Informacije MIDEM 27(1997)1, Ljubljana
UDK621.3:(53-l-54 + 621 +66), ISSN0352-9045
!. Zelinka, J. Diaci*, V. Kune, L. Trontelj, Faculty of Electrical Engineering, University of Ljubljana, Slovenia ^Faculty of Mechanical Engineering, University of Ljubljana, Slovenia
Keywords- MST MicroSysTems, definitions, simulations, nondifferential capacitive measurements, bipolar measuring ranges, capacitive micromechanical'sensors, CAST, Custom Application Specific Technology, development trends, mechanical analysis, SPICE model, actuating capacitors, measuring capacitors
Abstract- In the paper Microsystem (MST) definition and development trends are described. Modeling of a capacitive micromehanical sensor is presented. Verification of dynamical behavior is analized. Mechanical analysis and the SPICE model of the mechanical part of the sensor are shown.
Ključne besede- MST mikrosistemi, definicije, simulacije, meritve kapacitivne nediferencialne, območja merilna dvosmerna, senzorji mikromehanski kapacitivni, CAST tehnologija specifična uporabniško aplikacijska, smeri razvoja, analiza mehanska, SPICE model, kondenzatorji aktivatorski, kondenzatorji merilni
Povzetek' Opisana je definicija mikrosistema (MST) in razvojni trendi. Prikazano je modeliranje kapacitivnega mikromehanskega senzorja in analizirano je dinamično obnašanje sistema. Podana je mehanska analiza in SPICE model mehankega dela senzorja.
1 INTRODUCTION
The basic difference between ICs and microsystems is sliown on Fig, 1. Wliile ICs mostly handle information, MSTs usually deal with energy. They always represent the complete system required to perform the desired function.
CQniniunicaiion
üglu Pressure
a tu re	
	Sensors
i	
Signal Processing
Vkchatucal
Electrical
Optical
Packaged MST
Fig. 1: Microsystem definition
The introduction of microsystems followed the same basic rules which promoted the development of ICs. They are small and require low power. A large number of them can be manufactured simultaneously, thus offering lower costs and greater reproducibility. In addition, the ratio of performance versus price is far superior to that of the lumped versions.
Two basic differences in comparing ICs and MSTs are essential: only few atoms are required to handle information in a well optimized and carefully designed IC, while the dimension of MST depends on the amount of energy to be manipulated. Therefore the same scaling rules as well as Moore's Law do not apply.
Development trends of ICs are still widely governed by the development of optical lithography. We see the advent of 0.18 jim custom application specific technology (CAST) for volume production and a substantial increase of the diameter of silicon wafers. Tools for the development of photoplates capable to be used together with the advanced imaging techniques are emerging.
On the contrary, the smaller and finer geometries in MSTs are not vital or even possible considering the amount of energy to be handled in specific application. Therefore the MST related activities are reserved for those environments of design and production which are not able to compete in the every day financially more demanding new equipment procurement and refined fab environment associated with the deep submicron technologies. Therefore, it is viable that the Laboratory for Microelectronics (LMFE) aggressively entered the new exciting field of MSTs, offering new applications in the fields of data storage, displays, communications, IR imaging, biochips, micromachines, and microinstruments.
Although there exist remarkable simulation tools, which offer great support to a designer confronted with specific design problems in the field of electrical/electronic or mechanical engineering, there's a very acute lack of simulation software which would allow efficient solutions to coupled electromechanical problems, which are commonly encountered in the field of MSTs. The gap between the two engineering disciplines seems to be too large in any practical situation requiring a solution of coupled electromechanical problems to allow a
microsystems designer to benefit from a coherent use of existing mechanical and electronic design packages.
Different schemes exist to construct a micromechanical part of the sensor. However, one which uses a cantilever seems to be the most promising, offering the largest sensitivity for a given size /2/. In the paper we present a non differential capacitive MST sensor which also has definite production advantages over the two capacitor version, but it requires more effort to model it properly. We have adapted the equations describing the micromechanical part of the sensor in a form acceptable as an input to the standard electronic analog simulator. This gives us the ability of prediction of a closed loop behavior of both parts of the system.
In the paper we present the analysis and modeling for the chosen MST.
The elastic element of the sensor acts as one piate of the sensing and actuating capacitor. Deformation of the elastic element, due to external loads (related to the measured physical quantities), are counteracted by the electronic servosystem, which consists of a capacitive sensor, actuator and signal processing electronics. In the dynamic equilibrium, the actuating electrical force equals the external load. From the parameters influencing the actuating force the external load and the related physical quantities can be determined.
2 STATIC ANALYSIS
The configuration of a single capacitor model is on Fig. 2.
D !

