06202515 Current Distribution on Finite Helix[Limiting Current at Ends]

IEEE Antennas and Propagation Magazine, Vol. 54, No. 1, February 2012 95

The Complete Surface-Current Distribution in a Normal-Mode Helical Antenna

R. A. Abd-Alhameed1, K. N. Ramli1, and P. S. Excell2

1Mobile and Satellite Communications Research CentreUniversity of Bradford

Bradford, BD7 1DP, UKE-mail: [email protected]; [email protected]

2Glyndwr UniversityWrexham LL11 2AW, Wales, UKE-mail: [email protected]

Abstract

An investigation of the surface-current distribution in a normal-mode helical antenna (NMHA) is reported. This enables precise prediction of the performance of normal-mode helical antennas, since traditional wire-antenna simulations ignore important details. A Moment-Method formulation was developed, using two geometrically orthogonal basis functions to represent the total nonuniform surface-current distribution over the wire of the helix. Extended basis functions were used to reliably treat the discontinuity of the current at the free ends. A surface kernel was used all over the antennas structure.

The surface-current distribution was computed for different antenna geometries, such as dipoles, loops, and helices. For helices, the currents were investigated for different pitch distances and numbers of turns. It was found that the axially-directed component of the current distribution around the surface of the wire was highly nonuniform, and that there was also a signi cant circumferential current ow due to inter-turn capacitance, both effects that are overlooked by standard lamentary current representations using an extended kernel. The impedance characteristic showed good agreement with the predictions of a standard lamentary-current code, in the case of applied uniform excitation along the local axis of the wire. However, the power-loss computations of the present technique produce signi cantly different results compared to those well-established methods when the wires are closely spaced.

Keywords: Moment methods; helical antennas; normal mode helical antenna; wire antennas; current distribution

1. Introduction

When parallel wires are close together, the surface-current distribution becomes nonuniform. This effect has been previously investigated, subject to certain approxi-mations. Smith [1, 2] and Olaofe [3] assumed that the average current fl owing in a set of parallel wires was equal, which means that the cross-sectional distribution of surface current remains constant along the wires. Tulyathan [4] used a more-general treatment, but still neglected the possibility of a circumferential component in the surface current: it is intui tively obvious that such a component must be present when there is signifi cant displacement-current fl ow in the inter-wire capacitance. A more-general detailed solution by Abd-Alhameed and Excell [5] included the modeling of two sur face-current components at any point on the wires surface, subject to certain geometry

constraints. More recently [6-8], two parallel dipoles and loop antennas were investigated for the existence of the nonuniform surface currents, and the antenna power losses were fully covered in [8].

In addition, most of the methods used for analysis of wire antennas of arbitrary shape (including the possibility of closely parallel wires) assume a uniform surface-current dis tribution across the cross section (e.g., Djordjevic et al. [9], Burke and Poggio [10], Richmond [11]). Hence, surface resis tive losses and reactive effects that may be augmented by the nonuniform surface current will not be correctly predicted.

This problem is particularly signifi cant for resonant coiled electrically small antennas, such as the normal-mode helical antenna (NMHA: see Figure 1), in which the surface-current

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IEEE Antennas and Propagation Magazine, Vol. 54, No. 1, February 2012

distribution has a critical effect on the effi ciency and Q factor. A Moment-Method (MoM) formulation that uses two orthogonal basis functions on the surface of the wire, includ ing its ends, was thus developed to investigate this problem in detail. A more generalized theory and results of the work done by the present authors was described in [6]. It should be noted that there are more-advanced commercial codes now avail able, which, in particular, implement patch modeling more effectively (e.g., FEKO [12], CST [13], HFSS [14], and IE3D [15]).

Basically, the original motivation for this work was to assess the degree of benefi t that would be obtained if an antenna of this type were to be realized in a high-temperature superconductor. Electrically small antennas have a low radia-tion resistance that is easily swamped by ohmic-loss resis-tance, resulting in a low effi ciency. Superconductors have the potential to remove much of the loss, and hence signifi cantly raise the effi ciency. There is then the possible disadvantage that the inherent Q factor of the antenna may become very high: whether this is a real disadvantage depends on the nature of the system into which the antenna is proposed for deploy ment.

