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Looks like the pen got carried away....
Update for March 2019... here's a daughter page that describes a free spreadsheet download for designing H-tree feeds!
An antenna feed is a network that connects many antenna elements or a single or multiple connection to a radar or communications system. In a monopulse antenna, hundreds of elements might be fed to four connections (the quadrants) for example.
The H-tree is often used in antenna feeds, as it offers equal lengths to each element (that is the key product characteristic). The H-tree is a pattern that you can continue ad infinitem, but most antenna arrays don't exceed a few thousand elements. H-tree feeds are used in phased arrays as well as fixed arrays, and date back to before World-War II.
Check out Wikipedia's H-tree page if you want to consider its Hausdorff space. Not something an engineer needs to get paid to worry about, but it might help you stave off Alzheimer's for a few more minutes.
The easiest antenna array to create is uniformly fed in amplitude, however, if you are creative you can apply an amplitude taper to the feed using unequal split power dividers. Why would you want an unequal feed? Antenna arrays are often tapered to reduce sidelobes...
If you really want to get creative, try applying an H-tree to a circular aperture some time. If someone can send us an example of this that we can post they can count on a free gift!
It is always better to use terminated power splitters such as the Wilkinson, in order to isolate antenna ports from each other. However, many antenna are fed from reactive splitters.
Although the H-tree is naturally set up for a square (or rectangular) grid, you can bend the final elements to create a triangular lattice. You can also have different spacing along the two axes, to create a rectangular array which will provide more gain in the broader dimension.
Update April 2019: when we first wrote this, we did not consider that the element spacing is often different from a half-wavelength. Our H-tree feed allows you to enter any fraction of a wavelength indepedently in X and Y. Afte you skim through the material below, go to this page that describes a spreadsheet you can download to analyze your own feed netwoork.
The H-tree feed is binary. Depending on how many splits you have, the number of elements goes up as 2, 4, 8, 16, 32.... To get to 1024 elements you will need to perform 10 splits. The length of each split is the key to how much loss it creates, as transmission lines have a uniform attenuation at a fixed frequency. At the element level, the length of the first split is often 1/4 of as wavelength, as the antenna spacing is 1/2 wavelength in many arrays (in particular, for phased arrays.) By examining the H-tree pattern for a square array, the lengths proceed to increase as you go further back in the feed, in a very distinct pattern of lengths. The first two splits are 1/4 wavelength, then the next two are 1/2 wavelength, then the next two are 1 wavelength... you can prove this to yourself by sketching it out (more on that below).
Depending on whether you are in transmit or receive, a very different thing happens to signal level, as shown below. The "cumulative combiner gain" is merely 10xlog(elements). Every time you split you have 3.01 dB gain. In receive, the signal would go up by 3.01 dB gain, for each split, reduced by the attenuation due to the length of the split. In transmit, the signal is reduced by the split as well as the attenuation. Got that?
The lengths of the segments are a sequence, which starts at 1/4 wavelength (the antennas are 1/2 wave apart).
Now it is time to reveal what may be a shocking truth to the engineering masses... if you don't regularly make sketches on "engineering paper", you probably suck at engineering.
Below the H-tree is sketched from 4 to 64 elements. You see, with a little thought those light squares not only help you make straight lines but also can help you create an H-tree that is correct.
This was done on a TOPS pad, not to be confused with AMPAD Evidence paper. Ampad was a US company that was drained by Mitt Romney's Bain Capital and left for dead. Learn how Bain's "success story" Staples helped screw the life out of Ampad while Mitt Romney served on the Staples board of directors. Note that each page costs about six cents, this is important for people that carry their coins in a coin purse so they can keep track of them and not worry about losing them in the couch.
Keep at it and you will arrive at an element per box on the engineering paper. Those boxes are 200 mils, if you live in Europe you will surely have paper that is in millimeters.