REFERENCES

1. G.P. Gauthier, L.P. Katehi, and G.M. Rebeiz, W-band finite groundcoplanar waveguide (FGCPW) to Microstrip line transition, IEEEMTT-S Dig, 1998, pp. 107–109.

2. T.J. Ellis, J.P. Raskin, L.P. Katehi, and G.M. Rebeiz, A widebandCPW-to-microstrip transition for millimeter-wave packaging, IEEEMTT-S Dig, 1999, pp. 629–632.

3. A.M.E. Safwat, K.A. Zaki, W. Johnson, and C.H. Lee, Novel designfor coplanar waveguide to microstrip transition, IEEE MTT-S Dig,2001, pp. 607–610.

4. G.F. Engen and C.A. Hoer, Thru-reflect-line: An improved techniquefor calibrating the dual six port automatic network analyzer, IEEETrans Microwave Theory Tech 27 (1979), 987–993.

5. J.P. Mondal and T.-H. Chen, Propagation constant determination inmicrowave fixture de-embedding procedure, IEEE Trans MicrowaveTheory Tech 36 (1988), 706–713.

6. R.B. Marks, A multiline method of network analyzer calibration, IEEETrans Microwave Theory Tech 39 (1991), 1205–1215.

7. H.-J. Eul and B. Shieck, Robust algorithms for TXX VNA self-calibrations procedures, IEEE Trans Instrumentation and Measure-ment 43 (1994), 18–23.

8. F. Purroy and L. Pradell, New theoretical analysis of the LRRMcalibration technique for vector network analyzers, IEEE Trans Instru-mentation and Measurement 50 (2001), 1307–1314.

9. C. Wan, B. Nauwelaers, and W.D. Raedt, A simple error correctionmethod for two-port transmission parameter measurement, IEEE Mi-crowave Guided Wave Lett 8 (1998), 58–59.

10. A. Pham, J. Laskar, and J. Schappacher, Development of on-wafermicrostrip characterization techniques, ARFTG Conf, 1997, pp. 85–94.

11. W. Wiatr, Coplanar-waveguide-to-microstrip transition model, IEEEMTT-S Dig, 2000, pp. 1797–1800.

12. A.C. Ng, L.H. Chua, G.I. Ng, H. Wang, J. Zhou, and H. Nakamura,Broadband characterisation of CPW transition and transmission lineparameters for small reflection up to 94 GHz, Asia-Pacific MicrowaveConf, 2000, pp. 311–315.

13. J. Grzyb and G. Troster, Characteristic impedance de-embedding ofprinted lines with the probe-tips calibration, 32nd Euro MicrowaveConf Proc, 2002, pp. 1069–1072.

© 2003 Wiley Periodicals, Inc.

AN ITERATIVE METHOD FOR SOLVINGA LARGE DENSE MATRIX IN THEMETHOD OF MOMENTS SOLUTION OFAN ELECTROSTATIC PROBLEM

Ritu Singh and Surendra SinghDepartment of Electrical EngineeringThe University of TulsaTulsa, Oklahoma 74104

Received 8 May 2003

ABSTRACT: This paper illustrates the application of Wynn’s vector �-al-gorithm to solve a system of equations arising in the method of moments(MoM) solution of an electrostatic problem. Since the method is iterative, itdoes not require inversion of a matrix. The degree of accuracy of the solu-tion can be controlled by specifying a convergence factor. © 2003 WileyPeriodicals, Inc. Microwave Opt Technol Lett 39: 378–380, 2003;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.11223

Key words: integral equations; method of moments; iterative method;solution of dense matrix; Wynn’s vector algorithm

1. INTRODUCTION

Obtaining the numerical solution of electromagnetic radiation orscattering problems by using the method of moments (MoM) leadsto a system of linear equations, which can be solved by matrixinversion, Gaussian elimination, or iterative techniques. However,an inherent limitation of the MoM is that it results in the generationof a dense matrix, which makes the solution process very timeconsuming, specifically, for electrically large structures. To over-come this limitation and to extend the application of MoM to thehigh-frequency regime, several alternative techniques have beensuggested. The methods suggested so far include the multilevelfast-multiple algorithm [1], the impedance-matrix localization [2],the application of specially constructed adaptive-basis functions[3], and iterative methods such as the conjugate-gradient method[4] and the method of steepest descent [5].

