IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 8, AUGUST 2001
1235
Letters__________________________________________________________________________________________
The Input Impedance and the Antenna Gain of the
Spherical Helical Antenna
H. T. Hui, K. Y. Chan, and E. K. N. Yung
Abstract—The input impedance and the antenna gain of the spherical
helical antenna are obtained theoretically and experimentally. Results indicate the better performance of the 3-turn antenna over the 7-turn one in
terms of the impedance bandwidth and the stability of the antenna gain.
The theoretical current distribution is also obtained and its salient characteristics are discussed.
Index Terms—Antenna gain, input impedance, spherical helical antenna.
I. INTRODUCTION
Fig. 1.
The coaxially fed spherical helical antenna.
Spherical helical antennas possess many distinct but very useful
characteristics from convention cylindrical helical antennas. A
spherical helical antenna was shown to radiate very pure circular
polarization field over a wide beam width [1]. The radiation patterns
of a spherical helical antenna are free from sidelobes with a relatively
high gain in the axial direction [1], [2]. These characteristics make it
a very suitable element for antenna arrays constructed for high-gain
receptions. Although this antenna has been proposed for several years,
research on it was mainly focused on its radiation characteristics and
very little was found on its input characteristics. In this letter, we focus
a study on the input impedance and the antenna gain of this antenna.
We will investigate this antenna both theoretically and experimentally.
II. THEORETICAL AND EXPERIMENTAL RESULTS
Consider a coaxially fed spherical helical antenna with a perfectly
conducting ground plane as shown in Fig. 1. Different from the cylindrical helix, the spherical helix with equal spacing between turns has
only two dimensional parameters: the radius a and the number of turns
N . The equation of the spherical helix in spherical coordinates is given
by [2]
r =a
(1a)
= cos01
N
01
0
2N:
(1b)
The length of the helix L can be obtained by performing the following
integral:
L=
a 1 + (N )2 sin4 d:
(2)
0
The length of the straight wire connecting the spherical helix to the
coaxial line is denoted by h and the radius of the wire is rw , equal to
the radius of the inner conductor of the feeding coaxial line. The radius
of the outer conductor of the coaxial line is b. As the overall size of the
antenna does not vary with the number of turns of the spherical helix,
the input impedance and the antenna gain will not vary in the same
way as for the cylindrical helical antenna. In fact, the increase in the
number of turns of the spherical helix will even have an adverse effect
Manuscript received March 28, 2000; revised November 8, 2000.
The authors are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: eehthui@cityu.edu.hk).
Publisher Item Identifier S 0018-926X(01)03593-1.
Fig. 2. The current distribution of a 7-turn spherical helical antenna at C=
1:17 with a = 1:95 cm, r = 0:04 cm, h = 0:4 cm, and b = 4:5r .
=
on the input impedance and the antenna gain, as will be shown later. The
spherical helical antenna studied in this research are made by winding
a copper wire on the surface of a polystyrene sphere with a relative
permittivity very close to one. The ground plane is a square copper plate
with a side length of 30 cm, equal to 2.45 wavelength at a frequency of
2.45 GHz. The ground plane with this size can actually be considered
as infinitely large [1]. In the theoretical study, the moment method [3]
together with the Galerkin matching procedure is used to obtain the
input impedance and the current distribution. The magnetic-current frill
model is employed to simulate the coaxial aperture on the ground plane.
The current distribution of a 7-turn spherical helical antenna
is shown in Fig. 2 at a frequency of 2.87 GHz, corresponding to
C= = 1:17 (where C is the circumference of the sphere). Compared
with that of the cylindrical helical antenna [4, p. 267], the current
magnitude of this antenna is very different in two aspects. First, only
two, instead of three, regions can be identified: the gradually decaying
region (labeled “D” in Fig. 2) near the input end and the standing-wave
region (labeled “S” in Fig. 2) near the open end. Secondly, rapid local
variations in the gradually decaying region are observed. The current
phase in Fig. 2 shows that a dominant traveling wave is propagating
in region D while a strong standing wave exists in region S. For a
shorter 3-turn spherical helical antenna, it is found that the current
distribution is quite similar to that of a short cylindrical helical antenna
0018–926X/01$10.00 © 2001 IEEE
1236
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 8, AUGUST 2001
Fig. 3. The input impedance of a 3-turn spherical helical antenna with
1:95, r = 0:04 cm, h = 0:1 cm, and b = 4:5r .
a
=
Fig. 4. The input impedance of the 3-turn spherical helical antenna with the
same dimensions as in Fig. 3 but with different values of h.
