromechanical resonator array. Experimentally determined natural frequencies and the frequency-response functions at very low frequencies and at frequencies near the magnitude peaks are the input data of calculations. The model describes the behaviour of the electromechanical system more accurately then the model with predicted parameters. Experimental improvement of models including time-consuming parameter calculations (e.g., terms to describe crosscoupling of array cells) seems to be an appropriate way o f getting a base for further simulations of s i m i l ~ mechanical devices.
Acknowledgements The authors would like to acknowledge the support by the DFG (Deutsche Forschungsgemeinschaft) under Sonderforschungsbereich 379 and the Federal Ministry for Education and Research (con tract 16SV397 / 8 ).
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Biographies Steffen Kurth was born in Stollberg, Germany, in 1965. He received the Dipl.-Ing. degree in 1991 and the Ph.D. degree in electronic engineering from the Technical University Chemnitz-Zwickau, Germany, in 1995. He is currently w o r k ing as a research assistant in the Department of Electronic Engineering and Information Technology at the same university, where he is engaged in research on micromechanical sensors and light-deflecting actuators. W o l f r a m DOtzel was born in Erfurt, Germany, in 1941. He received the Dipl.-Ing. degree in electrical and precision engineering from Technical University Dresden in 1966 and the Dr.-Ing, degree from the Technical University K a r l - M a r x Stadt (Chemnitz) in 1971. In 1973 he worked at the Energetic Institute in M o s c o w on the reliability o f electromechanical systems. From 1974 to 1986 he worked in the field o f peripheral computer equipment. Since 1987 he has been involved with the research and development of m i c r o m e c h a n i c a l c o m ponents. Since 1993 he has been a professor of m i c r o s y s t e m and device technology at the Technical University ChemnitzZwickau. His current work is focused on design and simulation of micromechanical structures and their application, especially in precision engineering.