- Email: [email protected]
Simulation model for micromechanical angular rate sensor
A
ELSEVIER
Sensors and Actuators A 60 {1997) 11~121
Simulation model for micromechanical angular rate sensor Timo Veijola ~'*, Heikki Kuisma b, Juha Lahdenper/i b Tapani Ryhfinen ¢ Helsinki Univer,~ity of Tec~mology. Circuit Theory ~boratory. . PO Box 3000. FIN-02015 Hut. Finland ~' VTi Hamlin Oy, PO Box 9. FIN-O0421 Helsmki. Finhmd c Nokia ]relecomnumications. PO Box 33. FIN-02601 Espoo. Finland
Abstract A simulation model for an angular rate sensor, a gyroscope, is presented. The device is based on a micromechanical dual torsional mass system which is actuated electrostaticaily and sensed capacitively. Model equations describing a dynamic, non-linear system are first presented and then realized as an electrical equivalent circuit. The vibrational modes of the system are modelled with coupled resonator circuits. The electrostatic and Coriolis forces as well as variable capacitances in the small air gaps are modelled with non-linear controlled current sources. External forces, torques and electrical actuation can act as inputs to the device. The model presented allows numerical sensor simulations concurrently with the interfacing electronics in the time and frequency domains. The model is verified by comparing its simulation results to measured frequency responses and capacitance-voltage characteristics. Keyword~: Angular rate sensors; Circuit simulation; Mictomechanics
1. Introduction In recent years there has been a great interest in angular rate sensors, especially for use in vehicles. These sensors are used to measure the yaw rate of a vehicle, or angular velocity about the vehicle's vertical axis, for chassis dynamic contrel. In the control system the steering-wheel angle is used as a set point indicating the driver's intentions. The hardware that exists for anti-skid braking control is used to control the yaw of the vehicle by individually adjusting the braking forces of the four wheels [ i ]. Other potential applications for angular rate sensors are listed in Ref. [2]. Vibrating rate gyros were introduced in the early 1950s tbr the replacement of rotating gyros in aerospace applications with the advantage of no wearing parts [31. After 40 years they have found their way to the consumer market. In topmodel passenger cars, vibrating angular rate sensors based on a metal cylinder [4], a metallic tuning lbrk [51 and a piezoelectric quartz tuning fork [61 are being used as of the mid-1990s. It was already predicted in Ref. [31 that the performance of a vibrating angular rate sensor will not be affected by miniaturization, contrary to the rotating sensor. Small microma~,nined angular rate sensors based on a tuning fork [7] and a vibrating ring [81 have been introduced. A * Corresponding author. Tel.: + 358 9 451 22 93. Fax: + 358 9 451 48 18. E-mail: [email protected]. 09244247/97/$17.00 © 1997ElsevierS~ienceS.A. All rights reserved Pil S0924-4247 (97) 01382-4
review article [ 21 describes the operating principles of various possible designs. A vibrating angular r; te sensor differs from most other micromechanicai sensors in the complexity of its operation. The sensor does not give any output signal without mechanical excitation. In fact, this sensor is a true microelectromechanical system with equally important sensing, actuation and signal-processing functions. To operate a vibrating ral,,' sensor one has to excite it to a vibratory (either linear or angular) mGvement. Mechanical resonance is used in most cases to increase the amplitude or decrease the need for drive force, scarce in the case of an electrostatic drive. Secondly, one has to measure the coupled vibration in an orthogonal direction. The signal to be measured is typically very small, less than 100 ppm of the excited vibration. Therefore mechanical amplification is often required in the form of resonance. It is required to match properly the excitation and detection resonance frequencies. Matching by manufacturing tolerances may be impossible. For an electrostaticaily actuated vibrator, a bias voltage can be used for frequency tuning as described in Ref. [ 8 ]. The system to control the device is always quite complex with multiple amplitude locks, phase locks and servo loops as described in Ref. [81. Therefore, a simulation tool with a correct description of the etectromechanical interactions is needed for the system design. In this paper we present the elementary equations for a dual torsional vibrator which is
114
T. Veijola et al. / Sensors and Acnmtors A 60 (1997) 113-121
actuated electrostatically and sensed capacitively. The dynamic, non-linear equations are realized as an electrical equivalent circuit and solved numerically by a circuit simulation program. An electrical equivalent circuit has been successfully applied in constructing a micromechanical accelerometer model [9-111. The sensor model presented here uses a similar approach in building the electrical equivalent circuit model. Mechanical resonance modes and coupling between the modes are realized with electrical coupled resonators. Electrostatic forces, variable capacitances in the air gaps and Coriolis force are realized with non-linear voltage-controlled currer, t sources (VCCS). The model has been implemented in the circuit simulation progiam APLAC [ 12] as a userdefinable module written in APLACs input language. The resulting model is a non-linear large-signal model that includes all essential static and dynamic properties of the sc:,sor. It allows numerical sensor system simulations concurrently with the interfacing electronics in the time and frequency domains.
2. Structure and model of the sensor
The structure of the mieromechanical sensor presented in Fig. I is being developed by VTI Hamlin Oy ( formerly Vaisala Technologies Inc.), The device is fabricated based on three silicon wafers, which are anodically bonded together. The centre wafer ct, ntains the beam-frame-spring structure. The outer wafers are composed of siliicon and borosilicate glass with electrode autd feedthrough structures. The gap capacitances are formed ih~small air gaps between the outer and centre wafers. Design details of the outer wafers are not shown. The size is 7.5 mm × 2.5 mm × 1.7 mm (L × W × H). Thousands of prototype devices have been successfully fabricated. The structure is sealed at low pressure. The remaining gas in the air gaps causes damping of the mass motion. The gas pressure is in practice so small that the spring forces caused by the squeezed gas can be neglected here. The damping due
to the dissipation in the springs is small enough also to be neglected in this model. 2.1. Model for dual torsional mass system Fig. 2 shows the dual torsional system consisting of two vibrating masses. The outer mass, the frame, is attached Io the sensor body by elastic torsional springs. The inner mass, the beam, is supported by the frame with another pair of torsional springs. In the norm'haloperating mode the frame and beam masses are excited with an electrical attractive force to vibrate about the x axis. If the structure is rotated about the z axis, the motion of the inner beam mass is coupled to the perpendicular axis y by Coriolis force. The motion of the beam mass is sensed capacitively in four air gaps. To create a compact but functional model for the mechanical structure several assumptions must first be made, Both masses are assumed to be rigid and the vibrating modes are defined by the torsional and flexurat springs only. The higherorder vibrational modes due to antisymmetric bending of the springs are included in the model. In practice there are other modes too, but only the first torsional and linear modes have importance in practice. The moving mass system is assumed to be linear and thus the modes of motion can be treated independently. However, this is true only when the mass vibration amplitude is relatively small, because the damping term caused by the squeezed gas film is a non-linear function of the gap height ( = d -3) [l I l, and when the deformation of the spring is small enough not to cause coupling between modes. The three modes of motion implemented in the model are shown in Table 1. All modes are modelled with similar coupled differential equation pairs. The spring lbrces acting on the frame mas~ depend on both the frame angle 0F (or position z~) and the difference 0~- 0a between the frame and beam angles (or positions z~- zn). The spring forces acting on the beam mass depend only on the difference OF-- 0a (orzF - - Z B ) . The equations modelling the vibrating motion of the frame and beam about the x axis are (oscillating mode .~")
Torsional spring
Dual vibrating n|;.lS~system [:Faille aCttlation contact.~
Beamsensing electrode,, ~ Fig, 1. Structureof angularratesensorKingdevelopedby VT!HamlinOy. The centresiliconwafercontainsthe mechanicalsystem.
F r -. a l ~n e Beani l
Fig.2. Mechanicalsystemconsistingof frameand beamstructures.
T. Veijola et al. / Sensors and Actuators A 60 (1997) 113-12.1
I 15
Table ! Coupled vibratin modes of the frame m',d the beam masses
Osc. mode
l'rame
Beam
.4 Vibrating fi'ame
Vibrating beiu ! l
1
C Linear
Table 2 Mechanical constants in the harmonic motion equations and their electrical equivalents
.~/Vibrating frame .& Vibrating beam F' Linear Electrical equivalent
momenta of inertia I~.,. in., momenta of inertia is.,, ia.~ mass m~. ma capacitance CF. Ca
torsional viscosity AE,, ,in_, torsional viscosity Av.,, ,lB., viscosity ~. 7~
torsional elasticity k~.,, kav., torsional elasticity ks.~,kas.~ elasticity as. Kas
conductance GF. GB
inductance- t lily, ! ILav
d2ZF +
('F4
mF d/2
CFt
(_.:.2 N~
"'1:
w,~ ' ' ~ 0
= v
()
-'~a ,nB
('F,~
('F'~
tF Fig. 3. CROSSsection oftbe frame mass and the air gaps. d "~ dt
d20a.x
Aa.f-~'-t "FkaF.,:(Or.~--Oa.,) = ~'B.~
2.2. Air-gap modelfor frame 12)
Torsional motion about the y axis is defined by (oscillating mode ~') d"0Ev+
6.,.-d7 _
d20a.v
IB.r-~
(6)
~
dOa.s
h.x-~ "-+
dzs + 7B--dTt+ KBF(Z~-- Z.) =FB
(5)
The constants used in these equations of harmonic motion are described in Table 2. The electrical equivalents that will be used later are also shown in Table 2. On the right-hand side of the equations the torques and forces act as inputs to the mechanical system. Each of them is a sum of excitations due to external acceleration (F¢.,, %.0, internal excitations due to the electrical attractive forces in the air gaps (F~t, r~t), and excitation due to the Coriolis torque r, or.
