CN113962084A - Frequency response characteristic analysis method of surface acoustic wave resonator based on dimension reduction PDE model - Google Patents
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Abstract
The invention belongs to the technical field of numerical calculation of surface acoustic wave filters, and particularly relates to a frequency response characteristic analysis method of a surface acoustic wave resonator based on a dimensionality reduction PDE model, which comprises the following steps: acquiring structural data of the surface acoustic wave resonator; inputting data into a dimensionality reduction PDF model to obtain a partial differential equation set representing the surface acoustic wave resonator; solving the partial differential equation set by adopting a finite element method to obtain a system matrix representing the surface acoustic wave resonator; eliminating the freedom degree processing which does not need attention by adopting a finite element freedom degree compression method and a reduction technology, and eliminating the freedom degree of a left boundary and a right boundary by adopting a periodic boundary condition to obtain a system matrix equation after the freedom degree is eliminated; solving a system matrix equation to obtain the frequency response characteristic of the surface acoustic wave resonator; the invention eliminates irrelevant degrees of freedom by adopting a finite element degree of freedom compression method and a reduced order technology, and can realize uniform residual sound field in a directional region.
Description
Technical Field
The invention belongs to the technical field of numerical calculation of surface acoustic wave filters, and particularly relates to a frequency response characteristic analysis method of a surface acoustic wave resonator based on a dimensionality reduction PDE model.
Background
Surface Acoustic Wave (SAW) devices are used as resonators in electronic systems, primarily for applications. Due to its excellent performance (low insertion loss, high isolation, low cost, small size, etc.), it has been mass-produced and widely used in the fields of the current defense military industry, mobile communication, internet of things, and automotive electronics. With the requirements of 5G mobile communication and high-precision national defense military, increasingly strict requirements are put on the performance of the SAW device. This has prompted the emergence of a number of new SAW devices, such as temperature compensated TC-SAW devices, high performance IHP SAW devices, ultra high frequency XBARSAW devices. In order to meet the increasing demand for high-performance SAW devices, new SAW devices having more complicated multilayer thin film structures are emerging. Accordingly, the field of SAW devices is constantly searching for solutions to obtain more versatile, fast and accurate simulation tools for characterizing SAW resonators.
Currently, the detailed information of the resonators can be obtained by performing an accurate calculation on the finite-length resonators using FEM/BEM or pure FEM-based Hierarchical Concatenation Technique (HCT). However, due to current computer memory and speed limitations, there are too many examples of even the latest GPU-HCT technology that show that the characteristics of finite length resonators cannot be characterized quickly and efficiently. Pure two-dimensional and three-dimensional finite element methods are generally numerical computation methods that are efficient and fairly flexible for handling SAW devices such as arbitrary material cuts, electrode shapes, and multilayer substrates. However, in the finite element calculation, in order to ensure the accuracy of numerical simulation, a sufficient number of nodes and degrees of freedom are required. However, the more nodes and degrees of freedom, the more consumed computer resources are required.
The periodic analysis of the surface acoustic wave filter based on the two-dimensional or three-dimensional finite element method still needs to calculate all degrees of freedom, consumes a large amount of computer memory, and is difficult to realize the rapid characterization of the surface acoustic wave filter characteristics. The periodic analysis method of the surface acoustic wave filter based on the periodic Green function is feasible for the traditional surface acoustic wave filter, and for the novel surface acoustic wave filter with a multilayer complex structure, the establishment of the Green function with the multilayer complex structure becomes quite difficult, the consumed calculation time is greatly increased, and the rapid and accurate characterization of the characteristics of the surface acoustic wave filter cannot be realized.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a method for rapidly analyzing the frequency response characteristics of a surface acoustic wave resonator based on a dimension reduction PDE model, which comprises the following steps: acquiring structural data of the surface acoustic wave resonator; preprocessing the data, and inputting the preprocessed data into a dimensionality reduction PDF model to obtain a partial differential equation set representing the surface acoustic wave resonator; solving the partial differential equation set by adopting a finite element method to obtain a system matrix representing the surface acoustic wave resonator; eliminating the degree of freedom of the system matrix by adopting a finite element degree of freedom compression method and a reduction technology; eliminating the left and right boundary freedom degrees of the system matrix by adopting a periodic boundary condition to obtain a system matrix equation with the freedom degrees eliminated; and solving the system matrix equation after the degree of freedom is eliminated to obtain the frequency response characteristic of the surface acoustic wave resonator.
