# No. 28: Beta Coils and Hairpins

### L. B. Cebik, W4RNL

In a number of columns, I have mentioned the beta match. Let's take a brief look at what it is and what it does.

The beta match appears to be simply a small coil or hairpin placed across the terminals of an antenna, most often a Yagi. Some folks mistake the coil for an RF choke, while others mistake the hairpin for a short circuit.

Actually, the beta coil or hairpin is one part of an impedance matching circuit, where the remaining elements are invisible, if you do not know what to look for. Many Yagi antennas have feedpoint impedances in the 20 to 35 Ohm range, somewhat low for feeding directly with coaxial cable. We need to raise the impedance to 50 Ohms--and that is what the beta match system does.

The coil is not the only element in the circuit. There is also a capacitor--or, more correctly, some capacitive reactance. We get that part of the circuit from the antenna element itself.

Fig. 28-1 shows how we move from a resonant driven element to a beta match. Let the resonant antenna impedance be low, say about 25 Ohms. If we shorten the element, the resistance does not change significantly, but the antenna becomes capacitively reactive, as the middle part of the figure shows.

If we shorten the element by the right amount, we get the right capacitive reactance in series with the antenna resistance to go together with an inductance across the coil to make an L-circuit. An L-circuit is one of the fundamental impedance transformation circuits, and in this case, -Xa and XL together change the 25- Ohm antenna resistance to 50 Ohms.

We can calculate the needed values if we know the antenna feedpoint resistance (Ra). (We know the coax has a characteristic impedance (Ro) of 50 Ohms.) First we calculate a value called "delta" by some and "working Q" by others. Delta = the square root of [(Ro/Ra)-1]. Now we can easily calculate the necessary values of capacitive reactance in the antenna (- Xa) and of inductive reactance to place across the terminals (XL). Xa = Delta times Ra. XL = Ro / Delta.

Since these values are given as reactances, you need to convert the inductive reactance into a component value. The capacitive reactance will be developed by simply shortening the antenna element until the beta match gives us 50 Ohms.

For reference, here is a small table of values we commonly encounter with beta matches with 50-Ohm coax for various values of antenna feedpoint resistance (Ra):

```Ra               33       25       17
Delta            0.7      1.0      1.4
Xa               23.6     25.0     23.6
XL               70.7     50       35.4```

Notice that the capacitive reactance reaches a peak when delta = 1, while the inductive reactance gets smaller as the feedpoint resistance gets smaller.

We have not yet converted these inductive reactances (XL) into a component value, because there are two distinct ways to achieve the required reactance across the coil. Fig. 28-2 below shows them both:

The beta inductor is simply a coil with the value of inductance that provides the inductive reactance at the operating frequency. If you divide the required inductive reactance by the product of the operating frequency (in Hz) and twice pi, you get the right inductance.

The hair pin version of the beta inductor is actually a small shorted stub of parallel transmission line. Rather than go through the calculation procedure, I shall simply once more recommend that you obtain a recent copy of HAMCALC, a suite of handy ham calculation programs in GW Basic. Among the selections on the disk is an excellent program that will calculate the dimensions of a hairpin for the match. It was written by Thomas Cefalo, Jr., WA1SPI. The program will also tell you the equivalent inductance in case you want to wind a coil. Other programs in HAMCALC will help you wind an accurate coil.

Many antenna builders use the experimental technique of adjusting the driven element for a beta match. After calculating the beta coil or hair pin, they install it and then adjust the element length for a low SWR. Antenna modelers tend to determine the required element length in advance from their software and save some time fumbling for the right element length.

Either way, the beta match results in a very low loss match. For inductor Qs over 100 (easy to obtain, but some maintenance is required to maintain the Q), losses will be well under 1%--and even less for the hairpin.

If you like to build antennas, you should become familiar with the beta match. Some folks actually avoid the beta match because it is "too simple to really work." However, it does work, and very well indeed for antennas with moderately low feedpoint impedances. Since there are easy-to-use utility programs for calculating everything you need, there is no need to avoid either the beta match or antennas that require one.

Updated 7-1-2000. © 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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