Research ArticleMulti-Field-Coupled Model and Solution of Active ElectronicallyScanned Array Antenna Based on Model Reconstruction

Dan Han ,1 Jin Huang ,1 Jinzhu Zhou ,1MeiWang,2 Sen Xu,1Haitao Li ,1 and Shen Li1

1Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi’an 710071, China2Structure Department, No. 38 Research Institute of CETC, Hefei 230031, China

Correspondence should be addressed to Jin Huang; jhuang@mail.xidian.edu.cn

Received 19 June 2018; Accepted 30 July 2018; Published 27 November 2018

Academic Editor: Francesco D'Agostino

Copyright © 2018 Dan Han et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Active electronically scanned array antenna (AESA antenna) is capable of controlling the radiation pattern by controlling thefeeding phase of the radiating elements. It has good performance and plays an important role in radar systems. With thedevelopment of AESA antenna towards high-frequency bands and high-density arrays, the structural-electromagnetic-thermal(SET) coupling becomes increasingly significant. It seriously restricts the realization of high performances of AESA antennas.However, the previously reported theoretical multi-field-coupled model for the coupling problem ignores the effect of thedeformations of the feed system and array elements on the electrical performance. It only considers the positional deviations ofthe array elements in the coupling field. As a result, the accuracy of the numerical solution by the theoretical model is reduced.To overcome the above problems, this paper first establishes the field-circuit coupling model by introducing the deformationerrors of the feed system into the existing theoretical model. Secondly, this paper proposes a new numerical solution for themulti-field-coupled problem of AESA antennas based on model reconstruction. And the model reconstruction includes thefollowing: the NURBS (nonuniform rational B-spline) surface fitting algorithm that completes the mapping from finite elementmodels to geometric models by the surface equations established by the node information and the local model reconstructionalgorithm that determines the local geometric models by the positions and the directions. The NURBS surface fitting algorithmguarantees the accuracy of both the positions and shapes of array elements. The local model reconstruction algorithm ensuresthe accuracy of the amplitudes and phases of feed connectors. Finally, the numerical solution was applied to the 32-elementAESA antenna and the simulations are close to the measurements.

1. Introduction

Active electronically scanned array antenna (AESA antenna)has high reliability, multiple functions, strong detectioncapability, and good stealth capability [1, 2]. In the past sev-eral decades, AESA antennas played an important role incivilian operations and the military arena [3]. To obtain bet-ter performance, the AESA antenna is developing towardshigher frequency and higher density. The technologicaldevelopments of meter wave and centimeter wave detectionradars are relatively mature. Millimeter waves are widelyused in communication systems. The frequency of high-speed and short-distance transmission equipment can reachup to dozens of gigahertz (GHz). The research of higherterahertz application has already begun. The frequency band

of radio waves observed in radio astronomy can reach upto hundreds of gigahertz (GHz) [4]. Moreover, the assemblydensity of AESA antennas continues to increase with the tran-sition from two-dimensional assembly to three-dimensionalassembly. Accordingly, the volume of equipment must be-come increasingly small. For example, the size of radio fre-quency systems has been reduced from 0.03 m3 to 0.001 m3

[4]. Thus, the structural-electromagnetic-thermal (SET) cou-pling becomes more and more significant. It causes the gainloss, sidelobe level upgrade, and inaccurate beam pointing[5, 6], thus restricting the development of high-performancearray antennas. Therefore, the multifield coupling is a crucialproblem that needs to be solved [7].

There are two major types of numerical solutions previ-ously reported for the multifield coupling of AESA antennas.

