CN104063587B - The method that panel mismachining tolerance is affected on electrical property is calculated based on block form - Google Patents

Abstract

The invention relates to a method for calculating the influence of panel processing errors on electrical performance based on the block form, which is used to guide the processing of the panel surface and the simulation analysis and evaluation of antenna electrical performance. The invention expresses the far field affected by the surface error as an ideal field and error The sum of fields can simplify the error power between different panels to zero when calculating the average power of the antenna, which reduces the amount of calculation; calculate the influence of surface processing errors on electrical performance according to the actual block form of the antenna, which is closer to engineering Practical; use accurate quadruple numerical integration to solve the average error power of a single panel, avoiding the defect that the traditional integral time interval vector Δ cannot be fully represented; through simulation analysis, the upper limit of the root mean square of the surface error is obtained, avoiding the need to rely on Experience puts forward high precision requirements, which reduces the difficulty of processing and meets the electrical performance indicators at the same time.

Description

Method of Calculating the Influence of Panel Processing Errors on Electrical Properties Based on Block Form

technical field

The invention belongs to the technical field of antennas, and specifically relates to a method for calculating the influence of panel processing errors on electrical performance based on block form, which is used to guide panel surface processing and antenna electrical performance simulation analysis and evaluation.

Background technique

As the most commonly used microwave and millimeter wave high-gain antenna, the reflector antenna is widely used in radar measurement and control, satellite communication, radio astronomy and other fields. It may also be used as a microwave weapon and a tool for energy transmission in the future. Since it works in a higher frequency band, the electromagnetic performance is susceptible to various errors. There are two main types of surface errors for reflector antennas: systematic errors and random errors. The system error mainly includes the deformation of the antenna due to its own weight, the thermal deformation caused by uneven sunlight, and the deformation caused by other environmental loads; the random error comes from the processing error of the panel surface and the installation uncertainty during splicing, which has a significant impact on the performance of the antenna. Impact. For a long time, the Ruze formula has effectively guided the structural design of the antenna to a certain extent, and he himself is recognized as the first person to systematically study the influence of small phase errors on the gain loss and sidelobe level of reflector antennas. However, the Ruze theory assumes that the phase error of the entire aperture surface is distributed in a Gaussian hat shape with a relative radius c, and does not consider the Gaussian hat cross-border calculation problem in the reflective surface formed by splicing panels. Although subsequent scholars further improved the Ruze formula, they also did not calculate the impact of surface manufacturing errors on antenna performance according to the panel splicing form.

With the development of reflectors in the direction of large aperture, high frequency band, and high gain, still using the Ruze formula as a guide will put forward extremely stringent requirements on surface accuracy, so that structural engineers will be able to maintain the shape of the back frame, panel processing and assembly. Facing huge challenges. However, in fact, it may not be necessary to achieve such a high precision to meet the electrical performance requirements. Therefore, more accurate and detailed performance prediction is particularly important.

Contents of the invention

The purpose of the present invention is to propose a method for calculating the influence of panel processing errors on electrical properties based on block form, so as to provide reasonable requirements for panel surface processing accuracy through accurate simulation analysis, thereby guiding panel processing.

The technical solution for realizing the purpose of the present invention is a method for calculating the influence of panel processing errors on electrical performance based on block form, which is characterized in that it includes at least the following steps:

(1) The phase influence term in the far-field integral expression of a single panel is first-order Taylor expanded, and the far field is expressed as the sum of the ideal field and the error field;

(2) Determine the random error correlation radius of each panel according to the processing technology, then obtain the error correlation function at different positions of the same panel, and finally obtain the normal random error γ _n,i (ρ ₁ ) of any two points on each panel The mean value of the product with γ _n,i (ρ ₂ )<γ _n,i (ρ ₁ )γ _n,i (ρ ₂ )>;

(3) Use the relationship between the normal error and the phase error to obtain the mean value of the product of the phase error δ _n,i (ρ ₁ ) and δ _n,i (ρ ₂ ) at any two points on each panel<δ _n,i (ρ ₁ )δ _n,i (ρ ₂ )>;

