I'll assume that readers know the characteristics and use of Beverage antennas, or at least how they're constructed, and perhaps even the theory of operation. There is ample material elsewhere, and I have several books and papers on my shelf that I've collected over the years. As always seems to be the case the most useful book for amateur radio use is ON4UN's Low-band DXing.
The characteristic impedance of a Beverage is nominally 500 Ω. The true value depends on its length, height and ground quality, with a typical range between 450 Ω and 550 Ω. At the far end a (non-inductive) resistor of this value terminates the travelling wave from the reverse direction, ensuring no reflection and thus the maximum F/B. The transmission line connects to the near end.
Since we will be using a coaxial cable with a characteristic impedance of 50 or 75 Ω it is necessary to transform the Beverage impedance, approximately 10:1 or 6.5:1, respectively. This is typically done with a broadband transformer since the Beverage is inherently broadband; within reasonable frequency bounds there is no other tuning required on several bands, but with wavelength-dependent patterns. Transformers of 9:1 and 6:1 are typically used, which are easy to build and close enough to the ideal to be effective under most circumstances.
Building the transformer
I am no expert on ferrite cores and RF transformers. I rely on the recommendations of the experts. From the ample experimental data presented in ON4UN's book I chose the BN-73-202 binocular ferrite core. It is inexpensive and an excellent choice for the bands from 160 to 40 meters, where it is most likely to be used. Its only negative point from my perspective is its small size.
The core is rectangular and only slightly more than ½" long and wide. The holes are especially small, with a diameter of only 0.15". Small gauge wire is mandatory. This proved a challenge since the best design for my purpose requires a 3:1 turns ratio -- square the ratio to get the 9:1 impedance transformation -- with 2 turns for the 50 Ω winding and 6 turns for the 500 Ω winding.
I stripped the cover off some old telephone quad cable for the 4 insulated 22 AWG conductors it contains. The spec sheet linked to above includes a calculator for the length of wire required, included pigtails. I used 1" pigtails to ease interconnection in my test setup.
The two-turn winding was easy. The final four turns on the second winding were more difficult. When the wires are pulled through the holes they quite naturally arc toward the far wall of the hole midway through the core. These must be pressed down to make room for subsequent passes of the wire. I used a jeweller's screwdriver. Some care is needed to avoid damage to the wire insulation and core (ferrite will only tolerate a moderate amount of abuse before breaking).
Double check the number of windings when done since it's easy to lose count while fighting for space within those narrow holes. I know I did. Remember that there are two passes of the wire through the core in every winding turn. Use different coloured wires for the windings.
Testing the transformer
A selection from an of old stock 470 Ω carbon composition resistors serves as a facsimile of the Beverage impedance. I chose this value since carbon composition resistors tend to increase in value as they age. Measure the resistance before use. I chose one with a resistance of 505 Ω. It terminates the 6-turn winding.
The coax winding is directly connected to an antenna analyzer. That adapter with the wire jacks on one end and a BNC connector on the other is one I ordered from Elecraft at the time I purchased the KX3 at the end of 2012 when I first got back on the air. I no longer use a random wire antenna so it has found new life on my workbench.
As you can see the match is excellent at the high end of 160 meters, as it is elsewhere across the band. The stray reactance (inductive in this instance) is a combination of residual error in the analyzer, core and winding properties, and those pigtails plus resistor leads. How much they each contribute I cannot say.
The transformer is not for only 160 meters. It must also work well on 80 meters, and it would be nice if it worked on 40 meters as well. I adjusted the analyzer to plot R and X over the range 1 to 11 MHz. In the next photo you can see that the transformer is very flat right across all the bands from 160 to 30 meters. X remains so small that it doesn't visibly deviate from the zero line (however the scale is deceiving).
This confirms that I correctly built the transformer. I had little doubt that ON4UN or W8JI would lead me astray!
The impedance does depart from this excellent performance on higher bands. This is almost certainly due to the increasing contribution of the stray reactances in my test setup. These lengths of wire on 160 and 80 meters are tiny relative to wavelength. Not so as the frequency increases. When boxed for deployment the construction technique will be better.
Clearly I need to improve my skill of winding transformers on these small binocular cores. What I did worked but is not recommended. For example, there is a danger of nicking the insulation on the wires when they are pulled hard against the resistance through the filled narrow holes. Ferrite is a ceramic matrix and ceramic is very hard and can easily cut plastic (or the coating on enamelled wire). Second, thin gauge wire will break when pulled too hard.
This experiment gave me confidence that I can construct the chokes and transformers I may need in the future. I have many more of these ferrite cores to experiment with, and I intend to do so. While I didn't time myself it took no more than an hour to acquire the wire, build the transformer, construct the test setup and make the measurements with the analyzer.
Don't be afraid to give this a try yourself. The materials are cheap and the results are very satisfying. Not many of us are in a position to put up Beverage antennas, but transformers of this type can be found in a variety of low-noise receive antennas and in QRP equipment. Similar transformers using much larger ferrite cores are used to handle a kilowatt when stacking yagis.
I learned to do something new and useful this weekend. That's time well spent.