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 3161928, 12 pageshttps://doi.org/10.1155/2018/3161928

One is by the theoretical multi-field-coupled model [8, 9].However, the model ignores the effect of the deformationsof the feed system and array elements on the electrical perfor-mance. It only considers the positional deviations of the arrayelements in the coupling field [9], thus reducing the accuracyof the numerical solution. The other is by simulations. How-ever, between different software in coupling fields, the datamodels are heterogeneous and the mesh of the finite elementmodel does not match because of different analysis purposesand tools. To overcome the above problems in simulation,two approaches came into being. One approach is importingthe heterogeneous mesh into the simulation software directly[10, 11]. However, the imported mesh must be simplified,refined, and homogenized to meet the analysis requirementsand the process is complicated [12]. The other approach isfitting a new solid model from the heterogeneous mesh andthen remeshing on the new model [13–15]. However, newerrors are introduced in the fitting process and the accuracyof the result of the electromagnetic analysis is reduced [12].

For the above problems, this paper first establishes thefield-circuit coupling model by introducing the deforma-tion errors of the feed system into the existing theoreticalmodel. Secondly, this paper proposes a new simulationmethod based on model reconstruction that precisely reflectsboth the positional deviations and self-deformations of thearray elements. One of the key benefits of the method ispassing the solid model between different simulation toolsin coupling fields and remesh on the new model, whichavoids the tedious meshing process. Moreover, the NURBS(nonuniform rational B-spline) surface fitting algorithmguarantees the accuracy of both the positions and shapes ofarray elements. And the local model reconstruction algo-rithm ensures the accuracy of the amplitudes and phases offeed connectors. Thus, the accuracy of the electromagneticanalysis is improved. Figure 1 shows the structure of theAESA antenna. And the antenna array and feed networkthat this paper concerns about are shown in the figure.

2. Multi-Field-Coupled Model and Solution ofAESA Antennas

The multifield coupling of the AESA antenna is primarilythe sequential coupling among the temperature field, thestructural displacement field, and the electromagnetic field.As shown in Figure 2, structural deformations are caused

by environmental loads and changes of structural parame-ters, also known as structural displacement field (path ①).Secondly, the structural displacement field affects the tem-perature field and the electromagnetic field through theabove deformations (path ②). Finally, the influence of thetemperature field has two forms: one is the impact on theperformance of electromagnetic devices (such as T/R compo-nents) (path ③) and the other is the impact on the electro-magnetic field by the changes of the structural displacementfield (path ④) [16].

2.1. Field-Circuit Coupling Model. For the above three-field coupling process of AESA antennas, in the traditionalthree-field coupling model [8, 9], as shown in (1), the excita-tion source Imn′ by the feed system was ideally processed,often by applying a unit excitation source or an excitationsource that satisfies the Taylor distribution. However, in thecoupling field, the feed system also deforms. Moreover, theelectrical performance is affected by the changed characteris-tic impedance and power ratio of the feed network, therebyaffecting the excitation current. In addition, in the feed net-work, the λ/4 impedance converter has a narrow line widthbecause of the high transmission impedance. And in orderto reduce the coupling effect, the spacing of its lines is large.Therefore, in the coupling field, the sensitivity of deforma-tion error of the λ/4 impedance converter is large. In thispaper, the relation between the length variation Δl of theλ/4 impedance converter in the feed network and the excita-tion current Imn′ is deduced. And this relation is introducedinto the existing coupling model to establish the field-circuit coupling model. Figure 3 shows the spatial relationof the AESA antenna.

E θ, ϕ = 〠M−1

m=0〠N−1

n=0Emn′ θ, ϕ Imn′

× exp jk mdx + 〠m

i=0Δxij δ, T f x θ, ϕ

+ ndy + 〠n

j=0Δyij δ, T f y θ, ϕ

+ Δzij δ, T f z θ + jSmn T ,

1

Antennaarray

Load-bearingframework

T/R circuits andbeam control

circuitsEncapsulation

shell

�e partof feed

network

Figure 1: Structural configuration of the AESA antenna.

