Method for fast prediction of electrical performance of flexible antenna one-dimensional deformation

Abstract

This invention discloses a method for rapid prediction of the one-dimensional deformable electrical performance of flexible antennas, specifically including the following steps: Step 1, given the shape parameters of the original model of the air dielectric patch antenna; Step 2, establishing the original model of the microstrip antenna in Feko based on the shape parameters and feed position given in Step 1; Step 3, dividing the original model established in Step 2 into triangular meshes using RWG basis functions; Step 4, obtaining the nodes of each triangle in the triangular mesh and their corresponding node coordinates in Feko; Step 5, calculating the surface current J of the microstrip antenna based on the triangular mesh nodes and their corresponding node coordinates obtained in Step 4; Step 6, calculating the electromagnetic field based on the surface current J obtained in Step 5. This invention solves the problems of long time consumption and low efficiency in existing prediction methods.

Description

Technical Field

[0001] This invention belongs to the field of flexible antenna electrical performance analysis technology, and relates to a method for rapid prediction of the one-dimensional deformable electrical performance of flexible antennas. Background Technology

[0002] With the development of wireless communication technology, electronic equipment is gradually trending towards higher frequency, smaller size, and lighter weight. Simultaneously, the contradiction between the spatial environment requirements and electrical performance demands of antennas is becoming increasingly apparent. To address this issue, flexible antennas break through the structural limitations of traditional rigid materials, being stretchable, bendable, small in size, and lightweight, leading to their increasingly widespread application in special operations, smart wearables, and aerospace. Flexible patch antennas, with their advantages of low profile, easy conformal design, and simple structure, have become a new research hotspot. However, structural flexibility also increases the difficulty of antenna electrical performance analysis and compensation. Because the electrical performance of high-frequency passive devices is extremely sensitive to changes in structural dimensions, the dynamic deformation of flexible antennas under operating conditions significantly affects their current distribution and radiation field. For this dynamic and complex electromagnetic problem involving close mechanical and electrical connections, traditional rigid antenna analysis methods require a large number of transient sample simulations, resulting in low analysis efficiency. However, utilizing electromechanical coupling (EMC) theory can achieve accurate and efficient analysis. Summary of the Invention

[0003] The purpose of this invention is to provide a method for rapid prediction of the one-dimensional deformable electrical performance of flexible antennas. This method takes into account the influence of the antenna's own deformation on its electrical performance and greatly reduces the number of sample points required for prediction, thus solving the problems of long time consumption and low efficiency of existing prediction methods.

[0004] The technical solution adopted in this invention is a method for rapid prediction of the one-dimensional deformable electrical performance of flexible antennas, which specifically includes the following steps:

[0005] Step 1: Given the shape parameters of the original model of the air dielectric patch antenna, including: the excitation method is probe feeding, the feed position is given, the voltage V of the feed source is V, and the frequency is f;

[0006] Step 2: Based on the shape parameters and feed location given in Step 1, establish the original model of the microstrip antenna in Feko;

[0007] Step 3: Divide the original model established in Step 2 into triangular meshes using RWG basis functions;

[0008] Step 4: Obtain the nodes and corresponding coordinates of each triangle in the triangular mesh in Feko, and transfer the obtained nodes and node coordinates to Matlab;

[0009] Step 5: Calculate the surface current J of the microstrip antenna based on the triangular mesh nodes obtained in Step 4 and the corresponding node coordinates.

[0010] Step 6: Calculate the electromagnetic field based on the surface current J obtained in Step 5.

[0011] The invention is further characterized by:

[0012] In step 3, the RWG basis function is defined as follows:

[0013]

[0014] Among them, l n Let n be the length of the common side of the nth pair of triangles. A- n Let the area be the triangle. V n - is the position vector of the triangle vertex. Let r be a vector pointing from the vertex of a triangle to point r. Outside the triangle pair, the basis function value is 0.

