Rapid calculation method, device and equipment of surface wave propagation constant, medium and program product

Abstract

The invention relates to the field of computational electromagnetics, and provides a method, a device, equipment, a medium and a program product for rapidly calculating a surface wave propagation constant, the method comprises the following steps: step 1, inputting a dielectric constant of a medium material uniformly coated above an infinite metal surface; 2, calculating a surface wave propagation constant for the input dielectric constant by adopting an explicit solution expression of the surface wave propagation constant; and step 3, outputting the calculated surface wave propagation constant. The calculation precision of the surface wave propagation constant is higher than that of an existing approximation method.

Description

Technical Field

[0001] The present invention relates to the field of computational electromagnetics, and more particularly, to a method, apparatus, device, medium, and program product for rapidly calculating the propagation constant of surface waves. Background Art

[0002] Solving the propagation constant of surface waves in a homogeneous dielectric material coated on an infinite metal surface is a fundamental problem in fields such as microstrip antenna design, surface wave antenna design, active standing wave analysis of antenna arrays, and radiation calculation of sources in layered dielectrics.

[0003] For a dielectric layer model as shown in Figure 1 with a thickness of d and a dielectric constant of ε above an infinite conductor surface, the dispersion equation of the surface wave propagation constant is: r k

[0004]

[0005] k c tan(k c d)=ε r h

[0006] where k0 is the propagation constant in air, h is the attenuation constant of the surface wave in air, k c is the transverse (x - direction) cut - off wavenumber of the surface wave, and β is the propagation constant of the surface wave along the z - direction.

[0007] Currently, there are various methods for studying the propagation constant in a dielectric - covered layer on a conductor surface. The graphical method (see reference: Pozar, D.M., Microwave Engineering, Third Edition, translated by Zhang Zhaoyi et al. Beijing: Electronic Industry Press, 2006), the approximate method (see reference: A. Kobyakov, “Analytical form of the scan blindness condition in an infinite phased array of printed dipoles with a superstrate”, Proc. European Conference on Antennas and Propagation (EuCAP‘22), Mar. 27 - Apr. 1, 2022), etc.

[0008] The graphical method can obtain a relatively accurate theoretical solution, but it cannot obtain an explicit solution expression for the TM - mode surface wave propagation constant in a homogeneous dielectric material coated on an infinite metal surface analytically. It can only be solved by numerical methods, and the solving speed is slow; while the existing approximate methods are only effective for dielectrics with a small electrical thickness. When the dielectric constant is high, the dielectric layer is thick, or the frequency is high, the solving error will be rapidly amplified.

[0009] Patent CN114942490B discloses a calculation method for solving the propagation constant in an optical fiber, but it cannot calculate the propagation constant in the dielectric layer on the metal surface. SUMMARY OF THE INVENTION

[0010] In view of the problems of complex calculation of the TM surface wave propagation constant in the homogeneous dielectric material coated on the infinite metal surface and low calculation accuracy under the condition of the explicit expression of the electrical thickness, the present invention provides a fast calculation method, device, equipment, medium and program product for the surface wave propagation constant.

[0011] In a first aspect, the present invention provides a fast calculation method for the surface wave propagation constant, including:

[0012] Step 1, input the dielectric constant of the homogeneous dielectric material coated on the infinite metal surface;

[0013] Step 2, for the input dielectric constant, calculate the surface wave propagation constant by using the explicit solution expression of the surface wave propagation constant; the explicit solution expression of the surface wave propagation constant is as follows:

[0014]

[0015] where ε r is the dielectric constant of the homogeneous dielectric material coated on the infinite metal surface, k0 is the propagation constant in air, and θ is the argument in the conformal transformation;

[0016] Step 3, output the calculated surface wave propagation constant.

