LOG-PERIODIC DIPOLE ARRAY OPTIMIZATION Y. C. Chung and R. Haupt Utah State University Electrical and Computer Engineering 4120 Old Main Hill, Logan, UT 84322-4160, USA Abstract-The element lengths, spacings and radii of log-periodic dipole arrays (LPDA) are optimized with a genetic algorithm (GA), Nelder-Mead downhill simplex optimization method, and GA & Nelder-Mead hybrid method. The design results of the three methods linked with NEC are compared. The gain and VSWR are superior to current commercial designs. 1. INTRODUCTION '. The log-periodic antenna is used for linearly polarized electromagnetic interference (EMI) measurements over a frequency range of 200 MHz to 1 GHz. Antenna components are proportional in size to each other and to the wavelength. Its geometry forces the antenna impedance and radiation properties to repeat periodically as a logarithm of frequency. These antennas come in many different shapes, including a wire dipole construction called the log-periodic dipole array (LPDA) shown in Figure 1. Adding antenna elements increases the bandwidth. The LPDA was introduced by Isbell [1], and improved using techniques in references [2-5]. The performance of a LPDA is a function of the number of elements as well as element length,
1270 Figure 1. LPDA geometry. spacing, and diameter. Antenna element lengths and spacings are proportionally related by a scale factor and spacing factor a where, the Ln is the length of the nth element, and dn is the spacing between the nth and (n + 1)th element. The longest element is approximately a half wavelength at the lowest frequency. Similarly, the shortest element is approximately a half wavelength at the highest frequency. This antenna is fed with a voltage source at the high frequency end. Figure 1 shows a uniformly spaced LPDA with an angle, a, which bounds all the dipole lengths. The angle a is calculated from T and a /1 \, At low frequencies, the larger antenna elements are active. As the frequency increases, the active region moves to the shorter elements. When an element is approximately one half wavelength long, it is resonant. Reference [6] has a plot of the gain of a well-designed uniformly spaced LPDA as a function of T and u. The voltage standing wave ratio is given by, {1 I W
1271 where, p is the reflection coefficient given by Za = antenna impedance, Zo = transmission line impedance Most antenna books give equations for the design of log-periodic antennas. Unfortunately, these design equations do not include the complex interactions between the dipoles that compose the antenna. Also, the equations do not give the input impedance as a function of frequency. Consequently, a numerical model and numerical optimization of that model are important for developing realistic designs. 2. COMPUTER MODEL OF THE LPDA Wire antennas, like the LPDA, are accurately modeled with the Numerical Electromagnetics Code (NEC). This FORTRAN code is widely used to design wire antennas. The geometry of the LPDA is specified in terms of wire segments. A method of moments (MOM) solution [7] is used to find the antenna impedance as a function of frequency and the antenna gain as a function of angle and frequency. Each wire is broken into segments, and the segments are specified in an input file that the program reads. Using the geometry and induced voltage source, the currents are calculated for each of the wire segments. Once the currents are known, the program calculates the antenna impedance and radiation characteristics. Each wire model of the LPDA is defined by the (x, y, z) coordinates of its two end points and its radius. A wire segment is defined by the coordinates of the element containing the segment and the number of segments in the element. A segment should be less than and larger than 0.001A [8]. The center of the last element is excited. In addition, the 50 ohm crossed line transmission line is connected between the center segment of each element, so there is always an odd number of segments. 3. NUMERICAL OPTIMIZATION OF A LPDA Our numerical optimization routines are written in MATLAB. A genetic algorithm (GA) and/or a Nelder-Mead downhill simplex optimization algorithm couples the optimization routines to a compiled version of the NEC2 code. The algorithms vary the element length, spacing and radius of each element in order to optimize the gain and
1272 Figure 2. Flowchart of the computer models. VSWR. A Nelder-Mead method and a GA are well-known optimization tools, and they are described in the references [9] and [10]. The flow chart of the antenna design with a GA, a Nelder-Mead algorithm, and a hybrid genetic/nelder-mead algorithm is shown in Figure 2 [11]. A Nelder-Mead downhill simplex local optimization method does not require the calculation of derivatives. The Nelder-Mead method forms a simplex or the most elementary geometrical figure for a given dimension. The idea is to move and expand the simplex until it surrounds the minimum point. Once it surrounds the minimum, then the diameter of the simplex gets smaller, and the method stops when the diameter reaches a specific tolerance. The GA begins by creating an initial random population of continuous parameters for element size, spacing and radius. This set of variables for a LPDA is placed in a vector called a chromosome. The values of the variables are limited to those that are physically feasible for a LPDA. A chromosome is shown in equation (6), where Li and Rl are the length and radius of the 1st element, respectively, and dl is the spacing between the 15t element and the 2nd element. The cost function of the adaptive genetic algorithm linked with NEC evaluates the gain and VSWR corresponding to the element size, spacing and radius settings. The cost function includes the mean and standard deviation of the gain and the VSWR of the LPDA, and it is shown in equation (7), where a, b, c and d are constants determined
