FULL-WAVE ANALYSIS OF FINITE LARGE PRINTED DIPOLE ARRAYS USING THE CONJUGATE GRADIENT- FFT METHOD Yuan Zhuang, Keli Wu, Chen Wu, and John Litva Communications Research Laboratory McMaster University Hamilton, Ontario, Canada L8S 4K1

KEY TERMS CC-FFT method. discrete image technique, full-wave analysis, finite lurge microstrip dipole urrays

ABSI'RACT The firll-wave unalysis of Jinite large printed dipole arrays is per- formed by the conjugate gradient-FFT method with full-wave dis- crete image iechnique. The CG-FFT method has (he merits of saving computer memory and computing time. So it is very suituble for the efficienr analysis of large printed dipole arrays by combining with the discrete imuge technique which can convert the spectrum-domain problem into the spatial domain wirhout losing uny full-wave infor- mution. Severul types of finite large printed dipole urrays ure unu- /?zed and discussions ure made according to the numerical results. b: IYY.? J o h n Wilev & Sons, Inc.

INTRODUCTION The well-known method of moments (MOM) has the disad- vantage of generating and storing the impedance matrix and its inverse, which necessitates large computer memory and long computing time, severely restricting the method to lim- ited scale application. In this work, the full-wave analysis of large printed dipole arrays is presented, where the intent is to achieve a highly efficient simulation of finite large printed arrays. We employ the conjugate gradient method for the solution of the resulting system. In the CG procedure, the FFT is applied for the calculation of the convolutional inte- grals involved. Within the process, the full-wave discrete im- age technique is utilized to calculate efficiently the Sommer- feld-type Green's function. This technique can convert a spectral-domain problem into a spatial problem without losing any full-wave information. The computing time required for generating the discrete image is negligible compared to the implementation of the CG-FFT. This approach reduces the storage to O(N) and in addition, enhances the computational efficiency significantly (0(4N( 1 + log2(N)))). In this article two finite large dipole arrays are analyzed. Parameters such as current distributions and radiation patterns are calculated. The results shown that the CG-FFT method is an efficient approach for the analysis of finite large printed dipole arrays.

DESCRIPTION OF THE METHOD The general integral equation describing a microstrip antenna can be written as

-Ez x Einc(P) = ri x G(F, P') . J(F') dS (1) 1- with the notation of the conducting surface S, the current J , the incident field Einc, the unit v e c t z normal to the surface i t , and the dyadic Green's function G . For the conventional MOM, to solve the above equation, the computer memory and computing time are O(N2) and O(N3), respectively. Var- ious kinds of iterative methods can be used to reduce the

requirement, and the CG-FFT method (1) is the most popular one among them.

Equation (1) can be written in an operator equation form: E = L(J) . The scheme of the CG algorithm starts with an initial guess J O , then follows the iterative cycle described in [I] to get a new J . The iterations minimize the functional F ( J ) = (r, r') = llr1l2, where r = LJ - E and (r, r) denotes the inner product. In the operation of each cycle, the key point is how to use FFT to compute the operation LV and L*V (L": adjoint operator; V : J or r).

We can express Eq. (1) as

fE:nc(P) + PE,p(P) = - jw

+j1

After expanding the current with rooftop basis functions and testing the equation with a blade-razor function [2], we are ready to find IS, 61 (@: convolution):

LJ = L(J,i + J , j ) = E,,,(i, j ) f + E,.(t(i, j ) j , (3)

where c, = jwpl4rr and c2 = 1 / 4 m ~ . The E,,,t(i, j ) can be obtained in a similar way. By carefully rearranging J.&(i, j ) and Gl(i, j ) (such as padding zeros and wrapping around storage), FFT can be applied to evaluate the convolution. Then Eq. ( 2 ) becomes

I 1 1 + - (1 - 8 : ) j p . . (8, - l)G,, + c,jpCI f AY

I + - (1 - SY*)j,f (6, - l)G/, + C , j . f G I } j , (8) 1

AY

where 8, = exp(@m/N), 8, = exp(j2m/N). The S ( u , u ) , j f ( u , u ) , GJu, u ) , G,(u, v).are the N-point DFT of J.?(i, j ) , J?i, j ) , GJi, j ) , Gl(i, j ) . With the definition of adjoint op- erator (f, W ) = (LaI, J ) , the LnJ can be found, which is in a similar form to that of (9).

