Novel surface-wave antenna

G.Fikioris R.W.P. King T.T. Wu

Indexing terms: Antenna arrays, Surface-wave antennas, Field patterns, Surface-wave generators

Abstract: A circular array of a large number of electrically short vertical dipoles of which only two are driven is described and analysed. The relevant parameters include the length and radius of the identical elements, the distance between adjacent elements and their number. A judicious choice of the parameters can establish one of many possible resonant modes for the current amplitude distribution around the array and generate a horizontally omnidirectional field pattern with very high vertical directivity. Applications as a surface-wave generator are discussed. Although the array is not a broadband device, the possibility of operating at several discrete frequencies using the same construction may be an advantage for some applications.

1 introduction

Recent theoretical and experimental studies [l-51 hat shown that properly dimensioned large circular arrays of electrically short vertical dipoles possess very narrow resonances when only one element is driven and the rest are parasitic. At each resonance, the currents on all elements are large and distributed as a standing wave around the circle. The driving-point reactance is zero. The associated field pattern consists of many pencil- like beams.

In other studies [6, 71, the complete electromagnetic field generated by a vertical eleciric dipole located in the air above planar earth (salt and lake water, wet and dry earth) has been formulated in simple integrated expressions. Included is the special case when both the vertical dipole and the observation point are on or close to the surface of the earth.

In applications such as broadcast, ground-wave over- the-horizon radar [8] and shore-to-ship communication, it is necessary to generate a significant electromagnetic field (surface wave) close to the air-earth boundary at 0 = ~ 1 2 . In addition to the field close to 0 = d 2 , a typ- ical transmitting antenna generates a significant field at 0 IEE, 1996 IEE Proceedings online no. 19960160 Paper first received 5th July 1995 and in revised form 3rd November 1995 G. Fikioris was formerly with the Gordon McKay Laboratory, Harvard University, Cambridge, MA 02138-2901, USA, and is now with the Rome Laboratory, Electromagnetics and Reliability Directorate, Hanscom AFB,

R.W.P King and T.T. Wu are with the Gordon McKay Laboratory, Harvard University, Cambridge, MA 02138-2901, USA

MA 01731-3010, USA

smaller angles 0. This field is unwanted. Furthermore, the upward generated field may reflect off the iono- sphere and interfere with the surface wave propagating near 0 = 7d2. It is the purpose of this paper to describe a novel, structurally simple antenna array that is espe- cially suited to generate an omnidirectional surface wave. Instead of directing the outward-travelling elec- tromagnetic field upward towards the ionosphere, the array directs the field along the surface of the earth in a pancake-shaped field pattern. The array is a large cir- cular array like the one in [l], but wiith two elements driven instead of one. Each driven element has a driv- ing-point impedance that is purely resistive. The description and analysis of the array are followed by a determination of its complete far field both when the array is in free space and when it is over planar earth. The generation of a pancake-shaped field pattern by a large circular array with many parasitic elements was first proposed in [9].

2 Omnidirectional array

The top view of a large circular array is shown in Fig. 1. The N elements (1, 2, ..., N) are placed on a cir- cle of radius R; in the Figure N = 96. 'The elements are numbered counter-clockwise from the x-axis. Their centres lie on the z = 0 plane. They are vertical (z- directed) dipoles of half-length h and radius a. The dis- tance between adjacent elements is d = 2Rsin(dN). The operating frequency is f = clh = 0127~; a time depend- ence e-1wt is assumed.

25

Fig. 1 The two driven elements of the array of Table 1 are shown as solid dots

Top view of circular array of N = 96 elements

Suppose that element 1 is centre-driven and the rest are parasitic and unloaded. The midpoint currents nor- malised to the single driving voltage are the self- and mutual admittances Y,,l = GI,/ - iB1,l. 1 = 1, 2, ..., N .

