Coaxial Cable Twelfth-wave Transformer Construction

Several members of the North Fulton Amateur Radio League engaged in a discussion on how best to match 75 ohms to 50 ohms for a feed line from a dipole. Various matching transformers were found, but one, for example, was a quintifilar-wound one, and I didn’t want to get into that, as the wire thickness and toroid size for QRO (100 watt) operation would mean thick wire and a fairly large, and somewhat expensive toroid (I like free solutions, or nearly so, anyway). I did some research and found a very simple solution for matching a 75 Ohm dipole feed line to a 50 Ohm line to the transmitter. The solution is based on an article on Darrel Emerson, AA7FV’s website at http://ourworld.compuserve.com/homepages/demerson/twelfth.htm. Darrel states that this is a summary of the article published in QST for June 1997. He references an original article published in 1961, "A Convenient Transformer for Matching Coaxial Lines" in Electronic Engineering, Vol. 33, pp 42-44. Why bother, you say, to match 75 to 50 ohms? Well, one reason is that I have several of the old Heathkit single-band transceivers, and the maximum SWR rating is 1.5:1. Those boatanchor rigs use sweep tube finals, and the output is not the familiar Pi net, rather an L-network (only one capacitor, not the “Plate” and Load” caps found in a Pi network).

The 75 to 50 Ohm transformation is of particular interest to me because I have some quad-shielded, direct-bury RG-6U 75 ohm stuff with an attenuation of 1.6 dB at 55 MHz and 4.15 dB @400MHz, slightly better than regular Belden 8237 RG-8/U. I'd like to use as much of that as possible and run the last few feet into the shack with a 50 Ohm type coax. High quality RG-6 can be obtained very cheaply, and sometimes free from cable installers who have tail ends they can't use.

The 12th wave technique is similar to using a quarter-wave matching section of coax between two other line impedances, except this calculation uses two twelfth-wave sections of the same impedances as the two lines. For example, matching 75 Ohm feed line to the antenna (a dipole's impedance is typically about 72 Ohms at the center, if it is a flat-top) to a section of 50 Ohm coax to enter the shack and connect to the rig would require two matching sections, one of 75 Ohms, and the other of 50 Ohms. At 28.470, using (arbitrarily) a type of RG-59U which has a velocity factor of 0.83 and RG-58U at 0.66 Vf, the matching section lengths are 28.0 inches of RG-59U and 22.3 inches of RG-58U. If the Vf's of the two lines are the same, the matching section lengths will be the same, also. The velocity factors of the main feed line sections themselves are irrelevant.

The matching sections are placed in the feed line between the 75 Ohm and 50 Ohm feed line sections, but reversed in order. In other words, the 75 ohm feed line from the dipole, then the 50 ohm 12th-wave matching section, and then the 75 ohm 12th-wave matching section, and from there, the 50 ohm feed line to the transmitter. The matching section pair does not have to be at any particular position in the overall feed line. Any length of 50 and 75 Ohm lines can be used for the main feed line sections; the matching section pair is simply inserted between them. The SWR bandwidth is very broad, similar to a quarter-wave matching system. Any ratio of coaxial feed lines may be matched this way, but the author points out that 10:1 may be a recommended practical limit.

At the North Fulton Amateur Radio League table at the Kennehoochee hamfest, Jim, W4QO, and I were discussing the 12th-wave matching sections which was the subject of the post I made to the NFARL Yahoo group. Jim asked the question "The match is good only at the one frequency, correct?" I first said yes, but thinking about it, and rechecking my notes, the match is very broadband. It actually covers the band for (75 to 50 Ohm matching) from DC to 150% above design frequency with SWR less than or equal to 1:5 to 1. At a design frequency on 6 meters, matching 75 to 50 Ohms for example, the SWR will be 1:5 to 1 or less from 0% (DC) to 150% above the target frequency in the 6 meter band. The source article shows a graph for this case. So, for a design frequency of 50.4 MHz, the SWR will be below 1:5 to 1 from DC to 75.6 MHz. For 3.750 MHz, the 1:5 SWR point extends from DC to 5.625 MHz. For other transformation ratios, the SWR curve is considerably steeper. The transformation ratio corresponds fairly close to the SWR at DC and 150%. A transformation ratio of 4:1 results in an SWR at DC of 4:1 and 4.4:1 at 150% design frequency.

