Analysis of a class of symmetrical thin-platetriangular antennas

Prof. B.D. Popovic, DScB.M. Kolundiija, MSc

Indexing terms: Antennas {theory), Antennas {wire), Numerical analysis

Abstract: A method is presented for the analysisof a class of symmetrical thin-plate triangularantennas. The antenna current distribution isobtained by solving the two-potential equation bymeans of the point-matching method with a poly-nomial approximation for the current distribution.The edge effect is taken into account by intro-ducing line currents at the edges. To obtain anaccurate and stable solution, point-matchingequations and conditions for currents at the dis-continuities are used to form an overdeterminedsystem of complex linear equations which is thensolved in the least-squares sense. Theoretical andexperimental results are presented, showing goodagreement and illustrating the antenna properties.

1 Introduction

Composite wire and plate planar antennas, i.e. antennasassembled from thin, flat, metallic plates and wires situ-ated in one plane, have a number of useful properties,namely light weight, simple construction, low aero-dynamic profile, etc. For example, they are used frequent-ly as aircraft antennas.

It appears that the design of such antennas has beendone mostly experimentally. However, modern designprocedures tend to be computer oriented. This paper isaimed at developing a numerical method for the analysisof a class of antennas of the type described.

Antennas dealt with in this paper belong to a class ofradiating systems consisting of highly conducting bodiesthe dimensions of which do not exceed greatly the wave-length of the radiated wave. Most approaches for theanalysis of such systems have used the integral-equationformulation in conjunction with the method of moments.In early works in this area the body surface was modelledas a wire mesh (the so-called wire-grid model [1]).However, such modelling is not well suited for calcu-lating quantities such as the near field, surface currentsand antenna impedance [2]. As shown in recent papersthese difficulties can be overcome if the body surface ispartitioned into smooth or piecewise-smooth patches (theso-called surface-patch model).

Several methods using surface-patch modelling havebeen reported. Knepp and Goldhirsh [3], and Albertsen

Paper 5222H (Ell), first received 7th August and in revised form 20thNovember 1986

The authors are with the Faculty of Electrical Engineering, Universityof Belgrade, Bulevar Revolucije, 73, P.O. Box 816, 11001 Belgrade,Yugoslavia

et al. [4] employed the magnetic field integral equation(MFIE) to closed surfaces. However, it is very difficult toapply this equation to open surfaces, as can be concludedfrom Rahmat-Samii and Mitra [5]. This is why for opensurfaces the electric field integral equation (EFIE) ismostly used. In what follows only papers using EFIE willbe mentioned.

Wang et al. [6] modelled body surfaces by means ofplanar rectangular patches with a sinusoidal basis func-tion. A similar method was developed by Singh andAdams [7], but they proposed the use of planar quadri-lateral patches instead of rectangular ones. Newman andPozar [8] extended the method of Wang et al. for sur-faces with attached wires. Glisson and Wilton [9] parti-tioned the surface of the body into planar rectangularpatches with the so-called rooftop basis function. Asimilar method was proposed by Rao et al. [10], but theyused triangular patches instead of rectangular ones. In allthese approaches continuity of charges at the junctions ofpatches was not required, and the Galerkin method wasused for obtaining a stable and accurate solution. In Ref-erences 11, 12 and 13 a polynomial approximation forthe current distribution on a flat rectangular plate wasadopted, in which case continuity of charges was satisfiedover the whole plate and a reasonably accurate solutionwas obtained by using the point-matching method.

Previous works [3-13] indicate clearly that there arestill several general problems which need to be solved inthe analysis of composite wire and plate antennas, suchas the choice of the optimal integral equation, the choiceof the surface patch, the choice of the expansion functionand the choice of the test function. In addition, particularcare needs to be devoted to the excitation region and todiscontinuities (junctions and ends). This paper attemptsto solve some of these problems for a simple, but typical,class of such antennas. According to the best knowledgeof the authors, this appears to be the first method for theanalysis of a thin metallic plate considered as an antenna(not as a ground plane, reflector or scatterer) by means ofan integral equation.

