2D-FDTD MODELING OF WIREANTENNAS NEAR ARTIFICIALIMPEDANCE SURFACES

Mikko Karkkainen and Sergei TretyakovRadiolaboratoryDepartment of Electrical and Communications EngineeringHelsinki University of TechnologyP.O. Box 3000FIN-02015 Hut, Finland

Received 2 January 2002

ABSTRACT: This Letter deals with wire antennas radiating in the vi-cinity of an impedance surface. The use of impedance surfaces has beensuggested, for instance, in the context of antenna reflectors. In this Let-ter a numerical surface impedance model is introduced. Some radiationpatterns are calculated to validate the model. © 2002 Wiley Periodicals,Inc. Microwave Opt Technol Lett 34: 38–40, 2002; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10366

Key words: surface impedance; antenna; FDTD

1. INTRODUCTION

Novel artificial impedance surfaces have been introduced in someapplications: In antenna reflectors to double the field of a wireantenna [1], in mobile antennas [2], and in microwave filters [3],to name but a few. The two-dimensional model of a wire antennapositioned near an artificial impedance surface is studied here withthe use of using finite-difference–time-domain technique. To cal-culate the radiation patterns of the antenna, an FDTD model withPML absorbing boundary conditions is constructed to truncate thelattice. A line-source excitation is located just above the imped-ance surface. An efficient and robust update scheme for the fieldcomponents on the impedance surface is developed, and near-fieldradiation patterns are calculated. The TMz case is considered,where the sinusoidal line source is directed parallel to the surface.The main motivation for this study was the fact that the simpleanalytical model used previously in [2] could not take into accountthe finite cross-section size and shape of the antenna body. Thiscan be done numerically with the use of FDTD simulations, andwe will show in this article that the main conclusions of theanalytical study hold true also for limited-size impedance surfaces.

2. SURFACE IMPEDANCE MODEL

The classical surface impedance relation on the impedance surface,that is,

Et � Zsn � Ht, (1)

where index t marks the tangential field components, and n is theunit vector normal to the surface, is used. Isotropic artificialimpedance sheets can be modeled with the following frequency-dependent surface impedance [1]:

Zs �jL

1 � 2LC. (2)

The update equation for electric field components on the imped-ance surface will be derived from these equations. The parameters

L (surface inductance) and C (surface capacitance) just describethe properties of the structure introduced in [1].

2.1. Update Equation for Electric Field on the SurfaceLet the impedance surface lie in the xz plane. The goal is to derivethe update equation for the electric field one cell above the xzplane. From Eqs. (2) and (1), with the use of the phasor notationfor the fields,

�jLHx � Ez � j2LCEz. (3)

is obtained. This can be easily transformed into the time domainwith the use of the inverse Fourier transform:

�L�Hx

�tx, y, t � Ezx, y, t � LC

�2Ez

�t2 x, y, t. (4)

Equation (4) From t � 0 to t � n �t is integrated with the use ofthe trapezoidal rule. In discrete form, just above the impedancesurface,

Hx�i,1/ 2n � �

�t

2L �l�1

n

Ez�i,1n�l�1/ 2 � Ez�i,1n�l�1/ 2

��t

4LEz�i,1n�1/ 2 � Ez�i,1n�1/ 2 �

C

�tEz�i,1n�1/ 2 � Ez�i,1n�1/ 2. (5)

In order to have a simple updating scheme, a half-cell spatial errorbetween Hx and Ez is neglected in the discretization. Equation (5)requires the whole time history of Ez. To avoid storing the wholetime history of Ez the sum is calculated recursively in the follow-ing manner. Let

��i,1n � �

�t

2L �l�1

n

Ez�i,1n�l�1/ 2 � Ez�i,1n�l�1/ 2. (6)

The auxiliary variable � can be updated as follows:

��i,10 � 0,

��i,11 � �

�t

2LEz�i,11/ 2,

��i,1n � ��i,1

n�1 ��t

2LEz�i,1

n�1/ 2 � Ez�i,1n�3/ 2. (7)

Now, the Maxwell–Ampere law can be used in integral form toobtain a time-stepping relation for electric field just above thesurface to get

Ez�i,1n�1/ 2 � Ez�i,1

n�1/ 2 � Hx�i,1/ 2n � Hx�i,3/ 2

n �t

�0�y

� Hy�i�1/ 2,1n � Hy�i�1/ 2,1

n �t

�0�x. (8)

Finally, Eqs. (5)–(8) are combined, and the update equation for theEz-component is obtained:

Ez�i,1n�1/ 2 �

4�0�yL � 4LC � �t2

4�0�yL � 4LC � �t2 Ez�i,1n�1/ 2Contract grant sponsors: Nokia Research Center; Nokia Foundation; Fil-

tronic LK; TEKES

38 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002

�4L�t

4�0�yL � 4LC � �t2 ��i,1n

�4L�t

4�0�yL � 4LC � �t2 �

� � Hy�i�1/ 2,1n � Hy�i�1/ 2,1

n�1 �y

�x� Hx i,3/ 2

n � . (9)

If the surface impedance approaches zero, then the electric fieldcomponents are zero all the time, corresponding to the PEC bound-ary condition. If the surface impedance tends to infinity, then the

update equation reduces to the normal update equation for electricfield in free space except that the tangential magnetic field is zeroat the boundary. Hence the FDTD algorithm that uses Eq. (9)remains stable for all values L from zero to infinity, provided thatC is a reasonable number. PML absorbing boundary conditionswith quadratic polynomial grading are used to truncate the com-putational zone.

