Dielectric coated wire antennas

Radio Science, Volume 11, Number 1, pages 13-20, January 1976

Dielectric coated wire antennas

J. H. Richmond and E. H. Newman

ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University, Columbus, Ohio 43212

(Received July 22, 1974.)

The problem considered is an electrically thin dielectric insulating shell on an antenna composed of electrically thin circular cylindrical wires. The solution is a moment method solution and the insulating shell is modeled by equivalent volume polarization currents. These polarization currents are related in a simple manner to the surface charge density on the wire antenna. In this way the insulating shell causes no new unknowns to be introduced, and the size of the impedance matrix is the same as for the uninsulated wires. The insulation is accounted for entirely through a modification of the symmetric impedance matrix. This modification influences the current distribution, impedance, efficiency, field patterns, and scattering properties. The theory will be compared with measurement for dielectric coated antennas in air.

1. INTRODUCTION

For a wire antenna in a conducting medium the radiation efficiency can often be improved by insulating all or part of the wire from the medium. This is accomplished with a thin dielectric layer coated on the wire surface. This paper considers the electromagnetic modeling of the dielectric layer or shell.

The problem of a circular insulating shell on a thin wire antenna has received considerable atten-

tion in the literature [ Wu et al., 1973; King, 1964; Iizuka, 1963; King et al., 1974]. Unfortunately, much of this work is restricted as to the geometry of the antenna or the case where the complex permittivity of the exterior medium is much greater than that of the insulating layer. A simple theory is presented here which models the insulating shell by equivalent volume polarization currents, and which is restricted to electrically thin insulating layers covering electrically thin wire antennas or scatterers [Richmond and Newman, 1974; Rich- mond, 1974a]. Unlike previous theoretical and experimental work, the insulating shell is considered to terminate at the ends of the wire structure.

The solution presented is a modification of the piecewise-sinusoidal reaction formulation [Rich- mond, 1974a] for bare or uninsulated wires. Thus, a brief review of the theory for bare wires is presented first. Next it is shown how the theory for the bare wires can be modified to account for

the insulating layer. Finally, a comparison is made between measured and calculated admittance.

2. BARE WIRE STRUCTURE

This section briefly reviews some aspects of the piecewise-sinusoidal reaction formulation for uninsulated thin-wire antennas or scatters in a homo-

geneous conducting medium. A more complete treatment is available elsewhere [Richmond, 1974a].

Let S denote the closed surface of the wire

structure, which is composed of a number of straight segments, and let V denote the interior volumetric region. In the presence of the wire, an external' source (Ji,Mi) generates the field (E,H). When radiating in the homogeneous medium (ix, •) without the wire, this source generates the incident field (E •, It •). (E "•, It "•) denotes the field of an electric test source located in V and radiating in the homogeneous medium (ix, •). All sources and fields are considered to be time-harmonic with the same

frequency. The time dependence e io, t is suppressed. In the wire structure, let each segment have a

circular cylindrical surface. At each point on the composite cylindrical surface of the wire, it is convenient to define a right-handed orthogonal coordinate system with unit vectors (fi, •, •) where • is the outward normal vector, •'is directed along the wire axis and

Copyright ̧ 1976 by the American Geophysical Union. 4, = •' x • (•)

13

14 RICHMOND AND NEWMAN

Thus (fi,•,•) correspond directly with the unit vectors (f•,4>, •) usually employed in the circular- cylindrical coordinate system.