2 Outline of method

The antenna considered is a monopole, consisting of ashort wire segment with a triangular plate attached to it,situated in a vacuum above an infinite, perfectly conduct-ing ground plane and fed by a coaxial line, as sketched inFig. 1. The whole structure is symmetrical with respect tothe y-axis.

As the first step, the coaxial-line excitation is modelledby a TEM magnetic-current frill. The influence of cur-rents and charges induced on the ground plane can be

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 2, APRIL 1987 205

replaced by the antenna image, and thus the equivalentsystem is obtained in the form of a symmetrical dipolewith the frill doubled in magnitude.

The analysis method is based on the two-potentialintegral-differential equation for current distribution inthe equivalent system.

Fig. 1 Sketch of thin-plate triangular antenna above ground plane

2.1 Two -potential equation for antenna currentdistribution

The antenna is assumed to be made of a perfect conduc-tor, and the tangential component of the total electricfield on its surface should be zero,

(E, = 0 (1)

where E, is the known impressed excitation field due tothe magnetic-current frill, and E is the electric-field vectordue to the antenna surface currents and charges.

To calculate the electric field due to the wire segmentof the antenna, the surface current distribution is approx-imated by a current filament of intensity It (Fig. 2), andthe extended boundary condition along the wire axis isimposed instead of eqn. 1. The electric field due to thecurrent filament can be evaluated by using expression1.14 in Reference 14:

E = -471

where sx is the local co-ordinate along the filament, sl0and slb are the co-ordinates of the beginning and endpoints of the filament, and reference direction of currentIt is from sla to sib.

To evaluate the electric field due to the plate, the cur-rents on the two sides of the plate are first approximatedby a single current sheet. It is known that, asymp-totically, the current-density component parallel to thesheet edge tends to infinity when approaching the edge asthe reciprocal of the square root of the distance from theedge. To simplify the numerical analysis, this singlecurrent sheet is further represented as a sum of two cur-rents; one of finite intensity and existing over the wholeplate (Jsp) and one of unbounded intensity parallel to theplate edges and existing only in a narrow strip near theedges, in the region shown hatched in Fig. 2. The second

component of the plate current is next approximated bycurrent filaments (J2 and 73) along the plate edges, asindicated in Fig. 2. The electric field due to current fila-ments I2 and 73 can be evaluated in a similar manner to

Fig. 2 Sketch of one quarter of symmetrical equivalent of antenna inFig. 1 showing antenna

the electric field due to current filament / x , i.e. by usingeqn. 2, whereas the electric field due to surface currents isobtained as in References 11 and 12:

,-jkr

4n

1 \ ex < I i Jsp + — grads divs Jsp ds

)s \ K j r

1 f e~jkr

- T T Q ) div,J,n d\ x n

where Etang is the component parallel to the plate, C isthe edge contour of the surface S, n is the unit vectornormal to S, and grads and divs are surface operators.

Substituting the expressions for the electric field ofcurrent filaments and surface currents into the boundarycondition and the extended boundary condition, oneobtains the so-called two-potential integral-differentialequation for the antenna current distribution. Essentially,this equation is an EFIE which, as mentioned, is suitablefor composite wire and open surfaces. Note that thisequation cannot be applied near the plate edges.

2.2 Solution of equation for current distributionThe two-potential integral-differential equation for theantenna current distribution can be solved only numeri-cally. One of the simplest and fastest methods for thispurpose is the point-matching method [15]. Essentially,it consists of stipulating that the equation for current dis-tribution be satisfied at a number of suitably locatedpoints. In order that the method be efficient and suffi-ciently stable, one has to choose the expansion functionsproperly, i.e. the approximation of the unknown current

206 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 2, APRIL 1987

distribution as well as the locations of the matchingpoints.

