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CN111553046A - Antenna radiation calculation method based on spherical wave expansion and source reconstruction - Google Patents

Antenna radiation calculation method based on spherical wave expansion and source reconstruction Download PDF

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CN111553046A
CN111553046A CN202010173747.2A CN202010173747A CN111553046A CN 111553046 A CN111553046 A CN 111553046A CN 202010173747 A CN202010173747 A CN 202010173747A CN 111553046 A CN111553046 A CN 111553046A
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spherical
antenna
current
huygens
expression
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CN111553046B (en
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赵华鹏
刘畅
胡俊
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University of Electronic Science and Technology of China
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Abstract

The invention provides an antenna radiation calculation method based on spherical wave expansion and source reconstruction, which comprises the steps of firstly calculating a radiation field of any point on a data spherical surface with a smaller radius by using a spherical wave expansion method and only needing a radiation field of a small number of sampling points on a sampling spherical surface with a larger radius, then reconstructing an equivalent source distributed on a Huygens surface by using the calculated radiation field on the data spherical surface, and finally calculating a radiation field generated by the equivalent source in a space to obtain a radiation field actually generated by an antenna to be measured in the space. The method combines the spherical wave expansion method and the source reconstruction method by introducing the data spherical surface, fully exerts the advantages of less sampling points, small calculated amount, large calculated area of the source reconstruction method and strong applicability of the spherical wave expansion method, and is used for solving the problems that part of areas of the spherical wave expansion method cannot be calculated, the calculated amount of the source reconstruction method is too large, and the calculation time is too long.

Description

Antenna radiation calculation method based on spherical wave expansion and source reconstruction
Technical Field
The invention relates to the field of computational electromagnetism, in particular to an antenna radiation calculation method based on spherical wave expansion and source reconstruction.
Background
In antenna testing, in order to meet the requirements of application scenarios, it is desirable to know the electromagnetic radiation generated by the antenna so as to further improve and optimize the antenna. In order to obtain the electromagnetic radiation generated by the antenna in the surrounding space, the most direct method is to sample and measure the electromagnetic field in the space at certain intervals, so as to obtain the electromagnetic field distribution in the whole space. However, the direct measurement has obvious disadvantages, a great deal of labor cost and time cost are consumed, and the electromagnetic field of the unsampled point can only be obtained approximately through interpolation, and large errors can be caused. To improve the drawbacks of this method, the electromagnetic field is typically sampled at only a small fraction of points in space, and then calculated using computational electromagnetic methods to obtain the electromagnetic field at other points in space. There are two common methods: spherical wave expansion and source reconstruction.
The spherical wave expansion method is derived from a vector wave equation (Helmholtz equation), and since the electromagnetic field of the passive region satisfies the vector wave equation, a series of spherical wave eigenmodes can be obtained by solving the vector wave equation on any spherical surface of the passive region. The radiation electromagnetic field in the passive region can be expanded by utilizing the eigenmode of spherical waves, and the wave expansion coefficients of the same eigenmode of spherical waves are the same after the electromagnetic fields at different field points are expanded. Since the working frequency and the size of the antenna to be tested are known, the eigenmode of the spherical wave at any point in space can be directly calculated. After the electromagnetic fields at a plurality of sampling points are subjected to mode expansion, the wave expansion coefficients can be calculated by utilizing the characteristics of the same wave expansion coefficients of the eigenmodes of the same spherical wave. After the wave expansion coefficient is obtained, the wave expansion coefficient is substituted into a spherical wave mode expansion expression of the electromagnetic field at any point in space, and the electromagnetic field at the point can be directly calculated. The spherical wave expansion method has the disadvantages that only the electromagnetic field distribution in the region outside the minimum spherical surface surrounding the antenna can be calculated, the electromagnetic field distribution in the region between the antenna and the minimum spherical surface cannot be calculated and can only be obtained through measurement, but when the measuring probe is closer to the antenna, the probe can generate larger interference on the electromagnetic field distribution, so that the measuring result error is larger, and the electromagnetic field distribution in the minimum spherical surface is difficult to accurately obtain.
The basis of the source reconstruction method is the equivalent source principle. In the electromagnetic field problem, the distribution characteristics of real sources are usually unknown or very complex, and in this case, in order to solve the electromagnetic field in the space, an equivalent source can be introduced instead of the real source, and the electromagnetic field generated by the equivalent source and the electromagnetic field generated by the real source are the same, so the electromagnetic field generated by the equivalent source can be calculated to obtain the electromagnetic field distribution in the space, which is the equivalent source principle. Since the equivalent source is composed of some current and magnetic current, the calculation amount and the calculation difficulty of the radiation electromagnetic field are greatly reduced. The equivalent sources can be divided into a surface source and a source according to the distribution characteristics of the equivalent sources. In the source reconstruction method, the equivalent sources are distributed on the huygens plane surrounding the real source, and thus are plane equivalent sources. The source reconstruction method is to reconstruct an equivalent source distributed on a Huygens surface surrounding a real source by using an electromagnetic field at a sampling point, and then calculate an electromagnetic field generated by the equivalent source to obtain the actual electromagnetic field distribution in space. Since the surface equivalent source is derived from the boundary conditions of the electromagnetic field, there is no requirement for the shape of the huygens surface surrounding the real source, and thus the huygens surface can be of any shape. If the shape of the Wheatstone surface is changed according to the shape of the outer surface of the antenna to be detected, so that the Wheatstone surface is tightly attached to the outer surface of the antenna, the electromagnetic field distribution of the whole space outside the antenna to be detected can be obtained through calculation. The source reconstruction method has the disadvantages of large calculation amount, huge data amount and long calculation time when the size of the antenna to be measured is large or the distance between a sampling surface and the antenna is long, and a computer is required to have a large memory, so that the feasibility of the method is not high.
