Self-adaptive frequency sweeping rapid simulation method for surface acoustic wave filter

Abstract

The invention provides a surface acoustic wave filter self-adaptive frequency sweeping rapid simulation method, which comprises the following steps of performing three-dimensional modeling on a surface acoustic wave filter, setting parameter information of each solving region, and extracting a corresponding calculation model; performing discretization processing on the surface acoustic wave filter calculation model by using a hexahedral mesh; a piezoelectric coupling discrete process is deduced, a piezoelectric coupling spectrum element frequency domain solver is constructed, the discrete domain and solving parameters of the surface acoustic wave filter are imported into the solver, and the solver is initialized; and constructing a self-adaptive frequency sweeping algorithm, updating a frequency point needing to be calculated in a self-adaptive updating manner, calling a piezoelectric coupling frequency domain solver to calculate admittance of the frequency point, and finally obtaining a frequency-admittance characteristic curve of the surface acoustic wave filter model under the broadband. Based on the spectral element method and the self-adaptive frequency sweeping algorithm, the method has the advantages of small calculation memory and high solving efficiency under the condition that the solving precision is met.

Description

Technical Field

[0001] The invention belongs to the technical field of piezoelectric effect modeling and simulation, and in particular relates to a surface acoustic wave filter adaptive frequency sweeping fast simulation method. Background Art

[0002] With the development of semiconductor technology and the widespread application of 5G communication technology, electronic communication systems are developing towards high speed and miniaturization. As the core of communication, RF filter devices need to meet the needs of high frequency band and small volume at the same time. Among them, surface acoustic wave (SAW) filters are widely used in 5G communication systems due to their excellent performance characteristics to solve the problem of signal interference between different frequency bands and communication systems. Its basic principle is to convert electrical signals into vibrations of particles based on the piezoelectric effect, filter invalid vibration frequencies through a specific structure, and finally generate electrical signals at the output end through the inverse piezoelectric effect to achieve filtering function. Since the wavelength of elastic waves at the same frequency is much smaller than the wavelength of electromagnetic waves, compared with traditional electronic filters, SAW filters have extremely small volumes while meeting the same filtering effect.

[0003] Numerical simulation is usually used in engineering to design SAW filter products. The basic method is to build a product model and parameter characteristics, solve the partial differential equation corresponding to piezoelectric coupling through numerical methods, and finally calculate the product performance parameters. This method can reduce the number of physical prototype trials, thereby reducing design costs and shortening the design cycle. The key step in the simulation design of SAW filters is to obtain the wide-band admittance response characteristics, so it is necessary to perform a frequency sweep analysis on the model, calculate the admittance response at each frequency, and finally obtain the frequency-admittance characteristic curve.

[0004] The efficiency of SAW filter sweep analysis is usually restricted by two factors: 1. The computational efficiency of the piezoelectric effect solver at a single frequency point; 2. The number of frequency points required for wide-band sampling. For the frequency domain analysis of the piezoelectric effect at a single frequency point, the finite element method (FEM) is currently the most widely used method due to its mesh flexibility and high accuracy. However, with the increase in the complexity of the simulation model, the traditional finite element solution method requires a lot of solution time and memory. As a special finite element method, the spectral element method (SEM) inherits the flexible geometric adaptability of the former, and is significantly superior to the finite element method in terms of frequency domain solution memory usage due to the high-order diagonal nature of its mass matrix. For the acquisition of the frequency-admittance characteristic curve under a wide band, the widely used scheme is to obtain the uniformly distributed sampling frequency based on the Nyquist sampling theorem. Although this scheme can achieve a high solution accuracy, the sampling frequency is too high, resulting in extremely low solution efficiency. In order to reduce the number of frequency points required for sampling, a method based on polynomial interpolation is currently widely used. A small number of sampling points are selected, the responses of these frequency points are calculated, and then the swept frequency characteristic curve is obtained through polynomial interpolation. Although polynomial interpolation can significantly improve the computational efficiency of frequency sweep analysis, wideband frequency response usually exhibits large sharp changes and fluctuations, and often has numerical instability. For example, the increase in the number of interpolation polynomials will lead to the Jung phenomenon, which results in large errors in polynomial fitting. Therefore, for the simulation analysis of SAW filters, a new frequency sampling point acquisition scheme needs to be proposed to improve computational efficiency while taking into account the frequency sweep accuracy.