[0015] The specific process of step 5 is as follows:

[0016] Step 5.1, transform the integral equation in formula (2) and substitute it into the L operator equation in formula (3) to obtain formula (4):

[0017]

[0018] in, Let η be the unit vector normal to the antenna surface, η be the wave impedance in free space, and E be the unit vector normal to the antenna surface. i For the incident electric field, the expression for the L operator is:

[0019]

[0020] Where k is the wave number in free space, and g is the Green's function. r is the field point, and r′ is the source point;

[0021]

[0022] Where R = |r - r'| represents the distance s = T from the field point to the source point. + +T - ;

[0023] Step 5.2, Solve formula (4) using the Galerkin method;

[0024] Step 5.3: Combine Chebyshev and Melly approximations to calculate the surface current at individual locations of the antenna, and derive the surface current under all curvatures of the antenna based on the surface current.

[0025] The specific process of step 5.2 is as follows:

[0026] Step 5.2.1, the surface current J is approximately expanded using the RWG basis functions as follows:

[0027]

[0028] Where N is the total number of common edges in the model's triangular mesh, α n The current expansion coefficient is unknown;

[0029] Step 5.2.2, substitute formula (5) into formula (4), and use the RWG basis function as the test function to obtain:

[0030]

[0031] in, All are RWG basis functions;

[0032] The matrix elements in formula (6) are:

[0033]

[0034] Z mn Substitute the left side of the equation (6) with V in the following equation (8). m Substitute the right side of the equation (6):

[0035]

[0036] Step 5.2.4, introduce the voltage model in the following formula (9) to solve for V in step 5.2.3. m :

[0037] E i =Vδ(y)n y (9);

[0038] Where δ(y) is a delta function, n y V is the unit vector in the Y direction, and V is the voltage.

[0039] If the gaps in the model shown in formula (9) are replaced with the common edges of the antenna grid, then the excitation model is applied only on the inner edges of the feed grid. Let the nth edge be the feed excitation edge, then the expression for the voltage matrix V is as follows:

[0040]

[0041] The discretized matrix equation of EFIE is obtained through formulas (7) and (10), and the coefficients α are obtained by matrix inversion. n , will α n Substituting into formula (5), the surface current J can be calculated.

[0042] The specific process of step 5.3 is as follows:

[0043] Step 5.3.1, in order to represent the bending state of the microstrip antenna, a one-dimensional bending deformation model is introduced, as shown in the following formula (11):

[0044]

[0045] Where R is the radius of the curved arc, and assuming the original model has a length of l and is placed horizontally on the plane z = 0, then the arc length x after deformation is... l = l / 2, θ is the central angle corresponding to point P, x0 is the arc length corresponding to the central angle θ, x p Let Z be the x-coordinate of point P. p Let P be the ordinate of point P;

[0046] Step 5.3.2: Following step 5.3.3, the coordinates of the antenna surface nodes with different curvatures are obtained. Given that the current coefficient matrix I is a one-dimensional function of R, a Chebyshev approximation is first performed. To satisfy the domain requirement of the Chebyshev expansion, a coordinate transformation is first performed to obtain R'. The range of values ​​for the given curvature variation R is R∈[R...]. a R b R` is shown in formula (12) below:

[0047]

[0048] When R = R a When R' = R, R' = -1; when R = R b When R' = 1, i.e., R' ∈ [-1, 1], then the function I(R) is expressed as:

[0049]

[0050] Step 5.3.3: Calculate the Chebyshev zeros, where Q is the order of the Chebyshev polynomial expansion. A Chebyshev polynomial of order Q+1 has Q+1 zeros. The calculation formula is as follows:

[0051]

[0052] Then, according to formulas (12) and (14), the Chebyshev sampling node R can be obtained. i The calculation formula is:

[0053]

[0054] Step 5.3.4, introducing the Chebyshev expansion, we obtain:

[0055]

[0056] in,

[0057] Among them, T l The Chebyshev polynomial recurrence formula is:

[0058] T0(x)=1, T1(x)=x, T2(x)=2x 2 -1 (17);

[0059] T l+1 (x)=2xT l (x)-T l-1 (x) (18);

[0060] The surface current of the microstrip antenna under different curvatures can be obtained by formula (16);

[0061] Step 5.3.5: Use the Meryl approximation to match the Chebyshev polynomial expansion; the substitution formula is:

[0062]

[0063] In the formula, n is the number of unknowns, and L and M are the order of the expansion. Generally, L = M or L = M + 1.