1273 by the user. The first and third items in the cost function, mean of gain and VSWR, give low cost values to the high average gain and low average VSWR of a LPDA over the frequency range. The second and fourth items reduce fluctuations in gain and VSWR of a LPDA in the frequency range. For the hybrid genetic/nelder-mead downhill simplex algorithm, the parameters optimized by a GA are transferred to a Nelder-Mead local optimization method, and single or multiple runs of the Nelder- Mead algorithm refine the solution. 4. RESULTS The GA, Nelder-Mead, and hybrid GA/Nelder-Mead methods linked with NEC are applied to three LPDA designs. The first design is a 7 element LPDA (LPDA 1) with a frequency range of 800 to 1600 MHz. The second design is a 20 element LPDA (LPDA2) with a frequency range of 200 to 1300 MHz. The mean and standard deviation of the gain and VSWR are included in the cost function in equation (7). The constants in equation (7) are different for each LPDA design, and are problem dependent. The results of the 2 designs element lengths, spacings, radii, gain and VSWR are shown in the following subsections. LPDAL: 7 elements over 800 to 1600 MHz The 7 element lengths and radii and 6 spacings (20 variables) for LPDAI are optimized with the 3 methods. The constants of equation (7) are also carefully selected, and they are a = 1, b = 1, c = 2, and d = 1. The parameters of the GA are initial population = 48, discard rate = 0.5 and mutation rate = 5%. The maximum number of evaluations is 6000 for all 3 methods. The gain and VSWR are sampled every 80 MHz from 800 MHz to 1600 MHz. The element lengths, spacings, radii, mean and standard deviation of gain and VSWR over the frequency range are shown in Table 1. The average gain over the frequency range improved from 8.0673 db to 8.7606 db using the Nelder-Mead method, and to 8.6597 db using a GA with very low variation less than 0.33. In addition, the gain improves to 8.9040 db with a low variation over the entire frequency range when the hybrid method is used. The VSWR of LPDAL improved from 1.8232 to 1.1051 (Nelder-Mead), 1.0497 (GA) and 1.0114 (hybrid GA/Nelder-Mead) with very low variation around 0.05. All VSWRs are very closed to 1. The boom length of the initial
1274 Table 1. The 7 element LPDA1 for 800 MHz to 1600 MHz: Element lengths, spacings and radii (in meter), and gain & VSWR. design is 0.2997 m, and the boom length of the LPDAs optimized by the Nelder-Mead, GA, and hybrid GA & Nelder-Mead algorithms are 1.1%, 1.8% and 1.2% larger than initial boom length. The initial and optimized gain and VSWR vs. frequency are shown in Figures 3 and 4. In Figure 4, the highest VSWR over the frequency range for Nelder- Mead, GA and hybrid GA/Nelder-Mead are 1.23, 1.0497 and 1.013, in that order.
1275 Figure 3. Gain of 7 element LPDA1 for 800 to 1600 MHz. Figure 4. 4 VSWR of 7 element LPDA1 for 800 to 1600 MHz. The hybrid method gives the best gain, VSWR, standard deviation of gain and VSWR compared to the results of the other methods, while the Nelder-Mead method gives better gain than the GA for the 7 element LPDA 1 (800 to 1600 MHz) optimization. LPDA2: 20 elements over 200 to 1300 MHz The 20 element lengths and radii and 19 spacings (59 variables) for LPDA2 are optimized with the 3 methods. The constants of equation
1276 Table 2. The 20 element LPDA2 for 200 MHz to 1300 MHz: Initial element lengths, spacings and radii (in meter), and optimized gain & VSWR.
1277 Figure 5. Gain of 20 element LPDA2 for 200 to 1300 MHz. (7) are chosen to be a = 1.5, b = 1, c = 1.5, and d = 1. The parameters of the GA are initial population = 24, discard rate = 0.5 and mutation rate = 10%. The large mutation rate, 10%, is selected to avoid getting stuck in a local minimum due to the large number of parameters. The maximum number of evaluations is 3000 for all the algorithms. The number of evaluations is reduced since the smaller population size and the higher mutation rates on real continuous parameter generate a better design quickly with a given fixed number of evaluations. The Nelder-Mead algorithm requires more iterations due to the large number of variables to optimize. The gain and VSWR are sampled every 50 MHz from 200 MHz to 1300 MHz. The gain, VSWR and element lengths, spacings and radii of the initial 20 element LPDA are shown in Table 2. The optimized gain, VSWR, standard deviation of gain and VSWR are compared in Table 2. The gain over the frequency range improved from 8.5017 to 9.5809 db using the Nelder-Mead method, to 10.6255 db using a GA, and to 10.7943 db using a hybrid GA/Nelder-Mead method. The averaged VSWR of the 20 element LPDA2 improved from 1.4158 to 1.0618 (Nelder-Mead), 1.2926 (GA), and 1.2271 (hybrid GA/Nelder- Mead). The initial and optimized gain and VSWR vs. frequency are shown in Figures 5 and 6. In Figure 5, most of optimized gains over the frequency range by a GA and a hybrid are larger than the initial gain, and the gain optimized by a Nelder-Mead method. The optimized VSWRs by the three methods are low compared to the initial VSWR.