2.2.1 Approximation of current distribution and addi-tional equations for currents at discontinuities: Weapproximate the current distribution along each currentfilament by power series in the form

h = (4); = o

where s, is the local co-ordinate and n, is the desireddegree of the polynomial approximation along the ;th fil-ament. The surface-current density vector has both x-and jz-components which depend on both x- and y-co-ordinates. Each of the components we approximate by atwo-variable polynomial:

Jspy=

I 5>i=l j=0(2)

I 2i = 0 j =(2)

(5)

(6)

equivalent), as sketched in Fig. 3. The matching pointsare equidistantly distributed along the wire segment, thedistance between them being twice the distance betweenthe segment end and the closest matching point. For

where nx and ny are the desired degrees of the polynomialapproximations. The summation in eqn. 5 and that ineqn. 6 are performed over odd and even indices of x only,respectively, to take into account the antenna symmetry.The polynomial expansion for current was adoptedbecause it is very flexible and can approximate accuratelyboth smooth functions and their lower derivatives. Inaddition, polynomials are simple and fast from a compu-tational point of view.

The matching points should not be placed near dis-continuities, since point-matching equations at thesepoints are undefined. To obtain a stable and accurate sol-ution, discontinuity behaviour must be determined first,and then incorporated in the current expansion.

In this paper discontinuity behaviour is not incorpor-ated into the current expansion directly (as can be seenfrom eqns. 4-6, but indirectly by imposing the followingconditions that current distribution must satisfy at dis-continuities :

(a) The charge density at adjacent points on the twosides of the magnetic-current frill is determined from theboundary conditions in terms of the coaxial-line dimen-sions and voltage. This implies that the first currentderivative at the antenna driving point is discontinuous,the discontinuity being determined by the coaxial-lineparameters [14].

(b) The total current intensity is required to be contin-uous at the wire-to-plate junction.

(c) The current-density component normal to an edgeshould vanish as the square root of the distance from theedge. Since polynomials cannot follow this law, it is onlyrequired that this current-density component is zero atthe edges.

(d) At the junction of two edge current filaments thecurrent intensity and its first derivative are required to becontinuous.

(e) Edge current filament 73 (see Fig. 2) is forced to bezero at the y-axis; this follows from symmetry consider-ations.2.2.2 Choice of matching points and junction-fieldconstraint: Owing to symmetry, the matching pointsneed to be distributed over one half of the monopoleantenna (i.e. over one quarter of its symmetrical

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c 2 d v

c 2 d u

Fig. 3 Positions of matching pointsArrows denote the electric-field components

a = 0°, i.e. for a strip of constant width, the matchingpoints are uniformly spaced along x- and y-axes over onehalf of the plate, the distance between them along oneaxis being C times the distance between the edge normalto the axis considered and the closest matching point. Bynumerical experiments it was found that optimal valuesfor C are the following: 1 to 2 for the edge at the wire-to-plate junction and 2 to 4 at the other edges. With increas-ing a, the matching points are spread out in the samemanner.

Polynomials 5 and 6 cannot approximate properlycharge distributions due to the component of the surface-current density vector normal to the edges in the vicinityof the edges. Since this charge distribution creates intenselocal electric field, which has a component normal to theedges, only the equation for the field component parallelto the edges is imposed. Exception to this rule is made inthe case of matching points placed in the corners of theplate. In the upper corner equations for both field com-ponents are imposed, whereas in the lower corner noequation is imposed. Thus, boundary conditions aresatisfied to a high degree in the central part of the plate.Finally, for matching points distributed along the j/-axis,because of symmetry, the equation for the x-componentof the field can be omitted.