To quantify the reduction in loss, and hence the improve-ment in effi ciency that might accrue from the use of a super-conductor, it is necessary to quantify the surface loss, sP

[ 2W/m ]:

2

ss

s

JP

= [ 2W/m ], (1)

where sJ is the surface-current density [A/m], and s is the surface conductivity [].

Figure 1. The basic geometry of the helical antenna driven by a voltage source at its center. The directions of the orthogonal basis or test functions are shown on the right, and represent the source or observations points on the wires surface and its ends.

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The self-resonant helix was already identifi ed as a con-venient design for electrically small antennas about which quite interesting results were reported: for example, broadband V-helical antennas [16], circular normal-mode helical anten nas [17], double-pitch normal-mode helical antennas [18], and multiple-pitch normal-mode helical antennas [19]. However, for realization in a high-temperature superconductor, the superconducting element may be left electrically isolated. The detailed quantifi cation of sJ in this particularly complex case was thus the main original objective of the work. The very detailed modeling procedure that has been developed has much wider uses, particularly in the accurate modeling of normally-conducting normal-mode helical antennas, which see extensive use in mobile telecommunications.

Complete validation of the predictions of the procedure poses considerable diffi culties, since it would require meas-urement of the surface-current distribution on a wire. This matter is an important topic for future work, but an adequate degree of validation can be claimed for the results that have been presented in this work from this type of modeling proc ess

2. Moment-Method Formulation

Initially, the normal MoM approach was followed, but no attempt was made to approximate the surface current or the scattered-fi eld observation points to a single point on the wires cross section. Instead, both were allowed to be com pletely general points on the wires surface, and the surface current was allowed to have components both parallel to, and transverse to, the wires axis. This leads to an equation of the form

( ) ( )j j ij

L = I J E , (2)

where jI is a basis function for the surface current jJ , iE is the incident electric-fi eld strength, and L is the integro-differ-ential operator given by

( ) ( )tanL j = +J A . (3)

If a set of testing functions, mW , is defi ned, Equation (1) may be rewritten as

( )I , ,j m j m ij

L = W J W E for 1, 2,...,j N= , (3)

where

( ) ( ),m j m j mjs s

L L ds ds Z

= = W J W J , (5)

( ),m i m i ms

ds V

= =W E W E , (6)

where ds and ds are the differential areas on the wires sur face



for the source and the observation points, respectively; 1, 2,...,m N= is the index of the testing functions; and Z and V

are the conventional abbreviations for the interaction matrix and excitation vector terms in the Method of Moments.

2.1 Evaluation of Impedance-Matrix Elements

The impedance-matrix elements (Equation (5)) can be rewritten using the closed surface-integral identity [20] as follows:

( )( ) ( )21

mj j m j ms s

Z j G R ds dsk

= J W J W ,

(7)

where ( )G R is the free-space Greens function, and R is the distance between the observation and source points on the wires surface.

The xyz coordinates for a point on the surface of the helixs wire can be given by

( ) ( ) ( ), cos sinx x y = ,

( ) ( ) ( ), sin cosy x y = + , (8) ( ) ( ) ( )2, sin cos

Pz a = + ,

where

( )cosx b a = + ,

( ) ( )sin siny a = , (9)

( )tan2

Pb

= ,

and a is the radius of the helixs wire, b is the radius of the helix, P is the pitch distance between the turns, is the azi muth angle of the circumferential cross section of the wire, and is the pitch angle. Equation (8) is the exact coordinates of the helixs geometry. Now, defi ning the two orthogonal directions on the surface of the helixs wire as shown in Fig ure 1, the unit vectors of the curvilinear surface patches in both directions are as follows:

( ) ( ) ( ) ( ) ( ) sin cos cos cos sinx y za a a a = + + ,

( ) ( ) ( ) ( ) ( ) sin cos cos sin sincs xa a = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin sin cos cos sin cos cosy za a + +

(10)

where a and csa are the unit vectors on the axial and the circumferential surfaces of the wire, as shown in Figure 1.

