Trapped Energy Resonators Research Articles

An analysis of thickness-extensional trapped energy resonant device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration

It has recently been shown that highly uniform thin layers can be etched in a small well-defined region of a silicon wafer and that good quality thin piezoelectric films such as zinc-oxide can be deposited along with the electrodes to form high frequency trapped energy resonant device structures. The accompanying analytical work has been for pure thickness vibrations only. In this work an analysis of essentially thickness-extensional trapped energy modes in the thin piezoelectric film on silicon composite structure is performed. It is shown that for small wavenumbers along the plate the dispersion equation is isotropic in the plane of the plate even though the silicon is severely anisotropic in that plane. From the resulting dispersion relation an asymptotic differential equation describing the mode shape along the surface of the composite plate vibrating in the vicinity of a thickness-extensional resonance is obtained along with the associated edge conditions. Since the mode is essentially thickness-extensional, trapping does not usually occur in the electroded regions for the fundamental mode of the flat composite plate. However, by appropriately thickening the silicon outside the electroded regions trapping in the electroded regions can be made to occur for the fundamental mode. Furthermore, in the case of zinc-oxide on silicon, trapping in the electroded regions can be made to occur for the fundamental mode of a flat plate simply by making the zinc-oxide sufficiently thicker than the etched silicon. The aforementioned asymptotic differential equation and edge conditions are applied in the analyses of the steady-state vibrations of trapped energy resonators and acoustically coupled two-pole resonator filters, both with rectangular electrodes. Lumped parameter representations of the admittance in the case of the single-pole device and admittance matrix in the case of the two-pole device, which are valid in the vicinity of resonances, are obtained. The analyses hold for the first few thickness-extensional modes and their accompanying lateral overtones.

Energy Trapping of Width-Shear Vibrations in Thin Piezoelectric Strips

This paper presents theoretical analyses and experiments of energy trapping of width-shear vibrations in thin piezoelectric strips. By the use of width-shear vibrations, miniaturization of trapped-energy resonators is achieved in the medium-frequency range below several megahertz. The width-shear-mode trapped-energy resonators have some advantages, such as: 1) Excitation electrodes are deposited only on the one side of the plate. 2) The ratio of capacitances of the resonator can widely be varied by the electrode spacing. 3) A specified higher-harmonic mode can easily be suppressed by special electrode arrangement related to stress distribution of the mode. In addition, trapped-energy single-mode filters using the width-shear modes are proposed, and small-sized filters of 455 kHz are constructed.

Thickness-shear vibrations of a beveled AT-cut quartz plate

Thickness-shear vibration modes of a beveled AT-cut quartz plate have been analyzed from an equivalent three-dimensional equation of motion. This equation has recently been applied in the analysis of plano-convex contoured quartz crystal resonators by Tiersten and Smythe, and has been derived from the dispersion relation obtained in the course of an investigation on overtone modes of coupled thickness–shear and thickness–twist vibrations of trapped energy resonators with rectangular electrodes. Calculations based on the analysis have been shown to have excellent agreement with experimental results. The analysis accounts for the anisotropy, piezoelectricity, and mass loading effects due to the electrodes. The solution for a beveled quartz strip is expressed in terms of Bessel functions of the order ±1/3, and the eigenfrequencies are obtained for harmonic thickness and anharmonic lateral overtones for a given bevel angle and thickness-to-length ratio of the linearly tapered crystal strip. Both the clamped and free-edge boundary conditions are considered.

An analysis of contoured crystal resonators operating in overtones of coupled thickness shear and thickness twist

A previous treatment of overtone modes in trapped energy resonators is extended to the case of plates with slowly varying thickness. The resulting single scalar equation is applied in the analysis of plano-convex contoured quartz crystal resonators, and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. The analysis holds for the fundamental and anharmonic overtones of the fundamental and each harmonic overtone thickness mode. The influence of piezoelectric stiffening, electrode mass loading, and electrical shorting is included in the analysis. No adjustable parameters are required in the theory. Although the basic piezoelectric differential equation employed here is quite different from the ones that have been employed in similar applications heretofore, the analysis accounting for the contouring has appeared in the literature. It is shown that calculations based on the analysis agree extremely well with experimental results obtained with contoured AT-cut quartz crystal resonators.

Analysis of overtone modes in monolithic crystal filters

A previous treatment of overtone modes in trapped energy resonators is extended to the case of two-pole monolithic crystal filters. The asymptotic dispersion relations for the fundamental and odd overtone coupled thickness-shear and thickness-twist waves near cutoff along with the simple approximate edge conditions at junctions, which were employed in the earlier treatment of resonators, are applied in the analysis of two-pole monolithic crystal filters. Both thickness-shear and thickness- twist coupling directions are considered. The influence of piezoelectric stiffening, electrode mass loading, and electrical shorting is included in the analysis. A lumped parameter representation of the admittance matrix for the two-port device is obtained. Numerical results are presented for a few geometrical configurations.