In this work, we present an iterative technique for solving theMoM matrix problem. The method is based on the vector �-algo-rithm proposed by Wynn [6]. The application of the method isillustrated by applying it to find the charge distribution on aconducting strip using the MoM. Since this method is iterative,matrix inversion is not needed. The solution process can bestopped when a predefined convergence criterion is met. This isadvantageous, as higher accuracy in the solution can be obtainedby making the convergence criterion more stringent. The storagerequirements needed in the solution process are considerably mod-est. Typically, to solve an N � N matrix problem, the methodrequires a working space equivalent to two N � p matrices wherep � N. The numerical results presented here illustrate the useful-ness of the method. The advantage of Wynn’s vector �-algorithmover the biconjugate gradient method has been pointed out byBrezinski [7]. The biconjugate-gradient method’s limitation is thatit may have a breakdown, which would cause the algorithm tostop, or a near breakdown, which would cause the propagation ofround-off errors, resulting in a failure of convergence. In Wynn’salgorithm, such a breakdown or near breakdowns are curable.Brezinski has shown that under certain conditions, Wynn’s algo-rithm can be used to solve a system of linear equations, AX � b,where A is a real, singular, square matrix [6].

2. WYNN’S VECTOR �-ALGORITHM

Let us solve, say, the following matrix problem given by thesystem of linear equations:

AX � b, (1)

where A is a known N � N MoM matrix A � [aij], X is anunknown N � 1 column vector, X � ( x1, x2, . . . , xN)T, and bis a known N � 1 forcing function vector b � (b1, b2, . . . , bN)T.Wynn’s algorithm provides an estimate of X without the use ofmatrix inversion.

In the �-algorithm, the starting vectors {��1(0) } and {�0

(q)}, eachof length N, are constructed from the initial values:

���1�q� � � �0�, q � 1,2,. . . (2)

��0�0�� � �S�0�� � �0�,

��0�q�� � �S�q��, q � 1,2,. . . (3)

The initial vectors, S(q) � (s1(q), s2

(q), . . . , sN(q)), can be calculated

using the Gauss–Seidel relaxation [8]:

378 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 5, December 5 2003

si�q� � �bi � �

j�1

i�1

aijsj�q� � �

j�i�1

N

aijsj�q�1�� � aii,

i � 1,2, . . . Nq � 1,2, . . .

(4)

Once we have at least two initial vectors S(0) and S(1), then theWynn’s algorithm can be started to obtain (r � 1)th-order iterates,which are given by

�r�1,i�q� � �r�1,i

�q�1� �1

��r,i�q�1� � �r,i

�q��

q � 0, 1,2, . . .i � 1,2, . . . N (5)

The value of index r in the above equation takes on values of 0, 1,2, . . . and is bounded by one less than the maximum value of q. InEq. (5), the even-order vectors {�2

(q)}, {�4(q)}, . . . provide an

estimate of the solution vector X. To check for convergence, wedefine a convergence criterion. We calculate the two-norm of theerror vector AX � b, defined as

Norm � ��AX � b��2 � �AX � b�T�AX � b�1/ 2. (6)

This norm is compared with a predefined convergence factor Cf,which is also a measure of the error in the solution. The compu-tations are stopped when the following condition is met:

Norm � NCf, (7)

where N is the number of unknowns or elements of solution vectorX. The above condition will, to a great extent, guarantee that eachelement of the solution vector X converges to within the conver-gence factor Cf. The maximum value of q in Eq. (5) at which theconvergence criterion is met is termed qmax. Note that the value ofqmax signifies the number of initial vectors, defined in Eq. (4),upon which Wynn’s algorithm is applied. The storage require-ments and computational efforts are directly proportional to qmax.

