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

Abstract

The present invention relates to calculate the method that panel mismachining tolerance is affected on electrical property based on block form, for instructing the processing and antenna electric performance simulation analysis and evaluation of panel surface, the present invention is expressed as ideal field and error field sum the far field affected by surface error, error power between different panels can be reduced to zero process when antenna mean power is sought, reduce amount of calculation；Impact according to the actual block form gauging surface mismachining tolerance of antenna to electrical property, closer to engineering reality；The average error power of monolithic panel is solved using accurate quadruple numerical integration, it is to avoid be spaced during traditional quadrature vector Δ cannot complete representation defect；The root mean square higher limit of surface error is drawn by simulation analysis, it is to avoid propose too high required precision by rule of thumb, difficulty of processing is reduced while and meeting electrical performance indexes.

Description

The method that panel mismachining tolerance is affected on electrical property is calculated based on block form

Technical field

The invention belongs to antenna technical field, specifically calculating panel mismachining tolerance based on block form affects on electrical property Method, for instruct panel surface processing and antenna electric performance simulation analysis with evaluate.

Background technology

Reflector antenna is widely used in radar control, satellite and leads to as the most frequently used microwave and millimeter wave high-gain aerial The fields such as letter, radio astronomy, in the future it is also possible to as microwave weapon and the instrument of energy transmission.Due to its be operated in it is higher Frequency range, electromagnetic performance are vulnerable to the impact of various errors.Reflector antenna surface error mainly has two kinds：Systematic error and random Error.The thermal deformation and other environment that systematic error mainly causes including the deformation induced by gravity of antenna, sun uneven irradiation is carried The deformation that lotus causes；Random error then comes from the installation uncertainty when mismachining tolerance of panel surface and splicing, to day Can linearly there is obvious impact.For a long time, Ruze formula effectively direct the structure of antenna to a certain extent and set Meter, he or she are also acknowledged as what the small phase error of systematic study was affected on reflector antenna gain loss and minor level It is the first.But the phase error in the whole bore face of Ruze theory hypothesis is distributed for the Gauss hat shape of c in correlation radius, not Consider by the transboundary computational problem of the Gaussian hat in the reflecting surface of panel splicing.Scholar after which is although further perfect Ruze formula, but carry out impact of the gauging surface foozle to antenna performance also without according to the splicing form of panel.

As reflecting surface develops towards the direction of heavy caliber, high band, high-gain, still will as guidance with Ruze formula Extremely harsh requirement is proposed to surface accuracy, makes Structural Engineering personnel in terms of backrest guarantor's type, panel processing and assembling by face Face huge challenge.But actually perhaps and the requirement that so high precision can just meet electrical property need not be reached, therefore, More accurate and detailed performance prediction is particularly important.

The content of the invention

The purpose of the present invention is to propose to a kind of calculate the method that panel mismachining tolerance is affected on electrical property based on block form, Will pass through the reasonable request that accurate simulation analysis provide panel surface machining accuracy, so as to instruct panel to process.

The technical scheme for realizing the object of the invention is to calculate what panel mismachining tolerance was affected on electrical property based on block form Method, is characterized in that：At least comprise the steps：

(1) the phase effect item in the integral expression of monolithic panel far field is carried out first order Taylor expansion, far field is represented For ideal 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 difference position is then obtained The error correlation function at place is put, every piece of panel any two points normal direction random error γ is finally tried to achieve_n,i(ρ₁) and γ_n,i(ρ₂) product Average<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) with 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 monolith surface Average error power expression formula of the plate 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 both sides：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 will be by product letter Number is expressed with integration variable, then obtains monolithic panel average error power by quadruple numerical integration；

(7) average error power by all panels in far field is superimposed, acquired results and antenna work(in the ideal case Rate is sued for peace, and finally gives average power pattern of the whole antenna containing Surface Machining error in far field.

The far field of every piece of panel is expressed as ideal field and error field sum described in step (1), is carried out according to the following procedure：

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

In formula,For antenna aperture field amplitude distribution function, wherein B+C=1, by changing The shape of the controllable aperture field amplitude distribution function of P values, usual 1≤P≤2, ρ are the radius of antenna point on bore face, and a is day The radius of line.K is 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

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

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 Arrive error correlation function between the two

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

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 of the target relative to coordinate axess The relation of cosine is：

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 angle of pitch of impact point and azimuth；cosα_x,cosα_y,cosα_zIt is mesh The direction cosines of mark direction unit vector.

(6b) obtain the dot product of the vector of any two points and direction of observation unit vector on bore face

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

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

(6d) the average error power expression formula of monolithic panel is

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

Step (7) is carried out according to the following procedure：

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

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

In formula, E⁰For it is error free when whole reflecting surface field strength pattern；Assume that the surface on each piece of panel is random It is 0 that error obeys average in correlation radius, 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, average that the error field intensity between different panels is multiplied etc. In the product of respective average, 0 is similarly, i.e.,

Therefore only it is left the superposition with the block panel power of itself in far field, therefore whole antenna is in the average radiation work(in far field Rate directional diagram is

The present invention compared with prior art, has the advantage that：

1. the far field affected by surface error is expressed as ideal field and error field sum, can when antenna mean power is sought Error power between different panels is reduced to into zero process, amount of calculation is reduced；

2. the impact according to the actual block form gauging surface mismachining tolerance of antenna to electrical property, closer to engineering reality Border；

3. the average error power of monolithic panel is solved using accurate quadruple numerical integration, it is to avoid the traditional quadrature time Every vector Δ cannot complete representation defect；

4. the root mean square higher limit of surface error is drawn by simulation analysis, it is to avoid propose that too high precision will by rule of thumb Ask, difficulty of processing is reduced while and meeting electrical performance indexes.

