case of the tetrahedral basis functions. The limitations im-posed on our procedure are the same as those imposed onthe method of moments, i.e., the speed and memory limita-tions of computers available today.

ACKNOWLEDGMENT

ŽThis work was supproted by the CNPq Conselho Nacional de.Desenvolvimento Cientıfico e Tecnologico and FAPESP´ ´

Ž .Fundacao de Amparo a Pesquisa do Estado de Sao Paulo .˜ ` ˜

REFERENCES

1. L.S. Mendes and S.A. Carvalho, Scattering of EM waves byhomogeneous dielectrics with the use of the method of momentsand 3D solenoidal basis functions, Microwave Opt Technol Lett

Ž .12 1996 , 327]331.2. D.R. Wilton and R. Mittra, A new numerical approach to the

calculation of electromagnetic scattering properties of two-dimen-sional bodies of arbitrary cross section, IEEE Trans Antennas

Ž .Propagat AP-20 1972 , 310]317.3. R.F. Harrington, Time-harmonic electromagnetic fields, McGraw-

Hill, New York, 1961.4. R.J. Schilling and H. Lee, Engineering analysis, Wiley, New York,

1988.5. J.J.H. Wang, Generalized moment methods in electromagnetics,

Wiley, New York, 1991.6. A.L. Aden and M. Kerker, Scattering of electromagnetic waves

Ž .from two concentric spheres, J Appl Phys 22 1951 , 1242]1246.7. C.A. Balanis, Advanced engineering electromagnetics, Wiley, New

York, 1989.

Q 1999 John Wiley & Sons, Inc.CCC 0895-2477r99

MATRIX INTERPOLATION TECHNIQUEIN THE CALCULATION OF MICROSTRIPANTENNA IMPEDANCEFan Yang1 and Xue-Xia Zhang11 Department of Electronic EngineeringTsinghua UniversityBeijing 100084, P.R. China

Recei ed 24 March 1999

ABSTRACT: The matrix interpolation technique is used for the analysisof microstrip antenna impedance. The positions and number of interpo-lation points are discussed. Some complicated antennas are used fordemonstration. The time of the simulation can be greatly reduced by thismethod. Q 1999 John Wiley & Sons, Inc. Microwave Opt TechnolLett 23: 46]49, 1999.

Key words: matrix interpolation; input impedance; microstrip antenna

INTRODUCTION

The microstrip antenna is now widely used because of itsmany advantages. Input impedance is an important parame-ter of a microstrip antenna. There are many methods tocalculate the impedance. Among them, the method of mo-

w xments is an effective and popular way 1 . However, whenusing the method of moments, the input impedance is ob-tained at a certain frequency. If a wideband impedancefeature is needed, the method of moments has to be usedagain and again: filling the matrices Z and V in every

Figure 1 Z-matrix elements varying with frequency

different frequency, and then calculating the impedance.Normally, it takes a long time to fill the matrices directlyfrom the integral equation. In this paper, a matrix interpola-tion technique is adopted. First, it calculates the Z- andV-matrix in some selected frequencies directly from the inte-gral equation. Then it fills the matrices in other frequenciesby interpolating them from those previously obtained. Be-cause the interpolation is faster than direct calculation, it cansignificantly reduce the simulation time of wideband analysis.

ANALYSIS OF THE MATRIX INTERPOLATION TECHNIQUE

Matrix Interpolation Technique. The method of moments iswidely used in the calculation of microstrip antennaimpedance. As an example, a microstrip antenna with lengths 48 mm, width s 48 mm, height s 4.8 mm and e s 1 isrcalculated. The antenna is fed by a coaxial probe at positionŽ . Ž .x, y s 36, 24 mm. In the method of moments, we use thetriangular surface function as the base function, and choose

w xthe test function to be the same as the base function 2 .Figure 1 shows some elements in the Z-matrix varying with