To simplify the integral equation, we assume the wire radius a is much smaller than the wavelength h, and the wire length is much greater than the radius. Furthermore, we shall neglect the integrations over the flat end surfaces of the wire, neglect the circumferential component J. of the surface- current density, and consider the axial component J• to be independent of {b. (For thick wires, a more detailed treatment is essential for the (b-dependent current modes and the integrations over the junction regions and the open ends of the wire. A more elaborate formulation may also be required if one wire passes within a few diameters of another, or if a wire is bent to form a small acute angle.) In view of these approximations the reaction integral equation [Rumsey, 1954] reduces to

I(l)(E["- ZsH•) dl = V m (2)

where /is a metric coordinate measuring position along the wire axis, L denotes the overall wire length, Zs is the surface impedance for exterior excitation, I(l) is the total current (conduction plus displacement), and

ß H m ) dv (3)

(4)

(5)

The sinusoidal reaction formulation for thin wires

is based on the integral equation (2). In this equation the known quantities are E re,It m, V,,, and Z s. The current distribution I(l) is regarded as an unknown function. To permit a solution for the current distribution, suitable test sources and expansion modes are now defined.

For a test source we choose a filamentary electric dipole with a sinusoidal current distribution. This is not a wire dipole, but merely an electric line source in the homogeneous medium. The sinusoidal dipole is probably the only finite line source with simple closed-form expressions for the near-zone fields. Furthermore, the mutual impedance between two sinusoidal dipoles is available in terms of

z)l

Zl z 2 Z3

Fig. 1. A linear test dipole and its sinusoidal current distribution. The endpoints are at Z l and z3 with terminals at z2.

exponential integrals, and the piecewise-sinusoidal function is evidently close to the natural current distribution on a perfectly conducting thin wire. These factors governed the choice of test sources. To simplify integrations in equations 4 and 5, we choose to locate the test dipole on the wire axis.

For the linear test dipole illustrated in Figure 1, the current distribution is I (z) = F (z) where

F(z) = [• P• sinh •/(z - z•)]/sinh•/d•

+ [ • P2 sinh •/(z3 - z)]/sinh •/d 2 (6)

P1 (Z) is a pulse function with unit value for z l < z < z2 and zero value elsewhere. The pulse function P2 (z) has unit value for z2 < z < z3 and vanishes elsewhere. The segment lengths are dl= z2 - z 1 and d2 = z3 - z2. The current distribution on a V test dipole is

F(/) = [[• P• sinh•/(/- l•)]/sinh•!d•

+ [ [2 P2 sinh •/(/3 - /)]/sinh •/d 2 (7)

In (6) and (7), •/denotes the complex propagation constant of the homogeneous exterior medium:

• = jto (•)'/2 (8)

It is only with this value for •/that the sinusoidal test sources have the advantages mentioned earlier.

A typical problem requires not just one but several test dipoles located at different positions along the wire axis to form an overlapping array. Using N test dipoles, (2) is enforced for each one. Thus, (2) represents a system of N simultaneous integral equations with m = 1, 2, ..., N. In other words, (2) requires each test dipole in the array to have the correct reaction with the true source. Since the

test dipoles and expansion modes both form an

overlapping array, wire junctions require no special treatment.

The current distribution on the wire structure is expanded in a finite series as follows'

N

I (l) -- Z InF n (l) n=l

(9)

where the normalized expansion functions F,(/) are the same as the test-dipole current distributions in (7). Since each expansion function extends over just a two-segment portion of the wire structure, these functions are subsectional bases. Since N is finite, (9) may be considered either as an expansion or an approximation, depending on the context. In (9), the coefficients I, are complex constants which represent samples of the current function I(l).

By inserting (9) into (2), we obtain the following system of simultaneous linear algebraic equations:

N

•InZmn---V m where m= 1,2,...,N (10)

Z,.. = - f F.(/)(E•"- ZsH•)dl (11) In (tt), the integral extends over the two segments in the range of the expansion mode F•. Equation t0 can be expressed in matrix form as ZI = V where Z denotes the symmetric sqt•are impedance matrix, I is the current column and V is the voltage column. This matrix equation can be solved for the current column I. Inserting the components of I into (9) yields an approximation for the current on the wire. structure. Once the current is known, it is straightforward to determine impedance, far- field patterns, or other quantities of interest.

3. INSULATED WIRE STRUCTURE

In this section it is shown how the matrix equation for the bare wire structure can be modified to account for the presence of a thin dielectric insulating layer. A typical cross section for an insulated wire is shown in Figure 2. Initially the insulating layer is considered to be homogeneous and with a circular cross section of radius b.