The differential length in both directions is

d b d = , (11)

dcs ad= , (12)

where

2

22Pb b

= +

,

( ) 22 cosb b a = + .

d and d are the differential lengths in and , respec-tively.

Now, the x, y, and z coordinates of the starting end sur-faces of the helix at 0 = are given by

( ) ( ), cosx r b r = + ,

( ) ( ) ( ), sin siny r r = , (13)

( ) ( ) ( ), sin cosz r r = ,

where 0 r a .

Hence, the unit direction vectors of the basis function on the end surface can be given by

( ) ( ) ( ) ( ) ( ) cos sin sin sin cosr x y za a a a = + , (14)

( ) ( ) ( ) ( ) ( ) sin cos sin cos cosce x y za a a a = + , (15)

where ra and cea are the unit vectors in the radial and the circumferential directions on the end surface of the wire, as shown in the Figure 1. The differential area on the end surface can be given by

enddA rdrd= . (16)

The unit direction vectors, coordinates, and differential area on the other end of the helix can be similarly defi ned.

Now, assume the surface-current density over the wires surface can be expressed by two orthogonal current compo-nents in a and csa (similarly, at the surfaces end directions,

97


IEEE Antennas and Propagation Magazine, Vol. 54, No. 1, February 2012ISSN 1045-9243/2011/$26 2011 IEEE

ra and cea ). If the surface current is expanded over the wires surface using triangular basis functions in which the diver gence of the current continuity is fi nite [21], then, as an exam ple, these functions in the axial direction can be given by

0

0

( )1

ff

f

+ == =

for 00 and 1 2 , (17)

( )0

( )f f

= = for 00 and 1 2 , (18)

where ( )f is the derivative of ( )f , and 0 is the axial length of the curvilinear patch presented in Figure 1 in the direction of for all angular values of from 1 to 2 . Similar basis functions in the directions of cea , ra , and csa can be given. The testing functions were chosen to be identical to the expansion basis functions (Galerkins method), yielding a symmetric impedance matrix. Hence, by substituting Equa-tions (8-17) into Equation (7), the impedance matrix elements can be found. As an example, the impedance element for the basis and testing functions in the axial direction can be stated as follows:

Z

( ) ( ) ( ) ( ) ( )1 2S S

j f f f f G R ds dsk

=

a a

(19)

where ds ab d d = . The other self- and mutual-impedance elements for all other basis directions can be obtained in a similar way.

3. Simulation and Results

Initially, simple antenna geometries, such as dipoles and loop antennas, were investigated and discussed as special cases of more-complex geometries, such as the helix. The antenna geometries of the parallel dipoles and loops are shown in Figure 2. A similar procedure as for the dipoles of placing an orthogonal basis distribution over the wires surface and the wires ends was used. A computer program was written to implement the analysis given in the previous section. The sur-face-patch subdivision was automatically generated by the

orthogonal directions. The impressed fi eld, iE , was modeled

the helix. For the loop, it was placed at 0 = . A simple axial excitation (in the direction) was considered. The impressed fi eld can thus be given by

( )1 2i c aa =E for 0 2 , (20)

where cantenna wires for all geometries were assumed to be perfectly conducting and surrounded by free space. Several examples

by the formulation, as follows.

The response of the input impedance of two parallel

both dipoles were centrally fed as presented in Equation (20). The axial and circumferential lengths were subdivided into 16 curvilinear patches of equal lengths in both directions. The attachment basis modes between the wire ends and the wire

the results agreed well with those calculated using NEC [10]

NEW would be less reliable, as less detail in the behavior of the

Figure 2a. Geometrical models of two parallel dipoles, including the directions of the basis or test functions used.

Figure 2b. Geometrical models of two parallel loop anten-

used.

98





wire was taken into account. It is worth noting that Tulyathan and Newman [4] observed this behavior on half-wavelength dipoles when they ignored the circumferential surface-current component.

For the same antenna geometry, the input impedance at 300 MHz (equivalent to half-wavelength dipoles with 0.005-wavelength wire radius) as a function of the separation dis tance between the dipoles is shown in Figure 4. It was clearly seen that there was good agreement between the results of the present work and those obtained from NEC, except for the reactance values for closely spaced distances. However, the methods were completely different in their numerical solu tions.