Ž . Ž . Ž .frequency. We choose elements Z 1, 1 , Z 1, 23 , Z 1, 130 asrepresentative because they demonstrate three different casesof the Z-matrix: the self-impedance, and the impedance ofdifferent base functions with middle distance and far dis-tance. Figure 2 shows some elements in the V-matrix varying

Ž . Ž . Ž .with frequency. We choose V 1 , V 39 , V 130 as representa-tives because they demonstrate different distances betweenthe feed point and base functions. From the figures, it can befound that the elements in the Z- and V-matrices varysmoothly with frequency. Thus, if the element value in somefrequencies is obtained, the element value in other frequen-cies is easy to obtain: just interpolated from those alreadycalculated.

In this paper, the polynomial function is used for interpo-Ž .lation. If N base frequencies f , f , . . . , f are selected and1 2 N

the element values in these frequencies are obtainedŽ Ž . Ž . Ž . Ž . Ž . Ž ..Z f , V f , Z f , V f , . . . , Z f , V f , themn 1 m 1 m n 2 m 2 m n N m N

Ž Ž . Ž ..element values in other frequencies Z f , V f can bemn m

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 23, No. 1, October 5 199946

Figure 2 V-matrix elements varying with frequency

calculated by the following formulas:

N

Ž .f y fŁ jNjs1, j/iŽ . Ž . Ž .Z f s Z f 1Ýmn mn i N

is1 Ž .f y fŁ i jjs1, j/i

N

Ž .f y fŁ jNjs1, j/iŽ . Ž . Ž .V f s V f . 2Ým m i N

is1 Ž .f y fŁ i jjs1, j/i

It should be emphasized that the matrices Z and V can beinterpolated because they are not affected by the resonanceeffect. As a comparison, matrix Y, the inversion of matrix Z,is not suitable for interpolation because of its resonance

w xcharacteristics 3 .

Positions of the Basic Frequency Points. To interpolate accu-rately, there are two things that should be considered: thepositions of the basic frequencies, and the number of thebasic frequencies. We first discuss the effect of the basicfrequency positions. The number of basic frequencies here isset to three, and different kinds of basic frequency positionsare analyzed. The frequency band is from 1 to 5 GHz. Thefirst and the last basic frequencies are chosen to be the startand stop frequencies of the frequency band. The middlefrequencies are chosen according to the following three for-mulas:

ŽŽ Ž . Ž .. . Ž .case 1: f s exp ln f q ln f r2 32 1 3

Ž . Ž .case 2: f s f q f r2 42 1 3

ŽŽ Ž . Ž .. . Ž .case 3: f s ln exp f q exp f r2 . 52 1 3

From the formulas, the middle frequencies of the three casesare 2.236, 3, and 4.325 GHz, respectively. Figure 3 shows theresults of three different interpolation cases and the directcalculation results. In the first case, the middle frequency isnear the start frequency. Thus, the impedance values be-tween the start frequency and the middle frequency are moreaccurate than those in other cases, but the values betweenthe middle frequency and the stop frequency are less accu-rate than those in other cases. On the other hand, the resultin the third case is just the opposite. If we are more con-cerned about the whole band and the resonant character, thesecond case is the most accurate.

Number of the Basic Frequency Points. Another question ishow many frequencies should be chosen as the basic frequen-cies. The more frequencies are chosen, the more accurate theresult will be. However, the calculation time will increase ifthe number of basic frequencies increases. Figure 4 shows acomparison between the direct calculation and the interpola-tion calculation. Three interpolation cases are analyzed: three

Ž . Ž .basic frequencies case 1 , four basic frequencies case 2 , andŽ .five basic frequencies case 3 . Here, the uniform distribution

of the basic frequencies is adopted. From the figure, it can be

Figure 3 Comparison of the direct calculation results and interpolation calculation results. The number of basic frequencies is three.Three kinds of basic frequency positions are compared.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 23, No. 1, October 5 1999 47

Figure 4 Comparison of the direct calculation results and interpolation calculation results. The uniform distribution of the basicfrequency position is adopted. The number of basic frequencies is three, four, and five, respectively

Figure 5 Calculation times of direct calculation and interpolationcalculation

found that the real parts of the input impedance are almostthe same, and the image parts of the input impedance have

Žsome discrepancies. The results of case 3 five-point interpo-.lation are almost the same as the direct calculation results,

Ž .and the results of case 1 three-point interpolation havesome discrepancies with the direct calculation results.