actuating and measuring capacitor
d i

"^rr
Fig. 2: Cantilever with one capacitor for actuating and measuring
The basic equation for deflection w of the cantilever loaded with distributed load q is /3, 4/:
tr, d'w dx^
(1)
where E is Young's modulus, I is the area moment of inertia, w=w(x) and x is measured from the tip towards the clamped end of the beam. The boundary conditions are: at clamped end: w(L)=0 w'(L) = 0 at free end: w"(0)=0 w"'(0)=0
For the special case of point force F load we take q = F5(0). The deflection of the beam depends on loads. We
consider the beam (cantilever) loaded with one distributed external load qa, and point electrostatic force Fe. We can assume electrostatic force as a point force if capacitor length (U) is less than 10% of beam length (L) /5/.
For distributed load Wq or point force wp we have the following equations describing the deflection /3/:
wjx) = -
Wp x =
qL^	3-4		-h	
24EI		.L,		UJ
PL3				'xV
	2-3		4-	
6EI		J.		
(2)
(3)
Of special importance for the analysis are the deflections of the beam tip:
w,(0) = Wp(0) =
8EI
fk' 3EI
(4)
(5)
According to the principle of superposition, the total deflection w(x) under combined loads is the sum of the two contributions:
w(x) = Wq(x) + Wp(x)
(6)
In order to examine the stability of the system, we assume the beam loaded with one distributed load qa and point electrostatic force Fe. With introducing new variables kF=3EI/L2 and kq = 8EI/L2, which represent stiffness of the beam, we can write eq.(6) for the deflection of the beam tip
Wo = Wf
- w„
fE ^qJi kp k.
(7)
In general, Fe is a sum of the electrostatic forces of actuation and measuring. Therefore this equation is valid for an open loop system (no actuation voltage, electrostatic force only due to read-out voltage) and for a closed loop (voltage driven) with one or two capacitors. We seek solutions from the above equation for wo subject to the obvious restriction wo < h.
By inserting
FE=Fe
(h^Wo
(8)
where Fe=1/2 e A (U/h)^ in eq.(7) and by introducing dimensionless variables

K =
kph '
W„ =
ma
we can write eq.(7) in form
Wo=K
(1-Wo)
■W„
(9)
There is exactly one value Wo5 where we can fit the required load range amin<a<amax without any waste of the dynamic range. To calculate Wo5, we equate the lower and the higher bound value for Wo in eq.(12) and insert the ratio: ra=amin/amax=Wa/Wa. The result is:
To find the solution, we rewrite the above equation in the form of a cubic:
Wo' - (W„ + + (1 + 2W„ )Wo = K + W„ (10)
With finding the maximum of the l.h.s. of expression (10) we get the stability limits:
^ K<±(1-Wj (11)
The system is stable when either of the above inequalities hold. With the additional condition K>0 we get a range of possible solutions:
W„.<W„<Iif^ o w„<w„<^(12)
Wo5=1/(3-2r„,
(13)
Example: for the symmetrical bipolar range we have ra=-1 and Wo5=1/5.
If we select our Wo above Wo8 , then the corresponding Wa value is smaller than optimal and we have to increase the beam stiffness/mass ratio kq/m (increase thickness D) to be able to measure the required amin. We will waste some dynamic range on the positive side then, because the corresponding mass, stiffness and Wcxmax would allow the measurement of higher max. load than required by amax- The opposite happens when we select Wo below the optimal value. We have to design beam thickness according to amax and thus waste some dynamic range below amin. We can summarize this discussion with the following formulae:
Fig. 3:
0,2 OA Wo - rel. preload deflection
Re!, measured load (Wa) vs. rel. preloading (Wo) - shaded area shows the useful range for bipolar measurement
Wo>WO5:W„„,=(3W„-1)/2=>D =
3pLV.„
^Eh(3Wo-1)
W<W'W =W„	=
^ "'OS- '"amax "'O
I 2EhW„
(14)
(15)
There are, of course, practical limitations to D; therefore we shouldn't expect to be able to realize the beam when the selected Wo is close to 1/3 or 0.
3 DYNAMIC ANALYSIS
In general, we can have a system with two capacitors that are not located on the same plane. For the analysis of dynamic behavior, we use the system configuration shown on Fig.(4).
The capacitor sizes estimated by using the point capacitor models show that in practice both capacitors (meas-
POSITIONING OF THE BEAM
Since the electrostatic force can not change polarity, we have to preload the beam with a static actuating force Feo and move the beam tip by wo below its initial distance h if we want to measure load in both directions. The stability diagram on Fig.(3) shows that in principle we can perform bipolar measurement for any relative preloading in the range 0<Wo<1/3. To accommodate the required range amin<a<amax at a selected Wo we have to select the appropriate beam stiffness (via thickness change). We can select Wo according to different criteria. We will take a closer look at just one - Maximization of dynamic range.
imrnmsmsff'^^
Fig. 4:Cantilever with two distributed capacitors
uring and actuating) should have lengths larger than 10% of the length of the beam. For this size, the inter-plate spacing variations within each capacitor are not negligible and we have to treat the capacitors as distributed along the beam. We model the electrostatic forces as distributed loads:

