Transduction magnitude and phase for COM modeling of SAW devices

An analytic formulation for transduction parameter required by the coupling-of-modes analysis of surface acoustic wave (SAW) transducers is derived by comparing solutions obtained using the coupling-of-modes analysis and an equivalent quasi-static analysis using the Green function method. This analysis results in the derivation of a convenient representation for the transduction strength of periodic SAW transducers. Using the formulation, analytic solutions are obtained for the transduction magnitude and phase of regular SAW transducers. Nonregular transducers are also modeled with the help of an electrostatic field analysis of the transducer's structure.     

A P-matrix-based model for the analysis of SAW transversely coupled resonator filters, including guided modes and a continuum of radiated waves

When designing transversely coupled resonator filters, unexpected spurii are often observed on the high-frequency side of the transfer function. These spurii cannot be described using only the classical waveguide model. Discrete transverse modes inside the grating can be identified if one assumes that the modes have exponential decay outside the grating; however, a continuum of solutions exist in the case of propagating waves outside the grating. A large part of the source excitation may be coupled to these radiated waves. To include this phenomena in the model, a decomposition on the above mentioned continuum was performed. We describe our P-matrix-based model for transversely coupled structures. This model takes into account all guided modes and the continuum. It allows the use of an arbitrary number of acoustical layers and electrical ports. A comparison of the measured and simulated frequency responses is presented for different filters and different metallization thickness showing an excellent agreement.     

Elementary solutions of coupled model equations in the kinetic theory of gases

The method of elementary solutions is employed to solve two coupled integrodifferential equations sufficient for determining temperature-density effects in a linearized BGK model in the kinetic theory of gases.Full-range completeness and orthogonality theorems are proved for the developed normal modes and the infinite-medium Green's function is constructed as an illustration of the full-range formalism.The appropriate homogeneous matrix Riemann problem is discussed, and half-range completeness and orthogonality theorems are proved for a certain subset of the normal modes. The required existence and uniqueness theorems relevant to the H matrix, basic to the half-range analysis, are proved, and an accurate and efficient computational method is discussed. The half-space temperature-slip problem is solved analytically. and a highly accurate value of the temperature-slip coefficient is reported.     

Proton-exchanged wet etching of recessed-structure SAW filter

An improved method for fabricating recessed-structure surface acoustic wave (SAW) filters is reported. The relation between proton-exchanged duration and etched depth are studied, as well as the relation between internal reflection, etched depth, and thickness of the aluminum electrode. Experimental results show that the devices have superior performance, and ideas for further improvement of the fabricated SAW device are discussed.     

Derivation of COM equations using the surface impedance method

The surface impedance method is used for the consistent derivation of coupling of modes equations which describe the interaction of SAW with a periodical system of electrodes of finite thickness. The exact analytic solution of the electrostatic problem in the presence of an arbitrary external electric field for a plane system of electrodes is applied to the calculation of the charge and electric field distributions. Mechanical perturbations are taken into account to first order of the thickness of the electrodes. As a result the scalar self-consistent equation for the electric potential of acoustic waves in the gratings is obtained. For the periodic structure this equation is reduced to the form of COM equations for slowly varying amplitudes. Analytical expressions for all coefficients of the COM equations connecting them with geometrical and material parameters are found. The NSPUDT effect can be considered. Dissipation and energy storage terms can be introduced empirically. The solution of the COM equations is represented in the form of a P matrix with elements written in a convenient form. A simple formula for calculating the location of maximum transducer frequency response is proposed, The balance of energy is considered. Some new relations among the elements of P matrix are found     

An iterative approach to design a SAW filter having desired response

Present work proposes an iterative technique to estimate reflection weighting function and transduction weighting function simultaneously to achieve a desired single-phase unidirectional transducer (SPUDT) response. The technique uses the p-matrix formulation in order to describe characteristics of a SPUDT. Simulation results are presented to show the validity of the approach.     

