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Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2012, Article ID 120208, 4 pagesdoi:10.1155/2012/120208

Research Article

A Volume-Surface Integral Equation Solver for Radiation fromMicrostrip Antenna on Anisotropic Substrate

Yuhuang Ye, Jiade Yuan, and Kaixiong Su

College of Physics and Information Engineering, Fuzhou University, Fuzhou, Fujian 350108, China

Correspondence should be addressed to Jiade Yuan, [email protected]

Received 4 March 2012; Revised 22 July 2012; Accepted 24 July 2012

Academic Editor: Tat Yeo

Copyright © 2012 Yuhuang Ye et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A volume-surface integral equation (VSIE) solver is presented for the calculation of electromagnetic radiation from arbitraryshaped microstrip antenna on anisotropic substrate. The method of moments (MoM) is used to convert the integral equation intoa matrix equation, where the equivalent volume current and surface current are expanded into a finite series of SWG and RWGbasis function, respectively. A simple strip model is incorporated in the VSIE to simplify the analysis of the probe-fed microstripantenna. The present approach is sufficiently versatile in handling microstrip antenna with arbitrary shaped anisotropic dielectricsubstrate. Numerical results indicate the reliability and accuracy of the proposed method.

1. Introduction

During recent years, great interests have been shown in usingmicrostrip antenna deposited on anisotropic substrate sincethe substrate anisotropy could have important applicationson the operation of microstrip antennas [1–4]. With theincreasing complexity of geometry and material property,designing these antennas requires more and more dedicatedand sophisticated computer-aided-design (CAD) tools topredict the characteristics. The method of moments (MoM)has been proven to be one of the most powerful CAD toolsfor solving this class of problems. By now, a number ofmicrostrip antennas with anisotropic substrate have beeninvestigated using the MoM-based spectral domain analysismethod [1–3]. However, in the MoM-based approaches forthe analysis of microstrip antenna, the volume-surface inte-gral equation (VSIE) [5–9] solver is more suitable to analyzecomplex structure targets and has a number of advantagesover the other MoM-based approaches [2, 10], such as theapplicability to various inhomogeneous materials, the samesimple form regardless of the complexity of the materials andno special treatment for problems with junctions. It shouldbe noted that the VSIE solver has been applied to calculatethe electromagnetic scattering by the composite anisotropicmaterial and metal targets in previous work [11, 12].

In this paper, the VSIE solver is extended and appliedfor the computation of electromagnetic radiation from the

microstrip antenna on an anisotropic substrate. A simplestrip model is incorporated in the VSIE to simplify theanalysis of the probe-fed microstrip antenna. Radiationproperties of both the plane and the cylindrical conformalmicrostrip antenna with uniaxial substrate are calculatedusing the VSIE solver.

2. Theory

The microstrip antenna is discretized into triangular patchesfor metallic surfaces and tetrahedral cells for dielectricvolume, respectively. The unknown surface currents on themetallic surface and the volume currents in the anisotropicdielectric substrate can be obtained by solving the followingcoupled volume-surface integral equation [12]:

(Ei(r) + Es(r)

)∣∣∣tan= 0, r ∈ S,

εr−1 ·Dv(r) = Ei(r) + Es(r), r ∈ V ,

(1)

where S and V denote the metallic surface and the dielectricbody, respectively. Dv(r) is the electric flux density in V ,while Ei(r) and Es(r) are the electric field due to the appliedsource and the scattered field from the induced currents onS and in V , respectively. The metallic surface and dielectricbody are meshed into triangular patches and tetrahedralcells, and then RWG and SWG basis and test functions areapplied to expand the unknown surface current Js(r) and

2 International Journal of Antennas and Propagation

d

ln

x

12

3

w

(a) (b)

Figure 1: A strip model in the excitation source of antenna. (a) Wire-surface junction. (b) Surface-surface junction model.

x

yz

a

b

Figure 2: Geometrical structure of the planar microstrip antenna.

6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4

0

VSIE-MoMCalculated in [4]

S11

(dB

)

Frequency (GHz)

−25

−20

−15

−10

−5

Figure 3: S-parameter of the microstrip patch antenna solved byMoM based VSIE solver, and compared with the results calculatedin [4].

electric flux density Dv(r), respectively. εr is the permittivitytensor of the dielectric material and given as

εr =⎡⎢⎣εxx 0 00 εyy 00 0 εzz

⎤⎥⎦. (2)

The volume current Jv is related to the total electric fluxdensity Dv(r) by

Jv = jωκ(r) ·Dv(r), (3)

where κ(r) is the contrast ratio tensor defined as

κ(r) =(εr − I

)· εr−1

. (4)

The hybrid integral equation (1), togther, together with (2)–(4), can be solved by using Galerkin’s MoM, and then RWGand SWG basis and test functions are applied to construct theMoM matrix, respectively. The detail derivation of the MoMmatrix equation can be found in [11].