Therefore, the traditional methods have certain limitations, the spherical wave expansion method cannot obtain the electromagnetic field distribution in the minimum spherical surface, the source reconstruction method cannot solve the problem of large scale, and the two methods are difficult to meet the requirements in actual antenna tests.
Disclosure of Invention
The invention aims to provide an antenna radiation calculation method based on spherical wave expansion and source reconstruction aiming at the limitations of the two methods, the spherical wave expansion method and the source reconstruction method are combined, the advantages of small calculated amount of the spherical wave expansion method and wider calculation range of the source reconstruction method are fully utilized, the limitations of the two methods are avoided, and a method with the advantages of the two traditional methods is obtained, so that the method is higher in feasibility and wider in application range, and the requirement of antenna testing is better met.
In order to realize the method, the invention provides an antenna radiation calculation method based on spherical wave expansion and source reconstruction, which comprises the following steps:
step 1: sampling the radiation field of the antenna to be measured on a spherical sampling surface by adopting a near-field measuring probe, wherein the radius of the spherical sampling surface is r, the sampling interval is determined by the size of the antenna to be measured, and the measured radiation field is recorded as
Figure BDA0002410105890000021
And calculating the wave expansion coefficient a of each spherical wave eigenmode according to the spherical wave expansion expression of the radiation field generated by the antenna to be measured, wherein the spherical wave expansion expression is as followspqAnd bpq：
Figure BDA0002410105890000022
Wherein,
Figure BDA0002410105890000023
is the vector of the positions of the sample points,
Figure BDA0002410105890000024
and
Figure BDA0002410105890000025
is a spherical eigenmode, apqAnd bpqThe wave expansion coefficients of the corresponding spherical wave eigenmodes are represented, p and Q are orders of the spherical wave eigenmodes, different orders represent different spherical wave eigenmodes, and Q is a truncation number of the spherical wave eigenmode;
step 2: the calculated wave expansion coefficient apqAnd bpqSubstituted into the wave expansion expression of the radiation field on the spherical data surface, the radius of the spherical data surface is r0(r0R) calculating the radiation field on the spherical data surface, and recording as
Figure BDA0002410105890000026
And step 3: shaping the radiation field on a spherical data surface
Figure BDA0002410105890000027
Substituting the equivalent source into an expression of a radiation field generated by the equivalent source to reconstruct the equivalent source distributed on the Huygens surface surrounding the antenna to be tested, wherein the equivalent source is a surface current
Figure BDA0002410105890000028
Magnetic flow of dough kneading
Figure BDA0002410105890000029
Composition, surface current
Figure BDA00024101058900000210
Magnetic current of dough mixing
Figure BDA00024101058900000211
The expression of the generated radiation field is:
Figure BDA0002410105890000031
wherein,
Figure BDA0002410105890000032
is the position vector of the data point on the spherical data surface,
Figure BDA0002410105890000033
is the position vector of the current source point, omega is the working angular frequency of the antenna to be measured, mu is the magnetic conductivity, k is the wave number, S is the Huygens surface,
Figure BDA0002410105890000034
is a green function, and the expression is:
Figure BDA0002410105890000035
and 4, step 4:the reconstructed surface current
Figure BDA0002410105890000036
Magnetic flow of dough kneading
Figure BDA0002410105890000037
Substituting the equivalent source into the expression for generating the radiation field, and calculating to obtain the radiation field generated by the reconstructed equivalent source and recording the radiation field as
Figure BDA0002410105890000038
Due to the fact that
Figure BDA0002410105890000039
The electromagnetic field distribution generated by the antenna to be tested is consistent, so that the radiation field generated by the antenna to be tested at any spatial position is obtained.
The antenna radiation calculation method based on spherical wave expansion and source reconstruction provided by the invention combines two traditional methods, namely a spherical wave expansion method and a source reconstruction method, sufficiently exerts the advantages of the two methods, overcomes the defect of insufficient calculation range of the spherical wave expansion method by utilizing the advantage of wide calculation range of the source reconstruction method, and solves the problem of overlarge calculation amount when the source reconstruction method is used for processing a large-scale problem by utilizing the advantages of few sampling points and small calculation amount of the spherical wave expansion method, thereby obtaining a new method with the advantages of wide calculation range and small calculation amount. In addition, the data surface is introduced between the sampling surface and the source surface, the spherical wave expansion method is firstly used for calculating the radiation field on the data spherical surface with smaller radius by using the radiation fields of fewer sampling points on the sampling spherical surface, and the defects of excessive sampling points, excessive data volume and overlong calculation time caused by the overlarge radius of the sampling spherical surface when the equivalent source is reconstructed by the source reconstruction method are avoided. Compared with two traditional methods, namely a spherical wave expansion method and a source reconstruction method, the method has the advantages of higher feasibility and wider application range, and can better meet the requirements in antenna test.
Drawings
FIG. 1 is a schematic diagram illustrating the principle of the antenna radiation calculation method based on spherical wave expansion and source reconstruction
FIG. 2 is a diagram for assisting in explaining RWG basis functions
FIG. 3 is a schematic diagram of a quinary dipole antenna array to be tested in the embodiment
FIG. 4 is a schematic diagram of the positions of the antenna to be tested, the Huygens surface and the verification surface in the embodiment
FIG. 5 is a comparison graph of the radiation field actually generated by the antenna to be tested on the verification surface and the radiation field calculated by the present invention on the verification surface
Detailed Description
The invention is described in further detail below with reference to the following figures and examples, which are intended to illustrate, but not to limit the invention.