41
dr"
dZF
~'F"~t+ '~ZF-- ~Bd Z~-- Z.) = FF
d0F v
;%T['+k~..~.:.-kB~.,.(O~.:,- 0~.,.) =~'~.,. (3)
d0n ,. + An,x---~ff + kaF v( OFv-- Oa.y) = ¢a ~.
(4)
The linear motion in the direction of the z-axis is modelled here with the following equations of harmonic motion (oscillating mode ~')
The cross section of the frame mass is shown in 7ig. 3. YF is the distance from the rotation origin to the position (/F+ bF)/4. In this structure the distance Yr is much larger than the distance forming the capacitance. When the motion of the mass is assumed to be relatively small compared with the static gap height, motion perpendicular to the air-gap surfaces can be assumed. The electrostatic force in each of the four air gaps is FF,-'- 2 (dFo-Zi) 2' i = 1,2,3,4
(7)
116
T. Veijola et al. / Sensors and Actuators A 60 (1997) 113-121
where u~, is the voltage across the frame air gap having a static gap height dFo and a displacement z,. A is the area of the air gap, and ~ is the dielectric constant of the gas. Due to the geometry of the gap structure, the area is reduced from w(l~-br)/2, where w is the gap width. The displacements z, consist of linear displacements and displacements due to rotation in mode ,<,v'. Large movement in mode ~ changes the frame forces and capacitances, but their contribution is assumed to be negligible in this model. The displacements at YF are zt =zF+ 0~.
(8)
z2 = - ZF-- 01~,_V~
(9)
Z~= --ZF+ 0~.,','F
(10)
Z,,= Z~- 0t~,_VF
(II)
The gap height is small compared with the distance y~, and thus the rotation angle is so small that 0v., can be used instead of sin0F.~. The total electrostatic force in the direction of the z axis acting on the frame mass centre is Fr:xl = FFt - Fv2-Fe.~+F~4
to cause negligible change in the torque. The exact expression for the torque function is [9,101
ewu2 ~
f'("'d'°) =ST
l Od
Cld-(b+)Olld-bO]
[if, 7}
-in d
+1)0
<15)
where u is the voltage across the gap, d is the gap height and 0 is the rotation angle, w is the gap width (in the direction of the y axis), l= ( I n - b a ) / 2 is the gap length and b=bn/2 is the length of the area without metallization. Unfortunately, this function is singular when the rotation angle 0 is zero and a Taylor series expansion at 0 = 0 must be used when the angle is small. For simplicity, the model presented is based on the series expansion. It is accurate only when the mass displacement is small compared with the static gap height. The total torque acting on the beam mass due to the electrostatic force is a sum of four terms rfi.y.el =f~-( uB i ,dno - Zli, On.y) - f T( Igl~2,dno + zn, - 0n.y)
+f,(ua3,dlio + Zn,0il.:.) -f~-( l/na,dllo- Zn, - 0B..v)
(12)
(16)
and the electrostatic torque is (13)
where dao is the static gap height. The total force acting on the beam mass centre in the direction of the z axis is
Again, assuming perpendicular motion to the air-gap surfaces, the capacitances in the air gaps are
Fn,cl =.IT(unl,dao - Zn,0n,~.)]xn -f~( iin:,dno + Zn, - On.y)/Xn
~'F.,.~I= ( Fv, - FF, + FF3- FF4)yF
-f,(un3,dao + Za,0a.~.)/Xa+f,( ua4,dao- Zn, - On.r)/rn Cri = ~ + CFo, i = 1,2,3,4 dro - z~
(14)
(17)
where the displacements z, are given in Eqs. (8)-( 11 ). C~o is a constant stray capacitance of the air gaps.
The exact capacitance of the beam air gap is also a function of the linear displacement and the rotation angle [9,10]
2.3. Air-gap model for beam
fc(d,O)
Fig. 4 shows the cross section of the beam mass. The gap displacement is measured at referencedistance xa = In~4. The assumption of motion perpendicular to the air-gap surfaces is not applicable when modelling air-gap effects on the beam. The electrical attractive forces and gap capacitances are functions of the rotation angle in mode <~ and the linear displacement of the beam. The rotation angle in mode ~ is assumed
In this model, a three-term Taylor expansion of the capacitance at 0 = 0 is used. The beam capacitances are the sum of a varying capacitance and a constant stray capacitance Cao:
Clt~l
~:
('H1
ew r d - b O ] ="-~ln[d~~-~) (~J
(18)
Cal =fc( dao- z,, On.~.)+ Cat)
(I9)
Ca: =fc (dBo + Za, - On.:.)+ CBo
(20)
CB3=fc ( dao + za, OB.~.)+ Cacl
(21 )
Ca, =fc( dno- za, - 0a.:.) + CBo
(22)
2.4. Coriolis force When external influences change the angu!m,"moment of a rigid body the counteracting torque is equal to
Cll3
C1~2
,=(d__n) ~.dt,
- ~
~--
bl~ '
Fig. 4. Cross section of the beam mass and the air gaps.
i.~ni~l
(23)
where H = loJ is the angular momentum and I is the matrix of inertia. In rotating body coordinates this becomes [ 13,14 ]
7". Veijola et al. / Sensors and Actttators A 60 t 1997) 113 -12I
t 17
iB
\ dt/body q" O.1X []
(24)
u~.,>~:TC'BI_
By applying Euler's angles to Eq. (24), assuming that o~:.... is much smaller than ton.,, and neglecting higher-order terms, one finds for the component of the torque in the y-direction r,a.~...... = ( - ln~, +/n.~. + In~.) ~.~(o. ....
L~FIiL~')UB,£L::iLBF k l
.... "
( 25 )
In the case of a vibrating sensor m,, is periodically time dependent and hence the resulting torque -q.is periodic. Other Coriolis force terms are not implemented in this model.
=
-
2.
5_
Fig.5. Equivalentcircui!forcoupledparallelresonators.Thiscircuitmodels both linearand torsionalmodesof motion. [.
.
.
.
.
.
.
.
. . . . . . .
3. Electrical simulation model and its implementation in APLAC
The equations presented in the previous section describe a non-linear dynamic system. A versatile circuit simulator, such as APLAC [ 12 ], is a good alternative when the system response is to be solved numerically in the frequency and time domains, especially when the device has an electrical interface. The system must be constructed of electrical components that are connected to each other building an electrical equivalent circuit. In APLAC all static and dynamic components are ir,ternally built of voltage-controlled current sources (VCCS) [15], and the nodal formulation to create the system matrix is used. For a simple and efficient model the use of VCCSs and parallelly connected components in constructing the equivalent circuit are required. All three vibrational modes of the system are modelled w:th coupled parallel L C R resonators, see the equivalent circuit in Fig, 5. The motion of the beam mass is coupled to the motion of the frame and thus the inductance that models the coupling spring force term is connected to the frame resonator, The fluxes ~;nthe inductances are equivalent to mechanical angles (displacements). They are converted to voltages using current-controlled voltage sources as shown in Fig. 5. The voltage u~ is equivalent to flux ~ = LFiLF in inductance LF and the voltage u~ is equivalent to flux ~b~ = ~ + LBFiLB~ in both inductances. The coupled resonator circuiE realizes the following equations c d2~bF _ d ~ F
1 .
1
.
:-~-+o ~ + y o ~ - ~ ( ~ - ~) = ~
" I CB~+G,d.~ +Ln~ .-'-( ~-~,)=i~ fit~ Ot
(~6) (27)
The similarity with Eqs. ( i )-(6) is evident. Table 3 shows the equivalences between the mechanical and electrical sig-
I
I mF.x t Resonator A I ~
I
'~
~ I Resonator
B O,.
