Control of a drop-ejector used as photo-resist dispenserP.—E. Rochea, A. Hanssonb and B. T. KhuriYakubL

a L. Ginzton Laboratory, Stanford UniversityStanford, CA 94305, USA

binformation Systems Laboratory, Stanford UniversityStanford, CA 94305—9510, USA

ABSTRACTPhoto-resist dispensers traditionally apply a generous amount of resist on the wafer and then spin the wafer to reacha uniform desired thickness. With this technique, over 95% of expensive and hazardous liquid is wasted. The goalof this project is to reduce the waste by using a drop-on-demand ejection technology to apply the photo-resist. Inpractice, a controller turns out to be necessary to compensate for the variability of the performance of the ejectors,to insure the stability of the ejection, and to speed up the transient regime for drop-on-demand operation. Thispaper reports on the design and simulation of this controller.

Keywords: Smart Structures, Control, Piezoelectric, Drop-Ejector, Modeling, Non-Linear

1. INTRODUCTIONA key step in ICs' and nanostructures' fabrication is photo-resist coating. Present technology for this coating isverywasteful and does not allow spatial discrimination of the coated region. In this work is explored an alternativewayof performing this task by adapting drop-ejection technology. This technology is currently used for printing.1

The ejector is built around a circular membrane glued at the bottom end of a cylinder. The resulting cavityacts as a photo-resist tank. When the membrane is perforated at its center and vibrated at resonance, drops ofphoto-resist are ejected. A flat piezoelectric ring, glued on the bottom side of the membrane and driven by a voltagesource provides the membrane actuation. The ejector has previously been described in a couple of references.2'3 Amicro-machined version is also being developed.4'5 Similar devices have been developed by others.68 Research onpiezoelectric flexural transducers has a long history.9

Due to imprecise manufacturing and aging there is a certain variability in the behavior of the micro-machined/macrodrop-ejectors. The objective is to use feedback control to compensate for this variability. A crucial point is to obtainan adequate model suitable for model based controller design. The model used is a third order non-linear differentialequation,1° where especially the non-linear part is uncertain. It will be seen that is possible to synthesize a controllerthat makes the closed loop robust with respect to the uncertainty. The controller has been implemented in analogelectronics and successfully tested on a drop-ejector.

The dynamics and control of piezoelectric laminated circular plates is an active are of ' Piezoelectricactuators are also used for control in other areas such as gust alleviation control of wings,'4 position and force controlfor micro-robotics,15 and vibration control of flexible space structures,16 flexible manipulators,'7"8 and cantilever

19

Tue remaining part of the paper is organized as follows. In Section 2 the model is derived. In Section 3 thecontroller is synthesized and evaluated in simulations. Finally, in Section 4 some concluding remarks together withsuggestions for future research are given.

P.-E. Roche was on leave from CRTBT CNRS - 25 avenue des Martyrs, BP 166 - 38042 Grenoble Cedex 9, France.Send correspondence to Anders Hansson, E-mail: andersh©isl.stanford.edu.

Part of the SPIE Conference on Mathematics and Control in Smart Structures • San Diego, California . March 1998446 SPIE Vol. 3323 . 0277-786X/98/$10.oo

2. MODELThe ejector has been characterized experimentally under realistic conditions by itselectrical and mechanical frequency

response within the frequency range 3 kHz—30 kHz and with a driving voltage amplitude of 10—20 Volts. It appearsthat its behavior is mostly linear and can be viewed as a succession of responses of isolated modes. For ejection

purposes, the device is operated at one of its peaks of resonance. For controller design it is sufficient to model the

response around this single resonance peak. Similar devices have been modeled in the literature.20'21

A schematic figure of the drop-ejecting device and the driving electronics is shown in tothe left in Figure 1 . The

driving voltage is denoted by u, the voltage over the piezoelectric by y, and the displacement of the membrane by z.

It is assumed that the driving electronics has an impedance of R . Then y , and u are related to the current i coming

from the input voltage viaU = R2i + y

It is assumed that the drop-ejector dynamics in response to an applied voltage is notaffected by the formation and

ejection of drops. This assumption is justified by the small amount of kineticand potential (surface tension) energyassociated with each drop in comparison with the kinetic and potential (elastic) energy exchanged duringoscillations

of the loaded membrane and the amount of energy dissipated during one oscillation of theloaded membrane.

When a voltage y is applied across the piezoelectric ring, two effects take place. Electrical energy is stored

between the electrodes as in any capacitance, denoted by C0, and a strain field proportional to the applied voltageis generated in the piezoelectric layer and forces the membrane to flex.

