Six-Element Yagi Array Designs Using Central Force Optimization with Pseudo Random Negative Gravity

A six-element Yagi-Uda array is optimally designed using Central Force Optimization (CFO) with a small amount of pseudo randomly injected negative gravity. CFO is a simple, deterministic metaheuristic analogizing gravitational kinematics (motion of masses under the influence of gravity). It has been very effective in addressing a wide range of antenna and other problems and normally employs only positive gravity. With positive gravity the six element CFO-designed Yagi array described here exhibits excellent performance with respect to the objectives of impedance bandwidth and forward gain. This paper addresses the question of what happens when a small amount of negative gravity is injected into the CFO algorithm. Does doing so have any effect, beneficial, negative or neutral? In this particular case negative gravity improves CFO’s exploration and creates a region of optimality containing many designs that perform about as well as or better than the array discovered with only positive gravity. Without some negative gravity these array configurations are overlooked. This Yagi-Uda array design example suggests that antennas optimized or designed using deterministic CFO may well benefit by including a small amount of negative gravity, and that the negative gravity approach merits further study.


Introduction
The Yagi-Uda array (Yagi) has been around for nearly one hundred years [1] [2], undoubtedly because of its ability to provide excellent performance across a wide range of antenna measures, always in a simple geometry and often in a compact one as well [3]. The basic Yagi comprises a single driven dipole element (DE) flanked by a single parallel parasitic reflector (REF) on one side and any number of parasitic parallel directors on the other (D i ). A typical six-element geometry is shown in Figure 1 (red dot marking DE feedpoint). The X-axis is the Yagi's "boom", and the array's elements are an arranged along it in the X-Y plane parallel to each other and to the Y-axis as shown. REF, the element that is closest to the Y-axis, is closer to DE than the first director and longer than DE while the D i are shorter, but these characteristics do not by any means constitute absolute design requirements, as they simply are typical geometry that produces appropriate phase and current relationships between the array elements.
A complete Yagi design must include the lengths and diameters of each element and the inter-element spacings. A six-element array thus has seventeen design parameters (six element lengths, six diameters, and five spacings). If the feedpoint input impedance is considered a design variable as well (discussed below), then the design/optimization (D/O) problem comprises eighteen dimensions. Many Yagis have far more than six elements, and every additional one adds three more design parameters (variables) to the problem.
There is no accurate analytical approach to Yagi array design using models that assume purely sinusoidal element current distributions and that neglect their mutual couplings. These assumptions lead to inaccuracies, and as a result many design approaches are inherently approximate or based entirely on empirical data [3] [4] [5]. The difficulty in analytically calculating a Yagi's element currents is highlighted by the fact that there is no known formula for expressing the current distribution on even the simplest of radiating elements, a single center fed dipole in free space [6].

Background
Over the past two decades or so, in response to the limitations described above, Yagi D/O largely has been done using metaheuristics, that is, algorithms that provide acceptable solutions in reasonable times without being exact or having a  π-fractions also were used to determine the degree of injected negative gravity.
As a result, and as expected, the target and actual levels of negative gravity are not precisely equal. Even though CFO is deterministic, it is reasonable to speculate that CFO may benefit from the inclusion of a pseudo random component. A pseudo random variable (prv) is a number known precisely by enumeration or calculation, in contrast to a true random variable (rv) whose value is a priori unknowable because it must be calculated from a probability distribution. With respect to any D/O problem, the prv is "random" in the sense that it is uncorrelated with the problem's topology ("landscape"). Additionally, the prv's themselves must be uniformly distributed and uncorrelated. Pseudo randomness is included here using π-fractions generated by the Bailey-Borwein-Plouffe (BBP) algorithm. A detailed discussion of π-fractions and their use in a sample GSO algorithm, πGASR (Genetic Algorithm with Sibling Rivalry), appears in [40].

CFO Run Setup
A detailed description of the basic CFO algorithm is in the Appendix Part 2. Run parameters and the decision space boundaries for the basic CFO implementation used in this paper appear in Table 1 and Table 2 and are discussed in detail in the Appendix. For this particular problem the Yagi element radius was fixed at 0.00635 m (1/2 inch diameter to facilitate easier fabrication). This simplification results in eleven geometric variables instead of seventeen. The twelfth variable to be determined by CFO is Z 0 , the Yagi's feedpoint impedance. Variable Z 0 (VZ 0 ) is a patented "product-by-process" technology in which the antenna's feedpoint impedance is treated as an optimization variable rather than as a fixed parameter as usually is the case. At this time VZ 0 is publicly available for use by anyone who wishes to use it (U.S. Patent No. 8,776,002). CFO pseudo code appears in Figure 2, and the source code used to generate the data presented here is available on request to the author (rf2@ieee.org).