[0017] In some embodiments, the method for obtaining the explicit solution expression of the surface wave propagation constant includes:

[0018] Step 21, perform conformal transformation and simplification on the surface wave transcendental equation to obtain a first equation;

[0019] Step 22, use Taylor approximation for the first equation and write it in the form of an arctangent function to obtain a second equation;

[0020] Step 23, perform electrical thickness approximation on the second equation to obtain an algebraic equation about the argument of the conformal transformation;

[0021] Step 24, solve the algebraic equation, and generate the explicit solution expression of the surface wave propagation constant according to the calculation formula of the surface wave propagation constant and the relationship of the conformal transformation.

[0022] In some embodiments, step 21 specifically includes:

[0023] Step 211, perform conformal transformation on the surface wave transcendental equation: k c = rcosθ, h = rsinθ, transform the surface wave transcendental equation into: where r is the radius of the circle equation in the conformal transformation, which is eliminated after the conformal transformation, θ is the argument in the conformal transformation, h is the attenuation constant of the surface wave in air, d is the thickness of the medium material uniformly coated above the infinite metal surface, k0 is the propagation constant in air, and k c is the transverse cut-off wave number of the surface wave;

[0024] Step 212, let the intermediate substitution quantity Simplify the surface wave transcendental equation into the first equation tan(acosθ) = ε r tanθ.

[0025] In some embodiments, step 22 specifically includes:

[0026] Perform a first-order Taylor approximation on tanθ and cosθ in the first equation and write it in the form of an arctangent function to obtain the second equation

[0027] In some embodiments, step 23 specifically includes:

[0028] Adopt the first-order Taylor approximation of the large-argument arctangent function Obtain an algebraic equation about the conformal transformation argument θ:

[0029] In some embodiments, in step 24, solve the algebraic equation of the conformal transformation argument θ according to the root formula of the cubic equation of one variable to obtain the solution of the conformal transformation argument θ:

[0030] where Both p and q are intermediate substitution quantities used to simplify the expression.

[0031] In a second aspect, the present invention provides a device for quickly calculating the surface wave propagation constant, including:

[0032] An input unit for inputting the dielectric constant of the medium material uniformly coated above the infinite metal surface;

[0033] A processing unit for calculating the surface wave propagation constant by using the explicit solution expression of the surface wave propagation constant for the input dielectric constant; the explicit solution expression of the surface wave propagation constant is as follows:

[0034]

[0035] where ε ris the dielectric constant of the medium material uniformly coated above the infinite metal surface, k0 is the propagation constant in air, and θ is the argument in the conformal transformation;

[0036] An output unit, configured to output the calculated surface wave propagation constant.

[0037] In a third aspect, the present invention provides an electronic device, including:

[0038] At least one processor; and a memory communicatively connected to the at least one processor;

[0039] Wherein, the memory stores instructions executable by the at least one processor, and the at least one processor, by executing the instructions stored in the memory, causes the at least one processor to execute the above method.

[0040] In a fourth aspect, the present invention provides a computer-readable storage medium, which is used to store instructions, and when the instructions are executed, the above method is implemented.

[0041] In a fifth aspect, the present invention provides a computer program product, which, when called by a computer, causes the computer to execute the above method.

[0042] In summary, due to the adoption of the above technical solutions, the beneficial effects of the present invention are:

[0043] Aiming at the problems of complex calculation of the TM surface wave propagation constant in the medium material with uniform electric thickness coated above the infinite metal surface and low calculation accuracy under the explicit expression electric thickness condition at present, the present invention uses the explicit solution expression of the surface wave propagation constant to calculate the surface wave propagation constant, which can improve the calculation accuracy of the surface wave propagation constant and has a faster solution speed. Description of the Drawings

[0044] Figure 1 It is a schematic diagram of a single-layer dielectric structure on the conductor surface.

[0045] Figure 2 It is a flowchart of a fast calculation method for the surface wave propagation constant provided by an embodiment of the present invention.

[0046] Figure 3 In the embodiment of the present invention, when d = 3mm and ε r = 6, it is a graph of the calculation result of the surface wave propagation constant.