The invention provides an antenna radiation calculation method based on spherical wave expansion and source reconstruction, and a calculation model is shown in figure 1.
The specific operation steps are as follows:
step 1: sampling sphere omega using vector network analyzer in microwave darkroomsThe upper radiation field is uniformly sampled to obtain a spherical surface omegasHas a radius of rsThe sampling interval is determined by the size of the antenna to be measured, and specifically, as shown in fig. 1, the minimum spherical surface Ω surrounding the antenna to be measuredmRadius rminThen in sampling sphere omegasUpper theta and
Figure BDA0002410105890000041
the maximum sampling interval for a direction is:
Figure BDA0002410105890000042
q is the truncation number of the eigenmode of the spherical wave, and the expression of Q is as follows:
Q＝krmin+10
wherein k is a wave number, which can be calculated from the working frequency f of the antenna to be measured:
Figure BDA0002410105890000043
where λ is the wavelength and c is the speed at which the electromagnetic wave propagates in space.
Therefore, after the working frequency of the antenna to be measured and the medium of the measurement space are determined, the spherical surface omega is sampledsThe number of the upper sampling points is only equal to the minimum spherical surface omega surrounding the antenna to be measuredmRadius r ofminRelated to sampling the spherical surface omegasRadius r ofsIrrelevant, therefore, the sampling point number is not along with the sampling sphere omegasRadius r ofsIs increased to ensure that the amount of data is maintained at a lower level.
Recording and sampling spherical surface omegasThe radiation field to be up-sampled is
Figure BDA0002410105890000044
Using spherical wave expansion method pair
Figure BDA0002410105890000045
Unfolding was carried out to obtain:
Figure BDA0002410105890000046
wherein,
Figure BDA0002410105890000047
and
Figure BDA0002410105890000048
is a spherical eigenmode, apqAnd bpqThe wave spreading coefficients for the corresponding eigenmodes of the spherical wave are expressed as follows:
Figure BDA0002410105890000049
Figure BDA00024101058900000410
wherein,
Figure BDA00024101058900000411
is the vector of the positions of the sample points,
Figure BDA00024101058900000412
the coordinates of the sampling point in the spherical coordinate system,
Figure BDA00024101058900000413
in order to have the function of the tie-lengder,
Figure BDA00024101058900000414
for the second class of spherical Hankel functions, p and q are the orders of eigenmodes of the spherical wave, different orders represent different eigenmodes of the spherical wave, k is the wave number,
Figure BDA00024101058900000415
representing unit vectors, C, in a spherical coordinate systempqThe expression of (a) is:
Figure BDA00024101058900000416
to obtain
Figure BDA00024101058900000417
And
Figure BDA00024101058900000418
wave expansion coefficient of (a)pqAnd bpqNeed to utilize
Figure BDA00024101058900000419
And
Figure BDA00024101058900000420
orthogonality of (a):
Figure BDA0002410105890000051
Figure BDA0002410105890000052
Figure BDA0002410105890000053
wherein,
Figure BDA0002410105890000054
and
Figure BDA0002410105890000055
are respectively
Figure BDA0002410105890000056
And
Figure BDA0002410105890000057
the conjugate of (a) to (b),
Figure BDA0002410105890000058
and
Figure BDA0002410105890000059
is the eigenmode of the spherical wave, p 'and q' are the order of the eigenmode of the spherical wave, ΔpqThe expression of (a) is:
Figure BDA00024101058900000510
in the expression
Figure BDA00024101058900000511
The left and the right sides are respectively multiplied by
Figure BDA00024101058900000512
And
Figure BDA00024101058900000513
and for theta and
Figure BDA00024101058900000514
the integration yields:
Figure BDA00024101058900000515
Figure BDA00024101058900000516
will be provided with
Figure BDA00024101058900000517
And
Figure BDA00024101058900000518
substituting the expression of (A) into the two formulas and finishing to obtain:
Figure BDA00024101058900000519
Figure BDA00024101058900000520
wherein,
Figure BDA00024101058900000521
and
Figure BDA00024101058900000522
are respectively as
Figure BDA00024101058900000523
Of theta component and
Figure BDA00024101058900000524
and (4) components.
Thereby, the sampled radiation field can be utilized
Figure BDA00024101058900000525
Calculating to obtain the eigenmode of the spherical wave
Figure BDA00024101058900000526
And
Figure BDA00024101058900000527
wave expansion coefficient of (a)pqAnd bpq。
Step 2: the calculated wave expansion coefficient apqAnd bpqSubstitution data sphere omegadIn the expression of the spherical wave expansion of the upper radiation field, the expression and expression
Figure BDA00024101058900000528
Same form, data sphere omegadHas a radius of rd(rmin≤rd＜rs) Calculating the spherical data surface omegadA radiation field of
Figure BDA00024101058900000529
Figure BDA00024101058900000530
Is a data sphere omegadThe position vector of the upper data point.
Data sphere omegadThe radiation field of any point can be calculated, and in an actual test, the radiation field of data points with proper number and position can be calculated according to the requirement of reconstructing an equivalent source in the step 3.
