Method-of-moments analysis of electrically large circular-loop antennas: Nonuniform currents L.-W.Li, C.-PLim and M.-S.Leong Abstract: A method-of-moments analysis is carried out so as to obtain, in closed form, the nonuniform current distributions, and their resulted radiation patterns in both near and far zones, of circular loop antennas with electrically larger circumferences. An oblique incident field in its general form is considered in the formulation of the nonuniform current distributions. In Galerkin’s method- of-moments analysis, the Fourier cosine series is considered as the full-domain basis-function series. As a result, the current distributions along the circular loops are expressed analytically in terms of the azimuth angle for various diameters of large loops. Finally, the radiated electromagnetic (EM) fields and their power pattern in both near and far zones are determined by applying the dyadic Green’s function (DGF) in spherical co-ordinates and plotted using a commercial software package, respectively. 1 Introduction The current distribution and admittance of thin circular- loop antennas have been studied extensively over the last five decades. The recognised contributions to the field have been made by many researchers, e.g. [l-201. Also, the radi- ation characteristics of the circular-loop antennas are read- ily available from textbooks such as [21, 221. In their formulations of the current distribution along the circular wires, many of the above classical studies, e.g. [l, 3, 8, 141, have neglected the higher-order modes of the Fourier series, which contribute sigmfkantly to the radiated fields elsewhere when the loop radii arc electrically large. Although some of them considered the hgher-order modes of the Fourier-series expansion, the analytical expressions of the Fourier-series-expansion coeficients have not been derived so that the field expressions in closed form become impossible. Furthermore, note that none of the literature has shown so far, in both near and far zones, the radiation patterns of thin-loop antennas with circumferences larger than three free-space wavelengths. Therefore, this paper aims at providing the current distri- butions in closed form in terms of the azimuth angle 4 for dfferent circumferences of the loop antennas. At the same time, this paper seeks to gain an insight into the radiation power patterns of these electrically large circular-loop antennas in both the near and far zones. From the three- dimensional expression of the current distributions in terms of the continuous variation of the angle 4 and the discrete change of the loop circumference a, the current distribution along the circular wire can be generated for any circular- loop antenna whose circumference falls (inclusively) within six wavelengths. OIEE, 1999 IEE Proceedhzgs online no. 19990784 DO1 10.1049/ipmap:19990784 Paper first received 25th January and in revised form 16th September 1999 The authors are with the Communications & Microwave Dvision, Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 inc jkr EO e Y J Fig. 1 Geomtry of u thin circular-loop antenna 2 General formulation of current distributions Fig. 1 shows the geometry of a thin circular-loop antenna located at z = 0 and fed at & = 0 with the delta-function generator Vac?(@). The loop current satisfies the integral equation [l, 5, 101 as follows: 7t V06(4) + uE;yu, 4) = - K(4 - 4’)1((b’)d$b’ 47r -7r (1) jr10 .I where EGmc denotes the incident field, a stands for the radius of the loop and q, = 120n Q is the free-space wave impedance. The current I(@) can be expressed by 00 I($) = (2 - 6mo)lm cos(m4) (2) m=O where Sm0 denotes the Kronecker symbol and Zm stands for the series-expansion coefficients. The integral kernel in eqn. 1 is represented by the Fourier series expansion 00 ~ ( 4 - 4’) = 1 (2 - 6mO)grn COS m(4 - 4‘) (3) m=O 416 IEE Proc.-Microw. Antennas Propug., Vol. 146, No. 6, December 1999
Transcript
Method-of-moments analysis of electrically large circular-loop antennas: Nonuniform currents
L.-W.Li, C.-PLim and M.-S.Leong
Abstract: A method-of-moments analysis is carried out so as to obtain, in closed form, the nonuniform current distributions, and their resulted radiation patterns in both near and far zones, of circular loop antennas with electrically larger circumferences. An oblique incident field in its general form is considered in the formulation of the nonuniform current distributions. In Galerkin’s method- of-moments analysis, the Fourier cosine series is considered as the full-domain basis-function series. As a result, the current distributions along the circular loops are expressed analytically in terms of the azimuth angle for various diameters of large loops. Finally, the radiated electromagnetic (EM) fields and their power pattern in both near and far zones are determined by applying the dyadic Green’s function (DGF) in spherical co-ordinates and plotted using a commercial software package, respectively.