[0064] in, Written in matrix form, we get:

[0065]

[0066]

[0067] According to formula (19), the current matrix I of the Merley solution under different curvatures is obtained. Then, by substituting the current matrix I back into formula (5), the antenna surface current J under different curvatures can be obtained.

[0068] The specific process of step 6 is as follows:

[0069] Introducing the dipole model, the dipole moment m is equal to the product of the effective dipole current and the effective dipole length:

[0070]

[0071] in: The midpoint of the negative triangle of RWG basis functions. It is the midpoint of the equilateral triangle of the RWG basis functions;

[0072] The vector expression for the radiation field strength of an infinitesimal dipole at a point r in free space is:

[0073]

[0074] Where r is the vector from the source point to the field point, and r is the distance from the source point to the field point;

[0075]

[0076] .

[0077] The beneficial effects of this invention are as follows:

[0078] 1. Existing technologies neglect the impact of changes in the antenna's own structure on electrical performance when deforming array antenna elements. This invention addresses this deficiency by considering the impact of bending deformation of the flexible antenna itself on electrical performance, providing a basis for dynamic compensation analysis of electrical performance and improving the accuracy of the analysis.

[0079] 2. Existing technologies using the MOM distance method for antenna deformation prediction require calculating Z(r), V(R), and I(R) for each deformation interval to obtain the desired antenna electrical performance prediction. However, the BURA extrapolation algorithm only needs to calculate Z(R), V(R), and I(R) at the sampling points, and extrapolates the required I(R) for other points using information from a few sampling points to ultimately obtain the desired antenna electrical performance prediction. Furthermore, it meets the required computational accuracy. Attached Figure Description

[0080] Figure 1 This is a diagram of the FEKO antenna model used in the rapid prediction method for the one-dimensional deformable electrical performance of flexible antennas in this invention.

[0081] Figure 2 This is a microstrip antenna mesh model diagram in the method for rapid prediction of the one-dimensional deformable electrical performance of flexible antennas in this invention;

[0082] Figure 3 This is a diagram of the RWG basis function model in the fast prediction method for the one-dimensional deformable electrical performance of flexible antennas in this invention;

[0083] Figure 4 This is a model diagram of the centroid segmentation method in the fast prediction method for one-dimensional deformable electrical performance of flexible antennas in this invention;

[0084] Figure 5 This is a δ voltage source model diagram in the fast prediction method for the one-dimensional deformable electrical performance of flexible antennas in this invention;

[0085] Figure 6 This is a bending model diagram of the antenna in the fast prediction method for the one-dimensional deformable electrical performance of the flexible antenna of the present invention;

[0086] Figure 7 This is a diagram of the dipole equivalent model in the fast prediction method for the one-dimensional deformable electrical performance of flexible antennas in this invention.

[0087] Figures 8(a) and (b) are mesh models of different bending radii R in the fast prediction method of one-dimensional deformable electrical performance of flexible antennas of the present invention; wherein, R = 1 in Figure 8(a) and R = 1.05 in Figure 8(b);

[0088] Figures 9(a) and (b) are comparison diagrams of radiation pattern and gain analysis results in the fast prediction method of one-dimensional deformable electrical performance of flexible antenna of the present invention; wherein, Figure 9(a) is the gain error; and Figure 9(b) is the radiation pattern when R=0.1.

[0089] Figures 10(a) and (b) are antenna array mesh models with different bending radii R in the fast prediction method of one-dimensional deformable electrical performance of flexible antennas in this invention; wherein, R = 1 in Figure 10(a) and R = 1.06 in Figure 10(b);

[0090] Figures 11(a) and (b) are comparison diagrams of the radiation pattern and gain results of the bent array in the fast prediction method of the one-dimensional deformable electrical performance of the flexible antenna of the present invention.

[0091] Figure 12 This is a diagram showing the change in radiation pattern during the bending process of the antenna array in the fast prediction method for the one-dimensional deformation electrical performance of flexible antennas in this invention. Detailed Implementation

[0092] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0093] The present invention provides a method for rapid prediction of the one-dimensional deformable electrical performance of flexible antennas, which specifically includes the following steps:

[0094] Step 1: Given the shape parameters of the original model of the air dielectric patch antenna; the excitation method is probe feeding, and the feeding position, voltage V and frequency f of the feed source are given.