2
h(x)-w(x]
with: n(x,L,„Li2) =
f1
0 elsewhere
for:L;, <x<Li2
(16)
where i = a for the actuating and i = c for the measuring capacitor. B and Uri represent the width of the beam and applied voltage respectively. For the analysis of dynamic behavior dumping and moment of inertia have to be considered also. Inserting the two loads into the basic eq.(1) we describe the deflection w(x,t) of the beam by the following boundary problem:
is the elevation of the measuring capacitor plate above the actuator plate.
The damping force qd(w,w,x,t) is strictly speaking the solution of a special squeezed-film air-flow boundary problem. We find that it would be quite impractical to try to solve it by means of electrical analogies using an electronic simulation codes, such as SPICE. Instead of that, we suggest the use of an approximate analytical solution of the varying gap squeezed-film boundary problem /6, 7/:
q^ (w, w, X, t) = b( w, x) w(x, t)
where
b(w,x) = 12|aBL'
(20)
cosh(7l2(L-x)/B cosh(Vl 2L/B
h(x)-w(x,t)
(21)
pDBw(x, t) + q^ (w, w, X, t) -F El
3V(x,t)
= qa(w,x,t) + q,(w,x,t)-t-q„(t)	(17)
w(L,t) =0 w'(L,t) =0 w"(0,t) =0 w"'(0,t) =0
where pDB w(x,t) represents the moment of inertia and qd(w,w,x,t) dumping, qa and qc are substituted with eq.(16) and ma(t)/L=pDBa(t).
Electrical inputs to the mechanical part (from the electronic part of sensor) are voltages on actuating capacitor (Ua(t)) and measuring capacitor (Uc(t)). The output for the electronic part is measuring capacitance C(t) of the air gap capacitor:
4 MODELING
Usually approaches to modeling /8,9,10/ are based on substituting mechanical elements with equivalent electrical elements (Fig. 5).
(a)
(b)
			
