On the problem of dielectric-coated thin-wire antennas

Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 431 On the Problem of Dielectric Coated Thin Wire Antenna A. Adekola1 , A. I. Mowete1 , and A. Ogunsola1, 2 1 Department of Electrical and Electronics Engineering, Faculty of Engineering, University of Lagos Lagos, Nigeria 2 Parsons Group International, Rail Treansit Divison, London, United Kingdom Abstract— This paper addresses the problem of thin-wire antennas coated with thin layers of dielectric material, using a moment-method approach. First, it establishes that implicit in certain other moment-method solutions published in the literature is the assumption that the coated wire may be regarded as being essentially infinite in extent; and then further shows that another earlier solution, which removes this restriction, is limited by a related assumption, concerning the model for the dielectric insulation. The paper then reformulates the problem, using a quasi-static moment-method model, in which the two assumptions alluded to in the foregoing are eliminated. Computational data obtained for the input impedance and admittance of a reactively-loaded, coated thin-wire dipole, display features that are consistent with those reported in the literature, as being characteristic of this antenna type. 1. INTRODUCTION Thin wire antennas are often coated with dielectric materials to avoid direct contact between the metallic wire and the surrounding medium. They were first proposed for use in highly conducting media, exemplified by the utilisation of coated loops as low-frequency communication antennas mounted on submarines in sea water environments [1]. Dielectric coating is also useful as a means of electrically lengthening an antenna while maintaining the frequency characteristics and physical length [2, 3]; and in addition, they find applications in the determination of the equivalent cylindrical antennas for antennas whose cross-sections are non-circular [4], as well as in the treatment of cancer with microwave hyperthermia [1]. Numerous researchers have expended efforts in developing theoretical and experimental understanding of the physics underlying the electromagnetic behavior of dielectric coated thin wire antennas and scatterers. Bretones et al. [5], for example, utilised an existing DOTIG3 computer code for a time-domain investigation of the response of coated wire antennas to excitation by electromagnetic pulses. Their formulation suggested that for thin coatings with materials characterised by relatively low permittivities, a quasi-static model consisting of a radially directed polarisation current within the dielectric, and two layers of polarization charges, one each on the inner and outer cylindrical surfaces of the coating, becomes applicable. Further analytical development discarded contributions from the polarization current, to obtain a time-domain electric field integral equation, in which the effects of the coating were taken into account by the polarisation charges only. Moore and West [3], following the earlier treatment of the problem by Popovic and Nesic [4], developed a quasi-static approximation, through which electrically-thin coatings of either the dielectric or mixed dielectric-ferrite varieties, are transformed into equivalent magnetic coatings. In the moment-method solution provided by Richmond and Newman [6], the dielectric layer is modeled by an electric volume polarisation current, which derives from a radially directed electric field in the dielectric region, due to the distribution of current along the axis of the bare-wire antenna. Lee and Balmain [8], extended Richmond and Newman’s formulation, by introducing a magnetic volume polarisation current to include cases for which the insulating layer’s material has both electric and magnetic losses. Adekola and Mowete [7], pointed out that the expressions employed in [6] and [8], as approximations for the electromagnetic field in the dielectric region, are the same as those for static charge and steady current distributions of infinite extents, respectively. Based on that physical interpretation, they reformulated the approach suggested by Richmond and Newman [6], by modeling the dielectric insulation with an electric volume polarisation current, described in terms of the electric field of a uniform static charge distribution of the same extent as the physical length of the bare wire: that reformulation however, implicitly suggested that only the radial component of the volume polarization current need be considered. This paper also follows the earlier approach adopted by Adekola and Mowete [7], but advances arguments to suggest that not only is the contribution of the axial component of the electric PIERS Proceedings, Moscow, Russia, August 18–21, 2009 432 polarisation current dominant, but also that its radial component, as indicated by Bretones et al. [5], may be neglected. Numerical results obtained and described in graphical formats, for the input impedance and admittance of a coated dipole antenna display features, which have been reported in the literature [2, 6, 8], as being characteristic of coated thin-wire dipole antennas. 