And step 3: using a data sphere omegadUpper radiation field
Figure BDA0002410105890000061
Reconstructing the equivalent source distributed on the Wheatstone surface S, wherein the Wheatstone surface S is a closed curved surface tightly attached to the outer surface of the antenna to be measured, and in FIG. 1, the antenna to be measured is located in a cuboid, so that the Wheatstone surface S is the outer surface of the cuboid with a slightly larger length, width and height. Equivalent source is composed of surface current distributed on S
Figure BDA0002410105890000062
Magnetic flow of dough kneading
Figure BDA0002410105890000063
The reconstruction of the equivalent source is to calculate any point on the Huygens surface S
Figure BDA0002410105890000064
Surface current of
Figure BDA0002410105890000065
Magnetic flow of dough kneading
Figure BDA0002410105890000066
Need to calculate
Figure BDA0002410105890000067
And
Figure BDA0002410105890000068
and their direction needs to be indicated.
From the data sphere omegadUpper radiation field
Figure BDA0002410105890000069
Equal to the current on Huygens surface S
Figure BDA00024101058900000610
Magnetic flow of dough kneading
Figure BDA00024101058900000611
The resulting radiation field can be constructed as follows:
Figure BDA00024101058900000612
wherein,
Figure BDA00024101058900000613
is the position vector of the data point on the spherical data surface,
Figure BDA00024101058900000614
is the position vector of the current source point, omega is the working angular frequency of the antenna to be measured, mu is the magnetic conductivity, k is the wave number, S is the Huygens surface,
Figure BDA00024101058900000615
is a Green function, expression of whichComprises the following steps:
Figure BDA00024101058900000616
due to surface current
Figure BDA00024101058900000617
And surface magnetic current
Figure BDA00024101058900000618
And
Figure BDA00024101058900000619
the relation of (A) is unknown, and the expression of the integral term is too complex to obtain
Figure BDA00024101058900000620
And
Figure BDA00024101058900000621
so that any point on the Huygens surface S can be expressed
Figure BDA00024101058900000622
Surface current of
Figure BDA00024101058900000623
Magnetic current of dough mixing
Figure BDA00024101058900000624
Is set as unknown quantity, then is substituted into the surface integral expression of the equation, and the approximate result of the surface integral is calculated by using a numerical integration method, wherein the approximate result contains the surface current
Figure BDA00024101058900000625
And surface magnetic current
Figure BDA00024101058900000626
And finally, substituting an approximate result containing the unknown quantity into an equation to solve the unknown quantity.
Due to the benefits ofThere are infinite points on the wheatstone surface S, and therefore, a specific method is required to discretize the entire wheatstone surface S into finite points, and it is required that the approximation result of the surface integral obtained by numerical integration using the finite points has a small error. To reasonably select the position of discrete points, RWG basis function is introduced to induce surface current
Figure BDA00024101058900000627
And surface magnetic current
Figure BDA00024101058900000628
Characterization was performed by the following area current
Figure BDA00024101058900000629
For the purpose of illustration, surface magnetic flow
Figure BDA00024101058900000630
The characterization of (2) is similar.
As shown in FIG. 2, RWG basis functions are used to describe any point in a pair of co-bounded triangles
Figure BDA00024101058900000631
Surface current of
Figure BDA00024101058900000632
In the size and direction of (1), after the huygens surface S is divided by using the triangular mesh, assuming that the divided triangular mesh has N sides in total, any point on the whole huygens surface S
Figure BDA00024101058900000633
Surface current of
Figure BDA00024101058900000634
Can be expressed as:
Figure BDA0002410105890000071
wherein n is the serial number of the common edge in the triangular mesh, anTo correspond toThe spreading factor of the surface current at all points within the triangle of the common side with the serial number n,
Figure BDA0002410105890000072
is at any point
Figure BDA0002410105890000073
The current basis function corresponding to the surface current of (a), which is defined as follows:
Figure BDA0002410105890000074
Figure BDA0002410105890000075
Figure BDA0002410105890000076
as shown in figure 2 of the drawings, in which,
Figure BDA0002410105890000077
and
Figure BDA0002410105890000078
two triangles representing a common side, LnIs the length of the common side n,
Figure BDA0002410105890000079
and
Figure BDA00024101058900000710
are respectively triangular
Figure BDA00024101058900000711
And
Figure BDA00024101058900000712
the area of (a) is,
Figure BDA00024101058900000713
and
Figure BDA00024101058900000714
is defined as follows:
Figure BDA00024101058900000715
Figure BDA00024101058900000716
wherein,
Figure BDA00024101058900000717
and
Figure BDA00024101058900000718
are respectively triangular
Figure BDA00024101058900000719
And
Figure BDA00024101058900000720
the position vector of the vertex opposite the common edge.
Similarly, any point on the whole Huygens surface S can be used
Figure BDA00024101058900000721
Surface magnetic current of
Figure BDA00024101058900000722
Can be expressed as:
Figure BDA00024101058900000723
wherein, bnThe expansion coefficients of the surface magnetic flow at all points in the triangle with the same side corresponding to the side with the serial number n are shown,
Figure BDA00024101058900000724
to an arbitrary point
Figure BDA00024101058900000725
The magnetic flow basis function corresponding to the surface magnetic flow.
