However, we also began to develop a more visual and hopefully a more intuitively correct understanding of the behavior of resistance and reactance along a transmission line. As well, we saw from the models that we used to explore this territory the ways in which the line length, even apart from line losses, can affect resistance and reactance as we move away from the design frequency toward band edges. Along the way, we encountered some interesting uses of feedline transformers that are an odd multiple of a quarter wavelength.
In this episode, we shall do more of the same, but with a difference. Part of our survey will involve looking at different antennas, including a 1 wavelength doublet, a single quad loop, a three-element Yagi, and an extended double Zepp (EDZ). The first 3 of these antennas are resonant at the design frequency, but the EDZ is inherently non-resonant. As well, we shall examine further the quarter wavelength of feedline as an impedance transformer used to effect a match between an antenna and a main feedline that do not initially match. This exploration will lead us to look at at least two other matching systems: the beta or hairpin match and the match line-and-stub system. In each case, we shall be interested in the behavior of resistance and reactance as we place these systems between the antenna terminals and the main feedline.
Our main goal is to refine our working understanding of resistance, reactance, and SWR behavior on a feedline relative to the same parameters when measured at the antenna terminals. Granted that all of these items can be calculated by the use of handbook equations: in fact, we shall be using NEC-4 to perform our calculations (on lossless lines). Nevertheless, graphical analysis can potentially add some clarity and vividness to the numbers so that we may gradually develop more correct expectations of the situations that present themselves to us.
However, for our work, we shall use a more precisely cut 1 wavelength doublet: it will be 133.4' long and use AWG #12 copper wire. We shall place this 7.15-MHz (40-meter) antenna at 50' above average ground. Fig. 1 shows the outlines of our simple antenna system.
The top portion of the sketch shows our baseline antenna with no feedline. We shall throughout these notes always begin at a comparable situation. Fig. 2 outlines the resistance and reactance behavior across 40-meters. (Since the resistance and reactance are both so high, SWR curves would be useless, at least at this stage in our exploration.)
The feedpoint impedance at the design frequency is 4784 + j0.42 Ohms. The increments along the left Y-axis of the graph cover a sizable range of values, so the shallow curves actually are sharper than any of those that we encountered in Part 1. Especially notable is the fact that the reactance goes from being considerably inductive to being considerably capacitive across the 40-meter band. This curve is precisely the opposite of the reactance behavior of a 1/2 wavelength dipole. However, in the vicinity of an integral multiple of a wavelength, the reactance must reverse its curve and be inductive when the antenna is short (under 1 wavelength) and be capacitive when the antenna is long (more than 1 wavelength). This phenomenon occurs only for a small range either side of the 1 wavelength (or any integral wavelength) mark. Indeed, the toughest job in creating a 1 wavelength antenna will be finding the exact length--for the materials, height, and environment--at which the transition occurs.
A common question regarding 1 wavelength doublets is how to feed them. Why not use common 50-Ohm coaxial cable? Of course, we shall need a means of transforming 4784 Ohms to 50 Ohms (both resistive), but that is a task for the quarter wavelength matching section of feedline. Since the antenna is resonant (by modeling design), we can use the simple equation to find the required matching section impedance. Multiply the two impedances (input and output) together to get 239,200 and take the square root: 489.1 Ohms. Essentially, this is the geometric mean between the two impedance values.
A 489-Ohm impedance transmission line is fairly easy to build. Two lengths of AWG #16 spaced 1.5" apart or two lengths of AWG #10 spaced 3" apart will do the job just about perfectly. There are, of course, equations and utility programs to perform the calculations, or you can arrive at a very workable approximation from some graphs in "Some (Old) Notes on Home-Brew Parallel Transmission Lines". Let's build such a line and use a perfect 1/4-wavlenegth section between the antenna terminals and the measuring point or source. (As we did in Part 1, we shall assume that the velocity factor is 1.0, since we can always trim our open-wire home-made line if the value is slightly lower.) Fig. 3 shows the outcome for the resistance, reactance, and the 50-Ohm SWR.