Preferably, the acquiring of the structural data of the surface acoustic wave resonator includes: structural data in the propagation direction of the surface acoustic wave resonator, structural data in the aperture direction of the surface acoustic wave resonator, structural data in the depth direction of the surface acoustic wave resonator, and material data of the surface acoustic wave resonator.
Preferably, the surface acoustic wave resonator is characterized by a double-finger structure.
Preferably, the process of processing data by using the dimension-reduced PDF model includes: inputting the material data of the surface acoustic wave resonator into a dimensionality reduction PDF model to realize the conversion from a crystal axis coordinate system to a Cartier coordinate system; representing a partial differential equation set of the surface acoustic wave resonator according to the parameters after coordinate conversion; the acoustic surface wave resonator material data comprises resonant frequency, resonator calculation parameters, electrode thickness calculation parameters and piezoelectric material parameters.
Preferably, the processing of the partial differential equation system by using the finite element method includes: and carrying out grid division on the topological structure of the surface acoustic wave resonator, defining the physical property of the resonator, and extracting a system matrix representing the surface acoustic wave resonator according to the defined physical property.
Further, the system matrix includes a stiffness matrix, a damping matrix, and a mass matrix.
Preferably, the process of eliminating the degree of freedom of the system matrix by using the finite element degree of freedom compression method and the order reduction technology comprises: reclassifying and sorting the system matrix for representing the surface acoustic wave resonators, sequentially taking the degree of freedom X of the left edge of the propagation direction of the resonatorsLRight edge free degree XRInternal degree of freedom XIAnd electrode structure boundary degree of freedom V; for internal degree of freedom X without attentionIEliminating to make the system matrix representing the surface acoustic wave resonator only contain the left edge freedom degree XLRight edge free degree XRAnd an electrode structure boundary degree of freedom V.
Preferably, the process of eliminating the left and right boundary degrees of freedom of the system matrix by using the periodic boundary condition includes: using periodic boundary conditions, for the left edge containing only degree of freedom XLRight edge free degree XRAnd eliminating the degree of freedom again by the system matrix of the electrode structure boundary degree of freedom V, so that the system matrix representing the surface acoustic wave resonator only comprises the left edge degree of freedom XLAnd an electrode structure boundary degree of freedom V.
The invention eliminates irrelevant degrees of freedom by adopting a finite element degree of freedom compression method and a reduced order technology, and can realize uniform residual sound field in a directional region.
Drawings
FIG. 1 is a surface acoustic wave resonator of the present invention;
fig. 2 is a grid division of the saw resonator topology of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
A frequency response characteristic analysis method of a surface acoustic wave resonator based on a dimension reduction PDE model comprises the following steps: preprocessing the structural data of the surface acoustic wave resonator, and inputting the preprocessed data into a dimension-reduction PDF model to obtain a partial differential equation set representing the surface acoustic wave resonator; solving a partial differential equation set by adopting a finite element method to obtain a system matrix representing the surface acoustic wave resonator, wherein the system matrix comprises a rigidity matrix, a damping matrix and a mass matrix; eliminating the freedom degree processing without attention by adopting a finite element freedom degree compression method and a reduction technology, and eliminating the freedom degrees of left and right boundaries by adopting a periodic boundary condition, so that the freedom degree required to be solved is greatly reduced; and solving the system matrix equation after the freedom degree is eliminated to obtain the frequency response characteristic of the surface acoustic wave resonator.