[0047] Figure 4 In the embodiment of the present invention, when d = 3mm and ε r = 10, it is a graph of the calculation result of the surface wave propagation constant.

[0048] Figure 5In the embodiment of the present invention, d = 3 mm, ε r When = 15, it is a calculation result diagram of the surface wave propagation constant.

[0049] Figure 6 In the embodiment of the present invention, when d = 3 mm, different dielectric constants ε r It is a comparison diagram of the calculation accuracy of the surface wave propagation constant under different conditions.

[0050] Figure 7 In the embodiment of the present invention, d = 6 mm, ε r When = 3, it is a calculation result diagram of the surface wave propagation constant.

[0051] Figure 8 In the embodiment of the present invention, d = 10 mm, ε r When = 3, it is a calculation result diagram of the surface wave propagation constant.

[0052] Figure 9 In the embodiment of the present invention, d = 15 mm, ε r When = 3, it is a calculation result diagram of the surface wave propagation constant.

[0053] Figure 10 In the embodiment of the present invention, ε r When = 3, it is a comparison diagram of the calculation accuracy of the surface wave propagation constant under different thicknesses d.

[0054] Figure 11 It is a schematic structural diagram of a fast calculation device for the surface wave propagation constant provided in the embodiment of the present invention.

[0055] Figure 12 It is a schematic structural diagram of an electronic device provided in the embodiment of the present invention. Detailed implementation manners

[0056] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. Usually, the components of the embodiments of the present invention described and illustrated herein can be arranged and designed in various different configurations.

[0057] Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0058] Aiming at the problems of complex calculation of the propagation constant of TM surface waves in a homogeneous dielectric material coated on an infinite metal surface and low calculation accuracy under the condition of the explicit expression of the electrical thickness, such as Figure 2 shown, an embodiment of the present invention proposes a fast calculation method for the propagation constant of surface waves, including the following steps:

[0059] Step 1, input the dielectric constant of the dielectric material uniformly coated above the infinite metal surface;

[0060] Step 2, for the input dielectric constant, calculate the propagation constant of the surface wave by using the explicit solution expression of the propagation constant of the surface wave;

[0061] Step 3, output the calculated propagation constant of the surface wave.

[0062] In some embodiments, the method for obtaining the explicit solution expression of the propagation constant of the surface wave includes:

[0063] Step 21, perform a conformal transformation on the surface wave transcendental equation and simplify it to obtain a first equation; specifically:

[0064] Step 211, perform a conformal transformation on the surface wave transcendental equation: k c = rcosθ, h = rsinθ, and transform the surface wave transcendental equation into:

[0065] where r is the radius of the circle equation in the conformal transformation, which is eliminated after the conformal transformation, θ is the argument in the conformal transformation, h is the attenuation constant of the surface wave in the air, d is the thickness of the dielectric material uniformly coated above the infinite metal surface, k0 is the propagation constant in the air, and k c is the transverse cut-off wave number of the surface wave;

[0066] Step 212, let the intermediate substitution quantity and simplify the surface wave transcendental equation to the first equation tan(acosθ) = ε r tanθ.

[0067] Step 22, perform a Taylor approximation on the first equation and write it in the form of an arctangent function to obtain a second equation; specifically:

[0068] Perform a first-order Taylor approximation on tanθ and cosθ in the first equation and write it in the form of an arctangent function to obtain the second equation

[0069] Step 23, perform an electrical thickness approximation on the second equation to obtain an algebraic equation about the argument of the conformal transformation; specifically:

[0070] Adopt the first-order Taylor approximation of the arctangent function of a large argument An algebraic equation for the argument θ of the conformal transformation is obtained:

[0071] Step 24: Solve the algebraic equation, and generate an explicit solution expression for the surface wave propagation constant according to the relationship between the surface wave propagation constant, the dielectric constant, the propagation constant in air, and the transverse cutoff wavenumber of the surface wave, as well as the relationship of the conformal transformation. Specifically:

[0072] Solve the algebraic equation of the argument θ of the conformal transformation according to the root formula of the cubic equation of one variable to obtain the solution of θ: Wherein, Both p and q are intermediate substitution quantities used to simplify the expression.