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 4, March 20, 1993 235

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From the above discussion, it is clear that G,(i, j ) , GJi, j ) must be calculated before the CG-FFT procedure. This involves a Sommerfeld-type Green’s function [3] , such as

Due to the complexity of Sommerfeld-type integration as in (9). it is inefficient to calculate Gr(i, j ) , G,,(i, j ) on a large array structure by using conventional numerical integration procedures. The full-wave discrete image technique [3] pro-

vides a powerful tool for carrying out this task. The idea underlying the technique is the use of the Prony method to approximate the spectral-domain Green’s function by the sum of a series of complex exponential functions: {aiehlk:}, i = 1, 2, . . . , N ; . Usually, Ni = 4 is accurate enough.

Once the complex coefficients a,, h, are determined by optimization for a given substrate, the inverse Fourier trans- form of (9) can be easily carried out analytically by using the Sommerfeld identity, The complicated Sommerfeld-type Green’s function then becomes a series of simple free-space- type Green’s functions without losing any full-wave infar-

1 ! ELECTRIC FIELD PATTERNS FOR TEE M I C R O S " DIPOLE ARRAY 1 i

o in w M to IM 120 i(o I W im P M (I) co IW im I (O 18) im

H-plane of Rect. Array H-plane of Ellipse Array - --___

I Ib-----

Figure 3 Comparison of the radiation patterns (H plane) between the two arrays (Icft: rectangular; right: ellipse)

mation. The computing time required for the generation of the discrete image is negligible compared to the CG-FFT procedure, and for a given substrate (single or multilayer) only one set of images has to be found, regardless of the antcnna's configuration and layout. So, the technique is one of the most suitable tools for large array simulation.

NUMERICAL RESULTS In our work, the printed dipole arrays are analyzed. For these simulations, all the formulations given above are applicable by a very simple modification. For instance, for an X-directed dipole array

To verify our solution, some microstrip dipole arrays in [4] are calculated. The input impedances are in good agree- ment with those in (4). To demonstrate the efficiency of this approach two printed dipole arrays are analyzed (Figure 1: 47 x 17 rectangular and ellipse layouts). The current distri- butions and the radiation pattern are calculated and shown in Figures 2 and 3. In our calculation, 9-7 basis functions are used to expand the current on each dipole. The number of unknowns of these arrays are 5593,4105. For the CG-FFT, the computer memories are 0(5593),0(4105) and the com- puting times are 56,41 minutes (on SunSpac), respectively. For the conventional MOM, we can predict that the computer memories will be U(3.13 x 107),0(1.7 x lo7), and the com-

puting times at least 100 times those for CG-FFT. So it is very hard for the conventional MOM to attack large array prob- lems. In Figure 2 , the current on the central row (ninth) of the arrays are drawn and the space between the dipoles on the row are omitted in the plot. From the figures, we can find the differences of current distributions of dipoles at different positions within an array and the differences of the current distributions and radiation patterns of arrays with different layouts. These show the edge effect of array layout on the current distribution and other parameters of an array, which cannot be found by infinite large array analysis.

CONCLUSIONS Finite large printed dipole arrays are analyzed by using the CG-FIT and full-wave discrete image technique. The results show that this is a very efficient approach to analyze finite large array structures, which are very difficult or even im- possible to analyze using MOM. From the simulation results, it is found that the change of the array's layout may influence the current distribution on array elements, and change the performance of the arrays. So it is necessary to implement finite large array analysis without using the infinite large array approximation.

REFERENCES 1. T. K. Sakar, E. Arvas, and S . S . Rao, "Application of F l T and

the Conjugate Gradient Method for the Solution of Electromag- netic Radiation from Electrically Large and Small conducting Bod- ies," IEEE Trans. Antennas Propagat., Vol. AP-34,1986, pp. 635- 640.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 6, No. 4, March 20, 1993 237

2.

3.

4.

5.

6.

J . R. Mosig and F. E. Gardiol. “General Integral Equation For- mulation for Microstrip Antennas and Scatterers,” IEE Proc., Pt. H. Vol. 132, 1985, pp. 424-432. D. G. Fang, J. J . Yang, and G. Y . Delisle, “Discrete ImagcTheory for Horizontal Electric Dipoles in a Multilayer Medium,” IEE Proc., Pt. H, Vol. 135, 1988, pp. 297-303. D. M. Pozar. “Analysis of Finite Phased Arrays of Printed Di- poles,” IEEE Trans. Antennas Propagat., Vol. AP-33, 1985, pp.

