CN109782069B - Method for measuring mutual impedance between antennas - Google Patents

Method for measuring mutual impedance between antennas Download PDF

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CN109782069B
CN109782069B CN201910145332.1A CN201910145332A CN109782069B CN 109782069 B CN109782069 B CN 109782069B CN 201910145332 A CN201910145332 A CN 201910145332A CN 109782069 B CN109782069 B CN 109782069B
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antenna
spherical
electric field
spherical harmonic
expansion coefficient
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CN109782069A (en
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袁浩波
刘宏伟
贾建生
董欣欣
周虹光
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Xidian University
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Xidian University
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Abstract

The invention discloses a method for measuring mutual impedance among antennas, which belongs to the technical field of antenna measurement and is characterized in that a second electric field intensity and a second magnetic field intensity of a second antenna on a second spherical surface are obtained; determining a first spherical harmonic expansion coefficient and a second spherical harmonic expansion coefficient of a first antenna according to forward spherical harmonic transformation, wherein n and m are integers; calculating a first electric field intensity and a first magnetic field intensity of the first antenna on a second spherical surface by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficient; calculating a mutual impedance value Z between the first antenna and the second antenna according to the first electric field strength, the first magnetic field strength, the second electric field strength and the second magnetic field strength21. The mutual impedance between any two pairs of antennas can be analyzed quickly, accurately and stably, and the technical effect of universality is achieved.

Description

Method for measuring mutual impedance between antennas
Technical Field
The invention relates to the technical field of antenna measurement, in particular to a method for measuring mutual impedance among antennas.
Background
Mutual coupling between two antennas is of great engineering significance. The effective measurement of mutual coupling between antennas is a fundamental means of antenna system design, troubleshooting, and electromagnetic compatibility design. The magnitude of the mutual coupling can be generally described by parameters such as isolation, coupling coefficient, mutual impedance, and the like. The parameters can be mutually converted by a simple formula, and the coupling magnitude between the two antennas can be accurately described by only obtaining one of the parameters.
Currently, the classical method of calculating mutual coupling is proposed by Yaghjian. Firstly, taking a reaction surface S as an infinite plane between two antennas, then obtaining tangential electric fields of the two antennas on the S by using a plane near-field measurement method, expanding the tangential electric fields into a plane wave spectrum, and finally substituting the wave spectrum into a reaction integral to obtain a coupling coefficient between the two antennas. Since the contribution of the attenuated spectral components in the plane spectrum to the integral is small, it can be neglected. The remaining spectrum is linear with the far field of the antenna, so the coupling coefficient can be calculated by substituting the far field into the reaction integral. The disadvantage of this method is that ignoring the attenuation spectrum introduces large errors in many cases, making the measurement inaccurate. Another existing method is to use the near field instead of the far field to calculate the reaction integral. The method generally adopts a spherical surface as a reaction surface and adopts spherical harmonic transformation to spread the near-zone electromagnetic field of the antenna. The theoretical basis is a set of transmission formulas for coupling between two antennas, which is given by Hansen in 1988. The formula is strictly accurate without any approximation. But involves translation and rotation calculations of the spherical harmonics. Both of these calculations are extremely complex and unstable. Especially when the order of the spherical harmonics is high, the translation calculation cannot be numerically realized at all. Therefore, the existing algorithm based on the spherical harmonic transformation can only be applied to the coupling analysis of small antennas, and the analysis of the antenna coupling with larger size is impossible.
Disclosure of Invention
The invention provides a method for measuring mutual impedance between antennas, which is used for solving the technical problems that algorithms based on spherical harmonic transformation in the prior art can only be applied to coupling analysis of small antennas and cannot analyze the coupling of antennas with larger size, achieves the technical effects of quickly, accurately and stably analyzing the mutual impedance between any two pairs of antennas and has universality.