2 International Journal of Antennas and Propagation

where E θ, ϕ is the pattern function, Emn′ θ, ϕ is the patternfunction of the radiating elements in an antenna array, Imn′ isthe amplitude of excitation current, T is the structure tem-perature, δ β1, β2,… , βR is the structural displacement, βii = 1, 2,… , R are the structural design variables, f x θ, ϕ ,f y θ, ϕ , and f z θ are functions of element position anddirection determined by the array arrangement form, suchas the hexagonal AESA antenna and rectangular AESAantenna, Δxij δ, T , Δyij δ, T , and Δzij δ, T are the devia-tion values of the element position determined by T and δ,Smn T is the array phase difference controlled by the phaseshifter affected by T , and dx and dy are the intervals of theradiating elements.

Figure 4(a) shows the circuit parameters of the powerdivider in the feed network. Next, the relation between Δland Imn′ is deduced from two aspects. First, the deformationof the microstrip causes impedance mismatching of thepower divider, which results in the nonzero reflection coeffi-cient at the power input end Pin.

Pin = P0 1 − Γin 2 2

Secondly, the deformation of the microstrip causes theuneven distribution of the power on each branch, assumingthat the input power on one of the branches PinB = KPin.

y

N

dy

dxM-1M

o

x

z

z

O

𝜃

𝜙

r0

Far-field P

y

𝛼y

x

11 2

Figure 3: Spatial relation of the AESA antenna.

Electromagneticfield

Temperaturefield

Structuraldisplacement

field

2

44

2

13

dB (gain total)0.00Phi (deg)20.00

30.00

60.00

90.00120.00

120.00

150.00

180.00210.00

240.00

270.00

300.00

330.00

60.0090.00

120.00 150.00 100.00150.00

60.0030.00

20.00

10.00

0.00

−10.00

−20.00

−30.00

−40.00

−50.00

Temperature(ce1)

7.0741E + 0017.0628E + 0017.0515E + 0017.0402E + 0017.0289E + 0017.0289E + 0017.0064E + 0016.9951E + 0016.9838E + 0016.9725E + 0016.9613E + 0016.9500E + 0016.9387E + 0016.9274E + 0016.9161E + 0016.9048E + 001

13.649 Max12.13310.6169.09957.58296.06634.54983.03321.51660 Min

ax

Figure 2: Three-field coupling process of the AESA antenna.

3International Journal of Antennas and Propagation

And the K was deduced according to the impedance con-verter theory as

K = ZinA

ZinA + ZinB, 3

and thus,

PinA = 1 − K Pin 4

At the same time, considering that there is also the reflec-tion between the λ/4 impedance converter and the load, thus,the load power

PA = PinA 1 − ΓinA 2 ,

PB = PinB 1 − ΓinB 25

Finally, according to the relation between the current andthe power, the element excitation current is obtained as

where P0 is the total power at the input of the power divider,Z0 is the characteristic impedance of the microstrip at theinput end, Zl is the impedance of the array element, Zmnand Zmn′ (also ZinA or ZinB in the figure) are the impedanceof the λ/4 impedance converter looking into the load fromthe input end (the arrow pointing), and Zmn′ is the imped-ance of the λ/4 impedance converter looking into the sourcefrom the output end (the opposite direction of the arrow).Zmn is shown as

Zmn = Z2Zl + jZ2 tan β l + Δlmn

Z2 + jZl tan β l + Δlmn7

Zmn′ and Zmn′ are the same, where Z2 is the characteristicimpedance of the impedance converter. And the meaning ofn′ in the formula is as follows:

n′ =n + 1, n is an odd number,n − 1, n is an even number

8

If the above power divider is equivalent to the three-portnetwork, as shown in Figure 4(b), the cascaded microwavefull matrix is

b1b2b3b4

b7

s11s21s31s41

s71

s12s22s32

s13s23s33s43

s73

s14

s34s44

s74

s17

s37s47

s77

a1a2a3a4

a7

… … … … …

… …

…

… … …

…

…

…

… ……

= 9

Thus, the scattering parameter matrix of the three-portnetwork is

S = SΙ Ι + SΙ ΙΙ β − SΙΙ ΙΙ−1SΙΙ Ι, 10

where SΙ Ι, SΙ ΙΙ, SΙΙ Ι, and SΙΙ ΙΙ are the blocked submatrix.

