[Published in Transducers ‘99, The 10th International Conference on Solid-State Sensors and Actuators, Sendai, Japan, June 7-9, 1999, pp. 474-477.]
DYNAMICS AND CONTROL OF PARALLEL-PLATE ACTUATORS BEYOND THE ELECTROSTATIC INSTABILITY Joseph I. Seeger and Bernhard E. Boser Berkeley Sensor and Actuator Center, 497 Cory Hall, University of California, Berkeley, CA 94720-1774
[email protected] ABSTRACT A switched-capacitor circuit, consisting of one operational amplifier and one capacitor, is proposed for controlling the charge on a gap-closing actuator and stabilizing it against voltage pull-in. The circuit requires voltage slightly higher than the actuator pullin voltage in order to extend the stable range of motion. Using charge control, ideal parallel-plate actuators are stable for all deflections, but non-ideal effects, such as a significant amount of parasitic capacitance, can lead to a “charge pull-in” instability. INTRODUCTION Typically, electrostatic, gap-closing actuators are limited to small motion or bistable operation because large motion can lead to the “pull-in” instability in which the actuator snaps closed. There are techniques for increasing the stable deflection of gap-closing actuators, but there is a tradeoff between voltage and complexity. The simplest approach limits the actuator to small, stable deflections and uses leverage to magnify the motion [1]. A second approach adds capacitance in series with the actuator to stabilize it against pull-in [2]. The most complex approach stabilizes the actuator through the use of position feedback [3]. In this paper, a switched-capacitor circuit is presented for controlling the charge on the actuator and increasing the stable range. The circuit is an extension of the series capacitor approach, and it reduces the voltage requirement by a factor of three to five. The circuit design is preceded by a study of the electromechanical dynamics of a gap-closing actuator. ACTUATOR MODEL An electrostatic actuator is a nonlinear, dynamic capacitor. In general it can be modeled by a transfer function from voltage V to charge q: 1 ∂C a 2 md˙˙ + bd˙ + k ( d – d 0 ) = – --- ---------- V 2 ∂d q = C a ( d )V ,
(1) (2)
where C a , the actuator capacitance, depends on gap d, nominally d0, and d˙ is the time derivative of d. Figure 1 is a model of a gap closing actuator that consists of an ideal parallel-plate actuator, C, plus parasitic capacitance Cp along with top-and-bottom-plate parasitic capacitors. The parasitic capacitors represent,
Ctop
V1 q
m k
d V
b
C
A
V2
Cp Cbot
Figure 1. Parallel-plate actuator including parasitic capacitors. Actuator capacitance, Ca=C+Cp. Parallel-plate capacitance, C=εA/d, nominally C0=εA/d0. Plate area, A; gap d, nominally do; mass, m, is constrained to 1-D motion and subject to a linear spring k, a damper b, and an electrostatic attractive force due to charge q on the actuator and voltage V=V1-V2. for example, fringing field capacitance, capacitance to a ground plane or substrate, or electrode overlap capacitance. In general, all of the parasitic capacitors vary with actuator displacement, but in this model the parasitic values are assumed to be constant. The transfer function for the gap-closing actuator is 1 C0d0 2 md˙˙ + bd˙ + k ( d – d 0 ) = – --- ------------V 2 d2
(3)
C0d0 q = ------------ + C p V . d
(4)
If either V2 or V1 is connected to ground, the top or bottom plate capacitors will have to be included in Cp. DYNAMICS Equilibrium and Low Frequency Behavior In equilibrium ( d˙ = d˙˙ = 0 ), equations (1) and (3) reduce to the balance of spring force and electrostatic force. The relationship between the actuator voltage and gap in equilibrium is plotted in Figure 2. The maximum voltage for which equilibria exist is Vpi, the pull-in voltage. Independent of parasitic capacitance, the pull-in voltage Vpi corresponds to gap d/d0=2/3. The relationship between charge and gap in equilibrium is plotted in Figure 3. In the case of Cp
