IEEE TRANSACTIONS ON Et ECTROMAGNETIC COMPATIBILITY, VOL. EMC-28, NO. 2, MAY 1986

derived from such a measurement will improve the accuracy of themeasurement.A conservative approach to making a measurement in a nonideal

environment would be to always add the absolute value of anycorrection factors derived from site-attenuation measurements. Ap-plying correction factors in this way would not necessarily improvethe accuracy of the measured results, but it would help insure thatmeasurement error was not making noncompliant units appear tomeet the requirements.

REFERENCES[1] FCC Methods of AMeasurement of Radio Noise Emissions from

Computing Devices, FCC-OST MP-4, GPO, Washington, DC, Dec.1983.

[2] Characteristics of Open Field Test Sites, FCC-OST 55, GPO,Washington, DC, Aug. 1982.

[3] A. A. Smith, Jr., R. F. German, and J. B. Pate, "Calculation of siteattenuation from antenna factors," IEEE Trans. Electromagn. Com-pat., vol. EMC-24, no. 3, Aug. 1982.

/

/ _/

-h

ILa

I11ZZ1 ////7

K7 7

Fig. 1. Vertical electric dipole above a plane metallic grid at zcondition rO << a << X is presumed in the analysis.

Image Theory for Dipole Excitation of Fields Above andBelow a Wire Grid with Square Cells

ISMO V. LINDELL, SENIOR MEMBER, IEEE, VALERI P. AKIMOV, ANDESKO ALANEN, STUDENT MEMBER, IEEE

Abstract-The exact imnage theory, recently introduced by two of thepresent authors for the Sommerfeld problem, is applied to the problem ofa vertical electric dipole above a plane metallic grid with rectangular cells.The mesh size is assumed small compared to the wavelength and the wirediameter small compared to the mesh size, whence the Kontorovichaveraged boundary condition concept is applicable. The image currentconsists of two exponential-line-current waves located in complex space.The theory is seen to produce simple asymptotic expressions for the fieldclose to the grid surface. Also, an integral expression for the calculationof the change in the vertical-electric-dipole impedance due to the grid isconstructed.Key Words-Vertical electric dipole, impedance, plane metallic grid,

image current, near field.Index Code-12d, Illd.

I. INTRODUCTION

The metallic wire grid is widely used in shielding, grounding, andreflecting applications when material saving or small wind resistanceis of concern. The reflection and transmission of plane waves withrespect to planar wire grids has been thoroughly studied both inwestern and Soviet literature [1]. The existing studies also involvesurface-wave transmission along the grid, as well as problems of twoparallel grids or a grid above the ground [2], [3]. The presentproblem of a dipole above the grid has, however, to our knowledge,

Manuscript received May 14, 1984. This work was supported in part by theAcademy of Finland.

I. V. Lindell and E. Alanen are with the Department of ElectricalEngineering, Helsinki University of Technology, Otakaari 5A, SF-02150,Espoo 15, Finland. (90/ + 358)460144.

V. P. Akimov is with the Radiophysical Faculty, Leningrad PolytechnicalInstitute, Politekhnitsheskaja ulitsa 29, Leningrad, USSR.IEEE Log Number 8607914.

been neglected. In this study, the Kontorovich method of averagedboundary conditions [4]-[7] is applied for the problem of an electricvertical dipole above a horizontal wire grid. The problem isformulated by applying an image theory similar to that recentlydiscussed by two of the present authors for the problem of a dipoleabove the ground [8], [9]. The image current is located in complexspace for best convergence, and, in the present case, it is seen to becomposed of two exponential currents, in terms of which the fieldscan be calculated with a simple algorithm. As examples of applica-tion, we obtain explicit expressions for the field above and below thegrid surface due to the dipole, and an integral expression for theinput-impedance change of the vertical electric dipole due to the grid.