2.2. Numerical Solution for Multifield Coupling Problem.The multifield coupling of the AESA antenna is primarily

PA, IA, Z1

P0, I0, Z0

PB, IB, Z1

Z2

ZinA ZinB

. . .

Pin

(a) Diagram of circuit parameters

[S]1a1

b1

a5

a4

b4

b5[S]2

[S]3a7

a6

b6

b7

a2

b2

a3

b3

(b) Three-port network

Figure 4: Diagram of the power divider.

Imn′ = P0 × 1 − Zmn Δlmn Zmn′ Δlmn − Z0 Zmn Δlmn + Zmn′ Δlmn

Zmn Δlmn Zmn′ Δlmn + Z0 Zmn Δlmn + Zmn′ Δlmn

2

× 1 − Zmn′ Δlmn − Zl

Zmn′ Δlmn + Zl

2

× Zmn′ Δlmn

Zmn Δlmn + Zmn′ Δlmn

1/2

× Zl−1/2,

6

4 International Journal of Antennas and Propagation

sequential coupling. Thus, the solution is performed bysequential decoupling [16]. In the traditional method bytheoretical model, the actual model was simplified duringthe numerical solution, which only considers the positionaldeviations of the array elements and ignores its self-deformations. This paper performs the numerical solutionby using advanced simulation software, which is based onthe model reconstruction that takes into account the defor-mation of the array element. This approach avoids the com-plicated meshing process and overcomes the problem oflarge fitting errors in the traditional numerical simulationusing simulation software. Figure 5 shows the flow chart ofthe numerical solution for the multifield coupling problem,and the specific steps are as follows:

(1) The finite element model of the AESA antenna isestablished. And the thermal load is applied to themodel in the thermal analysis software. The temper-ature field and the deformed model of the antennaare obtained

(2) The deformed model in (1) is introduced into thestructural analysis software using the model recon-struction method in this paper (explained in Section3), and it is applied as the external load. The struc-tural displacement analysis is performed. And thestructural displacement field and the deformedmodel superimposed with (1) are obtained

(3) The model superimposed in (2) is introduced intothe electromagnetic analysis software still using themodel reconstruction method in this paper, and thetemperature field in (1) is applied to perform the finalelectromagnetic performance

3. Model Reconstruction

The key to solve the multifield coupling problem of AESAantennas using the simulation software is to overcome theproblems that the data models are heterogeneous and themesh does not match caused by the different analysis tools.In this paper, the model reconstruction of antenna arrayincludes the reconstruction of array elements and the

reconstruction of SMA connectors. Figure 6 shows thestructure of the antenna array. And the array elements andfeeds are shown in the figure. For the above, the NURBS(nonuniform Rational B-Spline) surface fitting algorithmthat guarantees the accuracy of positions and shapes of arrayelements and the local model reconstruction algorithm thatensures the accuracy of amplitudes and phases of feed con-nectors are proposed. The flow chart of the reconstructionalgorithm is shown in Figure 7.

3.1. Array Element Reconstruction Based on NURBS SurfaceFitting. The AESA antenna undergoes various forms ofdeformation via the different external loads during service.The NURBS surface provides the unified mathematical formfor the surfaces representing arbitrary shapes, thereby facili-tating the transfer for models through the surface equationsand avoiding the geometric modeling errors because of theheterogeneous data models between different coupled fields.At the same time, NURBS technology has been adopted bymany excellent CAD/CAM software and has extremelyhigh accuracy for surface fitting [17], thereby improvingthe fitting accuracy of array element surfaces in the modelreconstruction of AESA antennas. The expression of thep × q-ary NURBS surface is [18]

S u, v =∑m

i=0∑nj=0wi,jdi,jNi,p u N j,q v

∑mi=0∑

nj=0wi,jNi,p u N j,q v

, 11

where di,j, i = 0, 1,… ,m ; j = 0, 1,… , n are the control verti-ces that correspond to a topological rectangular array thatforms a control mesh, wi,j is the weight associated with thevertices, Ni,p u and Nj,q v are the canonical B-spline basis

Finite element model

Thermal analysis softwareThermal analysis Temperature field

and the deformed model

Model reconstruction

Structural analysis softwareStructural analysis Structural displacement field

and the deformed model

Model reconstruction

Electromagnetic analysis softwareElectromagnetic performanceElectromagnetic

analysis

Analysissoftware Coupling

fields

Figure 5: The flow chart of the numerical solution.