			
		f	
C(Q)
Fig. 5: Mechanical system and (b) equivalent eiec-trical circuit

(18)
For solving the upper equation we need function h(x), describing the initial form of the beam. Assuming a uniform distribution of the residual stress along the beam results in a parabolic form and the height function is defined as:
h(x) = h, + (h„ - h J(1 - X / Lf - h„,n(x,L,„L,
(19)
The equation describing a single mass mechanical system
mx -f bx -t- kx = Fg -I- F(, H- q
(22)
is similar to the equation describing electrical circuit on Fig. (5b)
L-^-t-Ri + -Jidt = V dt C^
where ho and hi are the initial heights of the tip and the clamped end respectively, measured relative to the base line, defined by the actuator plate, and where hac
where
x = i x^Jidt m = L b^R k = f
The equation describing the behavior of the cantilever beam (eq. 17) can be substituted with eq.(22) only when the mechanical system with the single degree of freedom is assumed. Modeling (substitution) of our cantilever beam with single mass mechanical system on Fig.(5a) does not give satisfactory accuracy.
Our modeling of differential equation follows the work of Herbert /11/ and Pelz /12, 13/. All state variables in the equation, e.g. velocity, deflection are represented by node voltages.
Nonlinear dependent voltage controlled voltage sources are used to determine the state variables of the highest time derivatives and the algebraic equations. Simple integrators calculate the values for the lower derivatives.
For example, the equation for distributed load qa (load due to electrostatic force of actuating capacitor) from eq.(16) can be written in HSPICE by using Behavioral voltage source in form:
E_q_a q_a 0 vol = '((0.5*eps*B^(v(Urefa)**2))/ /((v(hx)-v(w))**2))'
where eps and B are defined as parameters. In this way, systems of algebraic and ordinary differential equations can be solved.
The output from the mechanical part and the input for the electronic part is capacitance of the measuring capacitor:
dx

(24)
In order to solve the upper equation, the initial form (height) of cantilever h(x) has to be known and the deflection w(x,t) calculated. For h(x) we take eq.(19) where hac=0
h(x) = hL+ (ho-hL)(1 -x/L)'
(25)
To get the w(x,t) we have to solve the boundary problem in the eq. (17), which we rewrite in order to get the highest time derivative on the l.h.s.
w(x,t) = -
El rrw(x,t) pDB dx"
1
pDB
q,(w,w,x,t)
/14, 15/, the spatial variables of partial differential equations are discretized and an algebraic equation is inserted for each node. Discretization schemes for spatial derivatives up to the fourth order /14, 15/ are:
ax	- 12h
a^w(x,t)	1
	12h
3V(x,t)	1
ax^	12h
a'w(x,t)	1
Xi-2
-w,,^^^
(27)

(28)


(29)
(30)
Each equation describes the behavior of the respective slot i, by regarding itself and some of its neighbours in both directions. In our case (eq. 26), w(x,t) is the function to be derived, and x is the spatial variable (for 0 < x < L). The descretization step is h, and n is the number of discretization steps (h = L/n).
All equations containing the term d^\N{x,t)/d x*^, where k is the order of the derivative, have to be duplicated n times. The number of discretization steps n is the number of nodes. Eq. (26) for slot i is written in form for HSPICE using E source:
E_i w_i_tt 0 vol =
■(-konsti *(v(w_(i-2))-4*v(w_(i-1)) + -f-6*v(wJ)-4*v(w_(i + 1))+v(w_(i-l-2)))
-konst2*v(q_d_i) 4- konst2*v(q_a_i) 4--f-konst2*v(q_cJ)-t-v(a))'
Values for konsti ,konst2 are calculated and defined as parameters. Deflection w(x,t) is calculated with a simple integrator:
XINTEGRATOR_bi wj_tt wJJ INTEGRATOR XiNTEGRATOR_ai w_i_t w_i INTEGRATOR Index i represents respective slot (0 < i < n)
+ ^q„(w,x,t) + ^q,(w,x,t) + a(t) (26)
The proposed method of modeling does not allow direct modeling of partial differential equations or integration over spatial variable. With the implementation of some mathematical approximations, we can extend this work and solve the system of equations describing our cantilever beam. With the method of finite differences (EDM)
i-2 i-1
1+1 i+2
Fig. 6: Discretization of spatiai variabie x
■990.0
400,0
380,0
300.0
?80.0
200.0
ISO.O
0.0 BO.O
WJD
Fig. 7: Deflection of the beam for different discretization nodes
C.aoc
70.0;
M.Oi
tB.O,
U.Ol