[0005] In summary, the present invention proposes a SAW filter adaptive frequency sweeping fast simulation method, which improves the frequency sweeping analysis efficiency of the SAW filter by applying SEM and adaptive interpolation scheme to the numerical analysis of the piezoelectric coupling equation, thereby reducing the time of SAW filter product design. Summary of the invention

[0006] In order to solve the problems of large amount of sweep frequency analysis calculation and long solution time in the existing surface acoustic wave filter simulation design process, the present invention provides an adaptive sweep frequency fast simulation method for surface acoustic wave filters to accurately and efficiently predict the wide-band response in piezoelectric effect simulation and improve the design efficiency of surface acoustic wave filters.

[0007] The present invention comprises the following steps:

[0008] S1, three-dimensional modeling of the surface acoustic wave filter, setting parameter information of each solution area, and extracting the corresponding calculation model;

[0009] S2, performing hexahedral meshing on the surface acoustic wave filter calculation model to obtain a discrete domain;

[0010] S3, deriving the piezoelectric coupling discrete process, constructing a piezoelectric coupling spectral element frequency domain solver, and importing the above-mentioned surface acoustic wave filter discrete domain and solution parameters into the solver, and initializing the solver;

[0011] S4. Construct an adaptive frequency sweep algorithm, adaptively update the frequency points that need to be calculated, and call the piezoelectric coupling frequency domain solver to calculate the admittance of the frequency points, and finally obtain the frequency-admittance characteristic curve of the surface acoustic wave filter model under wide bandwidth.

[0012] In S1, the constructed geometric model should include a substrate made of non-conductive material, a substrate made of piezoelectric material and an electrode made of metal conductive material; the calculation model proposed in step S1 includes: the length, width, thickness, material properties of the substrate, the substrate and the metal conductive area, the number of metal conductive areas, the boundary conditions of the substrate, the substrate and the metal conductive area, and the circuit excitation parameters connected to the metal conductive area, wherein the circuit includes impedance.

[0013] In S2, the computational model generated by S1 is input to construct a discrete domain, which should include the following information: node coordinates; node numbers constituting the hexahedral mesh; material numbers corresponding to the hexahedral mesh; plate numbers corresponding to the hexahedral mesh; node numbers constituting the surface mesh; and boundary condition numbers corresponding to the surface mesh.

[0014] In S3, the solver initialization process is to input the solution parameter information and discrete domain, use the spectral element method to construct the weak form of the piezoelectric coupling differential equation, and establish the mapping relationship between frequency points and admittance.

[0015] Preferably, the solution parameter information should include: the solution areas corresponding to the electrostatic field and the elastic wave respectively; the starting frequency and the ending frequency of the frequency sweep analysis; the electrode area corresponding to each electrode; the parameters of the external circuit connected to the electrode; the boundary condition parameters of each area; and the material parameters of each area.

[0016] Preferably, the piezoelectric coupled differential equation includes the elastic wave equation and the electrostatic field equation:

[0017]

[0018] Where ρ is the density of the piezoelectric material, ω is the operating frequency, u is the deformation, τ is the stress tensor, f is the elastic wave excitation term, D is the electric displacement vector, ρ e is the external charge density. The constitutive relation of piezoelectric coupling is as follows:

[0019] T=-e·E+C:∈

[0020] D=∈0∈ r E+e:∈

[0021] Among them, e is the piezoelectric coupling coefficient, E is the electric field intensity, C is the elastic coefficient matrix, ∈ is the strain tensor, ∈0 is the vacuum dielectric constant, ∈ r is the relative dielectric constant of the material.

[0022] Preferably, the piezoelectric coupled differential equation is discretized using Legrange basis functions, and Gauss-Lobatto-Legendre (GLL) points are selected as interpolation points for numerical integration. In the reference unit, the GLL basis function of the i-th node of the N-order unit is defined as:

[0023]

[0024] Among them, L N (ξ i ) and L′ N (ξ i ) are the Nth-order Legrange polynomial and its first-order derivative, respectively.

[0025] Preferably, to establish a mapping relationship between frequency and admittance, it is necessary to extract the potential at the plate, calculate the current at the plate through circuit analysis, and calculate the admittance of the surface acoustic wave filter using Ohm's law, and finally obtain the following function:

[0026] Y = f(ω).