Description of the drawings

Fig. 1 is the flow chart that the present invention calculates that based on panel block form mismachining tolerance is affected on electrical property；

Fig. 2 is the reflecting surface schematic diagram of the panel splicing composition used by the present invention；

Fig. 3 is the integral and calculating parameter schematic diagram of the monolithic panel used by the present invention；

Fig. 4 is the schematic diagram of target and the space geometry relation of coordinate system；

Fig. 5 is simulated example figure of the present invention；

Fig. 6 is the presence error and electrical property comparison diagram ideally of simulation result of the present invention.

Specific embodiment

Referring to the drawings the present invention is described in further detail.

With reference to Fig. 1, the present invention's comprises the following steps that：

The far field of monolithic panel is expressed as ideal field and error field sum by step one；(1a) with reference to Fig. 2, spelled by panel The reflecting surface for connecing is divided into N rings, and the n-th ring (n=1,2 ..., N) can be divided into K again_nIndividual sub-block.All can on the surface of every piece of panel There is foozle, for (n, i) block panel, if the phase error that the random error at its surface point causes is δ (ρ), then its far-field pattern is represented by

In formula,For antenna aperture field amplitude distribution function, wherein B+C=1, by changing The shape of the controllable aperture field amplitude distribution function of P values, usual 1≤P≤2, ρ are the radius of antenna point on bore face, and a is day The radius of line.K is 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

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

Step 2, calculates monolithic panel any two points normal direction random error γ_n,i(ρ₁) and γ_n,i(ρ₂) product average< γ_n,i(ρ₁)γ_n,i(ρ₂)>。

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

(2b) with reference to Fig. 3, if the surface error of any two points on monolithic panel at intervals of Δ is respectively γ_n,i(ρ₁), γ_n,i(ρ₂), obtain error correlation function between the two

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

Step 3, calculates monolithic panel any two points random phase error δ_n,i(ρ₁) and δ_n,i(ρ₂) product average<δ_n,i (ρ₁)δ_n,i(ρ₂)>。

Mating surface error can be obtained with the relation of spatial phase errors

In formula, λ is antenna operating wavelength.

Step 4, is derived from average error power expression formula of the monolithic panel in far field.

The error field intensity of every piece of panel is

Therefore average error power is

Step 5, determines the integrating range of monolithic panel.

With reference to Fig. 3, the strong integrating range of monolithic panel error field is：Radially [a₁₁,a₁₂], circumferential [α₁₁,α₁₂]；Error field The integrating range of conjugate is by force：Radially [a₂₁,a₂₂], circumferential [_α21,α₂₂]。

Step 6, integral and calculating monolithic panel average error power.

(6a) with reference to Fig. 4, the direction (θ, φ) that target setting is located relative to coordinate system O-xyz is expressed as with direction cosines (cosα_x,cosα_y,cosα_z), and according to target and the space geometry relation of coordinate system, obtain folder of the target relative to coordinate axess Angle with the relation of direction cosines is：

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

(6b) with reference to Fig. 3 and Fig. 4, obtain the dot product of the vector and direction of observation unit vector of certain point on bore face

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

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

(6d) by various substitution monolithic panel average error power expression formula in step 5, six, obtain

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

Step 7, calculates the average radiating power of whole reflector antenna.

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

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

In formula, E⁰For it is error free when whole reflecting surface field strength pattern.Assume that the surface on each piece of panel is random It is 0 that error obeys average in correlation radius, 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 summation (i.e. the above formula Section 4) to all error powers, average that the error field intensity between different panels is multiplied etc. In the product of respective average, 0 is similarly, i.e.,

Therefore only it is left the superposition with the block panel power of itself in far field, therefore whole antenna is in the average radiation work(in far field Rate directional diagram is

Step 8, substitutes into above formula different panel surface mismachining tolerance root-mean-square, simulation analysis its shadows to electrical property Ring, and then propose that rational surface processing accuracy is required.

Advantages of the present invention can be further illustrated by following emulation experiment：

1. simulated conditions

It is Matlab by the method establishment affected on electrical property based on block form calculating panel mismachining tolerance of the present invention Program, carries out electrical property simulation analysis on one 35 meters of piecemeal reflector antennas.

As shown in figure 5, its reflecting surface is divided into 9 rings, 1～4 ring is equal per ring for the partitioned organization schematic diagram of the reflector antenna It is made up of 16 pieces of panels, the 5th ring is divided into 32 pieces, 6～9 rings are divided into 96 pieces again per ring.Operating frequency of antenna is 10GHz, burnt footpath ratio For F/D=0.35, edge illumination taper is ET=-10dB, and aperture field illumination function isSurface Mismachining tolerance correlation radius L_p=0.4m.

2. simulation result

If the surface error root-mean-square of every piece of panel is ε=λ/30, using the Matlab programs of establishment to model above Being emulated, being obtained antenna directional diagram contrast when there is surface foozle and ideally, as shown in Figure 6.From figure In as can be seen that when there is less surface foozle, the main lobe broadening of antenna, nearly secondary lobe affect less, but far field secondary lobe Raise substantially.Knowable to simulation analysis result, can be used to instruct the panel of large-scale reflector antenna to add using the method for the present invention Work manufacture and the A+E of electrical property.

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

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