No. 22: The Handy Quarter-Wave Matching Section

L. B. Cebik, W4RNL

In the quad project we looked at last time, we specified the use of a 1/4 wavelength matching section of 70- to 75-ohm coax between the antenna feedpoint and the 50-ohm coax run to the shack. Let's find out why we needed it and how the matching section does its work. These types of matching sections are handy and easy to make, so you may find them useful in future antenna projects.

The Raw vs. the Matched Antenna: Let's start by comparing the feedpoint impedance of our quad both with and without the matching section. The figures are based on a quad beam that is self-resonant just below 28.25 MHz. The 1/4 wavelength section of 75-ohm, 0.66 velocity factor coax was cut for 28.5 MHz and turned out to be 68.3" long. All SWR figures are relative to 50 ohms.

Frequency   Without matching section    With matching section
in MHz           Feed Z       SWR           Feed Z       SWR
            (R+/-jX Ohms)              (R+/-jX Ohms)
28.00        70.6 - j29.2     1.81      66.9 + j27.1     1.73
28.25        96.8 + j 3.4     1.94      58.1 - j 2.7     1.17
28.50       125.1 + j25.9     2.63      43.1 - j 9.1     1.28
28.75       150.1 + j39.4     3.23      34.9 - j 8.5     1.51
29.00       168.6 + j47.7     3.67      30.7 - j 7.1     1.68
29.25       180.8 + j54.8     3.97      28.2 - j 6.0     1.81
29.50       188.2 + j63.1     4.22      26.6 - j 5.4     1.91
29.75       192.6 + j73.6     4.45      25.1 - j 5.2     2.02

Without the matching section, the SWR figures for the quad are high enough that the automatic shut down feature of most current solid state rigs would reduce rig output to almost nothing. With the matching section, the SWR figures within 10 meters permit normal operation.

As the unmatched impedances go up, the matched impedances go down. This gives us a clue to how the matching section operates. Every length of coax of any characteristic impedance (Zo) is an impedance transformer. For odd lengths, the transformation is complex. However, when a length of coax is exactly 1/4 wavelength long at a given frequency, the transformation is simple, especially if the impedance to be transformed is wholly resistive. We can use a calculator to handle this equation:

Zmatched = Zo2 / Zantenna

This little formula is only approximate in the real world. for example, with the quad, there is a reactive part of the antenna feedpoint impedance in every line of the table. Moreover, the matching section is only an exact 1/4 wavelength at 28.5 MHz and nowhere else. However, if the reactances are not too high and the frequency span is not to great, the simple equation makes a good approximation. As we look at the table, for a single ham band and for reactance values less than half the resistive values, the simple equation works well enough for antenna building.

Remember that for purely resistive impedances, a 2:1 50-ohm SWR accommodates an impedance range of 25 to 100 ohms. This resistive range shrinks when we combine reactances with resistance. However, note the 4:1 range of impedance that these SWR limits can handle. (Also remember that the 2:1 ratio is somewhat arbitrary as a set of limits. It's chief effect is noted by automated power reduction circuits in transceivers. Apart from this, there would be little difference in radiated power between, say, SWRs of 1.8 and 2.5.)

With a 1/4 wavelength 75-ohm matching section, again in purely resistive terms, we can take antenna feedpoint impedances between just above 56 ohms up to 225 ohms and transform them to values that fit the 50-ohm 2:1 SWR limits--again, a 4:1 range. Notice that our quad does not reach 225 ohms when the matched SWR exceeds 2:1, but notice also that there is considerable reactance that accompanies the resistive value at 29.75 MHz.

Likewise, at 28 MHz, we would expect the antenna impedance of 70.6 ohms to yield about an 80-ohm figure instead of the 66.9-ohm figure that actually emerges. However, not only do we have reactance at the antenna feedpoint, but as well the matching section is shorter than 1/4 wavelength at this frequency. Hence, the impedance does not undergo a full quarter wavelength transformation. (Likewise, above 28.5 MHz, the impedance undergoes more than a 1/4 wavelength transformation.)

These are the finer points of using a 1/4 wavelength matching section that affect the matching range by just a little bit and throw the actual impedances somewhat off the calculated results from the simple formula. But the simple formula works well enough for most ham antennas. To be on the safe side, if you have a range of antenna feedpoint impedances from about 80 to 200 ohms, then a 1/4 wavelength section of 75-ohm coax will transform them to values appropriate to a 50-ohm feedline and transceiver system.

Other Applications: 50-ohm and 75-ohm coax cables are the ones most easily obtained by hams, even though other values are available from manufacturers. However, this fact does not limit us to matching only values above 50-ohms to our 50-ohm system. If you cut 2 lengths of 75-ohm cable to 1/4 wavelength and connect them in parallel (center conductor to center conductor and braid to braid at both ends), you have a 37.5- ohm cable. If we plug this value into the simple equation, we find that we can match impedances values below 50-ohms up to values within the 2:1 50-ohm SWR limits. This is useful for Yagis and other antennas that often have feedpoint impedances in the 20-35 ohm range. The double line can be a bit bulky, but that is about its only significant disadvantage over other matching methods.

Consider another situation: At certain wire antenna heights below 1/2 wavelength, the feedpoint impedance of a dipole is not 70 ohms, but more like 80-95 ohms. The 75-ohm matching section would transform these values to a 70-60 ohm range. However, we can broaden the range over which these values apply by first running a section of 50-ohm cable that is 1/2 wavelength long or a multiple of 1/2 wavelength (allowing, of course, for the cables velocity factor). Cut the 50-ohm cable for a frequency at the band center, such as 7.15 MHz for a 40-meter dipole. Since the cable is short at the low end of the band, the impedance will be higher than at the antenna at the same frequency. Equally, since the cable is long at the high end of the band, the impedance will also be higher than at the antenna terminals. The result will be band edge values closer to 100-120 ohms.

Now, if we plug in our 1/4 wavelength 75-ohm matching section, we have lower SWR values across the band than we would have by placing the 75-ohm matching section at the antenna terminals. In fact, such a system can, with some dipole heights on 80 meters, cover more than 4/5 of the band with under 2:1 SWR.

75-ohm quarter wavelength matching sections (and derivatives) make up a quite flexible array of methods for adapting 50-ohm transmission line to antennas that do not present 50-ohms at their feedpoint terminals. However, they do have some major limitations. Because a length of coax is 1/4 wavelength long at only one frequency, this technique is for monoband antennas only. If you have a multiband antenna, you will have to use some other method of matching your 50-ohm coax/transceiver system to the antenna.

Likewise, the transformations become far more complex the higher the reactance at the antenna feedpoint. Hence, the quarter wavelength matching system is also only for low- reactance matching situations, such as the one shown in the table for the quad. If you have higher reactances, you may need a different matching system.

But where the 1/4 wavelength matching section is suited to the task, it is simple, inexpensive, low-loss, and effective. Those are pretty good credentials.

Updated 1-24-99. © L. B. Cebik, W4RNL. Data may be used for personal purposes, but may not be reproduced for publication in print or any other medium without permission of the author.

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