IEE Proc-Microw. Antennas Propag., Vol. 143, No. I , February 1996

They are related to the phase-sequence admittances I") = Gb) ~ iB(P) by

where 1 p = O o r p = N / 2

2 otherwise The resonant and directive properties of the array in this case were first reported in [2], where the numerical results come from the original two-term theory of (101. Subsequently, the improved kernels of Freeman and Wu [l 1, 121 were incorporated into the theory in a sim- ple 'modified' form and a systematic presentation of the array's properties was made [l]. The m-th 'phase- sequence resonance' occurs at f = f,. At this frequency, the term with p = m in eqn. 1 becomes very large. As a result, the Y1,[ follow the standing-wave distribution cos[2n(l- l)m/N] around the array. Phase-sequence res- onances can occur only if the parameters a, h and d are properly chosen and if the condition dlh < m1N 5 !12 is satisfied. Near resonance, one has approximately [ 1, 31

J m

where Q is the quality factor of the resonance curve. At resonance, the field pattern is adequately described by an array factor. It has the form of 2m pencil-like beams.

The purpose of the present paper is to design a reso- nant array with two driven elements, which has a trav- elling-wave distribution of current, namely

instead of a standing-wave distribution. The discussion is based on the properties of the resonant array with a single driven element and on numerical calculations from the full formulas in [l]. If elements 1 and n are the two driven elements (see Fig. l), it is first shown that eqn. 3 is possible, provided certain simple condi- tions are satisfied by N, m and n.

2. I driven elements The midpoint currents 1x0) on the dipoles when ele- ments 1 and n are driven by voltages VI and Vn, respec- tively, can be found from the self- and mutual admittances q,* and Yl,n by the principle of superposi- tion, namely

Excitation of a travelling wave with two

Z=1, . . . , N (4)

where the self- and mutual admittances Ylj satisfy yi,l = YIJ, = Yl,l and q,l = Ylj-l+l = Ylj-I+Ifl. The last equality follows from the symmetry of the circular array with one element driven. At or near a narrow res- onance (as long as the self-conductance is large), the admittances follow a standing-wave distribution. Thus, eqn. 4 becomes

1 = 1,. . t ,N ( 5 )

Defining tmn = 2n(n - l)m/N, it is seen that if the ratio V,lVl is chosen so that

then the currents satisfy the desired eqn. 3 with the cur- rent on element l given by

(6)

I ~ ( o ) = -~1~ l , l i s in t , , e~~ . - " ( 7 )

V - -yIeit"" n -

It follows that the choice in eqn. 6 is not sufficient for a travelling-wave distribution of current around the array; in addition, the condition

must be satisfied. This is a restriction on the choice n of the second driven element for given number N of ele- ments and number m of phase-sequence resonance.

The two driving-point admittances are given by:

sint,, # 0 (8)

Y1,in = G1,m - iB1,in II(O)/VI = WmnY1,1 (9)

K, in = Gn,in - ZBn,in = In(O)/Vn = wL,y1,1 (10) where w,, = U,, - Zv,, = sin2 t,, - i sint,, cost,, (11) and the asterisk denotes the complex conjugate. It fol- lows from U, > 0 that the total power supplied to the array, namely

Ptotai,in= -Gi,inlVi 2 lz+-Gn,i~lKn12 2 = 2(G1,zn+Gn,tn)lV1 1' (12)

is positive as one would expect. However, the individ- ual powers supplied may include one that is negative. This means that it is necessary to extract power from one of the elements to excite a travelling-wave distribu- tion of current. Although this may be done by centre- loading the element (as opposed to driving it by a gen- erator), a case like this is undesirable and can be avoided as will be shown below.

As in the case with one driven element, the current distributions along the length of all elements are the same [l], and the far-field pattern is adequately described by the product of the field of a single antenna and the array factor A(")(B,+). With the cur- rents in eqn. 3

A(") (0,4) = e-

1 1 1

N i [2 . rr ( l -~)m/NIe~i lcRs in B cos($-$i) (13)

1=1 where (R, n12, C p l ) is the location of element 1 in spheri- cal co-ordinates. This is evaluated asymptotically for large N in the Appendix. The result is

or

A(")(@, 6) - N g m ( 0 ) e - z m ~ + NgN-m(B)ez((N-m)6 (15) where g,(8) = (-i)"J,[N(d/h)sinO].