II. THEORY

The problem of a source above a planar grid with rectangular cellsat z = 0, Fig. 1, can be studied in terms of average boundaryconditions, which are valid for grids with the wire diameter muchsmaller than the grid spacing, and this much smaller than thewavelength. Thus the grid can be replaced by a continuous partiallytransparent surface whose surface current J, satisfies

(1)

Here, HI is the magnetic field at the grid on the side of positive zand H2 on the side of negative z. From the Kontorovich theory, thetransverse electric field at the grid can be written as

Et =jK(J +(1/2k2)VV (2)

where we denote

K=(a/X) ln (a/27rro) (3)

with a = grid spacing and ro = wire radius. For K -O 0, the gridbecomes a conducting plane, and for K -° 00, it becomes very thin.

The method of average boundary conditions is valid for grids withsmall interwire spacing a, and is asymptotically exact up to the term(a/X)2. However, it turns out to produce good results for practicalgrids even with the ratio a/X as high as 0.25. The thin-wireapproximation ro < a must be valid at the same time. All values from0 to oo are possible for K within the range of approximation.

0018-9375/86/0500-0107$01.00 1986 IEEE

0. The

I^f w

. z- 1 IF -y 7sr z' J1 f-1

pfle-_ ,,.rI - L-w

7z4 7 'AY 7f - ~ e, 74 f

107

21-0

J, = uz x (Hi H2).

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-28, NO. 2, MAY 1986

The boundary condition for the electric field at the grid is

uz x (El - E2) -0 (4)

or the tangential component is continuous through the grid.In the present analysis, we perform a Fourier transformation in the

xy plane. The Fourier variable is denoted by K, which is a two-dimensional vector with no component parallel to the z axis. Theprevious equations can be written in the following form with thecurrent J eliminated:

El t = E2t=-jK(I- KK/2k2)xu (H1 - H2)

Imon '

(5)

which can be identified as an impedance-sheet condition [10]. Here, Iis the unit dyadic.

Let us now consider the problem of a vertical electric dipole at theheight z = h above the grid. The problem can be formulated in termsof a single-component Hertzian vector Hl, satisfying the equation

(V2 + k2)f1(r) = (jIL/cE)6(r - ugh) (6)

in the half-space above the grid. Fourier transformation of (6) leavesus with

HI" (z) + 032H(z) = (jIL/cE)6(z - h) (7)

if differentiation with respect to z is denoted by a prime and : =

Ik2 _ K2The impedance-sheet condition (5) for the grid can be formulated

for the Hertzian potential as follows:

1H' = - K[(k2 + 2)/2k](Hl - H2)- (8)

Now the problem of (7) and (8) can be solved for the Hertzianpotential. The solution in the region z > 0 reads

l=A [exp (-jz- hI)+R exp (-j1(z+h))] (9)

with A - IL/2wcfE, and

R = -(jkf3/K)/(32 +k22jkf/K). (10)

The inverse transformation can be performed by representing R (f)as a Laplace transform of a function f(t) which is very straightfor-wardly written as

f(t)-B[31 exp (jf10t) -12 exp (1f2t)]

djB [exp (jit)--exp (i02t)] (11)

dt

with

B- j/ 1±4K2

13,2=(k/2K)(1 ±1± 4K2).

The function f(t) is closely related to the concept of image currentof the dipole in the grid. It is seen that the image source is not a

dipole, but consists of two exponential line currents. The potentialcan be written as the inverse transformation of (9). If R is expressedas the Laplace transform of f(t), the contribution due to the grid isseen to reduce to the integral

Hg(r) = C f(t)G(r+uz(h -jt)) dt (12)

with G(r) = exp( -jkd)/47rd, d = X < arg (d) < 0,C

jIL/cE. When (12) is written as the potential of a line current I(z'),

Re{IY-

h

we I~~~~~

DS-hquo

Fig. 2. Locations of the image-line currents in the complex z' plane.

we may identify

(13)I(z')- -jILf(-j(z'+h))t- j(z' + h).