Ultra-steep cut-off double mode SAW filter and its application to a PCS duplexer

This paper describes the double mode surface acoustic wave (DMS) filter design techniques for achieving the ultra-steep cut-off characteristics and low insertion loss required for the Rx filter in the personal communications services (PCS) duplexer. Simulations demonstrate that the optimal combination of the additional common ground inductance Lg and the coupling capacitance Cc between the input and output terminals of the DMS filters drastically enhances the skirt steepness and attenuation for the lower frequency side of the passband. Based on this result, we propose a novel DMS filter structure that utilizes the parasitic reactance generated in bonding wires and interdigital transducer (IDT) busbars as Lg and Ce, respectively. Because the proposed structure does not need any additional reactance component, the filter size can be small. Moreover, we propose a compact multiple-connection configuration for low insertion loss. Applying these technologies to the Rx filter, we successfully develop a PCS SAW duplexer.     

A refined theory of transversely isotropic piezoelectric plates

Summary. A refined theory for transversely isotropic piezoelectric plates is derived from the general solution of three-dimensional transversely isotropic piezoelasticity by means of Lure operator method. As a special case, the governing differential equations for transversely isotropic elastic plates are obtained directly.     

Passband widening of transversely coupled resonator filters using the fundamental symmetric and antisymmetric modes

This paper describes a method for widening the passband of transversely coupled resonator filters (TCF) using only the fundamental symmetric and antisymmetric modes. The coupling of modes analysis in the transverse direction is applied to the TCF design to investigate the passband width. As a result, it is found that the passband width can be increased by making the surface acoustic wave (SAW) velocity difference between the interdigital transducer (IDT) region and the resonator gap region smaller. It is proposed that a grating structure be applied to the common ground bar, instead of the uniform metal, to reduce the SAW velocity difference. Using the grating-type common ground bar, filters are fabricated on ST-quartz substrate. The passband of a single filter with a center frequency of 248 MHz is widened up to 410 kHz without any increase of the insertion loss. The effect of the impedance mismatch at the junction of two cascaded devices is investigated. It is shown that the filter performance is improved by reduction of the small parasitic capacitance existing at the cascade point. Experimentally, the capacitance formed between the bus bar of the IDT and the bottom surface of the ceramic package is reduced. The insertion loss is reduced by 0.0 dB, and 3 dB passband is widened by 8 kHz for a filter with a center frequency of 248 MHz. On the basis of these two improvements, cascaded TCFs are fabricated. For a filter with a center frequency of 248 MHz, an insertion loss of 5.5 dB and a 3-dB passband width of 270 kHz are obtained     

General solutions of the coupled diffusion equations

New finite integral transform and the corresponding infinite series are introduced, which brings the solution of the coupled diffusion equations within the realm of integral transform theory. The formulation of this transform leads to an eigenvalue problem which is not of the conventional Sturm-Liuoville type and therefore a special integral condition was derived which serves as an orthogonality relation. The solution obtained can be applied when studying diffusion in a tubular reactor, heat transfer by a turbulenty flowing fluid-solids mixture in a pipe.     

The propagation of stress discontinuities according to the coupled equations of thermoelasticity

Summary The method of the characteristics is used to study the influence of thermomechanical coupling on the magnitudes of discontinuities at the wavefronts of dilatational waves. A set of unified equations is employed, which is applicable to plane, cylindrical and spherical waves. The thermal conductivity is taken as a function of the space coordinate. For cylindrical and spherical symmetry the discontinuity at the wavefront of a diverging wave is subjected to both material damping and decay due to geometry. The sharp wavefront is attenuated over a very short distance. For converging waves the material damping is counteracted by an increase in magnitude due to geometry. It is shown that material damping is predominant, except at extremely small distances from the center which fall outside the realm of continuum theory.

---

Zusammenfassung Die Charakteristikenmethode wird benützt, um den Einfluß der thermomechanischen Kopplung auf die Größe der Unstetigkeiten an den Fronten von Dilatationswellen zu studieren. Ein System vereinheitlichter Gleichungen, das auf ebene, zylindrische und sphärische Wellen anwendbar ist, wird aufgestellt. Die Wärmeleitfähigkeit wird als Funktion des Ortes angenommen. Bei Zylinder- und Kugelsymmetrie ist die Unstetigkeit der Front einer divergierenden Welle sowohl der Materialdämpfung als auch einer geometrisch bedingten Abnahme unterworfen. Die steile Wellenfront wird entlang eines sehr kurzen Weges abgemindert. Bei konvergenten Wellen wirkt die geometrisch bedingte Amplitudenvergrößerung der Materialdämpfung entgegen. Es wird gezeigt, daß die Materialdämpfung überwiegt; ausgenommen sind extrem kleine Gebiete um das Wellenzentrum, die außerhalb des Geltungsbereiches der Kontinuumstheorie liegen.

