The Twelfth-Wave Matching Transformer


Darrel Emerson.

(This is a summary. Last modified May 25 1997)

The full article appeared in QST, Vol 81, No.6,
 June 1997,pp.43-44, published by the ARRL. 

The Twelfth-Wave Transformer is often a more convenient alternative to the more well-known quarter-wave transformer. It is not a new concept, being first publilshed in 1961, but it is relatively unknown in amateur radio circles.

With the quarter-wave transformer, two impedances Z1 and Z2 are matched by using a quarter-wave of transmission line of characteristic impedance sqrt(Z1.Z2). This works well, but often requires a non-standard characteristic impedance. For example, to match a 50-ohm load to 75-ohm cable, a quarter-wave transformer needs a length of cable of characteristic impedance 61.2 ohms.

With the twelfth-wave transformer, two lengths of cable are used in series, each electrically nearly one twelfth-wavelength, but of characteristic impedances equal to the two impedances Z1 and Z2 being matched. The figures below illustrate the difference between the twelfth-wave and a quarter-wave transformer.

Illustration of the twelfth-wave transformer

The above figure illustrates the Twelfth-Wave transformer. To match impedances Z1 and Z2, two lengths of cable are needed, each of length L close to one electrical twelfth-wavelength. The characteristic impedance of one length is Z1, of the other Z2.

Illustration of a quarter-wave transformer

As illustrated above, with the quarter-wave transformer, only one matching cable is required, of electrical length Lq=one quarter wavelength, but this is very often a non-standard characteristic impedance.

The precise length of a twelfth-wave section

The precise electrical lengths needed in the twelfth-wave transformer are just slightly less than an exact twelfth. When matching impedances Z1 and Z2, if B=Z1/Z2, then the precise electrical length of the "twelfth"-wave section is given by:

L = [arctan(sqrt(B/(B^2 + B +1)))]/(2.pi)

where the arctan function is assumed to return an angle in radians. L is the electrical length of each section of matching cable, measured in wavelengths.

The length L is plotted below as a function of the impedance matching ratio, Z1/Z2. The horizontal dashed line corresponds to an exact twelfth length (0.0833333).


The SWR bandwidth of the twelfth-wave transformer is very broad, and is comparable to the quarter-wave transformer. The plot below shows the resulting SWR as a function of frequency, where a twelfth-wave transformer is used to match impedance ratios Z1/Z2 of 1.5, 2, 3 and 4. The frequency axis extends from D.C. to 50% higher than the nominal design frequency.

SWR bandwidth  of a twelfth-wave transformer


(1) A twelfth-wave matching transformer for 50 MHz

A transformer for 50 MHz, matching 75 to 50 ohms: From the figure, or the equation, the required length of matching section is 0.0815 wavelengths. At 50 MHz (=6-meter wavelength) this becomes 0.489 meters. Allowing for a velocity factor of 0.66, the physical length bcomes 0.323 meters, or 12.7 inches. The figure below illustrates the complete transformer.

Example for 50 MHz

(2) Matching several equal impedances Z0, in parallel, to Z0 impedance cable

Suppose we wish to match two elements of a phased array, individually fed by a matched 50-ohm cable. Putting the two feeders to the individual elements in parallel gives a combined impedance of 25 ohms. To match this 25-ohm impedance to conventional 50-ohm cable, we need a length of 50-ohm, and a length of 25-ohm cable. The 25-ohm length can be constructed by putting two 50-ohm lengths in parallel. The figure below shows the general arrangement.

Matching two 50-ohm antennas to a single 50-ohm feeder

In the above figure, since we are matching an impedance ratio of 2:1, each length L is (from the above equation or the figure given the required length of matching section for a given transformation ratio) 0.0781 wavelengths. If this were to be used at 28 MHz (10.7 meters) the length becomes 0.836 meters. Allowing for a velocity factor of 0.66, this becomes a physical length of 0.552 meters or 21.7 inches. All lengths L would be 21.7 inches, and all cables are of the same (e.g. 50 ohm) characteristic impedance..

Note that this is a perfectly general way of solving the matching problem of putting N feeders, each of impedance Z0, in parallel. The required matching section impedances are Z0, and Z0/N. It is always possible to make a feeder of characteristic impedance Z0/N simply by putting N sections of impedance Z0 in parallel.


Although relatively unknown in amateur radio circles, the twelfth-wave transformer was first described in 1961, when it was in use for matching 200 MHz components at the CERN accelerator. See:

"A Convenient Transformer for Matching Coaxial Lines"
by B. Bramham, in Electronic Engineering, Vol 33, pp.42-44,
January 1961.

Feedback from readers on the "Twelfth-Wave Transformer"

Since the appearance of the article in the June 1997 QST, I have received some correspondence from QST readers asking for further clarification on the use of the Twelfth-Wave Transformer. In the hope that it may help others, replies given to readers can be found on this link.

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