CN112446152B - Antenna far-field directional pattern analysis method based on infinitesimal small dipole model deformation array - Google Patents
Antenna far-field directional pattern analysis method based on infinitesimal small dipole model deformation array Download PDFInfo
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Abstract
The invention discloses a method for analyzing a far-field directional pattern of a deformed array antenna based on an infinitesimal dipole model, which comprises the steps of determining a geometric model of a radiation unit of the deformed array antenna and acquiring near-field data; establishing an infinite small dipole initial model and an optimization model; optimizing by using a quantum particle group algorithm to obtain an infinitesimal dipole model of the actual radiation unit; using an infinitesimal dipole model to substitute and deduce a transadmittance expression between each radiation unit and each radiation unit; researching the influence of array surface deformation on the mutual admittance among the array elements and obtaining an admittance matrix of the deformation array antenna; and finally, revealing the actual excitation value of each radiating element, and calculating a far-field directional diagram of the deformed array antenna. The method can accurately analyze the radiation performance of the deformed array antenna, and has theoretical guidance significance for accurately analyzing the actual electrical performance of the array antenna in engineering.
Description
Technical Field
The invention belongs to the technical field of antennas, and particularly relates to a computational analysis method of a far-field directional diagram of a deformed array antenna based on an infinitesimal dipole model, which can provide a guiding idea for electrical performance analysis of the array antenna in practical application.
Background
The array antenna can form arrays in different forms according to the number of units, the spatial position distribution of each unit, the excitation amplitude and the excitation phase of each unit to obtain different radiation directional performances, and has great flexibility, thereby being widely applied. In practical engineering, due to environmental loads such as self gravity and wind power factors, the array surface of the array antenna is deformed, so that the position of an array element is shifted (assuming that the array element is not deformed), the radiation performance of the array antenna is greatly reduced, and the array antenna is deviated from an ideal array radiation directional diagram, so that the analysis of the electrical performance of the array antenna after the array surface is deformed is also very important.
However, the current methods for analyzing the electrical properties of the array antenna after the wavefront deformation are very limited: the traditional directional diagram product theorem is simple and easy to realize, but mutual coupling among antennas cannot be considered, so that the accuracy of a solution result is greatly reduced, and the sidelobe level and the cross polarization level cannot be accurately calculated; although mutual coupling is considered in subarray extrapolation, the subarray extrapolation can only be applied to a planar array and cannot consider the existence of deformation factors; the full-wave numerical solutions provided by commercial software, while accurate in result, are very costly and grow exponentially as the complexity and scale of the problem increases.
Disclosure of Invention
Aiming at the defects and defects existing in the existing electric performance strategy for analyzing the deformed array antenna, the invention provides a calculation and analysis method for a far-field directional pattern of the deformed array antenna based on an infinitesimal small dipole model. The method strictly considers the mutual coupling effect among the array elements on the basis of carrying out near field equivalence on the single array element, thereby accurately calculating the far field directional diagram of the deformable array antenna. Compared with the traditional method, the method has clear physical concept and high calculation efficiency, and can be suitable for antennas with any shapes.
The invention is realized by the following technical scheme.
The method for analyzing the far-field directional pattern of the array antenna based on the infinitesimal small dipole model deformation comprises the following steps:
(1) determining array antenna unit parameters according to the type of the array antenna; extracting a radiation unit, modeling and simulating in full-wave analysis software to obtain near-field data of the radiation unit;
(2) establishing a group of infinitesimal dipoles according to near-field data of an actual radiation unit, setting initial parameter intervals of the infinitesimal dipoles, and deducing a radiation field of the infinitesimal dipoles;
(3) establishing an optimization model according to the near field data of the actual radiation unit and the radiation field of the infinitesimal dipole, so that the radiation field of the infinitesimal dipole is matched with the radiation field of the actual array radiation unit, and dipole moment amplitude, phase and bit vector parameters of the infinitesimal dipole are obtained;
(4) according to each parameter of the infinitesimal dipole model, replacing each radiation unit of the array antenna with the infinitesimal dipole model, and deducing a transadmittance expression;
(5) analyzing the change of an infinitesimal small dipole model according to the position of the array element after the array surface of the array antenna is deformed, and deducing the mutual admittance after the array surface is deformed;
(6) according to the mutual admittance among the position array elements, the input admittance is considered, a mutual admittance matrix of the deformation array antenna based on the infinitesimal small dipole model is obtained, and the actual excitation value of each radiation unit is determined;
(7) and calculating a far-field directional diagram of the deformed array antenna according to the space phase of the radiation unit and the actual excitation parameter of the radiation unit.