The normalized magnitudes of the axial and circumferen-tial surface-current components of two parallel dipoles sepa-rated by 15 mm, for the same wire radius as above, are shown in Figure 5 as a function of (the azimuth of the circumferen-tial cross section of the wire) at 6.25 cm and 4.6845 cm, respectively, considered from the bottom of the dipoles of their local axes (equivalent to length measured from the bottom of the dipole), for different operating fre quencies. It was very interesting to note that the nonuniform variations of these currents over different frequencies had small marginal differences. The maximum ratio of the axial component to the circumferential component was around 34:1. Similarly, these currents at 300 MHz (equivalent to half-wavelength dipoles) for different separation distances are pre sented in Figure 6a. It should be noted that the actual magni tudes of the circumferential component are inversely propor tional to the distance between the dipoles, in spite of their fi xed variations shown in Figure 6b. The axial component was also still nonuniform, even when the separation distance between the dipoles was 100 mm (0.1 wavelength).

The normalized surface currents for a thicker wire, with a radius of 10 mm (0.01 wavelength), for the same antenna geometry as above as a function of the separation distances between the dipoles are shown in Figure 7. Comparing Fig-ures 6a and 7a, the nonuniform effects on the axial compo nents could be strongly seen on the thick wires, for example, at a separation distance of 100 mm.

It was clear from Figures 3 and 4 that the average current along the local axis of the dipoles was similar to that com puted

NEC [6]; tance from the present work).

Figure 4. The input impedance, at a 300 MHz operating frequency, of two parallel dipoles with a length of 50 cm and a wire radius of 5 mm, as a function of the separation distance between them ( , + + + : the present work, and : NEC).

100 IEEE Antennas and Propagation Magazine, Vol. 54, No. 1, February 2012

Figure 5. The normalized magnitudes of the axial (a, top) and circumferential (b, bottom) surface-current components of the antenna geometry given in Figure 2, separated by 15 mm, as functions of at 6.25 = cm (for axial) and

4.6845 = cm (for circumferential) from the bottom of the dipoles for different operating frequencies: : 100 MHz, : 300 MHz, : 500 MHz.

Figure 6: The normalized magnitudes of the axial (a, top) and circumferential (b, bottom) surface-current components as func tions of at similar locations as in Figure 5, for different separation distances between the dipoles, at 300 MHz (equivalent half-wavelength dipoles with 0.005 wire radius) ( : 15 mm; : 30 mm, : 50 mm, + + + : 100 mm, : 500 mm).


Figure 7. The normalized magnitudes of the axial (a, top) and circumferential (b, bottom) surface-current components as a func tion of at similar locations as in Figure 5, for different separation distances between the dipoles, at 300 MHz (equivalent to half-wavelength dipoles with 0.01 wire radius) ( : 30 mm, : 50 mm, : 100 mm).

using NEC. The expected fi eld pattern was thus similar, and is not reproduced here.

The ratio of the power losses predicted from the nonuni-form surface-current distribution to those predicted from the average (or uniform) current distribution was considered as an equivalent measure of the improvement in modeling verisi-militude when using the new method. However, since the losses were small in most cases, it was possible to assume that the antennas wire was perfectly conducting and surrounded by free space. The losses could then be predicted by taking them to be proportional to the surface-current density squared. The variation of the power loss ratio as a function of the sepa ration distance between two parallel half-wavelength dipoles, for two wire thicknesses, is shown in Figure 8. It could be noticed that for large separation distances, the power loss con verged to a value of unity, as expected. However, the varia tions showed a signifi cant power loss for closely spaced dipole antennas.

For the following two antenna geometries, we restrict our discussion to the presence of the non-uniformity of the surface currents that clearly matched the variations of the power loss ratios.

The normalized magnitudes of the axial and circumferen-tial surface-current components for a single and two parallel loops as functions of for different operating frequencies are

Figure 8. The power-loss ratio of two parallel half-wave-length dipoles for various separation distances: : wire radius of 0.005 ; : wire radius of 0.01 .


