Log-Period Dipole Array Optimization

Log-Period Dipole Array Optimization You Chung Chung University of Nevada, Reno Electrical Engineering Reno, NV 89557 775-784-6927 youchung @unr.edu Randy Haupt Utah State University Electrical and Computer Engineering 4120 Old Main Hill Logan, UT 84322-4120 435-797-2840 haupt (@ ieeo.org Abstract-The log periodic dipole array is a commonly used broadband antenna. Design equations exist for calculating the size and spacings of the wires that make up the antenna. These equations fail to take into account the electromagnetic interactions between the wires, though. The Numerical Electromagnetics Code is an excellent wire modeling code for analyzing the full electromagnetic interactions of a wire antenna. In this paper we use the Nelder-Mead downhill simplex algorithm and a genetic algorithm to optimize the design of a log periodic dipole array. Modeling results show that a hybrid genetic a1gorith"elder-mead downhill simplex algorithm produce superior antenna designs to traditional analytical approaches or using the genetic algorithm or Nelder-Mead algorithm alone. 1. 2. 3. 4. 5. TABLE OF CONTENTS INTRODUCTTON COMPUTER MODEL OF LPDA NUMERICAL OPTIMIZATION AF'PROACH Rl5SULTS CONCLUSIONS 1. INTRODUCTION The log-periodic antenna finds use in many broadband communications applications. 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 the 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. Increasing the bandwidth of the antenna requires adding more antenna elements. Figure 1. LPDA geometry. I I t The LPDA was introduced by Isbell [I], and improved using techniques shown in references [2-51. The performance of a LPDA is a function of number of elements as well as element length, spacing, and diameter. Antenna element lengths and spacings have proportionality factors given by a scale factor and spacing factor 2=- Ll,+1 where, the L,, is the length of nth element, and d, is the element spacing between nth and (n+l)*. 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 L" 0-7803-5846-5/00/$10.00 0 2000 IEEE 449

bounds the dipole lengths. The angle a is calculated from z and G by cx=2tan -1(z) I-r in order to model a variety of antenna structures. The antenna model can have nonradiating networks and transmission lines connecting parts of the structure, perfect or imperfect conductors, and lumped element loading. 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 z and G. The voltage standing wave ratio is given by, vswr = (1 + IPI) where, p is the reflection coefficient given by Z, = antenna impedance 2, = transmission line impedance Most antenna books give equations for the design of logperiodic antennas. Unfortunately, these design equations do not include the complex interactions between the dipoles that compose the antenna. Consequently, a numerical model and numerical optimization of that model are important for developing realistic designs. 2. COMPUTER MODEL OF LPDA Wire antennas, like the log-periodic dipole array (LPDA), are accurately modeled with the Numerical Electromagnetics Code (NEC). This FORTRAN code is widely used and forms the basis for commercial and noncommercial wire modeling codes. The geometry of the log-periodic antenna 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 as shown in Figure 2. The locations of 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. (4) Figure 2. Wire segments for NEC. An integral equation formulation works best for antennas with dimensions of up to several wavelengths. Beyond that size, the MOM requires a rather large matrix equation, since the matrix order increases relative to a wavelength. Hence, modeling very large structures may require more computer time and file storage than is practical on a particular machine. In such cases high-frequency codes based on approximations such as geometrical optics, physical optics, or geometrical theory of difhction are more appropriate. The NEC program uses an electric-field integral equation (EFIE) and a magnetic-field integral equation (MFIE). The EFIE works best for thin-wire structures of small conductor volume while the MFIE (doesn t work for the thin-wire case) works best for voluminous structures with large smooth surfaces. The EFIE can be used to model very thin surfaces. Although the EFIE is specialized to thin wires in NEC, it is frequently used to represent surfaces may be modeled by wire grids with reasonable success for far-field quantities but with variable accuracy for surface fields. 