## 1. Introduction

[2] Log-periodic (LP) antennas are important with their ability to display nearly frequency-independent characteristics over wide bands of frequency. This type of antenna has a long history since the late 1950s, when DuHamel and Isbell introduced the first log-periodic structures [*DuHamel and Isbell*, 1957; *Isbell*, 1958; *DuHamel and Ore*, 1958]. After that, further studies focused on improving the performance and geometric properties of the LP antennas, as well as developing and inventing novel structures, such as the LP dipole arrays introduced in 1960 [*Isbell*, 1960].

[3] LP dipole arrays have been very popular with their detailed analysis given by Carrel [*Carrel*, 1961] and developed later in various studies [*De Vito and Stracca*, 1973; *De Vito and Stracca*, 1974; *Butson and Thompson*, 1976; *Peixeiro*, 1988]. In those reports, the antenna problem is modelled by a network of mutually coupled dipoles. On the basis of Carrel's work, straightforward and widely accepted design procedures for the LP dipole arrays are readily available in many famous textbooks [*Stutzman and Thiele*, 1981; *Kraus*, 1988; *Balanis*, 1997] and well-known references [*DuHamel and Chadwick*, 1984; *Mayes*, 1988]. Those design recipes, which will not be repeated here for sake of brevity, have passed the test of time by providing remarkably successful solutions to the LP dipole array problem.

[4] Geometric simplicity of the LP dipole arrays leads to an important advantage in terms of numerical modeling, which facilitates the investigations and simulations prior to their hardware realizations. Numerous reports introducing new analysis techniques for the LP dipole arrays [*Jones and Mayes*, 1969; *Wolter*, 1970; *Paul and Gupta*, 1981], investigating their operational properties [*Kyle*, 1970; *Gong and Balmain*, 1986; *Hilbert et al.*, 1989; *Baker and Reuss*, 1990; *Hassan et al.*, 1991], and developing novel configurations for improved performance [*Oakes and Balmain*, 1973; *Tranquilla and Balmain*, 1983; *Wakabayashi et al.*, 1999; *Excell et al.*, 1999] have appeared in the literature. The same degree of success has not been obtained in the study of other LP structures with more complicated geometries, resulting in relatively fewer reports of detailed analysis [*Bell et al.*, 1960; *Liang and Lo*, 1968; *Lee and Mei*, 1970; *Hall*, 1986; *Smith and Mayes*, 1991] based on measurements and approximate numerical techniques.

[5] In this paper, our goal is to focus on the nonplanar trapezoidal tooth LP antennas (Figure 1) and investigate their important features, such as, frequency independence, pattern, and directivity, both individually and in a two-element array environment. At the same time, we demonstrate that accurate numerical simulations can bridge the gap between the approximate designs of LP structures obtained with the theoretical design recipes and the more precise performance requirements. This kind of an improvement supplied by the simulation can be very valuable for LP antennas that have more complicated geometries than the LP dipole arrays [*Tammen et al.*, 1993]. Our simulation environment, which is described in section 2, is capable of handling arbitrary geometries, including complicated LP antennas. In section 3, we focus on the nonplanar trapezoidal tooth LP antennas due to genuine practical needs since this work is part of our pursuit of developing arrays of LP antennas with beam-steering ability [*Gürel and Ergül*, 2004]. Along this direction, frequency independence and beam-steering abilities of a two-element array of trapezoidal tooth LP antennas are also reported in section 3. Optimization of more populous arrays using genetic algorithms will be reported elsewhere [*Gürel and Ergül*, 2004]. To our knowledge, this is the first investigation of the beam-steering abilities of arrays of these broadband log-periodic antennas.

[6] It is imperative to note that the numerical results obtained by a blind use of the simulation environment cannot and should not replace the theoretical and analytical design procedures. Simulation results can provide improvements only by suggesting corrections to fundamental designs on the basis of thorough theoretical understandings.