Exploration vs. Exploitation in CFO
Positive gravity causes CFO's probes always to move toward greater fitnesses, never away, and consequently to some degree perhaps to impede CFO's exploration. CFO often converges very quickly [42], which is a favorable attribute, but not if it is at the expense of under-sampling DS, which may be the case. This paper speculates that adding some negative gravity causing probes to fly away from each other may improve CFO's exploration because probes that otherwise would coalesce will explore further by flying into DS regions that have been under-sampled or perhaps not sampled at all. The effect of negative gravity is well illustrated in the discussion and figures in Section 6 of [43], and parameter values that insure CFO's convergence on maxima are discussed in [44].
With respect to G's sign, + or -, the specific question is, Does making it negative benefit CFO's performance, and if so, why? Or does it impede it, and if so, why? To quote: "'Exploration and exploitation are the two cornerstones of problem solving by search.' For more than a decade, Eiben and Schippers' advocacy for balancing between these two antagonistic cornerstones still greatly influences the research directions of evolutionary algorithms (EAs)..." [45], emphasis added. This issue also is discussed in [46] [47] [48]. Like all GSO algorithms CFO is subject to the inescapable tension between exploration (adequately sampling DS) and exploitation (quickly converging on global maxima). The test case reported here shows that a small amount of negative gravity indeed does benefit CFO's performance, ostensibly because it enhances CFO's exploration while retaining the algorithm's ability to exploit already located maxima. At each step negative gravity was pseudo randomly injected into CFO using π-fractions whose values are input from an external file containing the BBPcomputed fractions π k , 1 ≤ k ≤ 215829 (k is the π-fraction index). In order to avoid correlations between the π-fractions (see [40]), k is incremented by 5 at each step. If k exceeds 215829 it is reset to max(k-215827,3). The parameter 0 ≤ P neg ≤ 100 specifies the target amount of negative gravity in percent, for example, P neg = 6.5 sets the target at 6.5%. At each step the then current value of π k is tested against P neg to determine whether or not gravity is negative at that step. Specifically, if π k ≤ P neg /100 then G G = − (negative gravity), otherwise G G = (positive gravity). Because prv's are used, the nature of gravity at each step is a priori known precisely and is uncorrelated with the outcome at any other step.
Also because prv's are used, the desired (target) level of negative gravity and the actual level will not be precisely equal. As the CFO run includes more and more steps the difference between actual and target levels of G ≤ 0 grows smaller with a limiting value of zero because the π-fraction prv's are uniformly distributed.   [49] whose accuracy is well established. Another version of NEC was used in the program 4nec2 (below) to compute and visualize performance data for the two CFO-optimized arrays at 0% and 6% G < 0 over a very wide frequency range (data in Appendix Part 1).

Fitness Function
The performance of every candidate antenna design was measured by the following simple fitness function (of course, the choice of fitness function is entirely up to the algorithm designer): This approach embraces array current distributions that otherwise would be excluded because they fail to adequately match a predetermined value of Z 0 . Whether or not the CFO-returned value is feasible and desirable is an engineering and economic judgment, and more often than not it is worth impedance-matching the "non-standard" Z 0 because the antenna's performance is better, often much better [50]. At each of the three frequencies L, M, U, NEC-4 returned the Yagi's maximum gain and feedpoint impedance. Appendix Part 1 contains additional performance data computed with the freely available program 4nec2 that facilitates visualization of its computed data [51].