For simplicity, let the dielectric shell have the same permeability as the ambient medium. From the volume equivalence theorem, the dielectric shell may be replaced with ambient medium and an

DIELECTRIC COATED WIRE ANTENNAS 15

iIGHLY CONDUCTING WIRE

Fig. 2. A cross section of a dielectric coated wire.

equivalent source with electric current density

J = ./to (•: - •) E (12)

where E denotes the electric field intensity in the shell and E2 and E are the complex pe•:mittivities of the shell and the ambient medium, respectively. From (12), the current J vanishes outside the region of the dielectric shell.

Let (E,H) denote the field generated by (Ji, Mi) in the presence of the insulated wire. Outside the wire, this field may also be generated by (Ji,M•), (J •,M•), and J, radiating in the homogeneous medium. These sources, radiating in the homogeneous medium, generate a null field in the interior region of the wire. The surface currents (J •,M •) are located on the surface of the wire and are related to the field (E,H) by [Schelkuno[[, 1939]:

J s = h x H (13)

Ms=EXfi (14)

For the insulated wire, the reaction integral equation (2) is modified by replacing J i with J i + J. The current J may be regarded as an additional source which plays much the same role as the impressed source J•. However, J i is considered to be a known source whereas J is unknown because E is unknown. If the dielectric shell is thin, J may be regarded as a dependent unknown function because it is simply related to the current distribution on the wire.

Using the equation of continuity [Harrington, 1961], the charge density on the wire surface is related to the current by

Os = -I' (l)/•o•2•ra (15)

in which I' denotes dI/dl. For perfectly conducting wires, the O component of the electric flux density, D o, at the surface of the wire is equal to O s. We use the approximation that Dp(o = a)= 0s for


wires with high but not necessarily perfect conductivity. In this case the electric field on the surface of the wire is approximated by

E(p--- a) --- • PsiS2 --- -• I' (l)/]co2•ae 2 (16)

The t and • components of E are considered to be negligible in the dielectric shell. To extrapolate from this value at the wire surface to points in the insulating layer, we use a 1 /p radial dependence and obtain from (16):

E = -b I' (/)/jto2'rrE2p

From (12) and (17),

(17)

J = -(E2 - ½)15 I' (/)/2'rre2p (18)

For an insulated wire, each expansion mode, F, (l), has associated with it a shell of radial electric current

J. Thus, from (9) and (18), the mutual impedance Z.•. between the filamentary test dipole m and the tubular expansion dipole n has an additional term given by

ff (•2- •) A Zmn -'- • F'. (l) E•"(p, l) dp dl (19) n •2

where the integration extends through the dielectric shell in the range of the expansion dipole n. In deriving (19), the integration on • was performed with the assumption that J and E•' are independent of •. In integrating on p, the limits are a and b which denote the inner and outer radii of the

dielectric shell.

In the dielectric shell, the test-dipole field E•" may be approximated by

Em -_. t • - F m (l) / 2•r ]toep (20)

In (19) and (20), O denotes distance from the axis of dipole n or m, respectively. Furthermore, the vector direction of the field component E•" in (19) generally differs from that in (20) unless dipoles m and n are colinear. If (20) is employed, (19) reduces to

P f - F• (l) F'. (l) dl (21)

where (m,n) denotes the region of l shared by dipoles m and n, and the dimensionless parameter P is given by

• (•2 - •) (•2 - •) P = • dp= • In (b/a) (22) a E2P E2

Note that the elements of AZ,•, are very easy and fast to evaluate on a digital computer since the integration in (21) is available in closed form and in terms of simple functions (hyperbolic sines and cosines).