(4) Conjugate the error field strength δE _n,i of the panel in the far field to its conjugate Multiply and average to derive the average error power expression of a single panel in the far field

(5) Determine the integral interval parameters of each panel, including radial parameters, namely: inner and outer diameters of a single panel: a ₁₁ , a ₁₂ , a ₂₁ , a ₂₂ ; circumferential parameters, namely: two Angular coordinates of sides: α ₁₁ , α ₁₂ , α ₂₁ , α ₂₂ ;

(6) Substitute the <δ _n,i (ρ ₁ )δ _n,i (ρ ₂ )> obtained in step (3) into the average error power expression, and express the integrand as an integral variable, and then pass four The average error power of a single panel is obtained by multiple numerical integration;

(7) The average error power of all panels in the far field is superimposed, and the result is summed with the power of the antenna under ideal conditions, and finally the average power pattern of the entire antenna in the far field including surface processing errors is obtained.

The far field of each panel described in step (1) is expressed as the sum of the ideal field and the error field, as follows:

(1a) The reflective surface spliced by panels is divided into N rings, and the nth ring (n=1,2,...,N) can be divided into K _n sub-blocks; there will be manufacturing errors on the surface of each panel, For the (n,i)th panel, assuming that the phase error caused by a random error at a certain point on its surface is δ(ρ), its far-field pattern can be expressed as

In the formula, is the antenna aperture field amplitude distribution function, where B+C=1, the shape of the aperture field amplitude distribution function can be controlled by changing the P value, usually 1≤P≤2, ρ is the radius of a certain point of the antenna on the aperture surface, and a is the antenna of the radius. k is the wave number of electromagnetic wave propagation, is the unit vector of the target direction;

(1b) Perform first-order Taylor expansion on e jδ(ρ) in the above formula, e ^{jδ(ρ) ≈1} +jδ(ρ), and then get

In the formula, is the field strength pattern of the ideal panel in the far field, and δE _n,i is the error field strength caused by random manufacturing errors on the surface.

Step (2) is carried out as follows:

(2a) Determine the relative radius L _p of the random error on the surface of each panel according to the processing technology;

(2b) Suppose the surface errors of any two points with an interval of Δ on a single panel are γ _n,i (ρ ₁ ) and γ _n,i (ρ ₂ ), respectively, and the error correlation function between them can be obtained

In the formula, is the surface mean square error of the (n,i)th panel, and Δ is the modulus of the vector Δ.

Step (6), proceed as follows:

(6a) Set the direction (θ, φ) of the target relative to the coordinate system O-xyz to be expressed as (cosα _x , cosα _y , cosα _z ) by the direction cosine, and obtain the target according to the spatial geometric relationship between the target and the coordinate system The relationship between the angle relative to the coordinate axis and the direction cosine is:

cosα _x = sinθ cosφ, cosα _y = sinθ sinφ, cosα _z = cosθ;

In the formula, the meaning of each parameter is: θ, φ are the pitch angle and azimuth angle of the target point; cosα _x , cosα _y , cosα _z are the direction cosines of the target direction unit vector.

(6b) Obtain the point product of the vector of any two points on the aperture surface and the unit vector of the viewing direction

In the formula, φ′ ₁ and φ′ ₂ are the angular coordinates of ρ ₁ and ρ ₂ respectively. then there is

(6c) The distance Δ between any two points on each panel is expressed by the integral variable as

(6d) The average error power expression of a single panel is

The above equation is a quadruple integral, which can be solved by numerical integration.