The invention provides a method for measuring mutual impedance among antennas, which comprises the following steps: obtaining a second electric field intensity of the second antenna on the second spherical surface
Figure BDA0001979823500000021
And a second magnetic field strength
Figure BDA0001979823500000022
Determining a first spherical harmonic expansion coefficient a of a first antenna according to forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mWherein n and m are integers; calculating the second spherical harmonic expansion coefficient of the first antenna by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficientFirst electric field intensity on spherical surface
Figure BDA0001979823500000023
And a first magnetic field strength
Figure BDA0001979823500000024
According to the first electric field intensity
Figure BDA0001979823500000025
First magnetic field intensity
Figure BDA0001979823500000026
Second electric field intensity
Figure BDA0001979823500000027
And a second magnetic field strength
Figure BDA0001979823500000028
Calculating a mutual impedance value Z between the first antenna and the second antenna21。
Preferably, the first spherical harmonic expansion coefficient a of the first antenna is determined according to forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mThe method specifically comprises the following steps:
the first electric field intensity
Figure BDA0001979823500000029
Is composed of
Figure BDA00019798235000000210
The first magnetic field intensity
Figure BDA00019798235000000211
Is composed of
Figure BDA00019798235000000212
Wherein pi is the circumference ratio, N is the spherical harmonic truncation order, N and m are integers,
Figure BDA00019798235000000213
for the first vector-sphere harmonic function,
Figure BDA00019798235000000214
is a second vector sphere harmonic function;
Figure BDA00019798235000000215
Figure BDA0001979823500000031
wherein the content of the first and second substances,
Figure BDA0001979823500000032
j is the sign of an imaginary number,
Figure BDA0001979823500000033
is the spherical coordinates of a point in space,
Figure BDA0001979823500000034
and
Figure BDA0001979823500000035
is a unit vector of the spherical coordinate system,
Figure BDA0001979823500000036
is an associated Legendre function, Zn(kr) is a spherical Bessel function, and
Figure BDA0001979823500000037
exp is an exponential function;
obtaining the radius r of the first antenna on a second spherical surfaceminAnd the spherical radius r at a predetermined point0Wherein r is0＞rmin；
Obtaining a tangential electric field of the first antenna at the predetermined point
Figure BDA0001979823500000038
Calculating the first spherical harmonic expansion coefficient a according to the tangential electric fieldn,mAnd a second spherical harmonic expansion coefficient bn,mWherein, in the step (A),
Figure BDA0001979823500000039
Figure BDA00019798235000000310
preferably, the pitch angle Δ α of the sampling points satisfies the Nyquist sampling theorem Δ α ≦ λ/(2 r)0)。
Preferably, the first electric field strength of the first antenna on the second spherical surface is calculated by using an FFT interpolation method
Figure BDA00019798235000000311
The method specifically comprises the following steps: using FFT acceleration pairs
Figure BDA00019798235000000312
Summing Fourier series in direction, wherein the summation formula is
Figure BDA00019798235000000313
Wherein the content of the first and second substances,
Figure BDA00019798235000000314
Figure BDA00019798235000000315
preferably, the formula (10) is calculated, specifically: in the theta direction, aiming at the preset point theta0Dividing the spherical surface to obtain a plurality of circular rings; for each and theta0Corresponding circular rings, when m and n are different, all are calculated
Figure BDA00019798235000000316
Will preset a point theta0Is arranged in a ring
Figure BDA00019798235000000317
Uniformly subdividing in the direction to enable the length of each section of arc to be less than 0.5 lambda; the value of the formula (10) is obtained by calculation.
Preferably, after the calculating obtains the value of the formula (10), the calculating further includes: calculating the corresponding near field value of each point by adopting an FFT acceleration method for all the uniformly distributed points on the circular ring; and obtaining the near field values of the plurality of circular rings, and obtaining the field values of all discrete points on the second spherical surface according to the near field values of the plurality of circular rings.
Preferably, the method further comprises the following steps: a mutual impedance value Z between the first antenna and the second antenna21Is composed of
Figure BDA0001979823500000041
Wherein S is a reaction surface surrounding the second antenna, I11Current value for exciting said first antenna, I21Current value for exciting said second antenna
One or more technical solutions in the embodiments of the present invention at least have one or more of the following technical effects:
the method for measuring the mutual impedance between the antennas provided by the embodiment of the invention obtains the second electric field intensity of the second antenna on the second spherical surface
Figure BDA0001979823500000042
And a second magnetic field strength
Figure BDA0001979823500000043
Determining a first spherical harmonic expansion coefficient a of a first antenna according to forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mWherein n and m are integers; calculating a first electric field intensity of the first antenna on a second spherical surface by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficient
Figure BDA0001979823500000044
And a first magnetic field strength
Figure BDA0001979823500000045
According to the first electric field intensity
Figure BDA0001979823500000046
First magnetic field intensity
Figure BDA0001979823500000047
Second electric field intensity
Figure BDA0001979823500000048
And a second magnetic field strength
Figure BDA0001979823500000049
Calculating a mutual impedance value Z between the first antenna and the second antenna21. Therefore, the technical problems that in the prior art, algorithms based on spherical harmonic transformation can only be applied to coupling analysis of small antennas and cannot analyze coupling of antennas with larger sizes are solved, mutual impedance between any two pairs of antennas can be analyzed quickly, accurately and stably, and the technical effect of universality is achieved.
