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DESIGN ISSUES IN ELECTROMECHANICAL FILTERS WITH PIEZOELECTRIC TRANSDUCERS by Michael P. Dmuchoski B.S. in M.E., University of Pittsburgh, 2000 Submitted to the Graduate Faculty of School of Engineering in partial fulfillment of the requirements for the degree of Master of Science University of Pittsburgh 2002 UNIVERSITY OF PITTSBURGH SCHOOL OF ENGINEERING This thesis was presented by Michael P. Dmuchoski It was defended on December 11, 2002 and approved by Dr. Jeffrey S. Vipperman, Professor, Mechanical Engineering Department Dr. Marlin H. Mickle, Professor, Electrical Engineering Department Dr. William W. Clark, Professor, Mechanical Engineering Department Thesis Advisor ii ________ ABSTRACT DESIGN ISSUES IN ELECTROMECHANICAL FILTERS WITH PIEZOELECTRIC TRANSDUCERS Michael P. Dmuchoski, MS University of Pittsburgh, 2002 The concept of filtering analog signals was first introduced almost one hundred years ago, and has seen tremendous development since then. The majority of filters consist of electrical circuits, which is practical since the signals themselves are usually electrical, although there has been a great deal of interest in electromechanical filters. Electromechanical filters consist of transducers that convert the electrical signal to mechanical motion, which is then passed through a vibrating mechanical system, and then transduced back into electrical energy at the output. In either type of filter, electrical or electromechanical, the key component is the resonator. This is a two-degree-of-freedom system whose transient response oscillates at its natural frequency. In electrical filters, resonators are typically inductor-capacitor pairs, while in mechanical filters they are spring-mass systems. By coupling the resonators correctly, the desired filter type (such as bandpass, band-reject, etc.) or specific filter characteristics (e.g. center frequency, roll-off, ripple, etc.) can be realized. Even though mechanical filters are in general more complex than electrical filters, given the required transducers and the additional fabrication steps, they are desirable because of the extremely good resonator characteristics of mechanical systems, which can result in superior filter characteristics. Existing mechanical filter technology could be considered to be “macro-scale” (centimeters and up), and the design process has been somewhat of an art. There is interest in developing micro-scale filters that are well- iii integrated with electronics. This thesis discusses the overall design process of mechanical filters, bringing together information from the filter design literature that is somewhat spread out. An example of the design and analysis of a narrow-band mechanical filter with piezoelectric transducers is offered. This design was constructed and tested, and the results are presented. Finally, the thesis also presents implications for designing mechanical filters at the micro-scale. iv ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. William W. Clark, for his guidance and assistance throughout this tenure, as well as for the opportunity to work for him as a graduate student. I also express my gratitude to the members of the Vibration and Control Laboratory. Your help has been invaluable. Finally, I would like to thank all of my friends and family. I especially want to thank Alexandra Wood for always being there even when you couldn’t. And yes Mom, it’s really finished this time. The author also gratefully acknowledges the long time support of Aerotech Inc., dating back to when I was an undergraduate. v TABLE OF CONTENTS 1.0 INTRODUCTION ................................................................................................................... 1 1.1 Outline of Thesis.................................................................................................................. 2 2.0 GENERAL FILTER BACKGROUND ................................................................................... 4 2.1 Description of What They Are and What They Do ............................................................. 4 2.2 Filter Responses................................................................................................................... 4 2.2.1 Low-pass Filters............................................................................................................ 6 2.2.2 High-pass Filters ........................................................................................................... 7 2.2.3 Bandpass Filters............................................................................................................ 7 2.2.4 Band-reject Filters......................................................................................................... 9 2.2.5 All-pass Filters.............................................................................................................. 9 3.0 LITERATURE REVIEW ...................................................................................................... 10 3.1 Electrical and Electromechanical Filter Background Information .................................... 10 3.2 Electromechanical Filters................................................................................................... 11 3.2.1 Knowledge Involving All Electromechanical Filter Sizes.......................................... 12 3.2.1.1 Types of Electromechanical Filters ..................................................................... 12 3.2.1.1.1 Crystal and Ceramic Filters .......................................................................... 12 3.2.1.1.2 Surface Acoustic Wave (SAW) Filters......................................................... 14 3.2.1.1.3 Mechanical Filters......................................................................................... 15 3.2.1.2 Electromechanical Transducers ........................................................................... 16 vi 3.2.1.2.1 Types of Transducers.................................................................................... 16 3.2.1.2.1.1 Magnetostrictive Transducers................................................................ 16 3.2.1.2.1.2 Piezoelectric Transducers ...................................................................... 17 3.2.1.3 Resonators............................................................................................................ 18 3.2.1.3.1 Geometries, Vibrational Modes, and Boundary Conditions......................... 19 3.2.1.3.2 Cantilever Beam Example ............................................................................ 20 3.2.2 Macro-scale Mechanical Filters.................................................................................. 22 3.2.2.1 Mathematical Models........................................................................................... 22 3.2.2.1.1 Pictorial Form ............................................................................................... 22 3.2.2.1.2 Electromechanical Analogy.......................................................................... 23 3.2.2.1.2.1 Conventional Analogy ........................................................................... 24 3.2.2.1.2.2 Mobility Analogy................................................................................... 25 3.2.2.1.2.2.1 Schematic Form Using Mobility Analogy...................................... 26 3.2.2.1.2.2.1.1 Mechanical Schematic ............................................................. 26 3.2.2.1.2.2.1.2 Electrical Schematic................................................................. 27 3.2.2.2 Coupling Wires .................................................................................................... 27 3.2.2.2.1 Effects and Development.............................................................................. 28 3.2.2.2.2 Flexural Beam Coupling Example................................................................ 28 3.2.2.2.3 Bridging ........................................................................................................ 30 3.2.2.3 Fabrication ........................................................................................................... 31 3.2.3 Micro-scale Mechanical Filters................................................................................... 32 3.3 Electrical Bandpass Filter Design...................................................................................... 33 3.3.1 Wide-band Bandpass Filter Design ............................................................................ 33 vii 3.3.2 Narrow-band Electrical Bandpass Filters ................................................................... 34 3.3.2.1 All-pole Network ................................................................................................. 34 3.3.2.2 Low-pass Prototype Circuit ................................................................................. 36 3.3.2.3 Filter Approximations.......................................................................................... 36 3.3.2.3.1 Butterworth Approximation.......................................................................... 36 3.3.2.3.2 Chebyshev Approximation ........................................................................... 37 3.3.2.3.3 Legendre Approximation.............................................................................. 38 3.3.2.4 Normalized Filter Elements ................................................................................. 38 3.3.2.4.1 Butterworth Approach
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