Step (7) is carried out as follows:

(7a) The power pattern of the whole reflector antenna in the far field is

The average value of the above formula is the average power pattern of the antenna, that is,

In the formula, E ⁰ is the field strength pattern of the entire reflector without error; assuming that the surface random error on each panel obeys a normal distribution with a mean value of 0 and a variance of σ ² within the relevant radius, then the error field strength the mean value of

(7b) Since each block panel is manufactured independently, the surface random errors between different panels are not correlated, so when summing all error powers, that is, the fourth term of the above formula, the error field between different panels The mean of the strong multiplication is equal to the product of the respective means, which is also 0, that is

Therefore, only the superposition of the power of the same panel in the far field remains, so the average radiation power pattern of the entire antenna in the far field is

Compared with the prior art, the present invention has the following advantages:

1. The far field affected by the surface error is expressed as the sum of the ideal field and the error field, and the error power between different panels can be simplified to zero when calculating the average power of the antenna, which reduces the amount of calculation;

2. According to the actual block form of the antenna, the influence of the surface processing error on the electrical performance is calculated, which is closer to the actual engineering;

3. Use accurate quadruple numerical integration to solve the average error power of a single panel, avoiding the defect that the traditional integration time interval vector Δ cannot be fully represented;

4. Through the simulation analysis, the upper limit of the root mean square of the surface error is obtained, which avoids the excessive precision requirements raised by experience, reduces the processing difficulty and meets the electrical performance indicators at the same time.

Description of drawings

Fig. 1 is the flow chart of calculating the influence of processing error on electrical performance based on panel block form in the present invention;

Fig. 2 is a schematic diagram of a reflective surface formed by panel splicing used in the present invention;

Fig. 3 is the integral calculation parameter schematic diagram of the single panel used in the present invention;

Fig. 4 is a schematic diagram of the spatial geometric relationship between the target and the coordinate system;

Fig. 5 is the emulation example figure that the present invention adopts;

Fig. 6 is a comparison chart of the existing error of the simulation result of the present invention and the electrical performance under ideal conditions.

detailed description

The present invention will be described in further detail below with reference to the accompanying drawings.

With reference to Fig. 1, concrete steps of the present invention are as follows:

Step 1, the far field of a single panel is expressed as the sum of the ideal field and the error field; (1a) Referring to Figure 2, the reflective surface formed by splicing panels is divided into N rings, the nth ring (n=1,2 ,…,N) can be divided into K _n sub-blocks. There will be manufacturing errors on the surface of each panel. For the (n,i)th panel, assuming that the phase error caused by a random error at a certain point on the surface is δ(ρ), its far-field pattern can be expressed as

In the formula, is the antenna aperture field amplitude distribution function, where B+C=1, the shape of the aperture field amplitude distribution function can be controlled by changing the P value, usually 1≤P≤2, ρ is the radius of a certain point of the antenna on the aperture surface, and a is the antenna of the radius. k is the wave number of electromagnetic wave propagation, is the unit vector of the target direction;

(1b) Perform first-order Taylor expansion on e jδ(ρ) in the above formula, e ^{jδ(ρ) ≈1} +jδ(ρ), and then get

In the formula, is the field strength pattern of the ideal panel in the far field, and δE _n,i is the error field strength caused by random manufacturing errors on the surface.

Step 2: Calculate the mean value of the product of random error γ _n,i (ρ ₁ ) and γ _n,i (ρ ₂ ) of any two points on a single panel<γ _n,i (ρ ₁ )γ _n,i (ρ ₂ )>.

(2a) Determine the relative radius L _p of the random error on the surface of each panel according to the processing technology;

(2b) Referring to Fig. 3, suppose the surface errors of any two points with an interval of Δ on a single panel are γ _n,i (ρ ₁ ) and γ _n,i (ρ ₂ ) respectively, and the error correlation between them can be obtained function

In the formula, is the surface mean square error of the (n,i)th panel, and Δ is the modulus of the vector Δ.

Step 3: Calculate the mean value of the product of random phase error δ _n,i (ρ ₁ ) and δ _n,i (ρ ₂ ) at any two points on a single panel<δ _n,i (ρ ₁ )δ _n,i (ρ ₂ ) >.

Combined with the relationship between the surface error and the spatial phase error, we can get

In the formula, λ is the working wavelength of the antenna.

Step 4, derive the expression of the average error power of a single panel in the far field.

The error field strength of each panel is

So the average error power is

Step five, determine the integration interval of the single panel.