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 O.lh and larger than 0.001h [SI. 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 OFTIMEATION APPROACH NEC numerically solves integral equations for the currents Our numerical routines are written in induced on antennas by voltage sources. This code combines MATLAB. A GA andor a Nelder-Mead local an integral equation for smooth surfaces with one for wires 450

algorithm couples the optimization routines to a compiled version of the NEC2 code. The algorithms vary the element length, spacing and radius in order to optimize the gain and VSWR. A Nelder-Mead downhill simplex local optimization method and a GA are well-known optimization tools, and they are described well in the references [9] and [lo]. The Nelder-Mead method is executed after the GA finds a reasonably good design for the LPDA. The flow chart of the antenna design with a genetic algorithm & Nelder-Mead is shown in Figure 3. The initial population is evaluated by NEC. Chromosomes in the population are ranked from the best fit to the worst fit based on their costs. The bottom 50% of the initial population is discarded, and the remaining top half is called the parents. Two partners are randomly selected from the parents, and they mate. A random number, p, and a random crossover point are selected. The variables at the right side of the random crossover point are swapped, and the variables at the crossover point are calculated with p. The sample of a chromosome and the parents having variables for an LPDA, and their mating and offspring are shown as follows in Figure 4. mom1 = [L,,,I... L, dmlidm2i dm3... d,.l RmI... R,] dad 1 = [Ldl... Ld, ddi ddi dd3... ddn-l&l... &,,I JN Figure 4. Continuous chromosome mating with a random GA optimization crossover point in a GA. The total number of new offspring is the same as the number of discarded population members. A small percentage of variables in the new offspring and their parents are randomly I Local optimization I I Great design Figure 3. Flowchart of the computer model. 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 RI are the length and radius of the lst element, respectively, and dl is the spacing between the lst element and the 2nd element. chromosome=[ll L2... Ln dl d2... dn-1 Ri R2... Rn1 (6) 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 by the user. Cost = -axmean( Gain(dB)) + bxstd( Gain(dB)) + cxmean{vswr) + dxstd{vswr) (7) switched. This process is called mutation, and a high percentage of mutation gives more freedom to search new LPDA designs. There is no mutation of the best chromosome on each generation. The new population is evaluated, and these processes repeat until the GA satisfies the results or reaches a given maximum generation. For the hybrid GA and Nelder-Mead downhill simplex method, 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 five element LPDA (LPDAl) with a frequency range of 300 to 400 MHz. The second design is a 7 element LPDA (LPDA2) with a frequency range of 800 to 1600 MHz. The third design is a 20 element LPDA (LPDA3) with a frequency range of 200 to 1300 MHz. The mean and standard deviation of gain and VSWR are included in the cost function in equation (7). The constants in equation (7) are different for each LPDA design. The results of the 3 designs-lement lengths, spacings, radii, gain and VSWR-are shown in the following subsections. LPDAl: 5 elements over 300 to 400 MHz 45 1