[0095] Step 2: Based on the shape parameters and feed location given in Step 1, establish the original model of the microstrip antenna in Feko (e.g., ...). Figure 1 ).

[0096] Step 3: Divide the original model established in Step 2 into triangular meshes using RWG basis functions (e.g., Figure 2 The RWG basis function is defined as follows:

[0097]

[0098] like Figure 3 As shown, where l n Let n be the length of the common side of the nth pair of triangles. Let the area be the triangle. This is the position vector of the triangle vertex. Let be a vector pointing from a vertex of the triangle to point r. Outside the triangle pair, the basis function value is 0. This definition determines that the current will flow from... Triangular flow direction Triangles, and the positive and negative signs of triangles are relative. In the same basis function, if one triangle is marked as "+", then the other is automatically marked as "-".

[0099] Step 4: Obtain the nodes of each triangle and their corresponding node coordinates in Feko, and then transfer them to Matlab for the next step of electrical performance calculation.

[0100] Step 5: Based on the triangular mesh nodes obtained in the above steps, as well as the corresponding node coordinates and excitation methods, the feed frequency f and voltage U can be used to calculate the microstrip antenna surface current J.

[0101] Solving the electric field integral equation (EFIE): To calculate the surface current J;

[0102] in, Let η be the unit vector normal to the antenna surface, η be the wave impedance in free space, and E be the unit vector normal to the antenna surface. i For the incident electric field, the expression for the L operator is:

[0103]

[0104] Where k is the wave number in free space, and g is the Green's function. r is the field point, and r′ is the source point;

[0105] Integral equation of electric field Substituting the transformation into the L operator, we get:

[0106]

[0107] Where R = |r - r'| represents the distance s = T from the field point to the source point. + +T -

[0108] To solve this equation using the Galerkin method, the surface current J is first approximately expanded using the RWG basis functions as follows:

[0109]

[0110] N is the total number of common edges in the model's triangular mesh, α n For unknown current expansion coefficients

[0111] Substituting formula (2) into (1) and using the RWG basis function as the test function, we get:

[0112]

[0113] All are RWG basis functions;

[0114] The matrix element Z in formula (3) mn :

[0115]

[0116] The integral calculation uses the centroid partitioning method to trisect each side of the original triangular mesh, transforming the integration of the impedance matrix into an accumulation operation, thus dividing it into nine sub-triangles of equal area, such as... Figure 4 As shown, r is the midpoint of the mother triangle. Let Z be the midpoint of a sub-triangle, and take the midpoint of the sub-triangle as the integration point. mn (The left side of equation (3) represents the contribution of the triangle pair m (edge ​​element m) to the current on the triangle pair n (edge ​​element n) through the radiation field; the right side of equation (3) uses V m The alternative is shown in the following formula:

[0117] ;

[0118] To calculate V m like Figure 5 Introducing a voltage model:

[0119] E i =Vδ(y)n y

[0120] Where δ(y) is a delta function, n y V is the unit vector in the Y direction, and V is the voltage.

[0121] Replacing the gaps in the model with common edges of the antenna mesh, the excitation model is applied only to the inner edges of the feed mesh, with a voltage V amplitude of 1, while the excitation electric field on other edges is 0. Let the nth edge be the feed excitation edge; then the expression for the voltage matrix V is as follows:

[0122]

[0123] Based on the discretized matrix equation Z of EFIE mn and V m α can be obtained by finding the matrix inverse. n α n

[0124] The coefficient set is obtained, which is the current matrix I. nThus, the surface current J can be obtained. The MOM method of moments can be used to obtain the surface current of the model when it is not curved. Similarly, the surface current under a single curvature can also be obtained. However, the efficiency of obtaining the surface current under many curvature states is very low when the antenna is continuously changing. The following will combine Chebyshev approximation and Meryl approximation to extrapolate the surface current under all curvatures by obtaining the surface current at several positions.