1 Introduction
The current distribution and admittance of thin circular- loop antennas have been studied extensively over the last five decades. The recognised contributions to the field have been made by many researchers, e.g. [l-201. Also, the radi- ation characteristics of the circular-loop antennas are read- ily available from textbooks such as [21, 221.
In their formulations of the current distribution along the circular wires, many of the above classical studies, e.g. [l, 3, 8, 141, have neglected the higher-order modes of the Fourier series, which contribute sigmfkantly to the radiated fields elsewhere when the loop radii arc electrically large. Although some of them considered the hgher-order modes of the Fourier-series expansion, the analytical expressions of the Fourier-series-expansion coeficients have not been derived so that the field expressions in closed form become impossible. Furthermore, note that none of the literature has shown so far, in both near and far zones, the radiation patterns of thin-loop antennas with circumferences larger than three free-space wavelengths.
Therefore, this paper aims at providing the current distri- butions in closed form in terms of the azimuth angle 4 for dfferent circumferences of the loop antennas. At the same time, this paper seeks to gain an insight into the radiation power patterns of these electrically large circular-loop antennas in both the near and far zones. From the three- dimensional expression of the current distributions in terms of the continuous variation of the angle 4 and the discrete change of the loop circumference a, the current distribution along the circular wire can be generated for any circular- loop antenna whose circumference falls (inclusively) within six wavelengths.
OIEE, 1999 IEE Proceedhzgs online no. 19990784 DO1 10.1049/ipmap:19990784 Paper first received 25th January and in revised form 16th September 1999 The authors are with the Communications & Microwave Dvision, Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
inc jkr EO e
Y J Fig. 1 Geomtry of u thin circular-loop antenna
2 General formulation of current distributions
Fig. 1 shows the geometry of a thin circular-loop antenna located at z = 0 and fed at & = 0 with the delta-function generator Vac?(@). The loop current satisfies the integral equation [l, 5, 101 as follows:
7t
V06(4) + u E ; y u , 4) = - K ( 4 - 4’)1((b’)d$b’ 47r -7r
(1)
jr10 .I where EGmc denotes the incident field, a stands for the radius of the loop and q, = 120n Q is the free-space wave impedance. The current I(@) can be expressed by
00
I($) = ( 2 - 6mo)lm cos(m4) ( 2 ) m = O
where Sm0 denotes the Kronecker symbol and Zm stands for the series-expansion coefficients. The integral kernel in eqn. 1 is represented by the Fourier series expansion
For rn = 0, one obtains the following intermediate No:
1 N~ = -ln (F)
7r
2ka 2 k a
and for rn z 0, one obtains N,, as follows:
Nm = 1 7r {K. (?) I , (?) + c m }
2 k a
where m-l ~
1 c m = 1 n 4 m + y - 2 C ~
2i + 1 i=O
sin(zsin8 - m8)dO 0
being the Lommel-Weber function, b denotes the radius of the wire, J, is the Bessel function of the first kmd, I , identi- fies the modified Bessel function of the first kind, KO repre- sents the modified Bessel function of the second kind, and y = 0.5772 stands for Euler's constant. The incident field E T at the angle 4 is given by
E r c = EAnc {cos$ cos($ - 40)
+ sin $I sin($ - 4 0 ) cos e } x exp (jkacos(q5 - $ 0 ) sine} (9)
It can also be expressed in terms of the Fourier series as follows:
7n=--oo
where
f m = E;"" { jm-' cos $e3m60 ?J&(ka sin 6 )
m J , (ka sin 0) ka sin 0
+ j m sin $ cos Oejrn4o
(11) To obtain maximum electric and magnetic responses, the loop orientation is made at I) = 0, 8 = id2 and ik, = 0 which simplifiesf;,, and I , to
1 f m = jmV1 { Jm-l ( k a ) - Jm+l ( k a ) } (12)
and
3
The dyadic Green's functions of electric and magnetic kinds in free space and in the regions r 2 a that are sepa- rated by the circular-loop antenna are gjven by [19, 231.