Beam air gaps
••e I
UBI g/B2
~
AB
~, F , , . j - w
j
.A,
•
Fig. 6. Blockdiagramof the angularrz:iesensormodel, nals used in the equivalent circuit. The absolute values of the voltages u~ and u~ are very small compared to the voltages across the gap capacitances and the voltages in the surrounding electrical circuit. The model has been implemented in APLAC as a userdefinable module. It can be connected to an electrical circuit in a similar way to any other component model. The block diagram of the model is shown in Fig. 6. The electrical attractive forces in air gaps are modelled with static, non-linear VCCSs (iF.~l, iB.~) controlled by the air-gap voltages, mass displacements and rotation angles defined in Eqs. (12) and (13) for frame and Eqs. (i6) and (17) for beam. In this model, four controlled current sources altogether are used. The charges of the eight air-gap capacitances are realized with dynamic, non-linear VCCSs also controlled by the airgap voltage, mass displacement and rotation angle. The frame capacitances Eq. (14) and electrical attractive forces dedend
Table 3 Equivalencesbetweenelectricaland mechanicalvariablesin the resonatorcircuits Vibratingframe Vibratingbeam ~: Linear Electricalequivalent
angularvelocityo#_,,,~., angularvelocityo,,v.,co~., velocityre, voltageUF,US
torque rr.~, re., tO~lUel"v.~, %.y force FF, Fe
rotation angle 8F:, On., rotation angZe 8F.~,~.~, displacement zr, ze
cunrnt iF, in
flux ~, ~ , u~, u~
! 18
7". Veijola et al./Sensors andActuators A 60 (1997) 113-121
on zF and 0~.~, whereas the beam capacitances Eqs. ( I 9 ) (22) and electrical attractive forces depend on zn and 0B.y. The Coriolis force acting on the beam mass depends on angular velocities that equal voltages in this model. A VCCS connected in parallel with the mode ~ resonator having two controlling voltages realiz...~ the torque in Eq. (25), see block . ~ in Fig. 6. An external parallel resonator circuit modelting the rotation of the sensor system about the z axis can be connected directly to node to~. The component model has six fixed nodes, the nodes of the resonator circuits and 23 optional nodes. The capacitance nodes are optional. The model is simplified greatly when the force current terms and non-linear gap capacitances are left out if the corresponding nodes are not needed. The 28 model parameters include the constants shown in Table 2 and several other parameters needed in calculating the electrical attractive forces and gap capacitances.
4. S i m u l a t i o n
and measurement
results
4.1. Simulating with APLAC Simulation tasks in APLAC are defined as programming language style descriptions that consist of, e.g., circuit definition, simulation and output commands. In the first simulation the frequency response of the sensor, electrostatically excited at the frame, is first simulated using a.c. analysis. In the a.c. analysis the non-linear circuit is first linearized at the operating point defined by the d.c. sources. Then the response to a.c. sources is solved from the linearized circuit at each fiequency. The following circuit description in APLAC's input language generates the response shown in Fig. 7. #include "gyro.i" Gyro "gl" NvF NvB NvFy NwBx NvF NvB + MODEL "V'rlproto" + NF1 n f l NF2 nf2 Volt "ul" nfl 0 AC
NZF4 nzf4 1 DC 20
Volt "u2" nf2 0 AC -I De 20 Sweep "" + LOOP 600 FREq LOG Ik lOOk + ¥ "Amp" "dB" -100 20 + GRID BIGSCREF~ Disr!,y Y
Display
""
PHASE ""
MagDB(Vac(nZF4))
Pha(Vac(nZF4))
EndSveep The circuit described in the listing consists of one angular rate sensor model called Gyro and two voltage sources ul and u2. The model definition and its parameters are included from file gyro.i using the #include macro shown in the beginnine" of the listing. The parameters of an early prototype manufactured by VTI Hamlin Oy are specified using a
A°
APLAC7,00User:HUTCitcui!TheoryLab.Sat Sep 7 1996 ........... 1~ '~1"~ I" Pl Torsionalframe+beam It
* B't
I ii
'"'
" i '-Ilk
i l
180 PHASE
ill
Illll '°
-12e
180 I0 30 100 f [kHzl Fig. 7. Simulated frequency response of the angular rate sensor. Relative amplitude (solid line) and phase (dashed line) responses, t
3
MODEL definition specified in the included file. The sensor dimensions, masses and momenta of inertia in the model parameters are directly the design dimensions. The damping coefficients, spring constants and amplitudes of frequency responses are fitted experimentally to measurement results discussed later. In the listing three optional parameters are used in defining optional nodes for frame air gaps 1, 2 and 4. The voltage sources ul and u2 consist of a.c. components in opposite phases and symmetrical 20 V d.c. voltages. The non-symmetrical a.c. signal connected to air gaps [ t,nd 2 excites both torsional and linear modes of m3tion. The Sweep block defines the a.c. simulation and the graphical output. A logarithmic frequency sweep is requested in 600 points from ! to 100 kHz to include all resonances of the system. The lower resonance peak at 3.2 kHz is ca',sed by t.",~ vibrating system in torsior;al mode, and me resonanc~ at 9.5 kHz is due to the linear motion of both masses. At 17 kHz the linear and torsional motions compensate each other, resulting in zero movemera at gaps 3 and 4. The resonances at 54 and 72 kHz are due to frame and beam masses working at opposite phases in the linear and torsional modes, respectively. In the second simulation example the sensor sensitivity to an external constant angular velocity (to~.,~,,= 10 rad s- i) is simulated. The beam mass is actuated to vibrate at its resonance frequency, and the beam gap d.c. voltage is swept to tune its resonance frequency to match the frame's resonance. Frame mass is actuated with a biased 5 V p--p square-wave voltage applied to opposite air gaps in reverse phases. This is modelled in this a.c. analysis with a d.c. source of 2.5 V and an a.c. source equal to the fundamental component of the square wave, that is 10/~'V. The simulation results in Fig. 8 show the vibration amplitudes for the beam and the frame. Note the decrease of the frame amplitude caused by energy transfer to the beam at the resonance.
4.2. Measured and simulated capacitance-voltage curves Capacitance-voltage measurements were performed on one capacitor at time, the other seven being shorted. The
119
T. Veijolaet al. I Sensors and Actuators A 60 ¢1997) I 13-121 APLAC 7.00 Ua¢c tIUT Circuit TheoO"Lab. MortAug 26 1996
~
IO0
Amp "- ~
75
~
i
APLAC 7.00 User: llUT Ci~¢mlT~c¢~ Lab Tu¢ ,q~ 3 1996
- 180
" " ! ....
PHASE
--"
.. -
135
5O
~
22
45
32
33
34
-20
APLAC 7.00 User. HUT Circuit Theo~ Lab. Mort Sop 02 1996
2 z 1.5 [uml I
\ G.5
0
T%
I L
°-~
- - " 0 "0" -I0
0
tO
2O
Bt~s IV]
Fig. 8. Sensor response to constant rotation as a functionof beam bias voltage.Beamvibrationamplitude(solid line) and phase (dashedline) and frame(-.-) amplitude(dividedby 20) at theirreferencepositions.
-150
T
........... I...........
2,2.
35
Bi a."iIV I
-300
1
'-' "~-~----
0 3t
2,-
12
150
300
Bias IV I
Fig.9. Capacitance-voltagecharacteristicsof theframemass.The simulated (solid line) and measured([] }capacitancesandthe simulateddisplacement (dashedline}. equipment used was an HP LCR bridge 4274A with an external bias source Keithley 487 decoupled from the capacitance test signal by an external RC network. When a non-symmetrical d.c. bias voltage is applied in the air gap, the mass will be displaced. This will cause a measurable change in the frame capacitances as shown in the measured and simulated responses in Fig, 9. The bias voltage across air gap I was varied from - 3 0 0 to + 300 V. The simulated mass displacement is also shown in the Figure. A similar measurement and simulation was performed for the beam mass, shown in Fig. 10. The d.c. bias voltage source was swept in this measurement from - 2 0 to + 20 V. The measured and simulated capacitances are shown in the Figure together with the simulated displacement of the reference position. Curve fitting was performed by varying the stray capacitance only. The minimum of the C - V curve is not at zero volts but at about 0.5 V. This due to the work-function difference between the silicon beam and the Ti/AI metallization. This value is used throughout this work when comparing predictions with experimental data.
Fig. t0. Capacitance-voltagecharacteristicsof the beammass The simulated {solid line} ,andmeasured(t-'l} capactmncesand the simalaleddisplacement{dashedline). half-bridge detection on the second similar opposite electrode pair, see Fig. I 1. A I MHz test signal (Vs) was used on the bridge and the amplifier output was tuned by an inductor. Linear voltage-force (aud torque) conversion is obtained with a d.c. biPs voltage (VD and opposite phase a.c. drive. The frequency response was measured with an NF 5020 analyser. The model presented includes an output for mass displacements at each air gap and thus the simulation through the varying capacitance is not needed. This simulation could also be done with the harmonic balance techniques of APLAC, used widely in r.f. design. The low-pass filter in the meas,.a'ement circuit is modelled with a second-order Butterworth filter with a cut-off frequency of 15 kl-lz (block LP in Fig. !!}. The damping coefficients were extracted experimentally from measurement results because the gas pressure inside the sensor is unknown. The dimensions of the torsional springs were in the range where textbook formulae are not valid. The spring constants were also extracted from measured frequency responses and capacitance-voltage curves. Using the extracted parameters, an empirical design formula for the elasticity was developed. The parameter extraction was performed using APLAC's built-in optimization capabilities [161.
£ T
4.3. Measured and simulated frequency responses
Analyzer The frequency response was measured using electrostatic excitation on one top-bottom electrode pair and capacitance
Fig. 1I. Measurementset-upfor frequencyresponses.
Veijolaet al. / Sensors and Actuatorx A 60 (1997)
T.