Figure 9. The normalized magnitudes of the axial (a, top) and circumferential (b, bottom) surface-current components as a func tion of at 0 = for the axial component and

33.75 = for the circumferential component, for a single loop antenna for different operating frequencies. The loop radius was 3 cm and the wire radius was 5 mm (: 400 MHz, : 600 MHz, + + + : 800 MHz, : 900 MHz).

Figure 10. The normalized magnitudes of the axial (a, top) and circumferential (b, bottom) surface-current components as a func tion of at 0 = for the axial component and

33.75 = for the circumferential component, for two par-allel loops separated by 15 mm. Each had a radius of 3 cm and a wire radius of 5 mm. The normalized magni tudes are shown as a function of different operating fre quencies ( + + + : 400 MHz, : 600 MHz, : 800 MHz, : 900 MHz).


shown in Figures 9 and 10, respectively. For both fi gures, the loop radius and wire radius were 3 cm and 5 mm, respectively. In the case of the parallel loops, the separation distance was selected to be 15 mm. The loops were fed by a simple delta excitation source at 0 = . The axial component was taken at the source location, whereas the circumferential component was considered at 33.75 = (angles are simply used here to defi ne the locations of the circumferential cross section wire, and for this particular angle, the length was 0.0177 cm). It should be noted that the variations of the currents for the two parallel loops were taken for the bottom loop as shown in Fig-ure 2b. It was clearly shown that the maximum variations of the axial currents for frequencies less that the expected paral lel-resonance frequency of the antennas structure were always pointed inside the loops geometries (i.e., 180 = ). The circumferential component for the single loop antenna was similar to that computed on the two parallel dipoles, and its ratio compared to the axial component was found to be 41:1. However, the same component for two parallel loops was reduced to a minimum around 90 = , whereas its ratio to axial component was 28:1.

Moreover, the normalized magnitudes of the axial and circumferential surface-current components for two half-wavelength parallel loops, each of radius 0.0796 wavelength and with a wire radius of 0.013 wavelength, as a function of for various separation distances, d, are shown in Figure 11. The locations of these currents were similar to those used in Figures 9 and 10. The axial component reserved its variations for most of the distances considered in this example as in the case of the single loop antenna, except when at a very close distance. The variations of the circumferential component were also eliminated around 90 = , even when the separa tion distance was 20a .

The normalized magnitudes of the axial and circumferen-tial surface-current components of a half-wavelength single-turn helix antenna as a function of at different positions from the fi rst end of the helix are shown in Figure 12. The radius of the helix was 0.0796 , the radius of the wire was 0.013 , and the pitch distance was 3a . It was shown that the strong effects of the axial component were pointed inside the helix for all locations presented. These results might help to approximate the equivalent of these variations into one curve, which might be taken along all the local length of the helix to assess the total loss power in the axial direction. Similar variations were observed in the circumferential component, except that the strong effect shifted from 90 = to 135 = for the locations presented. However, considering the loca tions of the strong effects of these currents, regardless of the locations of their maxima, and taking into account the simi larities in these variations, it could be concluded that this indi cated the approximate contribution of this current to the total loss power.

Figure 11. The normalized magnitudes of the axial (a, top) and circumferential (b, bottom) surface-current components as a func tion of 0 = for the axial component and

33.75 = for the circumferential component, for two half-wave length parallel loops, each with a radius of 0.0796 wave length and a wire radius of 0.013 wavelength, as a func tion of the separation distance, d ( + + + : 3d a= ; :

6d a= ; : 20d a= ; : 100d a= ).


Figure 12. The normalized magnitudes of the axial and circumferential surface-current components of a half-wavelength single-turn helical antenna as a function of at different positions from the rst end of the helix. The helix radius was 0.0796 , the wire radius was 0.013 , and the pitch distance was 3a . (a, top) axial component: : 0.031 ; : 0.125 ; + + + : 0.25 ; (b, bottom) circumferential component: : 0.0156 ; : 0.109 ; + + + : 0.234 .