The 5 element lengths and radii and 4 spacings (14 variables) for LPDAl are optimized with the 3 methods. The constants of equation (7) are a=1.5, b=l, c=1.5, and d=l for the CA. The parameters of the GA are initial population=48, discard rate=0.5 and mutation rate =5%. The maximum number of function evaluations is 5000 for all 3 methods. The gain and VSWR are sampled every IOMHz. The element lengths, spacings, radii, mean and standard deviation of the gain and VSWR over the frequency range are shown in Table 1. The average gain over the frequency range improved from 7.4561dB to 8.691dB (Nelder-Mead), 8.4285 (GA), and 8.8126 (hybrid). In addition, the average VSWR of LPDAl improved from 1.90 to 1.1585 (Nelder- Mead), 1.3336 (GA), and 1.1278 (hybrid). The boom length of initial the design is 0.5937m, and the boom length of the optimized LPDAs by Nelder-Mead, GA and hybrid are 11. I%, 39% and 11.4% larger than the initial boom length. The initial and optimized gain and VSWR vs. frequency are shown in Figures 5 and 6. The highest VSWR over the frequency range for Nelder-Mead, CA and CA & Nelder- Mead are 1.24, 1.7 and 1.23, in that order. The 7 element lengths and radii and 6 spacings (20 variables) for LPDAl are optimized with the 3 methods. The constants of equation (7) are a=l, b=l, c=2, and d=l for the GA. The parameters of GA are initial population=48, discard rate=0.5 and mutation rate =5%. The maximum number of evaluations is 6000 for all 3 methods. 11, I' I 0 Initial gain 105 Gain by GA 10 * Gain by Nelder + Gain by GA & Nelder 95 m U 9,; ; 851. 8-. 8 * * *. * * * i &. * I Table 1. The 5 element LPDAl for 300 to 400MHz: Element lengths, spacings and radii, and gain & VSWR. VSWR 2.0 - VSWR by GA 2.6-0 VSWR by Nelder VSWR by GA & Nelder 24-1 ; Initial + _, 4 320 340 360 380 400 Figure 5. Gain of 5 element LPDAl for 300 to 400 MHz. R1 L2 R2 L3 R3 L4 R4 L5 R5 SI s2 s3 Nelder 0.00041 0.0058 0.0050 0.0054 0.4860 0.3794 I 0.2168 0.1929 0.00032 0.0040 0.0050 0.0053 0.4374 0.3424 0.1998 0.1778 0.00032 0.0072 0.0050 0.0021 0.3937 0.3826 0.1914 0.1687 0.00032 0.0009 0.0050 0.0054 0.3543 0.3774 0.1846 0.1827 0.00032 0.0001 0.0048 0.0014 0.1728 0.090 0.2500 0.1290 0.1555 0.1750 0.2266 0.1382 0.1397 0.1997 0.1521 0.1579 s4 0.1257 0.1953 1 0.2008 0.2363 I Gain I 7.4561 I 8.6891 I 8.4285 I 8.8126 STD(gain) I 0.7610 I 0.2983 I 0.3974 I 0.3966 \",, VSWR 1 1.900 STD(VSWR) I 0.4129 1.1585 1.3336 1.1278 0.0899 0.1821 0.0851 The GA & Nelder method gives best gain, VSWR, standard deviation of gain and VSWR compared to the results of other methods, while Nelder-Mead method gives better results than the CA for 5 element LPDA optimization. LPDAZ: 7 elements over 8300 to 1600 MHz 2.21 0 0 do0 320 340 360 380 400 Figure 6. VSWR of 5 element LPDAl for 300 to 400 MHz. The gain and VSWR are sampled every 80MHz from 800MHz to 1600MHz. The element lengths, spacings, radii, mean and standard deviation of gain and VSWR over the frequency range are shown in Table 2. The average gain over the frequency range improved from 8.0673dB to 8.7606dB by the Nelder-Mead method, and to 8.0781dB by a CA. In addition, the gain is improved to 8.9040dB with low variation all over the frequency range by the hybrid method. The VSWR of LPDA2 improved from 1.7974 to 1.1051 (Nelder-Mead), 1.0864 (GA) and 1.0114 (hybrid). All are very closed to 1. The boom length of initial design is 0.2997m, and the boom length of optimized LPDAs by Nelder-Mead, GA, and hybrid 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 7 and 8. In Figure 7, the highest VSWR over the frequency range for 452

~ ~ Nelder-Mead, GA and GA & Nelder-Mead are 1.23, 1.14 and 1.013, in order. Table 2. The 7 element LPDA2 for 300 to 400MHz: Element lengths, spacings and radii, and gain & VSWR. Initial 7 element LPDA design. z=0.9 and o=o. 17 I I I Ontimized I 281 ~ ~ $ ~ ~ ~ ~ 26 VSWR by Nelder VSWR by GA & Nelder 24 2 2c 0 1 1 R5 0.004 0.0039 0.0054 I 0.0039 L6 I 0.1107 I 0.0924 I 0.0950 1 0.0924 1 R6 ~. 