120
First a simulated frequency response using the extracted parameters is comp,tred with measurement results of the frame mass displacement. A non-symmetrical electrostatic force signal was used to excite the mass in both linear and torsional modes. The frame mass was actuated with an a.c. voltage from 1 to 20 kHz and a symmetrical 200 V d.c. bias voltage Vh was applied to air gaps I and 2. The measured and simulated frequency responses are compared in Fig. 12. The background noise level in the measurement results is clearly seen in the frequency response. Fig. t3 shows the measured and simulated frequency responses of the beam mass. The measurement was similar to the frame a.c. response measurement with the exception that the bias voltage is 10 V. The extracted torsional and linear viscosities were used to estimate the remaining gas pressure inside the sealed structure. Applying Eq. (7) for the effective viscosity and Eq. (19) in Ref. { I I ], the estimated pressure was about 100 Pa. The measured and simulated frequency responses ;., Fig. 14 show how the increasing symmetrical d.c. bias effectively changes the spring constants through the non-linear force function Eq. (15) and shifts the beam resonance to a lower frequency. This measurement is similar to the previous one with the exception that the bias voltage is varied ( 1, 3, 10, 20 and 30 V). APLAC 40-
7.00 User: HUT Circui! Tlumry Lab. Tue Sep 3 1996 ~t ~,
Amp [a~l
- .... ~
20 .
.
.
.
.
.
-
%
.
.
. PHASE
.
90
'il
I ,-180
J143-
1
- 180 I
10
3
[va~z! Fig. 12. Relativeamplitude(solid line) and phase (dashedline) responses of simulatedand measured([]) frequencyresponsesof framemass. f
APLAC "/,IX)U~t-:.HUT Ci~uit Theov/Lab, Tue Sop 3 1996 Amp
o
PHASE
Ill N,,o 1o
f [kHzl Fig. 13. Relative a m p l i t u d e ( s o l i d line) and phase (dashed line) responses of simulated and measured (I-q) frequency responses o f b e a m mass.
50Amp taB] 37.5.
113-121
APLAC 7.00 User: llt/T Citcuil Theory Lab. Sol $ep 7 1996
A3°"i
/
20V
J \A
12.5-
02
3
4
5
6
f lkHz]
Fig. 14. Simulated(--) and measured([]) frequencyresponseof the beam masswithdifferentbias voltages.
5. Conclusions The complex etectromechanical interactions present, in a microelectromechanical system (case vibrating angular rate sensor) can be effectively modelled and analysed with a circuit simulation program. Our simulations gave results which are close to the experimental data. Non-trivial side effects like the change of the effective spring constant and resonance frequency by a bias voltage were also correctly predicted. The model was used here also to extract values for parameters that could not be accurately estimated in the design. With the model presented a large rumber of different simulations can be performed on the sensor combined with external electrical or mechanical equivalent circuits. All model parameters can be variables in an optimization process [ 161, where unknown model parameters are extracted by comparing simulation results with measurements. A statistical simulation (Monte Carlo), where the tolerances of the parameters are included, can predict the variations in sensor response. Temperature dependencies can also be simulated if the model parameters are written as a function of temperature. The model presented covers the standard operation conditions of the sensor. The cases where the mechanical or electrical actuation is so large that one of the vibrating masses will hit the sensor body are not included in this model. Geometrical non-idealities like non-orthogonality and asymmetry that cause mechanical cross-talk are not included in the model. It is rather straightforward to add these couplings between the two modes. Electrical cross-talk and stray capacitances can be added external to the model. The gas damping is modelled with separate conductances in the resonance circuits. This is justified when the displacement and lhe gas p~essure are small. A model that is valid for higher pressures and displacements consists of a non-linear damping circuit, as presented in Ref. [ 1 ! 1, common to all modes of motion. The higher-order vibrational modes included in this model did not show any importance in the simulations, verifying the successful device design. However, if this modelling approach is applied to other similar microelectromechanical
T. Veilolaet al. / Sensors and Actuawrs A 60 f1997) 113-121
systems, the higher-order modes can reveal mismatch between the torsional and flexural stiffness of the supporting springs.
References [ I ] T.P. Mathues, Extending the scope of ABS, Automotive Eng.. 102 (1994) 15-17. [2] J. S6derkvist, Micromachined gyroscopes,Sensors and Actuators A, 43 (1994) 65-71. [ 31 J.B. Chatterton, Some general comparisonsbetweenthe vibratoryand conventional rate gyro, J. Aeronaut Sci., 9 (1955) 633--638. [41 A. Reppichand R. Willig, Yaw rate sensorfor vehicledynamicscontro! system, SAE Paper 950537, 1995, pp. 67~76. [51 T. Ichinose and J. Terada, Angular rate sensor for automotive applicat,on, SAE Paper 950535, 1995,pp. 49-56. [6] S.D.Orloskyand HD. Morris, A quartz rotational ratesensor,Semors, 2 (1995) 27-31. 171 J. Berastein, S. Cho, A.T. King, A. Kourepenis, P. Maciel and M. Weinberg, A micromachinedcomb-drivetuning fork l,..e gyroscope. Proc. MEMS '93, Fort Landerdale, FL, US4, 1993, pp. 143-148. [ 81 J.D. Johnson. S.R. Zarabadiand D.R. Sparks, Surfacemicromachined angular rate sensor, SAE Paper 950538. 1995,pp. 77-83. [91 T. Veijola, Accelerometermodel in APLAC, Helsinki Universityof Technology, Circuit Theory Laboratory,Report CT-18, Feb. 1994. [ 101 T. Veijola and T. Ryhtinen, Model of capacitive micromechanical accelerometerincluding effect of squeezed gas film, Proc. ISCAS "95, Seattle. WA, USA. 1995, Vol. !. pp. 664--667. [ I I 1 T. Veijola, H. Kuisma, J. Lahdenpe~ and T. Ryhanen, Equivalent circuit model of the squeezed gas film in a silicon accelerometer, Sensors and Actuators A, 48 (1995) 239-248. [121 M. Valtonen et al., APLAC ~ An Object-Oriented Analog Circuit SimulatorandDesign Tool, HelsinkiUniversityofTechnology,Circuit Theory Laboratory and Nokia Research Center. Hardware Design Technokgy, 7.0 Re[erenceManual and 7.0 User's Manual, Otaniemi, Jan. 19t36. [ 13] E. Miihlenfeld, Rotor-Schwingkreisel zur Messung der Winkelgeschwindigkeit, lngenieur-Archiv, Vol, 38, Springier, Berlin, 1969, pp. 58-66. [ 141 A.L. Fetter and JD. Walecka, Theoretical Mechanics of Particles and Cm~tinua, McGraw-Hill,New York, t980. [15] P. Heikkilti, Object-oriented approach to numerical circuit analysis, PhD. Thesis, Helsinki Universityof Technology,Jan. 1992. [ 16] R. Niutanen, M. Valtonen and K. Mannersalo, Optimization methods in APLAC, Helsinki University of Technology. Circuit Theory Laboratory,Report CT-15, Dec. 1992.
Biographies Timo Veijola was born in Helsinki on February 4, 1954. He received the Diploma Engineer (M.Sc.) and the Licen-
121
tiate in Technology degrees in electrical engineering from the Helsinki University of Technology, Finland, in 1980 and 1986, respectively. Since 1981 he has been working as laboratory manager in the Circuit Theory Laboratory at the Helsinki University of Technology, where he has been a member of the APLAC development team since 1988. His current research interests are computer-aided circuit simulation and modelling of electromechanical and electrothermal devices and systems. Heikki Kuisma was born in Kalvola on August 3, 1953. He received the Diploma Engineer degree (M.Sc.) in electrical engineering from the Helsinki University of Technology, Finland, in 1978. Since 1980 he has been working at Vaisala Oy and Vaisala Technologies Inc. doing research and development on silicon micromechanics and microsensors. He holds several patents on silicon microsensors. Currently he is leading the sensor research group at VTI Hamlin Oy (formerly Vaisala Technologies, lnc). Juha Lahdenperd was born in Liperi on March 30, 1959. He received the Diploma Engineer degree (M.Sc.) in applied physics from the Helsinki University of Technology in 1986. Since 1986 he has been working as a research and development engineer at Vaisala Oy working in micromechanical pressure and acceleratinn sensor development and from I991 on at VTI Hamlin Oy (formerly Vaisala Technologies, Inc). His current activities are the design of micromechanical acceleration sensors and micromeehanieal process development. Tapani Rvhdnen was born in Heisinki on lu!)' 7, 1959. He received the Diploma Engineer degree (M.Sc.) in technical physics and the Doctor of Technology degree in applied dectronics from the Helsinki University of Technology, Finland in 1986 and 1992, respectively. He has authored several publications on the theory, design and characterization of ultra-low-noise superconducting thin-film magnetometers. From 1992 to 1995 he worked as a research and development engineer at Vaisala Technologies Inc. in design of micromechanical pressure and rotation rate sensors, development of new measurement systems for microm,:chanical sensors and the modelling of micromechanical structures. In I995 he joined Nokia Telecommunications as a research and development manager.