The foregoing description is only an overview of the technical solutions of the present invention, and the embodiments of the present invention are described below in order to make the technical means of the present invention more clearly understood and to make the above and other objects, features, and advantages of the present invention more clearly understandable.
Drawings
FIG. 1 is a flowchart illustrating a method for measuring mutual impedance between antennas according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of an applied field between a first antenna and a second antenna according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a minimum spherical surface and a measurement spherical surface between a first antenna and a second antenna according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of the reaction sphere subdivision in the embodiment of the present invention;
FIG. 5 is a view of the ring of FIG. 4
Figure BDA0001979823500000051
Schematic diagram of direction subdivision;
FIG. 6 is a schematic diagram of discrete points in a reaction sphere in an embodiment of the present invention;
FIG. 7 is a schematic diagram of Taylor arrays of two planar antenna arrays according to an embodiment of the present invention;
FIG. 8 is the Taylor array 3d pattern of FIG. 7;
fig. 9 is a transimpedance diagram of the antenna array of fig. 7 at different tilt angles;
fig. 10 is another transimpedance diagram of the antenna array of fig. 7 at different tilt angles.
Detailed Description
The embodiment of the invention provides a method for measuring mutual impedance among antennas, which is used for solving the technical problems that algorithms based on spherical harmonic transformation in the prior art can only be applied to coupling analysis of small antennas and cannot analyze coupling of antennas with larger sizes.
The technical scheme in the embodiment of the invention has the following general idea:
the method for measuring the mutual impedance between the antennas provided by the embodiment of the invention obtains the second electric field intensity of the second antenna on the second spherical surface
Figure BDA0001979823500000052
And a second magnetic field strength
Figure BDA0001979823500000053
Determining a first spherical harmonic expansion coefficient a of a first antenna according to forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mWherein n and m are integers; calculating a first electric field intensity of the first antenna on a second spherical surface by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficient
Figure BDA0001979823500000054
And a first magnetic field strength
Figure BDA0001979823500000061
According to the first electric field intensity
Figure BDA0001979823500000062
First magnetic field intensity
Figure BDA0001979823500000063
Second electric field intensity
Figure BDA0001979823500000064
And a second magnetic field strength
Figure BDA0001979823500000065
Calculating a mutual impedance value Z between the first antenna and the second antenna21. The mutual impedance between any two pairs of antennas can be analyzed quickly, accurately and stably, and the technical effect of universality is achieved.
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Examples
Fig. 1 is a method for measuring mutual impedance between antennas according to an embodiment of the present invention, as shown in fig. 1, the method includes:
step 1: obtaining a second electric field intensity of the second antenna on the second spherical surface
Figure BDA0001979823500000066
And a second magnetic field strength
Figure BDA0001979823500000067
Step 2: determining a first spherical harmonic expansion coefficient a of a first antenna according to forward spherical harmonic transformationn,mAnd a second ballCoefficient of harmonic expansion bn,mWherein n and m are integers.