Around each of its modes of resonance, the structure behaves like a one-dimensional oscillator and can be modeledwith an equivalent spring, clamped at one end and loaded by a point mass at its other end. The displacementof the

mass equals the displacement at the center of the membrane. The underlying assumption is that the mode shapedoes not change within the frequency window of interest. This is a fair assumption around the resonance of isolated

modesThe damping is supposed to be linear. Recall that the damping associated with the loss of dropshas been

neglected. Physically the main loss mechanisms are mechanical: damping in the fluid and in the bulk of themembrane, and damping and radiation in the support. Measurements of the real part of theelectrical impedanceof the devices confirm that the electrical losses are negligible. Even if the damping is not viscous, i.e. dampingforce proportional to the velocity, it can always be approximated by a viscous one within a limited frequency range.The quality factor of the devices being above say 50, this viscous hypothesis is always a fair assumption within theresonance bandwidth of an isolated mode.

The distributed force generated on the structure, and consequently the equivalent point force f on the spring-massequivalent system, is proportional to the voltage y across the piezoelectric layer, i.e. f = cy for some coefficient u.

+U

+ ye

+y

II

+U

Figure 1. Ejecting device and driving source (left) and electrical equivalent circuit (right).

+VL

447

The dynamical equation for z is:

d2z dzm+dT+kz=f (1)

where m is the point mass of the equivalent oscillator, k is the stiffness of the equivalent spring, and d is the viscousdamping coefficient.

It is assumed that the force acting on the structure is such that the power transmitted to the mechanical oscillatoris fully provided by the electrical input. The conservation of energy can then be written

. dy dz dy dzyi = yCo + f = yCo + yand after division by y

. dy dzi=Co+a (2)

where i is the current through the electrical input, yCo is the power in the capacitance and f =cy the powerstored/dissipated in the mechanical oscillator:

The large deflection of the membrane during its resonant operation entails some non-linearities in its elasticbehavior. These non-linearities are assumed to be square and cubic terms in the expression of the spring elastic forceas a function of displacement. They can be added to the model by replacing kz with kz + k2z2 + k3z3.

It is possible to derive an equivalent electric circuit by replacing the mechanical oscillator with an electricaloscillator consisting of a resistance R = d/a2, an inductance L = rn/a2 and a capacitance C = a2/k in series as seento the right in Figure 1. Then with

a . dzv = =

where v is the voltage over the capacitor and i1 is the current through it, it holds thatdx = Ax + Bu + Bwy = Cxz=Czx

w = 0(z)where

o 0 0 0A= — — ; B= 0 ; B= —1; x=

0 —- 0 0 y

C = [0 0 1]; C = 0 0]; 0(z) = - (kiz2 + k2z3)

448

Figure 2. Open loop system block diagram (left), and closed loop system block diagram (right).

Figure 3. Bode diagrams for the open loop system.

The parameter values for the devise used are given by R = 5Oohm, Co = O.954nF, R = 20270ohm, L = 28.111,C = 18.3pF, and a = O.0019N/V. Define o = 1/V'L = 44.lrad/s and the Q = Lwo/R = 61.1 which are goodapproximations of the resonance angular frequency and quality factor. The model can also be writtenin terms of its

transfer functions as

z _ P1 wl [C A — rB B Wy Pw Pyuj U]

—

Lc] (s — ) i w uj

w=Ø(z)

which is depicted in Figure 2. This is the standard form for robustness analysis.22 The Bode diagrams for thetransfer functions are shown in Figure 3.

Simulations with u(t) = 6.82 sin(wot)V for the case when k1/(oL) = k2/(cL) = 0 and for the case whenk1 /(cL) = 0 and k2/(aL) = 0.8 x 1016 are shown in Figure 4. Notice how the nonlinear term yields a loweramplitude.

3. CONTROLLERIn this section the controller will be designed. It is composed of a non-liner observer in conjunction with linear statefeedback from the estimated states. The design has not been optimized, but this could be done using results fromH control theory together with harmonic balance analysis.

449

I

4. 4.1$

F,sq.ncy(r.Wse) F,qusncy(r.W3.C)

-001

-0.015

-002

—0.025

Faqw0y(redl$eC)

2

Ereqancy(sd/.c)

:—: ___________ -0005 001 0015 0 0005 001 0015

.me (s) I,mo e)

10riiiiii.i

001 00150 0005 001 00,5 0 0005 001 0015 0 0005h,ne (s) l,m () Sn',e (e)

Figure 4. Simulation of the open ioop system when w = 0 (left) and when o 0 (right).

The observer for estimating the state z with is given by

=A±+ Bu + B11,n + K(y -

th =

where the estinate of the nonlinear term is given by

Not ice that the estimation error satisfies

All stabilizing controllers are given by

= + /32Z

=

to — L11.i + V

V =

= y —

= +&r +

ti' =

>

10 r—

'H'—5

-Ic0005 001

I,mo (s0005 001 0015 0

t,me (e)

C I0

where Q is a stable transfer function with realization

= AQ + BQJ

v = CQ + DQ

The closed ioop system is then given by

450

Nyqust Diagrams

0.3

02

-0.2

—0.4-.1 -0.5 c,.4

Figure 5. Nyquist plot of the ioop transfer function C(s)P(s).

where— [xT eT T]T ; = [wT

T

A -BL CQ BU(LU + DQC)A= 0 AQ BQC

0 0 A-KCBuir B 0L= 0 ; E= 0 00 B-B

c_1cz 0 0 • -1(z)z —LCZ ° —Cz ' (z) - [O()