ELSEVIER
Sensors and Actuators A 60 {1997) 11~121
Simulation model for micromechanical angular rate sensor Timo Veijola ~'*, Heikki Kuisma b, Juha Lahdenper/i b Tapani Ryhfinen ¢ Helsinki Univer,~ity of Tec~mology. Circuit Theory ~boratory. . PO Box 3000. FIN-02015 Hut. Finland ~' VTi Hamlin Oy, PO Box 9. FIN-O0421 Helsmki. Finhmd c Nokia ]relecomnumications. PO Box 33. FIN-02601 Espoo. Finland
Abstract A simulation model for an angular rate sensor, a gyroscope, is presented. The device is based on a micromechanical dual torsional mass system which is actuated electrostaticaily and sensed capacitively. Model equations describing a dynamic, non-linear system are first presented and then realized as an electrical equivalent circuit. The vibrational modes of the system are modelled with coupled resonator circuits. The electrostatic and Coriolis forces as well as variable capacitances in the small air gaps are modelled with non-linear controlled current sources. External forces, torques and electrical actuation can act as inputs to the device. The model presented allows numerical sensor simulations concurrently with the interfacing electronics in the time and frequency domains. The model is verified by comparing its simulation results to measured frequency responses and capacitance-voltage characteristics. Keyword~: Angular rate sensors; Circuit simulation; Mictomechanics
1. Introduction In recent years there has been a great interest in angular rate sensors, especially for use in vehicles. These sensors are used to measure the yaw rate of a vehicle, or angular velocity about the vehicle's vertical axis, for chassis dynamic contrel. In the control system the steering-wheel angle is used as a set point indicating the driver's intentions. The hardware that exists for anti-skid braking control is used to control the yaw of the vehicle by individually adjusting the braking forces of the four wheels [ i ]. Other potential applications for angular rate sensors are listed in Ref. [2]. Vibrating rate gyros were introduced in the early 1950s tbr the replacement of rotating gyros in aerospace applications with the advantage of no wearing parts [31. After 40 years they have found their way to the consumer market. In topmodel passenger cars, vibrating angular rate sensors based on a metal cylinder [4], a metallic tuning lbrk [51 and a piezoelectric quartz tuning fork [61 are being used as of the mid-1990s. It was already predicted in Ref. [31 that the performance of a vibrating angular rate sensor will not be affected by miniaturization, contrary to the rotating sensor. Small microma~,nined angular rate sensors based on a tuning fork [7] and a vibrating ring [81 have been introduced. A * Corresponding author. Tel.: + 358 9 451 22 93. Fax: + 358 9 451 48 18. E-mail: [email protected]. 09244247/97/$17.00 © 1997ElsevierS~ienceS.A. All rights reserved Pil S0924-4247 (97) 01382-4
review article [ 21 describes the operating principles of various possible designs. A vibrating angular r; te sensor differs from most other micromechanicai sensors in the complexity of its operation. The sensor does not give any output signal without mechanical excitation. In fact, this sensor is a true microelectromechanical system with equally important sensing, actuation and signal-processing functions. To operate a vibrating ral,,' sensor one has to excite it to a vibratory (either linear or angular) mGvement. Mechanical resonance is used in most cases to increase the amplitude or decrease the need for drive force, scarce in the case of an electrostatic drive. Secondly, one has to measure the coupled vibration in an orthogonal direction. The signal to be measured is typically very small, less than 100 ppm of the excited vibration. Therefore mechanical amplification is often required in the form of resonance. It is required to match properly the excitation and detection resonance frequencies. Matching by manufacturing tolerances may be impossible. For an electrostaticaily actuated vibrator, a bias voltage can be used for frequency tuning as described in Ref. [ 8 ]. The system to control the device is always quite complex with multiple amplitude locks, phase locks and servo loops as described in Ref. [81. Therefore, a simulation tool with a correct description of the etectromechanical interactions is needed for the system design. In this paper we present the elementary equations for a dual torsional vibrator which is
114
T. Veijola et al. / Sensors and Acnmtors A 60 (1997) 113-121
actuated electrostatically and sensed capacitively. The dynamic, non-linear equations are realized as an electrical equivalent circuit and solved numerically by a circuit simulation program. An electrical equivalent circuit has been successfully applied in constructing a micromechanical accelerometer model [9-111. The sensor model presented here uses a similar approach in building the electrical equivalent circuit model. Mechanical resonance modes and coupling between the modes are realized with electrical coupled resonators. Electrostatic forces, variable capacitances in the air gaps and Coriolis force are realized with non-linear voltage-controlled currer, t sources (VCCS). The model has been implemented in the circuit simulation progiam APLAC [ 12] as a userdefinable module written in APLACs input language. The resulting model is a non-linear large-signal model that includes all essential static and dynamic properties of the sc:,sor. It allows numerical sensor system simulations concurrently with the interfacing electronics in the time and frequency domains.
2. Structure and model of the sensor
The structure of the mieromechanical sensor presented in Fig. I is being developed by VTI Hamlin Oy ( formerly Vaisala Technologies Inc.), The device is fabricated based on three silicon wafers, which are anodically bonded together. The centre wafer ct, ntains the beam-frame-spring structure. The outer wafers are composed of siliicon and borosilicate glass with electrode autd feedthrough structures. The gap capacitances are formed ih~small air gaps between the outer and centre wafers. Design details of the outer wafers are not shown. The size is 7.5 mm × 2.5 mm × 1.7 mm (L × W × H). Thousands of prototype devices have been successfully fabricated. The structure is sealed at low pressure. The remaining gas in the air gaps causes damping of the mass motion. The gas pressure is in practice so small that the spring forces caused by the squeezed gas can be neglected here. The damping due
to the dissipation in the springs is small enough also to be neglected in this model. 2.1. Model for dual torsional mass system Fig. 2 shows the dual torsional system consisting of two vibrating masses. The outer mass, the frame, is attached Io the sensor body by elastic torsional springs. The inner mass, the beam, is supported by the frame with another pair of torsional springs. In the norm'haloperating mode the frame and beam masses are excited with an electrical attractive force to vibrate about the x axis. If the structure is rotated about the z axis, the motion of the inner beam mass is coupled to the perpendicular axis y by Coriolis force. The motion of the beam mass is sensed capacitively in four air gaps. To create a compact but functional model for the mechanical structure several assumptions must first be made, Both masses are assumed to be rigid and the vibrating modes are defined by the torsional and flexurat springs only. The higherorder vibrational modes due to antisymmetric bending of the springs are included in the model. In practice there are other modes too, but only the first torsional and linear modes have importance in practice. The moving mass system is assumed to be linear and thus the modes of motion can be treated independently. However, this is true only when the mass vibration amplitude is relatively small, because the damping term caused by the squeezed gas film is a non-linear function of the gap height ( = d -3) [l I l, and when the deformation of the spring is small enough not to cause coupling between modes. The three modes of motion implemented in the model are shown in Table 1. All modes are modelled with similar coupled differential equation pairs. The spring lbrces acting on the frame mas~ depend on both the frame angle 0F (or position z~) and the difference 0~- 0a between the frame and beam angles (or positions z~- zn). The spring forces acting on the beam mass depend only on the difference OF-- 0a (orzF - - Z B ) . The equations modelling the vibrating motion of the frame and beam about the x axis are (oscillating mode .~")
Torsional spring
Dual vibrating n|;.lS~system [:Faille aCttlation contact.~
Beamsensing electrode,, ~ Fig, 1. Structureof angularratesensorKingdevelopedby VT!HamlinOy. The centresiliconwafercontainsthe mechanicalsystem.
F r -. a l ~n e Beani l
Fig.2. Mechanicalsystemconsistingof frameand beamstructures.
T. Veijola et al. / Sensors and Actuators A 60 (1997) 113-12.1
I 15
Table ! Coupled vibratin modes of the frame m',d the beam masses
Osc. mode
l'rame
Beam
.4 Vibrating fi'ame
Vibrating beiu ! l
1
C Linear
Table 2 Mechanical constants in the harmonic motion equations and their electrical equivalents
.~/Vibrating frame .& Vibrating beam F' Linear Electrical equivalent
momenta of inertia I~.,. in., momenta of inertia is.,, ia.~ mass m~. ma capacitance CF. Ca
torsional viscosity AE,, ,in_, torsional viscosity Av.,, ,lB., viscosity ~. 7~
torsional elasticity k~.,, kav., torsional elasticity ks.~,kas.~ elasticity as. Kas
conductance GF. GB
inductance- t lily, ! ILav
d2ZF +
('F4
mF d/2
CFt
(_.:.2 N~
"'1:
w,~ ' ' ~ 0
= v
()
-'~a ,nB
('F,~
('F'~
tF Fig. 3. CROSSsection oftbe frame mass and the air gaps. d "~ dt
d20a.x
Aa.f-~'-t "FkaF.,:(Or.~--Oa.,) = ~'B.~
2.2. Air-gap modelfor frame 12)
Torsional motion about the y axis is defined by (oscillating mode ~') d"0Ev+
6.,.-d7 _
d20a.v
IB.r-~
(6)
~
dOa.s
h.x-~ "-+
dzs + 7B--dTt+ KBF(Z~-- Z.) =FB
(5)
The constants used in these equations of harmonic motion are described in Table 2. The electrical equivalents that will be used later are also shown in Table 2. On the right-hand side of the equations the torques and forces act as inputs to the mechanical system. Each of them is a sum of excitations due to external acceleration (F¢.,, %.0, internal excitations due to the electrical attractive forces in the air gaps (F~t, r~t), and excitation due to the Coriolis torque r, or.
