(Last updated June 16 1997)
I have received some letters from readers of the Twelfth-Wave Transformer article in the June 1997 QST. Included here are some notes from that correspondence, in case it may be of interest to other readers. I wish to thank all those who have contacted me with questions about the article and with helpful suggestions.
One reader asked what length he should make the twelfth-wave sections if the velocity factors of the cables with different impedance are different.
The only thing that matters is the electrical length. For example, to match 50 ohms to 75 ohms, the electrical length of cable is (from the graph or the equation in the article) 0.0815 wavelengths.
If the 75-ohm cable has a velocity factor of, say, 0.80, and the 50-ohm cable has a velocity factor of, say, 0.66, then in making the transformer the short 75-ohm length needs to be 0.0815*0.80 = 0.065 wavelengths, and the 50-ohm section needs to be 0.0815*0.66 = 0.054 wavelengths.
A reader asked if the transformer would still work if the antenna were mismatched, such that on the antenna side of the transformer there were a relatively high standing wave ratio.
A properly designed and constructed Twelfth-Wave Transformer will neither solve nor exacerbate the standing wave problem. For example, supposing the transformer is designed and properly constructed to match 75-ohm cable to the antenna to a 50-ohm cable from the transmitter. Suppose that on the 75-ohm cable there is a 2:1 standing wave ratio (SWR). This means that on the 50-ohm cable the other side of the transformer, there will also be a 2:1 SWR.
This a general property of all linear, properly matched transformers, not anything particular to the Twelfth-Wave Transformer. It is easy to see that this result (same SWR either side of the transformer) has to be true. Think of the forward and the reflected waves separately. Each wave will pass through the transformer, in opposite directions, without reflection and without attenuation. On both sides of the transformer the ratio of the power in the forward wave to the power in the reflected wave will be the same; this is exactly equivalent to saying that the SWR will be the same on both sides of the transformer. The relative phases of the forward and reflected wave will be changed after passing through the transformer, but the relative amplitude of each wave will be unchanged; so, the SWR is the same.
A reader asked if the operation of the transformer would be dependent on exactly where along the cable from a mismatched antenna it is placed.
As explained in the above answer, the SWR after the transformer will be the same as the SWR before the transformer. Suppose again, as an example, that the transformer is used between a 75-ohm line going to a mismatched antenna, with a 2:1 SWR. The other side of the transformer a 50-ohm cable goes to the transmitter. As above, the line to the transmitter will also suffer a 2:1 SWR.
The precise complex impedance seen looking into the transformer from the transmitter side WILL depend on exactly where in the 75-ohm cable to the antenna the transformer is inserted. The complex impedance seen on the transmitter side of the transformer will depend on the phase of the standing wave present on the antenna side of the transformer. The phase of the standing wave on the transmitter side of the transformer will also depend on the phase of the standing wave on the antenna side - i.e. on how long is the 75-ohm cable between the transformer and the antenna. However, the numerical standing wave ratio (SWR) will be the same on both sides of the transformer. The phase of the standing wave will change, but its amplitude (SWR) will be unchanged.
In other words, it does not matter how long the cables are on either side of the transformer. It does not matter where in the mismatched antenna cable the transformer is inserted - although of course it is always at the junction of the 2 cables of differing characteristic impedance.
One reader asked why I had only shown plots in the article up to a transformation ratio of 4:1. Does the transformer work for higher transformation ratios?
The transformer will certainly work for higher transformation ratios. The diagrams in the article were limited to examples of 4:1 or lower just for clarity.
As the transformation ratio becomes higher, several things happen:
(1) The SWR bandwidth of the transformer becomes less, as seen in Figure 2 of the June 1997 QST article, p.44.
(2) The loss will increase, depending on the loss of the original cable used to make the transformer.
(3) The maximum power handling capability will be reduced somewhat, a result of the higher peak voltage or peak current in any transmission line supporting a high SWR.
As a rough guide, I suggest that the transformer be limited to transformation ratios of 10:1 or less, which will probably cover most amateur needs. The calculation of precise bandwidth and insertion loss in a given situation may be the subject of a later article.
In his notes to me, Albert Weller, WD8KBW (see below) points out that the bandwidth of the twelfth-wave (and quarter-wave) transformer may be increased a great deal by carrying out the impedance transformation in steps. For example, from WD8KBW, in transforming from impedance Z1 to impedance Z2, an arrangement could be:
... Z1 (any length), Z2 (La), Z1(Lb), Z2(Lb), Z1(La), Z2 (any length) ...
If Z1 is 50 ohms and Z2 is 75 ohms, then the electrical length La is 0.0392 wavelengths, and Lb is 0.10385 wavelengths.
The algebra to work the lengths out is a little more complicated than for the simple twelfth-wave transformer, but has been calculated by WD8KBW.
A reader asked if the twelfth-wave transformer could be adapted to matching an arbitrary antenna impedance.
Yes it can, but this is really outside the scope of the article. See the references  and  to "series section matching transformers," given in the original article. (Regier, QST July 1978, and the ARRL Antenna Handbook.)
On page 44 of the June 1997 QST article, in the last paragraph of "An Example," an extra sentence was mistakenly added in the final editing. The text may be easier to understand if that complete sentence, "At the transmitter end, the 75-ohm cable ... finally to the 50-ohm transmitter" is omitted. Thanks to Paul Atkins, K2OZ for pointing this out.
Other references to the transformer have been drawn to my attention. In particular, in the October 1971 issue of "RadCom," the journal of the RSGB, in the "Technical Topics" column edited by Pat Hawker, G3VA, there is a description of the twelfth-wave transformer by G3KYH. This is quoted in "Hints and Kinks from Abroad" on p.42-43 of the January 1980 QST, edited by Doug DeMaw, W1FB. The same formula, although in a very slightly different form, is quoted for the precise length of the twelfth-wave transformer sections.
Albert Weller, Jr., WD8KBW has sent me a very nice treatment of the topic that he had written, called "Series Section Transmission Line Transformers." As far as I am aware, this has not yet been published.