[0027] In S4, the process of adaptive frequency sweep analysis is as follows:

[0028] S41, set the minimum value f of the target frequency band min and the maximum value f max , converted to the minimum value of angular frequency ω min and the maximum value ω max , the initial number of frequency points N = N0, the initial iteration step k = 0, the maximum iteration error e max , the maximum number of iteration steps k max ;

[0029] S42. Calculate the sampling frequency W of the initial iteration step according to the parameters set in S41. (0) ;

[0030] S43, call the piezoelectric coupling spectral element frequency domain solver to calculate the admittance matrix Y corresponding to each initial frequency point (0) :

[0031] S44, use a rational function to fit the sampling frequency W corresponding to the k-th iteration step ( k ) and the admittance matrix Y (k) , and get the fitting function

[0032] S45, calculate each sampling frequency point W (k)Among them, the admittance matrix Y (k) and the fitting function The root mean square error between them is used as the global error e of the current iteration g . If the global error e g is less than the maximum allowable error, or the number of iteration steps is greater than the maximum allowable number of iterations, end the iteration; otherwise, proceed to the next step;

[0033] S46. Calculate the band error e between every two sampling frequency points l,n . The calculation method is that in the k-th iteration step, for the frequency band between point n and point n + 1 (n < N - 1), define the integral

[0034]

[0035] where i and j are the row index and column index of the matrix respectively.

[0036] S47. Select the interval with the largest value, add a node at its central position, calculate its angular frequency ω and admittance matrix Y, and insert them into the corresponding positions of the sampling frequency points W (k) and the admittance matrix Y (k) to obtain the sampling frequency points W (k+1) and the admittance matrix Y (k+1) for the next iteration, and return to S44.

[0037] As can be seen from the above implementation steps of the present invention, compared with the prior art, the beneficial effects of the present invention are as follows:

[0038] The piezoelectric effect adaptive frequency sweeping method proposed by the present invention reduces the memory required for solving by using a spectral element method solver for piezoelectric effect analysis during the frequency point solving process; by selecting fewer initial frequency points, error analysis is performed during the frequency sweeping process, and the number of frequency points is dynamically increased in an adaptive manner, improving the numerical stability of interpolation and ensuring accurate interpolation while reducing the number of frequency points that need to be accurately analyzed and the calculation time of iteration. In summary, the present invention improves the design efficiency of surface acoustic wave filters by shortening the calculation time of the frequency-admittance characteristic curve and is easy to implement. Description of the Drawings

[0039] The drawings are included to provide a further understanding of the embodiments and are incorporated into and constitute a part of this specification. The drawings illustrate the embodiments and, together with the description, are used to explain the principles of the present invention. Other embodiments and many of the expected advantages of the embodiments will be readily apparent as they become better understood by reference to the following detailed description. The elements of the drawings are not necessarily to scale with each other. Like reference numerals refer to corresponding like parts.

[0040] Figure 1 It is a flow chart of an adaptive frequency sweeping fast simulation method of a surface acoustic wave filter proposed by the present invention;

[0041] Figure 2 is a geometric model of a specific example of the surface acoustic wave filter of the present invention;

[0042] Figure 3 is a discretized model of a specific example of the surface acoustic wave filter of the present invention;

[0043] Figure 4 It is the adaptive frequency sweeping flow chart proposed by the present invention;

[0044] Figure 5 The present invention is used to compare the admittance results obtained by using the adaptive frequency sweeping fast simulation method and the discrete frequency sweeping solution. DETAILED DESCRIPTION

[0045] The present invention proposes an adaptive frequency sweeping fast simulation method for a surface acoustic wave filter, wherein a discrete domain of the surface acoustic wave filter is constructed using hexahedral units, the admittance at each frequency point is calculated using a spectral element method, and an adaptive frequency sweeping method is used to obtain the number of frequency points that need to be accurately analyzed, and finally a frequency-admittance characteristic curve of a specific frequency band is obtained to guide the design and optimization of the surface acoustic wave filter product.

[0046] The features of various aspects of the present invention will be described in detail below with reference to examples:

[0047] S1. Perform three-dimensional modeling on the surface acoustic wave filter, set parameter information of each solution area, and extract the corresponding calculation model.