41
dr"
dZF
~'F"~t+ '~ZF-- ~Bd Z~-- Z.) = FF
d0F v
;%T['+k~..~.:.-kB~.,.(O~.:,- 0~.,.) =~'~.,. (3)
d0n ,. + An,x---~ff + kaF v( OFv-- Oa.y) = ¢a ~.
(4)
The linear motion in the direction of the z-axis is modelled here with the following equations of harmonic motion (oscillating mode ~')
The cross section of the frame mass is shown in 7ig. 3. YF is the distance from the rotation origin to the position (/F+ bF)/4. In this structure the distance Yr is much larger than the distance forming the capacitance. When the motion of the mass is assumed to be relatively small compared with the static gap height, motion perpendicular to the air-gap surfaces can be assumed. The electrostatic force in each of the four air gaps is FF,-'- 2 (dFo-Zi) 2' i = 1,2,3,4
(7)
116
T. Veijola et al. / Sensors and Actuators A 60 (1997) 113-121
where u~, is the voltage across the frame air gap having a static gap height dFo and a displacement z,. A is the area of the air gap, and ~ is the dielectric constant of the gas. Due to the geometry of the gap structure, the area is reduced from w(l~-br)/2, where w is the gap width. The displacements z, consist of linear displacements and displacements due to rotation in mode ,<,v'. Large movement in mode ~ changes the frame forces and capacitances, but their contribution is assumed to be negligible in this model. The displacements at YF are zt =zF+ 0~.
(8)
z2 = - ZF-- 01~,_V~
(9)
Z~= --ZF+ 0~.,','F
(10)
Z,,= Z~- 0t~,_VF
(II)
The gap height is small compared with the distance y~, and thus the rotation angle is so small that 0v., can be used instead of sin0F.~. The total electrostatic force in the direction of the z axis acting on the frame mass centre is Fr:xl = FFt - Fv2-Fe.~+F~4
to cause negligible change in the torque. The exact expression for the torque function is [9,101
ewu2 ~
f'("'d'°) =ST
l Od
Cld-(b+)Olld-bO]
[if, 7}
-in d
+1)0
<15)
where u is the voltage across the gap, d is the gap height and 0 is the rotation angle, w is the gap width (in the direction of the y axis), l= ( I n - b a ) / 2 is the gap length and b=bn/2 is the length of the area without metallization. Unfortunately, this function is singular when the rotation angle 0 is zero and a Taylor series expansion at 0 = 0 must be used when the angle is small. For simplicity, the model presented is based on the series expansion. It is accurate only when the mass displacement is small compared with the static gap height. The total torque acting on the beam mass due to the electrostatic force is a sum of four terms rfi.y.el =f~-( uB i ,dno - Zli, On.y) - f T( Igl~2,dno + zn, - 0n.y)
+f,(ua3,dlio + Zn,0il.:.) -f~-( l/na,dllo- Zn, - 0B..v)
(12)
(16)
and the electrostatic torque is (13)
where dao is the static gap height. The total force acting on the beam mass centre in the direction of the z axis is
Again, assuming perpendicular motion to the air-gap surfaces, the capacitances in the air gaps are
Fn,cl =.IT(unl,dao - Zn,0n,~.)]xn -f~( iin:,dno + Zn, - On.y)/Xn
~'F.,.~I= ( Fv, - FF, + FF3- FF4)yF
-f,(un3,dao + Za,0a.~.)/Xa+f,( ua4,dao- Zn, - On.r)/rn Cri = ~ + CFo, i = 1,2,3,4 dro - z~
(14)
(17)
where the displacements z, are given in Eqs. (8)-( 11 ). C~o is a constant stray capacitance of the air gaps.
The exact capacitance of the beam air gap is also a function of the linear displacement and the rotation angle [9,10]
2.3. Air-gap model for beam
fc(d,O)
Fig. 4 shows the cross section of the beam mass. The gap displacement is measured at referencedistance xa = In~4. The assumption of motion perpendicular to the air-gap surfaces is not applicable when modelling air-gap effects on the beam. The electrical attractive forces and gap capacitances are functions of the rotation angle in mode <~ and the linear displacement of the beam. The rotation angle in mode ~ is assumed
In this model, a three-term Taylor expansion of the capacitance at 0 = 0 is used. The beam capacitances are the sum of a varying capacitance and a constant stray capacitance Cao:
Clt~l
~:
('H1
ew r d - b O ] ="-~ln[d~~-~) (~J
(18)
Cal =fc( dao- z,, On.~.)+ Cat)
(I9)
Ca: =fc (dBo + Za, - On.:.)+ CBo
(20)
CB3=fc ( dao + za, OB.~.)+ Cacl
(21 )
Ca, =fc( dno- za, - 0a.:.) + CBo
(22)
2.4. Coriolis force When external influences change the angu!m,"moment of a rigid body the counteracting torque is equal to
Cll3
C1~2
,=(d__n) ~.dt,
- ~
~--
bl~ '
Fig. 4. Cross section of the beam mass and the air gaps.
i.~ni~l
(23)
where H = loJ is the angular momentum and I is the matrix of inertia. In rotating body coordinates this becomes [ 13,14 ]
7". Veijola et al. / Sensors and Actttators A 60 t 1997) 113 -12I
t 17
iB
\ dt/body q" O.1X []
(24)
u~.,>~:TC'BI_
By applying Euler's angles to Eq. (24), assuming that o~:.... is much smaller than ton.,, and neglecting higher-order terms, one finds for the component of the torque in the y-direction r,a.~...... = ( - ln~, +/n.~. + In~.) ~.~(o. ....
L~FIiL~')UB,£L::iLBF k l
.... "
( 25 )
In the case of a vibrating sensor m,, is periodically time dependent and hence the resulting torque -q.is periodic. Other Coriolis force terms are not implemented in this model.
=
-
2.
5_
Fig.5. Equivalentcircui!forcoupledparallelresonators.Thiscircuitmodels both linearand torsionalmodesof motion. [.
.
.
.
.
.
.
.
. . . . . . .
3. Electrical simulation model and its implementation in APLAC
The equations presented in the previous section describe a non-linear dynamic system. A versatile circuit simulator, such as APLAC [ 12 ], is a good alternative when the system response is to be solved numerically in the frequency and time domains, especially when the device has an electrical interface. The system must be constructed of electrical components that are connected to each other building an electrical equivalent circuit. In APLAC all static and dynamic components are ir,ternally built of voltage-controlled current sources (VCCS) [15], and the nodal formulation to create the system matrix is used. For a simple and efficient model the use of VCCSs and parallelly connected components in constructing the equivalent circuit are required. All three vibrational modes of the system are modelled w:th coupled parallel L C R resonators, see the equivalent circuit in Fig, 5. The motion of the beam mass is coupled to the motion of the frame and thus the inductance that models the coupling spring force term is connected to the frame resonator, The fluxes ~;nthe inductances are equivalent to mechanical angles (displacements). They are converted to voltages using current-controlled voltage sources as shown in Fig. 5. The voltage u~ is equivalent to flux ~ = LFiLF in inductance LF and the voltage u~ is equivalent to flux ~b~ = ~ + LBFiLB~ in both inductances. The coupled resonator circuiE realizes the following equations c d2~bF _ d ~ F
1 .
1
.
:-~-+o ~ + y o ~ - ~ ( ~ - ~) = ~
" I CB~+G,d.~ +Ln~ .-'-( ~-~,)=i~ fit~ Ot
(~6) (27)
The similarity with Eqs. ( i )-(6) is evident. Table 3 shows the equivalences between the mechanical and electrical sig-
I
I mF.x t Resonator A I ~
I
'~
~ I Resonator
B O,.