With respect to the above technical solution, the present invention has a further preferable solution:
in the step (1), the array antenna unit parameters include a unit form and an excitation voltage, and the near field data include coordinates of the near field observation point in a rectangular coordinate system and components of field intensity of each observation point along each coordinate axis.
In the step (2), according to the near field data of the actual radiation unit, selecting the number of infinitesimal dipoles and a dipole moment interval; and determining the space domain where the infinitesimal dipole is located according to the actual geometrical model of the radiation unit.
In the step (3), an optimization model is established and optimized, and the method comprises the following steps:
(3a) constructing an objective function according to the near field data of the actual radiation unit, wherein the near field intensity difference between the actual radiation unit and the infinitesimal dipole model at an observation point is taken as the objective function, and establishing an optimization model;
(3b) according to the established optimization model, the quantum particle swarm optimization is used for optimizing to obtain a parameter χ i Thereby obtaining the absence of a real antennaPoor dipole model.
In the step (4), according to each parameter of the infinitesimal dipole model, each radiation unit of the array antenna is replaced by the infinitesimal dipole, so that the mutual coupling relationship between any two radiation units is converted into the mutual admittance between two groups of infinitesimal dipoles, and the mutual admittance between any two radiation units is obtained after equivalence is carried out by using the infinitesimal dipoles.
In the step (5), the step of calculating the mutual admittance between the array elements after the array surface deformation is as follows:
(5a) when the infinitesimal dipole model changes along with the position of the array element, the parameters
(5b) for the parameter [ x i y i z i ]The change of the dipole is obtained through a coordinate transformation matrix, and the position vector of the deformed infinitesimal dipole in a global coordinate system is obtained;
(5c) and (5) substituting the transformed parameter results of the steps (5a) and (5b) into a mutual admittance calculation formula to obtain the result.
The step (6) comprises the following specific steps:
expanding the self-admittance to an array admittance matrix according to the result obtained in the step (5c) by considering the input self-admittance; and exciting the voltage V by each array element port to obtain the actual exciting current of each array element.
In the step (7), a far-field directional diagram of the array antenna is calculated according to the spatial phase of the radiation unit and the actual excitation parameter of the radiation unit: determining a spatial phase factor of each radiation unit according to the coordinates of the central position of each array element of the deformed array antenna, and further determining a matrix; and calculating a far field directional diagram of the deformed array antenna.
Due to the adoption of the technical scheme, the invention has the following beneficial effects:
1. in practical engineering application, array antenna can generate array surface deformation due to self weight or environmental load and the like, so that the electrical property of the array antenna is influenced, and the existing analysis methods have a few defects and drawbacks. The invention provides a calculation and analysis method of a far-field directional diagram of a deformed array antenna based on an infinitesimal small dipole model, which is characterized in that each radiation unit of the array antenna is equivalently replaced by a group of infinitesimal small dipoles, the mutual coupling effect between two array elements is converted into the mutual admittance calculation between two groups of infinitesimal small dipoles, the physical significance is clear, then the mutual admittance of the two infinitesimal small dipole models under different position postures is analyzed and deduced, the mutual admittance matrix of the whole deformed array antenna is obtained, the actual excitation current of each radiation unit is revealed, and finally the far-field directional diagram of the deformed array antenna is obtained according to an antenna far-field directional diagram calculation formula.
2. Compared with the existing method for calculating the deformed array antenna, the infinite small dipole model is used, so that the maxwell equation set is prevented from being repeatedly solved in a large quantity, the calculation time and the calculation cost are greatly reduced, the mutual coupling expression among the radiation units is clear, the concept is clear, and the method has strong thought guiding significance and engineering significance for the analysis of the electrical performance of the array antenna in actual work.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principles of the invention:
FIG. 1 is a flow diagram of a method for deformable array antenna far field pattern analysis based on an infinitesimal small dipole model;
FIG. 2 is a schematic diagram of the equivalent principle of the infinitesimal dipole model, where (a) is the electromagnetic field of the actual antenna and (b) is the electromagnetic field of the infinitesimal dipole model of the actual antenna;
FIG. 3 is a schematic diagram of mutual coupling between radiating elements and their mutual admittance of an infinitesimal dipole model;
FIG. 4 is a schematic diagram of array element position shift;
FIG. 5 is a schematic diagram of the positive direction of rotation of an array element;
FIG. 6 is a schematic diagram of ideal and deformed half-wave oscillator linear arrays;
FIG. 7 is a comparison of FEKO simulation results with the calculation results of the present invention.