[0048] Figure 2 A specific example of the geometric model of the surface acoustic wave filter of the present invention is shown. The calculation area of ​​the model includes: two electrodes made of metal conductive materials, a substrate made of piezoelectric materials, and a base made of two non-conductive materials. The parameter information set includes: the length, width, thickness and material properties of the electrode, substrate and base; the terminal external circuit of the electrode; and the boundary conditions of the electrode, substrate and base.

[0049] In a preferred embodiment, the size parameters of the electrode are: 0.5um*15um*0.16um, and the material used is aluminum; the size parameters of the substrate are: 2um*15um*0.6um, and the material used is lithium tantalate; the size parameters of the base 1 are 2um*15um*0.5um, and the material used is silicon dioxide; the size parameters of the base 2 are 2um*15um*4um, and the material used is silicon.

[0050] For the boundary conditions, in the elastic wave part, the substrate and both sides of the base are periodic boundary conditions, the front and back and the bottom are set as absorption boundary conditions, and the rest are set as free boundaries; in the electrostatic field part, the substrate and both sides of the base are periodic boundary conditions, and the rest are zero charge boundaries. It should be noted that in order to simplify the calculation amount, this preferred model extracts the local geometric features of the actual model for modeling, and uses periodic boundaries to make the solution more consistent with reality.

[0051] For the excitation parameters, the external circuit of electrode 1 is set to a voltage source U s =1V plus R=50Ω resistor; electrode 2 is set to ground.

[0052] S2, performing hexahedral meshing on the surface acoustic wave filter calculation model to obtain a discrete domain;

[0053] Figure 3 It is a hexahedral mesh discretization domain of a preferred embodiment. In a specific implementation, for the preferred embodiment proposed by the present invention, the degree of freedom of the discretization domain is 289972.

[0054] S3. Derive the piezoelectric coupling discrete process, construct a piezoelectric coupling spectral element frequency domain solver, and import the above-mentioned surface acoustic wave filter discrete domain and solution parameters into the solver to initialize the solver.

[0055] During the frequency sweep analysis, it is necessary to use the piezoelectric coupled spectral element frequency domain solver to calculate the potential distribution at a specific frequency point, and then calculate the admittance of the surface acoustic wave filter.

[0056] Specifically, the admittance calculation of the surface acoustic wave filter involves the coupling between elastic waves and electrostatic fields, and it is necessary to construct a piezoelectric coupling spectral element frequency domain solver based on the following equation.

[0057] The governing equation for elastic waves is:

[0058]

[0059] Among them, ρ is the density of the piezoelectric material, ω is the operating frequency, u is the deformation, τ is the stress tensor, and f is the elastic wave excitation term.

[0060] The governing equation for the electrostatic field is:

[0061]

[0062] Where D is the electric displacement vector, ρ e is the external charge density. When considering piezoelectric coupling, the constitutive relationship between τ and D is as follows:

[0063] τ=-e·E+C:∈

[0064] D=∈0∈ rE+e:∈

[0065] Among them, e is the piezoelectric coupling coefficient, E is the electric field intensity, C is the elastic coefficient matrix, ∈ is the strain tensor, ∈0 is the vacuum dielectric constant, ∈ r is the relative dielectric constant of the material.

[0066] There is the following relationship between the electric field strength E and the electric potential U:

[0067]

[0068] The strain tensor and displacement have the following relationship:

[0069]

[0070] Using the Galerkin method to discretize the piezoelectric coupling control equation, we get:

[0071]

[0072] Among them, φ i，j are the basis functions corresponding to the i-th and j-th interpolation points respectively; N is the number of discrete points obtained after meshing; v is the volume where the integration domain is located; s is the surface where the boundary condition is located; C m is the tensor obtained by transforming the elastic coefficient matrix, e m is the tensor converted from the piezoelectric coupling coefficient.

[0073] In the spectral element method, the Legrange basis function is used, and the Gauss-Lobatto-Legendre (GLL) points are selected as interpolation points for numerical integration. In the reference unit, the GLL basis function of the i-th node of the N-order unit is defined as:

[0074]

[0075] Among them, L N (ξ i ) and L′ N (ξ i ) are N-order Legrange polynomials and their first-order derivatives. Through covariant mapping, the basis function and its gradient of any hexahedral unit can be obtained:

[0076]

[0077] Where J is the Jacobian matrix of the physical unit.

