Electromagnetic analysis method of metamaterial array antenna based on first-order approximation of dyadic green's function

Abstract

The application discloses a deformed array antenna electromagnetic analysis method based on dyadic Green's function first-order approximation, and specific steps are as follows: step 1, ideal mutual inductance calculation; step 2, deformed array element mutual inductance parameter approximate calculation; and step 3, array far field calculation. The method performs first-order Taylor expansion on the dyadic Green's function in the mutual inductance parameter in the existing analysis method, thereby constructing the incremental relationship between the deformed mutual inductance matrix and the ideal un-deformed mutual inductance parameter, so that the re-analysis process is avoided, and the deformed array antenna electrical performance can be quickly evaluated.

Description

Technical Field

[0001] This invention belongs to the field of antenna technology, specifically relating to an electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function. Background Technology

[0002] Array antennas are widely used in the military field due to their high gain, fast beam scanning, and other advantages. In engineering applications, the surface of array antennas inevitably deforms due to manufacturing errors, usage, or environmental factors. Taking active phased array antennas as an example, structural deformation can cause the position of array elements to shift, thereby affecting electrical performance and causing the entire array to deviate from its original design specifications. Therefore, rapidly evaluating the electrical performance of deformed arrays has practical engineering significance.

[0003] The infinitesimal dipole model is a method that uses a set of infinitesimal dipoles to represent an equivalent target antenna. Due to its simple principle and computational efficiency, it has attracted widespread attention from researchers. The infinitesimal dipole model can not only be used to analyze the radiation performance of a single antenna, but also, through equivalent array elements, for the analysis and design of array antennas.

[0004] Electromagnetic coupling, or mutual coupling, exists between any two elements in an array antenna. Based on the infinitesimal dipole model, the electromagnetic reaction theorem is used to calculate the self-admittance and mutual admittance parameters of the elements in the array environment to construct the admittance matrix, thereby characterizing the mutual coupling effect. However, for large arrays, the actual service environment leads to diverse deformation conditions, requiring multiple re-analysis sessions during use, wasting a significant amount of analysis time and resulting in low computational efficiency. Therefore, finding a way to save considerable re-analysis time while maintaining accuracy, and thus quickly assess its electrical performance, is of significant research importance. Summary of the Invention

[0005] The purpose of this invention is to provide an electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function. This method constructs the incremental relationship between the deformed mutual admittance matrix and the ideal, undeformed mutual admittance parameters by performing a first-order Taylor expansion of the dyadic Green's function in the mutual admittance parameters of existing analysis methods. This avoids the re-analysis process and enables rapid evaluation of the electrical performance of deformable array antennas.

[0006] The first technical solution adopted in this invention is an electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function, and the specific steps are as follows:

[0007] Step 1: Calculate the ideal mutual admittance;

[0008] Step 2, Approximate calculation of mutual admittance parameters of deformable array elements:

[0009] Step 3: Array far-field calculation.

[0010] The invention is further characterized in that,

[0011] Step 1 is as follows:

[0012] Step 1.1: Calculation of mutual admittance parameters of ideal array elements

[0013] The antenna is equivalently represented using a set of infinitesimal dipole models; it is assumed that the equivalent IDM of the array elements has been obtained. According to the electromagnetic reaction theorem, the mutual admittance of a two-element array is... Determined by formula (1), where the array elements of the binary array are numbered m and n:

[0014]

[0015] Where j is the complex unit, ω is the angular frequency, and μ is the spatial permeability of radiation; V m0 V is the free-space port excitation voltage of array element m; n0 N is the free-space port excitation voltage of array element n; d M is the number of dipoles. mi Let r′ be the dipole moment of the i-th equivalent dipole of array element m. mi M is the position vector of the i-th equivalent dipole; ns Let r′ be the dipole moment of the s-th equivalent dipole of array element n. ns Let be the position vector of the s-th equivalent dipole; Let be the electric dextral Green's function, representing the state located at r′. ns The unit current source at field point r′ mi The electric field generated.

[0016] Step 1.2: Calculation of the mutual admittance matrix of the ideal array

[0017] Assuming the number of array elements in the array is N, from equation (1) and the electromagnetic reaction theorem, the admittance matrix Y of the entire array is:

[0018]

[0019] In formula (2), matrix M s The expression for matrix Y0 is as follows:

[0020]

[0021]

[0022] Where matrix U is an N-ary identity matrix; matrix M s To obtain the Nth-order mutual admittance matrix considering the first-order mutual coupling effect, each mutual admittance parameter in the matrix can be obtained from equation (1); matrix Y0 is the self-admittance matrix without considering the mutual coupling effect, and each parameter y in the matrix... i(i = 1 to N) represents the port admittance of each array element in free space.