Array elementSupportframe

SMAconnector

Figure 6: Structure of the antenna array.

5International Journal of Antennas and Propagation

functions determined by the vector U = u0, u1,… , um−p+1of the direction u and the vector V = v0, v1,… , vn−q+1 ofthe direction v, respectively, according to the de Boor-Coxrecursion formula. The recursion formula of Ni,p u isdefined as

Ni,0 u =1, ui ≤ u ≤ ui+1 ;0,

Ni,p u = u − uiui+p − ui

Ni,p−1 u +ui+p+1 − u

ui+p+1 − ui+1Ni+1,p−1 u

12

Postulating 0/0 = 0, the recursion formula of Nj,q v issimilar.

The specific steps of NURBS surface fitting are as follows:

(1) Assuming that the array of the NURBS surface isp × q-ary and assuming that wi,j = 1

(2) The node information of the finite element modelwas extracted as the data points fitted to the NURBSsurface. Parameter values were assigned to eachdata point pi,j that determines the node vector Uand V [19]

(3) The cross-section curves were constructed fromthe cross-section data points pi,j on the nodal vec-tor U using the NURBS surface inverse algorithm[18], and the control vertices di,j i = 0, 1,… ,m ; j =0, 1,… , n − q + 1 were obtained

(4) The cross-section curves were constructed fromthe cross-section data points di,j on the nodal vectorV using the NURBS surface inverse algorithm, and

Select the fitting surface equation

Determine the surface boundary

Electromagnetic model

Determine the centerlocation of connector

Determine the direction ofcoaxial line

Node information

Geometric solidmodel

Node informationof substrate and patch

Node information ofconnector

Mechanical model

Compare rootmean square errors

Determine the fitting surfaceequation

Adjust connector model

Reconstruction ofarray element

Face number of substrateand patch

Line number of connector

Whether the error isless than required value

Reconstruction script of arrayelement

Reconstruction script ofconnector

Yes

No

Reconstruction ofSMA connector

Search the center positionof space circle

Search the normal directionof surface

Figure 7: Reconstruction algorithm flow chart.

6 International Journal of Antennas and Propagation

the control vertices di,j i = 0, 1,… ,m ; j = 0, 1,… , nwere obtained

(5) The surface was corrected locally and the newweight w∗

i,j was calculated by the weight changingmethod [19]

(6) The NURBS surface was obtained based on the calcu-lated control points di,j, the weights w∗

i,j, the nodevectors U and V , and the surface arrays p and q

After obtaining the fitting surface, the surface equation isinput into the next physics field in the coupling analysisthrough the programmed interface program and a new sur-face is created in the simulation software using the surfaceequation. In addition, the node information of the modelboundary extracted from the previous physical field is usedas the constraint condition of the surface equation, which isused to reconstruct the boundary of the array elementmodels. The reconstructed array element model is describedin Section 3.3.

3.2. Local Model Reconstruction for the SMA Connector. Inorder to ensure the accuracy of amplitudes and phases ofthe reconstructed feed model (SMA connector), this paperdiscusses the model reconstruction algorithm from twoaspects: the location and the pointing direction of SMAconnectors. Those two are the important factors affectingthe impedance matching between the radiating patch andthe feed lines and thus affecting the feeding amplitudesand phases.