M. F. Catcdra. .I. G. Cuevas, and L. Nuno, “A Scheme to Analyze Conducting Plates of Resonant Size Using the CG-FFT Method,” IEEE Trans. Antennas Propagat., Vol. AP-38, 1990, pp. 123-456. J . Jin and J. L. Volakis, “Biconjugate Gradient FFT Solution for Scattering by Planar Plates,” Electromagnetics, Vol. 12, 1992, pp.

1045- 1053.

123-456.

Received 10-16-92

Microwave and Optical Technology Letters, 614, 235-238 0 1993 John Wiley & Sons, Inc. CCC 0895-2477193

ON THE TIME-DOMAIN RESPONSE OF

A LOW-LOSS GROUNDED MICROSTRIP DIPOLES EMBEDDED IN

DIELECTRIC SLAB Renato Cicchetti Department of Electronic Engineering University of Rome ‘La Sapienza” Via Eudossiana 18 00184 Rome Italy

KEY TERMS Time domain, microstrip dipoles, microwave integrated circuits

ABSTRACT A space-time analysis of the field radiated from microstrip dipoles embedded in a low-loss grounded dielectric slab is proposed. The crnalysis is based on the dyadic Green’s function method. The radia- tion mechanism as well as the structure electrical and geometrical parameters influencing the rudiation phenomenon are analyzed. De- sign guidelines to reduce the amplitude of the radiated field, or its distortion, are suggested. 8 IYY3 John Wiley & Sotis, h c .

1. INTRODUCTION Planar microwave integrated circuits use wires, microstrip lines, or other elements to connect passive and/or active com- ponents. Microstrip elements are used in high-density inte-

Eo I Po /t 1

E l Y Po Y g L

grated circuits as well as in high-speed digital applications. When the excitation signals of these circuits are rapidly time varying, emission of electromagnetic energy from the inter- connecting conducting wires or strip lines appears. This is considered an undesirable effect in many practical applica- tions, such as in wideband digital and radio communications. In addition, the change of the time shape of the radiated field as a function of the observation angles is often considered a distortion when the structure works as an antenna.

In the short-time regime, where numerical methods be- come increasingly intractable, analytical approaches afford an important alternative t o the strict numerical methods. Of par- ticular importance are the analytical models, which parame- trize the radiative behavior of the structure in terms of phys- ical observables [l]. Observables here refer to distinctive features in the electric field signal such as spikes, dips, un- dulations, etc. These models also provide input toward un- derstanding the relevant wave mechanisms.

Current analyses of the transient electromagnetic field, radiated from microstrip structures, neglect the presence of the substrate and analyze the radiated field produced both by primary and/or secondary sources, using the free-space Green’s function [2-41. Only recently models for analyzing the radiated transient electromagnetic field from dipoles [ S ] and microstrip lines [6, 71 have been presented. In 151 the field was computed by solving the Maxwell’s equation in the spectral domain, while in 16, 71 the space-time electric dyadic Green’s function method was used.

In this letter, the model presented in [6] is extended in order to analyze the responses of microstrip dipoles embedded in a low-loss dielectric substrate. Diagrams of the transient field radiated from microstrip dipoles are proposed, and de- sign guidelines to reduce the amplitude of the radiated field, or its distortion, are suggested.

2. METHOD OF ANALYSIS Figure 1 shows an x-polarized horizontal microstrip dipole (HED) embedded in a low-loss grounded dielectric slab of thickness d , permittivity E , and conductivity g. For a volume source the space-time expression of the electric field is

e(r, f ) = C (r, t 1 r’ , t ’ ) - j (r ‘ , t ’ ) dr‘ d t ‘ , (1) I where G(r, t I r‘, t ’ ) is the electric dyadic Green’s function, j(r’, t ’ ) is the volume current density, and r,t and r’ , t ’ are the field and source points in the space-time domain, respectively.

f

Figure 1 z = - h in a low-loss grounded dielectric slab

(a) Vertical z-polarized (VED), and (b) horizontal x-polarized microstrip dipole (HED) embedded at x = 0, y = 0. and

238 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 6, No. 4, March 20, 1993