---

---

With 1 Figure

---

The work presented in this paper was supported by the Advanced Research Project Agency of the Department of Defense through the Northwestern University Materials Research Center.     

SAW/LSAW COM parameter extraction from computer experiments with harmonic admittance of a periodic array of electrodes

A novel numerical method for determining the surface acoustic wave and the leaky surface acoustic wave characteristics is proposed. The Green's function method is used to simulate an infinite periodic transducer driven by a periodic voltage. We show that the coupling of modes parameters and the dispersion relation can be extracted from the change in the admittance as the periodicity of the driving voltage is slightly shifted. The method first introduced here leads to significant savings in computing time.     

A bound on the error in the linear theory of transversely isotropic plate bending

Summary Energy inequalities are used to study the accuracy of solutions of Panc's component theory of elastic plates with large transverse shear deformability, compared with threedimensional elasticity theory solutions.     

Fractal coupled theory of drilling and wear

Drilling perforations and tool wear are intimately and mutually connected by fracture propagations at different size-scales. To study this interaction phenomenon, we propose an ad hoc developed fractal coupled theory. Describing the two processes in terms of drilling and wear velocities, the theory is able to predict the relation between these two quantities. The result is represented by a power law between wear and drilling velocities with exponent comprised between 2/3 and 3/2. Some experimental tests on different materials like mortar, concrete and reinforced concrete have also been performed. Theoretical predictions and experimental results agree satisfactorily.     

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

Longitudinal vibration of bars is usually considered in mathematical physics in terms of a classicalmodel described by the wave equation under the assumption that the bar is thin and relatively long. Moregeneral theories have been formulated taking into consideration the effect of the lateral motion of a relativelythick bar (beam). The mathematical formulation of these models includes higher-order derivatives in theequation of motion. Rayleigh derived the simplest generalization of the classical model in 1894, by includingthe effects of lateral motion and neglecting the shear stress. Bishop obtained the next generalization of thetheory in 1952. The Rayleigh–Bishop model is described by a fourth-order partial differential equation notcontaining the fourth-order time derivative. He took into account the effects of shear stress. Both Rayleigh’sand Bishop’s theories consider lateral displacement being proportional to the longitudinal strain. The Bishopmodel was generalized by Mindlin and Herrmann. They considered the lateral displacement proportional toan independent function of time and longitudinal coordinate. This result is formulated as a system of twodifferential equations of second order, which could be replaced by a single equation of fourth order resolvedwith respect to the highest order time derivative. To obtain a more general class of equations, the longitudinaland lateral displacements are expressed in the form of a power series expansion in the lateral coordinate. In thispaper, all of the above-mentioned equations are considered in the framework of a general theory of hyperbolicequations, with the aim of classifying the equations into general groups. The solvability of the correspondingproblems is also discussed.     

A refined buckling theory for the analysis of sandwich shells with transversely soft filler

A state of the art of the problem of buckling in sandwich structures is discussed and the shortcomings of some existing theories shown. A specified classification of the forms of stability is given and, in accordance with it, a refined theory for the study of the mixed forms of stability is formulated. Different models of the fillers are classified according to their stress–strain state. For the transversely soft model of the filler a set of geometrically nonlinear refined relations is derived. These relations are used to describe the subcritical instantaneous equilibrium of the sandwich plates in the case of both large and small changes in the shear stresses.     

Torsion of a transversely isotropic elastic layer bonded to a transversely isotropic half-space

Torsional stresses and displacement of a transversely isotropic elastic layer of finite thickness for which torsional shearing forces are prescribed on its boundary surface are considered. The solutions are given for a few particular cases.     

A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

The paper introduces a Galerkin method in the reproducing kernel Hilbert space. It is implemented as a meshless method based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs it leads to a system of ordinary differential equations. The method is used for solving the 2D nonlinear coupled Burgers' equations having Dirichlet and mixed boundary conditions. The numerical solutions for different values of Reynolds number (Re) are compared with analytical solutions as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Re in the case of Dirichlet boundary conditions.