04380569

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 11, NOVEMBER 2007 3093

of the antipodal Vivaldi antenna in [9] and the single layered printeddipole in [6].

IV. CONCLUSION

The design of a novel broadband double–layered printed dipole an-tenna with low cross-polarization level is presented. As compared tothe conventional single-layered dipole antenna, an extra layer of sub-strate with printed dipole is added to counteract the cross-polarizationcomponent. Experimental results show that the new dipole antennaachieves a bandwidth over 50% while reducing the cross-polarizationlevel to less than �30 dB within the frequency band. The double-lay-ered printed dipole antenna is easy to be expanded into a large-scalearray and is suitable for the applications with low cross-polarizationrequirement.

ACKNOWLEDGMENT

Z. Zhou would like to thank Mr. X. Zong for his constructive discus-sion and assistance during the experiment. The authors would also liketo acknowledge the anonymous reviewers, whose comments enhancedthe quality of this paper considerably.

REFERENCES

[1] K. Fujimoto and J. R. James, Mobile Antenna System Handbook.Boston, MA: Artech House, 1994.

[2] J. R. Bayard, M. E. Cooley, and D. H. Schaubert, “Analysis of infinitearrays of printed dipoles on dielectric sheet perpendicular to a groundplane,” IEEE Trans. Antennas Propag., vol. AP-39, pp. 1722–1732,Dec. 1991.

[3] E. Levine, S. Shtrikman, and D. Treves, “Double-sided printed arrayswith large bandwidth,” Proc. Inst. Elect. Eng. Microw. AntennasPropag., vol. 135, pp. 54–59, Feb. 1988.

[4] B. Edward and D. Rees, “A broad-band printed dipole with integratedbalun,” Microw. J., pp. 339–344, May 1987.

[5] F. Tefiku and C. A. Grimes, “Design of broad-band and dual-band an-tennas comprised of series-fed printed-strip dipole pairs,” IEEE Trans.Antennas Propag., vol. AP-48, pp. 895–900, Jun. 2000.

[6] U. K. Revankar and H. Harishchandra, “Printed dipole radiating ele-ments for broadband and wide scan angle active phased array,” in Proc.IEEE Antennas Propagation Soc. Symp., Boston, MA, Jul. 2001, pp.796–799.

[7] B. G. Duffley, G. A. Morin, M. Mikavica, and Y. M. M. Antar, “Awide-band printed double-sided dipole array,” IEEE Trans. AntennasPropag., vol. AP-52, pp. 628–631, Feb. 2004.

[8] [Online]. Available: http://www.ansoft.com/[9] J. D. S. Langley, P. S. Hall, and P. Newham, “Balanced antipodal Vi-

valdi antenna for wide bandwidth phased arrays,” Proc. Inst. Elect. Eng.Microw. Antennas Propag., vol. 143, pp. 97–102, Apr. 1996.

[10] [Online]. Available: http://www.parkelectro.com/

A Magnetic Frill Source Model for Time-DomainIntegral-Equation Based Solvers

Nan-Wei Chen

Abstract—A magnetic frill source model developed for time-domainintegral-equation-based solvers for accurate characterization of coaxiallydriven structures is presented. The MFS model is developed by repre-senting the coax as a magnetic frill current, and the evaluation of theexpressions for fields radiated by this current does not call for numericaldifferentiation of any kind. Also, the expression for the axial componentof the electric field only calls for one-dimensional numerical integra-tion. For model validation and demonstration of efficacy, the developedsource model is incorporated into a marching-on-in-time (MOT) schemefor broadband characterization of coaxially-driven antenna structurescomprised of arbitrarily shaped perfect electrically conducting surfacesand thin wires. The numerical results obtained from the MOT schemeincorporated with the developed source model are compared with valuesobtained by measurement, and with the traditionally used delta gap sourcemodel. Compared to the delta gap source model, the magnetic frill sourcemodel provides a relatively accurate characterization of the antennastructures over a wide band of frequency.

Index Terms—Delta gap source (DGS), magnetic frill source, marching-on-in-time (MOT), time-domain integral equation (TDIE).