3. NUMERICAL RESULTS

The algorithm is applied to the computation of charge distributionon a conducting strip. Consider a conducting strip of width 2W,which is charged to a known variable potential V( y). The strip isflat, vanishingly thin, and infinitely long. The integral equation forcharge per unit length u( y) is given by

�1

2��� ��W

W

u�y��ln�y � y��dy� � V�y�, y � ��W, W�. (8)

For simplicity, we take the permittivity of the medium �� � 1 F/m.We solve the above integral equation using pulse expansion and

Figure 1 Charge distribution on a conducting strip of width 2W � 1 mfor N � 50, Cf � 10�3, and V( y) � 1

Figure 2 Charge distribution on a conducting strip of width 2W � 1 mfor N � 100, Cf � 10�3, and V( y) � y

TABLE 1 Comparison of Wynn’s Algorithm for Different Convergence Factors with Matrix Inversion for N � 10

Pulse #

Wynn’s Algorithm

Matrix InversionCf � 10�2 Cf � 10�3 Cf � 10�4 Cf � 10�5 Cf � 10�6

1 8.836853692 8.742591917 8.761604450 8.758527744 8.758447351 8.7584545112 4.026968784 3.974868299 3.985881092 3.984095629 3.984062018 3.9840640183 3.428582465 3.376733167 3.387585696 3.386014902 3.386020519 3.3860159444 3.097261698 3.043013437 3.053556832 3.052507116 3.052505609 3.0525194885 2.966479502 2.910922897 2.921452299 2.921509946 2.921362707 2.9213631536 2.958942335 2.890926600 2.921119462 2.921351269 2.921368291 2.9213631537 3.060812836 3.068652163 3.056719505 3.052523488 3.052519439 3.0525194888 3.316275866 3.435567415 3.385046850 3.386014500 3.386015968 3.3860159449 4.036262227 3.980553798 3.984339815 3.984064747 3.984064001 3.984064018

10 8.753323516 8.759353075 8.758862845 8.758454032 8.758454539 8.758454511

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 5, December 5 2003 379

point matching to arrive at a system of linear equations of the typegiven in Eq. (1). We take 2W � 1 m.

For the first example, the strip is at a constant potential of 1 V,that is, V( y) � 1 in Eq. (8). The number N of pulse expansionfunctions used is 10. The results obtained from Wynn’s algorithmfor various convergence factors are shown in Table 1. For com-parison, the results obtained from solving the system of equationsusing matrix inversion are also given. We find that the computa-tional accuracy of the results obtained from Wynn’s algorithmimproves progressively as the convergence criterion becomesstringent. A closer examination of the results indicates that for aconvergence factor of 10�k, we obtain roughly k digits of accu-racy.

The storage requirements for implementing the algorithm aremodest. For Cf � 10�2 in the above example, we need twomatrices of order N � qmax and N � (qmax � 1), where N � 10and qmax � 5. Figure 1 shows the charge distribution, u( y), forV( y) � 1 V, Cf � 10�3, and N � 50. Results for variableexcitation V( y) � y and V( y) � y2 for Cf � 10�3 are presentedin Figures 2 and 3, respectively. The number of unknowns is N �100 for the result in Figure 2 and N � 200 for that in Figure 3.We find that in each case the results obtained from Wynn’s vector�-algorithm agree very well with those obtained from the matrixinversion. We have run the code for N � 1000 and Cf � 10�3.For this case, qmax came out to be 11. That is, only 11 initialvectors S(0), S(1), . . . , S(10) were needed to obtain the solution.The results were in agreement with those obtained from the matrixinversion.