It is seen from eqn. 14 that each term in eqn. 14 or eqn. 15 is proportional to the badiation field due to a continuous circular travelling wave of current. One is a clockwise travelling wave, the other a counter-clock- wise travelling wave. If m << Nl2, the first term domi- nates and the resulting radiation field has a pancake- shape with vertical directivity the same as in eqn. 26 in [I] so that lA[m)(B,Q)l = A(,)(B). In the extreme case m =

K E Proc -Microw Antennas Propag , Val 143, No 1, February 1996 2

N/2 (which is not allowed because eqn. 8 is not satis- fied), the second term would have the same magnitude as the first and the resulting radiation field would con- sist of 2m pencil-like beams.

2.2 Choice of N, n and m If N is chosen to be a multiple of four, then for certain n there exist values of m such that

cost,, = 0 U U,, = 0 and U,, = 1 (16) With such a choice of N, n and m, eqns. 9-11 give Yl,;, - Y,,,, = Y131 so that, at or near a narrow resonance, the two driving-point admittances are equal to the self- admittance and all desirable properties of the circular array with one element driven are preserved: The two driving-point susceptances become: zero at the same fre- quency and the driving-point conductances are very large at that frequency. Hence, it is desirable to use as a second driven element one that would have a very small current if only element 1 were driven.

If, furthermore, the second driven element is chosen to be a quarter-way around the circle, i.e. n = N/4 + 1, then U,, = 1 and v,, = 0 for all odd m. With such a choice of second driven element, it is possible to excite many phase-sequence resonances; the required voltage ratio is either or e-i7ci2.

The choice of m depends on various opposing fac- tors: If m is too large and the second term in eqn. 14 or eqn. 15 contributes, the field pattern will not have a true pancake-shape. However, the value of the ratio IgN~,(d2)/g,(7c/2)~ decreases very rapidly with decreas- ing m. The advantages of using a large value of m include a slight increase in directivity as well as smaller contributions from the other phase sequences; these contributions can cause departures from the omnidirec- tional pancake-like field pattern. For a given value of m, the array is not a broadband device. However, the possibility of using several discrete frequencies with the same construction may be an advantage for some applications.

-

0.05 i 1

close to ‘resonance’, that the driving-point susceptance becomes larger than the conductance, and that all val- ues are much smaller than the resonant self-conduct- ante G1,IVJ.

2.3 Choice of a, h and d The parameters a, h and d may be chosen based on the theory in [I] (Section 3). An example is given here for an operating frequency of about 30MHz. With N = 96, there are many combinations of m and n that satisfy eqn. 16. The choice a = 0.28m, h = 1.9m and d = 3.lm is appropriate for 30MHz. This choice of a, h and d corresponds to an approximate scaling of the N = 90 element experiment [3] with monopoles over a ground plane with one element driven by a coaxial line. The diameter of the proposed array is 2R - 95m. Table 1 shows the phase-sequence resonances that may be excited by using elements 1 and 25 (= N/4 + 1) as driven elements, the required voltage ratios V2,/V1, the theoretically predicted resonant frequencies f,, the val- ues of the driving-point conductance at resonance

= Zl(0)/Vl = Gn,lnVm) = GI,*Vm), the ratio of the two array-factor terms in eqn. 15 at 0 = d 2 and @ = 0, and the vertical directivity of the array factor D, as given by eqn. 26 in [l]. f, and Gl,lVm) were calculated using the full formulas of [l] including the two refine- ments in Section 6 of that work. It is seen that the sec- ond term in eqn. 14 or eqn. 15 will have a noticeable effect only in the first case of Table 1.