Further, the first spherical harmonic expansion coefficient a of the first antenna is determined according to the forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mThe method specifically comprises the following steps:
the first electric field intensity
Figure BDA0001979823500000068
Is composed of
Figure BDA0001979823500000069
The first magnetic field intensity
Figure BDA00019798235000000610
Is composed of
Figure BDA00019798235000000611
Wherein pi is the circumference ratio, N is the spherical harmonic truncation order, N and m are integers,
Figure BDA00019798235000000612
for the first vector-sphere harmonic function,
Figure BDA00019798235000000613
is a second vector sphere harmonic function;
Figure BDA0001979823500000071
Figure BDA0001979823500000072
wherein the content of the first and second substances,
Figure BDA0001979823500000073
j is the sign of an imaginary number,
Figure BDA0001979823500000074
is the spherical coordinates of a point in space,
Figure BDA0001979823500000075
and
Figure BDA0001979823500000076
is a unit vector of the spherical coordinate system,
Figure BDA0001979823500000077
is an associated Legendre function, Zn(kr) is a spherical Bessel function, and
Figure BDA0001979823500000078
exp is an exponential function;
obtaining the radius r of the first antenna on a second spherical surfaceminAnd a spherical radius r0 at a predetermined point, where r0＞rmin；
Obtaining a tangential electric field of the first antenna at the predetermined point
Figure BDA0001979823500000079
Calculating the first spherical harmonic expansion coefficient a according to the tangential electric fieldn,mAnd a second spherical harmonic expansion coefficient bn,mWherein, in the step (A),
Figure BDA00019798235000000710
Figure BDA00019798235000000711
further, the spacing angle delta α of the sampling points satisfies the Nyquist sampling theorem delta α ≦ lambda/(2 r)0)。
Specifically, as shown in fig. 2, the theoretical basis for calculating the mutual impedance between the two antennas is the following reciprocity theorem:
Figure BDA00019798235000000712
where S is the reaction surface surrounding the antenna 2.
Figure BDA00019798235000000713
When the first antenna is operated for the second antenna not operating (with current I)11The first antenna is excited), the electric field strength and the magnetic field strength generated in the space.
Figure BDA00019798235000000714
Is that the first antenna is assumed to be absent and the second antenna is operating (at current I)21The second antenna is excited), a field is generated in space. The exact values of these two fields are difficult to obtain and are generally approximated as the electromagnetic field strength radiated by the first antenna when the second antenna is not present.
Figure BDA0001979823500000081
Is the field that is generated in space when the second antenna is operating assuming that the first antenna is not present. Therefore, the first spherical surface is a spherical surface surrounding the first antenna, and the second spherical surface is a spherical surface surrounding the second antenna. Due to the fact that
Figure BDA0001979823500000082
And
Figure BDA0001979823500000083
of a physical quantity which is not at all relevant, so that the integral quantity in the above formula (1) has no practical physical significance, called the reaction integral, further wherein the second electric field strength
Figure BDA0001979823500000084
And a second magnetic field strength
Figure BDA0001979823500000085
Can be directly obtained through measurement in actual work. The radiation characteristics of the antenna can be measured, i.e. the near field strength and the magnetic field strength of the antenna.
Further, the spherical resonance transformation is a relatively mature algorithm in the near-field antenna measurement. Arbitrary in passive spaceThe time-harmonic electromagnetic field can be developed with spherical harmonics.
Figure BDA0001979823500000086
Figure BDA0001979823500000087
Wherein pi is a circumference ratio, N is a spherical harmonic truncation order, N and m are integers,
Figure BDA0001979823500000088
for the first vector-sphere harmonic function,
Figure BDA0001979823500000089
is a second vector sphere harmonic function; a isn,mAnd bn,mFor the expansion coefficients, the vector sphere harmonics are as follows:
Figure BDA00019798235000000810
Figure BDA00019798235000000811
Figure BDA00019798235000000812
wherein, among others,
Figure BDA00019798235000000813
j is the sign of an imaginary number,
Figure BDA00019798235000000814
is the coordinates of a sphere at a point in space,
Figure BDA00019798235000000815
and
Figure BDA00019798235000000816
is a unit vector of the spherical coordinate system,
Figure BDA00019798235000000817
is an associated Legendre function, Zn(kr) is a spherical Bessel function and is defined
Figure BDA00019798235000000818
exp is an exponential function; .
Further, as shown in fig. 3, a transmitting antenna is provided, and the minimum spherical radius surrounding the antenna isrmin. Assuming that the radius of the measuring sphere is r0(r0＞rmin) Measured tangential electric field of
Figure BDA00019798235000000819
Note that the pitch angle Δ α of the sampling points must satisfy the Nyquist sampling theorem Δ α ≦ λ/(2 r)0). The unknown coefficients in (2) can then be calculated using the following integration.