2.2. Validation of the ModelThe model has been validated by comparisons to the resultscalculated with the exact image theory [4] in [2]. In [2], the wireantenna was located above an infinite impedance surface. The

Figure 1 Near-field radiation patterns for an antenna over a wide screenwith Zs � j0.2�, where � � ��0/�0

Figure 2 Near-field radiation patterns for an antenna over a wide screenwith Zs � j�

Figure 3 (a) Near-field radiation pattern with Zs � j0.2 � �. (b)Near-field radiation pattern with Zs � j � �

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002 39

present model takes the impedance surface to be wide (w � 47.7cm) enough to enable comparison.

The results of the numerical simulations, together with theexact results, are presented in Figures 1 and 2. For simplicity thesimulations, take C � 0, and the value of L is changed to get thedesired surface impedance (� � ��0/�0 is the free-space waveimpedance). The excitation frequency is 1 GHz, and the fields areobserved at r � 4.77-cm distance (kr � 1, k � 2�/) from theorigin, which is taken to be on the surface just below the antenna.The antenna is located 4.77 mm from the origin. The radiationpatterns are calculated by dividing the observed electric fieldmagnitude by the corresponding quantity in free space. The nu-merically calculated radiation patterns in Figures 1 and 2 agreevery well with the exact patterns [2].

2.3. FDTD Modeling of a Finite-Size AntennaThe goal of this study is to calculate the near-field radiation patternof a more realistic antenna. As an example, a mobile terminalantenna can be modeled as a PEC box that is covered with anartificial impedance layer, modeled by a certain reactive imped-ance. In the present example, only the upper surface of the box is

covered by an impedance layer. The other three sides are ideallyconducting. When calculations are in two dimensions, the wireantenna is modeled as a single soft-sourced Ez component near themetal box. The fields are observed at a distance of 5 cm. The sizeof the box is 2 � 4 cm, and the frequency is 1 GHz. The radiationpatterns in this case for two different surface impedances arepresented in Figure 3. As in the case of a very wide impedancesurface, it is seen that the radiation is increased when the surfaceimpedance is increased. The fields are concentrated on one side onthe device, as was predicted in Reference 2. The currents inducedon the surface of the box can also be studied. The surface currentsJs are easily calculated from the magnetic fields on the surfaceaccording to

Js � n � H. (10)

Here, n is the unit normal vector pointing outward from thesurface. The surface current distributions for the same surfaceimpedance values are presented in Figure 4. The plotting indexruns clockwise around the box, starting from the lower left corner.The peaks in Figure 4(a) have a clear explanation: The surfacecurrents close to the antenna are largest for the case of smallimpedances, and there are current singularities at the box corners.Also, the current densities induced on the box are mainly concen-trated on one side of the device. In Figure 4(b), the peak in themiddle is clearly smaller, but the singularities in the current dis-tribution in the corners of the box (indices 15 and 47) becomemore clearly visible. This again corresponds to the conclusionsmade from the simple analytical model: In this case when theproperties of the surface are closer to magnetic screen, the nearfields and induced currents are less localized near the source.

3. CONCLUSIONS

An FDTD-model of a wire antenna near an impedance surface hasbeen developed. It has been shown that the results calculated withthe proposed FDTD model compare well with the exact solution tothe problem in the special case of an infinite screen when the exactsolution is available. The presented FDTD model allows the so-lution of more realistic and complicated problems where the in-troduced impedance relation is applicable. A finite-size two-di-mensional model of an antenna has been studied, and the resultsconfirm theoretical predictions made on the basis of the infinite-screen model. A detailed 3D simulation would be a natural exten-sion to the presented calculations.

REFERENCES

1. D. Sivenpiper, L. Chang, R.F.J. Broas, N.G. Alexopoulos, and E.Yablonovich, High-impedance electromagnetic surfaces with a forbid-den frequency band, IEEE Trans Microwave Theory Tech MTT-47(1999), 2059–2074.

2. S.A. Tretyakov and C.R. Simovski, Wire antennas near artificial im-pedance surfaces, Microwave Opt Technol Lett 27 (2000), 46–50.

3. F.R. Yang, K.P. Ma, Y. Qian, and T. Itoh, A novel TEM waveguideusing uniplanar compact photonic-bandgap (UC-PBG) structure, IEEETrans Microwave Theory Tech MTT-47 (1999), 2092–2098.

4. I.V. Lindell, Methods for electromagnetic field analysis, IEEE Press,New York, 1995.

© 2002 Wiley Periodicals, Inc.

Figure 4 (a) The surface current distribution with Zs � j0.2�. (b) Thesurface current distribution with Zs � j�

40 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002