The Beta Match: 2 Views

L. B. Cebik, W4RNL

The beta or hairpin match is so deceptively simple in appearance that many radio amateurs distrust it immediately. Surely adding a coil across the feedpoint terminals--and nothing else except the coax--must foul up the performance and produce dreadful results. The coil or the hairpin is so small that it must act like a short circuit.

Unlike a gamma match, which permits the builder to make a direct connection between the driver and the boom, the beta-matched driver requires insulation and isolation from a conductive boom. But, beyond that, the driver connections are simpler. In fact, we can use the same terminals to connect both the beta match and the coax. Every reduction in something that can degrade with time and weather is a blessing in home-built antennas.

The two common forms of the beta match appear in Fig. 1. The antenna terminals connect to a coaxial cable, usually 50 Ohms. The beta component is anything that produces a required inductive reactance. How much inductive reactance we need will appear shortly. The two most common ways to produce an inductive reactance are with a solenoid inductor or coil and with a shorted transmission-line stub. The stub form is responsible for the name "hairpin" match. Apparently, the term "beta match" was originally copyrighted by Hy-Gain, a company that has used the system extensively over the years. Sensitivity to that situation has led some folks to avoid the simple term "beta match" and to use only the expression "hairpin match." The result has been some odd locutions. For example, the solenoid or coil method of producing the required inductive reactance has been called a "hairpin inductor." Since I have acknowledged the potential copyright holder and have no commercial interests, I shall call the system by its simplest name.

The ARRL Antenna Book has long had a correct explanation of how a beta match works, along with the conditions for setting up the system. Part of what we shall see here parallels that account. However, there is also a second way to look at the beta match, one that takes us back to even more basic terms. By looking at the beta match in two ways, perhaps we can give you a way to avoid confusion about the system.

First things first: we call the system a matching system. This label implies that we have an antenna whose feedpoint impedance is not a good match for the ubiquitous 50-Ohm coaxial cable that we so often use as our feedline. We only use the beta match when the antenna feedpoint impedance is less than the cable impedance. For this case, we are talking of impedances well under 50 Ohms. Although we can use a beta match with impedances as high as 35 Ohms, we normally reserve the system for impedances of 25 Ohms or less.

If our driver is resonant at about 25 Ohms, then we do not need a beta, gamma, or Tee match. Instead, we can use a 1/4 wavelength section of 37-Ohm cable to effect an impedance match. Although there is a 35-Ohm coaxial cable available, it is not easy to find and runs about $3.00/foot. A simpler scheme is to connect 2 lengths of 70-Ohm cable in parallel, with the center conductors connected and the braids connected at both ends. A pair of RG-59 cables in parallel will just fit inside of the common male UHF connector shell without significant deformation. Hence, we can create and waterproof the required cable with fair ease. Remember to account for the velocity factor of the line that you actually use when constructing such a cable.

Of course, the line length will vary not only according to the line's velocity factor, but also as a function of the frequency that you use as your design frequency. The design frequency is normally the center frequency of the span of frequencies for the antenna (assuming a monoband antenna). If the passband (another name for the spread of frequencies used) is wide and the antenna has any odd properties, you may adjust the design frequency so that you achieve satisfactory performance at the edges of the span. For wider amateur bands, I often recommend that an antenna like the Moxon rectangle be designed for a point about 1/3 the way up from the low end of the passband. Other antennas may require different treatment.

I used to calculate the length of a wave at the design frequency from scratch. 299.7925/F in MHz gives a wavelength in meters. Divide that by 0.3048 for the length in feet. Multiply by 12 for the length in inches. Of course, if you need the length in cm or mm, then simply make an adjustment by 100 or 1000 to the length in meters. Nowadays, I simply keep my antenna modeling program running and let it do the calculating for me.

If we opt for a 1/4 wavelength matching section, then we have no need to use a beta match. That situation would mean that this little article would end right here. So let's assume that we do not have any 70-Ohm cable or do not want to cut a section out of our cable TV system. What we do have is some leftover house wiring material, specifically, some AWG #12 copper wire with a 0.0808" diameter. Ultimately, we shall apply it to a 3-element 10-meter Yagi. However, we first need a general procedure for calculating everything that we need for the beta match.