[0125] Step 6, to represent the bending state of the microstrip antenna, as follows: Figure 6 Introducing a one-dimensional bending deformation model, the expression is as follows:

[0126]

[0127] Where R is the radius of the curved arc, and assuming the original model has a length of l and is placed horizontally on the plane z = 0, then the arc length x after deformation is... l = l / 2, θ is the central angle corresponding to point P, x0 is the arc length corresponding to the central angle θ, x p Let Z be the x-coordinate of point P. p Let P be the ordinate of point P. This allows us to model the position coordinates of nodes in different bending states within MATLAB, which can then be used to calculate their surface currents.

[0128] The MOM impedance matrix Z(r), excitation vector V(r), and current coefficient matrix I(r) are related to the mesh node position vector. r Related, and r Furthermore, due to the influence of the bending radius R, Z(R), V(R), and I(R) can all be considered as functions of the radius R; here, Z(r), V(r), and I(r) represent the entire matrix, and Z in the above text... mn V is an element in Z(r). n For elements in V(r), I n For elements in I(r)

[0129] Step 7: Using the coordinates of the surface node positions of the antenna with different curvatures obtained in Step 6, and knowing that the current coefficient matrix I is a one-dimensional function of R, we first perform Chebyshev approximation. To satisfy the domain requirement of the Chebyshev expansion, we need to perform coordinate transformation to obtain R'.

[0130] Given the range of bending variation R, the range of values ​​R∈[R a R b ]

[0131]

[0132] When R = R a When R' = R, R' = -1; when R = R bWhen R' = 1, i.e., R' ∈ [-1, 1], then the function I(R) can be expressed as:

[0133]

[0134] To calculate the Chebyshev zeros, where Q is the order of the Chebyshev polynomial expansion, and a Chebyshev polynomial of order Q+1 has Q+1 zeros, the formula is as follows:

[0135]

[0136] Then, according to formulas (4) and (5), the Chebyshev sampling node R can be obtained. i The calculation formula is:

[0137]

[0138] R i That is, the current matrix I at the i positions to be determined;

[0139] Introducing the Chebyshev expansion, we obtain

[0140]

[0141] in,

[0142] Among them, T l The recurrence relation for Chebyshev polynomials is:

[0143] T0(x)=1, T1(x)=x, T2(x)=2x 2 -1;

[0144] T l+1 (x)=2xT l (x)-T l-1 (x);

[0145] The surface current of the microstrip antenna under different curvatures can be obtained by formula (6). However, since the Chebyshev solution is not accurate enough, the Merry approximation rationalization is performed to make the result more accurate.

[0146] Step 8: Use the Meryl approximation to match the Chebyshev polynomial expansion. The substitution formula is:

[0147] In the formula, n is the number of unknowns, and L and M are the orders of the expansion, generally L = M or L = M + 1.

[0148] in, Written in matrix form, we get:

[0149]

[0150]

[0151] Generally, b0 = 1 is taken. According to formula (7), the current matrix I of the Merley solution under different curvatures can be obtained, that is, α n Substituting back into formula (2), we can obtain the antenna surface current J under different curvatures.

[0152] Step 9: Electromagnetic field calculations can be performed using the surface current obtained in Step 8. To perform these calculations, a dipole model (such as...) is introduced. Figure 7 The dipole moment m is equal to the product of the effective dipole current and the effective dipole length.

[0153]

[0154] in: It is the midpoint of the negative triangle of the RWG basis functions. It is the midpoint of the equilateral triangle of the RWG basis functions;

[0155] The radiation field of an infinitesimal dipole can be calculated analytically. The vector expression for its radiation field strength at a point r in free space is:

[0156]

[0157] Where r is the vector from the source point to the field point, and r is the distance from the source point to the field point; The total radiated field strength of the antenna can be obtained by summing the field strength vectors of all infinitesimal dipoles.

[0158] Example 1

[0159] Single air dielectric patch antenna

[0160] 1. The total radiated power of an antenna can be calculated using the electric and magnetic field strengths. The formula for calculating the Poynting vector W at a point r in the far-field space is:

[0161]

[0162] Where E(r) is the electric field intensity at that point, and H * (r) is the complex conjugate of the magnetic field strength at that point.