Referring to Figure 3, the integral interval of the error field strength of a single panel is: radial [a ₁₁ , a ₁₂ ], circumferential [α ₁₁ , α ₁₂ ]; the integral interval of the conjugate value of the error field strength is: radial [a ₂₁ ,a ₂₂ ], circumferential direction [ _α2 1,α ₂₂ ].

Step six, integrally calculate the average error power of a single panel.

(6a) Referring to Figure 4, set the direction (θ, φ) of the target relative to the coordinate system O-xyz to be expressed as (cosα _x , cosα _y , cosα _z ) by the direction cosine, and according to the spatial geometry of the target and the coordinate system The relationship between the angle of the target relative to the coordinate axis and the direction cosine is obtained as:

cosα _x = sinθ cosφ, cosα _y = sinθ sinφ, cosα _z = cosθ;

(6b) Referring to Figure 3 and Figure 4, the point product of the vector of a certain point on the aperture surface and the unit vector of the viewing direction is obtained

In the formula, φ′ ₁ and φ′ ₂ are the angular coordinates of ρ ₁ and ρ ₂ respectively. then there is

(6c) The distance Δ between any two points on each panel can be expressed as

(6d) Substitute the formulas in steps 5 and 6 into the expression of the average error power of a single panel, and get

The above equation is a quadruple integral, which can be solved by numerical integration.

Step 7, calculate the average radiated power of the entire reflector antenna.

(7a) The power pattern of the whole reflector antenna in the far field is

The average value of the above formula is the average power pattern of the antenna, that is,

In the formula, E ⁰ is the field strength pattern of the entire reflector without error. Assuming that the surface random error on each panel obeys a normal distribution with a mean of 0 and a variance of ^σ2 within the relevant radius, then for the mean value of the error field strength,

(7b) Since each block panel is manufactured independently, the surface random errors between different panels are not correlated, so when summing all error powers (that is, the fourth term of the above formula), the error between different panels The mean value of the field strength multiplication is equal to the product of the respective mean values, which is also 0, that is

Therefore, only the superposition of the power of the same panel in the far field remains, so the average radiation power pattern of the entire antenna in the far field is

Step 8: Substitute the root mean square error of different panel surface processing into the above formula, simulate and analyze its impact on electrical performance, and then put forward reasonable surface processing accuracy requirements.

Advantages of the present invention can be further illustrated by following simulation experiments:

1. Simulation conditions

The method for calculating the impact of panel processing errors on electrical performance based on the block form of the present invention is compiled into a Matlab program, and the electrical performance simulation analysis is performed on a 35-meter block reflector antenna.

The block structure schematic diagram of the reflecting surface antenna is shown in Figure 5. The reflecting surface is divided into 9 rings. Each ring of the 1st to 4th rings is composed of 16 panels. The fifth ring is divided into 32 panels. The ring is further divided into 96 pieces. The working frequency of the antenna is 10GHz, the focal diameter ratio is F/D=0.35, the edge irradiation taper is ET=-10dB, and the aperture field irradiation function is Surface processing error-related radius L _p =0.4m.

2. Simulation results

Assuming that the root mean square of the surface error of each panel is ε=λ/30, the above model is simulated by using the compiled Matlab program, and the comparison of the antenna pattern when there is surface manufacturing error and the ideal situation is obtained, as shown in Figure 6 Show. It can be seen from the figure that when there is a small surface manufacturing error, the main lobe of the antenna is broadened, and the influence of the near side lobe is small, but the side lobe in the far area is significantly raised. It can be seen from the simulation analysis results that the method of the invention can be used to guide the panel processing and manufacturing of large reflector antennas and the analysis and evaluation of electrical performance.