A Beta Match: the Network Way

One of the conventional representations of a beta matched antenna element appears on the left in Fig. 2. Let's examine it more closely, because it tells us part of what we have to do to set up the antenna's driven element to receive a beta match.

Although the figure shows stubs to indicate that we are dealing with an antenna element, the load for the energy arriving via the coaxial cable is essentially between the terminals. Note that we must have a complex load. First, there is a series resistance (Rs). Next, there is a series capacitive reactance (XCs). That much alone tells us that the antenna element itself must not be resonant. If we lengthen the element beyond resonance, it will present a complex impedance composed of a series resistance and an inductive reactance. That condition is a possible one for a beta match, but by habit, amateurs want lighter elements, not heavier ones. So they traditionally use an element that is shorter than resonant. That physical situation results in a complex impedance consisting of a series resistance and a series capacitive reactance, as shown in the sketch. If we had started with a 1/2-inch resonant 10-meter element about 196.97" long, then we must reduce its length to about 192.2" to obtain a usable combination of resistance and capacitive reactance.

Before we look at why my measurements are so fussy, let's clear up another misconception, this one bred by the appearance of the left-hand sketch in Fig. 2. Because we have only 1 capacitor symbol, numerous newcomers tend to think of the beta match as an unbalanced system. Now the gamma match is inherently an unbalanced system, but the beta match is a balanced system. The center part of Fig. 2 shows a way of drawing the situation that gives us a balance of capacitor symbols. We have 1/2 of the capacitive reactance on each side of the resistor. (If we were using real capacitors as lumped components, we would make each one be twice the value of the virtual capacitor in the left-hand sketch, so that each one has 1/2 the total series reactance. But remember that we are not using a lumped component. Instead, we are using the reactance of an element that is shorter than its resonant length.)

Now we can return to the fussy element length measurements. How long or short we make the element is a function of the series resistance that it presents. If we know that value, then we can calculate everything else using some very simple equations that apply to an L-network when the energy source has a higher resistive impedance than the load (antenna element). To see how this works, look at the right-hand part of Fig. 2. There, I have redrawn the center sketch in the form of a balanced L-network. You calculate a balanced network the same way as an unbalanced one, except that you later divide the series capacitive reactance into 2 equal parts, one part for each side of the line.

The L-network consists of a series capacitive reactance (XCs) and a parallel inductive reactance (XLp) across the line on the source or cable side of the network. XLp is the beta-match component, either a solenoid inductor or a shorted transmission-line stub. (Note: XLp is also called a shunt component, but I shall stick to the term parallel. I do not want to confuse matters by using too many terms beginning with "s.")

There are several ways of calculating the required values of XCs and XLp, but at root, they depend on knowing the value of Rs, the resistive part of the antenna or load impedance. Let's create and define a term having the name "delta." Nowadays, we call this term the loaded Q, the working Q, or the network Q, but its original name was delta. In equations, we represent the name with the lower case Greek letter. We can show the definition of the delta of a down-converting L-network by a simple equation (and down-converting simply means that the source-end impedance is higher than the load-end impedance).

How do we find out the antenna feedpoint resistance. If we have our antenna set up for a field test, we can always measure the impedance with one of the several antenna analyzers on the market. Or we can go to an antenna modeling program and let it calculate the impedance. Let's suppose that we come up with 25 Ohms. The ratio of the 50-Ohm source end impedance to 25 Ohms is 2:1. Subtract 1 from that and you get the square root of 1, which is 1. That is the value of delta.

Next, let's figure out how much capacitive reactance we need in series with the 25-Ohm resistive value to satify the network requirements. This step is also an easy calculation.