I. INTRODUCTION

Mainly because of its simplicity in numerical implementation, thedelta gap source (DGS) model has been predominately used in fre-quency-domain integral equation (FDIE) [1]–[3] or time-domain in-tegral equation (TDIE) based solvers [4], [5] for representing the feedof coaxially driven structures, such as a monopole antenna mounted onan infinite ground plane. However, the DGS model is unable to takeinto account the capacitive loading effect of the gap and to generatenonzero field distribution in the vicinity of the aperture. These non-physical properties lead to inaccurate determination of the current dis-tribution around the feed. Hence, the evaluated input impedance of an-tenna structures is rendered inconsistent with the measured impedance,particularly at frequencies higher than their resonant ones [1], [2]. Var-ious methods, such as extended DGSs, have been proposed in [1] toalleviate these nonphysical properties. However, as pointed out in [1],the extended DGS model was limited by the assumption that only theTEM mode exists at the coaxial aperture, which leads to inconsistentinput impedance of a complex antenna structure.

To tackle the nonphysical issues, the magnetic frill source (MFS)model was originally proposed in [6] to provide a more accurate repre-sentation of the coax. In essence, the MFS model interprets the coax asan annular frill of magnetic current by exploiting the equivalence prin-ciple. Therefore, fields in close vicinity of the feed structure, which aresignificantly related to determination of the current distribution aroundthe feed, can be accurately represented as near-zone fields from the frillcurrent. Several studies have been performed to further improve the ac-curacy and efficiency associated with the numerical evaluation of thefields from the frill current distribution [7]–[9]. It appears that a sourcemodel of this sort is not restricted to the aforementioned TEM modeassumption. Technically speaking, the fields due to higher-order modefrill currents can be included in the same manner for modeling the ex-citation of the fundamental TEM mode. On the other hand, conjunction

Manuscript received February 8, 2007.The author is with the Department of Electrical Engineering, National Central

University, Jhongli 320, Taiwan, R.O.C. (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2007.908577

0018-926X/$25.00 © 2007 IEEE

3094 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 11, NOVEMBER 2007

of the MFS model and FDIE based solvers [10]–[12] has been shownto provide a relatively accurate evaluation of the input impedance ofthe coaxially-driven antenna structures, not only at their resonant fre-quencies but also at other frequencies of the band of interest. Contraryto its frequency-domain (FD) counterpart, to the best of the author’sknowledge, to date there has been no report in the literature on work onthe formulation of stable, computationally efficient time-domain (TD)field expressions for an MFS model. Indeed, the TDIE-based solverscall for this kind of source model to realize broadband, accurate char-acterization of the aforementioned structures in a single simulation.

Here, the derivation of numerical differentiation-free TD expres-sions for the field from the MFS, together with its incorporation into aTDIE-based solver for analyzing coaxially fed antennas, is presented.Specifically, the troublesome numerical differentiation is avoided in thederivation since it is often considered as a cause of instability that af-fects numerical techniques, such as the TDIE-based solvers. In addi-tion, the method that was originally proposed in [9] is adopted hereinto obtain a computationally efficient expression for the axial compo-nent of the electric field. For validation, the presented MFS model isincorporated into a marching-on-in-time (MOT) scheme [13]–[15] foranalyzing coaxial-driven antenna structures comprised of perfect elec-trically conducting (PEC) surfaces and wires. More specifically, theMFS-incorporated MOT scheme is applied to solve the TD electricfield integral equations pertinent to the analysis of two antenna struc-tures, viz. a quarter-wavelength monopole antenna mounted on a finiteground plane and a planar sector antenna. Both antennas were fabri-cated and measured. The accuracy and efficacy of the MFS-incorpo-rated MOT scheme is demonstrated by comparing the calculated inputimpedance and return loss of each antenna with the measured values,and to those obtained with the DGS model.

The paper is organized as follows. Section II details the derivationof the expressions for fields from the MFS. In addition, the devel-opment of the MOT scheme together with its conjunction with theMFS model is outlined. The numerical results demonstrating the ac-curacy and efficacy of the MFS-incorporated MOT scheme are pre-sented in Section III. Section IV summarizes our conclusions and fu-ture research.

II. FORMULATION

This section details the derivation of the expressions of the electricand magnetic fields radiated by a time-varying MFS. In addition, thedevelopment of the MOT scheme and its incorporation with the MFSmodel are described. In what follows, Section A contains a derivationof the TD integral expressions for field evaluation; Section B presentsthe development of the MOT scheme along with the incorporation ofthe MFS model into the MOT scheme.