[0023] Step 2 is as follows:

[0024] Step 2.1, First-order Taylor expansion of the scalar Green's function:

[0025] Let r′ be the position vector of the electromagnetic source point and r be the position vector of the field point. When the source point and the field point are offset by Δr′ and Δr respectively, we first perform a multivariate Taylor expansion on the deformed scalar Green's function G(r+Δr,r′+Δr′).

[0026] The first-order Taylor expansion of any multivariate function f(x) in the neighborhood of point x0 is as follows:

[0027]

[0028] When the source point and field point are subjected to small perturbations Δr′ and Δr respectively, the relative position vector changes Δ r-r′ Recorded as:

[0029] Δ r-r′ =Δr-Δr′ (6)

[0030] The scalar Green's function G(r+Δr,r′+Δr′) is approximated as follows, according to the gradient operation rule. It can be known that:

[0031]

[0032] In formula (7), The expression for G(r,r′) is as follows:

[0033]

[0034]

[0035] Where k is the free space wavenumber, G(r,r′) is the scalar Green's function, R = rr′, and its magnitude is R. The unit vector is denoted as . Operator It has an effect on the variable rr′.

[0036] Step 2.2, First-order Taylor expansion of the dyadic Green's function:

[0037] The expression for the dyadic Green's function in the ideal, undeformed case is as follows:

[0038]

[0039] in, For unit vector, This is the gradient operator.

[0040] The scalar Green's function expansion result of G(r+Δr,r′+Δr′) obtained in step 2.1 is then substituted into the dyadic Green's function. Perform analysis;

[0041] Transformed dyadic Green's function The expression is as follows:

[0042]

[0043] Among them, the operator Applying the effect to variable r, substituting formula (7) into formula (11) yields...

[0044]

[0045] Expanding and rearranging the right side of equation (12), we get:

[0046]

[0047] Further expanding and rearranging the dyadic operator within the parentheses on the right side, we get:

[0048]

[0049] In formula (15), G0(r,r′) and The expression is as follows:

[0050]

[0051]

[0052] Step 2.3 Incremental admittance analysis:

[0053] Based on the derivation in steps 2.1 and 2.2 above, the relationship between the mutual admittance parameters of the ideal array elements and the mutual admittance parameters of the deformed array elements when there is a position offset error is established.

[0054] To be consistent with formula (1), variable substitution is performed here.

[0055] r→r′ mi (17)

[0056] r′→r′ ns (18)

[0057] R→R ns,mi (19)

[0058] Δ r-r′ →Δr nm (20)

[0059] Among them, R ns,mi =r′mi -r′ ns , Δr nm =-Δr mn ;

[0060] According to equation (14), the approximate form of the dyadic Green's function can be obtained.

[0061]

[0062] Substituting equation (18) into equation (1), we can obtain an approximate relationship between ideal mutual admittance and deformed mutual admittance:

[0063] y′ mn =y mn +Δr mn ·Q mn (twenty two)

[0064] In formula (22), Q mn The expression is as follows:

[0065]

[0066] For an N-element array, when the element types are the same, we have

[0067]

[0068] Then, based on the mutual admittance parameters of equation (22), an admittance matrix in the form of equation (2) is constructed under the deformed condition.

[0069] In step 2.2, in formula (23), The expression is as follows:

[0070]

[0071]

[0072]

[0073] Step 3 specifically involves:

[0074] Step 3.1, Port Current Calculation:

[0075] Taking an N-element array as an example, based on step 2, the mutual admittance parameters y′ between array elements are obtained under the first-order Taylor approximation expansion. mn Then, considering the mutual coupling effect between array elements, the port currents are as follows:

[0076]

[0077] Where, y′ mn To construct the mutual admittance parameters of the mutual admittance matrix Y, V is , I i (i = 1 to N) represent the port excitation voltage, the element internal resistance, and the port excitation current, respectively.