[0073] According to the calculation formula of the surface wave propagation constant And k in the conformal transformation c = rcosθ, the explicit solution expression of the surface wave propagation constant is generated as: Using this explicit solution expression of the surface wave propagation constant, the dielectric constant of the medium material uniformly coated above the infinite metal surface can be input to calculate the surface wave propagation constant.

[0074] In order to verify the effect of the present invention, the dielectric constants ε of the medium material uniformly coated above the infinite metal surface with thickness d = 3mm are calculated respectively r To be 6, 10, 15, and the calculation results of the surface wave propagation constant are respectively as Figure 3 , Figure 4 , Figure 5 Shown, and the comparison of the calculation accuracy of the surface wave propagation constant is as Figure 6 Shown. The dielectric constant ε of the medium material uniformly coated above the infinite metal surface r = 3, and the thicknesses d are 6mm, 10mm, and 15mm respectively. The calculation results of the surface wave propagation constant are as Figure 7 , Figure 8 , Figure 9 Shown, and the comparison of the calculation accuracy of the surface wave propagation constant is as Figure 10 Shown. From Figure 6 , Figure 10 It can be seen that the calculation accuracy of the surface wave propagation constant by the method of the present invention is higher than that of the existing approximate methods.

[0075] Based on the same technical concept, as Figure 11 Shown, the embodiment of the present invention also provides a fast calculation device for the surface wave propagation constant, including:

[0076] An input unit for inputting the dielectric constant of the medium material uniformly coated above the infinite metal surface;

[0077] A processing unit, configured to calculate a surface wave propagation constant for an input dielectric constant by using an explicit solution expression of the surface wave propagation constant; the explicit solution expression of the surface wave propagation constant is as follows:

[0078]

[0079] where ε r is the dielectric constant of a dielectric material uniformly coated above an infinite metal surface, k0 is the propagation constant in air, and θ is the argument in the conformal transformation;

[0080] An output unit, configured to output the calculated surface wave propagation constant.

[0081] For each unit in the above device, reference may be made to the description in the method in the foregoing embodiments, and details are not described herein again.

[0082] Based on the same technical concept, an embodiment of the present invention further provides an electronic device, which can implement the fast calculation method flow of the surface wave propagation constant provided in the foregoing embodiments of the present invention. In one embodiment, the electronic device may be a server, or a terminal device or other electronic devices. As Figure 12 shown, the electronic device may include:

[0083] At least one processor, and a memory connected to the at least one processor. In the embodiments of the present invention, the specific connection medium between the processor and the memory is not limited. Figure 12 Here, it is taken as an example that the processor and the memory are connected through a bus. The bus is Figure 12 shown by a thick line in. The connection manners between other components are only for illustrative purposes and are not limiting. The bus may be divided into an address bus, a data bus, a control bus, etc. For the sake of convenience of representation, Figure 12 it is only shown by a thick line in, but it does not mean that there is only one bus or one type of bus. Alternatively, the processor may also be referred to as a controller, and the name is not limited.

[0084] In the embodiments of the present invention, the memory stores instructions executable by the at least one processor. The at least one processor can execute the instructions stored in the memory to execute a fast calculation method of a surface wave propagation constant described above. The processor can implement Figure 12 the functions of each module in the device shown.

[0085] Among them, the processor is the control center of the device, and can connect various parts of the entire control device through various interfaces and lines. By running or executing the instructions stored in the memory and calling the data stored in the memory, various functions of the device and process data, so as to perform overall monitoring on the device.

[0086] In an alternative design, the processor may include one or more processing units. The processor may integrate an application processor and a modem processor. Among them, the application processor mainly processes the operating system, user interface, application programs, etc., and the modem processor mainly processes wireless communications. It can be understood that the above-mentioned modem processor may not be integrated into the processor either. In some embodiments, the processor and the memory may be implemented on the same chip, and in some embodiments, they may also be separately implemented on independent chips.