A. Electric and Magnetic Fields From a Time-Varying MFS

Consider a free-space magnetic frill current of inner and outer radii,a and b, respectively, located on the plane z = zp (Fig. 1). The centerof the frill is on the z axis. The current distribution on the frill surfaceSfrill is of the form

M(r0; t) = u0

�v(t)

�0 ln(b=a)(1)

where the prime denotes source coordinates and v(t) represents its tem-poral signature. In Fig. 1, the electric vector potential F(r; t) observedat an observation point P (�; �; z) is

F(r; t) ="04� S

ds0M(r0; � )

R(2)

Fig. 1. MFS geometry.

where "0 is the free-space permittivity, � = t � R=c denotesdelayed time, c is the speed of light, and R = jr � r0j =[�2 + �02 + (z � zp)

2 � 2��0 cos(�� 0)]1=2 is the distance be-tween the source and observation points.

With the aid of (2), the electric field radiated by the current sourcecan be obtained from the expression

E(r; t) = �1

"0r� F(r; t): (3)

The corresponding components of the electric field are written as

Ez(r; t) =�1

"0

F�(r; t)

�+

@F�(r; t)

@�(4)

and

E�(r; t) =1

"0

@F�(r; t)

@z: (5)

In (4) and (5), the termsF�; @F�=@�, and @F�=@z associated with theevaluation of the electric field can be explicitly written as

F�(r; t)

="0

4� ln(b=a)

�b

� =a

2�

� =0

d�0d�0 cos(�� 0)v(�)

R(6)

@F�(r; t)

@�

="0

4� ln(b=a)

�b

� =a

2�

� =0

d�0d�0 cos(�� 0)@

@R

v(�)

R

@R

@�(7)

and@F�(r; t)

@z

="0

4� ln(b=a)

�b

� =a

2�

� =0

d�0d�0 cos(�� 0)@

@R

v(�)

R

@R

@z(8)

respectively. As shown in (4), the evaluation of Ez calls for the sum-mation of F� and @F�=@�. Similar to the formulation proposed in [9],integration by parts is first employed to rewrite F�(r; t) as

F�(r; t) ="0

4� ln(b=a)

�b

� =a

2�

� =0

d�0d�0 sin(�� 0)@

@R

v(�)

R

@R

@�0: (9)


As a result, the expression for F�(r; t) can be rewritten in a formsimilar to @F�(r; t)=@�. This manipulation significantly simplifies theevaluation of Ez , as shown below. First, the spatial derivatives of R in(7) and (9) are, respectively

@R

@�=

� � �0 cos(�� 0)

R(10)

and@R

@�0= �

��0 sin(�� 0)

R: (11)

Next, following the substitution of (10) and (11) into (7) and (9) respec-tively, one sums up the resulting F� and @F�=@� and finally obtains

Ez(r; t) =1

4� ln(b=a)

b

� =a

2�

� =0

d�0d�0

�@

@R

v(� )

R

�0� � cos(�� 0)

R

=1

4� ln(b=a)

�

b

� =a

2�

� =0

d�0d�0 @

@R

v(�)

R

@R

@�0

=1

4� ln(b=a)

�

2�

0

d�0 v(�)

R� =b

�v(�)

R� =a

: (12)

Obviously, since it only calls for one-dimensional integration, (12) iscomputationally more efficient than the one obtained by directly sum-ming up (6) and (7) without using the manipulation employed in (9).Moreover, the evaluation of (12) does not call for numerical differenti-ation of any kind.

As for the expression of E�, the spatial derivative @[v(� )=R]=@R in(8) is switched to a time derivative as

@

@R

v(�)

R= �

1

cR

@v(�)

@t+

v(�)

R2: (13)

Substituting (13) and

@R

@z=

z � zpR

(14)

into (8), one obtains

E�(r; t) =�1

4� ln(b=a)

b

� =a

2�

� =0

d�0d�0

�(z � zp) cos(�� 0)

R2

1

c

@v(�)

@t+

v(�)

R: (15)

Similarly, no numerical differentiation is required in the calculation ofE�(r; t) since @v(�)=@t in the bracket can be evaluated analyticallyupon the specification of v(t). Finally, the only component of the mag-netic field is

H� = �@

@tF�(r; t) =

�"04� ln(b=a)

�

b

� =a

2�

� =0

d�0d�0 cos(�� 0)

R

@v(�)

@t: (16)

Fig. 2. The PEC structure that is comprised of arbitrarily shaped surfaces andthin wires.