2. THEORITICAL FORMULATION The starting point of our moment-method formulation and solution of the coated thin-wire antenna problem is the proposal by [6] that the dielectric coating on the thin-wire structure may be modeled by a volume distribution of polarisation current, denoted by J and given by: J = jω (εd − ε0 ) E, (1) in which εd represents the permittivity of the dielectric material, and E, the electric field in the dielectric region. This electric field, according to [6], admits the approximate representation, in terms of the filamentary current distribution I(l) along the axis of the bare wire, given as: · ¸ ∂I(l)/∂l jωE = − ûρ , (2) 2πεd which, as pointed out by Adekola and Mowete [7], describes the electrostatic field of a uniform line charge distribution σ(l), of infinite extent, when it is recognised that the equation of continuity specifies that ∂I(l) = jωσ(l) (3) − ∂l It is a matter of interest to observe that Lee and Balmain [8], arrived at the same expression as (2), by suggesting that the dielectric coating may be modeled by H= I(l) ûφ , 2πρ (4) an expression, which as pointed out elsewhere [7], describes the static magnetic field of a uniformly distributed steady electric current carried by a conductor of infinite extent. In this analysis, we adopt a modification of the solution proposed in [6], by suggesting that the electric field of (2) derives from a filamentary charge distributed over the finite extent of the bare wire, and related to the filamentary current carried by the wire through (3), the equation of continuity; that is: ! Z à σ (l0 ) ρdl0 1 σ (l0 ) l0 dl0 (5) E= ûρ − ûl0 4πεd (l02 + ρ2 )3/2 (l02 + ρ2 )3/2 L I(l0 ) Because is an unknown, so is σ(l0 ); but the method of moments [9] offers an approximate solution to I(l0 ) in a technique that starts with its series expansion in terms of known functions Gk (l), and unknown expansion coefficients αk according to I(l) = N X αk Gk (l) (6) k=1 In this analysis, we choose the piece-wise linear functions described by Kuo and Strait [10], for which the nth member of the series expansion of (6) is given by: ( l−l2n−1 l2n−1 ≤ l ≤ l2n+1 l2n+1 −l2n−1 , Gn (l) = (7a) l2n+3 −l l2n+1 ≤ l ≤ l2n+3 l2n+3 −l2n+1 , so that we may then describe the charge distribution of (5) in terms of the piece-wise uniform functions given by ( 1 l2n−1 ≤ l ≤ l2n+1 ∂Gn (l) l2n+1 −l2n−1 , 0 Gn (l) ≡ = (7b) −1 ∂l l2n+1 ≤ l ≤ l2n+3 l2n+3 −l2n+1 , Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 433 A substitution of (7b) for Gn into (5) then yields: !  à l2n+1 l2n+3 R R ρdl0 ρdl0 1 1 − l2n+3 −l2n+1 ûρ  l2n+1 −l2n−1 (l02 +ρ2 )3/2 (l02 +ρ2 )3/2 −1 1  l2n−1 l2n+1  à ! En = l2n+1 l2n+3 jω 4πεd  R R 0 0 0 0  l dl 1 l dl + l2n+1−1 + l2n+3 −l ûl −l2n−1 2n+1 (l02 +ρ2 )3/2 (l02 +ρ2 )3/2 l2n−1       (8) l2n+1 so that with (1), we find that: ¶    ³ ´µ l2n+1 l2n−1 1    −  1/2 − 1/2  2 2   (l2n+1 −l2n−1 ) l2n+1 +ρ2 ) l2n−1 +ρ2 ) ( ( ûρ  µ ¶ ³ ´   ρ  l l 1 2n+3 2n+1     1/2 − 1/2   (l2n+3 −l2n+1 ) 2 2 2 2 (l2n+3 +ρ ) (l2n+1 +ρ ) − (εd − ε0 )  µ ¶    ³ ´ Jn =   1 1 1    2εd  1/2 − 1/2  2 2   l2n+1 −l2n−1 l2n+1 +ρ2 ) l2n−1 +ρ2 ) ( (  + ¶ µ ûl ³ ´    −1 1 1    −   1/2 1/2  l2n+3 −l2n+1 2 2 +ρ2 ) +ρ2 ) (l2n+3 (l2n+1                (9) According to Harrington and Mautz [11], for a thin dielectric shell with a large dielectric constant, the polarization current’s normal component may be neglected and the tangential component expressed in terms of a net surface current. Also, the quasi-static solution described by Bretones et al. [5], indicate that the normal (radial) component of the polarization current is negligible. Accordingly, for each expansion function, Gnw , defined for the bare wire by (7a), we defineRR a model for the dielectric insulation, a corresponding expansion function, Gnd , given as Gnd = ûl s Jl ρdρdφ, so that from (9), we find that ( ¡   ¢1/2 ¡ 2 ¢1/2 ) 2 2 b2 + l2n+1 − a + l2n−1 ¡ ¢ ¡ 2 ¢  ζn  2 2 − b2 + l2n−1 + a + l2n−1  0 0 (ε0 − εd )    Gn (l ) ( ) Gnd = (10) ¡ ¢ ¡ ¢   1/2 1/2 2 2 2εd b2 + l2n+3 − a2 + l2n+3   +ζn+1 ¡ ¢1/2 ¡ 2 ¢1/2 2 2 − b2 + l2n+1 + a + l2n−1 in which the Neumann functional symbolised by ζ is defined by: ½ 1, l2n−1 ≤ l0 ≤ l2n+1 ζn = 0, elsewhere Figure 1: Dielectric coated thin wire. (11) PIERS Proceedings, Moscow, Russia, August 18–21, 2009 434 so that ½ ζn+1 = l2n+1 ≤ l0 ≤ l2n+3 elsewhere 1, 0, (12) where “a” and “b” in (10) represents the radii of the thin wire and dielectric shell, respectively, as shown in the problem geometry of Fig. 1. For the composite coated-wire structure therefore, we define the total current in terms of the series expansion X IT = Iwn + Idn = Gwn + Gdn , (13) n with the subscripts “w” and “d” referring to bare wire and dielectric insulation, respectively, and in which Gwn and Gdn are given, respectively, by (7) and (10). Hence, the moment-method solution [9] to the problem emerges as: µ ww wd ¶−1 Zmn ∆Zmn [IT ] = [Vn ] (14) dw ∆Z dd ∆Zmn mn where · ¸ Z Z 0 ¡ ¢ 1 dGmw (l) dGnw (l0 ) e−jk|r−r | ww Zmn = dl0 dl jωµGmw (l) Gnw l0 + (15) jωε dl dl0 4π |r − r0 | axis C and Z wd ∆Zmn = Z dl 0 ¡ ¢¤ e−jk|r−r0 | £ dl jωµGmw (l) Gnd l0 4π |r − r0 | (16) C axis provided that “c” is a line on the surface of the bare wire parallel to the wire’s axis [10]. 