Acquiring structural data of the surface acoustic wave resonator includes: structural data in the propagation direction of the surface acoustic wave resonator, structural data in the aperture direction of the surface acoustic wave resonator, structural data in the depth direction of the surface acoustic wave resonator, and material data of the surface acoustic wave resonator.
A structure of a surface acoustic wave resonator, as shown in fig. 1, comprising: a Piezoelectric substrate, an IDT chip, a surface acoustic wave device SAW and a Reflector; the IDT chip, the surface acoustic wave device and the reflector are all arranged on the piezoelectric substrate; the surface acoustic wave device SAW is arranged on the IDT chip, and the IDT chip is connected with external voltage; the reflector is connected with the IDT chip to form a surface acoustic wave resonator.
Preferably, the surface acoustic wave resonator is of a double-finger structure, that is, the device comprises two reflectors; the two reflectors are similar in structure. A surface acoustic wave resonator of a two-finger structure, the device comprising: a piezoelectric substrate, an IDT chip, a surface acoustic wave device, and two reflectors; the IDT chip, the surface acoustic wave device and the two reflectors are all arranged on the upper part of the piezoelectric substrate; the IDT chip is externally connected with voltage, so that the device works normally; the output end of the IDT chip is connected with the input end of the surface acoustic wave device, and the surface acoustic wave device is arranged at the top of the IDT chip; the two reflectors are respectively disposed at both ends of the IDT chip, and reflect signals.
The Piezoelectric substrate is a semi-infinite Piezoelectric crystal, the upper surface of the crystal is covered with a metal layer, the metal layer is arranged on the surface of the crystal by means of evaporation or sputtering, and therefore the metal layer is an isotropic conductive layer. The parameters in the piezoelectric substrate crystal should satisfy the following conditions:
Tij＝cijklSkl-ekijEk
Di＝ekijSkl-εikEk
wherein, TijRepresenting stress of surface acoustic wave resonator, cijklExpressing the stiffness coefficient, S, of a piezoelectric substrate in a surface acoustic wave resonatorklRepresents strain, ekijDenotes the piezoelectric stress constant, EkRepresents an electric field; diRepresenting the electric displacement, epsilon, of the surface acoustic wave resonatorikRepresents the dielectric constant.
The process of processing the data by adopting the dimension reduction PDF model comprises the following steps: inputting the material data of the surface acoustic wave resonator into a dimensionality reduction PDF model to realize the conversion from a crystal axis coordinate system to a Cartier coordinate system; inputting the parameters after the coordinates are converted into a partial differential equation set representing the surface acoustic wave resonator; the acoustic surface wave resonator material data comprises resonant frequency, resonator calculation parameters, electrode thickness calculation parameters and piezoelectric material parameters.
Calculating a wave equation of the surface acoustic wave resonator according to the surface acoustic wave resonator material data including the resonant frequency, the resonator calculation parameter, the electrode thickness calculation parameter and the piezoelectric material parameter, wherein the expression is as follows:
wherein c represents an elastic constant, ρ represents a density, ε represents a dielectric constant matrix, u represents a displacement, e represents a piezoelectric stress constant matrix, e' represents a transposed matrix of the piezoelectric stress constant matrix,
represents the stress, phi represents the potential,
represents a Nabla operator expressed as
An operator matrix is constructed according to the direction of the characteristic sound surface propagation and a Nabla operator, wherein the operator matrix is as follows:
wherein x is1、x2、x3Respectively representing the directions characterizing the propagation of the acoustic surface,
representing the direction x of propagation of a surface acoustic wave along a characteristic acoustic surfaceiAnd (5) calculating partial derivatives. Transforming the operator matrix to obtain a new operator matrix and a Nabla operator
The expression is as follows:
according to the new operator matrix
Nabla operator
And expanding the wave equation of the surface acoustic wave resonator by the surface acoustic wave resonator material data to obtain a partial differential equation set of the surface acoustic wave resonator, wherein the matrix expression of the equation set is as follows:
where ρ represents density, ω represents vibration frequency, u represents vibration frequencyiRepresents an edge xiThe displacement in the direction of the displacement is,
denotes the Nabla operator,. phi.denotes the potential, cmnElement representing the mth row and nth column of the elastic constant matrix, eijRepresenting the element, ω, in the ith row and jth column of the piezoelectric stress constant matrixdfRepresenting the elements in the d-th row and f-th column of the permittivity matrix.