Claims (5)

1. the method that panel mismachining tolerance is affected on electrical property is calculated based on block form, be it is characterized in that：At least include following step Suddenly：

(1) the phase effect item in the integral expression of monolithic panel far field is carried out first order Taylor expansion, far field is expressed as into reason Think field and error field sum；

(2) the random error correlation radius of every piece of panel is determined according to processing technique, same block panel various location is then obtained Error correlation function, finally try to achieve every piece of panel any two points normal direction random error γ_n,i(ρ₁) and γ_n,_i(ρ₂) product it is equal Value<γ_n,i(ρ₁)γ_n,i(ρ₂)>；

(3) relation using normal error with phase error tries to achieve every piece of panel any two points phase error δ_n,i(ρ₁) and δ_n,i (ρ₂) product average<δ_n,i(ρ₁)δ_n,i(ρ₂)>；

(4) by panel far field error field intensity δ E_n,iIt is conjugated with whichIt is multiplied and is averaging, is derived from monolithic panel and exists The average error power expression formula in far field

(5) the integrating range parameter of every piece of panel, including radial direction parameter are determined, i.e.,：The inside and outside footpath of monolithic panel：a₁₁, a₁₂, a₂₁, a₂₂；Circumferential parameter, i.e.,：The angle coordinate of monolithic panel two ends string：α₁₁, α₁₂, α₂₁, α₂₂；

(6) step (3) is drawn<δ_n,i(ρ₁)δ_n,i(ρ₂)>Substitute in average error power expression formula, and integrand is used Integration variable is expressed, and then obtains monolithic panel average error power by quadruple numerical integration；

(7) average error power by all panels in far field is superimposed, and power of the acquired results with antenna in the ideal case is asked With finally give average power pattern of the whole antenna containing Surface Machining error in far field.

2. according to claim 1 that the method that panel mismachining tolerance is affected on electrical property is calculated based on block form, which is special Levy and be：The phase effect item in the integral expression of monolithic panel far field is carried out first order Taylor expansion described in step (1), will Far field is expressed as ideal field and error field sum, carries out according to the following procedure：

(1a) N rings are divided into by the reflecting surface of panel splicing, the n-th ring can be divided into K again_nIndividual sub-block, wherein, n=1,2 ..., N；Can all there is foozle in the surface of every piece of panel, for (n, i) block panel, if the random error at its surface point The phase error for causing is δ (ρ), then its far-field pattern is represented by

$E_{n,i}=\underset{{(n,i)}_{zone}}{\&Integral;\&Integral;}Q\left(\mathrm{\&rho;}\right)e^{j\mathrm{\&delta;}\left(\mathrm{\&rho;}\right)}{e}^{jk\mathrm{\&rho;},\hat{r}}{\mathrm{dS}}_{n,i}$

In formula,For antenna aperture field amplitude distribution function, wherein B+C=1, ρ go up to the sky for bore face The radius of line point, radiuses of a for antenna, k are electromagnetic wave propagation wave number,For the unit vector of target direction；

(1b) to the e in above formula^jδ(ρ)Make first order Taylor expansion, have e^jδ(ρ)≈ 1+j δ (ρ), then obtains

$\begin{array}{c}{E}_{n,i}\&ap;\underset{{\left(n,i\right)}_{zone}}{\&Integral;\&Integral;}Q\left(\mathrm{\&rho;}\right)\&lsqb;1+j\mathrm{\&delta;}\left(\mathrm{\&rho;}\right)\&rsqb;e^{jk\stackrel{\&RightArrow;}{\mathrm{\&rho;}}\&CenterDot;\hat{r}}{\mathrm{dS}}_{n,i}\\ =E_{n,i}^0+{\mathrm{\&delta;E}}_{n,i}\end{array}$

In formula,For ideal panel far field field strength pattern, δ E_n,iFor the error field intensity that surface random manufacturing error causes.

3. according to claim 1 that the method that panel mismachining tolerance is affected on electrical property is calculated based on block form, which is special Levy and be：Step (2) is carried out according to the following procedure：

(2a) the correlation radius L of every piece of panel surface random error is determined according to processing technique_p；

(2b) surface error for setting any two points on monolithic panel at intervals of Δ is respectively γ_n,i(ρ₁), γ_n,i(ρ₂), obtain two Error correlation function between person

$C_{n,i}\left(\mathrm{\&Delta;}\right)=\frac{<{\mathrm{\&gamma;}}_{n,i}\left({\mathrm{\&rho;}}_1\right){\mathrm{\&gamma;}}_{n,i}\left({\mathrm{\&rho;}}_2\right)>}{{\mathrm{\&epsiv;}}_{n,i}^2}=\exp \&lsqb;-{(\mathrm{\&Delta;}/L_p)}^2\&rsqb;$

In formula,For the surface mean square error of (n, i) block panel, modulus value of the Δ for vector Δ.