A simple equivalent strip model [5] is applied in thetreatment of the feeding probe, where the wire-surfacejunction [13, 14], as shown in Figure 1(a), is changed toa surface-surface junction, as shown in Figure 1(b). Thefeeding probe of diameter d is replaced by its equivalent stripstructure with the width w = 2d, as shown in Figure 1.The excitation source locates at the junction, and the Ei inequation can be established by using the delta-gap voltagemodel:

Ei = Vδ(l)nl, (5)

where V is the voltage across the gap and nl is normal to thejunction ln.

3. Numerical Results

In this section, two numerical results will be shown. For thefirst example, We consider a planar microstrip antenna, asshown in Figure 2. The geometric parameters are taken from[4] and are as follows: the dimension of the patch is a× b =12.45 mm × 16 mm, the substrate of the microstrip antennais 28.1 mm×32 mm×0.794 mm, and the permittivity tensoris chosen as εr = [2.31 0 0, 0 2.31 0, 0 0 2.19]. (Refer to[4], in Figure 5, when θ = 0.) The planar microstrip antennais discretized into 1384 tetrahedral cells and 568 triangularpatches, respectively, yielding a total of 4091 unknowns.

Figure 3 shows the S-parameter of the microstrip patchantenna solved by the presented method and comparedwith results in [4]. It is observed that good agreements areobserved between the two methods. Moreover, the E-planeradiation pattern of antenna at the frequency of 7.5 GHzis obtained from the MoM and compared with the finefinite-element method- (FEM)- based solution, as shown inFigure 4. Again, good agreements are exhibited. In addition,the CPU time for both the MoM and the FEM-based solutionis tested using the same computer, and the test results are 757seconds and 771 seconds, respectively.

The second example considers a cylindrical conformalmicrostrip antenna and discusses the effects of the uniaxialsubstrate on the input impedance. The geometric parameters

International Journal of Antennas and Propagation 3

MoMFEM solution

0

30

60

90

120

150

180

210

240

270

300

330

Figure 4: E-plane radiation pattern of microstrip antenna at f =7.5 GHz.

Patch

Feedline

εr h

θyz

x

Wg

Lg

W

L

Figure 5: Geometry of a conformal microstrip antenna.

are taken from [5], as shown in Figure 5. The cylindrical-rectangular patch has a dimension of L × W = 30 mm ×40 mm with a curvature radius of 50.795 mm, and thecylindrical-rectangular substrate is Lg ×Wg × h = 50 mm×60 mm × 0.795 mm. The feeding point is on the centerlineof the curved wide side and is 10 mm from the center ofthe patch, where the size of the rectangular strip feedline is2 mm×0.795 mm. The curved microstrip antenna is modeledwith 600 triangular patches and 1340 tetrahedral cells, result-ing in 4035 unknowns. The substrate is chosen the uniaxialanisotropic medium and the diagonalized permittivity tensoris taken in the form εr = [ε1 0 0, 0 ε1 0, 0 0 ε2].

Figure 6 shows the input impedance of the antennaversus different permittivity, where ε1 is 2.32, and ε2 ischosen as 1.8, 2.32, 2.8, respectively. It can be seen that thepermittivity ε2 has a significant influence on the impedanceversus frequency variation, which implies that increasing theε2 decreases the resonant frequency of the antenna.

0

20

40

60

80

100

120

140

Frequency (GHz)

−60

−40

−20

2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Inpu

t im

pude

nce

(Ω

)

ε2 = 1.8ε2 = 2.32ε2 = 2.8

Figure 6: Input impedance of the cylindrical conformal microstrippatch antenna (ε1 = 2.32, ε2 = 1.8, 2.32, 2.8, resp.).

4. Conclusion

The VSIE solver has been applied to analyze the elec-tromagnetic radiation from the microstrip antenna witharbitrary shaped anisotropic substrate. The targets aremodeled using tetrahedral volume elements for substrateand triangle face elements for metal patches. The coupledvolume-surface integral equations are derived by introducingtensor permittivity in conventional MoM. A simple stripmodel is used to simplify the analysis of the probe-fedmicrostrip antenna. Compared with the conventional MoM-based spectral domain analysis method, the VSIE solverapproach offers good flexibility to model arbitrarily shapedmicrostrip antenna structures while keeping a good accuracy.The presented approach is formulated using the free-spaceGreen’s function. This feature makes it easy to apply theMoM-based fast algorithms to reduce the computationalcomplexity of microstrip antenna with arbitrary shapedanisotropic substrate.