Equation (11) substituted in (13) gives us an image current layingin complex space. Starting from the mirror-image point z' - h, itgoes as z' = - h + jt with t: 0 *xo. Thus the distance function d

becomes complex as explained in [8]. The direction of the image-currentline in the complex z' plane can also be chosen otherwise (see Fig. 2). Tobe of any use, the integrand in (12) should be converging. It is easily seen

that, if the image current is chosen as above, along the imaginarydirection, it is oscillating, i.e., nonconverging and nondiverging, but the

Green function converges for large Iz' |. When 0 > 0, the imagecurrent is diverging, but the integrand is converging up to a certainvalue f0, depending on the value of the parameter K: 00 = arctan

(2K/(JI + 4K2 1). For greater values of X, the integrand diverges.For best convergence, we should choose = 0, with oscillatingimage current. In this case, the convergence of the integrand is like

exp (- kt). Thus the effective range of integration of (12) is from t =

0 to a certain value of t for which the function d(t) can be

approximated by -jt. This approximation depends, of course, on the

values of p and z + h (see Fig. 1).Another possibility for the image is obtained if we allow for two

separate line currents instead of only one. Since 31 > 0, the

corresponding current component exp (jOi3t) is exponentially con-

verging if we change the integration path along the real z' axis. In

fact, for - q = jt, where q = - (z' + h), the integration path goesfrom z' = - h(q = 0) to z' = -- oo(q = +oo) and the

corresponding image-current component is exponentially decaying as

exp (-01q). The second image component must be parallel to the

imaginary axis because 02 < 0, and the current would be divergentalong any other ray in the lower half-plane.

It can be seen from (11) that, if the parameter K approaches oo, the

image current disappears as (-jk/K) cos kt. This corresponds to the

disappearance of the grid in the averaged-boundary-condition ap-proximation. The other limit case K -- 0 does not seem that simple in

terms of (11). The resulting function can, however, be interpreted in

(14)

108

,*- -OD two

96

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-28, NO. 2, MAY 1986

terms of a known limit theorem mg(mx) -* 6(x) as m -s co andig(x)dx = 1 [11]. Hence, we have the limit f(t) -* b(t), whichcorresponds to a dipole at the mirror-image point z = -h. Thisresult is in accord with the perfect-conductor surface limit case of K

0.

III. FIELD CLOSE TO THE GRID

As an application of the theory, we consider the Hertzian potentialfield close to the grid surface. With the approximation d = p + (z +h - jt)2/2p, the integral (12) can be written, after some manipula-tions, in the form

FIg(p, z)D j/2'xrkpe-jkr2

[fl exp (w2) erfc (wI) -12 exp (w2) erfc (w2)J

wi=-2kp [k(z+h)-j0jp]

r2- \,p2+(z+h)2 (15)

Here, erfc(x) is the complementary error function and D =IL/4wev1 + 4K . This has a close resemblance to the field above thedissipative ground. In fact, (15) can be seen to be composed of adifference of two Sommerfeld-type TM-wave solutions, each with anumerical distance variable -jf2p/2k (assuming z + h = 0) [12],[13]. A closer look reveals that (15) contains one surface wave, oneleaky wave, plus a space-wave component. These could be identifieddirectly from the Fourier-transformed field (9) with the poles j]lI and12 corresponding to the leaky-wave and surface-wave contributions,respectively.As a check, we consider (15) for K Z 0, i.e., for an almost

perfectly conducting plane. Hence, we have approximately Al = k/Kand 12 kK, which imply w, =z j-kp/2K2 and w2vj/2kpk(z + h + jKP). At this moment, we consider distances psatisfying 2K2 < kp < 2/K2 and very small values of z + h, makingapproximations w, > 1 and w2 < 1 valid. The asymptoticapproximations erfc (wl) - exp (w2)/wv'-rand erfc(w2) -1 cannow be applied, whence (15) takes on the form

FIg(p, z) =z(ILq1/4)e -jkr2 j/2w7- kp

[1j V2/j7rkp + kK(Z±h) jK2kp 2]* (16)

The first term, arising from the exponential-image source corres-ponding to 01, is a typical space-wave term diminishing as l/p,whereas the second one (12 term) has the appearance of a surfacewave with 1/\/p dependence and exponential decay of the field abovethe surface. The leaky-wave component of the first source has alreadydied out before this distance.