Figure 13. The normalized magnitudes of the axial and circumferential surface-current components of the same antenna geometry given in Figure 11, except that the oper-ating frequency was half than that used in Figure 11 (i.e., the operating wavelength was 2 ). (a, top) axial component: : 0.031 ; : 0.125 ; + + + : 0.25 ; (b, bottom) circumferen tial component: : 0.0156 ; : 0.109 ; + + + : 0.234 .


Figure 14. The normalized magnitudes of the axial ((a, top), taken at 0.031 from the bottom end of the helix) and the circumferential ((b, middle), taken at the center of the helix; (c, bottom), taken at 0.023 from the bottom end of the helix) surface-current components of a half-wavelength two-turn helical antenna as a function of for different pitch distances. The helix radius was 0.04 and the wire radius was 0.006 ( : 3P a= ; + + + : 5P a= ; :

7P a= ; : 9P a= ).

Figure 15. The normalized magnitudes of the axial ((a, top), taken at 0.0294 from the bottom end of the helix; (b, middle), taken at the center of the helix) and circumferential ((c, bottom), taken at 0.022 from the bottom end of the helix) surface-current components of a half-wavelength three-turn heli cal antenna as a function of for different pitch dis tances. The helix radius was 0.0265 and the wire radius was 0.004 ( : 3P a= ; + + + : 5P a= ; :

7P a= ; : 9P a= ).


Figure 16. The normalized magnitudes of the axial surface-current components of a half-wavelength multiple-turn helical antenna as a function of for different pitch dis-tances: (a, top) four turns: the helix radius was 0.02 and the wire radius was 0.003 ; (b, bottom) ve turns: the helix radius was 0.015 and the wire radius was 0.002 ( :

3P a= ; + + + : 5P a= ; : 7P a= ; : 9P a= ).

However, for the same antenna geometry given in Fig-ure 12, the currents were computed at half the operating fre-quency (i.e., an operating wavelength of 2 ), as shown in Figure 13. It was clearly shown that the strong effects of these current variations were mostly similar to those presented in Figure 12. Moreover, a strong correlation could be found in the variations of the circumferential currents for different locations for this example. However, a one-turn helix is not suffi cient to permit comment on the current variations of a multi-turn helical antenna. The following examples were thus considered.

The normalized magnitudes of the axial and circumferen-tial surface-current components of a half-wavelength of two-, three-, four-, and fi ve-turn helical antennas as a function of for different pitch distances are shown in Figures 14, 15, and 16. It was very clear that the axial and circumferential compo-nents were nonuniform, even when the pitch distance between the turns of the helix was nine times the radius of the wire. Another interesting point was that the peak values of the axial component were pointed inside the helix (i.e., around 180 = ) for all helices presented. This was clearly shown in the variations of these currents at the feed points in Fig ures 14b, 15b, 16a, and 16b, and the helix turns in Figures 14a (fi rst turn) and 15a (fi rst turn). The similarities of these varia tions permit the approximate calculation of the effective power loss in that particular direction. It was also observed that the maximum variations of the circumferential component on the fi rst turn were confi ned between 0 = and 180, as shown in Figures 12b, 13b, 14c, and 15c. However, the maxi mum ratio of the axial component to the circumferential com ponent for all pitch distances was found to be between 15:1 and 40:1 for all helices of more than one turn.

General comments on the trends of these results are to predict the accurate or approximated equivalent power losses that are associated with nonuniform variations along the wires surfaces. This will hence affect the radiation effi ciency of this kind of antenna.

4. Conclusions

The surface-current distributions on structures with closely spaced parallel wires, such as dipoles, loops, and heli cal antennas, can be computed by using the Method of Moments with a general surface-patch formulation. The cur rent distribution varies substantially from the common assumption that it is uniform around the wires cross section. Transverse (circumferential) currents have been shown to be present. They are relatively weak on thin wires (around a wire radius of


0.01 ) excited by an axial component parallel to the local axis of the wire. The effect is still signifi cant when the wire-separation distance is relatively large.