Electromagnetic fields in the near and far zones
IEE Proc.-Microw. Antennas Propug., Vol 146, No. 6. December 1999
With current distribution expressed by eqn. 2, one can obtain the electromagnetic field expressions for the two regions as follows:
x P,"(cos e ) sin(m4) (14)
dP," (cos e ) m X d0 cos(,$) F -Pnm(cosO) sin 0
and
x P,"(cosQ) cos(m4) (17)
m dPF(cos8) x ---P,"(cos0) cos(m4) + sin 0 dB
dP," (cos 0) m X d0 sin(mq5) 7 -P,"(cos8) sin 6'
where jn(kor) and h,(')(kor) are the spherical Bessel and Hankel functions of the first kind, respectively, Pnm(cos8) is the associated Legendre function, and the normalisation coefficient D,7,, is given by
(2n + 1) (n - m)! ~~ Dmn = n(n + 1) (n + m)!
417
The coefficients of the EM fields, i.e. QCg;, @gz, and @:&, are expressed by
since
The associated Legendre function P,"(O) and its first-order derivative dPnrn(O)/dO are given by
dPp (0) - 2"+l sin { (n + r n ) ~ } r (T + 1) ~- - dB f i r (T)
(22 )
2m COS { $(n + W L ) ~ } r (v) P,"(O) = fir (7 + 1) (23)
The radiated power pattern is given by
A = 27rr2Re ( N r N t * - N f N f * ) (24) where
and the asterisk * denotes the complex conjugate.
4 Numerical results
The current distributions are obtained using the method of moments, whereas the gap voltage Vos(f) = 1 and the rela- tionship between the loop and wire cross-section radii is 21n(2m/b) = 10. The current distributions of loop antennas are plotted with respect to the azimuth angle 4 and the loop circumference, i.e. ka in terms of free-space wave- length A. Also, the analytical expressions of the current dis- tributions are provided in the Appendix (Section 8). Engineers who would design the loop antennas can use the formulas directly to calculate various antenna parameters. Certainly, convergence of the analysis is checked in detail. It is realised from the computation that only 11 terms of the Fourier series are necessarily taken in the summation to ensure the accuracy whde the 12th and hgher series can be neglected.
As shown in Figs. 2 and 3, the real and imaginary parts of the nonuniform current distribution are plotted, respec- tively. The first 11 harmonics in eqns. 1421 are applied to obtain the electromagnetic radiated fields. 20 terms are con- sidered for the numerical computation of the summation of the spherical Bessel and Hankel functions (i.e. with respect to the index n). It is clear from Figs. 2 and 3 that, when the loop size is electrically small (e.g. ku = 1 wavelength), the current distribution just follows a cosine functional varia- tion. This is expected and is commonly introduced in text- books on antenna theory/engineering. However, when the loop dimension gets larger, more terms of the cosine series are needed so as to achieve the accuracy and/or conver- gence. Certainly, when the loop size is larger, the current distribution becomes quite oscillating.
418
p"r"-,, reatpart
Zn ' Fig.2 Realpart of the current distribution
- imaginary p ~ r l
zx ' Fig. 3 Imuginury part of the current d&ibution
R o x 0
4 2
a - 6 2
b xi2 n12
-RI2
C
-x/z d
x O R 0
-RI2 e
X I 2
f
RI2
- 4 2 9
-n/2 h
Fig.4 ka = 3 , 4 , 5 m d 6 wavelengths in the nearfiki zones a, b: ka = 3; c, d: ka = 4; e,f ka = 5; g , h: ka = 6
Radiutwn patterns due to circuhr-loop untennas of cirm@rence
IEE Proc-Microw. Antennas Propug., Vol. 146, No. 6, December 1999
With the current distribution obtained, then it becomes easier to obtain the radiation patterns of circular-loop antennas of circumferences ka between 3 and 6 wave- lengths in the near and far zones. The various field patterns are obtained and depicted in Figs. 4 and 5. Note that all numerical results are normalised by their maxima.