[0078] Therefore, according to equation (28), the actual excitation current I at each element port of the entire array considering the mutual coupling effect can be obtained. i The current matrix I formed by this process is expressed as follows:

[0079] I = (U + YR) s ) -1 YV s (29)

[0080] In formula (29), R s and V s The expression is as follows:

[0081]

[0082]

[0083] Where R s V is the internal resistance matrix of the array element. s Given the excitation voltage matrix of the array elements, substituting equations (30) and (31) into equation (29) yields the current matrix, as shown in equation (32):

[0084] I = [I1 I2…I] N ] T (32)

[0085] Step 3.2, calculate the radiated electric field:

[0086] After obtaining the actual excitation current of each array element through equation (29), the calculation formula for the far-field radiation pattern E(θ,φ) of the current-driven array antenna is as follows:

[0087]

[0088] Among them, f i (θ,φ) (i=1~N) is the radiation pattern of the array element, r i For each array element's position vector, The unit vector for the direction of observation.

[0089] The beneficial effects of this invention are:

[0090] (1) This invention relates to an electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function. The change in the field-source relationship after deformation is approximated by a first-order Taylor approximation using a scalar Green's function, and then substituted into the dyadic Green's function to establish the incremental relationship between the ideal mutual admittance parameters and the deformed mutual admittance parameters. Relevant parameters can be pre-calculated and stored. Compared to re-analysis, this method saves a significant amount of analysis time while maintaining accuracy. In practical engineering, it can be used to guide the electrical performance analysis of deformable array antennas in actual work, and is of great significance for the rapid evaluation of the electrical performance of large deformable array antennas.

[0091] (2) Compared with traditional methods for analyzing the electrical performance of large deformable array antennas, the electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function in this invention has higher computational efficiency for both ideal and deformable arrays with the same elements. This is because the infinitesimal dipole model treats the target antenna as a fixed mode, meaning the current distribution does not change with the incident electromagnetic field affecting the array elements. It can improve the computational accuracy of the radiated field of deformable array antennas while ensuring computational precision. Attached Figure Description

[0092] Figure 1 This is a flowchart of the electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function according to the present invention;

[0093] Figure 2 This is a schematic diagram of the antenna's equivalent infinitesimal dipole model;

[0094] Figure 3 This is a schematic diagram of the mutual coupling effect between infinitesimal dipole models;

[0095] Figure 4 This is a schematic diagram of the relative positions of two-element arrays;

[0096] Figure 5 This is a schematic diagram of the equivalent circuit model of the radiating unit;

[0097] Figure 6 This is a schematic diagram of a linear array of half-wave dipoles;

[0098] Figure 7 The electric field pattern curve of the array on the E-plane under the first deformation condition is compared with that of the present invention and the method of moments.

[0099] Figure 8 The electric field pattern curve of the H-plane of the array under the first deformation condition is compared with that of the present invention and the method of moments.

[0100] Figure 9 The electric field pattern curve of the array on the E-plane under the second deformation case is compared with that of the present invention and the method of moments.

[0101] Figure 10The electric field pattern curve of the H-plane of the array under the second deformation case is compared with that of the present invention and the method of moments.

[0102] Figure 11 The electric field pattern curve of the array on the E-plane under the third deformation case is compared with that of the present invention and the method of moments.

[0103] Figure 12 The electric field pattern curve of the H-plane of the array under the third deformation case is compared with that of the present invention and the method of moments. Detailed Implementation

[0104] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0105] This invention provides an electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function. The analysis process is as follows: Figure 1 As shown, the specific steps are as follows:

[0106] Step 1, Calculation of Ideal Mutual Admittance:

[0107] Step 1 is as follows:

[0108] Step 1.1: Calculation of mutual admittance parameters of ideal array elements

[0109] like Figure 2 As shown, the antenna is equivalently represented using a set of infinitesimal dipole models (IDM), making the actual antenna electromagnetic field (E) on the observation surface... a H a The electromagnetic field (E) generated by the equivalent model d H d They are approximately the same.

[0110] The mutual coupling effect between array elements is characterized as the port mutual admittance between IDMs. Figure 3 This diagram illustrates the mutual coupling effect between infinitesimal dipole models. It assumes the equivalent IDM of the array elements has been obtained. According to the electromagnetic reaction theorem, the mutual admittance of a two-element array (element numbered m and n) is... Determined by the following expression:

[0111]

[0112] Where j is the complex unit, ω is the angular frequency, and μ is the spatial permeability of radiation; V m0 V is the free-space port excitation voltage of array element m; n0 N is the free-space port excitation voltage of array element n; d M is the number of dipoles. mi Let r′ be the dipole moment of the i-th equivalent dipole of array element m.mi M is the position vector of the i-th equivalent dipole; ns Let r′ be the dipole moment of the s-th equivalent dipole of array element n. ns Let be the position vector of the s-th equivalent dipole; Let be the electric dextral Green's function, representing the state located at r′. ns The unit current source at field point r′ mi The electric field generated.