Table 1: Phase-sequence resonances m that may be excited when the number of elements #U 96, radius a = 0.28111, half-length h = 1.9m, element separation d = 3.1 m, and the number of second driven element n 25

G1 Jfm) lg,,(n/2)/gm(d2)1 D, rn VzdV1 M H ~ mA/V fm

47 i 30.652 2 x 1O1O 0.1 4.78

45 -i 30.589 4 x I O 8 2 x I O 3 4.60

43 i 30.460 1 x107 3 x 1 0 5 4.41

41 -i 30.262 5 x I O 5 4 x 1 0 7 4.21

39 i 29.987 3 x I O 4 5 x I O 9 4.01

37 -i 29.624 2 x I O 3 6 x 10 3.78

35 i 29.154 3 x 1 0 2 6 ~ 1 0 ’ ~ 3.54

-0.01 -0.005 0 0.005 0.01 ( f - f,)lf,,

Fig. 2 Normalised driving-point conductance Gj,i?lf‘/GI If ) (solid line) and driving-point susceptance Bl,inU/Gl, (f, &she&&) of ele- ment I as function of relative frequency v-&Jgrn Q = 1000, N = 90, m = 43, and n = 24

Finally, it must be pointed out that when N, m and n are not chosen to satisfy eqn. 16., a very different fre- quency dependence can result. Fig. 2 shows the driving- point admittance of element 1 as calculated by eqns. 2 and 9-11 when Q = 1000, N = 90, rn = 43 and n = 24 (this choice of n corresponds to using as a second driven element one that would have the largest possible current if only element 1 were driven). It is seen that the driving-point conductance becomes negative very

IEE Proc-Microw. Antennas Propag., Vol. 143, No. 1, February 1996

rn; required voltage ratios V2$V1; predicted resonant frequen- cies fm; resonant driving-point conductance Gl,ffl(fm) = /l(0)/V~ = G25 Jf,); ratio lgwm(d2)/gm(d2)l of two array-factor terms in eqns.‘ 14 and 15 at 8 = 62; and vertical directivity Dv

In the first few cases of Table 1, the currents are pre- dicted to be extremely large. As explained in [l], the underlying reason for this is that the imaginary part of the modified kernel of the m-th phase-sequence integral equation is exponentially small in N: roughly, the reso- nant currents are inversely proportional to this expo- nentially small quantity. The imaginary part of the modified kernel is defined in eqn. B9 in [l] as the sum of N/2+1 terms of order 1, so that calculation of this quantity is very sensitive to roundoff error. Related dis- cussions are contained in [12]. The special numerical considerations that should be taken into account when using the two-term theory are discussed in [3]. For other approaches such as direct numerical solution of the integral equations for the currents, and especially for approaches implemented by general purpose com- puter programs, numerical difficulties will also be encountered. Here, it may not be possible to identify and remove the source of numerical difficulties. When

3

the resonant currents are smaller, the numerical diffi- culties are less pronounced.

Note also that the imaginary part of the original m- th phase-sequence kernel [ 101 is not exponentially small in N and is, in some cases, totally inadequate [l, 121. Therefore, it is likely that methods sufficient for con- ventional arrays will lack the necessary accuracy.

2.4 Far field of array in free space The far field of the omnidirectional array is the prod- uct of the field of a single isolated antenna multiplied by the array factor. The far field of a vertical dipole in space that is electrically short and has the effective half-length he is

Figs. 3 and 4 show the normalised far-field power pattern IEer(6,$)I2 in the plane 6 = d 2 of the dipoles' centres for the cases m = 37 and m = 45 of Table 1, respectively. Figs. 3 and 4 were obtained using eqn. 4 and the full formulas of [I] so that the effects of the rest of the phase sequences are included. The small oscillatory departure from the smooth omnidirectional field in the m = 31 case is due to the contributions from the rest of the phase sequences. The slight ripples in the m = 45 case are due to the contribution of the term corresponding to the second term in eqn. 14 or eqn. 15. The cases m = 41 and m = 43 of Table 1 appear as smooth circles and are not shown here. Fig. 5 shows the normalised far-field power patterns in the elevation plane $ = 0 as a function of the polar angle 6. The m = 45 case is seen to be slightly more directive.

The far field of the circular array is thus

@=RI2 iwpo2he1I (0) ezkro E; rz - ----A(")(8)sinB (18)

47r TO

where A")(6) is given by the magnitude Of the first term in eqn. 15.