Fig. 3. Pictorial description of the coax modeling.

Fig. 4. Geometry and dimension of the monopole antenna.

In summary, as shown in (12), (15), and (16), all the integrandscontain no spatial derivatives and only call for instantaneous and timederivative values of the current distribution.

B. Incorporation of the MFS Model Into the MOT Scheme

The above field expressions are used in the MOT scheme presentedbelow for accurately modeling the distributed excitation of coaxiallydriven structures. Here, the development of the MOT scheme is firstoutlined, after which the incorporation of the MFS model into the MOTscheme is described.

Consider a PEC body S (Fig. 2) that resides in free space and thatcomprises of arbitrarily shaped surfaces and thin wires. Assume that Sis excited by an electric field Ei(r; t)that is of the temporal spectrumvanishing for frequencies f > fmax and that Ei(r; t) (approximately)vanishes on S for t < 0.

The current density J(r; t) induced on S generates the scatteredelectric field Es(r; t) given by

Es(r; t) = �

@

@tA(r; t)�r�(r; t): (17)

The vector and scalar potentials A(r; t) and �(r; t) are expressed as

A(r; t) =�04� s

ds0 J(r0; � )

R(18)


Fig. 5. (a) The transient current at the feed. (b) Simulated and measured returnloss of the monopole antenna. (c) Simulated and measured real part of the inputimpedance of the monopole antenna. (d) Simulated and measured imaginarypart of the input impedance of the monopole antenna.

and

�(r; t) =1

4�"0 s

ds0 �(r0; � )

R: (19)

In (18) and (19), �0 is the free-space permeability and �(r; t) denotesthe charge density on S. The current and charge densities on S arerelated by the continuity equation

r � J(r; t) +@�(r; t)

@t= 0: (20)

Enforcing the temporal derivative of the total electric field Ei(r; t) +Es(r; t) tangential to S to vanish leads to the following TD integro-

differential equation for J(r; t)

@

@tEitan(r; t) =

@2

@t2�04� s

ds0J(r0; � )

R

�1

4�"0r

s

ds0r0 � J(r0; � )

Rtan

r 2 S (21)

where the subscript “tan” refers to vector components tangential to S.To solve (21) numerically by a classical MOT procedure, S is ap-

proximated by a relatively uniform mesh of planar triangular facetswith controlled aspect ratio for surfaces and a set of straight wire seg-ments for wires. Let Ns denote the total number of spatial basis func-tions defined on the discretized surfaces and wires, and let Nt denotethe total number of time steps for the simulation. It appears that J(r; t)is approximated as

J(r; t) �=

N

l=0

N

i=1

Ii;lf�i (r)Tl(t); � = s; w; sw: (22)

In (22), Ii;l are unknown expansion coefficients; f si (r); fwi (r), and

fswi (r) denote the Rao–Wilton–Glisson basis function [16] associated

with the triangular facets, the wire basis function [4] defined on thewire segments, and the junction basis function [17] associated with thesurface-wire junctions, respectively; Tl(t) = T (t� l�t) is a temporalbasis function. In this study,T (t) is the prolate-spheroidal function pro-posed in [18]. Also, an implicit scheme is employed, and the time stepsize is chosen as �t = 1=(�fmax), with 10 � � � 20, i.e., currentdensities are sampled 5 to 10 times over the Nyquist rate and indepen-dent of the geometrical discretization. Substituting (22) into (21), ap-plying spatial Galerkin testing at time t = tl = l�t, and extrapolatingthe future currents caused by the noncausal temporal basis functionsusing the present and past currents [18] yield a system of equationsthat can be represented in matrix form as

�Z0Il = Vl �

l�1

m=1

�ZmIl�m (23)

where Il is a vector of unknown current coefficients fIi;lg; �Zm is anNs by Ns matrix that describes fields generated by a space-time basisfunction associated with the (l�m)th time step and measured at timetl, andVl is a vector that measures the time derivative of the impressedfield at time tl. More specifically, the elements of theNs-vector Vl =[Vl;j ]j=1;N are

V �

l;j = �S

dsf �j (r)@

@tEitan(r; t) j t=t ; � = s; w; sw: (24)


Fig. 6. Geometry and dimension of the sector antenna.