4. CONCLUSION

We have applied Wynn’s vector �-algorithm to solve a system oflinear equations obtained in an MoM problem. The results shownfor the computation of charge distribution on a conducting strip,for various values of expansion function, indicate very good agree-ment with the matrix inversion method. Since this method isiterative, it does not require inversion of a matrix, and therebyallows the applications of method of moments to solve electricallylarge structures. It has been reported in the literature that thismethod may be utilized for obtaining a solution even when asingular or ill-conditioned matrix is involved. This is a topic forfurther research and investigation. Wynn’s algorithm method also

provides flexibility in obtaining different degrees of accuracy byspecifying a convergence factor. Currently, work is in progress toapply the algorithm for solving the system of equations arising inthe MoM solution of 2-D and 3D problems.

REFERENCES

1. J.M. Song and W.C. Chew, Multilevel fast-multipole algorithm forsolving combined field integral equations for electromagnetic scatter-ing, Microwave Opt Technol Lett 10 (1995), 14–19.

2. F.X. Canning, Improved impedance matrix localization method, IEEETrans Antennas Propagat 41 (1993), 659–667.

3. M.L. Waller and S.M. Rao, Application of adaptive basis functions fora diagonal moment matrix solution of arbitrarily shaped three-dimen-sional conducting body problem, IEEE Trans Antennas Propagat 50(2002), 1445–1452.

4. T.K. Sarkar and S.M. Rao, The application of the conjugate gradientmethod for the solution of electromagnetic scattering from arbitrarilyoriented wire antennas, IEEE Trans Antennas Propagat 32 (1984),398–403.

5. T.K. Sarkar and S.M. Rao, An iterative method for solving electrostaticproblems, IEEE Trans Antennas Propagat 30 (1982), 611–616.

6. C. Brezinski and A.C. Rieu, The solution of systems of equations usingthe �-algorithm, and an application to boundary-value problems, MathComputation 28 (1974), 731–741.

7. C. Brezinski and M.R. Zaglia, Extrapolation methods-theory and prac-tice, North-Holland, Amsterdam, 1991, 306.

8. G. Golub and C.F. Van Loan, Matrix computations, Johns HopkinsUniversity Press, Baltimore, MD, 1983, 354.

© 2003 Wiley Periodicals, Inc.

CONFIGURATION OF A STEPPEDIMPEDANCE BANDPASS FILTER

Xuedong Wang, Bin You, Wusheng Ji, and Ying LiDepartment of Communication and Information EngineeringShanghai UniversityShanghai, P. R. China

Received 3 May 2003

ABSTRACT: This paper describes the design of a compact stripline ce-ramic filter, using short-circuit stepped-impedance resonators (SIRs) to en-hance the harmonic stop-band response. Moreover, this filter has a verysmall size of 3.6 mm � 3.0 mm � 0.5 mm and a low insertion loss of 0.6dB in the passband. The filter’s response, with a design center frequency of2390 MHz and 3-dB bandwidth at 250 MHz (10.4%), shows an increase inthe first harmonic frequency to 9004 MHz. At three times the fundamentalpassband frequency (7160 MHz), stop-band attenuation is better than �40dB. The results demonstrate the usefulness of SIRs for improving the stop-band performance of bandpass filters. © 2003 Wiley Periodicals, Inc.Microwave Opt Technol Lett 39: 380–383, 2003; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11224

Key words: stripline filters; bandpass filters; stepped-impedance reso-nators

1. INTRODUCTION

The filter, which can select a desired band of frequencies, is one ofthe most important components in communication systems. Com-munications services and cellular markets have placed high de-mands on filter manufactures, including the need for low insertionloss, small size, and good harmonic rejection. A common filtertopology has quarter-wave uniform impedance resonators (UIRs)[1]. While the response of this filter is adequate for many appli-

Figure 3 Charge distribution on a conducting strip of width 2W � 1 mfor N � 200, Cf � 10�3, and V( y) � y2

380 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 5, December 5 2003