The wire-to-plate junction is a specific source ofnumerical instabilities because the approximation for thecurrent at this junction is not a good one, which results inapparent energy leakage. To avoid this the condition thatenergy leakage in the vicinity of the junction be zero isimposed. It can be easily shown, as was done for thewire-to-wire junction in Reference 14, that this conditionis approximately satisfied if the following junction-field

207

constraint is applied:

j + E) • d\ = 0 (7)

In this equation L is a path along the y-axis of equallength at both sides of the junction, as sketched in Fig. 4.

1L/8 "

L /A

L IU

L \U

L/8i

y

y'

(

\

7

^^—^

z

Fig. 4 Sketch of path used injunction-field constraint

By numerical trial-and-error method it was found that inmost cases the optimal path length is between 4 and 8wire radii. The integral in eqn. 7 can be approximatelycomputed by using the repeated midpoint rule, i.e. bysimply summing the integrand of midpoints of sub-segments of equal lengths into which the path is sub-divided. Eqn. 7 is thus reduced to summing severalpoint-matching equations. It was also found by numeri-cal experiments that it is sufficient to take only two inte-gration (matching) points per arm of the path. Thesepoints are denoted by asterisks in Fig. 4.

A system of complex linear equations is thus obtained,consisting of equations for currents at the discontinuities,point-matching equations and the equation according tothe junction-field constraint. The number of these equa-tions depends on the number of matching points and canbe made equal to or larger than the number ofunknowns.

2.2.3 Least-squares solution and estimate of optimalparameters for analysis: Because of the complexity ofthe numerical model, it does not appear easy, or evenpossible, to choose matching points in some uniquemanner so that the number of equations be equal to thenumber of unknowns. Therefore, an overdeterminedsystem of equations is constructed by adopting morematching points than the minimum and solving them inthe least-squares sense.

When an overdetermined system of equations is solvedin the least-squares sense, every equation is satisfied withsome error. However, errors that are numerically equalmight not have the same influence on the physicalsystem. To obtain a solution with errors that tend to beof similar value in both senses, some equations are multi-plied by weighting factors. By numerical trial-and-errormethod it was found that for best results it suffices tomultiply equations for currents at the discontinuities by afactor G ranging between 10 and 1000.

To obtain a stable and accurate solution, over-determination must be satisfied along each of the co-ordinate axes (x-, y- or local s-axis). By numericalexperiments it was found that it suffices to adopt thenumber of matching points along any axis to be equal tothe degree of the polynomial approximation for currentalong the same axis, plus one.

Accurate solutions can be obtained for various degreesof polynomial approximations for the currents (nun2 andn3 for the line currents Iu I2, / 3 , and nx and ny for thesurface current) and for various numbers of matchingpoints along each axis (mx along the wire axis, and mxand my along x- and y-axis over one half of the plate). Bynumerical trial-and-error method simple rules for obtain-ing n2, »3, mit mx and my, when nlf nx and ny are given,were found, and optimal values for nu nx and ny wereestimated. These rules are indicated in Table 1. Estimatedoptimal values for nu nx and ny and the correspondingnumbers of equations and unknowns of the system neand nu are summarised in Table 2. All examples presented

Table 1 : Rules for obtaining n2. n3, m,, mx and my when /?,,nx and ny are given

n2 = nY + yn3 = integer part of (nx + 1 )/2m, =/7, +1mx = n2 +1m = n , + 1

Table 2: Estimated optimal values for n,, nx and ny and thecorresponding values of nu and ne obtained using rules inTable 1

ny; polynomial degree of /, 3, 4, 5, 6nx, ny; polynomial degrees of Jsp 4, 5nu; total number of unknowns 38-54n • total number of equations 42-65

in the following Section were obtained with the values ofthe parameters in accordance with Tables 1 and 2.

Once the current distribution has been determined, theantenna admittance and its radiation pattern can be cal-culated easily.

3 Numerical and experimental results

In the examples presented in this Section all the param-eters of the analysis, with the exception of the plate anglea and frequency/, were adopted to have the same values,as follows: hw — 18 mm, hp = 90 mm, a = 3 mm, b =6.9 mm, nt = 3, nx = 5, ny = 4, C = 2, L = 12 mm andG = 1000.