In spite the strong variations of the axial and circumferen-tial currents, it was found that the input impedance and the average value of the axial surface current were in rea sonably good agreement with the results of thin-wire codes such as NEC, using an extended-kernel solution. The power loss ratio resulting from use of a nonuniform surface current, compared with the conventional uniform assumption of two parallel dipoles, showed a signifi cant increase of power loss when the dipoles were closely spaced. However, these current variations will dominate the radiation effi ciency when pre-dicting the accurate total power loss on these types of anten-nas. This can be important in some applications, e.g., highly resonant antennas and antennas realized in superconducting materials. As a matter of interest, computations showed that the maximum ratio of the variations of the axial component to the circumferential component on a half-wavelength helix of a few turns for different pitch distances was between 15:1 to 40:1. This behavior was expected, as the normal-mode helical antenna has a behavior that is a hybrid of that of a dipole and a loop.

The modeling method employed a two-dimensional elec tric surface-patch integral-equation formulation, solved by independent piecewise-linear basis function methods in the circumferential and axial directions of the wire. A similar orthogonal basis function was used on the end surfaces, and appropriate attachments with the wires surface were employed to satisfy the requirements of current continuity. The results were stable, and showed good agreement with less-comprehensive earlier work by others.

5. References

1. G. Smith, The Proximity Effect in Systems of Parallel Conductors and Electrically Small Multiturn Loop Antennas, Technical Report 624, Division of Engineering and Applied Physics, Harvard University, USA, 1971.

2. G. Smith, The Proximity Effect in Systems of Parallel Conductors, Journal of Applied Physics, 43, 5, 1972, pp. 2196-2203.

3. G. O. Olaofe, Scattering by Two Cylinders, Radio Sci ence, 5, 1970, pp. 1351-1360.

4. P. Tulyathan and E. H. Newman, The Circumferential Variation of the Axial Component of Current in Closely Spaced Thin-Wire Antennas, IEEE Transactions on Antennas and Propagation, AP-27, 1, January 1979, pp. 46-50.

5. R. A. Abd-Alhameed and P. S. Excell, The Complete Sur-face Current for NMHA Using Sinusoidal Basis Functions and Galerkins Solution, IEE Proceedings Science, Measurement

and Technology on Computational Electromagnetics, 149, 5, 2002, pp. 272-276.

6. R. A. Abd-Alhameed and P. S. Excell, Surface Current Distribution on Closely Parallel Wires Within Antennas, 2nd European Conference on Antennas and Propagation (EuCAP), Edinburgh, UK, November 11-16, 2007, pp. 1-4.

7. R. A. Abd-Alhameed and P. S. Excell, Accurate Power Loss Computation of Closely Spaced Radiating Wire Ele-ments for Mobile Phone MIMO Application, IEEE Interna-tional Conference on Signal Processing and Communication (ICSPC), Dubai, United Arab Emirates, November 24-27, 2007, pp. 412-415.

8. R. A. Abd-Alhameed and P. S. Excell, Non-Uniform Sur-face Current Distribution on Parallel Wire Loop Antennas Using Curved Patches in the Method of Moments, Science, Measurement and Technology, IET, 2, 6, November 2008, pp. 493-498.

9. A. R. Djordjevic, M. B. Bazdar, V. V. Petrovic, D. I. Olcan, T. K. Sarkar and R. F. Harrington, Analysis of Wire Antennas and Scatterers, Norwood, MA, Artech House, 1990.

10. G. J. Burke and A. J. Poggio, Numerical Electromagnet ics Code (NEC): Method of Moments, US Naval Ocean Systems Center, Report No. TD116, 1981.

11. J. H. Richmond, Radiation and Scattering by Thin-Wire Structures in the Complex Frequency Domain, NASA, Report No. CR-2396, 1974.

12. FEKO EM Software and Systems S. A., (Pty) Ltd, Stellenbosch, South Africa.

13. Computer Simulation Technology Corporation, CST Microwave Studio, Version 5.0.

14. HFSS v. 10, Ansoft: http://www.ansoft.com.

15. IE3D, Release 12, Zeland Software, Inc., Fremont CA, USA, 2007.

16. D. A. E. Mohamed, Comprehensive Analysis of Broad Band Wired Antennas, National Radio Science Conference (NRSC), 2009, pp. 1-10.

