We’ve all seen this number over and over again – the “magic number” that gives us the length of a half-wavelength dipole in feet from the dipole’s resonant frequency: L = 468/f. In free-space the wavelength in feet is 492/f, but a practical half-wavelength antenna is shorter so the constant is smaller. The number 468 is on the license exams and in the literature. It’s been there ever since I started reading about ham radio in the mid-1960s. It’s a pillar of amateur antenna theory. Every ham is expected to memorize it. And it’s wrong.
It would be more accurate to say that it’s rarely correct. There are certain instances where it’s close, but using it often leads to wasted wire. The usual instructions to a new ham are, “Calculate how much wire you need using 468/f and then add a couple of feet.” What that really means is the value 468 is too small and we compensate for the error by “adding a couple of feet”. If 468 isn’t right, why do we use it? Answering that question requires a trip along the paths of history.
Recently, I had the opportunity to spend a few days at ARRL Headquarters to plan upcoming writing and editing projects. The ARRL has a great Technical Library with every edition of ARRL publications and technical publications going back decades. (If you ever get close to Connecticut, it’s well worth dropping in on the ARRL for a tour!) I had some time one afternoon and decided to find out when and how the number 468 first appeared in the ham literature.
My first stop was the ARRL Antenna Book’s initial edition in 1939. Sure enough, on page 13 in the chapter on “Antenna Properties”, the familiar formula 468/f appears. The Antenna Book states that the “end effect” due to the attachment of insulators at the ends of the antenna results in the approximately 5% reduction in length from the free-space 492/f to 468/f. The text goes on to state that the percentage “varies slightly with different installations”, but doesn’t say how, nor is a citation provided to identify how the value of 468 was obtained.
Since it is unlikely that the value of 468 appeared in the Antenna Book without any “prior art”, I next turned to the ARRL Handbook’s first edition in 1926. That turned out to be a dry hole – no formula for antenna length and nothing in 1927 or 1928 either. Then, in the 1929 edition’s “Antennas” chapter on page 128, I hit pay dirt! The text defines natural wavelength as the highest wavelength (the lowest frequency) at which the Hertz antenna (a half-wavelength dipole) will resonate. It is stated that “The natural wavelength of the wire…will be its length in meters multiplied by 2.1” Hmmm…2.1 is 5% longer than would be the free-space value of 2. (Remember, the text is discussing wavelength, not frequency.) Farther down the page I saw, “Speaking in terms of feet, the natural wavelength of the antenna will be its length in feet divided by 1.56.” That equation translates to L = (300 x 1.56)/f and 300 x 1.56 is 468! Here were the headwaters of the mighty River 468!
Still, no background for the correction was given. Where does the use of a correction factor originate? Back to the stacks! Did I really want to go through all of the QST magazines until I found my answer? Well, not really, but inspiration struck in the form of the online QST archives. I logged into the ARRL Web site, brought up the QST archive search page, and…hit another roadblock. I couldn’t very well search for “468” because it was unlikely to be a keyword. “Dipole” would return hundreds of hits. Then I realized that in the early days, a half-wavelength dipole would have been referred to as a “Hertz antenna” or “Hertzian antenna”. I entered the former and scrolled down to the very earliest entries.
The oldest article on Hertz antennas was in the July 1925 issue by 9BXQ and titled “The Hertz Antenna at 20 and 40 Meters” but it didn’t discuss a formula for length. The next oldest article, October 1926’s “The Length of the Hertz Antenna” by G. William Lang, turned out to be what I was looking for. In the article, Lang (who was apparently not a ham, but worked in the Dept of Radio Operations for Radio Station WBZ in Boston) set up some Hertz antennas at amateur station 1KA and also measured antennas at station 1CK and 1KF. He used an oscillator and a wavemeter to determine the frequency at which the antenna resonated then measured the entire antenna - tip-to-tip, including the counterpoise. A table of correction values was derived, with the free-space wavelength in meters multiplied by an average value of 1.46 to get the antenna’s resonant wavelength in feet. This corresponds to an equation of L = 438/f. This is the first suggestion that the actual resonant length of a practical amateur antenna can be predicted by using a correction factor to a free-space wavelength.
The early experiments of 1925 and 1926 took place on or near 40 meters. In those days, CW operation on what we now call the “low bands” of 80 and 40 meters was the norm. At these wavelengths, a half-wavelength dipole was of a reasonable length. It could be made of ordinary copper wire, probably #8 to #14 AWG, and installed in the back yard at heights of 20 to 40 feet. For these antennas, 1/8th to 1/4th wavelengths above ground, a value of 468 is about right, resulting in the equation printed in the ARRL Handbook in 1929.
In truth, many variables affect the resonant frequency of a half-wavelength dipole, the two primary factors being the length-to-diameter ratio of the antenna conductor and most strongly, the antenna’s height above ground. These can combine to change the actual correction factor quite a bit! (Insulation can also affect an antenna’s electrical length.) In my November 2009 QST column, “Hands-On Radio: Antenna Height”, I modeled a typical 20 meter dipole made of #12 AWG un-insulated wire at heights from 1/8th to 2 wavelengths over realistic ground and calculated the correction factor at each height. It varied from 466 to 481 over that range! Clearly, using 468/f would lead to an antenna being too short most of the time.
If 468 is too small and rarely correct, what should you do? Realistically, you should expect to trim your dipole to get the resonant frequency you want. Instead of being frustrated that the calculations aren’t exact, learn to adjust the antenna’s length efficiently by using an instrument such as an antenna analyzer. Start with an estimated value based on a more realistic formula such as 490/f that results in a small amount of extra wire for attaching insulators. During tuning, twist the wire connections together or use clamps, then raise the antenna into position and measure. When it’s right, only then solder and weatherproof the connections. Recognize that every antenna’s circumstances are slightly different – height, ground conductivity, thickness of wire, nearby conductors, and so forth.
Another lesson to learn from this exploration is to realize that “magic numbers” in formulas have often been determined through experimentation under specific circumstances. As such, they likely depend on a variety of factors that you may not be able to replicate. They will only approximate what you actually encounter. If the assumptions behind the value are given – you can use that information by comparing it to your situation. If the assumptions are not known – you should allow for variations or try to find a more accurate model representative of your own circumstances.
I hope you’ve enjoyed reading about this journey as much as I enjoyed taking it, opening the covers of books nearly 80 years old and mapping the stream of knowledge back to its sources - finding there the footprints of wireless pioneers that set ham radio on the course we travel today.