Figure 5 shows the simulation times in different cases. Itcan be observed from the figure that the matrix interpolationtechnique can significantly reduce the simulation time. Thefrequency band is from 1 to 5 GHz, and the frequency step is0.05 GHz. It takes 1463 s to directly calculate the entire bandcharacter. However, when the matrix interpolation techniqueis adopted, the calculation times are 155, 174, and 210 s,respectively. The basic frequency number increases, and thecalculation time increases. The time of the first case is about1r10 of the direct calculation time, and the time of the thirdcase is about 1r7 of the direct calculation time.

Some Complicated Results. To demonstrate the capability ofthe matrix interpolation technique, some additional compli-cated examples are discussed. A dual-frequency microstripantenna and a wideband microstrip antenna are analyzed.

Figure 6 Input impedance of a dual-frequency microstrip antennaŽdashed lines: directly calculated, 1042 s; dotted lines: interpolation

.calculated, 73 s

Figure 7 Input impedance of a broadband microstrip antennaŽdashed lines: directly calculated, 1174 s; dotted lines: interpolation

.calculated, 87 s

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 23, No. 1, October 5 199948

The uniform distribution of the basic frequency positions isadopted, and the number of the basic frequencies is set tofour. Figure 6 shows the results of the dual-frequency an-tenna, and Figure 7 shows the results of the broadbandantenna. From the figure, it can be found that the resultscalculated by the matrix interpolation technique show goodagreement with the results calculated directly. The calcula-tion times are also recorded. The times of the interpolationcalculations are greatly reduced from the direct calculations.

CONCLUSION

The matrix interpolation technique in the calculation ofmicrostrip antenna impedance is proposed in this paper. Thistechnique interpolates both the Z- and V-matrices in thewhole frequency band from those at some previously calcu-lated frequencies instead of directly filling them from theintegral equation. The positions of the basic frequencies arediscussed, and the uniform distribution has the most accurateresults. The number of basic frequencies is also analyzed.This technique can greatly reduce the simulation time inwideband analysis. Complex calculation results have demon-strated the capability of this technique.

REFERENCES

1. E.H. Newman and P. Tulyathan, Analysis of microstrip antennasusing moment methods, IEEE Trans Antennas Propagat AP-29Ž .1981 , 47]53.

2. S.M. Rao, D.R. Wilton, and A.W. Glisson, Electromagnetic scat-tering by surfaces of arbitrary shape, IEEE Trans Antennas Prop-

Ž .agat AP-30 1982 , 409]418.3. Y. Rahmat-Samii and M.A. Jensen, Personal communication an-

tennas: Modern design and analysis techniques including humaninteraction, IEEE AP-S Symp, short course, 1998.

Q 1999 John Wiley & Sons, Inc.CCC 0895-2477r99

A NOVEL MULTIRESOLUTIONAPPROACH TO THE EFIE ANALYSISOF PRINTED ANTENNASP. Pirinoli,1 L. Matekovits,1 and M. Sereno Garino1, *1 Dipartimento de ElettronicaPolitecnico di TorinoI-10129 Torino, Italy

Recei ed 30 March 1999

ABSTRACT: A new approach is presented for the integral-equationanalysis of printed antennas and arrays. It is based on the definition ofmultiresolution ¨ector functions with properties similar to those of thescalar wa¨elets; in particular, they are both spatially and spectrallylocalized. The resulting impedance matrix shows a good conditioning thatallows its sparsification and the use of iterati e methods for the solutionof the linear system. Numerical results for the case of electromagneticallycoupled square patches are presented. Q 1999 John Wiley & Sons, Inc.Microwave Opt Technol Lett 23: 49]51, 1999.