4. according to claim 1 that the method that panel mismachining tolerance is affected on electrical property is calculated based on block form, which is special Levy and be：Step (6), is carried out according to the following procedure：

(6a) direction (θ, φ) that target setting is located relative to coordinate system O-xyz is expressed as (cos α with direction cosines_x,cosα_y, cosα_z), and according to target and the space geometry relation of coordinate system, obtain angle and direction cosines of the target relative to coordinate axess Relation be：

cosα_x=sin θ cos φ, cos α_y=sin θ sin φ, cos α_z=cos θ；

In formula, parameters are meant that：θ, φ are the pitching of impact point and azimuth；cosα_x,cosα_y,cosα_zIt is object vector Direction cosines；

(6b) obtain the relation of the vector and direction of observation unit vector of certain point on bore face

$\left\{\begin{array}{c}{\mathrm{\&rho;}}_1\&CenterDot;\hat{r}={\mathrm{\&rho;}}_1{\mathrm{sin\&theta;cos\&phi;cos\&phi;}}_1^{\&prime;}+{\mathrm{\&rho;}}_1{\mathrm{sin\&theta;sin\&phi;sin\&phi;}}_1^{\&prime;}\\ {\mathrm{\&rho;}}_2\&CenterDot;\hat{r}={\mathrm{\&rho;}}_2{\mathrm{sin\&theta;cos\&phi;cos\&phi;}}_2^{\&prime;}+{\mathrm{\&rho;}}_2{\mathrm{sin\&theta;sin\&phi;sin\&phi;}}_2^{\&prime;}\end{array}\right.$

In formula, φ '₁, φ '₂Respectively ρ₁, ρ₂Angle coordinate；Then have

$({\mathrm{\&rho;}}_1-{\mathrm{\&rho;}}_2)\&CenterDot;\hat{r}={\mathrm{\&rho;}}_{1}sin\mathrm{\&theta;}cos(\mathrm{\&phi;}-{\mathrm{\&phi;}}_1^{\&prime;})-{\mathrm{\&rho;}}_{2}sin\mathrm{\&theta;}cos(\mathrm{\&phi;}-{\mathrm{\&phi;}}_2^{\&prime;})$

(6c) on every piece of panel, the separation delta of any point-to-point transmission is expressed as with integration variable

$\begin{array}{c}\mathrm{\&Delta;}=\left|\mathrm{\&Delta;}\right|=\sqrt{{\left({\mathrm{\&rho;}}_1{\mathrm{cos\&phi;}}_1^{\&prime;}-{\mathrm{\&rho;}}_2{\mathrm{cos\&phi;}}_2^{\&prime;}\right)}^2+{\left({\mathrm{\&rho;}}_1{\mathrm{sin\&phi;}}_1^{\&prime;}-{\mathrm{\&rho;}}_2{\mathrm{sin\&phi;}}_2^{\&prime;}\right)}^2}\\ =\sqrt{{\mathrm{\&rho;}}_1^2+{\mathrm{\&rho;}}_2^2-2{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2\cos \left({\mathrm{\&phi;}}_1^{\&prime;}-{\mathrm{\&phi;}}_2^{\&prime;}\right)}\end{array}$

(6d) by various substitution monolithic panel mean power expression formula in step 2, three, obtain