Thus, any point on the Huygens surface S can be represented using the RWG basis function
Figure BDA00024101058900000726
Surface current of
Figure BDA00024101058900000727
And surface magnetic current
Figure BDA00024101058900000728
The unknown quantity to be solved by the equation is the current expansion coefficient anAnd magnetic current expansion coefficient bnThe number of unknowns is 2N. Will express the formula
Figure BDA00024101058900000729
Mid-plane current
Figure BDA00024101058900000730
And expression
Figure BDA00024101058900000731
Middle magnetic flow
Figure BDA00024101058900000732
Substitution of expression (c)
Figure BDA00024101058900000733
In the equation, the following is obtained:
Figure BDA00024101058900000734
wherein the integral field SnIs triangular
Figure BDA00024101058900000735
And
Figure BDA00024101058900000736
the area is located, thereby the whole benefit is betterThe surface integral over the gaussian surface S is converted into a superposition of the surface integrals over the N pairs of co-bounded triangles. The integral domain is changed from an irregular closed curved surface S into a regular triangle S with common edgesnAnd S isnIs much smaller than S, the result of the calculation using numerical integration is more accurate and has less error. Because a isnAnd bnIs a constant, so thatnAnd bnIt is mentioned that the foregoing of the integral term does not affect the calculation result, so the equation can be expressed as:
Figure BDA0002410105890000081
since the above equation is a vector equation, it is inconvenient to anAnd bnIs solved by using a Galois gold match to convert it to a scalar equation, i.e. to simultaneously sum the vectors on both sides of the equation
Figure BDA0002410105890000082
And (4) performing inner product, wherein the result of the vector inner product is a scalar, and then the equation is converted into:
Figure BDA0002410105890000083
wherein,
Figure BDA0002410105890000084
is defined by
Figure BDA0002410105890000085
Similarly, the data sphere Ω is subtended with a triangular meshdDividing, supposing that the divided triangular mesh has M sides, M is the serial number of the side in the triangular mesh,
Figure BDA0002410105890000086
represents the data sphere ΩdAny one data point above
Figure BDA0002410105890000087
RWG basis function at (a).
Order to
Figure BDA0002410105890000088
Figure BDA0002410105890000089
Then the above mentioned
Figure BDA00024101058900000810
The equation of (c) can be expressed as:
Figure BDA00024101058900000811
using cmConstructing a column vector c, c having dimension M × 1, using
Figure BDA00024101058900000812
Construction matrix Ze，ZeDimension of (D) is M × N, using
Figure BDA00024101058900000813
Construction matrix Zc，ZcDimension of (a) is M × N, using anConstruct the column vector a, with a dimension of N × 1, using bnConstructing the column vector b, with the dimension of N × 1, a data sphere Ω can be useddThe radiation field of the upper M data points constructs a matrix equation as follows:
Figure RE-GDA00025683218400000814
due to the fact that
Figure BDA00024101058900000815
And cmAll quantities in the computational expression of (a) are known, so the matrix Ze、ZcEach element in the sum column vector c can be directly calculated, and the matrix Z is obtainede、ZcAnd substituting the sum column vector c into the matrix equation to solve to obtain a column vector a and a column vector b. A is tonAnd bnRespectively substitute for
Figure BDA00024101058900000816
Mid-plane current
Figure BDA00024101058900000817
And
Figure BDA00024101058900000818
middle magnetic flow
Figure BDA00024101058900000819
In the expression (2), an arbitrary point on the Huygens surface S can be obtained
Figure BDA00024101058900000820
Surface current of
Figure BDA00024101058900000821
Magnetic flow of dough kneading
Figure BDA00024101058900000822
I.e. the surface current on the whole Huygens surface S is obtained
Figure BDA00024101058900000823
Magnetic flow of dough kneading
Figure BDA00024101058900000824
The equivalent source reconstruction is completed.
And 4, step 4: the current expansion coefficient a obtained in the step 3 is usednAnd magnetic current expansion coefficient bnAnd the position vector of the field point to be calculated
Figure BDA0002410105890000091
Substitution into
Figure BDA0002410105890000092
In the expression of the electric field calculation, obtain
Figure BDA0002410105890000093
Electric field of
Figure BDA0002410105890000094
The expression is as follows:
Figure BDA0002410105890000095
due to the fact that
Figure BDA0002410105890000096
Is any point in space except the huygens surface S, the distribution of the radiation field generated by the equivalent source in the whole space except the interior of the huygens surface S can be calculated. And because the radiation fields generated by the antenna to be tested and the equivalent source in the space outside the Huygens surface S are completely consistent, the distribution situation of the radiation fields generated by the antenna to be tested in the space is obtained.
Specific examples are given below.
In the embodiment, a five-element dipole antenna array working at 1GHz is taken as an example for modeling; as shown in fig. 3, the antenna has a size L1＝60mm，L2＝120mm，L3150mm, wherein L1Is the length from the feed end to one end of the dipole antenna, L2Length of dipole antenna, L3The distance between two adjacent dipole antennas; the feeding ends of the five dipole antennas are all positioned at the middle point of the five dipole antennas, the feeding voltage is 1V, and the voltage frequency is 1 GHz; sampling sphere omegasRadius r ofs6m, data sphere ΩdRadius r ofd0.8m, the huygens plane S is the outer surface of a cuboid surrounding the quinary dipole antenna array, the dimension of the cuboid is 30mm × 630mm × 150mm, the proof plane omegavPerpendicular to the x-axis, at a distance of 0.2m from the origin of coordinates, 640mm × 120mm in size, Huygens surface S, and verification surface omegavThe position relation with the antenna to be measured is shown in fig. 4; to sampling sphere omegasWhen sampling is performed, the sampling interval Δ θ in the θ direction is 10 °, and
Figure BDA0002410105890000097
sampling interval of direction
Figure BDA0002410105890000098
648 sampling points in total; reconstructing an equivalent source on a Huygens surface S by the method provided by the invention, calculating a radiation field generated by the equivalent source on a verification surface, comparing the radiation field with a radiation field actually generated by a five-membered dipole antenna array on the verification surface, wherein the calculation time is 1311 seconds, and the result is shown in figure 5; therefore, the method can calculate the omega positioned on the minimum spherical surfacemInner verification surface omegavThe method has the advantages of high radiation field distribution, high calculation precision and short calculation time, and effectively solves the problems that partial areas of the spherical wave expansion method cannot be calculated and the calculation time of the source reconstruction method is too long.