Analysis of nonlinear resonance in thickness-shear and trapped-energy resonators

Electroelastic equations containing terms up to cubic in the small mechanical displacement field, but no higher than linear in the electric variables, are applied in the analysis of nonlinear resonance in rotated Y-cut quartz oscillators. Both pure thickness-shear vibrators and essentially thickness-shear trapped-energy resonators are treated. This is a natural continuation of earlier work on intermodulation in the same resonators. Since in each equation each nonlinear term is negligible compared to an associated linear term, the solutions are obtained by employing an asymptotic iterative procedure and expanding in the eigensolutions of the associated linear problem and, in the vicinity of a resonance, retaining only that nonlinear term correcting the dominant eigensolution. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance and relate the amplitude of the dominant mode nonlinearly to the voltage across the crystal, are presented for both the pure thickness-shear and trapped energy thickness-shear problems. In each instance the expression for the current through the crystal is determined, the influence of the external circuitry is included in the analysis and, ultimately, an expression cubic in the mode amplitude and linear in the driving voltage is obtained. The analyses hold for the fundamental and odd overtone thickness-shear modes. Nonlinear resonance curves are presented for AT-cut quartz using the nonlinear coefficient γ determined in earlier work on intermodulation. Subject Classification: [43]40.24, [43]40.30; [43]85.52.

Analysis of intermodulation in thickness−shear and trapped energy resonators

Electroelastic equations containing terms up to cubic in the small mechanical displacement field, but no higher than linear in the electric variables, are applied in the analysis of intermodulation in rotated Y−cut quartz oscillators. Both pure thickness−shear vibrators and essentially thickness−shear trapped energy resonators are treated. In the linear part of the analysis of the trapped energy resonator, a closed−form asymptotic expression for the frequency wavenumber dispersion relation for the fundamental and odd overtone thickness−shear branches near cutoff is obtained from the three−dimensional linear equations. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance, are presented for the linear and nonlinear portions of both the pure thickness−shear and trapped energy thickness−shear problems. The influence of the driving and detecting circuitry is included and, in particular, in each case the relation between the intermodulation and driving voltage is obtained. Application of this relation to a number of AT−cut quartz fundamental and third overtone trapped energy resonators yields good agreement with experiment and an estimate of the fourth−order elastic constant cE46666 for the AT−cut. Subject Classification: 40.24, 40.20, 40.65; 85.32.

Piezoelectric trapped energy resonators vibrating in thickness-twist modes

In the present paper we investigate theoretically the phenomenon of energy trapping in piezoelectric devices, operating in thickness-twist modes. For a single resonator we have computed the input impedance as a function of both the frequency and the width of the electrode. In addition, we have obtained the fundamental resonant frequency of the device as a function of the width of the electrode and the mass-loading. This is achieved by employing the integral equation formulation of piezoelectric diffraction problems. For a resonator with massless electrodes, the resonant frequency of the fundamental lateral mode and the lowest two inharmonic overtones have been computed as a function of the width of the electrodes. Also the influence of the mass-loading on the resonant frequency of the fundamental lateral mode has been studied. When considering a coupled resonator system, the first resonator is connected with an electric generator, while the second is connected with a load impedance. For this configuration the transfer function has been computed. Next, the second resonator is short-circuited and the input impedance of the first resonator is determined. Finally, the resonant frequency of both the even and the odd lateral mode of the resonator assembly is computed as a function of the spacing between the two resonators. As piezoelectric material we have taken the piezoelectric ceramic PZT-4.

Probing of Acoustic Surface Perturbations by Coherent Light

The need for sensitive methods of detection and visualization of acoustic surface perturbations has grown with the increasing interest in such fields as acoustic holography, ultrasonic surface wave devices, and acoustic trapped energy resonators. One very sensitive detection method utilizes a coherent light beam as a probe for locally measuring phase and amplitude of the acoustic field. Several variants of this technique are possible, based on measuring phase, deflection, wave front curvature, and spatial frequency content of the reflected beam. Each one of these variants may be combined with a scanning motion of the beam in order to visualize the entire sound field. This paper will attempt to survey the present state of the art and compare the different techniques on the basis of their sensitivity and applicability to specific requirements.

Equivalent electromechanical representation of trapped energy transducers

Trapped energy resonators and transducers have attained a considerable importance in quartz crystal technology both as single-frequency resonators for the control of crystal oscillators and as drivers for the monolithic crystal filter which appears likely to have a wide use as a channel filter for the separation of voice frequency channels for long-distance carrier systems, microwave radio, and submarine cable systems. It is the purpose of this paper to derive the equations for trapped energy resonators of the thickness-shear and thickness-twist types and to calculate the ratios of capacitances for straight crested waves. It turns out that the ratios are lower (coupling higher) than are observed in practice. It appears that this difference is connected with the finite width of the plate which causes the motion at the edge of the plate to be somewhat smaller than the motion in the center of the plate. While no exact solution has been obtained for the finite plate, an approximation is made which is in good agreement with the experiment. The resonator on a plate is a symmetrical device, whereas a transducer for driving a monolithic filter is a dissymmetrical device since it is driving different impedances on its two boundaries. To represent this dissymmetry requires a distributed network representation which is somewhat similar to that found for a plane longitudinal or shear wave except that the propagation constant for a trapped wave replaces that of the plane wave. The representation also requires a transformer whose transformation ratio is a function of the frequency and two negative element terms. By transformations the negative elements can be made to disappear. These together produce an equivalent circuit whose values depend on the ratio of the electrode length to the crystal thickness.

The Monolithic FM Discriminator A New Piezoelectric Device

The concept of controlled acoustic coupling between thickness shear trapped energy resonators fabricated on a single quartz plate led to the development of the monolithic crystal filter. Recent work at the Bell Telephone Laboratories has resulted in the development of a three-port monolithic structure capable of providing the frequency-selective network required for an FM discriminator. This device is capable of providing good linearity over bandwidths ranging from about 0.005 percent to nearly 0.2 percent at center frequencies in the range from 5 MHz to 40 MHz. An analysis of the equivalent network is given, as well as the design procedure. Experimental results from a model having a 15MHz center frequency and 6-kHz linear bandwidth are discussed.