[0113] Step 1.2: Calculation of the mutual admittance matrix of the ideal array

[0114] Assuming the number of array elements in the array is N, from equation (1) and the electromagnetic reaction theorem, the admittance matrix Y of the entire array is:

[0115]

[0116] In formula (2), matrix M s The expression for matrix Y0 is as follows:

[0117]

[0118]

[0119] Where matrix U is an N-ary identity matrix; matrix M s To obtain the Nth-order mutual admittance matrix considering the first-order mutual coupling effect, each mutual admittance parameter in the matrix can be obtained from equation (1); matrix Y0 is the self-admittance matrix without considering the mutual coupling effect, and each parameter y in the matrix... i (i = 1 to N) represents the port admittance of each array element in free space.

[0120] Step 2, Approximate calculation of mutual admittance parameters of deformable array elements:

[0121] Step 2 is as follows:

[0122] Step 2.1, First-order Taylor expansion of the scalar Green's function:

[0123] As can be seen from formula (1), the mutual admittance parameters of the array elements are obtained through the dyadic Green's function. Related to the array element position vector, such as Figure 4 As shown, when the positions of the two array elements are r′ mi 、r′ ns Δr occurs respectively m , Δr n When the position shifts, its electrical properties will change accordingly. Therefore, it is necessary to derive the dextral Green's function when the relative positions of the array elements shift. Green's function with original dyadic function The relationship between them.

[0124] Without loss of generality, let r′ be the position vector of the electromagnetic source point and r be the position vector of the field point. When analyzing the mutual coupling effect, the field point is the location of another array element, not the far-field observation point. When the source point and the field point are offset relative to each other by Δr′ and Δr, respectively, we first perform a multivariate Taylor expansion of the deformed scalar Green's function G(r+Δr, r′+Δr′).

[0125] The first-order Taylor expansion of any multivariate function f(x) in the neighborhood of point x0 is as follows:

[0126]

[0127] When the source point and field point are subjected to small perturbations Δr′ and Δr respectively, the relative position vector changes Δ r-r′ Recorded as:

[0128] Δ r-r′ =Δr-Δr′ (6)

[0129] Therefore, the scalar Green's function G(r+Δr,r′+Δr′) can be approximated as follows, according to the gradient operation rules. It can be known that:

[0130]

[0131] In formula (7), The expression for G(r,r′) is as follows:

[0132]

[0133]

[0134] Where k is the free space wavenumber, G(r,r′) is the scalar Green's function, R = rr′, and its magnitude is R. The unit vector is denoted as . Operator It has an effect on the variable rr′.

[0135] Step 2.2, First-order Taylor expansion of the dyadic Green's function:

[0136] The expression for the dyadic Green's function in the ideal, undeformed case is as follows:

[0137]

[0138] in, For unit vector, This is the gradient operator.

[0139] The scalar Green's function expansion result of G(r+Δr,r′+Δr′) obtained in step 2.1 is then substituted into the dyadic Green's function. Perform analysis;

[0140] Transformed dyadic Green's function The expression is as follows:

[0141]

[0142] Among them, the operator Applying the effect to variable r, substituting formula (7) into formula (11) yields...

[0143]

[0144] Expanding and rearranging the right side of equation (12), we get:

[0145]

[0146] Further expanding and rearranging the dyadic operator within the parentheses on the right side, we get:

[0147]

[0148] In formula (15), G0(r,r′) and The expression is as follows:

[0149]

[0150]

[0151] Step 2.3 Incremental admittance analysis:

[0152] Based on the derivation in steps 2.1 and 2.2 above, the relationship between the mutual admittance parameters of the ideal array elements and the mutual admittance parameters of the deformed array elements when there is a position offset error is established.

[0153] To be consistent with formula (1), variable substitution is performed here.

[0154] r→r′ mi (17)

[0155] r′→r′ ns (18)

[0156] R→R ns,mi (19)

[0157] Δ r-r′ →Δr nm (20)

[0158] Among them, R ns,mi =r′ mi -r′ ns , Δr nm =-Δr mn ;

[0159] According to equation (14), the approximate form of the dyadic Green's function can be obtained.