While the invention has been described with reference to specific embodiments, any feature disclosed in this specification may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise; all of the disclosed features, or all of the method or process steps, may be combined in any combination, except mutually exclusive features and/or steps; any non-essential addition and replacement made by the technical characteristics of the technical scheme of the invention by a person skilled in the art belong to the protection scope of the invention.

Claims (4)

1. An antenna radiation calculation method based on spherical wave expansion and source reconstruction is characterized by comprising the following steps:
step 1: spherical omega is sampled by vector network analyzer to antenna to be measured in microwave darkroomsThe upper radiation field is uniformly sampled, and the spherical surface omega is sampledsHas a radius of rsThe sampling interval is determined by the size of the antenna to be tested, and the minimum spherical surface omega surrounding the antenna to be testedmRadius rmin；
Recording sampling spherical surface omega of antenna to be testedsThe radiation field to be up-sampled is
Figure RE-FDA0002568321830000011
Using spherical wave expansionsOpening method pair
Figure RE-FDA0002568321830000012
Unfolding was carried out to obtain:
Figure RE-FDA0002568321830000013
wherein,
Figure RE-FDA0002568321830000014
and
Figure RE-FDA0002568321830000015
is a spherical eigenmode, apqAnd bpqThe wave expansion coefficients for the corresponding eigenmodes of the spherical wave are expressed as follows:
Figure RE-FDA0002568321830000016
Figure RE-FDA0002568321830000017
wherein,
Figure RE-FDA0002568321830000018
is the vector of the positions of the sample points,
Figure RE-FDA0002568321830000019
the coordinates of the sampling point in the spherical coordinate system,
Figure RE-FDA00025683218300000110
in order to be accompanied by a legendre function,
Figure RE-FDA00025683218300000111
for the second class of spherical Hank function, p and q are orders of eigenmodes of the spherical wave, different orders representing different eigenmodes of the spherical waveIn the formula, k is the wave number,
Figure RE-FDA00025683218300000112
representing unit vectors, C, in a spherical coordinate systempqThe expression of (a) is:
Figure RE-FDA00025683218300000113
from this, the wave expansion coefficient a is calculatedpqAnd bpq；
Step 2: the calculated wave expansion coefficient apqAnd bpqSubstituting the data spherical surface omega of the antenna to be measureddIn the expression of the spherical wave expansion of the upper radiation field, the expression and expression
Figure RE-FDA00025683218300000114
Same form, data sphere omegadHas a radius of rdWherein r ismin≤rd＜rsAnd calculating the spherical data surface omega of the antenna to be measureddA radiation field of
Figure RE-FDA00025683218300000115
Figure RE-FDA00025683218300000116
Is a data sphere omegadA position vector of an upper data point;
and step 3: spherical omega using data of antenna to be measureddUpper radiation field
Figure RE-FDA00025683218300000117
Reconstructing an equivalent source distributed on a Huygens surface S, wherein the Huygens surface S is a closed curved surface tightly attached to the outer surface of an antenna to be detected, the antenna to be detected is positioned in a cuboid, so that the Huygens surface S is the outer surface of the cuboid with a large length, width and height, and the equivalent source is formed by surface currents distributed on S
Figure RE-FDA0002568321830000021
Magnetic flow of dough kneading
Figure RE-FDA0002568321830000022
The reconstruction of the equivalent source is to calculate any point on the Huygens surface S
Figure RE-FDA0002568321830000023
Surface current of
Figure RE-FDA0002568321830000024
Magnetic flow of dough kneading
Figure RE-FDA0002568321830000025
Need to calculate
Figure RE-FDA0002568321830000026
And
Figure RE-FDA0002568321830000027
the size of (2), their direction also needs to be represented;
from the data sphere omegadUpper radiation field
Figure RE-FDA0002568321830000028
Equal to the current on Huygens surface S
Figure RE-FDA0002568321830000029
Magnetic flow of dough kneading
Figure RE-FDA00025683218300000210
The resulting radiation field can be constructed as follows:
Figure RE-FDA00025683218300000211
wherein,
Figure RE-FDA00025683218300000212
is the position vector of the data point on the spherical data surface,
Figure RE-FDA00025683218300000213
is the position vector of the current source point, omega is the working angular frequency of the antenna to be measured, mu is the magnetic conductivity, k is the wave number, S is the Huygens surface,
Figure RE-FDA00025683218300000214
is a green function, and the expression is:
Figure RE-FDA00025683218300000215
because there are infinite points on the huygens surface S, the whole huygens surface S is dispersed into finite points by adopting a preset method, the approximate result of the surface integral obtained by using the finite points to carry out numerical integration has smaller error, and in order to reasonably select the positions of the discrete points, RWG basis functions are adopted to carry out surface current
Figure RE-FDA00025683218300000216
And surface magnetic current
Figure RE-FDA00025683218300000217
Characterization was performed by the following area current
Figure RE-FDA00025683218300000218
For the purpose of illustration, surface magnetic flow
Figure RE-FDA00025683218300000219
Is characterized by the similarity:
the RWG basis function is used for describing any point in a pair of common-edge triangles
Figure RE-FDA00025683218300000220
Surface current of
Figure RE-FDA00025683218300000221
In the size and direction of (1), after the huygens surface S is divided by using the triangular mesh, assuming that the divided triangular mesh has N sides in total, any point on the whole huygens surface S
Figure RE-FDA00025683218300000222
Surface current of
Figure RE-FDA00025683218300000223
Can be expressed as:
Figure RE-FDA00025683218300000224
wherein n is the serial number of the common edge in the triangular mesh, anThe expansion coefficients of the surface currents at all points within the co-bounded triangle corresponding to the side numbered n,
Figure RE-FDA00025683218300000225
is at any point