1 0.003 10.0027 10.0045 10.0027 1 I L7 I 0.0996 I 0.0934 I 0.0836 1 0.0934 I R7 1 0.003 I 0.0001 1 0.0017 I O.OOO1 s1 I 0.0638 I 0.0731 I 0.0607 1 0.0731 s2 1 0.0574 I 0.0404 1 0.0571 I 0.0404 s3 I 0.0516 I 0.0607 I 0.0516 I 0.0607 s4 1 0.0465 1 0.0413 I 0.0542 I 0.0413 s5 I 0.0428 0.0399 I 0.0346 1 0.0399 1 S6 I 0.0376 I 0.0481 I 0.0471 I 0.0481 Gain 1 8.0673 1 8.7606 I 8.0781 I 8.9040 4 I STD(gain) 1.0326 0.3206 0.3363 1 0.2816 ] VSWR 1.8232 I 1.1051 I 1.0864 I 1.0114 STD(VSWR) I 0.6935 10.0611 10.0415 IO.0110 0 Initial gain Gain by GA * Gain by Nelder + Gain by GA & Nelder * P * + * : * + * O O. O. * * 7.L-l 6.5 800 320 340 360 380 400 Figure 7. Gain of 7 element LPDA2 for 800 to 1600 MHz. 1 4 5 6 7 14 15 16 17 0 * * * : * * * * *, 320 340 360 380 400 Figure 8. VSWR of 7 element LPDA2 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 results than the GA for 7 element LPDA2 optimization. Table 3. The 20 element LPDA3 for 200 to 1300MHz: Initial element lengths, spacings and radii, and optimized gain & VSWR. Initial 20 element LPDA desi n, -0.9 and 0=0.16 0.75000 0.24000 0.67500 0.004 0.21600 0.60750 0.004 0.19440 0.54675 0.004 0.17496 0.49207 0.004 0.15746 0.44286 0.004 0.14171 0.39858 0.004 0.12754 8 0.35872 I 0.004 0.11479 9 0.32285 0.004 I 0.10331 I 10 0.29056 0.004 I 0.09288 I 11 I 0.26151 I 0.003 1 0.08368 12 0.23536 I 0.003 1 0.07531 13 I 0.21182 1 0.003 I 0.06778 0. I9064 0.003 0.06100 0.17 157 0.003 0.05490 0.15442 0.003 0.04941 0.13897 0.003 0.04447 18 I 0.12508 I 0.003 I 0.04002 19 I 0.11257 1 0.002 I 0.03602 20 1 0.10131 0.002 Gain / STD(Gain) 8.5017 db 10.9707 VSWR I STD(VSWR) 1.415810.5462 LPDA3: 20 eleinents over 200 to 1300 MHz The 20 element lengths and radii and 19 spacings (59 variables) for LPDA3 are optimized with the 3 methods. The constants of equation (7) are a=l, b=l, c=2, and d=l for the GA. The parameters of the GA are initial population=24, J 453

~~ ~ ~ discard rate=0.5 and mutation rate =lo%. The maximum number of evaluation is 2000 for the GA, and 3300 for the Nelder-Mead method. The Nelder-Mead requires more number of iteration due to the large number of variables to optimize. The gain and VSWR are sampled every 5OMHz from 200MHz to 1300MHz. The gain, VSWR and element lengths, spacings and radii of initial 20 element LPDA are shown in Table 3. The optimized gain, VSWR, standard deviation of gain and VSWR are compared in Table 4. Performance Gain Initial Optimized Design Nelder-Mead I GA 8.5017 90118 I 9 9798 STD(gain) I 0.9707 I 0.5472 I 0.6646 VSWR 1.4158 I 1.1495 1 1.2324 STD(VSWR) I 0.5462 1 0.1676 I 0.1462 The gain over the frequency range improved from 8.5017 to 9.0118dB by the Nelder-Mead method, and to 9.9798dB by a GA. The VSWR of the 20 element LPDA3 improved from 1.4158 to 1.1495 and 1.2411 by a Nelder-Mead and a GA, respectively. The initial and optimized gain and VSWR vs. frequency are shown in Figures 9 and 10. In Figure 9, most of optimized gains over the frequency range by a GA are larger than the initial gain, and the gain optimized by a Nelder-Mead method. In Figure 10, the highest VSWR over the frequency range for Nelder-Mead, and GA are 1.498 and 1.58, respectively. The optimized VSWR by both methods are low compared to the initial VSWR. Figure 11 compares the convergence speed of both methods. The convergence speed of the GA outperforms that of Nelder-Mead. The GA improves the gain of 20 element LPDA with reasonably low VSWR 111/ quickly, compared to the Nelder-Mead method. 1 '7 I I I I 1 f Initial gain Gain by GA 11 Gain by Nelde 10 5 element, and seven element LPDAs are optimized with the three methods, and the performance compared. The average gain of the five element LPDA improved from 7.4561dB to 8.8126dB an average VSWR of 1.1278 over the frequency range 300 to 400MHz. The seven element LPDA average gain improved from 8.0673dB to 8.9040dB with an average VSWR of 1.0114 over the frequency ranges 800 to 1600MHz. The GA & Nelder methods yield improved designs over the analytical methods. The Nelder-Mead method performs better than a GA for the five and seven element LPDAs. The total number of variables for five element LPDA and seven element LPDA are 14 and 20, respectively, and the range of element lengths, spacings and radii are limited for having a feasible size of antenna. Therefore, the local optimization method, Nelder-Mead