$=O n12
$=n/2 n12
n o n 0
-n/2 a
nl2
n 0
4 2 C
nl2
n 0
- X I 2 e
n12
4 2 b
nl2
4 2 d
nl2
n 0
-1112 f
4 2 9
4 2 h
Fig.5 ka = 3, 4 , 5 und 6 wavelengths in the far-field zone a, b ka = 3; e, d k a = 4; e, f k u = 5; g , h ka = 6
Rrrdicltwn patterns rlUe to circular-loop a n t m of circlanference
5 Conclusions
This paper presents a method-of-moments analysis of elec- trically large circular-loop antennas. From the analysis, the nonuniform current distributions and the radiated electro- magnetic fields, in both the near and far zones, of thin cir- cular-loop antennas of circumference between 1 and 6 wavelengths, are obtained. The cosine functions are chosen as the full-domain basis and weighting functions in Galer- kin’s approach. Also, the dyadic Green’s function in spher- ical co-ordinates is applied in the derivation of the analytical expressions of the electromagnetic fields in both near and far zones. The radiation patterns due to the derived current distributions are plotted in polar co-ordi- nates. The numerical results for radiation patterns in the far field are compared with experimental results obtained in Figs. 3 and 4 of [13] and a good agreement between the two results is obtained. This confirms partially the correct- ness of the present theoretical derivation and numerical algorithm. Some new results are also presented in the paper for the antenna patterns due to large antennas with circum- ferences between 3 and 6 wavelengths.
IEE Proc -Microw Antenna5 Propug, Vol 146, No 6, December 1555
6 Acknowledgment
The research work reported here is supported by the Min- Def-NUS Joint Projects 12 & 13/96 from the DSO National Laboratories. Useful discussions with Profs. P.S. Kooi and T.S. Yeo are acknowledged.
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References
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10 ESSELLE, K.P., and STUCHLY, S.S.: ‘Resistively loaded loop as a pulse-receiving antenna’, IEEE Trans., 1990, AF-38, pp. 1123-1126
11 ZHOU, G.P., and SMITH, G.S.: ‘An accurate theoretical model for thin-wire circular half-loop antenna’, IEEE Trans., 1991, AF-39, pp. 1167-1 177
12 TSAI, L.L.: ‘A numerical solution for the near and far fields of an annular ring of magnetic current’, IEEE Trans., 1972, A€-20, pp. 569- 576
13 RAO, B.R.: ‘Far field patterns of large circular loop antennas: Theo- retical and experimental results’, IEEE Trans., 1968, AF-16, pp. 269- 270
14 CHEN, C.L., and KING, R.W.P.: ‘The small bare loop antenna immersed in a dissipative medium’, IEEE Trans., 1963, AI-11, pp. 26&269
15 IIZUKA, K., and RUSSA, F.L.: ‘Table of the field patterns of a loaded resonant circular loop’, IEEE Truns., 1970, AT-18, pp. 41G 418
16 ABO-ZENA, A.M., and BEAM, R.E.: ‘Transient radiation field of a circular loop antenna’, IEEE Trans., 1972, A€’-20, pp. 38&383
17 OVERFELT, P.L.: ‘Near fields of the constant current thin circular loop antenna of arbitrary radius’, IEEE Trans., 1996, A P 4 , pp. 16G 171
18 WERNER, D.H.: ‘An exact integration procedure for vector potential of thin circular loop antennas’, IEEE Trans., 1996, A F 4 , pp. 157- 165
19 LI, L.W., LEONG, M.S., KOOI, P.S., and YEO, T.S.: ‘Exact solu- tions of electromagnetic fields in both near and far zones radiated by thin circular-loop antennas: a general representation’, IEEE Trans., 1997, AP-45, pp. 1741-1748
20 WERNER, D.H.: ‘Analytical and numerical methods for evaluating electromagnetic field integrals associated with current-canying wires pntennas’ in BARRETT, T.W., and GRIMES, D.M. (Eds.): Advanced electromagnetism: foundations, theory and applications’ (World Scientific, Singapore, 1995), pp. 682-762
21 BALANIS, CA.: ‘Antenna theory: analysis and design’ (Wiley, New York, 1997, 2nd edn.)
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In ths Appendix, the current distributions along the loop- antenna wires of various dimensions are provided in closed form. They are expressed in terms of the spherical azimuth angle 4 for different loop dimensions (where ka = 1 - 6 wavelengths). These formulas are extremely useful for sci- entists and engineers who would like to obtain full-wave characteristics of the loop antennas since they can use those formulas straightforwardly without repeating the method- of-moments analysis.