As the 1/4 wavelength lines did in Part 1, this new line reverses the direction of the reactance curve. Notice that the total spread of reactance across the band is just under 60 Ohms. Combined with the almost changeless resistance line, the reactance allows a 50-Ohm SWR curve that remains well under 2:1 from one band edge to the other. The 1 wavelength doublet is often discounted as a monoband antenna on the grounds that it is too hard to match to standard cables and too sharply tuned to use without an antenna tuner. Neither of these grounds is true, and the 1 wavelength doublet is an extremely cheap and relatively broadband 40-meter antenna--with a couple of dB of extra gain. Of course, add a 1:1 choke or balun at the junction of the match line and the feedline to suppress common-mode currents.
Let's change gears and move to a 1 wavelength single quad loop. For most wire sizes in common use, an HF self-resonant loop will require a cutting formula like this one: L(ft) = 1041/Fr(MHz). Of course, the exact length will vary with the wire size or element diameter and with the frequency, since the wire size as a fraction of a wavelength varies with frequency. However, accurate algorithms for designing a self-resonant square (or diamond) loop are available. See "Calculating the Length of a Resonant Square Quad Loop", for background: the utility program is available in various formats from various sources.
We shall look at a self-resonant quad loop for 10 meters, with a 28.5-MHz design frequency. The loop uses AWG #14 copper wire, and the model places it in free space for this exercise. The circumference of the loop is 36.52' Fig. 4 outlines the two ways in which we shall examine the loop: as a "bare" antenna and with a matching line.
The basic antenna provides resistance and reactance curves as they appear in Fig. 5. The 50-Ohm SWR curve is somewhat gratuitous, but does show that we do not have a close match between the antenna feedpoint impedance and our 50-Ohm feedline.
The impedance of the bare quad loop is 127.0 + j0.76 Ohms at 28.5 MHz. Unlike the 1 wavelength doublet, the reactance curve has a normal direction, since a quad loop is essentially two 1/2 wavelength dipoles with their ends bent toward each other until they touch. They form two 1/2 wavelength antennas in phase (except for wire losses) and spaced 1/4 wavelength apart.
If we wish to match the loop to a 50-Ohm coax main feedline, we shall need a 1/4 wavelength section of 79.7-Ohm line (SQRT (50 * 127)). 75-Ohm line is about the closest value that we might readily find. (Building a 75-Ohm line with air insulation would require square conductors spaced very close to each other. The lowest obtainable value using round wires and air insulation is about 80 Ohms, and it is impractical in terms of maintaining the spacing.) If we use this line in a 1/4 wavelength matching section--the right side of Fig. 4--we would obtain the impedance behaviors shown in Fig. 6.
The resistance and reactance curves are perfectly normal for the situation. Because the line is only 1/4 wavelength at 28.5 MHz, the resistance shows a peak value there with declining values at the band edges. The reactance curve is the reverse of the one for the bare antenna. Since the matching section is not a perfect geometric mean between the input and output impedances (as it was in the 1 wavelength doublet case), the SWR does not reach 1.0:1 at the center of the operating passband. However, even with the less than perfect value, it only rises to about 1.5:1 at the band edges.
2-element monoband quad beams very often have impedances in the same range as the single quad loop. Hence, the 1/4 wavelength 75-Ohm matching section has seen wide use with these arrays. The graphed results of the model suggest that we can handle a considerable range of impedances with 75-Ohm cable, which ideally would match 112.5 Ohms to a 50-Ohm coax main feeder. Antenna feedpoints from 100 to 130 Ohms have used them. However, two cautions are in order. First, it may pay to model the actual situation before cutting any 75-Ohm cable to 1/4 wavelength to confirm that acceptable results will emerge. Second, one may use lengths that are not 1/4 wavelength at the antenna design frequency, but somewhat longer or shorter. Very often, modeling will allow one to find the best length to effect a match with equal 50-Ohm SWR values at both band edges. For further details on this idea, see "When is a Quarter Wavelength Not a Quarter Wavelength?".