Fi .5 Noyuhedfar-fieldpower attern lEor(8, 0)l2 as function of the pozr angk 8 m the plane @ = 0 of t fe dipoles centresfor the m = 37 and m = 45 cases of Table I Each pattern 1s normalised to its maximum at 0 = ni2 $I =E12

2.5 Effect of ohmic losses Up to now, the analysis has assumed that the dipoles are perfectly conducting. In practice this would require superconducting elements. When the elements are imperfectly conducting, the evaluation of the field becomes more complicated and is currently under investigation. The effect of ohmic losses on the self- and mutual admittances, however, is fully discussed in [3]. In general, the effect of ohmic losses is more pro- nounced when m and N are larger and additional con- siderations must be employed to design an array for a given frequency. Results for copper elements show that for the first few cases of Table 1, the extremely large currents predicted by the theory for lossless elements are in fact smaller and that the radiation field is signif- icantly changed. However, the moderately high cur- rents of the m = 37 case as well as the associated omnidirectional radiation field are the same as if the

Fig.3 Normalised arfield power pattern IEe'Clr/z. @)I2 as fmctlon o j

case of Tabg I elements were perfectly conducting. The extremely large currents required for the m = 45 case can be real- ised in practice only with superconducting elements.

the polar an le i$ in td &ne 8 = d 2 of the dipoles centres for the m = 37

- cb=O

-0.51 i

3

Suppose that the resonant array is located in region 2 (air) over region 1 (salt water, wet earth, lake water, dry earth), at a small distance do over the air-earth boundary. The wave number in air is k2 = k. Medium 1 is characterised by a complex wave number kl = w d b ~ [ E ~ ~ E ~ + i(ol/m)], so that

Resonant array over earth or sea

where (a) for salt water, k22 = 2400

= 80 and o1 = 4 Sim so that lk112/

(b) for wet earth, clr = 12 and 6 1 = 0.4 S/m so that

(c) for lake water, = 80 and o1 = 0.004 S/m so that Ikl/2/k22 = 80

k22 = 25.

lk112/k22 = 240

Fig.4 Normalised ar field powev pattern IEOr(7d2, @ ) I 2 as function of thepolar an le @ in td>une 8 = n/2 of the dipoles centres for the m = 45 cuse of TabE I

(4 for dry earth, Elr = 8 and 0 1 = 0.04 s/m So that lk1I2/

IEE Proc.-Microw. Antennas Propag., Vol. 143, No. I , February 1996 4

Consider a vertical electric dipole in region 2 (air) over region 1. If the electrical distance k2p from the dipole to the point of observation satisfies k2p < 21k112/k22, then the term involving the Fresnel integral in eqn. 33 in [6] is negligible. It follows that the z-component of the electric field E2z in region 2 is the same as when region 1 is a perfect conductor. The range k2p < 21k1I2/ k22 includes both the intermediate and the near regions, the latter being defined [13] by the stricter condition k2p < 1. In the m = 37 case of Table 1, the largest ele- ment-to-element electrical distance is k2p = k2(2R) = 59.5. At least in cases (a)-(e), therefore, all elements in the array are in each other's intermediate region and it is correct to assume that region 2 is perfectly conduct- ing when estimating mutual-coupling effects. Note that these depend entirely on E2=.

To provide an estimate of the coupling between the array in region 2 and its perfect image in region I , the behaviour of the z-directed electric field of an array in free space will be investigated. From [I], the current distribution along the length of all elements in the reso- nant array is coskz - coskh. Two simplifying assump- tions are made: (a) the elements have infinitesimal thickness (b) the current distribution is sink(h - 121). Thus, the currents are taken to be

(20) I i ( z ) - sink(h - 121) 4 ( 0 ) sin kh --

where ILO) is given in eqn. 3. These approximations are adequate for our purposes since the dipoles are electri- cally short (kh < d2), and sink(h - lzl) and coskz - coskh are quite similar. Now use is made of the exact formulas for the field of an infinitesimally thin dipole with a sink(h - lzl) distribution [14]. The centre of cylin- drical co-ordinates (p, @, z) is placed at the centre of the array and the location of the centre of dipole I is (R, 0) = (R, 2z(Z - l)/N, 0) in cylindrical co-ordi- nates. From [I41 p. 528, it follows that the z-compo- nent elz of the electric field due to dipole I in the circular array is given anywhere by

) 4 ~ s i n k h ( k2rtl kzrbl k27-d

zwpJL(0) ezkzrtr e Z k Z r b l ezkzTcr els(p, 4, z ) = - - + - - 2 cos kh-

( 2 1 ) where rtl, rb[ and rcl are, respectively, the distances from the observation point to the top (z = h), bottom ( z = - h), and centre (z = 0) of dipole I in the array.