Solving a surface acoustic wave filter system equation by adopting finite elements to obtain a system matrix; specifically, a topological structure of the surface acoustic wave resonator is subjected to grid division, as shown in fig. 2, physical attributes of the resonator are defined according to the divided grids, and a system matrix representing the surface acoustic wave resonator is extracted according to the defined physical attributes; the system matrix includes a stiffness matrix, a damping matrix, and a mass matrix.
The system matrix of the surface acoustic wave filter comprises:
wherein, XLLeft boundary degree of freedom, X, for finite element model of single-finger structurelInternal degree of freedom, X, for finite element models of single-finger constructionRThe right boundary freedom degree of the finite element model of the single-finger structure, v is the electrode surface potential freedom degree, q is the electrode surface charge quantity, and RAStress, R, for left boundary of finite element model of single finger structurelStress of degree of freedom, R, inside finite element model of single-finger structureBStress of the right boundary of the finite element model of the single-finger structure.
Obtaining a rigidity matrix of a system matrix according to the system matrix of the surface acoustic wave filter, wherein the rigidity matrix is as follows:
where K represents an element in the stiffness matrix, L represents a left edge degree of freedom of the single-finger strip, L represents an internal degree of freedom of the single-finger strip, R represents a right edge degree of freedom of the single-finger strip, and V represents an electrode voltage degree of freedom of the single-finger strip.
The quality matrix of the system matrix is:
where M represents an element in the quality matrix.
Arranging the system matrix to obtain an arranged system matrix:
SYS＝K-ω2M
where SYS represents an element in the system matrix, K represents an element in the stiffness matrix, and M represents an element in the mass matrix.
SYS is compressed by finite element freedom degree and reduced order technologyIICan be eliminated in performing SYSIIAnd the acceleration is realized by the GPU equipment in the freedom elimination process. The process of eliminating the degree of freedom of the system matrix by adopting a finite element degree of freedom compression method and a reduction technology comprises the following steps: reclassifying and sorting the system matrix for representing the surface acoustic wave resonators, and sequentially dividing the system matrix into the degree of freedom X of the left edge of the propagation direction of the resonatorsLRight edge free degree XRInternal degree of freedom XlAnd electrode structure boundary degree of freedom V; for internal degree of freedom X without attentionlEliminating to make the system matrix representing the surface acoustic wave resonator only contain the left edge freedom degree XLRight edge free degree XRAnd an electrode structure boundary degree of freedom V.
The expression for eliminating the degree of freedom of the system matrix by adopting a finite element degree of freedom compression method and a reduction technology is as follows:
eliminating the freedom degree of the left and right boundaries of the system matrix by using a periodic boundary condition; specifically, using the periodic boundary condition, only including the left edge degree of freedom XLRight edge free degree XRAnd eliminating the degree of freedom again by the system matrix of the electrode structure boundary degree of freedom V, so that the system matrix representing the surface acoustic wave resonator only comprises the left edge degree of freedom XLAnd an electrode structure boundary degree of freedom V. Elimination of XRDegree of freedom, can
According to elimination of XRThe expression of the degrees of freedom can be tabulatedAnd (3) carrying out mark:
Y＝-iω[SYS”21XL+SYS”22v]/V
wherein Y represents admittance, ω represents vibration frequency, V represents voltage degree of freedom, and V represents electrode structure boundary degree of freedom.