The closed ioop system is depicted in Figure 2, where P is defined via

= [zw ] [] =C(sI-A)' ELAt this point H controller design theory together with harmonic balance analysis could be used to synthesize arobust controller. This will be the focus of future work. Here a more simplistic design method based on LinearQuadratic (LQ) control theory and Kalman filter theory will be used.23

If the nonlinear term in the controller is neglected, i.e. th = 0, then with A0 = A —BL — KC it holds that

d- = A0x+B1r + KyU = + lrr

The matrices L and K are chosen as

r —u

K = PCfr1

where S and P solve the two algebraic Riccati equations

ATS + SA — SBq1B7'S+ q1C"C =0PAT + AP - PCr'CP+ r11 =0

451

-0.6 -0.4 -0.2 0 0.2Raal Axis

L±T TiC(k)

Frsqusncy (r,&,.c)

1o 1o 1o• 1o

Fisquency(eadisc)

Figure 6. Bode diagrams of the controller transfer functions Cur(S) and C(s).

associated to the LQ control problem and the Kalman filter problem The larger q /q is made the faster is thedynamics of A = A — The larger r1/r2 is made the faster is the dynamics of A0 =A — KC. Notice that Aand A0 defines the closed loop dynamics. Hence q /q and r1 /r2 can be used to tune the controller and the resultingclosed loop performance.

It turns out that the observer dynamics given by the system matrix A0 = A — KC cannot be made too fastwithout sacrificing robustness. This is unfortunate, since the suppression of w will then not be very high. The valuesused in the final design were q1 /q = 0.01 and r1 /r2 = 2 resulting in

.

—179.4L = [0.0048 1.0496 0.0002] ; K = —6888

3.4

For these values the distance from the imaginary axis to the eigenvalues closest to the imaginary axis increased forA0 with a factor of 1.005 and for A with a factor of 6.2 as compared to A. Hence it is expected that the controllerwill speed up the transient response from reference value with a factor of 6, and that the suppression of disturbances,i.e. the nonlinearities will not be improved. In Figure 5 is seen the Nyquist diagram of the loop transfer function,i.e. the open loop plant transfer function P(s) in series with the transfer function of the feedback controller

C(s) = L,(sI — ACO) K

It is believed that the Nyquist curve is sufficiently far away form the point —1 to ensure robustness against un-modeleddynamics and the neglected non-linearities. The closed loop system is for w =0 given by

d [x A BuLul Ixl + Buir

L: [c o][f]

0

and hence the transfer function from r to z by C(sI —Ac)'Bulr. The value of 1,. has been chosen such that the gainof the closed loop systems for the frequency wç of the reference signal is unity, i.e. as the solution of the equation

C(iwoI — A + BuLu)1BulrI = 1

which yields 1r = 1.053 x i0. Bode diagrams of the controller transfer functions, i.e. C(s) as defined above, andCur (s) = L(sI — A0)1B + 1r, are shown in Figure 6. Simulations of the closed loop system are shown to theleft in Figure 7. The control signal has been saturated for amplitudes above 15V. The reference value is given by

452

______:h

Figure 7. Simulation of the closed ioop system with a nonlinear plant and a linear controller (left) and a nonlinear

controller (right).

4 x 10 sin ot in meters. Notice that the controller speeds up the transient response as compared to the open 1001)case shown in Figure 4. However, the response does not reach the target value for the amplitude. This is due to theimpact of the non-linearity in the plant. This behavior can be improved by using the nonlinear term in the observer.

Simulation results are shown to the right in Figure 7, where values of /3 = 0 and /32 = 10'6m2s2 have been used

in the controller and, values of k1 /(L) = 0 and k2/(aL) = 0.8 x 1016m2s2 in the plant. It is seen that the valueof the amplitude is improved. Because /32 > k2/(aL) the resulting amplitude is larger than the reference value. Ifthe two nonlinear terms were perfectly matched, the amplitude of the deflection would have reached the targetvalue

of 4 x 106m.

4. CONCLUSIONSIn this paper it has been shown how model based feedback control can be successfully applied to control a drop-ejector.The benefits of control are faster transient response and suppression of the non-linear behavior of the device.

The controller consists of a nonlinear observer in conjunction with a state feedback from the estimated statesand a feed-forward from the reference value. So far a simplistic LQ controller arid Kalinan filter design has beenevaluated in simulations.

The next step will be to make robust design based on H control theory and harmonic balance analysis. Also itrernaiiis to implement the controller and make experiments.

ACKNOWLEDGMENTSP.-E. Roche was supported by MURI. A. Hansson was supported by the Swedish Research Council for EngineeringSciences (under 95 838) and MURI (under F49620-95-1-0525). B.T. Khuri-Yakub was supported by AFOSR (underF49620-95-1-0525). F. L. Degertekin and G. Percin are gratefully acknowledged for their assistance.

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