Acknowledgments

This work was supported by the Natural Science Foun-dation of Fujian Province of China (2011J01348) and theScience and Technique Major Program of Fujian Province(2010HZ0004-1).

References

[1] D. M. Pozar, “Radiation and scattering from a microstrippatch on a uniaxial substrat,” IEEE Transactions on Antennasand Propagation, vol. 35, no. 6, pp. 613–621, 1987.

[2] C. S. Gurel and E. Yazgan, “Characteristics of a circular patchmicrostrip antenna on uniaxially anisotropic substrate,” IEEETransactions on Antennas and Propagation, vol. 52, no. 10, pp.2532–2537, 2004.

4 International Journal of Antennas and Propagation

[3] C. Zebiri, M. Lashab, and F. Benabdelaziz, “Rectangularmicrostrip antenna with uniaxial bi-anisotropic chiral sub-strate-superstrate,” IET Microwaves, Antennas and Propaga-tion, vol. 5, no. 1, pp. 17–29, 2011.

[4] A. P. Zhao, J. Juntunen, and A. V. Raisanen, “An efficientFDTD algorithm for the analysis of microstrip patch antennasprinted on a general anisotropic dielectric substrate,” IEEETransactions on Microwave Theory and Techniques, vol. 47, no.7, pp. 1142–1146, 1999.

[5] M. He, Q. Chen, Q. Yuan, K. Sawaya, and X. Xu, “A simplestrip model in the volume-surface integral equation for anal-ysis of arbitrary probe-fed conformal microstrip antennas,”IEEE Antennas and Wireless Propagation Letters, vol. 8, pp.530–533, 2009.

[6] N. Yuan, Tat-Soon Yeo, X. C. Nie, Y. B. Gan, and L. W. Li,“Analysis of probe-fed conformal microstrip antennas onfinite grounded substrate,” IEEE Transactions on Antennas andPropagation, vol. 54, no. 2, pp. 554–562, 2006.

[7] N. Yuan, T. S. Yeo, X. C. Nie, and L. W. Li, “RCS computationof composite conducting-dielectric objects with junctionsusing the hybrid volume-surface integral equation,” Journal ofElectromagnetic Waves and Applications, vol. 19, no. 1, pp. 19–36, 2005.

[8] X.-C. Nie, N. Yuan, L.-W. Li, Y.-B. Gan, and T. S. Yeo, “A fastvolume-surface integral equation solver for scattering fromcomposite conducting-dielectric objects,” IEEE Transactionson Antennas and Propagation, vol. 53, no. 2, pp. 818–824, 2005.

[9] W. B. Ewe, L. W. Li, and M.-S. Leong, “Fast solution of mixeddielectric/conducting scattering problem using volume-sur-face adaptive integral method,” IEEE Transactions on Antennasand Propagation, vol. 52, no. 11, pp. 3071–3077, 2004.

[10] S. M. Rao, T. K. Sarkar, P. Midya, and A. R. Djordevic, “Elec-tromagnetic radiation and scattering from finite conductingand dielectric structures: surface/surface formulation,” IEEETransactions on Antennas and Propagation, vol. 39, no. 7, pp.1034–1037, 1991.

[11] J. Yuan and C. Gu, “A volume-surface integral equation solverfor scattering from microstrip antenna on anisotropic sub-strate,” in Proceedings of the 3rd IEEE International Symposiumon Microwave, Antenna, Propagation and EMC Technologiesfor Wireless Communications (MAPE ’09), pp. 1083–1085,October 2009.

[12] J. Yuan, Z. Niu, Z. Li, and C. Gu, “Electromagnetic scatteringby arbitrarily shaped PEC targets coated with anisotropicmedia using equivalent dipole-moment method,” Journal ofInfrared, Millimeter, and Terahertz Waves, vol. 31, no. 6, pp.744–752, 2010.

[13] W. J. Zhao, J. L. W. Li, and L. Hu, “Efficient current-basedhybrid analysis of wire antennas mounted on a large realisticaircraft,” IEEE Transactions on Antennas and Propagation, vol.58, no. 8, pp. 2666–2672, 2010.

[14] W.-B. Ewe, L.-W. Li, C. S. Chang, and J.-P. Xu, “AIM analysisof scattering and radiation by arbitrary surface-wire configu-rations,” IEEE Transactions on Antennas and Propagation, vol.55, no. 1, pp. 162–166, 2007.

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