We simply multiply the value of delta by the value of the series resistance that is the load, and the answer appears. 1 times 25 gives us 25 Ohm reactance. Notice that in this network calculation, we do not obtain a + or - sign. The absence of a sign simply means that the reactance can be either inductive or capacitive under a special condition. The condition is that the parallel reactance that we will next calculate must be (for ordinary cases) the opposite type from the series reactance. Since we want a shorter element with capacitive reactances as the series reactance, we must use an inductive reactance for the parallel reactance.

The source-side or parallel reactance is simply the value of the source resistance (50 Ohms) divided by the value of delta. Since delta = 1, then the parallel reactance must be 50 Ohms. Remember that these steps apply to the situation in which we are matching a higher impedance (the 50-Ohm cable that is the source of emergy) to a lower value of resistive impedance (the value of Rs that is the resistive component of the antenna element impedance). When receiving, the situation is reversed. the antenna element is the source of energy at a lower resistive impedance that we up-convert to the 50-Ohm cable impedance. The same component values apply. However, to help keep things straight, we usually only think in terms of transmitting. So the cable end becomes our energy source and the antenna becomes the load.

That is all there is to doing the basic calculations for the beta-match as a down-converting L-network. We have not yet translated these reactance values into physical components. Of course, we have only one component needing the translation: the parallel inductive reactance across the terminals. The capacitive reactance is contained in the antenna feedpoint impedance. However, before we effect the translation, let's step back and take another look at what we just did.

A Beta Match: the Parallel-Circuit Way

One of the shortcomings of most handbooks designed for radio amateurs is that they do not give enough space to a fundamental fact about impedances. We receive the information about the antenna feedpoint impedance and do not stop to think about the orientation of the information. In our ideal case, the shorter antenna elements has an impedance of 25 - j25 Ohms. We normally forget that the values for each component of the impedance are series values.

Why is the qualification necessary? The answer lies in the fact that for every series impedance, there is an equivalent parallel combination of impedance values. Although the handbooks give the conversion equations in each direction, they do not give enough practical examples of using them. Our beta match presents us with a very significant use, since it may teach us something about impedances at the same time that it allows us to flex those long-dormant equations. What we shall do is convert the left side of Fig. 3 into a form that reflects the right side.

Note that the L-network consists of source and load resistances, with a combination of series and parallel reactances. Suppose that we converted the series antenna impedance components into their parallel equivalent values. For reference, here are the series-to-parallel equations. Note the similarities that let you simplify the actual key presses needed on a hand calculator.

Since both the series resistance and series reactance values had numerical values of 25, we can quickly arrive at the parallel equivalent resistance (50 Ohms) and the parallel equivalent reactance (-j50 Ohms). Now note that something amazing has occurred. The parallel resistive component of the load--with its associated capacitive reactance at work--yields a 50-Ohm resistance.

Now be amazed by a fact that is well known to people to match things up for a living: The series and parallel values of the resistive component of an impedance are the same when the is no series reactance or there is an indefinitely high parallel reactance. We cannot simply re-resonate the antenna element, because that would yield a 25-Ohm resistive impedance. We need a way to keep the capacitive reactance working while getting rid of the series reactance net value-- or driving the parallel reactance upward and off scale for all practical purposes.

One very practical way to accomplish this is to place a parallel reactance of the opposite type scross the initial parallel reactance value. Parallel reactances that are equal make up a resonant circuit. Parallel reactances operate in the same manner as parallel resistances with respect to arriving at a net value for 2 values.

Since the reactances are equal within this idealized case, the denominators add up to zero, because the two reactance values have opposite signs. Division by zero is not possible, of course. But we know that the value of the result becomes higher without limit as the denominator approaches zero. One common computer program technique of handling division by zero, when zero is within a range of possible values, is to set the result at an arbitrary but very high number. Typically, programs use 1E10. We may use that number here in order to see what has happened to the series values for R and X that we might encounter as the impedance at the coax terminals.