[0087] The processor may be a general-purpose processor, such as a CPU, a digital signal processor, an application-specific integrated circuit, a field-programmable gate array, or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, and can implement or execute the various methods, steps, and logic block diagrams disclosed in the embodiments of the present invention. The general-purpose processor may be a microprocessor or any conventional processor, etc. The steps of a method for quickly calculating the surface wave propagation constant disclosed in the embodiments of the present invention in combination can be directly embodied as being executed by a hardware processor, or executed by a combination of hardware and software modules in the processor.

[0088] As a non-volatile computer-readable storage medium, the memory can be used to store non-volatile software programs, non-volatile computer-executable programs, and modules. The memory may include at least one type of storage medium, for example, it may include flash memory, hard disk, multimedia card, card-type memory, random access memory (RAM), static random access memory (SRAM), programmable read-only memory (PROM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), magnetic memory, magnetic disk, optical disc, and so on. The memory is any other medium that can be used to carry or store the desired program code in the form of instructions or data structures and can be accessed by a computer, but is not limited thereto. The memory in the embodiments of the present invention may also be a circuit or any other device capable of implementing a storage function, for storing program instructions and / or data.

[0089] By programming the design of the processor, the code corresponding to the method for quickly calculating the surface wave propagation constant introduced in the foregoing embodiments can be solidified into the chip, so that the chip can execute the steps of the method in the above embodiments when running. How to program the design of the processor is a well-known technology to those skilled in the art and will not be elaborated here.

[0090] Based on the same inventive concept, an embodiment of the present invention further provides a storage medium storing computer instructions, which, when running on a computer, cause the computer to execute a fast calculation method of a surface wave propagation constant described above.

[0091] In some alternative embodiments, various aspects of the fast calculation method of the surface wave propagation constant of the present invention can also be implemented in the form of a program product, which includes program code that, when the program product runs on a device, is used to cause the control device to execute the steps in the fast calculation method of the surface wave propagation constant according to various exemplary embodiments of the present invention described above in this specification.

[0092] It should be noted that although several units or subunits of the device are mentioned in the above detailed description, this division is merely exemplary and not mandatory. In fact, according to the embodiments of the present invention, the features and functions of the two or more units described above can be embodied in one unit. Conversely, the features and functions of one unit described above can be further divided and embodied by multiple units. In addition, although the operations of the method of the present invention are described in a specific order in the drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. Additionally or alternatively, some steps may be omitted, multiple steps may be combined into one step for execution, and / or one step may be decomposed into multiple steps for execution.

[0093] Those skilled in the art should understand that the embodiments of the present invention can be provided as a method, a system, or a computer program product. Therefore, the present invention can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present invention can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0094] The present invention is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present invention. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of flows and / or blocks in the flowcharts and / or block diagrams, can be realized by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a server, such that the instructions executed by the processor of the computer or other programmable data processing devices generate for realizing in the process Figure 1 one process or multiple processes and / or blocksFigure 1 a device for the functions specified in one or more boxes

[0095] Program code for performing the operations of the present invention may be written using any combination of one or more programming languages, including object-oriented programming languages such as Java, C++, etc., and also including conventional procedural programming languages such as the "C" language or similar programming languages. The program code may be executed entirely on the user's computing device, partially on the user's device, executed as a stand-alone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server.

[0096] In the case of a remote computing device, the remote computing device may be connected to the user's computing device through any kind of network including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computing device (e.g., connected through the Internet using an Internet service provider).