[0160]

[0161] Substituting equation (18) into equation (1), we can obtain an approximate relationship between ideal mutual admittance and deformed mutual admittance:

[0162] y′ mn =y mn +Δr mn ·Q mn (twenty two)

[0163] In formula (22), Q mn The expression is as follows:

[0164]

[0165] In formula (23), The expression is as follows:

[0166]

[0167]

[0168]

[0169] For an N-element array, when the element types are the same, we have

[0170]

[0171] Then, based on the mutual admittance parameters of equation (22), an admittance matrix in the form of equation (2) is constructed under the deformed condition.

[0172] Step 3: Array far-field calculation.

[0173] Step 3 is as follows:

[0174] Step 3.1, Port Current Calculation:

[0175] Taking an N-element array as an example, based on step 2, the mutual admittance parameters y′ between array elements are obtained under the first-order Taylor approximation expansion. mn Then, when considering the mutual coupling effect between array elements, such as Figure 5 The general circuit model shown has the following port currents:

[0176]

[0177] Where, y′ mn To construct the mutual admittance parameters of the mutual admittance matrix Y, V i s , Ii (i = 1 to N) represent the port excitation voltage, the element internal resistance, and the port excitation current, respectively.

[0178] Therefore, according to equation (28), the actual excitation current I at each element port of the entire array considering the mutual coupling effect can be obtained. i The current matrix I formed by this process is expressed as follows:

[0179] I = (U + YR) s ) -1 YV s (29)

[0180] In formula (29), R s and V s The expression is as follows:

[0181]

[0182]

[0183] Where R s V is the internal resistance matrix of the array element. s Given the excitation voltage matrix of the array elements, substituting equations (30) and (31) into equation (29) yields the current matrix, as shown in equation (32):

[0184] I = [I1 I2…I] N ] T (32)

[0185] Step 3.2, calculate the radiated electric field:

[0186] After obtaining the actual excitation current of each array element through equation (29), the calculation formula for the far-field radiation pattern E(θ,φ) of the current-driven array antenna is as follows:

[0187]

[0188] Among them, f i (θ,φ) (i=1~N) is the radiation pattern of the array element, r i For each array element's position vector, The unit vector for the direction of observation.

[0189] 1. Simulation parameters

[0190] like Figure 6 The image shows a 10-element half-wave dipole array antenna array arranged along the x-axis. The half-wave dipoles are placed collinearly with the z-axis, the center operating frequency is 3 GHz, and the element spacing is 0.5λ, where λ is the operating wavelength. The following simulation example further illustrates this.

[0191] Given three arbitrary deformation modes, the effects of calculating using a first-order approximation method are analyzed. The array antenna deformation is shown in Table 1 (considering only the y-axis position shift of the array surface). A comparison of the electric field patterns of the array antenna under the three deformation conditions is shown below. Figure 7-12 As shown.

[0192] Table 1. Deformation information for three linear array structures (unit: mm)

[0193] Undeformed 0 0 0 0 0 0 0 0 0 0 0 Transformation 1 8 5 13 3 7 10 9 2 6 14 3.7 Transformation 2 3 9 2 -5 7 0 13 2 8 -3 5.3 Transformers 3 -8 20 -13 3 30 10 -9 2 6 -14 13.7

[0194] 2. Simulation Content and Results

[0195] Figure 7 The electric field pattern curves of the array on the E-plane under the first deformation condition are presented, comparing the present invention with the method of moments. Figure 8 The electric field pattern curve of the H-plane of the array under the first deformation condition is presented, comparing the present invention with the method of moments. Figure 9 The electric field pattern curves of the array on the E-plane under the second deformation case are presented, comparing the present invention with the method of moments. Figure 10 The electric field pattern curves of the H-plane of the array under the second deformation case are presented, comparing the present invention with the method of moments. Figure 11 The electric field pattern curve of the array in the third deformation case is presented, comparing the present invention with the method of moments. Figure 12 The electric field pattern curve of the array in the H-plane under the third deformation case is presented, comparing the present invention with the method of moments.