419
Case 1: ka = 1 wavelength
I(+) = (4.836 x - 8.061 x 10-5j)
+ 2 x (2.352 x + 3.821 x l O P 4 j ) cos($)
+ (4.012 x lov7 + 3.464 x l O P 4 j ) cos(24)
+ (9.566 x 10-lo + 1.509 x lo-") cos(34)
+ (1.428 x lo-'' + 1.506 x 10-4j) cos(44)
+ (1.483 x + 7.604 x 10-5j) cos(64)
+ (3.377 x lo-'' + 6.021 x 10-5j) cos(74)
+ (6.182 x + 4.963 x l O P 5 j ) cos(84)
+ (9.258 x + 4.213 x 10-5j) cos(94)
+ (1.152 x + 3.656 x 10-5j) cos(104))
Case 2: ka = 2 wavelengths
{
+ (5.169 x 10- l~ + 1.019 x 10-~j) cos(54)
(26)
I ( @ ) = (1.573 x - 3.918 x 10-4j)
+ 2 x { (3.806 x
+ (5.769 x l o r 7 + 4.896 x 10-4j) cos(34)
+ (3.101 x lo-' + 2.295 x 10-4j) cos(44)
+ (1.790 x lo-'' + 1.467 x 10-4j) cos(54)
+ (8.675 x + 1.066 x l O P 4 j ) cos(64)
+ (3.408 x + 8.314 x l O P 5 j ) cos(74)
+ (1.088 x 10-l' + 6.790 x 10-5j) cos(84)
+ (2.857 x + 5.729 x 10-5j) cos(94)
+ (6.253 x + 4.950 x l O P 5 j ) cos(lOc$)}
Case 3: ka = 3 wavelengths
- 7.118 x 10-4j) cos($)
+ (2.848 x 10-4 - 7.354 x 10-3j) C O 4 2 4 4
(27)
I (+) = (2.682 x lop5 - 2.575 x 10-4j)
+ 2 x { (9.612 x
+ (5.303 x - 5.423 x 10-4j) cos(24)
+ (2.733 x l o r 3 - 1.061 x 10-2j) cos(3$)
+ (4.658 x lop7 + 5.969 x l O P 4 j ) cos(@)
+ (3.556 x lo-' + 2.887 x l O P 4 j ) cos(54)
+ (3.211 x lo-'' + 1.891 x 10-4j) cos(64)
+ (2.591 x + 1.400 x l O P 4 j ) cos(74)
+ 9.173 x 10-5j) cos(94)
+ 7.821 x 10-5j) cos(104)}
- 3.861 x l O P 4 j ) cos(4)
+ (1.769 x 10- l~ + 1.109 x 10-~j) cos(84)
+ (1.013 x
+ (4.886 x
(28)
Case 4: ku = 4 wavelengths
I ( 4 ) = (3.480 x - 1.966 x 10-4j)
+ 2 x (1.065 x - 2.449 x 10-4j) cos(+)
+ (3.902 x - 3.127 x 10-4j) cos(24)
+ (3.648 x - 6.0570 x 10-4j) cos(34)
+ (1.780 x - 1.272 x 10-2j) cos(44)
+ (3.157 x + 6.960 x 10-4j) cos(54)
+ (2.928 x lo-' + 3.413 x 10-4j) cos(64)
+ (3.446 x lo-'' + 2.264 x 10-4j) cos(7$)
+ 1.355 x l O P 4 j ) cos(94)
+ 1.129 x 10-4j) cos(lOq5))
{
+ (3.799 x 10-l~ + 1.694 x 1 0 - ~ j ) cos(84)
+ (3.666 x
+ (3.045 x
Case 5: ka = 5 wavelengths (29)
I ( 4 ) = (3.763 x - 1.665 x 10-4j)
+ 2 x (1.102 x 10-~ - 2.192 x 10-~j) { + (3.299 x - 2.311 x 10-4j) cos(24)
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