[0097] These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, such that the instructions stored in the computer-readable memory produce a manufactured article including an instruction device that implements the functions specified in one Figure 1 one process or more processes and / or boxes Figure 1 a box or multiple boxes

[0098] These computer program instructions may also be loaded onto a computer or other programmable data processing device, such that a series of operational steps are performed on the computer or other programmable device to produce a computer-implemented process, and thus the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one Figure 1 one process or more processes and / or boxes Figure 1 a box or multiple boxes

[0099] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may have various changes and modifications. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A fast calculation method for surface wave propagation constant, characterized in that, Including: Step 1: Input the dielectric constant of the dielectric material uniformly coated above the infinite metal surface. Step 2: For the input dielectric constant, calculate the surface wave propagation constant using the explicit solution expression of the surface wave propagation constant. The explicit solution expression of the surface wave propagation constant is as follows: where ε r is the dielectric constant of the medium material uniformly coated above the infinite metal surface, k0 is the propagation constant in air, and θ is the argument in the conformal transformation; Step 3: Output the calculated surface wave propagation constant.

2. The rapid calculation method of surface wave propagation constant according to claim 1, characterized in that The method for obtaining the explicit solution expression of the surface wave propagation constant includes: Step 21: Perform a conformal transformation and simplification on the surface wave transcendental equation to obtain the first equation. Step 22: Use Taylor approximation on the first equation and write it in the form of an arctangent function to obtain the second equation. Step 23: Use the electrical thickness dimension approximation for the second equation to obtain an algebraic equation regarding the conformal transformation argument. Step 24: Solve the algebraic equation, and based on the calculation formula of the surface wave propagation constant and the relationship of the conformal transformation, generate the explicit solution expression of the surface wave propagation constant.

3. The rapid calculation method of surface wave propagation constant according to claim 2, characterized in that Step 21 specifically includes: Step 211, perform a conformal transformation on the surface wave transcendental equation: k c = rcosθ, h = rsinθ, and transform the surface wave transcendental equation into: where r is the radius of the circle equation in the conformal transformation, which is eliminated after the conformal transformation, θ is the argument in the conformal transformation, h is the attenuation constant of the surface wave in air, d is the thickness of the medium material uniformly coated above the infinite metal surface, k0 is the propagation constant in air, and k c is the transverse cut-off wavenumber of the surface wave; Step 212, let the intermediate substitution quantity Simplify the surface wave transcendental equation into the first equation: tan(acosθ) = ε r tanθ.

4. The rapid calculation method of surface wave propagation constant according to claim 3, characterized in that Step 22 specifically includes: Perform a first-order Taylor approximation on tanθ and cosθ in the first equation and write it in the form of an arctangent function to obtain the second equation 5. The rapid calculation method of the surface wave propagation constant according to claim 4, characterized in that Step 23 specifically includes: Adopt the first-order Taylor approximation of the arctangent function with a large quantity An algebraic equation about the conformal transformation argument θ is obtained as follows:

6. The rapid calculation method of surface wave propagation constant according to claim 5, characterized in that In Step 24, solve the algebraic equation of the conformal transformation argument θ according to the root formula of the cubic equation to obtain the solution of the conformal transformation argument θ: Among them, both p and q are intermediate substitution variables used to simplify the expression.

7. A fast calculation device for surface wave propagation constant, characterized in that Including: An input unit for inputting the dielectric constant of the dielectric material uniformly coated above the infinite metal surface. A processing unit for calculating the surface wave propagation constant using the explicit solution expression of the surface wave propagation constant for the input dielectric constant. The explicit solution expression of the surface wave propagation constant is as follows: where ε r is the dielectric constant of the medium material uniformly coated above the infinite metal surface, k0 is the propagation constant in air, and θ is the argument in the conformal transformation; An output unit for outputting the calculated surface wave propagation constant.

8. An electronic device, characterized in that, Including: At least one processor; And a memory communicatively connected to the at least one processor; Wherein, the memory stores instructions executable by the at least one processor, and the at least one processor, by executing the instructions stored in the memory, causes the at least one processor to execute the method according to any one of claims 1 - 6.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium is used to store instructions, and when the instructions are executed, the method according to any one of claims 1 - 6 is implemented.

10. A computer program product, characterized in that, When the computer program product is called by a computer, the computer is caused to execute the method according to any one of claims 1 - 6.

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