Beam air gaps
••e I
UBI g/B2
~
AB
~, F , , . j - w
j
.A,
•
Fig. 6. Blockdiagramof the angularrz:iesensormodel, nals used in the equivalent circuit. The absolute values of the voltages u~ and u~ are very small compared to the voltages across the gap capacitances and the voltages in the surrounding electrical circuit. The model has been implemented in APLAC as a userdefinable module. It can be connected to an electrical circuit in a similar way to any other component model. The block diagram of the model is shown in Fig. 6. The electrical attractive forces in air gaps are modelled with static, non-linear VCCSs (iF.~l, iB.~) controlled by the air-gap voltages, mass displacements and rotation angles defined in Eqs. (12) and (13) for frame and Eqs. (i6) and (17) for beam. In this model, four controlled current sources altogether are used. The charges of the eight air-gap capacitances are realized with dynamic, non-linear VCCSs also controlled by the airgap voltage, mass displacement and rotation angle. The frame capacitances Eq. (14) and electrical attractive forces dedend
Table 3 Equivalencesbetweenelectricaland mechanicalvariablesin the resonatorcircuits Vibratingframe Vibratingbeam ~: Linear Electricalequivalent
angularvelocityo#_,,,~., angularvelocityo,,v.,co~., velocityre, voltageUF,US
torque rr.~, re., tO~lUel"v.~, %.y force FF, Fe
rotation angle 8F:, On., rotation angZe 8F.~,~.~, displacement zr, ze
cunrnt iF, in
flux ~, ~ , u~, u~
! 18
7". Veijola et al./Sensors andActuators A 60 (1997) 113-121
on zF and 0~.~, whereas the beam capacitances Eqs. ( I 9 ) (22) and electrical attractive forces depend on zn and 0B.y. The Coriolis force acting on the beam mass depends on angular velocities that equal voltages in this model. A VCCS connected in parallel with the mode ~ resonator having two controlling voltages realiz...~ the torque in Eq. (25), see block . ~ in Fig. 6. An external parallel resonator circuit modelting the rotation of the sensor system about the z axis can be connected directly to node to~. The component model has six fixed nodes, the nodes of the resonator circuits and 23 optional nodes. The capacitance nodes are optional. The model is simplified greatly when the force current terms and non-linear gap capacitances are left out if the corresponding nodes are not needed. The 28 model parameters include the constants shown in Table 2 and several other parameters needed in calculating the electrical attractive forces and gap capacitances.
4. S i m u l a t i o n
and measurement
results
4.1. Simulating with APLAC Simulation tasks in APLAC are defined as programming language style descriptions that consist of, e.g., circuit definition, simulation and output commands. In the first simulation the frequency response of the sensor, electrostatically excited at the frame, is first simulated using a.c. analysis. In the a.c. analysis the non-linear circuit is first linearized at the operating point defined by the d.c. sources. Then the response to a.c. sources is solved from the linearized circuit at each fiequency. The following circuit description in APLAC's input language generates the response shown in Fig. 7. #include "gyro.i" Gyro "gl" NvF NvB NvFy NwBx NvF NvB + MODEL "V'rlproto" + NF1 n f l NF2 nf2 Volt "ul" nfl 0 AC
NZF4 nzf4 1 DC 20
Volt "u2" nf2 0 AC -I De 20 Sweep "" + LOOP 600 FREq LOG Ik lOOk + ¥ "Amp" "dB" -100 20 + GRID BIGSCREF~ Disr!,y Y
Display
""
PHASE ""
MagDB(Vac(nZF4))
Pha(Vac(nZF4))
EndSveep The circuit described in the listing consists of one angular rate sensor model called Gyro and two voltage sources ul and u2. The model definition and its parameters are included from file gyro.i using the #include macro shown in the beginnine" of the listing. The parameters of an early prototype manufactured by VTI Hamlin Oy are specified using a
A°
APLAC7,00User:HUTCitcui!TheoryLab.Sat Sep 7 1996 ........... 1~ '~1"~ I" Pl Torsionalframe+beam It
* B't
I ii
'"'
" i '-Ilk
i l
180 PHASE
ill
Illll '°
-12e
180 I0 30 100 f [kHzl Fig. 7. Simulated frequency response of the angular rate sensor. Relative amplitude (solid line) and phase (dashed line) responses, t
3
MODEL definition specified in the included file. The sensor dimensions, masses and momenta of inertia in the model parameters are directly the design dimensions. The damping coefficients, spring constants and amplitudes of frequency responses are fitted experimentally to measurement results discussed later. In the listing three optional parameters are used in defining optional nodes for frame air gaps 1, 2 and 4. The voltage sources ul and u2 consist of a.c. components in opposite phases and symmetrical 20 V d.c. voltages. The non-symmetrical a.c. signal connected to air gaps [ t,nd 2 excites both torsional and linear modes of m3tion. The Sweep block defines the a.c. simulation and the graphical output. A logarithmic frequency sweep is requested in 600 points from ! to 100 kHz to include all resonances of the system. The lower resonance peak at 3.2 kHz is ca',sed by t.",~ vibrating system in torsior;al mode, and me resonanc~ at 9.5 kHz is due to the linear motion of both masses. At 17 kHz the linear and torsional motions compensate each other, resulting in zero movemera at gaps 3 and 4. The resonances at 54 and 72 kHz are due to frame and beam masses working at opposite phases in the linear and torsional modes, respectively. In the second simulation example the sensor sensitivity to an external constant angular velocity (to~.,~,,= 10 rad s- i) is simulated. The beam mass is actuated to vibrate at its resonance frequency, and the beam gap d.c. voltage is swept to tune its resonance frequency to match the frame's resonance. Frame mass is actuated with a biased 5 V p--p square-wave voltage applied to opposite air gaps in reverse phases. This is modelled in this a.c. analysis with a d.c. source of 2.5 V and an a.c. source equal to the fundamental component of the square wave, that is 10/~'V. The simulation results in Fig. 8 show the vibration amplitudes for the beam and the frame. Note the decrease of the frame amplitude caused by energy transfer to the beam at the resonance.
4.2. Measured and simulated capacitance-voltage curves Capacitance-voltage measurements were performed on one capacitor at time, the other seven being shorted. The
119
T. Veijolaet al. I Sensors and Actuators A 60 ¢1997) I 13-121 APLAC 7.00 Ua¢c tIUT Circuit TheoO"Lab. MortAug 26 1996
~
IO0
Amp "- ~
75
~
i
APLAC 7.00 User: llUT Ci~¢mlT~c¢~ Lab Tu¢ ,q~ 3 1996
- 180
" " ! ....
PHASE
--"
.. -
135
5O
~
22
45
32
33
34
-20
APLAC 7.00 User. HUT Circuit Theo~ Lab. Mort Sop 02 1996
2 z 1.5 [uml I
\ G.5
0
T%
I L
°-~
- - " 0 "0" -I0
0
tO
2O
Bt~s IV]
Fig. 8. Sensor response to constant rotation as a functionof beam bias voltage.Beamvibrationamplitude(solid line) and phase (dashedline) and frame(-.-) amplitude(dividedby 20) at theirreferencepositions.
-150
T
........... I...........
2,2.
35
Bi a."iIV I
-300
1
'-' "~-~----
0 3t
2,-
12
150
300
Bias IV I
Fig.9. Capacitance-voltagecharacteristicsof theframemass.The simulated (solid line) and measured([] }capacitancesandthe simulateddisplacement (dashedline}. equipment used was an HP LCR bridge 4274A with an external bias source Keithley 487 decoupled from the capacitance test signal by an external RC network. When a non-symmetrical d.c. bias voltage is applied in the air gap, the mass will be displaced. This will cause a measurable change in the frame capacitances as shown in the measured and simulated responses in Fig, 9. The bias voltage across air gap I was varied from - 3 0 0 to + 300 V. The simulated mass displacement is also shown in the Figure. A similar measurement and simulation was performed for the beam mass, shown in Fig. 10. The d.c. bias voltage source was swept in this measurement from - 2 0 to + 20 V. The measured and simulated capacitances are shown in the Figure together with the simulated displacement of the reference position. Curve fitting was performed by varying the stray capacitance only. The minimum of the C - V curve is not at zero volts but at about 0.5 V. This due to the work-function difference between the silicon beam and the Ti/AI metallization. This value is used throughout this work when comparing predictions with experimental data.
Fig. t0. Capacitance-voltagecharacteristicsof the beammass The simulated {solid line} ,andmeasured(t-'l} capactmncesand the simalaleddisplacement{dashedline). half-bridge detection on the second similar opposite electrode pair, see Fig. I 1. A I MHz test signal (Vs) was used on the bridge and the amplifier output was tuned by an inductor. Linear voltage-force (aud torque) conversion is obtained with a d.c. biPs voltage (VD and opposite phase a.c. drive. The frequency response was measured with an NF 5020 analyser. The model presented includes an output for mass displacements at each air gap and thus the simulation through the varying capacitance is not needed. This simulation could also be done with the harmonic balance techniques of APLAC, used widely in r.f. design. The low-pass filter in the meas,.a'ement circuit is modelled with a second-order Butterworth filter with a cut-off frequency of 15 kl-lz (block LP in Fig. !!}. The damping coefficients were extracted experimentally from measurement results because the gas pressure inside the sensor is unknown. The dimensions of the torsional springs were in the range where textbook formulae are not valid. The spring constants were also extracted from measured frequency responses and capacitance-voltage curves. Using the extracted parameters, an empirical design formula for the elasticity was developed. The parameter extraction was performed using APLAC's built-in optimization capabilities [161.
£ T
4.3. Measured and simulated frequency responses
Analyzer The frequency response was measured using electrostatic excitation on one top-bottom electrode pair and capacitance
Fig. 1I. Measurementset-upfor frequencyresponses.
Veijolaet al. / Sensors and Actuatorx A 60 (1997)
T.