The current distribution on the wire in the presence of the insulating layer is found by solving the matrix equation

(Z + AZ) I = V (23)

where the elements of Z are given by (11) and the elements of AZ are given by (21). AZ,•, is zero if dipoles m and n do not share a segment. Also, that part of the integration in (21) which is over a portion of the wire which is not insulated is zero. Thus, (21) is sufficiently general to handle partly insulated and partly bare wires. The approximate result represented by (21) yields a matrix Z which is symmetric even when some of the wire segments are insulated and others are bare.

Equation 19 is sufficiently general to treat an inhomogeneous insulating layer where e2 varies as a function of p and I only.

If e2 is a function of •, the analysis will be complicated because the current densities J and J• will be functions of •. Therefore this case is not considered.

If e 2 is simply a function of p, then the only modification required is that e 2 in (22) be considered as a function of p. If e2 has a sufficiently simple radial dependence, then P can still be obtained in closed form. For example, consider an L-layered insulating shell. If we use the notation that the ith layer extends from p• to p•+• and has uniform permittivity e •, then (22) becomes

P= Z •ln(pi+l/P i) (24) i=1 •i

Thus by using (24) in (21) it is a simple matter to treat multilayered insulating shells.

4. NUMERICAL EXAMPLES

In this section numerical computations based on the theory presented above are compared with measurements. The computer program used for all computations and a user's manual describing these programs is available [Richmond, 1974b,c]. All


4O

3O

CALCULATED ß ß ß MEASURED BY LAMENSDORF

GROUND PLANE

- COAX "I',M / -- FEED I' 2o -

o.7s'•., ...... •.••,.,,,,_,•,• • . -

2o: 0.25" _

ß • 2b= 0.5" f = 600.0 MHz -

•a = 9-OEo

ß --

,I 0.2 0.3 0.4 0.5 0.6

MONOPOLE HEIGHT

Fig 3. The conductance of an insulated monopole plotted versus its height in wavelengths.

antenna feeds are modeled by delta-gap generators. The ambient medium for all measurements and

computations is free space. No more than N = seven current expansion modes were required for the computations to be presented.

The insert in Figure 3 illustrates the geometry for the dielectric coated monopole measured by Lamensdorf [ 1967]. His measurements were made at [ = 600 MHz and with a monopole of diameter 0.25 inches. The dielectric sleeve extended beyond the monopole to simulate an extension to infinity.

E I0 E

0

--I0 0.1 0.2 0.3 0.4 0.5 0.6

MONOPOLE HEIGHT h/x

Fig. 4. The susceptance of the insulated monopole, shown in Figure 3, plotted versus its height in wavelengths.

Lamensdor[ [ 1967] measured the monopole admittance for dielectric sleeves with diameters from

0.25 to 2.0 inches and relative dielectric constants

from 3.2 to 15.0. Generally speaking, our computations agree well with Lamensdor['s [1967] measurements for the thinner dielectric sleeves and the

smaller permittivities. The agreement deteriorates as the sleeve diameter increases or the permittivity increases. Figures 3 and 4 show the measured and calculated monopole admittance plotted versus the monopole height in wavelengths for the case 2b = 0.5 inches and E 2 = 9.0Eo. The agreement between experiment and theory is good for both the conductance, G, and the susceptance, B. As mentioned above, in the theoretical model the dielectric sleeve terminates at the tip of the monopole. Lamensdorf [1967] indicated that the extensions did affect the monopole's admittance, and its effect was greatest near resonance. Thus, it is possible that the discrepancies between our calculations and Lamensdorf's [1967] measurements for the thicker and denser dielectric layers resulted from the fact that our model does not include the extension of the dielec-

tric.

The remaining measurements were made at The Ohio State University. The antennas were con- structed by removing the outer insulation and copper braid from RG59/U coaxial cable. Then for the remaining examples the diameter of the copper wire is 2a = 0.025 inches, the diameter of the insulating layer is 2b = 0.146 inches, and the permittivity of the insulating layer is • 2 = 2.3•o. The measurements were made with monopoles, or half-loops mounted - on a two-ft square aluminum ground plane whose edges were terminated in six-inch diameter cylinders to reduce edge reflections. The inner and outer diameter of the coaxial feed was 1/8 inch and 3/8 inch, respectively. The calculations were made for dipoles and complete loops, and it is the dipole or loop admittance which is plotted.