Determining the location of the SMA connector, the out-put port of the connector is approximated as a plane circle.The maximum values and minimum values on the plane cir-cle by the coordinates x, y, and z of the nodes are searched, asshown in Figure 8. According to the similar triangle princi-ple, the median value is taken as the point position of theconnector. The approximate conditions of the feed port areas follows:

(1) The SMA connector has large rigidity and the actualdeformation in service is minimal

(2) Not considering the ductility of the material

Determining the pointing direction of the SMA connec-tor: the surface equation has been obtained in Section 3.1and written in the general form as F x, y, z = 0. Assumingthat the continuity condition is satisfied at P0 x0, y0, z0

and substituting the coordinates of the position of the SMAconnector (also the approximate coordinates of the centerof the circle), the normal direction equation of the surface is

x − x0Fx P0

= y − y0Fy P0

= z − z0Fz P0

13

The partial derivatives Fx, Fy, and Fz are constructedby substituting the coefficients of the surface equation intothe above formula. Moreover, the normal direction at P0x0, y0, z0 is obtained as the new pointing direction of theSMA connector. The reconstructed SMA connector modelis described in Section 3.3.

3.3. Results and Analysis. In the traditional numerical solu-tions, some scholars analyzed the multifield coupling ofAESA antennas by fitting the solid model in the analysis soft-ware and then remeshing on the newmodel. For example, theFinite Element Modeler (FEM) software is used to completethis work. Figure 9 shows the antenna element model fittedby the FEM software. In the fitting process by the software,the finite element model must first be converted into theform of patch in .x_t format through the surface informationof the model. Secondly, the entity model is sewn with thepatch, which is a necessary step for FEM to handle the model.In this process, some mesh information and connectioninformation between components are lost and the fittingerrors are introduced. Due to the above reasons, the accuracyof the final model is reduced as follows:

(1) As shown in the figure, the cylindrical surface ofthe SMA connector consists of multiple patches thataffect the integrity of the model. This is becauseof the sparse meshing in the previous structural anal-ysis. However, if the mesh is encrypted, the amountof calculations will be increased. When the model iscomplex or there are many components, the calcula-tion efficiency will be seriously affected. At the sametime, in order to take into account the fitting accuracyand the amount of calculation, the process of meshadjustment is extremely complicated

(2) As shown in the figure, the upper-end face of theSMA connector is recognized as nonplanar by theelectromagnetic analysis software, which makes itimpossible to apply the excitation current in the elec-tromagnetic analysis software. This is because of thedeformation of the antenna array

(3) As shown in the figure, the lower-end face of the SMAconnector and the substrate cannot be completelyattached to each other, which affects the feeding effect

In the model reconstruction of this paper, as long as thelocation and the pointing direction of the SMA connectorare accurately determined according to Section 3.2, the stan-dard cylinder in the electromagnetic analysis software can beused to complete the reconstruction of the SMA connectormodel. Therefore, the model is complete and regular and itslower-end surface is plane. In addition, the substrate and

P0

Z

Y

X

Figure 8: Determination of the location of the SMA connector.

7International Journal of Antennas and Propagation

the SMA connector need to be subtracted by the Booleansubtraction operation in the analysis software. After the sub-traction, the curvature of each point on the upper-end sur-face of the SMA connector coincides with the curvature ofeach point on the substrate and a perfect fit between thetwo is guaranteed. At the same time, the NURBS surface fit-ting algorithm ensures the accuracy of the array element

model. Figure 10(a) shows the antenna element model gener-ated by the model reconstruction method in this paper, andFigure 10(b) shows the side view.

The results of the electrical performance obtained by theFEM fitting method and the model reconstruction methodwere compared with the experimental test results, respectively.Figure 11(a) shows the radiation pattern and Figure 11(b)

Substrate

PatchArrayelement

SMA connector

Upper-end face

Lower-end face

Figure 9: Antenna element model fitted by the FEM software.

Upper-end face

Lower-end face

(a) Element model generated by the model reconstruction method

(b) Side view

Figure 10: Antenna element model generated by the model reconstruction method.