Figure RE-FDA00025683218300000226
The current basis function corresponding to the surface current of (a), which is defined as follows:
Figure RE-FDA00025683218300000227
Figure RE-FDA00025683218300000228
Figure RE-FDA00025683218300000229
wherein,
Figure RE-FDA00025683218300000230
and
Figure RE-FDA00025683218300000231
two triangles representing a common side, LnIs the length of the common side n,
Figure RE-FDA00025683218300000232
and
Figure RE-FDA00025683218300000233
are respectively triangular
Figure RE-FDA00025683218300000234
And
Figure RE-FDA00025683218300000235
the area of (a) is,
Figure RE-FDA00025683218300000236
and
Figure RE-FDA00025683218300000237
is defined as follows:
Figure RE-FDA0002568321830000031
Figure RE-FDA0002568321830000032
wherein,
Figure RE-FDA0002568321830000033
and
Figure RE-FDA0002568321830000034
are respectively triangular
Figure RE-FDA0002568321830000035
And
Figure RE-FDA0002568321830000036
a position vector of vertices opposite the common edge;
similarly, any point on the whole Huygens surface S can be used
Figure RE-FDA0002568321830000037
Surface magnetic current of
Figure RE-FDA0002568321830000038
Can be expressed as:
Figure RE-FDA0002568321830000039
wherein, bnThe expansion coefficients of the surface magnetic flow at all points in the triangle with the same side corresponding to the side with the serial number n are shown,
Figure RE-FDA00025683218300000310
is at any point
Figure RE-FDA00025683218300000311
The magnetic current basis function corresponding to the surface magnetic current;
thus, any point on the Huygens surface S can be represented using the RWG basis function
Figure RE-FDA00025683218300000312
Surface current of
Figure RE-FDA00025683218300000313
And surface magnetic current
Figure RE-FDA00025683218300000314
The unknown quantity to be solved is the current expansion coefficient anAnd magnetic current expansion coefficient bnThe number of unknowns is 2N; will express the formula
Figure RE-FDA00025683218300000315
Mid-plane current
Figure RE-FDA00025683218300000316
And expression
Figure RE-FDA00025683218300000317
Middle magnetic flow
Figure RE-FDA00025683218300000318
Substitution of expression (c)
Figure RE-FDA00025683218300000319
In the equation, the following is obtained:
Figure RE-FDA00025683218300000320
wherein the integral field SnIs triangular
Figure RE-FDA00025683218300000321
And
Figure RE-FDA00025683218300000322
the area is changed from an irregular closed curved surface S to a regular triangle SnAnd S isnThe area of (A) is much smaller than (S);
because a isnAnd bnIs a constant, so thatnAnd bnIt is mentioned that the foregoing of the integral term does not affect the calculation result, so the equation can be expressed as:
Figure RE-FDA00025683218300000323
since the above equation is a vector equationIs convenient for anAnd bnSo that it is converted into a scalar equation using Galois matching, i.e. simultaneously with the vector on both sides of the equation
Figure RE-FDA00025683218300000324
And (4) performing inner product, wherein the result of the vector inner product is a scalar, and then the equation is converted into:
Figure RE-FDA0002568321830000041
wherein,
Figure RE-FDA0002568321830000042
is defined by
Figure RE-FDA0002568321830000043
Similarly, the data sphere Ω is subtended with a triangular meshdDividing, supposing that the divided triangular mesh has M sides, M is the serial number of the side in the triangular mesh,
Figure RE-FDA0002568321830000044
represents the data sphere ΩdAny one data point above
Figure RE-FDA0002568321830000045
The RWG basis function of (c);
order to
Figure RE-FDA0002568321830000046
Figure RE-FDA0002568321830000047
Then the above mentioned
Figure RE-FDA0002568321830000048
The equation of (c) can be expressed as:
Figure RE-FDA0002568321830000049
using cmConstructing a column vector c, c having dimension M × 1, using
Figure RE-FDA00025683218300000410
Construction matrix Ze，ZeDimension of (D) is M × N, using
Figure RE-FDA00025683218300000411
Construction matrix Zc，ZcDimension of (a) is M × N, using anConstruct the column vector a, with a dimension of N × 1, using bnConstructing the column vector b, with the dimension of N × 1, a data sphere Ω can be useddThe radiation field of the upper M data points constructs a matrix equation as follows:
Figure RE-FDA00025683218300000412
due to the fact that
Figure RE-FDA00025683218300000427
And cmAll quantities in the computational expression of (a) are known, so the matrix Ze、ZcEach element in the sum column vector c can be directly calculated, and the matrix Z is obtainede、ZcSubstituting the sum column vector c into a matrix equation to solve to obtain a column vector a and a column vector b; a is tonAnd bnRespectively substitute for
Figure RE-FDA00025683218300000414
Mid-plane current
Figure RE-FDA00025683218300000415
And
Figure RE-FDA00025683218300000416
middle magnetic flow
Figure RE-FDA00025683218300000417
In the expression (2), an arbitrary point on the Huygens surface S can be obtained
Figure RE-FDA00025683218300000418
Surface current of
Figure RE-FDA00025683218300000419
Magnetic flow of dough kneading
Figure RE-FDA00025683218300000420
I.e. the surface current on the whole Huygens surface S is obtained
Figure RE-FDA00025683218300000421
Magnetic flow of dough kneading
Figure RE-FDA00025683218300000422
The equivalent source reconstruction is completed under the distribution condition of (1);
and 4, step 4: expanding the current expansion coefficient a obtained in the step 3nAnd magnetic current expansion coefficient bnAnd the position vector of the field point to be calculated
Figure RE-FDA00025683218300000423
Substitution into
Figure RE-FDA00025683218300000424
In the expression of the electric field calculation, obtain
Figure RE-FDA00025683218300000425
Electric field of
Figure RE-FDA00025683218300000426
The expression is as follows:
Figure RE-FDA0002568321830000051
due to the fact that
Figure RE-FDA0002568321830000052
The distribution of the radiation field generated by the equivalent source in the whole space except the interior of the huygens surface S can be calculated, and the distribution of the radiation field generated by the antenna to be tested in the space except the huygens surface S is obtained because the radiation field generated by the antenna to be tested and the equivalent source in the space except the huygens surface S is completely consistent.