The "bare" element treatment uses a 198" driver. With that length, the array has a feedpoint impedance of 25.71 - j0.93 Ohms at 14.175 MHz. The resistance, reactance, and 25.71-Ohm SWR appear in Fig. 8.
The SWR curve appears to tilt more toward the high end of the band because the impedance is essentially flat between 14.175 and 14.20 MHz. However, the SWR rises more slowly below the design frequency than above it. Hence, the band-edge SWR at 14 MHz is lower than at 14.35 MHz. Nonetheless, a good 25.71-Ohm SWR curve is not a good match for a 50-Ohm feedline.
Otherwise, the behavior of the resistance and reactance are normal--for a parasitic array with a director. The reactance shows a rising characteristic with frequency. The resistance, however, decreases with increasing frequency: that is the mark of a director-controlled parasitic array. Single wire antennas and even Yagis having a driver and reflector show the reverse resistance curve: rising resistance with increasing frequency. See, for example, the bare 40-meter dipole in Part 1 or the bare quad loop in Fig. 5 of this part.
We shall need to add a matching network of some kind to effect an impedance transformation to match our 50-Ohm main feedline. We have three main choices--although a few others are possible. First is the gamma match, often used when we want to directly connect all elements to a conductive boom. Due to modeling limitations, we shall pass over this matching system in this exercise. Although we can model a gamma in principle, using the same diameter for all wires in the system, normal gamma construction uses several different wire diameters. NEC-2 and NEC-4 tend to yield inaccurate results when we insist on using angular junctions of wires having different diameters.
A second option is to use a 1/4 wavelength matching section. The ideal characteristic impedance for this section would be the square root of 25.71 * 50, or 35.85 Ohms. We can approximate a 36-Ohm transmission line by paralleling two 1/4 wavelength sections of 72-Ohm cable, connecting together the two center conductors at both ends and the two braids at both ends. In fact, a pair of RG-59 cables will just fit inside a normal UHF coax connector without deformation, making such a matching section both easy to construct and easy to use.
The impedance behavior, graphed in Fig. 9, is strictly normal for a 1/4 wavelength matching section. Because the line is short at 14 MHz and long at 14.35 MHz, the resistance curve peaks at mid-band. The reactance curve has the opposite slope of its counterpart for the bare driver. The 50-Ohm SWR curve replicates the general appearance of the 25.71-Ohm curve for the bare element. Note that the SWR at 14 MHz is 1.66:1, while at 14.35 MHz, it is 1.75:1, a reasonably well-balanced curve for full 20-meter coverage.
An alternative matching scheme is the beta or hairpin match. For more detailed information on beta matches in general, see "Beta Coils and Hairpins". Essentially, a beta match is an L-network consisting of a series capacitive reactance on the antenna side and a shunt inductive reactance on the feedline side. To simplify construction, we provide the series capacitive reactance at the antenna terminals by shortening the driver element enough to provide the right reactance value. Instead of using the self-resonant length of 198", we can use a 193" driver to provide about 25 Ohms capacitive reactance.
To match a 50-Ohm line to a 25-Ohm load, we need, in addition to the 25-Ohm series capacitive reactance, a shunt inductive reactance of about 50 Ohms. We can provide the reactance using any means that shows inductive reactance. For example, we might place a coil across the antenna terminals, so long as the coil's inductive reactance at 14.175 MHz was 50 Ohms. Alternatively, we can use a shorted transmission line stub. Since we can derive more inductive reactance from a shorter stub using a higher characteristic impedance for the transmission line, most builders use parallel open-wire transmission line sections, with the end result resembling a large hairpin.
To produce 50 Ohms inductive reactance from a 600-Ohm transmission line, we must make it a little over 11" long at 14.175 MHz. So our first attempt at a beta match will use the 193" driver with an 11" 600-Ohm hairpin across the feedpoint terminals. The results appear in Fig. 10.