[0196] from Figures 7-12 As can be seen from the table, the first-order approximate deformable array analysis method of the present invention can analyze the electrical performance of the deformed array. By calculating the admittance increment, the numerical relationship between the ideal array and the deformed array is established, which can quickly improve the calculation efficiency while ensuring the calculation accuracy. Table 2 shows the comparison of the calculation time of the proposed method with that of the method of moments. The error is within a reasonable range when compared with the analysis results of the method of moments, which verifies the effectiveness of the method of the present invention.

[0197] Table 2 Comparison of Calculation Time (unit: seconds)

[0198] (CPU: 11th Gen Intel(R)Core(TM)i7-11700@2.50GHz, RAM: 16GB)

[0199]

[0200]

[0201] Example 1

[0202] The electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function comprises the following steps:

[0203] Step 1: Calculate the ideal mutual admittance;

[0204] Step 2, Approximate calculation of mutual admittance parameters of deformable array elements:

[0205] Step 3: Array far-field calculation.

[0206] Example 2

[0207] The electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function comprises the following steps:

[0208] Step 1: Calculate the ideal mutual admittance;

[0209] Step 1 is as follows:

[0210] Step 1.1: Calculation of mutual admittance parameters of ideal array elements

[0211] The antenna is equivalently represented using a set of infinitesimal dipole models; it is assumed that the equivalent IDM of the array elements has been obtained. According to the electromagnetic reaction theorem, the mutual admittance of a two-element array is... Determined by formula (1), where the array elements of the binary array are numbered m and n:

[0212]

[0213] Where j is the complex unit, ω is the angular frequency, and μ is the spatial permeability of radiation; V m0 V is the free-space port excitation voltage of array element m; n0 N is the free-space port excitation voltage of array element n; d M is the number of dipoles. mi Let r′ be the dipole moment of the i-th equivalent dipole of array element m. mi M is the position vector of the i-th equivalent dipole; ns Let r′ be the dipole moment of the s-th equivalent dipole of array element n. ns Let be the position vector of the s-th equivalent dipole; Let be the electric dextral Green's function, representing the state located at r′. ns The unit current source at field point r′ mi The electric field generated.

[0214] Step 1.2: Calculation of the mutual admittance matrix of the ideal array

[0215] Assuming the number of array elements in the array is N, from equation (1) and the electromagnetic reaction theorem, the admittance matrix Y of the entire array is:

[0216]

[0217] In formula (2), matrix M sThe expression for matrix Y0 is as follows:

[0218]

[0219]

[0220] Where matrix U is an N-ary identity matrix; matrix M s To obtain the Nth-order mutual admittance matrix considering the first-order mutual coupling effect, each mutual admittance parameter in the matrix can be obtained from equation (1); matrix Y0 is the self-admittance matrix without considering the mutual coupling effect, and each parameter y in the matrix... i (i = 1 to N) represents the port admittance of each array element in free space.

[0221] Step 2, Approximate calculation of mutual admittance parameters of deformable array elements:

[0222] Step 3: Array far-field calculation.

[0223] Example 3

[0224] The electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function comprises the following steps:

[0225] Step 1: Calculate the ideal mutual admittance;

[0226] Step 1 is as follows:

[0227] Step 1.1: Calculation of mutual admittance parameters of ideal array elements

[0228] The antenna is equivalently represented using a set of infinitesimal dipole models; it is assumed that the equivalent IDM of the array elements has been obtained. According to the electromagnetic reaction theorem, the mutual admittance of a two-element array is... Determined by formula (1), where the array elements of the binary array are numbered m and n:

[0229]

[0230] Where j is the complex unit, ω is the angular frequency, and μ is the spatial permeability of radiation; V m0 V is the free-space port excitation voltage of array element m; n0 N is the free-space port excitation voltage of array element n; d M is the number of dipoles. mi Let r′ be the dipole moment of the i-th equivalent dipole of array element m. mi M is the position vector of the i-th equivalent dipole; ns Let r′ be the dipole moment of the s-th equivalent dipole of array element n. ns Let be the position vector of the s-th equivalent dipole; Let be the electric dextral Green's function, representing the state located at r′. ns The unit current source at field point r′ miThe electric field generated.

[0231] Step 1.2: Calculation of the mutual admittance matrix of the ideal array

[0232] Assuming the number of array elements in the array is N, from equation (1) and the electromagnetic reaction theorem, the admittance matrix Y of the entire array is:

[0233]

[0234] In formula (2), matrix M s The expression for matrix Y0 is as follows:

[0235]

[0236]

[0237] Where matrix U is an N-ary identity matrix; matrix M s To obtain the Nth-order mutual admittance matrix considering the first-order mutual coupling effect, each mutual admittance parameter in the matrix can be obtained from equation (1); matrix Y0 is the self-admittance matrix without considering the mutual coupling effect, and each parameter y in the matrix... i (i = 1 to N) represents the port admittance of each array element in free space.