100 0 Input Resistance (Ohm) Input Resistance (Ohm) 800 600 400 200 0 0.2 0.4 0.6 0.8 1 -100 -200 -300 -400 -500 0.2 1.2 0.4 Normalized Length 0.6 (a) 1 1.2 1 1.2 (b) 10 Input Susceptance (mMho) 12 Input Conductance (mMho) 0.8 Normalized Length 10 8 6 4 5 0 2 0 0.2 0.4 0.6 0.8 1 1.2 -5 0.2 0.4 0.6 0.8 Normalized Length Normalized Length (c) (d) Figure 2: Comparison of the input impedance and admittance characteristics of the thin wire dipole antenna and its dielectric coated version (- - - bare wire, — coated wire). Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 435 3. RESULTS AND DISCUSSSIONS For the computation of numerical results, we used an adaptation of the FORTRAN computer program described by Kuo and Strait [10], for a bare reactively loaded thin-wire of length 0.5 m and radius 3.37 mm. The numerical results described in the ensuing discussions are for this same wire, when uniformly coated by a homogeneous, electrically lossy dielectric, of varying thickness. Displayed in Fig. 2 below are graphical representations of computation data obtained for the input impedance and admittance of the coated wire, for a dielectric material of relative permittivity εr = (4.6 − j4.7), and whose coating thickness b = 6a. It is readily observed from Fig. 2(a) that the input resistance values for the bare wire are less than those for the coated wire below first resonance. In addition, peak resistance value for the bare wire is less than that for the coated wire, and since the latter occurs before the former, it may be said that one effect of the coating is to lower resonant frequency, and these observations are consistent with those reported elsewhere [6, 8]. The input reactance curves of Fig. 2(b) also exhibit the same general behavior as those described in the foregoing, for input resistance. In this case, however, we observe that the difference between values of reactance for the coated and bare dipoles differ sharply for values of normalized length (l/λ) between 0.6 and 0.7. Input conductance characteristics for the bare and insulated dipole antennas are displayed in Fig. 2(c) from which we find, as noted by Richmond and Newman [6], that the effect of insulation is to shift conductance, such that it is between that of a bare wire in free space and a bare wire in a homogeneous medium, whose permittivity is the same as that of the insulating material. The curves of Fig. 2(d) for input susceptance exhibit the same general features as those for input reactance. Unlike input reactance, however, the sharp difference in input susceptance values for the bare and coated dipole antennas. The numerical results obtained for the reactively loaded coated thin-wire are displayed in the response curves of Fig. 3. 1200 500 Input Resistance (Ohm) Input Resistance (Ohm) 1000 800 600 400 0 -500 200 0 0.2 0.4 0.6 0.8 1 -1000 0.2 1.2 0.4 Normalized Length 0.6 (a) 1.2 1 1.2 10 10 Input Susceptance (mMho) Input Conductance (mMho) 1 (b) 12 8 6 4 2 0 0.2 0.8 Normalized Length 0.4 0.6 0.8 Normalized Length (c) 1 1.2 5 0 -5 0.2 0.4 0.6 0.8 Normalized Length (d) Figure 3: Comparison of the input impedance and input admittance characteristics of a reactively loaded thin wire dipole antenna and its dielectric coated version (- - - bare wire, — coated wire). PIERS Proceedings, Moscow, Russia, August 18–21, 2009 436 1200 400 1000 200 Input Resistance (Ohm) Input Resistance (Ohm) The dimensions of the wire and coating thickness remain the same as before, but in this case, the antenna was loaded with an inductive load of 1 kΩ, as described by Kuo and Strait [10]. It is clear from the curves that the effects of dielectric insulation are more pronounced in this case of the reactively loaded dipole that was the case with the unloaded dipole. A comparison of Figs. 2(a)–2(d) and 3(a)–3(d), for example, suggests that bandwidth is significantly reduced, when the antenna carries an inductive load. From Fig. 3(a), we find that apart from the reduced bandwidth, peak input resistance for the coated wire is now significantly much larger than peak input resistance for the bare wire. A similar observation is true for peak input conductance, which