The equations tell us how to back calculate into the more familiar series forms of resistance and reactance at the coaxial cable terminals. Remember that Rp is still 50 Ohms. Adding the square of 50 to the square of 1E10 has no effect. Any hand calculator will return 1E20 as the result because the square of 50 (2500) only appears at the far end of the result, outside the range of what the readout will show. The resistance numerator is 50 times 1E20. When we divide this by 1E20, we end up with 50 Ohms as the series resistance. The reactance conversion back into a series value has a quite different result. We end up with 2500 times 1E10 divided by 1E20. The result is 2.5E-7 or about 250 nano-Ohms. Effectively, the reactance is zero, just as we predicted.

The calculations that we have just performed are commonplace in matching situations. In another item at the site are ways to calculate the match-line-and-stub system of matching. We find the place along a transmission line at which the parallel resistive impedance is 50 Ohms. Then we add a parallel transmission-line stub to match the parallel reactance value at that point, but use a stub that produces a reactance of the opposite type. The result is a parallel resistive impedance in paralel with a very high value of parallel reactance. When we reconvert to series values, the series resistance equals its parallel counterpart, and the reactance goes to zero or very close to it. Then we can run coax from that point back to the transmitter. Of course, there is no magic in using 50 Ohms as the desired impedance. It just happens to be the most common line impedance to which we normal gear our matching efforts.

Although the series-parallel-series conversion equations seem more complex than the simplified L-network equations, they are in fact more fundamental. Indeed, you now have a more complete idea of what those network equations are actually calculating.

One of the main questions facing the would-be beta matcher is knowing what length to make the antenna element. One of the advantages of the L-network equations is that they provide guidance here, where the series-parallel conversions require that we almost hunt out the correct series reactance value by trial and error. Since we most often calculate for a 50-Ohm source resistance, let's make a small chart of beta-match reactance values using small increments of change in the series load resistance.

     Some Beta-Match Resistance and Reactance Values

Rso = 50 Ohms for all cases

Rs (Ohms)        35     25     17     12.5    10     8.3
Delta            0.65   1.0    1.4    1.7     2.0    2.2
Xs (Ohms)        22.9   25.0   23.6   21.7    20.0   18.6
Xp (Ohms)        76.4   50.0   35.4   28.9    25.0   22.4

One way to use this chart is as a guide to setting the length of the driven element in an array that might use a beta match. If you have antenna modeling software, you can try shorter elements relative to resonance to arrive at vaue very close to the optimum for the match. It is also likely that you might be able to interpolate between listings for the entire set of reactance values for your own installation.

Finishing the Job

Let's set up a worked problem so that we can track the techniques and finish the job of creating all of the beta-match components. We may use the 3-element Yagi shown in Fig. 4 as an example that is not far from the idealized numbers that we have used. The antenna uses 1/2" aluminum elements.
3-Element 10-Meter Yagi Dimensions in Inches
Design Frequency:  28.5 MHz     Elements: 0.5" aluminum
Element              Length        Spacing from Reflector
Reflector            206.28        ----
[Driver (resonant)   196.97        62.40]
Driver (shortened)   192.20        62.40
Director             185.33        134.54

The dimension table shows the original resonant driver. With this element, the feedpoint impedance was 25.7 Ohms. In the process of shortening the element for the beta match, the resistive component dropped a bit to 23.76 Ohms, with a capacitive reactance of -j24.27 Ohms. The numbers that I am citing are too precise for practical purposes, but they certainly cannot hurt our progression of calculations. Both the resistive and reactive components of the modeled feedpoint impedance fit the table nicely, being slightly below those in the column for a 25-Ohm resistive component.

To calculate delta, we would take the ratio of 50 Ohms, the cable or source resistive impedance, to the value of Rs at the antenna, 23.76 Ohms. The value for delta is (again, in overly precise terms) 1.051. Delta times Rs gives us the series reactance: -j24.97 Ohms or very close to the modeled value, and certainly too close to require any more trial changes of the antenna length. The source resistance divided by delta gives us the reactance of the parallel inductive component or j47.58 Ohms.