Fig. 5 shows the antenna admittance for a = 15° in thefrequency range 0.2-1.2 GHz. Also given in the Figureare experimental results for the antenna admittanceobtained with an antenna of plate thickness 0.5 mm. It isseen that agreement between theory and experiment issatisfactory.

Figs. 6a and 6b show real and imaginary parts of thesurface current density and line current intensity over onehalf of the antenna plate for a = 15°, at the frequency/ = 0.5 GHz which is very close to the antenna resonantfrequency. In both Figures the surface-current densityvector is given at points of the plate far from the edges,and the line-current vector is given at points along theedges. It is seen that current distribution on the antennais very complex, that the surface-current density vector is,in general, elliptically polarised, and that the main part of

208 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 2, APRIL 1987

the plate current is localised in the vicinity of the edges,as it should be, as a result of skin effect.

Figs, la and 1b show the antenna conductance G andsusceptance B in the range 0.2 to 1.2 GHz, for a = 0°,

Fig. 5 Antenna conductance G and susceptance B

numerical resultsO # experimental results

Fig. 6 Current distribution over one half of antenna platea Real partb Imaginary part

10°, 15°, 20°, 25° and 30°. The results for a = 0°, 15°and 30° were compared with experimental results ofReference 16. It was found that agreement betweentheory and experiment was satisfactory and that theresults for a = 0° practically coincided with those for theequivalent cylindrical monopole. It is seen that when a, isincreasing, maximum values for G and maximum positiveand negative values for B are also increasing, the reson-ant frequency is decreasing and broadband properties ofthe antenna in the vicinity of the first antiresonance areimproving.

The radiation pattern of the antenna with a = 0° ispractically identical to the radiation pattern of a thin-wire dipole of the same length. When a is increasing the

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 2, APRIL 1987

radiation pattern does change somewhat, but the generalshape of the pattern remains the same. Therefore onlyresults for the radiation pattern in the azimuthal planeare presented. Fig. 8 shows the directive gain in the zx-

50

Fig. 7 Antenna conductance and susceptance versus frequency forvarious plate angles

a Antenna conductanceb Antenna susceptance

30°

60°

90°.

Fig. 8 Directive gain in zx-plane at frequencies f' = 0.2 and 1.2 GHza = 30°a = 0°

plane of the antenna with a = 0° and 30°, at frequencies/ = 0.2 and 1.2 GHz. Fig. 9 shows the directive gain inthe directions of the x- and z-axes, gx and gz, respectively,in the range from 0.2 to 1.2 GHz, for a = 0°, 15° and 30°.It is seen that gz and gjgx are increasing when a or/areincreasing.

4 Conclusions

A method is presented for the analysis of symmetricalthin-plate triangular antennas. The analysis is based on

209

the two-potential equation which is solved by means ofthe point-matching method combined with a polynomialapproximation for the current distribution and an over-determined system of equations. Particular care isdevoted to modelling the currents at the discontinuities.

6 References

CD•O

N

1.5

02 0.5f .GHz

Fig. 9 Directive gain in direction of z- and x-axes versus frequency, fora = 0°, 15° and 30°

9z(x = 0°, 15°,30°)M « = 0°)<U« = 15°, 30°)

Theoretical and experimental results are presented,showing good agreement and illustrating antennaproperties. It is shown that the antenna current distribu-tion is very complex: the surface-current density vector is,in general, elliptically polarised and the main part of theplate current is localised in the vicinity of the edges.When the plate angle a is increasing, the resonant fre-quency is decreasing, broadband properties in the firstantiresonance region are improving and the radiationpattern changes only slightly.

The method can be extended with relative ease to theanalysis of more complex structures.

5 Acknowledgments

The authors would like to thank Professors M.B. Drag-ovic and A.R. Djordjevic for valuable discussions duringmany stages of this work.

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