If p grows large enough, kp > 2/K2, we can apply the asymptoticapproximation of w, also for w2 to obtain the result

j,IL e-jk2fig(p, z) -F

CE 47rr2

with

F Z±h± (17)

instead of (16). Equation (17) is, in fact, obtained for arbitrary valuesof K. It is readily verified that for K -- 00, F = 0 or the field due to thevanishing grid is zero. On the other hand, for K -- 0 we have F = 1,whence the field above a conducting plane (16) is seen to be in accordwith the conventional image concept, because r2 is the distance of thefield point from the mirror-image point. For K > (z + h)/p, we have

F z -j(z + h)/p, whence the potential field is seen to depend on pas p- 2and vanish with z + h - 0.An expression of the type (17) is also obtained for the far-field

approximation, i.e., if we approximate the distance function in theGreen function by d = r2 - j(z + h)t/r2. This is obviously valid forlarge r2, but not for very small values of z + h. In fact, substitutingthis in (12) leaves us with an integral, which can be interpreted as theLaplace transform of the function f(t), or the reflection coefficientR (1) for the parameter value 1 = k(z + h)/r2. The expression isknown as the reflection coefficient method (RCM) [12]. Evaluationof the coefficient F, in this case, yields the result

F=- (z + h)r2

(z+h)r2+jK(r2+(z+h)2) - (18)

This expression obviously matches (17) for small values of z + h.It should be noted in this context that the Kontorovich-averaged

boundary condition is valid when fields are calculated sufficiently farfrom the grid, the critical distance being the mesh size magnitude a.Thus the expressions based on (15) are not strictly valid at the grid z

0.

IV. CALCULATION OF THE DIPOLE IMPEDANCE

With the present image theory, we are able to derive an integralexpression for the change of the input impedance of a small verticalelectric dipole. The change AZ can be written as [14]

A Z/Ro = - (67r/k271 IL)Egz

RP= (ij/67r)(kL)j (19)

where Egz is the vertical electric-field component at the dipole (p =0, z = h) due to the grid. The electric field is obtained from theHertzian potential as

(20)

For the z component, we can write at p = 0

Egz = (k2+ d2/dZ2)H1g

0C (21)

The integral can be evaluated as an analytic expression containingexponential-integral functions Ei(z) of a complex argument. Toobtain a more convergent expression, however, it is wise to considerthe two-component image representation, whence the result for theimpedance change can be written as the following integrals:

AZ 3e- 2jkh ( 0

R = 1+4K24 tJfu| [(I +jkd)/(kd)3je(ik-fI)q dq

1-2 [(1 +jkd)/(kd)3] e(J2 - k) t dt} . (22)

Here, we denote d = 2h + q or = 2h - jt in the respectiveintegrals. Because the integrands are exponentially converging, theycan be integrated applying simple routines. As a check of the result(22), we may note that AZ 0 for the vanishing grid (K -° oo). Also,for the conductive plane (K 0), we have from (22) AZ/RO = 3j(1+ 2jkh) exp ( - 2jkh)/(2kh)3, which coincides with the result givenin [14]. A corresponding method for the Sommerfeld problem hasalso led to a simple procedure as was recently demonstrated byexamples [15].

109

E(r) = V x (VrI (r) x uz).

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-28, NO. 2, MAY 1986

V. TRANSMITTED FIELD

It is most straightforward to extend the present theory to give thefield on the other side of the grid. In fact, from (9) and the gridcondition (8), the Fourier-transformed potential for z < 0 is readilyobtained in the form

fl(z) =A (I -R)ej(z-). (23)

The field can now be thought of as arising from an image sourcelocated in the complex z' plane with Re(z') > 0. In fact, the in .-.ge iscomposed of a delta-function term (dipole image) plus a continuousimage, which equals the previous image with opposite sign:

I(z ') = IL6(r - uzh) +jILf( -jj(z ' - h)). (24)

The field below the grid can also be written through the reflectiontransformation of the scattered field above the grid:

1ig (p, z)=Fi(p, z) -Ilg+(P, -z) (25)

where Hi is the field without the grid.