[0163] Due to the relationship between radiation power density and spatial field point distance r 2 Since they are inversely proportional, the concept of radiation intensity is introduced to characterize the power per unit solid angle. The expression for radiation intensity U is: U = r 2 W;

[0164] Further, the antenna gain Gain can be obtained:

[0165] Among them, P in Total power at antenna input;

[0166] 2. As shown in Figure 8(a), the patch antenna is in its initial state with R=1; Figure 8(b) shows the patch antenna in its bent state with R=1.05. The ground plane is 6cm*6cm. The air dielectric patch antenna has a metal patch of 4cm*4cm. The value of R ranges from 1m to 1.05m, with sampling every 0.0095m. The unknown number n=1444, and the Chebyshev expansion order N. t =12, Merry approximation order L=M=4. Based on the extrapolated current coefficient matrix, the radiation pattern of the antenna is calculated. Figure 9(a) shows the maximum gain error between the extrapolation result and the point-by-point calculation by MOM during the entire parameter scanning process. The error is less than 10. -4 The magnitude and accuracy meet engineering requirements. Figure 9(b) shows a comparison of the radiation patterns calculated point-by-point by BURA and MOM when R = 1.1, and the results are in good agreement. Table 1 shows the time required for dynamic deformation analysis by different methods. It can be seen that, with the calculation platform unchanged, the time required by MOM-Maehly is only 11.1% of that required by MOM point-by-point calculation.

[0167] Table 1

[0168] method Number of unknowns Number of parameters Time required (s) MOM 1444 101 380 MOM-Maehly 1444 101 42

[0169] Example 2

[0170] Taking antenna arrays as an example:

[0171] Figure 10(a) shows the initial state of a 1*6 linear array with R=1 and a single antenna spacing of 0.07m. Figure 10(b) shows an example of the antenna array grid under bending conditions with R=1.06, the same as in Example 1. The bending deformation-related parameters remain unchanged, and the Chebyshev expansion order N is... t =12, Merry approximation order L=M=4, number of unknowns n=8664. Figure 11(a) shows the maximum error between the BURA extrapolation result and the MOM point-by-point calculation of the main lobe direction gain during the entire parameter scanning process. It can be seen that the error does not exceed 0.5dB. Figure 11(b) shows the comparison of the radiation patterns calculated by BURA and MOM point-by-point when R=0.069. The results are in good agreement. Table 2 shows the time required for different methods to perform array dynamic deformation analysis. Figure 12 The changes in the radiation pattern of the antenna array during bending are shown.

[0172] Table 2

[0173] method Number of unknowns Number of parameters Time required (s) MOM 8664 101 14443 MOM-Maehly 8664 101 1863

Claims

1. A method for fast prediction of one-dimensional deformation electrical performance of flexible antennas, characterized in that: Specifically comprising the following steps: Step 1, the shape parameters of the original model of the air medium patch antenna are given, including: the excitation mode is probe feeding, the given feeding position, the voltage V of the feed source, and the frequency f; Step 2, according to the shape parameters and the feeding position given in step 1, an original model of the microstrip antenna is established in feko; Step 3, the RWG basis function is used to divide the triangular mesh on the original model established in step 2; In step 3, the RWG basis function definition formula is: （1）； wherein, is the length of the n-th common edge of the triangles, is the area of the triangle, is the position vector of the n-th vertex of the triangle, is the vector from the n-th vertex of the triangle to the point and the basis function value is 0 outside the triangle pair. Step 4, the nodes of each triangle in the triangular mesh and the corresponding node coordinates are obtained in feko, and the obtained nodes and node coordinates are transferred to Matlab; Step 5, the surface current J of the microstrip antenna is calculated according to the triangular mesh nodes and the corresponding node coordinates obtained in step 4; the specific process of step 5 is: Step 5.1, the integral equation in formula (2) is transformed and brought into the L operator equation in formula (3) to obtain formula (4): （2）； wherein is the antenna surface normal unit vector, is the wave impedance in free space, E i is the incident electric field, L the operator expression is: （3）； where k is the wave number in free space, g is the Green's function r is the field point, s is the source point; （4）； wherein, represents the distance s = T from the field point to the source point + + T - ; Step 5.2, formula (4) is solved by using the Galerkin method; the specific process of step 5.2 is: Step 5.2.1, the surface current J is approximated and expanded into the RWG basis function as formula (5): （5）； where N is the total number of common edges in the model triangular mesh, are unknown current expansion coefficients; Step 5.2.2, formula (5) is substituted into formula (4), and the RWG basis function is used as the test function to obtain formula (6): （6）； wherein all are RWG basis functions; The matrix element in formula (6) is: （7）； Z mn Instead of the left side of equation (6), use V in equation (8) below m Instead of the right side of equation (6), use the following equation (9) （8）； Step 5.2.