Fitness Evolution
The reference Yagi design for this study is, of course, the one that CFO generates with zero negative gravity, the usual CFO implementation. Figure 4 differently with the 6% curve rising far more quickly than the 0% curve and with saturation taking place faster. This CFO implementation does not employ elitism (always including the best global fitness), so that here the best fitness at each step is a function only of the then current probe distribution. How the fitness varies with the amount of negative gravity is plotted in Figure  5. For values below about 6% the fitness drops appreciably from its 0% value, but then it quickly recovers reaching a value higher than at 0% G ≤ 0 and settling into a plateau-like region up to about 10% G ≤ 0. Thereafter the fitness decreases monotonically through the test range of about 20%. The fitness plateau between approximately 6% and 10% is marked the "~region of optimality" on the plot because in that range the fitness is more or less flat. Fitness varies from the maximum value of 49.1892 at 6% G < 0 to a minimum of 47.392 at 6.36% G < 0.
All other values are in this range, and consequently very similar one to the next and also to the 0% G < 0 value.   Returning to the fundamental question addressed in this paper, Has pseudo randomly adding a small amount of negative gravity improved CFO's exploration of the six-element Yagi's decision space? The fitness data clearly show that the answer is "yes." However, adding too much G < 0 prevents CFO from exploiting the solutions it has found. How much negative gravity is appropriate no doubt is problem-specific, but this array design example strongly suggests that every CFO implementation should experiment with some measure of G < 0.

Average Probe Distance
Step-by-step the normalized average distance from the probe with the best fitness to all other probes, denoted D avg , is a good measure of CFO's convergence (see Appendix Part 2 for details). In many cases it approaches zero meaning that all probes have very tightly coalesced around the probe with the best fitness (maximum value). In other cases, such as here, the probe distribution more or less stabilizes but on average at some distance from the best probe. In order to investigate D avg 's behavior in very long runs, two were made, both with N t = 10,000 steps, zero negative gravity and at a target value of G < 0 of 5%.
As discussed in Section 2.3 the actual and target levels of negative gravity are not equal except in the limit of an infinite number of steps. While both the 550-step Figure 6. (a) Average distance to best probe, zero neg. gravity; (b) Average distance to best probe, 6% neg. gravity. and 10,000-step runs targeted G < 0 at 5%, in the first case the actual value was 6% and in the second a much closer 5.22%. The D avg results, which are quite interesting, are plotted in Figure 7. In both cases, zero and 5.22% G < 0, D avg settles down to what appears to be a stable, uniform magnitude oscillation with much smaller amplitude in the 5.22% case. In fact, at 5.22% G < 0 D avg does not oscillate at all, and from steps #676-10,000 it is equal to 0.2352928. CFO's probe distribution is stable and no longer changes step-to-step. In contrast, for the zero gravity case D avg oscillates in an erratic but repetitive pattern as seen in the expanded plot for steps 9900 -10,000, Figure 8. It is reasonable to expect that these behaviors for cases zero and 5.22% G < 0 will continue indefinitely with an increasing number of steps. It is evident from these data that pseudo randomly injecting a small measure of negative gravity indeed does improve CFO's exploration and apparently eliminates the probe trapping seen in Figure 8 as well.

Maximum Yagi Gain
The variation of maximum Yagi gain with degree of negative gravity is plotted in Figure 9. The antenna's maximum gain occurs along the direction of its boom (the +X-axis) at NEC angles θ = 90˚, φ = 0˚ in its right-handed spherical coordinate system (see [49]). With zero negative gravity the gain is 11.37 dBi, but at 6% G < 0 it increases to 11.92 dBi, which is a significant improvement. The max gain curve mirrors the Figure 8 structure of the fitness curve in Figure 5. Their shapes are very similar, and the array's maximum gain is highly correlated with the array's fitness as measured by Equation (1). Figure 10 plots the relative frequency of maximum gain as the percentage deviation from the midband reference frequency of 299.8 MHz. Its shape is quite irregular showing no correlation with either the fitness or maximum gain curves. The frequency of maximum gain is close to 299.8 MHz only for actual G < 0 of ~1.3% and in two narrow bands near 6% and ~11.2%. Otherwise the deviations from midband show considerable variability, ranging as high as nearly +3% and as low as about −2.4%.    The array's relative gain bandwidth is plotted in Figure 11. Bandwidth (BW) is defined as the frequency range for which the gain falls to 3 dB below the maximum gain expressed as a percentage of the midband reference frequency of 299.8 MHz and plotted as a function of negative gravity level. Minimum BW coincides with maximum gain at about 6% G < 0, and it increases quickly and more or less monotonically to about 13% G < 0 after which it decreases monotonically and somewhat more slowly. With gain BW greater than ~0.1F c , BW is fairly large throughout the negative gravity range, as many Yagi arrays exhibit much lower bandwidths.