Figures 5 and 6 compare the measured and calculated admittance for a coated dipole of length L = 8 inches. Agreement between experiment and theory is good.

Figures 7 and 8 compare the measured and calculated admittance for a square loop of perimeter L = 8 inches. In Figures 9 and 10 the admittance of a loop identical to that shown in Figure 7, except that the insulation is removed from two arms, is shown. The good agreement between theory and experiment in Figures 7-10 illustrates the ability


18

16 /

DIELECTRIC COATED

DIPOLE

L =8" 20 =0.025" 2b =0. I46"

E z -- 2.3E o

CALCULATED

ß ß ß MEASURED

AT O.S.U.

0.1 0.2 0.:5 0.4 0.5 0.6 0.7

DIPOLE LENGTH L•),

Fig. 5. The conductance of an insulated dipole plotted versus its length in wavelengths.

of the computer programs to treat geometries other than simple dipoles as well as partially coated antennas.

IO /

E E

• 0

--2 /

m4

•6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

DIPOLE LENGTH L/x

Fig. 6. The susceptance of the insulated dipole, shown in Figure 5, plotted versus its length in wavelengths.

IELECTRIC COATED

i SQUARE LOOP

o

'- L -- 8" IE 5 E 2 a = 0,025"

•' 2b =0. 146"

o i Ez '- 2.3Eo CALCULATED .

ß ß ß MEASURED /

0 0.2 0.4 0.6 0.8

I

I I I.O 1.2

LOOP PERIMETER

I I I

Fig. 7. The conductance of an insulated square loop plotted versus its perimeter in wavelengths.

Calculations not shown here were made to

compare with measurements by Iizuka [1963] of insulated monopoles in water solutions. (In these calculations the feed was modeled by a magnetic frill current and not a delta-gap generator.) While the calculated and measured admittances generally followed the same locus, a significant resonant shift was noticeable. As was the case with Lamensdorf's [ 1967] measurements, the insulating sleeve in Iizu- ka's [ 1963] experiment extended beyond the rnon- opole. Thus the discrepancies between our calculations and Iizuka's [1963] measurements could be explained in the same manner as was suggested above to account for differences between our cal-

culations and some of Lamensdorf's [1967] measurements. A second possible explanation is that the theory presented here is inadequate to treat the case of insulating layers with permittivity vastly different from that of the ambient medium.

Comparing (21) and (22) or (24) it can be seen that the diameter and permittivity of the insulating layer affect AZ,• only through the dimensionless parameter P. Thus, to the approximations being used here, several insulating shells with different


10

2

--4-

16 .

--8-

'l

0 0.2 0.4 0.6 0.8 1.0 LOOP PERIMETER

I

Fig. 8. The susceptance of the insulated square loop, shown in Figure 7, plotted versus its perimeter in

wavelengths.

'l'l I ' i PARTIALLY COATED j• -

4 Ez = 2.3•o I 1 - ß ß ß ß ß MEASURED • • -

2

I

0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 .6

LOOP PERIMETER

Fig. 9. The conductance of a partially insulated square loop plotted versus its perimeter in wavelengths.

I0 i I I

0 0.2 0. 4 0.6 0.8 1.0 1.2 1.4 1.6 LOOP PERIMETER L/).

Fig. 10. The susceptance of the partially insulated square loop, shown in Figure 9, plotted versus its perimeter in wavelengths.

diameters and densities will have the same effect on an antenna if the value of P associated with each shell is identical. Figure 11 shows the resonant length and conductance of a dipole plotted versus P. Here it can be seen that the conductance at

resonance increases almost linearly with increasing P, while the resonant length decreases almost linearly.

0.50

19

L//o = 500 0.48 • o i.iJ z

z o <[ 0.46• z i.u

o •17

r• 0.44

• --16 o

•-,• 0.42 • 15----

0.40

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 P

Fig. 11. The resonant length and conductance at resonance for an insulated dipole plotted vs. the dimensionless parameter P.