17. W. G. Hong, Y. Yamada and N. Michishita, Low Profi le Small Normal Mode Helical Antenna, Asia-Pacifi c Micro-wave Conference (APMC), 2007, pp. 1-4.

18. H. Mimaki and H. Nakano, Double Pitch Helical Antenna, IEEE International Symposium on Antennas and Propagation, June 21-26, 1998, 4, pp. 2320-2323.

19. S. Ooi, Normal Mode Helical Antenna Broad-banding Using Multiple Pitches, IEEE International Symposium on Antennas and Propagation, June 22-27, 2003, 1, pp. 860-863.


20. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, Second Edition, New York, John Wiley & Sons, 1981.

21. M. I. Aksun and R. Mittra, Choices of Expansion and Testing Functions for the Method of Moments Applied to a Class of Electromagnetic Problems, IEEE Transactions on Microwave Theory and Techniques, 41, 3, March 1993, pp. 503-509.

Introducing the Feature Article Authors

Raed A. Abd-Alhameed is Professor of Electromagnet ics and Radio Frequency Engineering in the school of Engi neering, Design and Technology, Bradford University, UK, where he has worked since 1985. He specializes in computa tional modeling of electromagnetic-fi eld problems, antenna design, analysis of microwave nonlinear circuits, signal proc essing of pre-adaptation fi lters for adaptive antenna arrays, and simulation of active inductance. Prof. Abd-Alhameed is the Director of Mobile and Satellite Communications Research Centre, Leader of Communications Research Group, and the head of the Radio Frequency, Antennas, Propagation and Computational Electromagnetics Research Group. He was appointed as a research visitor for Wrexham University, Wales, UK in 2009. He has worked on several funded projects from EPSRC, the Department of Health, Mobile Telecommu nications and Health Research Programme, EU FP5, and a number of industrial KTPs. He has published over 300 techni cal journal and conference papers, including several book chapters. In addition, he holds two patents on RF antenna designs. He was invited as a keynote speaker for EPC01-IQ 2010, ITA 2009, Mobimedia 2010. He chaired the fi rst EERT 2010 workshop and several sessions at many international conferences. He was also appointed a guest editor for the IET SMT journal special issue on EERT for 2011. His current research interests include hybrid electromagnetic computa tional techniques, EMC, antenna design, low-SAR antennas for mobile handsets, bioelectromagnetics, RF mixers, active antennas, and MIMO antenna systems. Prof. Abd-Alhameed is a Fellow of the Institution of Engineering and Technology, a Chartered Engineer, and a Fellow of the Higher Education Academy.

K. N. Ramli was born in Batu Pahat, Johor, Malaysia, in 1974. He obtained a BEng in Electronic Engineering from the University of Manchester Institute of Science and Technology (UMIST), United Kingdom, in 1997, and the MEng at the Universiti Kebangsaan Malaysia (UKM), Malaysia, in 2004. Since 2007, he has worked within the Antennas and Applied Electromagnetic research group at Bradford University, United Kingdom, on a number of projects, concentrating on antenna design and computational electromagnetics. He is currently working towards his PhD. His main interests are in the fi elds of computational electromagnetics, antenna design, and radio-frequency circuit design. Mr. Ramli is a member of the IEEE.

Peter Excell is Professor of Communications and Dean of Arts, Sciences and Technology at Glyndwr University in Wrexham (Wales, UK). He obtained his BSc in Engineering Science from the University of Reading in 1970, and his PhD for research in electromagnetic hazards from the University of Bradford in 1980. From 1971 to 2007, he was with the Univer-sity of Bradford (UK), rising to Associate Dean for Research in the School of Informatics. His classical academic interests cover wireless technologies, electromagnetics, engineering computing, and antennas. However, he has also engaged in interdisciplinary initiatives, developing broader interests in mobile communications, their future content, and applications. He has published around 400 papers and holds three patents. He is a Chartered Engineer and Chartered IT Professional, a Fellow of the British Computer Society, the Institution of Engineering and Technology, and the Higher Education Academy, a Senior Member of the IEEE, and a member of the ACM.

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