$\begin{array}{c}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{n,i}^{*}}={\&Integral;}_{{\mathrm{\&alpha;}}_{11}}^{{\mathrm{\&alpha;}}_{12}}{\&Integral;}_{{\mathrm{\&alpha;}}_{11}}^{{\mathrm{\&alpha;}}_{12}}{\&Integral;}_{{\mathrm{\&alpha;}}_{21}}^{{\mathrm{\&alpha;}}_{22}}{\&Integral;}_{{\mathrm{\&alpha;}}_{21}}^{{\mathrm{\&alpha;}}_{22}}Q\left({\mathrm{\&rho;}}_1\right)Q^{*}\left({\mathrm{\&rho;}}_2\right)\&CenterDot;{\left(\frac{4\mathrm{\&pi;}}{\mathrm{\&lambda;}}\right)}^2{\mathrm{\&epsiv;}}_{n,i}^2\exp \&lsqb;-{\left(\mathrm{\&Delta;}/L_p\right)}^2\&rsqb;\&CenterDot;\\ e^{jk\&lsqb;{\mathrm{\&rho;}}_1\sin \mathrm{\&theta;}\cos \left(\mathrm{\&phi;}-{\mathrm{\&phi;}}_1^{\&prime;}\right)-{\mathrm{\&rho;}}_2\sin \mathrm{\&theta;}\cos \left(\mathrm{\&phi;}-{\mathrm{\&phi;}}_2^{\&prime;}\right)\&rsqb;}{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2{\mathrm{d\&rho;}}_2{\mathrm{d\&phi;}}_2^{\&prime;}{\mathrm{d\&rho;}}_1{\mathrm{d\&phi;}}_1^{\&prime;}\end{array}$

Above formula is a quad-slope integration, is solved using numerical integration.

5. according to claim 1 that the method that panel mismachining tolerance is affected on electrical property is calculated based on block form, which is special Levy is that step (7) is carried out according to the following procedure：

(7a) whole power radiation pattern of the reflector antenna in far field is

$\begin{array}{c}P=E\&CenterDot;E^{*}=\&lsqb;\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\left(E_{n,i}^0+{\mathrm{\&delta;E}}_{n,i}\right)\&rsqb;\&CenterDot;\&lsqb;\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}\left(E_{m,l}^{0*}+{\mathrm{\&delta;E}}_{m,l}^{*}\right)\&rsqb;\\ =\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}\left(E_{n,i}^0{E}_{m,l}^{0*}+E_{n,i}^0{\mathrm{\&delta;E}}_{m,l}^{*}+{\mathrm{\&delta;E}}_{n,i}{E}_{m,l}^{0*}+{\mathrm{\&delta;E}}_{n,i}{\mathrm{\&delta;E}}_{m,l}^{*}\right)\end{array}$

The meansigma methodss of above formula are exactly the average power pattern of antenna, i.e.,

$\stackrel{\&OverBar;}{P}=E^0\&CenterDot;E^{0*}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}{E}_{n,i}^0\&CenterDot;\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{m,l}^{*}}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}{E}_{m,l}^{0*}\&CenterDot;\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{m,l}^{*}}$

In formula, E⁰For it is error free when whole reflecting surface field strength pattern；Assume the surface random error on each piece of panel in correlation half It is 0 that average is obeyed in footpath, and variance is σ²Normal distribution, then have for the average of error field intensity

(7b) as each piecemeal panel is independently manufactured, the surface random error between different panels does not have dependency, therefore In the summation to all error powers, i.e. above formula Section 4, the average that the error field intensity between different panels is multiplied is equal to each From the product of average, 0 is similarly, i.e.,

$\begin{array}{c}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{m,l}^{*}}=\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1,m\&NotEqual;n}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{K_m}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{m,l}^{*}}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\underset{l=1,l\&NotEqual;i}{\overset{K_m}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{m,l}^{*}}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{n,i}^{*}}\\ =\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{K_n}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{n,i}^{*}}\end{array}$

Therefore only it is left the superposition with the block panel power of itself in far field, therefore whole antenna is in the average radiating power side in far field To figure it is

$\stackrel{\&OverBar;}{P}=E^0\&CenterDot;E^{0*}+\underset{n=1}{\overset{N}{\&Sigma;}}\underset{i=1}{\overset{K_n}{\&Sigma;}}\stackrel{\&OverBar;}{{\mathrm{\&delta;E}}_{n,i}\&CenterDot;{\mathrm{\&delta;E}}_{n,i}^{*}}.$