To check out work, lets convert the series impedance terms at the antenna terminals to their parallel equivalents. With the series-to-parallel value equations, we can use the modeled feedpoint series values to obtain equivalents. Rp = 48.55 Ohms. Xp = -j47.53 Ohms. To compensate for the capacitive reactance, we need an inductive reactance of j47.53 Ohms in parallel with the equivalent parallel capacitive reactance. This value is only 0.05 Ohm different from the value we calculated with the network equations. Quite frankly, I know of no one who could build the difference. However, let's use that very slight difference to calculate the net parallel reactance. We end up with a net reactance of j45,229.55 Ohms

We can now apply the parallel-to-series conversion equations to the new values: Rp = 48.55 Ohms and Xp = j45229.55 Ohms. The value for the new Rs is 48.549, that is, 48.55 Ohms. The final value for Xs is j0.052 Ohms, that is, so close to zero that it does not matter.

Our work can now go in two directions. We may construct a solenoid inductor or a shorted transmission-line stub to give us the required parallel inductive reactance. Let's begin with a coil. The relationship between a value of inductive reactance and the inductance that produces it at a given frequency is given in common equations.

Since we know the frequency (f = 28.5 MHz) and the inductive reactance (XL = j47.58 Ohms), we can use the right-side version to obtain an inductance: L = 0.2657 uH. I have purposely used too many decimal places for practical purposes, but they never hurt an example on paper.

The next step is to calculate a coil that has this inductance. Remember at the beginning of these notes, I specified that we had some AWG #12 copper house wire. We can build a coil from this material. However, when installing it, we must be certain that the copper does not touch the aluminum elements. Otherwise, we shall see corrsion from electrolysis. However, we can use stainless steel hardware (including washers) both to make a secure connection and to separate the two reactive materials.

There is an age-old equation for calculating the inductance of a coil from its construction, so long as the coil is a single-layer solenoid. Of course, we already know the inductance, and so we can turn the equation around to obtain the number of turns. We shall assume a coil diameter of 1" and a total length of 1". These number assure us of a reasonable Q for the coil. As well, the 1" coil length will coincide with the spacing between the connecting bolts and still leave room for a small gap between the element halves.

In the equations, L is the inductance in uH and n is the number of turns. The length of the coil is l, and the diameter is d, both in the same unit of measure. We are using inches. Since the values of l and d are both 1, the value of n is the square root of the inductance times 58 or the square root of 15.41. So n = 3.93 turns. Of course, wwe shall raise that to 4 turns so that the leads both head for the element terminals.

Note that the coil-winding equation does not take the wire size into consideration and does not account for the inductance in the coil leads. We shall, of course, keep the leads as short as feasible. Then we may fine tune the coil inductance by spreading or squeezing the turns as needed.

Next, let's replace the coil with a shorted transmission-line stub or hairpin. We shall use our AWG #12 wire for the hairpin, using the same precautions at the element connections that we applied to the coil. First, we need to determine the characteristic impedance (Zo) of the parallel line that we shall create. Let's suppose that we retain the 1" center-to-center spacing. The diameter of AWG #12 wire is 0.0808". We can calculate the Zo from another standard equation, where S is the center-to-center spacing and d is the diameter when both are in the same unit of measure.

The equation gives us 276 times the log of 2/.0808 or 384.64 Ohms. Again, the sense of precision in the 2 decimal places is spurious and only reminds us that we are working a paper example. We shall assume a velocity factor of 1.0 for our short line that will be self-supporting. However, we need to know how short--or long--the line must be. Since we want an inductive reactance, we shall use a shorted transmission line. Again, we have some standard equations that relate the inductive reactance (XL) to the line length (l).

Note that the equation requires that we use a trig function on our hand calculator. "Arctan" is simply the inverse of the "tan" function. So we divide 47.58 (XL) by 384.64 (Zo) to get 0.1237 and then take the inverse of the tangent of that number. The result is a line that is 7.052 degrees long. We are now only 2 short steps from the final length.

The first step is to divide the electrical length in degrees by 360 to find the fraction of wavelength that the line happens to be: 0.0196 wavelength. That is the equation on the left.