In contrast to the 1/4 wavelength matching section system of effecting the match, the beta match or L-network shows a large change in resistance across the 20-meter band. Conversely, the reactance shows very little change. The previous system showed just the opposite characteristics--a large change in reactance but only a small change in resistance. Nevertheless, the resulting 50-Ohm SWR curve has almost the same shape for both systems. With the beta match, the 50-Ohm SWR at 14 MHz is 1.65:1, and at 14.35 MHz, it is 1.79:1.
The use of a 600-Ohm hairpin inductively reactive stub raises a natural question: will any of the operating parameters change is we use a different characteristic impedance for the stub. For example, we might want to use a 50-Ohm stub, although it will have to be 104.1" long. The line is physically impractical and is likely to be lossy compared to the short 600-Ohm stub. However, for this small side-exercise, we can ignore losses and impracticality.
The graph for the impedance behavior of our replace stub appears in Fig. 11. The graph is virtually indistinguishable in every respect from Fig. 10, the behavior of the 600-Ohm stub. Indeed, even the bend-edge values of 50-Ohm SWR are the same in both cases.
Throughout this exercise, we have ignored actual gain and front-to-back values, since they are not germane to our interests. The 3-element Yagi used for the models appears in several guises (element diameter taper schedules) at my web site, for anyone interested. It is a very good monoband 3-element Yagi with its design origins in the work of Brian Beezley, K6STI. In fact, a version of the beam appears in his classic MININEC program, AO.
Let's examine a 10-meter version of the EDZ, 44' of AWG #14 copper wire, about the right length for 28.5 MHz as a somewhat arbitrary design frequency. The design frequency is arbitrary in the sense that we have no special marker to indicate an appropriate goal for that frequency. In all of our other antennas, we used resonance, the condition where reactance goes to zero, as our marker. With the EDZ, we shall always have reactance. At best, we can roughly optimize the wire length for maximum gain at the design frequency, but so long as we are close to maximum gain, we can use a convenient wire length. Hence, we shall use 44'. Our models will all be in free space.
After we examine the characteristics of a bare EDZ (with no feedline as part of our model), we shall explore a type of matching system sometimes called the match-line and stub system. Fig. 12 shows the two possible versions, one using a shorted stub, the other using an open stub. Both require a transmission line length between the antenna and the stub. The combination yields an impedance at the junction of 50 Ohms, so the portion labeled main feedline can be 50-Ohm coax. However, before we can build a mathing system, we must find the impedance that requires matching.
The bare EDZ shows an impedance of 142.90 - j690.50 at 28.5 MHz, as shown in Fig. 13. The resistance changes hardly at all across the 28-29-MHz span. The reactance shows a rising characteristic, being more capacitively reactive at the low end of the band and less so at the upper end of the operating passband. Of course, a 50-Ohm SWR curve would be useless at this stage of antenna system development.
For any impedance to be matched, whether simple or complex, there will be a length of transmission line for a pre-selected characteristic impedance such that the addition of a compensating stub will yield a desired impedance. For the 142.9 - j690.5 Ohm antenna terminal impedance, we must find the combination that will give us 50 Ohms to match out coaxial cable. (Not all transmission line characteristic impedances will provide a solution to this requirement, although most higher values will do the job. Hence, most match-line and stub systems use parallel transmission line for the task.)
In fact, there are at least 8 solutions to our problem. There will be two match-line lengths. Each of these line lengths will have two parallel stub solutions, one for an open stub and one for a shorted stub. As well, there will also be for each line length a pair of series stub solutions, although they are less commonly used. Parallel transmission line lends itself to the use of parallel stubs.