Figure BDA0001979823500000091
Figure BDA0001979823500000092
The double integration includes an inner integration and an outer integration. Inner layer integral is related to
Figure BDA0001979823500000093
The fourier integral of (a) can be obtained by Fast Fourier Transform (FFT). The outer integral is usually calculated by gaussian integration. Taking the near field of the antenna into consideration, the spherical harmonic coefficient obtained from (6) and (7) is called forward spherical harmonic transformation, and the calculation complexity is O (N)3). In contrast, with the spherical harmonic coefficients of the antenna known, the use of (2) to calculate the near field is referred to as an inverse spherical harmonic transform. If the field points are evenly distributed on a spherical surface surrounding the antenna itself, the sum term of the series in (2) can be accelerated by the FFT, so that the computational complexity is O (N)3)。
And step 3: calculating a first electric field intensity of the first antenna on a second spherical surface by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficient
Figure BDA0001979823500000094
And a first magnetic field strength
Figure BDA0001979823500000095
And 4, step 4: according to the first electric field intensity
Figure BDA0001979823500000096
First magnetic field intensity
Figure BDA0001979823500000097
Second electric field intensity
Figure BDA0001979823500000098
And a second magnetic field strength
Figure BDA0001979823500000099
Calculating a mutual impedance value Z between the first antenna and the second antenna21。
Further, the first electric field intensity of the first antenna on the second spherical surface is calculated by adopting an FFT interpolation method
Figure BDA00019798235000000910
The method specifically comprises the following steps:
using FFT acceleration pairs
Figure BDA00019798235000000911
Summing Fourier series in direction, wherein the summation formula is
Figure BDA00019798235000000912
Wherein the content of the first and second substances,
Figure BDA00019798235000000913
further, the calculating the formula (10) specifically includes:
in the theta direction, aiming at the preset point theta0Dividing the spherical surface to obtain a plurality of circular rings;
for each and theta0Corresponding circular rings, when m and n are different, all are calculated
Figure BDA00019798235000000914
Will preset a point theta0Is arranged in a ring
Figure BDA00019798235000000915
Uniformly subdividing in the direction to enable the length of each section of arc to be less than 0.5 lambda;
the value of the formula (10) is obtained by calculation.
Further, after the calculating obtains the value of the formula (10), the calculating further includes:
and calculating the corresponding near field value of each point by adopting an FFT acceleration method for all the uniformly distributed points on the circular ring.
And obtaining the near field values of the plurality of circular rings, and obtaining the field values of all discrete points on the second spherical surface according to the near field values of the plurality of circular rings.
Further, the method also comprises the following steps: a mutual impedance value Z between the first antenna and the second antenna21Is composed of
Figure BDA0001979823500000101
Wherein S is a reaction surface surrounding the second antenna, I11Current value for exciting said first antenna, I21A current value for energizing the second antenna.
Specifically, a first electric field intensity of the first antenna on a second spherical surface is calculated by using an FFT/interpolation method
Figure BDA0001979823500000102
And a first magnetic field strength
Figure BDA0001979823500000103
However, the calculation process of direct calculation is very complicated and the calculation amount is large because the calculation complexity is O (N)4) Resulting in a very slow speed. Thus, for simplicity, each component of the electric field strength may be abbreviated as:
Figure BDA0001979823500000104
wherein each term of the summation equation is the performance of a constant, a Bessel function, a conjugal legendre function, and an exponential function. As shown in FIG. 3, the viewpoint is on an eccentric sphere, so the coordinate r depends on
Figure BDA0001979823500000105
And theta. To pair
Figure BDA0001979823500000106
The summation of Fourier series in the direction is accelerated by FFT, and the summation order can be changed.
Figure BDA0001979823500000107
Wherein the content of the first and second substances,
Figure BDA0001979823500000108
obviously, if the field point is at
Figure BDA0001979823500000109
The directions are evenly distributed, as shown in fig. 5, then equation (9) can be accelerated using FFT. And then the near-zone electric field of the first antenna can be obtained. In the same manner, the near field in the formula (3) can be obtained. Finally, the mutual impedance is obtained by introducing the reciprocity theorem (1).
Further, calculating the formula (9) specifically includes: in the theta direction, the reaction spherical surface is divided by using a plurality of planes shown in fig. 4 to obtain a series of circular rings; for each and theta0Corresponding circular rings, all of which are calculated when m and n are different
Figure BDA0001979823500000111
As shown in fig. 5, the ring is placed at
Figure BDA0001979823500000112
The direction is evenly divided, so that the length of each section of arc is less than 0.5 lambda; is calculated by equation (10)
Figure BDA0001979823500000113
Calculating the near field value of all uniformly distributed points on the circle by using an FFT acceleration in an equation (9); after all circles have been calculated, the field for all discrete points in FIG. 6 can be obtained.