Using eqn. 3 and setting p = R, the total z-directed electric field of the resonant array is

The formulas for the distances rtl, i"b1 and r,l in the case where p = R are:

sin2 !?Z.& + ( h - z ) ~ (23) 2

The magnitude of the normalised z-component of the

IEE Proc.-Microw. Antennas Propag., Vol. 143, No. I , February 1996

electric field as calculated from eqns. 22-25 is plotted in Fig. 6 for the m = 37 case of Table 1 as the distance z varies from -12m to -2m. It is seen that E, decays rapidly and monotonically away from the array. The rate of decay is much more rapid than for a single iso- lated element. This phenomenon is related to the rapid decrease of the field (surface wave) observed in infinite linear arrays [15]. This rapid decrease indicates that the coupling between the resonant array and its image is negligible even if the array is placed at a small distance above the surface of the earth.

-12 -10 -8 -6 - 4 -2 =,m

Fig.6 Magnitude of normalised z-component /[(4n sin kh / i w I l ( 0 ) JE,(R, @, z)l of electric field as function of z for @ = 0 andJ?r the m = 37 case of Table I

4 over earth or sea

Field of vertical dipole and of resonant array

General formulas for the three cylindrical components of the electromagnetic field of a vertical electric dipole with unit electric moment are given in [6, 71 subject only to the condition that the wave number of air (k2) be small compared to the magnitude: of the complex wave number (k,) of the earth or sea. That is, lk121 >> k22 or Jkll 2 3k2. The formulas are valid at all points in the air, z > 0, with the dipole at any height do. When the three conditions k2p 2 81k121/k22, do2 << yo2, and Ik2do/klrol << 1 are satisfied, the procedure carried out to obtain the formuIas for the cylindrical components may be extended to the spherical component Eer(vo, e). The result is

E,'(ro,@) = u@heIo 3zkzro - sin 8 COS @+do/ro) cos(J62do COS 6')

2Tk2 (2 [' cos@+cio/ro+/c2//c1

I - i ( k z / k l - &/rO cos2 0) sin(k2do cosd)' cos 6' + do/ro + h / k l

(26) where ro = (p2 + z ~ ) ~ ' ~ , he is the effective half-length of the dipole and IO is the current at its centre.

An alternative form is useful when the vertical heights z of the observation point are small compared to the radial distance from the transmitter and the transmitter itself is close to the earth. Specifically, where

P x To TO

Iklz( < k g o , sin6 = -, cos6 := - << 1,

eqn. 26 reduces to

This formula shows that for observation points at any fixed height z < Ik2r0/kli, the incident electric field is proportional to lIro2. This includes both the surface- wave term and the space-wave term. Note that the lat- ter vanishes when z = 0 so that the entire field along the surface is due to the lateral wave.

The far field of the resonant circular array of dipoles at a small height do - 0 over the earth or sea is given by eqn. 26 or eqn. 28 multiplied by A("(8) as given by the magnitude of the first term in eqn. 15. The pancake- like pattern represented by A(m)(8) is enhanced by the low-altitude field represented by eqn. 26 or eqn. 28. Virtually no field is maintained at upward angles that are not close to 8 - n12.

5 Conclusions

A novel array of electrically short dipoles that provides an omnidirectional field pattern limited to angles close to 8 = n12 has been described. The complete far field ovcr carth or sea has been formulated in simple inte- grated expressions. Only two of the elements are cen- tre-driven; the two driving-point impedances are purely resistive. With its omnidirectional low-angle field, the array is especially suitable as a surface-wave generator.

The basic ideas concerning the excitation of a travel- ling wave of current also apply to resonant noncircular closed-loop arrays. Such arrays are of interest because of their possible superdirective properties [16, 11.