The length in inches will be this fractional length times the length of a wave in inches. At 28.5 MHz, a wavelength is 414.135". So our AWG #12 1"-wide stub will be 8.112" long. One reason that the excessive number of decimal places is spurious lies in the various ways in which we construct beta stubs. Fig. 1 showed a rounded end, which is commercially common, since it is easy to create a smooth curve by bending the wire around a cylinder. However, the equation for the electrical length assumes that the shorting wire has effectively zero length and plays no role in the functioning of the stub. Even a flat end to the stub will have a small effect on the inductive reactance, and so too will the connection eyes at the element end. Like the coil, we can also spread or squeeze the stub line into submission.

Nevertheless, we may test our calculation of the hairpin by adding a shorted transmission line across the terminals of the Yagi with the shortened driver. We shall specify in NEC's transmission line facility a Zo of 384.64 Ohms and a length of 8.112". (The actual TL command requires the measurement in meters. Many commercial implementations of NEC allow user entry in the same dimensions as the wire entries for the elements. The program then converts this value to 0.2060 meters before turning the core loose to perform its calculations.)

Remember that when we performed our series-to-parallel-to series calculations, we ended up with an impedance of 48.55 + j0.05 Ohms. With the hairpin in place within the model, we obtain a NEC-calculated impedance of 48.55 - j0.05 Ohms. Those values are too close (j0.1 Ohm apart) to be coincidental. Indeed, you may wish to run this exercise through a number of trial value combinations just to become familiar with the process and to get a firm grasp on the relationship of the network calculations to those we performed to convert the feedpoint situation into an equivalent pure parallel circuit.

Does a beta matched Yagi have an advantages or disadvantages relative to one with a resonant driver? Fig. 5 overlays the 27.5-Ohm SWR curve of the version with a resonant driver on the 50-Ohm SWR curve of the final beta-match version using a shorted transmission-line stub. As you can see, the SWR curves are almost indistinguishable. The Yagi will cover between 28.0 and almost 28.9 MHz with under 2:1 SWR at the feedpoint. The curve will be slightly broader at the transmitter end of the coax line.

Both versions of the antenna show a free-space gain of 8.11 dBi. There is a divergence in the 180-degree front-to-back ratio: 27.12 dB vs. 27.13 dB. The pattern in Fig. 4 applies equally to both versions of the antenna.

This small exercise has tried to accomplish 2 things at the same time. First, it has tried to outline the procedure for creating a beta match, including the initial determination of whether or not such a matching system is in order. We calculated the required element capacitive reactance and the required parallel inductive reactance. Then we converted the inductive reactance into a practical beta inductor and also into a beta shorted transmission-line stub.

Our second goal was to understand better how a beta match does its work. We began with the standard L-network treatment, which is very handy for calculating the required reactances once we know the resistive component of the driver feedpoint. However, we supplemented that perspective by converting the circuit into an equivalent purely parallel circuit so that we could see some basic facts about matching situations of all sorts. We seek a series combination of resistance and reactance that--when converted to their parallel equivalents--yields a parallel resistive component of 50 Ohms. Then we parallel the reactance with the opposite type having the same absolute value. When reconverted back into series values, the reactance goes to zero or very close to zero. Under those conditions, the series and parallel resistive components are, for all practical purposes, the same. In the case of the beta match, we obtained a resistive impedance too close to 50 Ohms to notice any fussy numerical difference.

One final caution: the Yagi that we used for our example is a very good design. However, the uniform 0.5" diameter elements are impractical In effect, we would need a combination of 5/8 and 1/2 inch elements or, for lighter duty, a combination of 1/2 and 3/8 inch elements. However, even those values, which seem so close to 1/2", will require a complete refiguring the the element lengths. As well, since the beta match results in a balanced set of terminals, use a common-mode supressor at the feedpoint. Something like a W2DU-type bead choke will work fine in the transition from the balanced terminals to the single-ended coaxial line.

Updated 02-02-2005. © 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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