Further, fig. 7 shows two planar antenna arrays operating at 3.0 Ghz. The left 30 x 30 array consists of quarter-wave dipoles with a cell pitch of 0.5 λ. The total input power is distributed to each dipole according to a taylor distribution. The near field distribution of each antenna on the surrounding spherical surface is obtained by adopting a high-order moment method for simulation, and then the near field distribution is introduced into the method to obtain the mutual impedance of the two antennas. Meanwhile, the calculated result of the method is compared with the mutual impedance directly obtained by a high-order moment method.
Figure 8 shows the left taylor pattern with sidelobe levels of-35 dB. The antenna array on the right is identical to the antenna array on the left. But the antenna array on the right is rotated in the y ' z ' plane by some angle alpha about the x ' axis. The distance between the geometric centers of the two arrays is d 15 λ. In the spherical harmonic transformation, the radius of the spherical surface surrounding each array is 15 λ and the truncation order is 114.
As can be seen from fig. 9, the results of this method almost completely agree with those of the high-order moment method. The maximum error and the maximum relative error of the method are 0.08 Ω and 0.43%, respectively. The calculation time required for the higher order moment method and the calculation time required for the method are 2242 seconds and 152 seconds, respectively, i.e. the latter is 15 times faster than the former.
Therefore, the method for measuring the mutual impedance between the antennas provided by the embodiment of the invention can quickly, accurately and stably analyze the mutual impedance between any two pairs of antennas, and has universality. The computational complexity is the same as that of the common spherical harmonic transformation, namely O (N)3). Numerical simulation shows that when the mutual impedance of two large antenna arrays is analyzed, the method is faster than a moment method in electromagnetic calculation by more than ten times, andthe relative error is less than 1%. Further, the method achieves accurate and reliable results, and is completely consistent with a high-order moment method; the speed is dozens of times faster than the high-order moment method; faster than traditional measurement methods; the type, size or position of the antennas are not limited, and the technical effect of analysis can be rapidly measured no matter what the coupling between the antennas.
One or more technical solutions in the embodiments of the present invention at least have one or more of the following technical effects:
the method for measuring the mutual impedance between the antennas provided by the embodiment of the invention obtains the second electric field intensity of the second antenna on the second spherical surface
Figure BDA0001979823500000121
And a second magnetic field strength
Figure BDA0001979823500000122
Determining a first spherical harmonic expansion coefficient a of a first antenna according to forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mWherein n and m are integers; calculating a first electric field intensity of the first antenna on a second spherical surface by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficient
Figure BDA0001979823500000123
And a first magnetic field strength
Figure BDA0001979823500000124
According to the first electric field intensity
Figure BDA0001979823500000125
First magnetic field intensity
Figure BDA0001979823500000126
Second electric field intensity
Figure BDA0001979823500000127
And a second magnetic field strength
Figure BDA0001979823500000128
Calculating a mutual impedance value Z between the first antenna and the second antenna21. Therefore, the technical problems that in the prior art, algorithms based on spherical harmonic transformation can only be applied to coupling analysis of small antennas and cannot analyze coupling of antennas with larger sizes are solved, mutual impedance between any two pairs of antennas can be analyzed quickly, accurately and stably, and the technical effect of universality is achieved.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. Therefore, it is intended that the appended claims be interpreted as including preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various modifications and variations can be made in the embodiments of the present invention without departing from the spirit or scope of the embodiments of the invention. Thus, if such modifications and variations of the embodiments of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to encompass such modifications and variations.