10

5

0

Dire

ctiv

ity (d

B)

−5

−10

−15−150 −100 −50 0

Theta (degree)50 100 150

ExperimentModel reconstructionFEM fitting

(a) Radiation pattern

0

−5

−10

S11

(dB)

−15

−20

−255.4 5.6 5.8

Freq (GHz)6.0 6.2

ExperimentModel reconstructionFEM fitting

(b) Return loss

Figure 11: Electrical performance results.

8 International Journal of Antennas and Propagation

shows the return loss. The results show that the model recon-struction method has higher analysis accuracy.

4. Simulations and Experiment Results

The following is the case of multifield coupling analysis of the32-element AESA antenna whose operating frequency is5.8GHz in C-band. The measurements and the comparisonof simulations by different methods were shown.

4.1. Simulation Process Based on Model Reconstruction

4.1.1. Analysis of the Structural Displacement Field andTemperature Field. As described in Section 2.2, the analysisof the structural displacement field and the temperature fieldwas performed in ANSYS. In this case, the left-end face of theantenna was fixed and the displacement of 50mm (close to awavelength of the 5.8GHz antenna) was applied to the right-end face. At the same time, the temperature load was appliedin the form of body load. Superimposing the force load andtemperature load, the structure displacement field distribu-tion and temperature field distribution of the antenna wereobtained. And the deformation of the antenna array wasextracted for the subsequent model reconstruction and

electrical performance analysis, as shown in Figure 12.Figure 13 shows the material parameter testing experiment.The element types and material properties are shown inTable 1.

4.1.2. Model Reconstruction and Electrical PerformanceAnalysis. According to the method described in Section 3,the model was reconstructed in the electromagnetic analysissoftware by the node information of the finite elementmodel in the temperature and structural displacement fields.Meshing and electrical performance analysis were per-formed on the reconstruction model. Figure 14 shows theantenna entity model reconstructed in the electromagneticfield. The model is complete and accurate and can be directlyused for simulation analysis. Figure 14(a) shows the frontview, where the reconstructed model of the substrate andthe patch is regular and accurate and the two are completelyattached to each other. Figure 14(b) shows the rear view,where the position of the SMA connector is accurate andthe direction of the coaxial line is the same as the normalof the surface. The analysis results of electrical performanceare described in Section 4.2.

4.2. Comparisons of Simulations and Experiment Results. Theelectrical performance of the antenna array was measured,as shown in Figure 15. Figure 16 shows the antenna arraymodel [20].

4.2.1. Analysis of Electrical Performance without ConsideringField Coupling. Without considering the field coupling, theradiation pattern obtained by the method in this paper wascompared with experimental test results, as shown inFigure 17. By comparison, the relative error of the gainbetween the two is 1.92%, the relative error of the beampointing is 0%, and the relative error of the sidelobe level isless than 9.90%, as shown in Table 2. The experimentalresults illustrate the accuracy of this method.

4.2.2. Analysis of Electrical Performance in the Coupling Field.In the coupling field, the results of the model reconstructionmethod and other traditional methods were compared withthe measured results as shown in Figure 18. These methodswere compared from the five aspects: analysis accuracy, cal-culation time, applicability for AESA antenna, automationcapability, and coupling analysis capability. First, Table 3shows that the model reconstruction method has the highestaccuracy compared with the other traditional methods. Asfollows, the relative error of gain is 2.24%, the relative errorof beam pointing (ratio relative to the beam width) is2.17%, and the relative error of the sidelobe is less than8.71%. Secondly, Table 4 shows the comparisons from otheraspects. It shows that the model reconstruction method issuperior to other methods: (1) The model reconstructionmethod avoids the tedious meshing process and saves muchtime in computing. (2) The model reconstruction methodhas good applicability for the structure of AESA antenna,which is capable to describe both connection of componentsand self-deformation of elements accurately. (3) The modelreconstruction method has great automation capability,thereby improving the efficiency of the solution. (4) The

0 0.0111110.005556 0.016667 0.027778 0.038889 0.05

0.022222 0.033333 0.044444

MX

Figure 12: Antenna array.