[0078] Based on the above weak form, the original differential equation is converted into the form of a system of equations S(ω)A=F. The above system of equations can be calculated by a matrix solver to perform piezoelectric coupling frequency domain analysis and obtain the potential U0 on the plate. After that, the external circuit equation is solved to obtain the plate current I, and the model's admittance Y is further calculated by Ohm's law:

[0079]

[0080] When the number of electrodes is greater than 1, the model admittance Y is in matrix form.

[0081] Based on the above derivation process, a frequency domain solver for piezoelectric coupling spectral elements is developed. The solver consists of two processes: initialization process and frequency domain analysis process. In the initialization process, the discretized model and simulation parameters are input, and after model setting, unit analysis and overall synthesis process, the mapping relationship between the frequency of a single frequency point of the model and the admittance matrix is ​​obtained:

[0082] Y=f(ω)

[0083] S4, build an adaptive frequency sweep algorithm, adaptively update the frequency points that need to be calculated, and call the spectrum element piezoelectric coupling frequency domain solver to calculate the admittance of the frequency points, and finally obtain the frequency-admittance characteristic curve of the surface acoustic wave filter model under wide bandwidth. The process is as follows Figure 4 shown.

[0084] S41, set the minimum value f of the target frequency band min and the maximum value f max , the initial number of frequency points N0, the initial iteration step k = 0, the maximum iteration error e max , the maximum number of iteration steps k max ;

[0085] For the preferred case, the adaptive frequency sweep parameters are set as follows: f min =0.5GHz, f max =2GHz, N0=5, e max =10 -2 , k max =250.

[0086] S42, set the number of frequency points N (0) =N0, calculate the sampling frequency W of the initial iteration (0) ;

[0087] S43, call Y=f(ω) to calculate the admittance matrix Y of each frequency point (0) ;

[0088] S44, use the following form of rational function to fit the sampling frequency W corresponding to the kth iteration step (k) and the admittance matrix Y (k) :

[0089]

[0090] Substitute \(s = j\omega\) into the above formula to obtain

[0091] S45. Calculate each sampling frequency point \(W\) (k) Among them, the admittance matrix \(Y\) (k) and the fitting function The root mean square error between them is used as the global error \(e\) of the current iteration g :

[0092]

[0093] If the global error \(e\) g is less than the maximum allowable error, or the number of iteration steps is greater than the maximum allowable number of iterations, end the iteration; otherwise, proceed to the next step

[0094] S46. Calculate the frequency band error \(e\) between every two sampling frequency points l，n , and this error is obtained by analyzing the relative errors of each sub - frequency band in the current iteration and the previous iteration. Specifically, in the \(k\) - th iteration step, for the frequency band between point \(n\) and point \(n + 1\) (\(n\lt N - 1\)), define the integral

[0095]

[0096] where \(i\) and \(j\) are the row index and column index of the matrix. The above integral can be calculated by numerical integration methods such as Gaussian integration. The calculation method of the error \(e\) between every two sampling frequency points l，n is as follows:

[0097]

[0098] S47. Select the interval with the largest [[value]], add a node at its middle position, calculate its angular frequency \(\omega\) and the admittance matrix \(Y\), and insert them into the corresponding positions of the sampling frequency points \(W\)( k ) and the admittance matrix \(Y\)( k ) to obtain the sampling frequency points \(W\) of the next iteration (k+1) and the admittance matrix \(Y\) (k+1) , and return to S44

[0099] The preferred model is calculated using the adaptive frequency sweeping fast simulation method of the surface acoustic wave filter and the discrete finite element frequency sweeping method proposed in the present invention, respectively, wherein the discrete frequency sweeping step is set to 0.001 GHz. Comparing the solution results of the adaptive frequency sweeping and the discrete frequency sweeping in the same frequency band, as shown in Table 1, it can be seen that the solution time and memory occupied by the adaptive frequency sweeping fast simulation method proposed in the present invention are significantly reduced while meeting the solution accuracy.

[0100] algorithm Iteration steps Solution time Memory Discrete frequency sweep 601 44min 20G Adaptive frequency sweep 20 2min 5s 3.2G

[0101] Table 1 Comparison of solution efficiency between discrete frequency sweep and adaptive frequency sweep

[0102] In summary, the fast method for adaptive frequency sweeping of a surface acoustic wave filter proposed in the present invention has significant advantages in the optimization design of the surface acoustic wave filter.