As for the coaxially driven structures, the incident fieldEi(r; t) is fromthe MFS and its time derivative can be found simply from (12) and (15)as

@Ez(r; t)

@t

=1

4� ln(b=a)

2�

0

d�0

�

1

R

@v(� )

@t� =b

�

1

R

@v(�)

@t� =a

(25)

and

@E�(r; t)

@t

=�1

4� ln(b=a)

b

� =a

2�

� =0

d�0d�0

�

(z � zp) cos(�� 0)

R2

1

c

@2v(�)

@t2+

1

R

@v(�)

@t: (26)

Finally, (23) can be solved by a standard MOT scheme to obtain currentvectors for all time steps. Since �Z0 is not diagonal, a nonstationaryiterative solver such as transpose-free quasi-minimal residual is usedto solve the system. Note that the vector Vl is updated at every timestep. Hence, the efficiency regarding the evaluation of the elements ofVl should significantly affect the efficiency of the MFS-incorporatedMOT scheme.

III. MODEL VALIDATION

In what follows, the MFS-incorporated MOT scheme is applied toanalyze two coax-fed antenna structures. In both examples, the coax fedis modeled as the magnetic frill current with the ratio b=a = 2:3, whichcorresponds to a 50 coax. The signature of the voltage is a Gaussianpulse parameterized as v(t) = cos(2�fct) exp[�(t� tp)

2=(2�2)],where fc is the center frequency of the signal, � = 6=(2�fbw), andtp = 8� with fbw termed the “bandwidth” of the signal. In both ex-amples fc = 2 GHz and fbw = 1 GHz are used, together with thetime step �t = 0:022 ns and the number of time steps Nt = 2048. Asdepicted in Fig. 3, the coaxial aperture is sealed with the PEC surfaceand the MFS is placed right on the sealed surface in the analysis. Theinduced current on the structure is first obtained via the MOT scheme,and the input impedance of the antenna is then calculated using the vari-ational formula proposed in [19]. Both antenna structures were fabri-cated and measured for validation. For each antenna structure, the cal-culated input impedance and return loss over a range of frequency arecompared with the measured values. Also, the values obtained with theDGS model are compared.

The first example is a quarter-wavelength monopole antennamounted on a finite square ground plane (Fig. 4). As depicted in Fig. 4,the side length D of the ground plane is 30 cm. The length L of thewire is 3.25 cm. The monopole is modeled as a thin wire of radius0.5 mm. The current on the whole structure is described in terms ofNs = 4727 spatial unknowns.

Fig. 7. (a) The transient current at the feed. (b) Simulated and measured returnloss of the sector antenna. (c) Simulated and measured real part of the inputimpedance of the sector antenna. (d) Simulated and measured imaginary part ofthe input impedance of the sector antenna.

The magnitude of the current at the surface-wire junction located atthe center of the ground plane is shown in Fig. 5(a), which demonstratesa stable, convergent current at the feed. In Fig. 5(b)–(d), the return loss,


real and imaginary parts of input impedance of the monopole antennaare respectively compared with the measured values and with thoseobtained with the DGS model.

Fig. 5(b)–(d) show a reasonably good correspondence between theresults obtained from the MFS model and measurement within thewhole band of interest. In Fig. 5(b), the calculated resonant frequency(2.02 GHz) obtained from the MFS model is slightly higher than themeasured one (1.99 GHz). In contrast, those obtained by the DGSmodel agree very well with the measurement at the lower frequenciesof the band, but are off from the measured values at frequencies higherthan the resonant frequency. The discrepancy arises from the low fre-quency assumption used in the DGS model.

The second example is a planar sector antenna (Fig. 6). As depictedin Fig. 6, the side length D of the square ground plane is 30 cm. Theside lengthS of the sector is 10.6 cm and its expanded angle is 90�. Thesector is 2 cm above the ground plane and is connected to the groundplane with a short wire. Again, the wire is modeled as a thin wire ofradius 0.5 mm in the analysis. The current on the whole structure isdescribed in terms of Ns = 5017 spatial unknowns.