Detailed Description
The present invention will now be described in detail with reference to the drawings and specific embodiments, wherein the exemplary embodiments and descriptions of the present invention are provided to explain the present invention without limiting the invention thereto.
Referring to fig. 1, the invention relates to a method for analyzing a far field pattern of a deformed array antenna based on an infinitesimal dipole model, which comprises the following specific steps:
step 1, determining geometrical parameters and near field data of array radiation units
Determining excitation voltage and a geometric model of a unit according to the arrangement of a single radiation unit of the array antenna; according to the radiation field of the unit, as shown in fig. 2, modeling simulation solution is performed in full-wave analysis software to obtain near-field data of the unit.
Determining the number of dipoles and a dipole moment amplitude interval in the infinitesimal small dipole model according to the field intensity of the near-field data of the actual radiation unit; and determining the upper and lower boundaries of the space domain where the infinitesimal dipole is located according to the geometric model of the radiation unit, thereby determining the optimized constraint boundary condition.
Step 3, determining an objective function, establishing an optimization model, and optimizing to obtain an infinitesimal dipole model
The method comprises the following specific steps:
(3a) constructing an objective function according to the radiation field calculation formula of the near field data of the actual radiation unit and the infinitesimal dipole, and establishing the following optimization model by taking the near field intensity difference of the actual radiation unit and the infinitesimal dipole model at an observation point as the objective function:
wherein, χ i The parameters of the ith infinitesimal dipole respectively represent the amplitude of the dipole moment, the phase of the dipole moment, the included angle between the dipole and the x axis, the included angle between the dipole and the y axis and the coordinate of the center position of the dipole; n is a radical of d The number of the radiating elements of the array antenna is; n is a radical of op The number of observation points is;
the field intensity value of the actual antenna under the nth observation point is obtained;
the field intensity value of the infinite small dipole model under the nth observation point is obtained; d is the dipole moment amplitude of the infinitesimal dipole; the value of D is generally different according to different equivalent antenna types and different excitation voltage values, for a half-wave oscillator antenna with the excitation voltage of 1V, the value of D is 0.001 A.m, and for an air microstrip antenna with the same excitation, the value of D is 0.0001 A.m; Γ is the physical boundary of the actual antenna;
(3b) according to the optimization model established in the step (3a), optimizing by using a quantum particle swarm algorithm to obtain a parameter χ i Thus, an infinitesimal dipole model of the actual antenna is obtained.
Step 4, replacing the actual radiation unit with an infinite small dipole model to deduce mutual admittance between array elements
For any two radiating elements, as shown in fig. 3, their mutual coupling relationship (a) can be expressed as:
wherein, V m ，V n Port excitation voltages of the m and n units respectively; e m ,H m An electromagnetic field generated for the m cell; j. the design is a square n ,M n N units of current and magnetic current; omega is the physical boundary of two actual antennas.
After performing equivalence by using infinitesimal dipoles, as shown in fig. 3, the relationship is converted into (b) a mutual admittance relationship between two sets of infinitesimal dipoles, and the mutual admittance can be expressed as:
wherein j is an imaginary unit; omega is angular frequency; mu is the radiation space magnetic permeability; n is a radical of d The number of infinitesimal dipoles; r is i ,r s The i th dipole of the array element m and the s th infinitesimal dipole of the array element n respectively,
respectively the complex dipole moment of the ith equivalent dipole in the array element m and the s equivalent dipole in the array element n, and the expression is
Wherein M is the complex dipole moment of the infinitesimal dipole; x, y and z are potential vectors of infinitesimal dipoles in a global coordinate system; alpha, beta and gamma are included angles between infinitesimal dipoles and x, y and z axes respectively.