2. The method for calculating antenna radiation based on spherical wave expansion and source reconstruction according to claim 1, wherein the sampling interval in step 1 is at sampling spherical surface ΩsUpper theta and
Figure RE-FDA0002568321830000053
the maximum sampling interval for a direction is:
Figure RE-FDA0002568321830000054
q is the truncation number of the eigenmode of the spherical wave, and the expression of Q is as follows:
Q＝krmin+10
wherein k is a wave number, which can be calculated from the working frequency f of the antenna to be measured:
Figure RE-FDA0002568321830000055
where λ is the wavelength and c is the speed at which the electromagnetic wave propagates in space.
3. The antenna radiation calculation method based on spherical wave expansion and source reconstruction according to any one of claims 1 or 2, wherein the wave expansion coefficient a is calculated in the step 1pqAnd bpqThe method comprises the following steps:
by using
Figure RE-FDA0002568321830000056
And
Figure RE-FDA0002568321830000057
orthogonality of (a):
Figure RE-FDA0002568321830000058
Figure RE-FDA0002568321830000059
Figure RE-FDA00025683218300000510
wherein,
Figure RE-FDA00025683218300000511
and
Figure RE-FDA00025683218300000512
are respectively
Figure RE-FDA00025683218300000513
And
Figure RE-FDA00025683218300000514
the conjugate of (a) to (b),
Figure RE-FDA00025683218300000515
and
Figure RE-FDA00025683218300000516
is the eigenmode of the spherical wave, p 'and q' are the order of the eigenmode of the spherical wave, ΔpqThe expression of (a) is:
Figure RE-FDA00025683218300000517
in the expression
Figure RE-FDA0002568321830000061
The left and the right sides are respectively multiplied by
Figure RE-FDA0002568321830000062
And
Figure RE-FDA0002568321830000063
and for theta and
Figure RE-FDA0002568321830000064
the integration yields:
Figure RE-FDA0002568321830000065
Figure RE-FDA0002568321830000066
will be provided with
Figure RE-FDA0002568321830000067
And
Figure RE-FDA0002568321830000068
substituting the expression of (A) into the two formulas and finishing to obtain:
Figure RE-FDA0002568321830000069
Figure RE-FDA00025683218300000610
wherein,
Figure RE-FDA00025683218300000611
and
Figure RE-FDA00025683218300000612
are respectively as
Figure RE-FDA00025683218300000613
Of theta component and
Figure RE-FDA00025683218300000614
a component;
thereby, the sampled radiation field can be utilized
Figure RE-FDA00025683218300000615
Calculating to obtain the eigenmode of the spherical wave
Figure RE-FDA00025683218300000616
And
Figure RE-FDA00025683218300000617
wave expansion coefficient of (a)pqAnd bpq。
4. The antenna radiation calculation method based on spherical wave expansion and source reconstruction as claimed in claim 3, wherein the antenna to be measured is a quintuple dipole antenna array working at 1GHz, and the size of the antenna to be measured is L1＝60mm，L2＝120mm，L3150mm, wherein L1Is the length from the feed end to one end of the dipole antenna, L2Length of dipole antenna, L3The distance between two adjacent dipole antennas is equal, the feeding ends of the five dipole antennas are all positioned at the middle point of the five dipole antennas, the feeding voltage is 1V, and the voltage frequency is 1 GHz; sampling sphere omegasRadius r ofs6m, data sphere ΩdRadius r ofd0.8m, the huygens plane S is the outer surface of a cuboid surrounding the quinary dipole antenna array, the dimension of the cuboid is 30mm × 630mm × 150mm, the proof plane omegavPerpendicular to the x-axis, at a distance of 0.2m from the origin of coordinates, and having a dimension of 640mm × 120mm, and omega for a sampling spheresWhen sampling is performed, the sampling interval Δ θ in the θ direction is 10 °, and
Figure RE-FDA00025683218300000618
sampling interval of direction
Figure RE-FDA00025683218300000619
There were 648 sampling points.
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