4. The voltage model in equation (9) below is introduced to solve Step 5.2.

3. Vm : （9）； wherein is a function, ny is the unit vector in the Y direction, and V is the voltage; Substitute the model gap shown in equation (9) for the common edge of the antenna grid, then only apply the excitation model on the inner edge of the feed grid, and let the nth edge be the feed excitation edge, then the voltage matrix The expression is as follows: （10）； The matrix equation of the EFIE discretization is obtained by formula (7) and formula (10), so that the coefficient is obtained by matrix inversion , and is substituted into formula (5), so that the surface current J is obtained; Step 5.3, the surface current of the antenna at each position is obtained by combining the Chebyshev and Meili approximation, and the surface current of the antenna at all curvatures is derived according to the surface current; Step 6, the electromagnetic field is calculated according to the surface current J obtained in step 5.

2. The method of claim 1, wherein the method is characterized by: The specific process of step 5.3 is: Step 5.3.1, in order to represent the bending state of the microstrip antenna, a one-dimensional bending deformation model is introduced, as shown in formula (11): （11）； wherein, is the bending radius, let the original model length be , and horizontally placed on the plane of , then the arc length after deformation , is the corresponding central angle of the circle at point P, is the central angle corresponding arc length, is the horizontal coordinate of point P, is the vertical coordinate of point P; Step 5.3.2, get the node position coordinates of the antenna surface with different curvatures by step 5.3.3, and given the current coefficient matrix I is a one-dimensional function of R, first perform Chebyshev approximation, in order to meet the definition domain requirement of Chebyshev expansion, first perform coordinate transformation to get R`, give the value range of the bending change range R , R` is shown in the following formula (12): （12）； When , , when , , i.e. , the function I(R) is represented as: （13）； Step 5.3.3, Calculation of Chebyshev zeros, Q is the expansion order of the Chebyshev polynomial, The Chebyshev polynomial of order Q has Q zeros, which are calculated by the formula: The Chebyshev polynomial of order Q has Q zeros, which are calculated by the formula: （14）； Then, according to formula (12) and formula (14), the Chebyshev sampling node R i The calculation formula is: （15）； Step 5.3.4, the Chebyshev expansion formula is introduced to obtain formula (16): （16）； wherein ; where T l The Chebyshev polynomial recurrence formula is: （17）； （18）； Through formula (16), the surface current of the Chebyshev solution of the microstrip antenna at different curvatures can be obtained; Step 5.3.5, use Maclaurin approximation to match Chebyshev polynomial expansion, substitution formula is: (19); In the formula, n is the number of unknowns, L and M are the orders of expansion, and generally L=M or L=M+1; wherein , written in matrix form, gives: （20）； （21）； According to formula (19), the current matrix I of the Meili solution at different curvatures is obtained, and the current matrix I is substituted back into formula (5) to obtain the surface current J of the antenna at different curvatures.

3. The method of claim 2, wherein the method further comprises: The specific process of step 6 is: Introducing the dipole model dipole moment is equal to the product of the effective dipole current and the effective dipole length: （22）； wherein: is the RWG basis function negative triangle midpoint, is the RWG basis function positive triangle midpoint; An infinitesimal dipole at some point in free space The vector expression for the radiation field strength at a given location is: （23）； Where, r is the vector from the source point to the field point, and r is the distance from the source point to the field point; （24）； （25） 。

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