Sensors and Actuators A 113 (2004) 106–117 Electrostatic analysis of a comb-finger actuator with Schwarz–Christoffel conformal mapping P. Bruschi a , A. Nannini a , F. Pieri a,∗ , G. Raffa a , B. Vigna b , S. Zerbini b a Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Pisa, via Diotisalvi, 2; 56126 Pisa, Italy b STMicroelectronics, via Tolomeo, 1; 20010 Cornaredo (Milan), Milan, Italy Received 15 June 2002; accepted 23 February 2004 Available online 17 April 2004 Abstract The design and the analysis of a comb-finger actuator require the evaluation of the coupling capacitances as well as the estimation of the electrostatic forces. Several approaches are possible: very rough approximations based on combinations of parallel plate capacitor elements, analytical or numerical Schwarz–Christoffel (SC) conformal mapping-based techniques, numerical solutions of the electrostatic field equations. Powerful numerical approaches can be very accurate, but heavy and time-consuming, while simple approaches are very inaccurate. A partially analytical approximate method based on SC conformal mapping is discussed. The comparison with results from standard simulators and experimental measurements demonstrates that the studied procedure is fast and satisfactorily accurate. © 2004 Elsevier B.V. All rights reserved. Keywords: Conformal mapping; Schwarz–Christoffel transformation; MEMS actuators; Comb-fingers; FASTCAP; Capacitance measurements 1. Introduction Capacitive microactuators are very important for the func- tioning of microresonators, integrated gyroscopes, optical and RF MEMS. In fact, in these devices it is necessary to move a part of the micromachined structure, by using a sim- ple and reliable mechanism. Capacitive actuation allows us to satisfy these requirements. Moreover, the MEMS designer may properly decide the geometry of the microstructure and, consequently, the driving direction without renouncing sim- plicity and reliability. In the last years different types of ca- pacitive microactuators were constructed, but our attention will be restricted to the most frequently encountered, which is well known in the literature as comb-finger actuator or comb drive [1–3]. In order to design and analyze a generic capacitive actu- ator, it is necessary to evaluate both the capacitive coupling values and the electrostatic forces, which may be estimated by exploiting different approaches. The parallel plate for- mula, for instance, provides a very simple method to analyze the considered actuator, but the results are often inaccurate. On the contrary, the field-solvers, such as the finite element ∗ Corresponding author. Tel.: +39-050-2217661; fax: +39-050-2217522. E-mail address: f.pieri@iet.unipi.it (F. Pieri). method (FEM) or boundary element method (BEM)-based simulators, allow us to obtain precise values, but are disad- vantageous as regards the simulation time. Thus, in order to carry out our capacitance calculations, we chose to uti- lize the Schwarz–Christoffel (SC) transformation technique, which is described in detail in Appendix A. By exploiting this particular kind of conformal mapping, we obtained an approximate procedure, which is faster than the field-solvers and, when three-dimensional effects are not dominant, pro- vides an accuracy comparable with that of the latter. SC transformation is applied in a large number of techni- cal fields to evaluate several electrical parameters in a closed form or numerically. SC is used as a synthesis tool as well as an analysis technique in the microwave field, both classical and monolithic [4–7], in magnetics [8,9], in measurement sciences [10,11], and in electrical transport [12] and power applications [13]. In particular, many authors utilized this type of conformal mapping to analyze two-dimensional electrostatic problems [14–27]. Among them Koç and Ordung [15] and also Stellari and Lacaita [16] proposed calculations based on conformal mapping to calibrate simulators for VLSI applications and to evaluate interconnect capacitances in integrated circuits, respectively. Costamagna and Fanni [17] utilized this tech- nique to study structures with rotational symmetry, which are useful to analyze multilayered circuits boards [18]. Johnson 0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.02.038