Voltage Standing Wave Ratio (VSWR)
The antenna's impedance bandwidth is measured by the frequency range over which a maximum or lower VSWR is maintained, typically VSWR ≤ 2:1 relative to the antenna's feedpoint impedance. As discussed above the usual target input impedance is 50 + j0 Ω ( 1 j = − ), that is, a purely resistive 50 Ω that is the industry standard. By contrast, Variable Z 0 technology, which was used here, treats Z 0 as just another optimization variable whose value is determined by the GSO algorithm. For example, the Yagi optimized with 6% G < 0 has a feedpoint impedance of Z 0 = 59.8 ohms with VSWR//59.8 ranging from 1.49 to 2.01 over the optimization frequency range 294.8 to 304.8 MHz. Once an optimized design was developed, the array's performance over a much wider frequency range was investigated using the NEC-2 based program 4nec2 [51]. Those data appear in the Appendix Part 1.  Figure 12 shows the 2:1 VSWR bandwidth in MHz. In the approximate region of optimality it ranges from about 14.5 MHz to just over 20 MHz with much higher values on either side. The tradeoff for increasing impedance bandwidth is maximum gain. The overall best performing arrays discovered by CFO lie in the region of optimality. Figure 13 shows the 2:1 VSWR as a percentage of the midband reference frequency of 299.8 MHz. In the region of optimality it ranges from about 5% to just under 7%, so it is not particularly sensitive to the level of injected negative gravity. Outside this region it does reach higher values, but, again, the tradeoff is with maximum array gain.

Conclusion
This paper reports the results of the D/O of a six-element Yagi-Uda array using Central Force Optimization with pseudo randomly injected negative gravity. Adding 6% G < 0 results in an array that out performs its 0% G < 0 counterpart and also discovers a range of designs with similar fitnesses. Negative gravity was inserted using π-fraction pseudo random variables thereby preserving CFO's determinism, which is an important consideration in real-world problems that require the formulation of a suitable fitness function. G < 0 has the effect of spreading apart CFO's probes because negative gravity is repulsive in nature instead of attractive. This wider dispersal improves CFO's exploration of the decision space without sacrificing the algorithm's demonstrated high level of exploitation. Injecting some level of negative gravity likely will improve CFO's exploration across the board, not only in the Yagi array D/O problem but in other applications as well. This particular problem is only one example of the benefit provided by injecting some small measure of negative gravity, and it does not (necessarily) provide specific guidance as to how much G < 0 should be used in any other particular case. The effect of negative gravity in CFO is a research area that should be pursued because the work reported here strongly suggests that there may be considerable benefit in doing so. Wireless Engineering and Technology

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
In the following screenshots 0% G < 0 is on the left, 6% G < 0 on the right.
where r is the distance between them and γ the "gravitational constant." This force always is attractive, never repulsive, and mass in the real Universe always is positive, never negative. The force of gravity is a central force because it acts only along the line connecting the mass centers. Mass 1 m experiences a vector acceleration due to mass 2 m that is given by where r is a unit vector that points toward 1 m along the line joining the masses.

A2.2. Problem Statement
The CFO metaheuristic solves the following problem: In a decision space (DS) defined by min max , 1, , is called the "fitness." CFO explores DS by flying metaphorical "probes" whose trajectories are governed by equations of motion drawn from gravitational kinematics.

A2.3. Constant Acceleration
The vector location of a mass under constant acceleration is given by the position vector [56] ( ) where ( ) R t t + ∆ is the position at time t t + ∆ . 0 R and 0 V , respectively, are the position and velocity vectors at time t, and the acceleration a is constant during the interval t ∆ . In standard three dimensional Cartesian coordinates R xi yj zk = + + , where , , i j k are the unit vectors along the , , x y z axes, respectively. The CFO metaphor analogizes Equations (A1)-(A3) by generalizing them to a decision space of d N dimensions.

A2.4. Probe Trajectory
CFO's probes in a typical three-dimensional DS are shown schematically in Fig-( ) ( ) ( ) tions of mass as well. One possibility, for example, might be a ratio of fitnesses similar to the "reduced mass" concept in gravitational kinematics.

A2.6. Total Acceleration and Position Vector for a Single Probe
Taking into account the accelerations produced by each of the other probes on probe p, the total acceleration experienced by p as it "flies" from position How closely this metric approaches zero is a good indicator of how CFO's probe distribution has evolved around a maxima. avg D also is useful in identifying potential local trapping because oscillatory behavior in a avg D plot appears to signal trapping at a local maxima.