[0238] Step 2: Approximate calculation of mutual admittance parameters of deformable array elements;

[0239] Step 2 is as follows:

[0240] Step 2.1, First-order Taylor expansion of the scalar Green's function:

[0241] Let r′ be the position vector of the electromagnetic source point and r be the position vector of the field point. When the source point and the field point are offset by Δr′ and Δr respectively, we first perform a multivariate Taylor expansion on the deformed scalar Green's function G(r+Δr,r′+Δr′).

[0242] The first-order Taylor expansion of any multivariate function f(x) in the neighborhood of point x0 is as follows:

[0243]

[0244] When the source point and field point are subjected to small perturbations Δr′ and Δr respectively, the relative position vector changes Δ r-r′ Recorded as:

[0245] Δ r-r′ =Δr-Δr′ (6)

[0246] The scalar Green's function G(r+Δr,r′+Δr′) is approximated as follows, according to the gradient operation rule. It can be known that:

[0247]

[0248] In formula (7), The expression for G(r,r′) is as follows:

[0249]

[0250]

[0251] Where k is the free space wavenumber, G(r,r′) is the scalar Green's function, R = rr′, and its magnitude is R. The unit vector is denoted as . Operator Effect on variable rr′;

[0252] Step 2.2, First-order Taylor expansion of the dyadic Green's function:

[0253] The expression for the dyadic Green's function in the ideal, undeformed case is as follows:

[0254]

[0255] in, For unit vector, This is the gradient operator.

[0256] The scalar Green's function expansion result of G(r+Δr,r′+Δr′) obtained in step 2.1 is then substituted into the dyadic Green's function. Perform analysis;

[0257] Transformed dyadic Green's function The expression is as follows:

[0258]

[0259] Among them, the operator Applying the effect to variable r, substituting formula (7) into formula (11) yields...

[0260]

[0261] Expanding and rearranging the right side of equation (12), we get:

[0262]

[0263] Further expanding and rearranging the dyadic operator within the parentheses on the right side, we get:

[0264]

[0265] In formula (15), G0(r,r′) and The expression is as follows:

[0266]

[0267]

[0268] Step 2.3 Incremental admittance analysis:

[0269] Based on the derivation in steps 2.1 and 2.2 above, the relationship between the mutual admittance parameters of the ideal array elements and the mutual admittance parameters of the deformed array elements when there is a position offset error is established.

[0270] To be consistent with formula (1), variable substitution is performed here.

[0271] r→r′ mi (17)

[0272] r′→r′ ns (18)

[0273] R→R ns,mi (19)

[0274] Δ r-r′ →Δr nm (20)

[0275] Among them, R ns,mi =r′ mi -r′ ns , Δr nm =-Δr mn ;

[0276] According to equation (14), the approximate form of the dyadic Green's function can be obtained.

[0277]

[0278] Substituting equation (18) into equation (1), we can obtain an approximate relationship between ideal mutual admittance and deformed mutual admittance:

[0279] y′ mn =y mn +Δr mn ·Q mn (twenty two)

[0280] In formula (22), Q mn The expression is as follows:

[0281]

[0282] For an N-element array, when the element types are the same, we have

[0283]

[0284] Then, based on the mutual admittance parameters of equation (22), an admittance matrix in the form of equation (2) is constructed under the deformed condition.

[0285] Step 3: Array far-field calculation.