wherein
Is an identity matrix; k is a propagation constant;
is a gradient operator; g (r) i ,r s ) The expression of (a) is:
The infinitesimal dipole model can be correspondingly changed along with the change of the position of the array element, so that the x-ray imaging system is formed i Middle [ alpha ] i β i x i y i z i ]The parameters are changed, so that the mutual admittance among the array elements is changed, as shown in fig. 4, the specific steps of calculating the mutual admittance among the array elements after the array surface is deformed are as follows:
(5a) for alpha i β i Each infinitesimal dipole is considered as a vector u, then a i β i Contained in a vector u, which is assumed to be around the coordinate axis
Rotation angle of alpha, rotationThe positive direction is indicated in FIG. 5, in which
As a unit vector, the vector after rotation can be obtained:
then the angle alpha between the infinitesimal dipole and the coordinate axis after deformation i ',β i ' may be derived from u rot Obtaining;
(5b) for the parameter [ x i y i z i ]The change of (2) can be obtained by a coordinate transformation matrix:
wherein, T x ,T y ,T z Are rotation matrices, x, corresponding to the x, y, z axes, respectively i ,y i ,z i And x i ',y i ',z i ' are respectively the position vectors of the infinitesimal dipoles before and after deformation in a local coordinate system, and alpha, beta and gamma are respectively the rotation angles of the infinitesimal dipole model around the x, y and z axes. Then the bit vector of the deformed infinitesimal dipole in the global coordinate system can be expressed as:
r d ＝r'+L d
wherein r' is the position vector of the deformed local coordinate system, L d The coordinates of the center of the actual radiation unit under the global coordinate system after deformation;
(5c) and (5) substituting the results of the parameters after deformation in (5a) and (5b) into a mutual admittance calculation formula to obtain:
wherein j is an imaginary unit; omega is angular frequency; mu is the radiation space magnetic permeability; n is a radical of d The number of infinitesimal dipoles;
complex dipole moment of infinitesimal dipole model after deformation;
is the dyadic green function after deformation.
Step 6, solving the array admittance matrix according to the mutual admittance expression, and revealing the actual excitation value
(6a) Expanding the self-admittance as an array admittance matrix according to the result obtained in step (5c) by considering the input self-admittance as follows:
wherein the matrix U is an identity matrix; matrix Y 0 Inputting a self-admittance matrix for the port; y is d To be a deformed array admittance matrix without considering port self-admittance, it is expressed as:
(6b) according to the result obtained in the step (6a), the voltage V is excited by each array element port, and the actual exciting current of each array element can be obtained:
I d ＝Y·V
wherein, Y is an array admittance matrix, and V is a port excitation voltage matrix.
Step 7, calculating a far-field directional diagram of the deformed array antenna according to the actual excitation and the unit directional diagram of each radiation unit
(7a) Determining the space phase factor of each radiation unit according to the coordinates of the center position of each array element of the deformed array antenna:
wherein, the unit of the imaginary number of j,k is the propagation constant of electromagnetic wave in free space, r d i Is the bit-vector of the ith radiating element,
is a radiation space unit vector;
(7b) from the calculation result of (7a), the following matrix is determined:
wherein N is d Is the number of the radiation elements of the array,
is the radiation pattern of the i-th radiation element, W i Is the spatial phase factor of the ith radiation unit;
(7c) and (7) calculating the far field pattern of the deformed array antenna according to the results of (6b) and (7b) by using the following formula:
wherein A is r Is given by (7b), I d The actual excitation current matrix for each radiating element of the deformed array antenna is given by 6 (b). .
The advantages of the present invention are further illustrated by the following simulation cases:
1. simulation parameters
And (3) taking the linear array antenna with the working frequency of 3GHz as an analysis case, and analyzing the radiation performance of the array antenna under the condition of complex deformation. The array antenna comprises 10 radiating units, the type of the radiating units is a half-wave oscillator, the units are ideally spaced at a half wavelength, the positions of the ideal and deformed half-wave oscillator antennas are schematically shown in fig. 6, the deformation parameters of the radiating units are shown in table 1, the radiating units are excited by a voltage source, the excitation value is 1V, namely V is [ 11] T (ii) a Optimizing by adopting quantum particle swarm optimization to obtain actual radiation unitThe infinitesimal dipole model of (c), the parameter settings are shown in table 2.
TABLE 1 deformation of each array element
TABLE 2 QPSO parameters and optimized model boundaries
2. Simulation content and results
By utilizing the method, the far-field directional diagram after the half-wave oscillator deformation array antenna is deformed is calculated and compared with the simulation result of full-wave analysis software full-wave numerical simulation software FEKO. As can be seen from fig. 7, the analysis of the radiation performance of the deformed array antenna by the method of the present invention is substantially consistent with the FKEO simulation result, which proves the correctness and feasibility of the implementation method of the present invention.
