7.2.2.1.1 Meander line

Figure 7.89 Input impedance of different stage of Koch monopole: (a) resistance and (b) reactance ([38], copyright C 2000 IEEE).

of wire antennas, meander line antenna (MLA) and helical antenna (NMHA), reveals that the Koch Fractal antenna has relatively higher resonance frequency compared with that of MLA and NMHA (Figure 7.90) [39], for the same (stretched) wire length l. This means that when the resonance frequency is adjusted to be equal for the three types of antenna, keeping the antenna height h the same, the wire length l of the Koch Fractal monopole becomes the longest among these three types of antenna, while the antenna performances, radiation resistance, and the bandwidth, are about the same, as Table 7.9 shows [39].

This fact suggests that the antenna specific geometry is not necessarily significant for small-sizing of an antenna, but the electrical length, not physical length, is essential.

Instead of a single wire structure, a given space is filled more efficiently by single Peano or Hilbert curve elements. Consider these two curve structures, Peano and Hilbert, previously illustrated in Figure 7.81 (a) and (b), respectively, embedded in a square of 3 × 3 cm [41]. Induced currents due to normally incident plane waves for these

Table 7.9 Comparison of antenna performances for three types at the same resonance frequency ([39], copyright C 2003 IEEE)

Figure 7.90 Comparison of resonance frequency for three types of antenna in terms of iteration number ([39], copyright C 2003 IEEE).

structures differ depending on the curve structure and direction of the incident wave. Subsequently, resonance frequency differs depending on the polarization of the incident wave and relates with the electrical length of wire (stretched) residing in the square. For instance, for the horizontally polarized incidence wave the resonance frequency of the Peano curve structure is about one third of that for the vertical polarization incidence, because the electrical wire length is longer in the horizontally polarized wave than that of the vertically polarized wave. The antenna performance of the Hilbert curve structure is similar to that of the Peano curve structure. A Peano curve can fill a space much more than a Hilbert curve for the same number of iterations. This can be understood from Table 7.10.

The Peano curve structure is applied to an antenna of a dipole type as shown in Figure 7.91, in which the feed point is shown as an example [41].

In the same way, the Hilbert curve structure is applied to a dipole type as shown in Figure 7.92 [42]. Input impedance of the 2nd-order Hilbert structure is illustrated in

Table 7.10 Total length (stretched) S for Peano and Hilbert curves with respect to the iteration number ([40], copyright C 2006 John Wiley and Sons Inc.)

Note: L is the linear side dimension of the curve.

Figure 7.91 Peano dipoles, orders n = 1 through 3 ([41], copyright C 2004 IEEE).

Figure 7.92 Hilbert dipoles; (a) 1st order, (b) 2nd order, (c) 3rd order, and (d) fourth order ([42], copyright C 2003 IEEE).

Table 7.11 Resonance frequency and input impedance of Hilbert antenna of 2nd and 3rd stages ([42], copyright C 2003 IEEE)

Figure 7.93 Input impedance of 2nd-order Hilbert antenna fed at the center, with and without substrate; width = 4 mm and other dimensions = 50 mm ([42], copyright C 2003 IEEE).

Figure 7.93. The input impedance can be adjusted to about 50 by selecting the feed position. Table 7.11 provides variation of resonance frequency and input impedance with respect to the iteration number, outer dimensions and width of strip. Current distributions along a 3rd-order Hilbert dipole printed on the substrate of εr = 4.3 and fed at the center at resonance frequency 582.087 MHz having the outer dimensions of 4.5 cm is shown in Figure 7.94. The current distribution is almost the same as that for the center-fed and very similar to that of a half-wave dipole resembling a sinusoidal wave, and does not vary significantly with change in the iteration number. Radiation patterns of the 3rd order Hilbert Dipole antenna is shown in Figure 7.95, which is nearly the same as that of a dipole antenna.

Figure 7.96 compares the size between a printed dipole antenna and the 2ndand the 3rd-order Hilbert dipoles. The figure shows that the 2ndand the 3rd-order Hilbert dipoles have dimensions of about 1/4 and about 2/11, respectively, of that of the dipole.