6 Acknowledgments

The authors thank M. Owens for her help in preparing the manuscript. This research was supported in part by the US Air Force under Rome Laboratories contract F 19628-9 1-K-0020 with Harvard University. Addi- tional support was provided by the Joint Services Elec- tronics Program under grant NOOO14-89-5-1023 with Harvard University and AFOSR support of Rome Laboratory.

References

FIKIORIS, G., KING, R.W.P., and WU, T.T.: 'A novel reso- pant circular array: improved analysis' in KONG, J.A. (Ed.): Progress in electromagnetics research - vol. 8' (EMW Publishing, Cambridge, MA, 1994), pp. 1-30 FIKIORIS, G., KING, R.W.P., and WU, T.T.: 'The resonant circular array of electrically short elements', J. Appl. Phys., 1990, 68, pp. 431439 FIKIORIS, G.: 'Resonant arrays of cylindrical dipoles: theory and experiment', PhD thesis, Harvard University, Cambridge, MA, 1993 SHEN, H.-M.: 'Experimental study of the resonance of a circular array', Proc. SPIE, 1991, 1407, pp. 30&315

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KING, R.W.P.: 'On the radiation efficiency and the electromag- netic field of a vertical electric dipole in the air above a dielectric or conducting half-space' in KONG, J.A. (Ed.): 'Progress in elec- tromagnetics research - vol. 4' (Elsevier, New York, 1990), pp. 1- 43 SKQLNIK, M.I.: 'Introduction to radar systems' (McGraw-Hill, New York, 1980) GROSSMANN, A., SCHWARTZ, J., and WU, T.T.: 'Antenne iilectromagnttique tmettrice ou rtceptrice trBs directive', French Patent 2 605 148, 1988 KING, R.W.P., MACK, R.B., and SANDLER, S.S.: 'Arrays of cylindrical dipoles' (Cambridge University Press, New York, 1968), chap. 4 FREEMAN, D.K., and WU, T.T.: 'An improved kernel for arrays of cylindrical dipoles', Proceedings of IEEE/AP-S sympo- sium, London, Ontario, June 1991, pp. 630-633 FREEMAN. D.K.. and WU. T.T.: 'Variational-orinciole formu- lation of the two-term theory for arrays of cylhdricd dipoles', IEEE Trans., 1995, AP-43, pp. 340-349 KING, R.W.P., and SANDLER, S.S.: 'The electromagnetic field of a vertical electric dipole in the presence of a three-layered region', Radio Sci., 1994, 29, pp. 97-113 KING, R.W.P.: 'Theory of linear antennas' (IIarvard University Press, Cambridge, MA, 1956) MAILLOUX, R.J.: 'Antenna and wave theories of infinite Yagi- Uda arrays', IEEE Trans, 1965, AP-13, pp. 499-506 KING, R.W.P.: 'Supergain antennas and the Yagi and circular arrays', IEEE Trans., 1989, AP-37, pp. 178-186

Appendix: Asymlptotic evaluation of A("')(8,@1

The asymptotic formulas, eqns. 14 and 15, for the array factor A@)(B,@) is derived here. First, write eqn. 13 as

N A(")(B, 4 ) = eCm6f(q5) = h($ - $ 1 ) (29)

where @ I = 2n(l- 1)/N and A(@) = ezm~e-zkRslnOcos~ is peri- odic with period 2n. Thus, f(@) is periodic with period 2nlN. Expansion off(@) into a Fourier series leads to

1=1

q=-m,

e-zkRsin B~os$'~-z(Nq+m)$'d4i i"" Evaluation of the integrals yields

A(" ) (B ,4 )=NC (-i)"q+"J~ ,+,(kRsin B)e-z(Nq+m)4

When N is large, only the two terms with q = 0 and q = -1 are significant as may be seen from the condition di h < mIN 5 v2 and the asymptotic expansion of the Bes- se1 functions. With kR --- Ndlh, eqns. 14 and 15 have been derived.

00

q=-00 (31)

6 IEE Proc.-Microw. Antennas Propag., Vol. 143, No. I , February 1996