Choosing an Antenna Mathematically

Transcription

1 Choosing an Antenna Mathematically Open a Spreadsheet, Adjust a Few Sliders, and the Best Choice Appears. Wm C Bauldry, NC4WB Dept of Math Sciences Appalachian State University Boone, NC NC4WB@arrl.net Abstract We demonstrate a simple, spreadsheet-based, procedure for choosing an antenna. The spreadsheet implements a sophisticated technique called the Analytic Hierarchy Process (AHP). Hams fill in a set of criteria, their possible antenna choices, and use sliders to set their preferences. Then the interactive spreadsheet chooses which antenna is best. Our example selects from among a 1/4 wave vertical, a 2 element Yagi, and a J-pole. Introduction Elmers tell new hams, Your antenna is a critical part of your station. But how does a new amateur choose an antenna? There are so many different kinds and configurations that finding the best seems impossible. A web search for choosing an amateur radio antenna can return over a million links. The amount of information is staggering; the choices overwhelming. Most people are good at choosing between two alternatives; however, as the number of options grows, our choice facilities quickly break down becoming unworkable. Math comes to the rescue! We can use the Analytic Hierarchy Process, or AHP, a powerful method from the branch of mathematics called analytics. AHP was developed by Prof. Thomas Saaty in the 1980s and has become a leading decision technique. Fortunately, a pre-built, interactive spreadsheet makes AHP simple and easy for any ham to use. AHP is based on answering a set of preference questions once we ve identified our criteria and choices. For choosing an antenna, these questions could include, On a scale of 1 to 9, which is more important to you: antenna gain or size? A spreadsheet puts the sophisticated computations under the hood letting us focus on answering the set of questions by setting sliders. As we adjust the sliders, the best antenna for our situation appears. Not only do we see the best choice highlighted,

2 Choosing an Antenna Mathematically pg 2 but we also get a ranking of our choices with percentages showing how much one option is better than another. We can easily change any slider s settings, doing what if s, to see how the new value affects which antenna is selected as best. Choosing an Antenna Let s run through an example. Like a lot of new hams, my first radio was a 2 meter FM transceiver. I wanted to build an antenna to complete my station. My criteria for selecting an antenna were: size, gain, and ease of assembly. My initial scenario came down to choosing from among three types: a 1/4 wave vertical, a 2 element Yagi, and a J-pole. We ve just completed the first two steps of AHP: define the criteria, and list the alternatives. Our next step is to determine the relative importance of our criteria. Open the Antenna AHP spreadsheet 1, click the Criteria tab, and enter our three criteria Size, Gain, and Assembly as the Priority Variables in cells B6, B7, and B8. The order doesn t matter. Enter the three antennas Vertical, Yagi, and J-Pole in the Alternatives Variables in cells B12, B13, and B14. These entries will propagate throughout the spreadsheet. Now adjust the three sliders indicating which is more important and by how much in each of the three possible pairings: Size v. Gain, Size v. Assembly, and Gain v. Assembly. The scale we use is 1 for equally important up to 9 for extremely more important. Figure 1 shows Gain set as 4 times more important than Size, Size set as 2 times more important than ease of Assembly, and Gain set as 4 times more important than Assembly. In the Results box we see our priority rankings listed under the Normalized Priority Vector heading as Size 21%, Gain 66%, and ease of Assembly the lowest at 13%. The percentages indicate that we value gain more than three times as much as the other two. Notice, at the bottom of the Results box, the Consistency Ratio is labeled Good with its numeric value colored green. This ratio being good indicates that our settings weren t contradictory. If we had chosen gain as more important than size, size as more important than assembly, and assembly as more important than gain, then we d have a circular and inconsistent set leading to bad, unusable results. The Consistency Ratio would now be labeled Suspect and the numeric value would turn red. Adjust the sliders to try it! The next step in AHP is to assess your impression of the criteria with respect to each alternative separately. That is, just considering one aspect, Gain for example, we set the sliders in that tab to 1 Available at wmcb/ahp Antennas/

3 Choosing an Antenna Mathematically pg 3 Figure 1: Setting the Criteria Priorities show how antenna pairings stack up. With respect to Gain, we assess the Vertical s gain compared to the Yagi s, the Vertical s compared to the J-Pole s, and last, the Yagi s compared to the J-Pole s. Go to the Size tab in the spreadsheet. In an apartment setting, smaller size could be a big advantage. Figure 2 shows our results after setting the sliders: a Vertical is 6 times better than a Yagi, only a little better (2) than a J-Pole, and a J-Pole is 5 times better than a Yagi, all with respect to size. In this case, we chose better to mean is more compact. We make similar comparisons for the three antenna pairings for Gain and then again for Assembly. Look at the Gain and Assembly tabs to see the settings. After all the sliders are set in the four tabs, Criteria, Size, Gain, and Assembly, click the Overall

4 Choosing an Antenna Mathematically pg 4 Figure 2: Assessing Size for Each Antenna Results tab. All the intermediate calculations have propagated, doing computations behind the scenes, and the winner with the highest Ranking is... the 2-element Yagi (Figure 3). Given our assessments of each of the pairings, a 2 element Yagi is the best antenna for our situation. The Relative Ranking box in Figure 3 tells us that a 1/4 wave vertical will be 86% as good as a 2 element Yagi for our criteria and preferences, while a J-pole will be only 36% as good. Just by entering our criteria and our alternatives, setting a set of sliders to reflect our assessments of relative importance of pairs, we have determined mathematically which antenna is the best for our situation. Now, go back to the Criteria tab, enter a different antenna or criterion, and restart the process. Set each of the sliders to your own preferences in the different tabs. Which antenna is best for you?

5 Choosing an Antenna Mathematically pg 5 Figure 3: The Final Antenna Rankings Conclusion The Analytic Hierarchy Process is a very powerful, complex method for choosing a best option given a set of criteria. A pre-built spreadsheet makes using AHP very easy. We have used AHP to choose from among three antennas basing our decision on merging three different criteria and our assessments of their relative importance. The AHP Antenna spreadsheet hid the sophisticated mathematical computations computing eigenvectors of positive reciprocal matrices woof! By making several simple comparisons, we could focus on what is important to us as hams choosing the best antenna. AHP can handle as many alternatives and criteria as we can identify. Modifying the AHP spreadsheet for larger problems isn t overly difficult. However, as new criteria or alternatives are added, the number of pairings grows quickly it s proportional to the square of the number of items being

6 Choosing an Antenna Mathematically pg 6 paired, and the computations become more involved. On the other hand, several software packages, including web-based AHP calculators, are easy to find online. We can tell just how important and useful a mathematical tool AHP is since Google Scholar lists over a million articles investigating the intricacies and theory of the Analytic Hierarchy Process. Bill Bauldry is a relatively new ham, usually found on 2 meter FM and on 70cm DMR, and an old mathematician, usually found at Appalachian State. When not on the air, he s teaching math classes, doing research, and promoting math modeling as Associate Director of COMAP s Mathematical Contest in Modeling. You can reach the author at NC4WB@arrl.net.