The present invention is not limited to the above-mentioned embodiments, and based on the technical solutions disclosed in the present invention, those skilled in the art can make some substitutions and modifications to some technical features without creative efforts according to the disclosed technical contents, and these substitutions and modifications are all within the protection scope of the present invention.
Claims (10)
1. A method for analyzing a far-field directional pattern of a deformed array antenna based on an infinitesimal dipole model is characterized by comprising the following steps of:
(1) determining array antenna unit parameters according to the type of the array antenna; extracting a radiation unit, modeling and simulating in full-wave analysis software to solve to obtain near-field data of the radiation unit;
(2) establishing a group of infinitesimal dipoles according to near-field data of an actual radiation unit, setting initial parameter intervals of the infinitesimal dipoles, and deducing a radiation field of the infinitesimal dipoles;
(3) establishing an optimization model according to the near field data of the actual radiation unit and the radiation field of the infinitesimal dipole, so that the radiation field of the infinitesimal dipole is matched with the radiation field of the actual array radiation unit, and dipole moment amplitude, phase and bit vector parameters of the infinitesimal dipole are obtained;
(4) according to each parameter of the infinitesimal small dipole model, replacing each radiation unit of the array antenna with the infinitesimal small dipole model, and deducing a transadmittance expression;
(5) analyzing the change of an infinitesimal small dipole model according to the position of the array element after the array surface of the array antenna is deformed, and deducing the mutual admittance after the array surface is deformed;
(6) according to the mutual admittance among the position array elements, the input admittance is considered, a mutual admittance matrix of the deformation array antenna based on the infinitesimal small dipole model is obtained, and the actual excitation value of each radiation unit is determined;
(7) and calculating a far-field directional diagram of the deformed array antenna according to the space phase of the radiation unit and the actual excitation parameter of the radiation unit.
2. The method for analyzing the far-field pattern of the array antenna based on the infinitesimal dipole model deformation is characterized in that in the step (1), the array antenna element parameters comprise element forms and excitation voltages, and the near-field data comprise coordinates in a rectangular coordinate system of near-field observation points and components of field strengths at the observation points along the coordinate axes.
3. The far-field side of the deformed array antenna based on the infinitesimal dipole model as claimed in claim 1
The graph analyzing method is characterized in that in the step (2), the number and the dipole moment interval of infinitesimal dipoles are selected according to the near-field data of the actual radiating element; and determining the space domain where the infinitesimal dipole is located according to the actual geometrical model of the radiation unit.
4. The method for analyzing the far-field pattern of the deformed array antenna based on the infinitesimal dipole model as claimed in claim 1, wherein in the step (3), an optimization model is established and optimized, and the method comprises the following steps:
(3a) according to the near field data of the actual radiation unit, constructing an objective function, wherein the following optimization model is established by taking the near field intensity difference of the actual radiation unit and the infinitesimal dipole model at an observation point as the objective function:
find：
min：
s.t.：
wherein, χ i The parameters of the ith infinitesimal dipole respectively represent the amplitude of the dipole moment, the phase of the dipole moment, the included angle between the dipole and the x axis, the included angle between the dipole and the y axis and the coordinate of the center position of the dipole; n is a radical of hydrogen d The number of infinitesimal dipoles; n is a radical of op The number of observation points is;
for the field strength value of the actual antenna at the nth observation point,
the field intensity value of the infinite small dipole model under the nth observation point is obtained; d is the dipole moment amplitude of the infinitesimal dipole; Γ is the physical boundary of the actual radiating element;
(3b) according to the optimization model established in the step (3a), optimizing by using a quantum particle swarm algorithm to obtain a parameter χ i Thus, an infinitesimal dipole model of the actual antenna is obtained.
5. The method for analyzing the far-field pattern of the deformed array antenna based on the infinitesimal dipole model as claimed in claim 1, wherein in the step (4), each radiation unit of the array antenna is replaced by the infinitesimal dipole according to each parameter of the infinitesimal dipole model, so that the mutual coupling relationship between any two radiation units is converted into the mutual admittance between two sets of the infinitesimal dipoles; for any two radiating elements, the mutual coupling relationship is expressed as:
wherein, V m ，V n A port excitation voltage; e m ,H m An electromagnetic field generated for the m cell; j. the design is a square n ,M n N units of current and magnetic current; Ω is the physical boundary of two actual antennas;
after equivalence is carried out by using infinitesimal dipoles, the transadmittance of any two radiating elements is expressed as:
wherein N is d The number of infinitesimal dipoles; j is an imaginary unit; omega is angular frequency; mu is the radiation space magnetic permeability; r is a radical of hydrogen i ,r s The i th dipole of the array element m and the s th infinitesimal dipole of the array element n respectively,
the complex dipole moments of the ith dipole of array element m and the s equivalent dipole of array element n,
is a dyadic green function.