Fig. 5. The input impedance of the 5-turn and the 7-turn spherical helical
antennas with a = 1:95 cm, r = 0:04 cm, b = 4:5r .
Fig. 6.
r
[5, p. 151]. The current decays smoothly along the helical wire until it
degenerates into a standing wave near the end of the helix. However,
the rapid local variations as seen in region D of the 7-turn spherical
helix disappear. The deviations of these current distributions from that
of the cylindrical helical antenna give rise to the rather significant
differences in the input impedance as discussed below.
The input impedance of a 3-turn spherical helical antenna is shown
in Fig. 3. The impedance is obtained at h = 0:1 cm. The length of h is
found to have a significant effect on the input impedance. (Except for
h and N , all other dimensions of the antennas throughout this study
are same as those for Fig. 3). Measurement results are also shown in
Fig. 3. A very obvious characteristic observed from Fig. 3 is a relatively
constant impedance at the range from C= = 0:95 to C= = 1:3,
corresponding to a frequency range from 2.33 GHz to 3.19 GHz. The
impedance is very close to 100 0 j 20 . This characteristic indicates
that the 3-turn spherical helical antenna is a wide-band one. As the
frequency goes up, the impedance begins to oscillate. The variation
of the input impedance with different values of h is shown in Fig. 4.
It can be seen that, as the value of h increases, the impedance varies
more quickly with frequency and this means that the useful bandwidth
The antenna gain of the spherical helical antenna with a = 1:95 cm,
= 0:04 cm, b = 4:5r , and two different numbers of turns.
decreases. The real part of the impedance goes up to a higher value for
a greater h. Thus, in the design of a spherical helical antenna, the value
of h cannot be ignored.
The input impedances of a 5-turn and a 7-turn spherical helical antennas are shown in Fig. 5. For the 7-turn antenna, both theoretical and
measured values are shown. Compared with Fig. 3, it can be seen that
as the number of turns increases, the impedance varies more rapidly
with frequency. This means that a constant impedance region is more
unlikely to obtain and hence the useful bandwidth of the antenna is reduced. Thus, an increase in the number of turns of the helix does not
favor the impedance bandwidth for a spherical helical antenna. This
characteristic makes a spherical helical antenna with a large number
of turns very different from a long cylindrical helical antenna whose
input impedance remains relatively constant with the increase in the
number of turns of the helix [5, p. 143]. In [2], the input impedance
of a 7-turn spherical helical antenna was also computed by the electromagnetic surface patch (ESP) code but the value of h was not specified.
Not withstanding this, the computed values are very different from our
computed and measured ones in the corresponding frequency range.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 8, AUGUST 2001
Our results show that the mean value of the input impedance (both real
and imaginary parts) remains relatively unchanged with frequency in
the range from C= = 0:95 to C= = 1:8, different from that computed in [2]. We notice from Fig. 5 that the computed and the measured
impedance for the 7-turn spherical helical antenna differ more significantly with increasing frequencies. This is due to the use of a fixed
number of divisions for the antenna wire in the numerical computation
which gives less accurate results at higher frequencies. This error can
be reduced by proportionately increasing the number of divisions at
higher frequencies but at the expense of a longer computation time.
The measured antenna gains (with respect to an isotropic circularly
polarized source) for a 3-turn and a 7-turn antennas are shown in Fig. 6.
It can be seen that a relatively stable gain (within 1 dB) can be obtained for the 3-turn antenna from C= = 1:1 to C= = 1:3. On the
other hand, the gain of the 7-turn antenna changes more rapidly with
frequency. The maximum gain of the 3-turn antenna is 11.1 dB measured at C= = 1:28 and that of the 7-turn antenna is only 9.9 dB at
C= = 1:18. The generally smaller gain of the 7-turn antenna is due
to its deeper mismatched impedance (Fig. 5) when compared with that
of the 3-turn antenna (Fig. 3).