The magnitude of the current at the surface-wire junction locatedat the center of the ground plane is shown in Fig. 7(a). Again, astable, convergent current at the feed is observed. The return loss,and the real and imaginary parts of the input impedance of the sectorantenna are compared with the measured values, together with thoseobtained with the DGS model in Fig. 7(b)–(d), respectively. In thisexample, the numerical results obtained by the MFS model are also ingood agreement with the measured ones. In Fig. 7(b), the calculatedresonant frequency (1.825 GHz) obtained from the MFS model isslightly higher than the measured one (1.775 GHz). Similarly, thevalues obtained by the DGS model are in good agreement withmeasurement at the frequencies lower than the resonant frequency,but are off from the measurement at the frequencies higher than theresonant frequency.

IV. CONCLUSION

An MFS model poised to be incorporated into TD integral-equa-tion-based solvers for accurate, broadband characterization of coaxi-ally driven structures is presented. The demonstrated expressions forfields from the MFS are free of numerical differentiation. The effi-cacy of the MFS model is demonstrated through the application ofthe MFS-incorporated MOT scheme to the analysis of two coax-fedantenna structures. Compared with the traditionally used DGS model,the MFS model provides a relatively accurate evaluation of the inputimpedance of antenna structures over a wide frequency range. Futureworks will focus on analysis of the antennas installed in complex plat-forms using the proposed source model with the inclusion of higherorder mode currents at the feed.

REFERENCES

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[2] G. P. Junker, A. W. Glisson, and A. A. Kishk, “Accurate impedancemodel for antiresonant monopoles on finite ground planes,” ElectronLett., vol. 32, no. 18, pp. 1632–1633, Aug. 1996.

[3] H. Jeuland, B. Uguen, G. Chassay, and E. Grorud, “Numerical andexperimental processes for the analysis of a wire antenna mounted on ametallic body,” Microw. Opt. Technol. Lett., vol. 15, no. 5, pp. 267–273,Aug. 1997.

[4] K. Aygün, S. E. Fisher, A. A. Ergin, B. Shanker, and E. Michielssen,“Transient analysis of multielement wire antennas mounted on arbi-trarily shaped perfectly conducting bodies,” Radio Sci., vol. 34, no. 4,pp. 781–796, Aug. 1999.

[5] K. Aygün, B. Shanker, A. A. Ergin, and E. Michielssen, “A two-levelplane wave time-domain algorithm for fast analysis of EMC/EMI prob-lems,” IEEE Trans. Electromagn. Compat., vol. 44, no. 1, pp. 152–164,Feb. 2002.

[6] G. E. Albert and J. L. Synge, “The general problem of antenna radiationand the fundamental integral equation, with application to an antennaof revolution, Part. I,” Quart. Appl. Math., vol. 6, no. 2, pp. 117–131,Jan. 1948.

[7] D. V. Otto, “The admittance of cylindrical antennas driven from acoaxial line,” Radio Sci., vol. 2, no. 9, pp. 1031–1042, Sep. 1967.

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[12] I. R. Capoglu and G. S. Smith, “The input admittance of a prolate-sphe-roidal monopole antenna fed by a magnetic frill,” IEEE Trans. An-tennas Propag., vol. 54, no. 2, pp. 572–585, Feb. 2006.

[13] R. Mittra, , L. B. Felsen and C. E. Baum, Eds., “Transient electro-magnetic fields,” in Topics in Applied Physics. New York: Springer-Verlag, 1976, vol. 10.

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[15] S. M. Rao and D. R. Wilton, “Transient scattering by conducting sur-faces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 39, no.1, pp. 56–61, Jan. 1991.

[16] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scat-tering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag.,vol. 30, no. 3, pp. 409–418, May 1982.

[17] M. F. Costa and R. F. Harrington, “Minimization of radiation fromcomputer systems,” in Proc. Int. Electrical Electronics Conf., Toronto,Canada, Sep. 1983, pp. 660–665.

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[19] H.-Y. Chao and W. C. Chew, “Quasi-static analysis of fringe capaci-tances for horizontal and vertical annual frills,” Microw. Opt. Technol.Lett., vol. 33, no. 1, pp. 61–64, Apr. 2002.

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