120
First a simulated frequency response using the extracted parameters is comp,tred with measurement results of the frame mass displacement. A non-symmetrical electrostatic force signal was used to excite the mass in both linear and torsional modes. The frame mass was actuated with an a.c. voltage from 1 to 20 kHz and a symmetrical 200 V d.c. bias voltage Vh was applied to air gaps I and 2. The measured and simulated frequency responses are compared in Fig. 12. The background noise level in the measurement results is clearly seen in the frequency response. Fig. t3 shows the measured and simulated frequency responses of the beam mass. The measurement was similar to the frame a.c. response measurement with the exception that the bias voltage is 10 V. The extracted torsional and linear viscosities were used to estimate the remaining gas pressure inside the sealed structure. Applying Eq. (7) for the effective viscosity and Eq. (19) in Ref. { I I ], the estimated pressure was about 100 Pa. The measured and simulated frequency responses ;., Fig. 14 show how the increasing symmetrical d.c. bias effectively changes the spring constants through the non-linear force function Eq. (15) and shifts the beam resonance to a lower frequency. This measurement is similar to the previous one with the exception that the bias voltage is varied ( 1, 3, 10, 20 and 30 V). APLAC 40-
7.00 User: HUT Circui! Tlumry Lab. Tue Sep 3 1996 ~t ~,
Amp [a~l
- .... ~
20 .
.
.
.
.
.
-
%
.
.
. PHASE
.
90
'il
I ,-180
J143-
1
- 180 I
10
3
[va~z! Fig. 12. Relativeamplitude(solid line) and phase (dashedline) responses of simulatedand measured([]) frequencyresponsesof framemass. f
APLAC "/,IX)U~t-:.HUT Ci~uit Theov/Lab, Tue Sop 3 1996 Amp
o
PHASE
Ill N,,o 1o
f [kHzl Fig. 13. Relative a m p l i t u d e ( s o l i d line) and phase (dashed line) responses of simulated and measured (I-q) frequency responses o f b e a m mass.
50Amp taB] 37.5.
113-121
APLAC 7.00 User: llt/T Citcuil Theory Lab. Sol $ep 7 1996
A3°"i
/
20V
J \A
12.5-
02
3
4
5
6
f lkHz]
Fig. 14. Simulated(--) and measured([]) frequencyresponseof the beam masswithdifferentbias voltages.
5. Conclusions The complex etectromechanical interactions present, in a microelectromechanical system (case vibrating angular rate sensor) can be effectively modelled and analysed with a circuit simulation program. Our simulations gave results which are close to the experimental data. Non-trivial side effects like the change of the effective spring constant and resonance frequency by a bias voltage were also correctly predicted. The model was used here also to extract values for parameters that could not be accurately estimated in the design. With the model presented a large rumber of different simulations can be performed on the sensor combined with external electrical or mechanical equivalent circuits. All model parameters can be variables in an optimization process [ 161, where unknown model parameters are extracted by comparing simulation results with measurements. A statistical simulation (Monte Carlo), where the tolerances of the parameters are included, can predict the variations in sensor response. Temperature dependencies can also be simulated if the model parameters are written as a function of temperature. The model presented covers the standard operation conditions of the sensor. The cases where the mechanical or electrical actuation is so large that one of the vibrating masses will hit the sensor body are not included in this model. Geometrical non-idealities like non-orthogonality and asymmetry that cause mechanical cross-talk are not included in the model. It is rather straightforward to add these couplings between the two modes. Electrical cross-talk and stray capacitances can be added external to the model. The gas damping is modelled with separate conductances in the resonance circuits. This is justified when the displacement and lhe gas p~essure are small. A model that is valid for higher pressures and displacements consists of a non-linear damping circuit, as presented in Ref. [ 1 ! 1, common to all modes of motion. The higher-order vibrational modes included in this model did not show any importance in the simulations, verifying the successful device design. However, if this modelling approach is applied to other similar microelectromechanical
T. Veilolaet al. / Sensors and Actuawrs A 60 f1997) 113-121
systems, the higher-order modes can reveal mismatch between the torsional and flexural stiffness of the supporting springs.
References [ I ] T.P. Mathues, Extending the scope of ABS, Automotive Eng.. 102 (1994) 15-17. [2] J. S6derkvist, Micromachined gyroscopes,Sensors and Actuators A, 43 (1994) 65-71. [ 31 J.B. Chatterton, Some general comparisonsbetweenthe vibratoryand conventional rate gyro, J. Aeronaut Sci., 9 (1955) 633--638. [41 A. Reppichand R. Willig, Yaw rate sensorfor vehicledynamicscontro! system, SAE Paper 950537, 1995, pp. 67~76. [51 T. Ichinose and J. Terada, Angular rate sensor for automotive applicat,on, SAE Paper 950535, 1995,pp. 49-56. [6] S.D.Orloskyand HD. Morris, A quartz rotational ratesensor,Semors, 2 (1995) 27-31. 171 J. Berastein, S. Cho, A.T. King, A. Kourepenis, P. Maciel and M. Weinberg, A micromachinedcomb-drivetuning fork l,..e gyroscope. Proc. MEMS '93, Fort Landerdale, FL, US4, 1993, pp. 143-148. [ 81 J.D. Johnson. S.R. Zarabadiand D.R. Sparks, Surfacemicromachined angular rate sensor, SAE Paper 950538. 1995,pp. 77-83. [91 T. Veijola, Accelerometermodel in APLAC, Helsinki Universityof Technology, Circuit Theory Laboratory,Report CT-18, Feb. 1994. [ 101 T. Veijola and T. Ryhtinen, Model of capacitive micromechanical accelerometerincluding effect of squeezed gas film, Proc. ISCAS "95, Seattle. WA, USA. 1995, Vol. !. pp. 664--667. [ I I 1 T. Veijola, H. Kuisma, J. Lahdenpe~ and T. Ryhanen, Equivalent circuit model of the squeezed gas film in a silicon accelerometer, Sensors and Actuators A, 48 (1995) 239-248. [121 M. Valtonen et al., APLAC ~ An Object-Oriented Analog Circuit SimulatorandDesign Tool, HelsinkiUniversityofTechnology,Circuit Theory Laboratory and Nokia Research Center. Hardware Design Technokgy, 7.0 Re[erenceManual and 7.0 User's Manual, Otaniemi, Jan. 19t36. [ 13] E. Miihlenfeld, Rotor-Schwingkreisel zur Messung der Winkelgeschwindigkeit, lngenieur-Archiv, Vol, 38, Springier, Berlin, 1969, pp. 58-66. [ 141 A.L. Fetter and JD. Walecka, Theoretical Mechanics of Particles and Cm~tinua, McGraw-Hill,New York, t980. [15] P. Heikkilti, Object-oriented approach to numerical circuit analysis, PhD. Thesis, Helsinki Universityof Technology,Jan. 1992. [ 16] R. Niutanen, M. Valtonen and K. Mannersalo, Optimization methods in APLAC, Helsinki University of Technology. Circuit Theory Laboratory,Report CT-15, Dec. 1992.
Biographies Timo Veijola was born in Helsinki on February 4, 1954. He received the Diploma Engineer (M.Sc.) and the Licen-
121
tiate in Technology degrees in electrical engineering from the Helsinki University of Technology, Finland, in 1980 and 1986, respectively. Since 1981 he has been working as laboratory manager in the Circuit Theory Laboratory at the Helsinki University of Technology, where he has been a member of the APLAC development team since 1988. His current research interests are computer-aided circuit simulation and modelling of electromechanical and electrothermal devices and systems. Heikki Kuisma was born in Kalvola on August 3, 1953. He received the Diploma Engineer degree (M.Sc.) in electrical engineering from the Helsinki University of Technology, Finland, in 1978. Since 1980 he has been working at Vaisala Oy and Vaisala Technologies Inc. doing research and development on silicon micromechanics and microsensors. He holds several patents on silicon microsensors. Currently he is leading the sensor research group at VTI Hamlin Oy (formerly Vaisala Technologies, lnc). Juha Lahdenperd was born in Liperi on March 30, 1959. He received the Diploma Engineer degree (M.Sc.) in applied physics from the Helsinki University of Technology in 1986. Since 1986 he has been working as a research and development engineer at Vaisala Oy working in micromechanical pressure and acceleratinn sensor development and from I991 on at VTI Hamlin Oy (formerly Vaisala Technologies, Inc). His current activities are the design of micromechanical acceleration sensors and micromeehanieal process development. Tapani Rvhdnen was born in Heisinki on lu!)' 7, 1959. He received the Diploma Engineer degree (M.Sc.) in technical physics and the Doctor of Technology degree in applied dectronics from the Helsinki University of Technology, Finland in 1986 and 1992, respectively. He has authored several publications on the theory, design and characterization of ultra-low-noise superconducting thin-film magnetometers. From 1992 to 1995 he worked as a research and development engineer at Vaisala Technologies Inc. in design of micromechanical pressure and rotation rate sensors, development of new measurement systems for microm,:chanical sensors and the modelling of micromechanical structures. In I995 he joined Nokia Telecommunications as a research and development manager.