downhill simplex method performs better than a GA for a limited number of variables. VSWR by GA 26 VSWR by Nelde 24 Figure 10. VSWR of 20 element LPDA3 for 200 to1300 MHz. 454 Figure 9. Gain of 20 element LPDA3 for 300 to 1300 MHz. 5. CONCLUSIONS This paper presented results for optimizing a complex antenna system with a GA, Nelder-Mead local optimization method, and hybrid GA & Nelder-Mead method. The five -7-8 500 1000 1500 2000 2500 3000 Number of Evaluation Figure 11. Cost vs. number of evaluation for 20 element LPDA3 by a GA and a Nelder-Mead. The twenty element LPDA is optimized with a GA and the Nelder-Mead method to compare the performance of the methods with a large number of variables. The total number of yariables are 59, including 20 element lengths and radii, and 19 spacings. The gain of twenty element initial LPDA

design improved from 8.5017dB to 9.01 18dB by Nelder- Mead and to 9.9798dB by a GA. In addition the VSWR is improved from 1.415 to 1.1495 and to 1.2324 with low standard deviation 0.1676 and 0.1462, respectively. The GA outperforms Nelder-Mead optimization method due to the large number of variables. In all cases, the hybrid GA & Nelder-Mead algorithm worked better than either the GA or Nelder-Mead algorithm alone. REFERENCES [l] D. E. Isbell, Log periodic dipole arrays, IRE Trans. Anteiirza Propagat., AP-8, pp. 260-267, May., 1960. [2] R. L. Carrel, Analysis and design of the log-periodic dipole antenna, Ph.D. Dissertation, Elec. Eng. Dept., University of Illinois, 1961, University microfilms, Inc, Ann Arbor, MI. [3] G. DeVito and G. B. Stracca, Comments on the design of log-periodic dipole antennas, IEEE Trans. Antenna Propagat., AP-21, pp. 303-308, May., 1973. 141 G. DeVito and G. B. Stracca, Further comments on the design of log-periodic dipole antennas, IEEE Trans. Arztenna Propagat., AP-22, pp. 714-718, Sep., 1974. [SI P. C. Butson and G. T. Thomson, A note on the calculation of the gain of log-periodic dipole antennas, IEEE Trans. Antenna Propagat., AP-14, pp. 105-106. Jan., 1976. [6] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, John Wiley & Sons, New York, NY, 1981. [7] R.F. Harrington, Field Computation by Moment Methods, Robert E. Krieger Publishing Company, Malabar, FL, 1985. [SI G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC2) User s Manual, Lawrence Livermore Lab., Jan., 1981. [9] J. A. Nelder and R. Mead, A simplex method for function minimization, Comput. Journal, vol. 7, pp. 308-313, 1965. Randy Haupt is Department Head and Professor of Electrical and Computer Engineering at Utah State University. He was Dept. Chair and Professor of Electrical Engineering at University of Nevada Reno, Professor of Electrical Engineering at the USAF Academy, research engineer at RADC, and project engineer for the OTH-B Radar Program. He has a PhD from the University of Michigan, MS from Northeastern University, MS from Western New England College, and BS from the USAF Academy. He is co-author of the book Practical Genetic Algorithms, has 8 antenna patents, is an IEEE Fellow, and is recipient of the 1993 Federal Engineer of the Year Award. You Chung Chung received his BS in electrical engineering from Jnha University, Inchon, Korea in 1990, and MSEE degree from University of Nevada, Reno (UNR) in 1994. He recently completed all requirements for the Ph.D. degree in electrical engineering at UNR and is working as a Post Doctoral Research Engineer at Utah State University. Since 1995, he has been employed as a teaching and research assistant in electrical engineering at UNR. His research interests include computational electromagnetics, optimized antenna and array design, conformal and fractal antennas, adaptive array processing, optimization techniques, and genetic algorithms. In 1996, he received an Outstanding Teaching Assistant Award from UNR. He also received an Outstanding Graduate Student Award in 1999. The NSF sponsored his 1999 IEEE AP-S paper presentation. [lo] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, John Wiley & Sons Inc., New York, NY, 1998. 455