6. A radical as claimed in claim 5The method for analyzing the far-field pattern of the deformed array antenna in the infinitesimal small dipole model is characterized in that the complex dipole moment of the ith dipole of an array element m and the s equivalent dipole in the array element n
Are respectively obtained by the following formula:
wherein M is the complex dipole moment of the infinitesimal dipole; x, y and z are bit vectors of infinitesimal dipoles in a global coordinate system; alpha, beta and gamma are included angles between infinitesimal dipoles and x, y and z axes respectively.
7. The method as claimed in claim 5, wherein the method comprises a step of combining green functions
And scalar Green function G (r) i ,r s ) The relationship of (c) is expressed as:
wherein the content of the first and second substances,
is an identity matrix; k is a propagation constant;
is a gradient operator; g (r) i ,r s ) Is a scalar Green function, the expression of whichThe formula is as follows:
8. the method for analyzing the far-field pattern of the array antenna based on the infinitesimal dipole model deformation is characterized in that in the step (5), the step of calculating the mutual admittance between the array elements after the array surface deformation is as follows:
(5a) when the infinitesimal dipole model changes along with the position of the array element, the parameters
then the angle alpha between the infinitesimal dipole and the coordinate axis after deformation i ',β i ' may be derived from u rot Obtaining;
(5b) for the parameter [ x i y i z i ]The change of (2) can be obtained by a coordinate transformation matrix:
wherein, T x ,T y ,T z Are rotation matrices, x, corresponding to the x, y, z axes, respectively i ,y i ,z i And x i ',y i ',z i ' are respectively the position vectors of the infinitesimal dipoles before and after deformation in the local coordinate system, and α, β, γ are respectively the rotation angles of the infinitesimal dipole model around the x, y, z axes, so the position vector of the infinitesimal dipole after deformation in the global coordinate system is expressed as:
r d ＝r'+L d
wherein r' is the position vector of the deformed local coordinate system, L d The coordinate of the center of the actual radiation unit under the global coordinate system after deformation;
(5c) and (5) substituting the transformed parameter results of the steps (5a) and (5b) into a mutual admittance calculation formula to obtain:
wherein:
complex dipole moment of infinitesimal dipole model after deformation;
is the dyadic green function after deformation.
9. The method for analyzing the far-field pattern of the deformed array antenna based on the infinitesimal dipole model as claimed in claim 1, wherein the step (6) comprises the following specific steps:
(6a) expanding the self-admittance into an array admittance matrix according to the result obtained in step (5c) by considering the input self-admittance:
wherein the matrix U is an identity matrix; matrix Y 0 Inputting a self-admittance matrix for the port; y is d To be a deformed array admittance matrix without considering port self-admittance, it is expressed as:
(6b) and (3) obtaining the actual exciting current of each array element by exciting the voltage V at each array element port:
I d ＝Y·V
wherein, Y is an array admittance matrix, and V is a port excitation voltage matrix.
10. The method for analyzing the far-field pattern of the deformed array antenna based on the infinitesimal dipole model as claimed in claim 1, wherein in the step (7), the far-field pattern of the array antenna is calculated according to the spatial phase of the radiation unit and the actual excitation parameter of the radiation unit, and the specific steps are as follows:
(7a) determining a spatial phase factor of each radiation unit according to the coordinates of the center position of each array element of the deformed array antenna:
wherein j is an imaginary number unit, k is a propagation constant of the electromagnetic wave in free space, r i d The modified i-th radiation element has a bit vector,
is a radiation space unit vector;
(7b) determining the following matrix according to the calculation result of the step (7 a):
wherein N is d Is the number of the radiation elements of the array,
is the radiation pattern of the i-th radiation element, W i The spatial phase factor of the ith radiation unit;
(7c) calculating a far-field pattern of the deformed array antenna according to the results of the steps (6b) and (7 b):
wherein, I d The current matrix is actually excited for each radiating element of the deformed array antenna.
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