For detailed information on the calculation of match lines and stubs, see the special appendix to "The EDZ Family of Antennas". The account provides a detailed analysis of the math behind the system. Previously, the system had in virtually all antenna handbooks been left to cut-and-try techniques or to visual means, such as a Smith chart. My motivation for looking more intensively at the match of the method arose from the fact that if the solution could appear on a Smith chart, then it also had to have a calculable foundation. However, should the system have any interest to future antenna projects, you need not replicate the calculations with pencil. The HAMCALC collection contains a version of the utility program that I wrote to simplify the work and to show at least all of the parallel stub solutions.
For a shorted-stub solution, we shall use 450-Ohm transmission line. The matchline length will be 5.302' long, while the shorted stub will be 1.348' long. The alternative matchline length is not much longer, but the shorted stub is considerably longer. In most cases of match-line and stub matching, we select the solution that results in the shortest combination of lines.
Fig. 14 provides a view of the resulting impedance behavior. The new feedpoint impedance at the junction of the lines and stub is 49.99 - j0.02 Ohms, a result that is not likely to be achieved in a real application of the system. The reactance shows a rising curve (like the bare-wire curve), while the resistance curve has the opposite slope. Both exhibit considerable variation. The result is a 50-Ohm SWR curve that is below 2:1 only from 28.1 MHz through 28.9 MHz. We might ask whether we would get better results from an open-stub combination. Let's continue to use 450-Ohm line for both the match line and the stub.
The match line for the open stub system is 5.758' long, while the stub length is 7.220'. I selected the second match-line length because its open stub is about 2.5' shorter than the one for the first match-line length. The combination results in a 28.5-MHz feedpoint impedance of 50.02 - 0.01 Ohms, once more, a pleasant mathematical outcome, but not one to expect of real wires and lines.
As shown in Fig. 15, the reactance curve has a rising characteristic, similar to the curve for the shorted stub. But, the range of reactance across the first MHz of 10 meters is about 30% less than for the shorted stub. The resistance curve for the open stub has the opposite slope from the one for the shorted stub. However, the total range of source resistance is about 20% greater than the range for the shorted stub version of the system.
The consequence of these contrary curve spans is a 50-Ohm SWR curve that is virtually identical in both cases. Once more, the passband with less than a 2:1 50-Ohm SWR extends from about 28.1 to 28.9 MHz. In most cases, the precise values of resistance and reactance will not make an operational difference to the use of the system. Hence, in principle, the two versions of the match-line and stub system are equivalent. However, the shorted stub version has a slight physical advantage in requiring a slightly shorter match line and a considerably shorter stub.
Connecting a single transmission line to a mismatched antenna terminal set--even with only a small mismatch--resulted in very regular behavior for the resistance and reactance curves. Indeed, it is possible to mentally catalog the general characteristics of the curve and to use the catalog to assist our expectations in real antenna work.
When we began the process of adding matching networks that used transmission lines, but not in simple ways, the curves became less regular and hence less predictable without using some form of calculating system. Our primary system has been antenna modeling software using NEC-2 or NEC-4 within the limits of that software. (Adhering to those limits means that there are some cases that we cannot effectively model in detail, as well as factors that will not appear in the results, such as transmission line losses.) The advantage of antenna modeling software (and adjunct graphing software) is that it provides both graphical and numerical outputs together. In this exercise, we have used EZNEC in combination with EZPlots to calculate and present the frequency sweeps. Other software will do an equal job.
With such software or its equivalent, we can inspect the behavior of the impedance components and derivatives as we add transmission lines to the antenna feedpoint. The need for rules of thumb disappears whenever we can actually calculate the result. The more we actually calculate, the more nearly correct will be our intuitions about a situation. When we see clearly the sort of result that we should obtain--even if we cannot mentally calculate the numbers involved--the mystery that surrounds transmission lines gradually disappears, replaced by understanding. The goal of these exercises has been a set of small steps toward understanding the impedance behavior of transmission lines when connected to antennas.
Updated 12-01-2004. © L. B. Cebik, W4RNL. The original item appeared in AntenneX for December, 2004. 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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