1237
Fig. 1. Derivation of the Sierpinski gasket from Pascal’s triangle. When those
numbers divisible by 2 are deleted, the mod 2 Sierpinski gasket is obtained.
0
REFERENCES
[1] J. C. Cardoso and A. Safaai-Jazi, “Spherical helical antenna with a
circular polarization over a broad beam,” Electron. Lett., vol. 29, pp.
325–326, 1993.
[2] A. Safaai-Jazi and J. C. Cardoso, “Radiation characteristics of a spherical helical antenna,” Inst. Elect. Eng. Proc. Microw. Antennas Propagat., vol. 143, pp. 7–12, 1996.
[3] R. F. Harrington, Field Computation by Moment Methods. New York:
Macmillan, 1968.
[4] J. D. Kraus, Antennas . New York: McGraw-Hill, 1988.
[5] H. Nakano, Helical and Spiral Antennas—A Numerical Approach. London, U.K.: Research Studies Press, 1987.
Generalized Sierpinski Fractal Multiband Antenna
Jordi Romeu and Jordi Soler
Mod
mod
Abstract—A new set of fractal multiband antennas called
Sierpinski gaskets is presented.
Sierpinski fractal antennas derive from the Pascal triangle and present a log-periodic behavior, which is a
Sierpinski fractal
consequence of their self-similarity properties.
antennas constitute a generalization of the classical Sierpinski antenna.
Mod
Index Terms—Antennas, fractals, multifrequency antennas.
Fig. 2. Two Sierpinski gaskets (a) mod
Sierpinski gasket.
0 3 Sierpinski gasket, (b) mod 0 5
self similarity. Self similarity is a property common to many fractals,
but in order to become a useful radiator it is necessary that the
fractal antenna meet the specifications at the desired frequencies.
The Sierpinski gasket in a monopole configuration has a good
matching to 50 at the resonance frequencies, a log-period band
spacing of 2, and a fairly invariant radiation pattern in all bands,
which is very similar to the pattern of a monopole. The Sierpinski
gasket dipole reported in [2], is just one special case of a more
general class of fractal objects called Pascal–Sierpinski gaskets [3].
Furthermore, the geometry of the Sierpinski dipole can be altered
by changing the flare angle [4]. It is even possible to modify it
in order to obtain a desired log-period band spacing [5]. In this
paper the properties of these generalized Sierpinski gasket antennas
are presented. Their main advantage is the possibility to obtain
log-periodic behavior with values of the log-period larger than 2.
II. THE SIERPINSKI GASKET AND ITS VARIATIONS
I. INTRODUCTION
The multiband behavior of the fractal-shaped antennas and of the
Sierpinski gasket dipole has been described in [1], [2]. The multiband
properties of the Sierpinski gasket dipole are a consequence of its
Manuscript received February 8, 2000; revised September 21, 2000. This
work was supported in part by CICYT and the European Commission under
Grant FEDER 2FD97-0135 and in part by the Department of Research and Universities of the Generalitat de Catalunya, Spain.
The authors are with Department of Signal Theory and Communications,
Telecommunication Engineering School of the Universitat Politecnica de
Catalunya, Spain.
Publisher Item Identifier S 0018-926X(01)05249-8.
The Sierpinski gasket is a well known fractal. The way it can be
constructed and its main properties can be found in [1], [2], and [6]. In
[3] it is shown that the Sierpinski gasket is a special case of a wider class
of fractals that can be derived from the well known Pascal’s triangle.
This class of fractals can be derived in the following way. Consider
an equiangular triangular grid whose rows shall be labeled by n =
1; 2; 3; . . .. Each row contains n nodes and to each node a number is
attached. This number is the coefficient of the binomial expansion of
(x + y)n01 . Now delete from this grid those nodes that are attached
to numbers that are exactly divisible by p, where p is a prime number.
The result is a self-similar fractal that will be referred as the mod 0 p
Sierpinski gasket [3]. In Fig. 1, this process is shown for the mod 0 2
0018–926X/01$10.00 © 2001 IEEE