Models for Resonant Sensors
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Institute for Micro and Sensor Systems, Otto-von-Guericke-University Magdeburg 2 Institute de Fisica, Universidade de Campinas 3 Department of Chemical Engineering, Stanford University
3.1 Introduction The quartz crystal resonator (QCR), as its acronym implies, is a resonant physical device. Many of its behaviors and properties can be understood physically by examining its resonant behavior. The basic principle of operation for a generic acoustic-wave sensor is a traveling wave combined with a confinement structure to produce a standing wave whose frequency is determined jointly by the velocity of the traveling wave and the dimensions of the confinement structure. The most basic way of resonator modeling consequently requires applying the theory of wave propagation thereby considering material properties and geometric dimensions of the resonator. As another successful way, there is an electrical equivalent circuit often used to characterize the resonance. For these reasons, a closer inspection of the phenomenon of resonance is useful.
3.2 The Resonance Phenomenon On certain physical systems, the phenomenon of resonance can be used to multiply the effects of a force applied to the system. There are examples in mechanical, electrical and optical systems. When energy in a system is exchanged periodically between two forms, then resonance occurs. For example, in the case of a weight hanging on a rubber band, when the band is stretched, there is potential energy stored in the extended band. Subsequently, as the weight moves, the stored potential energy is exchanged into the kinetic energy of the weight. Following this, the kinetic energy is then transferred back into the potential energy in the band itself. If there are no A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_3, © Springer-Verlag Berlin Heidelberg 2008
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Ralf Lucklum, David Soares and Kay Kanazawa
losses in the system, then this back and forth energy transfer would continue. This is the resonance phenomenon. The energy exchange occurs periodically and is characterized by a resonance frequency. Similarly in an electrical circuit with an inductor and a capacitor, electrical energy can be stored as voltage across the capacitor. This stored energy then produces a current that flows in the inductor, exchanging the energy stored in the electrical field into a stored magnetic energy associated with the magnetic fields produced by the current in the inductor. The equivalent analogy between mechanical systems and electrical systems has been used also to describe the phenomenon of resonance of quartz crystal resonators and other acoustic-wave based sensors. In the case of the thickness shear mode quartz crystal resonator, the application of alternating voltage across the crystal results in the generation of a shear acoustic wave, causing a distortion of the crystal. When the frequency of the alternating voltage is far from the resonant frequency, the distortion, as measured by the shear displacement of the surface of
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