Tilmans%20JMM%201997%20Transducers%202

Tilmans%20JMM%201997%20Transducers%202 - J Micromech...

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J. Micromech. Microeng. 7 (1997) 285–309. Printed in the UK PII: S0960-1317(97)83701-1 Equivalent circuit representation of electromechanical transducers: II. Distributed-parameter systems Harrie A C Tilmans CP Clare NV, Bampslaan 17, B-3500 Hasselt, Belgium Received 25 April 1997, accepted for publication 29 August 1997 Abstract. Distributed-parameter electromechanical transducers are examined theoretically with special regard to their dynamic electromechanical behaviour and equivalent circuits used to represent them. The circuits are developed starting from basic electromechanical transduction principles and the electrical and mechanical equations of equilibrium. Within the limits of the assumptions on boundary conditions, the theory presented is exact with no restrictions other than linearity. The link with lumped-parameter transducers, that were the subject of a previous publication will be indicated. Elementary electrostatic, electromagnetic, electrodynamic and piezoelectric transducers are used to illustrate the basic theory. Exemplary devices include a capacitive force (displacement) sensor, several types of a mechanical resonator, electromechanical filters, a piezoelectric thickness monitor and a piezoelectric acceleration sensor. 1. Introduction The advantages of using equivalent circuit representations for the small-signal time and frequency analysis of electromechanical systems has been extensively explained and illustrated with examples in part I [1] of this paper. Summarizing, it can be said that equivalent circuits (1) provide a single representation of a system operating in more than one energy domain (e.g., electric, mechanical and acoustical), (2) give a better understanding and visualization of the system, (3) facilitate further analyses of the system in order to investigate the effects of connecting subsystems, or of making modifications to the system, and, moreover, (4) allow system analysis using the powerful methods of electrical network theory, for instance by using circuit simulation software such as SPICE. In part I [1], the analysis has been limited to lumped-parameter electromechanical systems. In this paper, the theory will be extended to include distributed- (or continuous-) parameter systems. In a distributed- parameter system, as opposed to lumped-parameter systems, the mass, compliance, capacitance etc are not easily identifiable as lumped elements at individual points, but instead, are continuously distributed throughout the system [2–4]. It has been indicated in part I [1] that in lumped-parameter systems the wavelength of the signal is (much) greater than all physical dimensions of the system. In situations, however, where the dimensions of Tel.: + 32 11 300 887. Fax: + 32 11 300 882. E-mail address: [email protected] the physical device are comparable or greater than the wavelength, the problem must be solved using the general theory of the three-dimensional propagation of waves in solids (and fluids). From such an analysis it follows that the basic difference between lumped-parameter and
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