For 75 to 50 ohm matching, the SWR bandwidth chart shows an almost linear progression from the design frequency to plus 150%. Plus or minus 10 % is about 1.1:1, 20% is about 1.2:1. etc. Of course, at plus or minus 50% of the match frequency, where the SWR is 1.5:1, one might as well just be using the 75 ohm line direct to the transmitter.

Jim asked in a later email on the subject, “How can it be frequency useful from DC to 150% of the design frequency? If you think about it, why would you build one that is 1/12 L at say 4MHz when you could do one at 4 GHz that would be much shorter and a wider bandwidth”. I replied, by saying, “That does sound counterintuitive, doesn't it? Think about it this way. If you made the 1.5:1 ratio 12th-wave transformers for a near-infinite frequency, the SWR would be 1.5:1 down to DC. Therefore, the SWR would be 1.5 at 160m, 80m, 40m, 20m, etc. – but that would be the same as not even using the 12th-wave transformers (which are near-infinitely short). Essentially, you'd be connecting 75 Ohm line directly to 50 Ohm line, giving a 1.5:1 SWR”.

If you are even reasonably close to the target frequency, there is a real benefit. For example, assume a frequency, say 50.4 MHz, for the 1:1 SWR target frequency for cutting the lengths of matching cables. At 110% of 50.4 (55.44 MHz), and 90% (45.36 MHz), the SWR has only risen to 1.1:1. At lower frequencies, the numerical bandwidth for the 10% is more limited. On 80, with the 1:1 SWR frequency selected at 3.750 Mhz, the 1.1:1 SWR limits are 3.375 and 4.125, but that is the whole 80 meter band!

One thing I wondered about was the availability of coax fitting to join the four line sections. Darrell’s summary doesn't mention connectors; his drawings simply show them connected as tubular sections with lines for the center and shields. I could not find any F female to BNC female adapters at any of my usual vendor sites, so I decided to make my own. Three adapters are needed, all identical. I used F female union connectors and soldered them to BNC female PC-mount connectors, the kind with the four legs. Using a product that I found on Phil Salas’ ad5x.com site, Solder-It (for steel, in this case), I soldered the connectors together at the point where the BNC legs slide over the F female connector. I applied a line of Solder-It to the area of the F connector where the legs of the BNC would end up. I used a butane pencil torch from Harbor Freight. Using Phil's technique, I heated the connector until the solder started forming balls, and then briefly flashed the solder directly with the flame. After my first attempt overheated an F connector and the dielectric melted in the open end, I placed a TO-36 transistor finned heat sink on the opposite end of the F connector from that being soldered being soldered, and I screwed an F male connector installed on a short piece of RG-59 so the wire would help keep the F female socket centered if the dielectric got hot. Another tip is to file down the rounded end of the F connector until it reaches the cylindrical part so the pin from the BNC can slide all of the way in. I used a 1-inch bench sander, and it made short work of it. The photos show the progression of the assembly of the connectors. The first shows the connectors lined up to be connected; soldered connectors; next, some quick-setting epoxy applied to help strengthen the union; a resistance check (leads alone measured 0.6 ohms), the new connector with shrink tubing applied, and a connector between two sections of line. I smeared a dab of Goop near the inside ends of the connector prior to heat shrinking to help weatherproof the connectors. I didn’t stress test the connectors by pulling on them with the feed lines and matching sections assembled, as I assumed that they would probably be placed in a slack section of the overall feed line, possibly inside the shack.

The following photos show the sequence for assembly of the connectors. Photo 7 shows a connector between two sections. Click on a photo to enlarge.

Connector Construction

F female and BNC Female	Two Connectors Fitted	Connectors Soldered	Resistance Check
Epoxy Applied	Heatshrink Applied	Connector in Feed Line

To avoid having to do the math, which involves arctangents and the like, I developed a spreadsheet which is downloadable from this website. The spreadsheet includes a diagram of how the matching sections are placed in the feed line, so that the person doing the calculation entries have something to refer to.