Claims

1. An electromagnetic analysis method for deformable array antennas based on the first-order approximation of the dyadic Green's function, characterized in that, The specific steps are as follows: Step 1: Calculate the ideal mutual admittance; Step 1 is as follows: Step 1.1: Calculation of mutual admittance parameters of ideal array elements The antenna is equivalently represented using a set of infinitesimal dipole models; it is assumed that the equivalent IDM of the array elements has been obtained; according to the electromagnetic reaction theorem, the mutual admittance of a two-element array is... Determined by formula (1), where the element numbering of the binary array is... , : (1) in, For complex units, Angular frequency, The radiative spatial permeability; For array element The free space port excitation voltage; For array element n The free space port excitation voltage; The number of dipoles. For array element The The dipole moment of an equivalent dipole For the first The position vector of an equivalent dipole; For array element The The dipole moment of an equivalent dipole For the first The position vector of an equivalent dipole; Let be the electric dextral Green's function, indicating that it is located at... The unit current source at the field point The electric field generated; Step 1.2: Calculation of the mutual admittance matrix of the ideal array Assume the number of array elements is N From equation (1) and the electromagnetic reaction theorem, the admittance matrix of the entire array can be obtained. for: (2) In formula (2), the matrix and matrix The expression is as follows: (3) (4) Among them, matrix for N Meta-identity matrix; matrix To consider the first-order mutual coupling effect N The mutual admittance matrix is ​​of order 1, and the mutual admittance parameters in the matrix can be obtained from equation (1); the matrix For the self-admittance matrix that does not consider mutual coupling effects, the parameters in the matrix are... Let the port admittance of each array element in free space be denoted as , where ; Step 2, Approximate calculation of mutual admittance parameters of deformable array elements: Step 2 is as follows: Step 2.1, First-order Taylor expansion of the scalar Green's function: Let the electromagnetic source point position vector be... The field point vector is When the source point and the field point are respectively offset , First, for the transformed scalar Green's function... Perform a Taylor expansion of the multivariate function; Arbitrary multivariable functions At point The first-order Taylor expansion of the neighborhood is as follows: (5) When small disturbances occur at the source point and the field point respectively , When, the relative position vector changes Recorded as: (6) scalar Green's function The following approximation is made, based on the gradient operation rule. It can be known that: (7) In formula (7), and The expression is as follows: (8) (9) in, For free space wavenumber, For scalar Green's function, Its modulus is The unit vector is denoted as Operator For variables effect; Step 2.2, First-order Taylor expansion of the dyadic Green's function: The expression for the dyadic Green's function in the ideal, undeformed case is as follows: (10) in, For unit vector, For gradient operators; The following will be the result obtained in step 2.1 Substituting the scalar Green's function expansion result into the dyadic Green's function Perform analysis; Transformed dyadic Green's function The expression is as follows: (11) Among them, the operator For variables Substituting formula (7) into formula (11) yields the following result. (12) Expanding and rearranging the right side of equation (12), we get: (13) Further expanding and rearranging the dyadic operator within the parentheses on the right side, we get: (14) In formula (15), and The expression is as follows: (15) (16) Step 2.3 Incremental Admittance Analysis: Based on the derivation in steps 2.1 and 2.2 above, the relationship between the mutual admittance parameters of the ideal array elements and the mutual admittance parameters of the deformed array elements when there is a position offset error is established. To be consistent with formula (1), variable substitution is performed here. (17) (18) (19) (20) in, , ; According to equation (14), the approximate form of the dyadic Green's function can be obtained. (21) Substituting equation (18) into equation (1), we can obtain an approximate relationship between ideal mutual admittance and deformed mutual admittance: (22) In formula (22), The expression is as follows: (23) The expression is as follows: (24) (25) (26)； For a N In a meta-array, when the element types are the same, there are... (27) Then, based on the mutual admittance parameters of equation (22), an admittance matrix in the form of equation (2) is constructed under the deformation condition. Step 3: Array far-field calculation; Step 3 specifically involves: Step 3.1, Port Current Calculation: by N Taking the element array as an example, based on step 2, the mutual admittance parameters between array elements under the first-order Taylor approximation expansion are obtained. Then, considering the mutual coupling effect between array elements, the port currents are as follows: (28) in, To construct the mutual admittance matrix The mutual admittance parameter, , , These represent the port excitation voltage, the element internal resistance, and the port excitation current, respectively. ; Therefore, according to equation (28), the actual excitation current at the ports of each element of the entire array when considering the mutual coupling effect can be obtained. The current matrix formed Current matrix The expression is as follows: (29) In formula (29), and The expression is as follows: (30) (31) in The internal resistance matrix of the array elements. The current matrix is ​​obtained by substituting equations (30) and (31) into equation (29), as shown in equation (32): (32)； Step 3.2, calculate the radiated electric field: After obtaining the actual excitation current of each array element through equation (29), the far-field pattern of the current-driven array antenna is obtained. The formula for calculation is: (33) in, This is the radiation pattern of the array element. For each array element's position vector, The unit vector for the direction of observation, where .

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