Figure 7.94 Current distribution along a 3rd-order Hilbert antenna with substrate and fed at node 32 (a dot on the element) at resonance frequency (outer dimension is 4.5 cm and width is 2 mm) ([42], copyright C 2003 IEEE).

90

0.0035

0

18

36

54

72

270

Figure 7.95 Radiation pattern of the 3rd-order Hilbert antenna with substrate fed at node 24 at the resonance frequency ([42], copyright C 2003 IEEE).

7.2.2.1.2.2 PIFA using Peano curve elements

A Peano curve structure is used for miniaturizing a PIFA element, replacing the horizontal element with planar Peano curve elements. The antenna has two planar Peano elements to cover dual bands, which are designed for applications to GSM in 900 MHz band and PCS in 1900 MHz band [43]. The geometry is shown in Figure 7.97, in which

Table 7.12 Dimensions of optimized antenna shown in

Fig. 7.97 ([43], copyright C 2006 IEICE)

Figure 7.96 Comparison of the dimension of printed dipole and Hilbert antenna at the same resonance frequency ([42], copyright C 2003 IEEE).

Figure 7.97 Dual-band 2nd-order Peano PIFA ([43], copyright C 2006 IEEE).

dimensional parameters are given. Optimized antenna dimensions are listed in Table 7.12, in which A and B give parameters for an antenna with height = 8.25 mm and C gives for that h = 8 mm and the one side element is replaced with a rectangular patch. Computed VSWR of the antenna in the 900 MHz band and that of the antenna in the 1900 MHz band are illustrated in Figures 7.98 and 7.99, respectively. By using a Peano element, the PIFA can be miniaturized about 50% in volume compared with a conventional PIFA.

Figure 7.98 Computed VSWR for antennas A, B, and C near 900 MHz band ([43], copyrightC 2006 IEEE).

Figure 7.99 Computed VSWR for antenna A, B, and C near 1920 MHz band ([43], copyrightC 2006 IEEE).

7.2.2.1.3 Minkowsky and Sierpinsky Fractal monopoles 7.2.2.1.3.1 Planar Minkowsky Fractal monopole

A Minkowsky Fractal monopole is generated starting from a Euclidean square by replacing each of the four segments of the initial square with the generator shown, and repeating it for an infinite number of times as illustrated in Figure 7.100 [44], which shows only from the initiator to the third iteration structure. To utilize space-filling ability, a firstorder planar Minkowsky loop is fabricated (Figure 7.101) and its performances are reported [44]. The perimeter length lp is given by

Figure 7.105 Sierpinsky fractal structure ([47], copyright C 2003 IEEE).

Figure 7.106 The first four stages in construction of a conventional Sierpinsky gasket fractal ([47], copyright C 2003 IEEE).

f (GHz)

Figure 7.107 Return-loss characteristics relative to 50 ([47], copyright C 2003 IEEE).

remaining triangles, and so on to form a sieve of triangles as shown in Figure 7.105 [46]. The Sierpinsky sieve (or gasket) of triangles is, when applied to an antenna structure, characterized by a certain multiband behavior owing to its self-similar shape. A monopole antenna based on the Sierpinsky gasket has been shown to be an excellent candidate for multiband applications. The simulated and measured input reflection coefficient (S11) of a four-stage Sierpinsky structure (Figure 7.106) is depicted in Figure 7.107 [46]. The antenna dimensions are: height h = 88.9, hn/hn+1 = 2, flare angle α = 60◦, thickness of the substrate hs = 1.588, permittivity εr = 2.5, and the iteration number n = 4 (in mm for all dimensional parameters). The antenna parameters are provided in Table 7.13, in which frequency band, resonance frequency, bandwidth, reflection coefficient, ratio of iteration segment, and height of triangle for each segment in terms of wavelength, are provided. A five-order Sierpinsky monopole shown in Figure 7.108 [47] was fabricated and measured and the simulated reflection coefficient is shown in Figure 7.109

Table 7.13 Sierpinsky gasket fractal monopole parameters ([46], copyright C 2009 IEEE, and [47], copyright C 2003 IEICE)

80 cm

Figure 7.108 A five-order Sierpinsky monopole ([47], copyright C 2003 IEEE).

f (GHz)

Figure 7.109 Input reflection coefficient relative to 50 ([47], copyright C 2003 IEEE).