as seen from Fig. 3(c), has significantly larger values for the bare wire than for the insulated wire. Input reactance characteristics share certain features with input resistance characteristics including the fact that peak reactance at first resonance is larger for the bare wire than for the coated wire. After the first resonance, when reactance may be said to be capacitive, we find that reactance varies more smoothly in the case of reactive loading, than for the case of the unloaded antenna. And it is also of interest to observe that peak capacitive reactance values in Fig. (3b) occur at much closer values of normalized length (and hence frequency of operation) than was the case with Fig. 2(b). In order to examine the effects of coating thickness on input impedance and admittance characteristics, the numerical results displayed in Fig. 4 were compiled for the reactively loaded dipole, using three different values (b = 6a, b = 4a, and b = 2a) of coating thickness. Figure 4(a) reveals that before first resonance, input resistance increases with increasing coating thickness, or equivalently, increasing coating thickness shifts the input resistance curve to the left. It is noteworthy to observe that values of peak input resistance for b = 6a and b = 4a are close; and that they are both significantly greater than the value of peak input resistance for b = 2a. The input reactance response curves of Fig. 4(b) also indicate an increase in values with increase in coating thickness, before first resonance; though the differences are not as large as those recorded for input 800 600 400 200 0 0.2 0 -200 -400 -600 -800 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 -1000 0.2 1.2 0.3 0.4 Normalized Length 0.5 0.6 0.8 0.9 1 1.1 1.2 1 1.1 1.2 (b) 10 10 8 Input Susceptance (mMho) Input Conductance (mMho) (a) 12 8 6 4 2 0 0.2 0.7 Normalized Length 6 4 2 0 -2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized Length Normalized Length (c) (d) Figure 4: Variation of input impedance and input admittance with coating thickness (*** b = 2a; — b = 4a; - - - b = 6a). Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 437 resistance, possibly because of the 1 kΩ inductive load carried by the dipole. It is interesting to observe from the input conductance curve of Fig. 4(c) that whereas the general behavior is similar to that described in the foregoing for input resistance and reactance, peak input conductance for b = 2a is significantly larger than peak input resistance for both b = 4a and b = 6a, at first resonance. On the other hand, we find from Fig. 4(d) that peak input susceptance is virtually the same at first resonance for all three values of coating thickness considered in this paper. View publication stats 4. CONCLUSION This concludes our analysis of the problem of coated thin-wire antennas. It has been shown in the paper that certain published frequency-domain (Moment-Method) and time-domain solutions of the problem indicate that it admits a quasi-static solution in terms of a charge distribution that serves as model for the coated wire’s dielectric insulation. By giving a physical interpretation to the analytical frequency-domain solutions alluded to above, this paper reformulated the problem to suggest that the solution should be in terms of a quasi-static charge distribution, which derives directly from the approximate current distribution utilized for the bare wire, and whose extent is the same as the wire’s physical length. Numerical results were obtained for the input impedance and input admittance characteristics of a reactively loaded thin-wire dipole antennas, and the results were in agreement with findings in the open literature, that the effects of insulating a thin-wire antenna include lowering resonant frequency, increasing peak admittance, and narrowing bandwidth. Our paper also examined the effects of varying coating thickness, and the results obtained indicate that in general, whereas peak input resistance, reactance, and susceptance decrease with increase in coating thickness, peak input conductance increases with decrease in coating thickness. REFERENCES 1. Hertel, T. W. and G. S. Smith, “The insulated linear antenna-revisited,” IEEE Trans. Antennas Propag., Vol. 48, No. 6, 914–919, June 2000. 2. Lamensdorf, D., “An experimental investigation of dielectric coated antennas,” IEEE Trans. Antennas Propag., Vol. 15, No. 6, 767–771, 1967. 3. Moore, J. and M. A. West, “Simplified analysis of coated wire antennas and scatterers,” Proc. IEE Microw. Antennas Propag., Vol. 142, No. 1, 14–18, February 1995. 4. Popovic, B. D. and A. Nesic, “Generalisation of the concept of equivalent radius of thin cylindrical antenna,” IEE Proc., Vol. 131, No. 3, 153–158, 1984. 5. Bretones, A. R., A. Martin, R. Gomez, A. Salinas, and I. 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