Figure 13: Testing experiment of material parameters.

9International Journal of Antennas and Propagation

Table 1: Element types and material attribute parameters.

Component Material attribute Element type Elastic modulus Mpa Poisson’s ratio Density kg/m3

Framework Rogers5880 SOLID185 33,500 0.38 1130

Substrate Rogers4350 SOLID185 1200 0.36 565

Patch Copper SOLID185 1.08e11 0.33 8900

Feed core Copper SOLID185 1.08e11 0.33 8900

Feed housing Rogers5880 SOLID185 33,500 0.38 1130

(a) Front view of the reconstructed antenna array

(b) Rear view of the reconstructed antenna array

Figure 14: Reconstructed antenna array model.

Antenna arrayAbsorbingmaterial

Target pointphotogrammetry

Microwaveanechoic chamber

Force transducer

Figure 15: Electrical performance tester.

Figure 16: Antenna array model.

−60 −30 0 30 60

−20

−10

0

10

20

Dire

ctiv

ity (d

B)

�eta (degree)

Simulation resultMeasured result

Figure 17: Radiation pattern without considering field coupling.

Table 2: Comparison of measured and simulation results.

Gain (dB)Beam

direction (°)Sidelobe(dB)

Simulation 20.44 0 6.56 6.63

Experiment 20.84 0 7.28 6.91

Relative error (%) 1.92 0 9.90 4.05

−80 −60 −40 −20 0 20 40 60 80−30

−20

−10

0

10

20

Dire

ctiv

ity (d

B)

Theta (degree)

Measured resultModel reconstruction method

Software fitting methodSET model method

Figure 18: Radiation pattern in the coupling field.

10 International Journal of Antennas and Propagation

model reconstruction method could to be used to analyzeboth multifield coupling and field-circuit coupling.

5. Conclusion

In this paper, the field-circuit coupling model was establishedby introducing the deformation errors of the feed system intothe existing theoretical model. In addition, the new numeri-cal solution method for the multi-field-coupled problemof AESA antennas based on model reconstruction was pro-posed. In the model reconstruction, two algorithms were pre-sented: the NURBS (nonuniform rational B-spline) surfacefitting algorithm that completes the mapping from finite ele-ment models to geometric models by the surface equationsestablished by the node information and the local modelreconstruction algorithm that determines the local geometricmodels by the positions and the directions. The numericalsolution was applied to the 32-element AESA antenna. Andthe comparisons between the simulations and measuredresults shows the following: (1) Compared with the tradi-tional fitting simulation software, the proposed methodavoids the fussy meshing process and saves much time incomputing. (2) Compared with the traditional coupled SETmodel, the proposed method has high accuracy. This isbecause the NURBS surface fitting algorithm guarantees theaccuracy of both the positions and shapes of array elementsand the local model reconstruction algorithm ensures theaccuracy of the amplitudes and phases of feed connectors.Furthermore, the method proposed in this paper provides anew idea for the numerical solution of the multi-field-coupled problem of AESA antennas.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (nos. 51575419, 51775405, and51490664), the Joint Funds of Ministry of Education ofthe People’s Republic of China (no. 6141A02022107), andDefense Basic Research Program (no. JCKY2016210B002).Zhiheng Cai, Bo Tang, Le Kang, Zhanbiao Yang, and SiwenZhang from Xidian University provided experimental helpduring the project. Their efforts were crucial to the comple-tion of the research.

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Table 3: Comparison between solution results using different methods.

Gain (dB) Beam direction (°) Left sidelobe (dB) Right sidelobe (dB)

Measured result 19.68 7.46 6.43 5.74

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Relative error compared with experiment (%) 2.24 2.17 1.97 8.71

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Relative error compared with experiment (%) 5.69 3.06 4.98 17.60

Table 4: Comparison of different methods in multifield coupling analysis.

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Automation capability Fully automated Unable to be automatedUnable to be automated in theprocess of model simplification

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Difficult to analyze multifieldcoupling and unable on analyzing

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