5. CONCLUSIONS

A simple method is presented to account for a thin insulating shell on a thin-wire antenna or scatterer. The shell is modeled by equivalent volume electric polarization currents. These polarization currents are simply related to the current on the wire structure. The insulating layer is accounted for by modifying the impedance matrix for the bare wire. Thus no increase in computer storage is required except that which is required to describe the insulating layer and to evaluate (21). The modification is obtained in terms of simple functions and is therefore easy to program on a digital computer. If no advantage is taken of symmetries, then (21) must be evaluated only N times for a given problem. Thus the modifications require very little computer time. Because of the simple approximation used for the electric field in the insulating layer, the theory was shown to be applicable to certain inhomogeneous insulating layers. Two important examples of inhomogeneous insulation which were treated are multilayered coatings and partially coated antennas.

The qualitative effect of placing an insulating layer, with permittivity E 2, on an antenna is to shift its admittance (or impedance) so that it is between the admittance of the bare antenna and the admit-

tance of the antenna in a homogeneous medium with permittivity E2- Specifically, the effects of placing an insulating layer, with permittivity greater than that of the ambient medium, on an antenna are to (a) lower the resonant frequency, (b) increase the peak admittance, and (c) narrow the bandwidth. If the insulating layer has permittivity less than that of the ambient medium, the effects are opposite to those indicated above.

An important advantage of the method presented is that it can be applied to any antenna geometry which could be treated in the absence of the

insulating layer. Thus one can treat insulated di-

poles, loops, crossed wires, spirals, etc. Further, the wires may have finite conductivity or contain lumped loads.

Acknowledgments. This work was supported in part by grant NGL 36-008-138 between National Aeronautics and Space Ad- ministration, Langley Research Center, Hampton, Virginia, and The Ohio State University Research Foundation, Columbus, Ohio.

REFERENCES

Harrington, R. F. (1961), Time-Harmonic Electromagnetic Fields, p. 2, McGraw-Hill, New York.

Iizuka, K. (1963), An experimental study of the insulated dipole antenna in a conducting medium, IEEE Trans. Antennas Propagat., AP-11, 518-532.

King, R. W. P. (1964), Theory of terminated insulated antenna in a conducting medium, IEEE Trans. Antennas Propagat., AP-12, 305-318.

King, R. W. P., K. M. Lee, S. H. Mishra, and G. S. Smith (1974), Insulated linear antenna: Theory and experiment, J. Appl. Phys., 45, 1688-1697.

Lamensdorf, D. (1967), An experimental investigation of dielectric coated antennas, IEEE Trans. Antennas Propagat. , AP- 15, 767-771.

Richmond, J. H. (1974a), Radiation and scattering by thin-wire structures in the complex frequency domain, NASA Contr. Rep. CR-2396, available from National Technical Information Service, Springfield, Virginia 22151.

Richmond, J. H. (1974b), Computer program for thin-wire structures in a homogeneous conducting medium, NASA Contr. Rep. CR-2399, available from National Technical In- formation Service, Springfield, Virginia 22151.

Richmond, J. H. (1974c), Radiation and scattering by thin-wire structures in a homogeneous conducting medium, IEEE Trans. Antennas Propagat., AP-22, 365.

Richmond, J. H., and E. H. Newman (1974), Dielectric coated wire antennas, paper presented at the 1974 Spring Meeting of URSI, Atlanta, Georgia.

Rumsey, V. H. (1954), Reaction concept in electromagnetic theory, Phys. Rev., 94, 1483-1491.

Schelkunoff, S. A. (1939), On diffraction and radiation of electromagnetic waves, Phys. Rev., 56, 308-316.

Wu, T. T., R. W. P. King, and D. V. Giri (1973), The insulated dipole antenna in a relatively dense medium, Radio $ci., 8, 699-709.

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Dielectric coated wire antennas