The above-mentioned embodiments, which further illustrate the objects, technical solutions and advantages of the present invention, should be understood that the above-mentioned embodiments are only preferred embodiments of the present invention, and should not be construed as limiting the present invention, and any modifications, equivalents, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (8)
1. A frequency response characteristic analysis method of a surface acoustic wave resonator based on a dimensionality reduction PDE model is characterized by comprising the following steps: acquiring structural data of the surface acoustic wave resonator; preprocessing the data, and inputting the preprocessed data into a dimensionality reduction PDF model to obtain a partial differential equation set representing the surface acoustic wave resonator; solving the partial differential equation set by adopting a finite element method to obtain a system matrix representing the surface acoustic wave resonator; eliminating the degree of freedom of the system matrix by adopting a finite element degree of freedom compression method and a reduction technology; eliminating the left and right boundary freedom degrees of the system matrix by adopting a periodic boundary condition to obtain a system matrix equation with the freedom degrees eliminated; and solving the system matrix equation after the degree of freedom is eliminated to obtain the frequency response characteristic of the surface acoustic wave resonator.
2. The method of analyzing frequency response characteristics of a surface acoustic wave resonator based on a dimension-reduced PDE model according to claim 1, wherein obtaining structural data of the surface acoustic wave resonator comprises: structural data in the propagation direction of the surface acoustic wave resonator, structural data in the aperture direction of the surface acoustic wave resonator, structural data in the depth direction of the surface acoustic wave resonator, and material data of the surface acoustic wave resonator.
3. The method of analyzing frequency response characteristics of a surface acoustic wave resonator based on a dimension-reduced PDE model according to claim 1, characterized in that the surface acoustic wave resonator is characterized by a two-finger structure.
4. The method of claim 1, wherein the processing of the data using the dimension-reduced PDF model comprises: inputting the material data of the surface acoustic wave resonator into a dimensionality reduction PDF model to realize the conversion from a crystal axis coordinate system to a Cartier coordinate system; calculating a partial differential equation set representing the surface acoustic wave resonator according to the parameters after the coordinates are converted; the acoustic surface wave resonator material data comprises resonant frequency, resonator calculation parameters, electrode thickness calculation parameters and piezoelectric material parameters.
5. The method of analyzing frequency response characteristics of a surface acoustic wave resonator based on a dimension-reduced PDE model according to claim 1, wherein the processing of the system of partial differential equations using the finite element method comprises: and carrying out grid division on the topological structure of the surface acoustic wave resonator, defining the physical property of the resonator, and extracting a system matrix representing the surface acoustic wave resonator according to the defined physical property.
6. The method for analyzing the frequency response characteristics of the surface acoustic wave resonator based on the dimension-reduced PDE model as recited in claim 5, wherein the system matrix comprises a stiffness matrix, a damping matrix and a mass matrix.
7. The method of claim 1, wherein the process of eliminating the degree of freedom of the system matrix using the finite element degree of freedom compression method and the order reduction technique comprises: reclassifying and sorting the system matrix for representing the surface acoustic wave resonators, and sequentially dividing the system matrix into the degree of freedom X of the left edge of the propagation direction of the resonatorsLRight edge free degree XRInternal degree of freedom XIAnd electrode structure boundary degree of freedom V; for internal degree of freedom X without attentionIEliminating to make the system matrix representing the surface acoustic wave resonator only contain the left edge freedom degree XLRight edge free degree XRAnd an electrode structure boundary degree of freedom V.
8. The method of claim 1, wherein the process of eliminating the left and right boundary degrees of freedom of the system matrix using periodic boundary conditions comprises: using periodic boundary conditions, for the left edge containing only degree of freedom XLRight edge free degree XRAnd eliminating the degree of freedom again by the system matrix of the electrode structure boundary degree of freedom V, so that the system matrix representing the surface acoustic wave resonator only comprises the left edge degree of freedom XLAnd an electrode structure boundary degree of freedom V.
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