Claims (3)

1. A method of measuring mutual impedance between antennas, the method comprising:
obtaining a second electric field intensity of the second antenna on the second spherical surface
Figure FDA0002518749020000011
And a second magnetic field strength
Figure FDA0002518749020000012
Determining a first spherical harmonic expansion coefficient a of a first antenna according to forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mWherein n and m are integers;
calculating the second spherical harmonic expansion coefficient of the first antenna by adopting an FFT interpolation method according to the first spherical harmonic expansion coefficient and the second spherical harmonic expansion coefficientFirst electric field intensity on spherical surface
Figure FDA0002518749020000013
And a first magnetic field strength
Figure FDA0002518749020000014
According to the first electric field intensity
Figure FDA0002518749020000015
First magnetic field intensity
Figure FDA0002518749020000016
Second electric field intensity
Figure FDA0002518749020000017
And a second magnetic field strength
Figure FDA0002518749020000018
Calculating a mutual impedance value Z between the first antenna and the second antenna21；
The second spherical surface is a spherical surface surrounding the second antenna;
determining a first spherical harmonic expansion coefficient a of the first antenna according to the forward spherical harmonic transformationn,mAnd a second spherical harmonic expansion coefficient bn,mThe method specifically comprises the following steps:
the first electric field intensity
Figure FDA0002518749020000019
Is composed of
Figure FDA00025187490200000110
The first magnetic field intensity
Figure FDA00025187490200000111
Is composed of
Figure FDA00025187490200000112
Wherein pi is the circumference ratio, N is the spherical harmonic truncation order, N and m are integers,
Figure FDA00025187490200000113
for the first vector-sphere harmonic function,
Figure FDA00025187490200000114
is a second vector sphere harmonic function;
Figure FDA00025187490200000115
Figure FDA00025187490200000116
wherein the content of the first and second substances,
Figure FDA00025187490200000117
j is the sign of an imaginary number,
Figure FDA00025187490200000118
is the spherical coordinates of a point in space,
Figure FDA00025187490200000119
and
Figure FDA00025187490200000120
is a unit vector of the spherical coordinate system,
Figure FDA00025187490200000121
is an associated Legendre function, Zn(kr) is a spherical Bessel function, and
Figure FDA0002518749020000021
exp is an exponential function;
obtaining a half of the first antenna on a second spherical surfaceDiameter rminAnd the spherical radius r at a predetermined point0Wherein r is0＞rmin；
Obtaining a tangential electric field of the first antenna at the predetermined point
Figure FDA0002518749020000022
Wherein E isθThe electric field intensity in the theta direction at the preset point in the spherical coordinate system is obtained;
Figure FDA0002518749020000023
for the preset point in the spherical coordinate system
Figure FDA0002518749020000024
Directional electric field strength;
calculating the first spherical harmonic expansion coefficient a according to the tangential electric fieldn,mAnd a second spherical harmonic expansion coefficient bn,mWherein, in the step (A),
Figure FDA0002518749020000025
Figure FDA0002518749020000026
calculating the first electric field intensity of the first antenna on the second spherical surface by adopting an FFT interpolation method
Figure FDA0002518749020000027
The method specifically comprises the following steps:
using FFT acceleration pairs
Figure FDA0002518749020000028
Summing Fourier series in direction, wherein the summation formula is
Figure FDA0002518749020000029
Wherein the content of the first and second substances,
Figure FDA00025187490200000210
the method further comprises the following steps:
a mutual impedance value Z between the first antenna and the second antenna21Is composed of
Figure FDA00025187490200000211
Wherein S is a reaction surface surrounding the second antenna, I11Current value for exciting said first antenna, I21A current value for energizing the second antenna.
2. A method of measuring inter-antenna mutual impedance as claimed in claim 1, characterized in that said calculation of said formula (10) is in particular:
in the theta direction, aiming at the preset point theta0Dividing the spherical surface to obtain a plurality of circular rings;
for each and theta0Corresponding circular rings, when m and n are different, all are calculated
Figure FDA0002518749020000032
Will preset a point theta0Is arranged in a ring
Figure FDA0002518749020000031
Uniformly subdividing in the direction to ensure that the length of each section of arc is less than 0.5 lambda, wherein lambda is the wavelength;
the value of the formula (10) is obtained by calculation.
3. The method of measuring inter-antenna mutual impedance of claim 2, wherein after said calculating obtains the value of said equation (10), further comprising:
calculating the corresponding near field value of each point by adopting an FFT acceleration method for all the uniformly distributed points on the circular ring;
and obtaining the near field values of the plurality of circular rings, and obtaining the field values of all discrete points on the second spherical surface according to the near field values of the plurality of circular rings.
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US9053268B1 (en) * 2007-12-19 2015-06-09 The United States Of America As Represented By The Secretary Of The Navy Analytic antenna design for a dipole antenna
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US9053268B1 (en) * 2007-12-19 2015-06-09 The United States Of America As Represented By The Secretary Of The Navy Analytic antenna design for a dipole antenna
CN106771672A (en) * 2017-02-21 2017-05-31 北京邮电大学 The method and device of Far-Field antennas measurement system is carried out based on cubic spline interpolation

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