Claims

1. A surface acoustic wave filter adaptive frequency sweeping fast simulation method, characterized in that: The method comprises the following steps: S1, three-dimensional modeling of the surface acoustic wave filter, setting parameter information of each solution area, and extracting the corresponding calculation model; S2, performing hexahedral meshing on the surface acoustic wave filter calculation model to obtain a discrete domain; S3, deriving the piezoelectric coupling discrete process, constructing a piezoelectric coupling spectral element frequency domain solver, and importing the above-mentioned surface acoustic wave filter discrete domain and solution parameters into the solver, and initializing the solver; S4. Construct an adaptive frequency sweep algorithm, adaptively add the frequency points that need to be calculated, and call the piezoelectric coupling frequency domain solver to calculate the admittance of the frequency points, and finally obtain the frequency-admittance characteristic curve of the surface acoustic wave filter model under wide bandwidth.

2. The method for rapid simulation of surface acoustic wave filter adaptive frequency sweeping according to claim 1, characterized in that: The calculation model proposed in step S1 includes: the length, width, thickness, material properties of the base, substrate and metal conductive areas, the number of metal conductive areas, the boundary conditions of the base, substrate and metal conductive areas, and the circuit excitation parameters connected to the metal conductive areas, wherein the circuit includes impedance.

3. The method for rapid simulation of surface acoustic wave filter adaptive frequency sweeping according to claim 1, characterized in that: The solver initialization proposed in step S3 uses the spectral element method to construct a weak form of the piezoelectric coupled differential equation, including elastic waves and electrostatic fields, and the corresponding coupled constitutive equations.

4. The solver initialization method according to claim 3, characterized in that: It is necessary to extract the plate potential U0 from the discrete potential U, and obtain the plate current I after solving the circuit connected to the plate. The admittance matrix Y of the surface acoustic wave filter is calculated using Ohm's law. The mapping relationship Y=f(ω) between the admittance matrix Y and a single frequency point ω can be obtained for subsequent adaptive frequency scanning.

5. The method for rapid simulation of surface acoustic wave filter adaptive frequency sweeping according to claim 1, characterized in that: The adaptive frequency sweep algorithm proposed in step S4 includes the following initial parameters: the minimum value f of the target frequency band min and the maximum value f max , the initial number of frequency points N0, the initial iteration step k = 0, the maximum iteration error e max , the maximum number of iteration steps k max .

6. The surface acoustic wave filter adaptive frequency sweeping fast simulation method according to claim 1, characterized in that: The adaptive frequency sweep algorithm proposed in step S4 adopts the following process: Before the iteration starts, set the number of frequency points N (0) =N0, calculate the sampling frequency W of the initial iteration (0) , calculate the admittance matrix Y of each frequency point (0) ; During the iteration process, a rational function is used to fit the relationship between the admittance matrix and the sampling frequency point, and the fitting function is obtained. Calculate the global error e g , the global error e of the current iteration step g Greater than the maximum iterative error e max , or the iteration number k is greater than the maximum number k allowed max In the case of , the iteration ends; If the iteration is not finished, calculate the frequency band error e between every two sampling frequency points l,n , select the frequency band error e l The largest interval, insert the frequency sweep point at its center, calculate the admittance matrix of the frequency sweep point, and insert the new sampling frequency point and admittance matrix into W (k) and Y (k) In the example above, construct the sampling frequency W used in the next iteration. (k+1) and the admittance matrix Y (k+1) , update N (k+1) , restart the iteration process.

7. The adaptive frequency sweep algorithm according to claim 6, characterized in that: The global error e g Defined as each sampling frequency W (k) In the equation, the admittance matrix Y (k) With the fitting function The root mean square error between:

8. The adaptive frequency sweep algorithm according to claim 6, characterized in that: The error e between every two sampling frequency points l,n is obtained by analyzing the relative errors of each sub - frequency band in the current iteration and the previous iteration. In the k - th iteration step, for the frequency band between point n and point n + 1 (n < N - 1), the integral is defined Among them, i, j are the row index and column index of the matrix. The above integral can be calculated by numerical integration methods such as Gaussian integral; the error e between each two sampling frequency points l,n The calculation method is:

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