Key words: numerical methods; method of moments; printed antennas;multiresolution

1. INTRODUCTION

The use of multiresolution constructions has been increas-ingly applied to overcome the present limitations of the

*Present address: Omnitel S.p.A., I-10015 Ivrea, Italy.

Ž .integral equation}method of moments MoM approach tothe solution of electromagnetic problems, and notably, toallow for ‘‘sparsification’’ of the MoM matrices that are

w x Žtypically dense 2, 6 . However, while wavelet multiresolu-.tion approaches are already applied to integral equations for

Ž w x.scalar problems e.g., 1]5 , their application to vectorthree-dimensional problems in complex real-life geometries is

w xstill sporadic 6 . There are two main difficulties in this sense:the geometry of the structure, and the vector nature of thefields, which requires different representation along and

w xacross the field direction 7, 8 .In this work, a new approach is proposed, based on the

introduction of functions constructed using the concepts ofthe wavelet representation; not being directly wavelets, these

Ž .functions will be called multiresolution MR functions. Theygenerate a stable and robust representation that permits thesparsification of the impedance matrix and exhibits, like allwavelets, a certain degree of adaptivity. The stability of the

Ž .linear system conditioning makes convenient the use ofŽ .iterative solutions, e.g., the conjugate gradient CG method,

which permits us to handle a large number of unknowns forlarge structures like complex arrays.

w xIn 8, 9 , it has been shown that the matrix ‘‘regularization’’Ž .depends on the quasistatic terms, the solenoidal TE and the

Ž .nonsolenoidal quasi-TM, qTM in the following ones, thathave a different behavior; therefore, in order to obtain thisregularization, it is necessary to split the current into a TE

w xand a qTM part 8, 9 , and to use different bases for the twosubspaces. The two parts of the current can be represented interms of isotropic scalar functions that, unlike the scalarcomponents of the current, present the same properties of

w xregularity in all directions of space 7, 8 : the qTM part isrepresented by the charge, piecewise constant, while the TEpart is represented by a piecewise linear magnetic ‘‘potential’’

w xexpressed by scalar interpolating ‘‘pyramidal’’ functions 8 .For these reasons, we have defined two different sets of MRfunctions, based on isotropic scalar quantities.

2. DEFINITION AND PROPERTIES OF MR FUNCTIONS

The proposed scheme will be described for a structure ofarbitrary shape, conforming to a grid with rectangular cells,on which the interpolating vector linear functions, calledrooftops, are defined. The technique is not limited to thisparticular case, but its description is more cumbersome for atriangular mesh. The structure is first divided into the mini-mum possible number of rectangular subdomains, which formthe grid at the coarsest level, indicated by l s 0; in it, ‘‘wide’’rooftops are defined that serve as links between the subdo-mains. The MR basis functions are built inside each rectan-gular subdomain.

Ž .As far as the charge the qTM part is concerned, everydomain of level l s 0 is split into four cells, on which thethree functions, which represent the three possible combina-

w Ž .xtions with zero total charge, are built Fig. 1 a ; their directrepresentation in the rooftop basis generates the vector func-

w Ž .xtions for the current at level l s 1 Fig. 1 b . The threefunctions defined on each domain at the finer levels areobtained by the same technique. The functions for the TEpart are obtained via a discrete wavelet transformation of the‘‘pyramids’’ that define the basis. The sides of the domainsare divided into the desired number of cells, and on eachside, a scalar and unsymmetric Haar basis is defined, so thatthe two-dimensional wavelet transformation is obtained fromthe direct product of the two one-dimensional bases. The

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 23, No. 1, October 5 1999 49