1278 Figure 6. VSWR of 20 element LPDA2 for 200 to 1300 MHz. Figure 7. Cost vs. number of evaluation for 20 element LPDA2. Figure 7 compares the convergence speed of the three methods. The convergence speed of the GA is faster than the Nelder-Mead method for up to 2000 generations. The hybrid GA/Nelder-Mead method generates a better design and better cost with fast convergence speed than either a GA or a Nelder-Mead method.
1279 5. CONCLUSIONS This paper presents results for optimizing a complex antenna system with a GA, Nelder-Mead local optimization method, and hybrid GA/Nelder-Mead method. In all cases, the 1 N 2 db higher gain with very low VSWR (less than 1.2 for Lpdal and less than 1.7 for LPDA2) compared to the current commercial designs. The current similar designs of LPDAs have around 5 N 8 db, and 2:1 VSWR in the frequency range. The optimized LPDAs have very low standard deviation of gain and VSWR over the frequency range. In addition, the hybrid GA/Nelder-Mead algorithm worked better than either the GA or Nelder-Mead algorithm alone. Having the GA find the region of the local minimum uses the "global" nature of the GA. A local optimizer is much faster once the bowl of the local minimum is found. REFERENCES 1. Isbell, D. E., "Log periodic dipole arrays," IRE Trans. Antenna Propagat., Vol. AP-8, 260-267, 1960. 2. Carrel, R. L., "Analysis and design of the log-periodic dipole antenna," Ph.D. Dissertation, Elec. Eng. Dept., University of Illinois, University Microfilms, Inc., Ann Arbor, MI, 1961. 3. DeVito, G. and G. B. Stracca, "Comments on the design of log-periodic dipole antennas," IEEE Trans. Antenna Propagat., Vol. AP-21, 303-308, 1973. 4. DeVito, G. and G. B. Stracca, "Further comments on the design of log-periodic dipole antennas," IEEE Trans. Antenna Propagat., Vol. AP-22, 714-718, 1974. 5. Butson, P. C. and G. T. Thomson, "A note on the calculation of the gain of log-periodic dipole antennas," IEEE Trans. Antenna Propagat., Vol. AP-14, 105-106, 1976. 6. Stutzman, W. L. and G. A. Thiele, Antenna Theory and Design, 2nd Edition, John Wiley & Sons, New York, NY, 1997. 7. Harrington, R. F., Field Computation by Moment Methods, Robert E. Krieger Publishing Company, Malabar, FL, 1985. 8. Burke, G. J. and A. J. Poggio, Numerical Electromagnetics Code (NEC2) User's Manual, Lawrence Livermore Lab., Jan. 1981. 9. Nelder, J. A. and R. Mead, "A simplex method for function minimization," Comput. Journal, Vol. 7, 308-313, 1965. 10. Haupt, R. L. and S. E. Haupt, Practical Genetic Algorithms, John Wiley & Sons Inc., New York, NY, 1998. 11. Chung, Y. C. and R. L. Haupt, "Log period dipole array
1280 optimization," IEEE Aerospace Conference Digest, 10.02 track CD, March 2000. You Chung Chung is a Research Assistant Professor of Electrical and Computer Engineering at Utah State University. He is working in The for Self Organizing Intelligent Systems and Center of Excellence for Smart Sensors in Utah State University. He received the B.S. in electrical engineering from INHA University, Inchon, Korea in 1990, and M.S. and Ph.D. degrees from University of Nevada, Reno (UNR) in 1994 and 1999. His research interests include computational electromagnetics, optimized antenna and array design, conformal and fractal antennas, smart wireless sensors, optimization techniques, EM design automation tool development and genetic algorithm. In 1996, he received an Outstanding Teaching Assistant Award from UNR. He also received an Outstanding Graduate Student Award in 1999. He received the 3rd student paper award from URSI National Radio Science Meeting in 2000. Randy Haupt is Professor and Department Head of Electrical and Computer Engineering at Utah State University. He has a Ph.D. from the University of Michigan, M.S. from Northeastern University, and B.S. in Electrical Engineering from the the USAF Academy. Dr. Haupt was a project and antenna engineer for the USAF, Professor of Electrical Engineering at the USAF Academy, and Professor and Chair of Electrical Engineering at the University of Nevada Reno. He is an IEEE Fellow and was the Federal Engineer of the Year in 1993. He is co-author of the book Practical Genetic ' Algorithms, John Wiley & Sons, Jan. 1998.