VI. CONCLUSION

A novel image theory has been proposed for the problem of avertical electric dipole above a metallic planar grid applying theKontorovich-averaged boundary condition for the grid. In addition tobeing conceptually attractive, the theory is seen to produce simpleexpressions for the field close to the grid and for the impedancechange of the dipole due to the grid.

REFERENCES[1] J. R. Wait, "Theories of scattering from wire grid and mesh

structures," in Electromagnetic Scattering, P. L. E. Uslenghi, Ed.New York: Academic, 1978, pp. 253-287.

[2] D. A. Hill and J. R. Wait, "Electromagnetic surface wave propagationover a bonded wire mesh," IEEE Trans. Electromagn. Compat.,vol. EMC-19, no. 1, pp. 2-7, Feb. 1977.

[3] A. M. Barbosa, A. F. dos Santos, and J. Figanier, "Leaky wavessupported by a square wire mesh with bonded junctions above a reactivesurface," Radio Sci., vol. 17, no. 1, pp. 219-228, Jan.-Feb. 1982.

[4] B. Ya. Moizhes, "Electrodynamic averaged boundary conditions formetallic grids," Zh. Eksp. Teor. Fiz., vol. 25, no. 1, pp. 158-176,1955 (in Russian).

[5] M. I. Kontorovich, V. Y. Petrunkin, N. A. Yesepkina, and M. I.Astrakhan, "Reflection factor of a plane electromagnetic wave reflect-ing from a plane wire grid," Radiotekh. Elektron., vol. 7, no. 2, pp.239-249, 1962 (in Russian).

[6] M. I. Kontorovich, "Averaged boundary conditions on the surface of agrid with quadratic cells," Radiotekh. Elektron., vol. 8, no. 9, pp.1506-1515, 1963 (in Russian).

[7] M. I. Kontorovich and V. P. Akimov, "Averaged boundary conditionson the surface of a plane wire grid with nonorthogonal cells,"Radiotekh. Elektron., vol. 22, no. 6, pp. 1125-1135, 1977 (inRussian).

[8] I. V. Lindell and E. Alanen, "Exact image theory for the Sommerfeldhalf-space problem. Part I: Vertical magnetic dipole," IEEE Trans.Antennas Propagat., vol. AP-32, no. 2, pp. 126-133, Feb. 1984.

[9] I. V. Lindell and E. Alanen, "Exact image theory for the Sommerfeldhalf-space problem. Part II: Vertical electric dipole," IEEE Trans.Antennas Propagat., vol. AP-32, no. 8, pp. 841-847, Aug. 1984.

[10] T. B. A. Senior, "Approximate boundary conditions," IEEE Trans.Antennas Propagat., vol. AP-29, no. 5, pp. 826-829, Sept. 1981.

[11] D. S. Jones, The Theory of Generalised Functions. Cambridge,England: Cambridge Univ. Press, 1982, p. 79.

[12] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves.Englewood Cliffs, NJ: Prentice-Hall, 1973, pp. 509-510.

[13] T. K. Sarkar, "Analysis of arbitrarily oriented thin wire antennas overa plane imperfect ground," Arch. Elek. Ubertragung, vol. 31, no. 11,pp. 449-457, Oct. 1977.

[14] J. R. Wait, "Characteristics of antenna over a lossy earth," inAntenna Theory, Part 2, R. E. Collin and F. J. Zucker, Eds. NewYork: McGraw-Hill, 1969, ch. 23, pp. 329-435.

[15] E. Alanen and I. V. Lindell, "Impedance of vertical electric andmagnetic dipoles above a dissipative ground," Radio Sci., vol. 19, no.6, pp. 1469-1474, Nov.-Dec. 1984.

110