w.o
'so.o '	100.0	150.0	200.0	Z80.0	300.0	380.0	400.0
tlM	»S
Fig. 8: Cfianging of measuring capacitance with changing of ioad
■48«.0	BOO.O	880.0	«00,0
Spatial integrals are also discretized and substituted with algebraic equations following the Simpson's rule;
f ydx = -(Yo + 4y, + 2y2 + Ay	+ + yj
(31)
where h = (b-a)/n, with additional condition that n is an even number. So eq.(24) for capacitance of an air gap capacitor in form for HSPICE is as follows:
E_C_acc C_acc 0 vol = 'epsilon*B*( 1/(v(hx_0) -v(w_0)) + 4/(v(hx_1) - v(w_1)) + 2/(v(hx_2) - v(w__2)) + 4/(v(hx_3) - v(w_3)) + 1/(v(hx_4) -v(w_4)))'
The exact form of equation depends on the length of the measuring capacitor and the number of discretizations steps.
5 RESULTS
The results are shown on Fig. (7) and (8).
The calculation of the deflection for each node of discretized beam can be seen on Fig. (7) and the resulting capacitance of an air gap capacitor as an output of micromechanical part of sensor on fig. (8). The voltage representing capacitance can be transformed back to capacitance as input for the electronic part with use of Voltage controlled Capacitor.
/3/ E. P. Popov: "Engineering Mechanics of Solids", rentice Hall,
New Jersey, 1990 /4/ Marko Škerij: "Mehanika - trdnost", Fakulteta za strojništvo, Ljubljana, 1971
/5/ Francis Westo Sears: "Electricity and Magnetism", Addison-
Wesiey publ., 1958 /6/ K. Hacki, Technnische Univ. Graz, 1995, unpublished /7/ J. Diaci, Fakulteta za strojništvo, 1995, unpublished /8/ S. Marco, J. Samitier, O. Ruiz, A, Herms, J.R. Morante: "Analysis of electrostatic-damped piezoresistive silicon acceierome-ter". Sensors and Actuators, A, 37-38, pp. 317-322, 1993
/9/ J. Dominicus, i. Jntema, H.A.C, Tilmans: "Static and dynamic aspects of an air-gap capacitor". Sensors and Actuators, A35, 1992, pp. 121-128 /10/ D.E. Bergfried, B. Mattes, R. Rutz: "Electronic Crash Sensors for Restraint Systems", Proc. Int. Cong, on Transportation Electronics, Detroit, 1990, pp. 169-177 /11/ D.B. Herbert: "Simulating Differential Equations with SPICE2", Simulation and Modeling, Editor: Ping Yang, Circuit & Devices, Jan. 1992, pp. 11-14 /12/G. Pelz, J. Bielefeld, F.J. Zappe, G. Zimmer: "Simulating Micro-Electromechanical Systems", Circuits & Devices, March 1995, pp. 10-13 /13/ G. Pelz, J. Bielefeld, G. Zimmer: "Model Transformation for Coupled Electro-mechanical Simulation in an Electronics Simulator"
/14/ G. Pelz, J. Bielefeld, F.J. Zappe: "MEXEL-Model-Conversion:
Mechanics to Electronics", April 1995 /15/R.M. Gutkowski: "Structures, Fundamental Theory and Behaviour", Van Nostrand Reinhold Co., New York, 1981
6 CONCLUSIONS
The accuracy of a single mass model is not satisfactory for the selected micromechanical sensor. With the implementation of mathematical substitutions, we developed a model for a system with distributed mass and analysed the behaviour of the sensor with SPICES and HSPICE simulator.
A comparison of the results acquired by the simulation with HSPICE to those of the MATLAB shows, that an error introduced with mathematical substitutions is one order of magnitude smaller than the resolution of the sensor.
The described model allows us to predict the behavior of the micromechanical part, and to simulate close loop measurements.
References
/1/ L.Hermans: "Trends in Microsystems", IMEC, ARRM 1996 /2/ Ljubisa Ristic (editor): "Sensor Technology and Devices", Artech House, 1994
doc.dr. Janez Diaci, dipl. ing.
Facuity of Mechanical Engineering Aškerčeva 25, 1000 Ljubljana, Slovenia Tel.: +386 61 1771 429, Fax. : +386 61 218 567 e-maii: janez. diaci@fs.uni-lj. si
prof.dr. Lojze Trontelj, dipl. ing. dr. Vinko Kune, dipl. ing. e-mail: vinko@im.eunet.si dr. Igor Zelinka, dipl. ing. e-maii: zigor@lm.euneLsi Faculty of Electrical Engineering Tržaška 25, 1000 Ljubljana, Slovenia Tel.: +386 61 1768 337 Fax. : +386 61 126 46 44
Prispelo (Arrived): 06.02.1997 Sprejeto (Accepted): 25.02.1997