n the course of modeling rectangular arrays, co-author Handelsman was looking to see how high he could push the gain and front-to-back ratio (f/b). Some of the antennas were wonderful from this point of view but were very critical in their dimensions and were very narrow-band. Co-author Jefferies suggested that, from a hard-nosed practical engineer's point of view, the ultimate criterion of an antenna's performance should be whether it was "well-behaved". Apart from antennas owned by keen amateurs, such installations are put up and then forgotten about. The keen amateur on the other hand, may devote constant love and attention to his/her antenna installation, including taking some trouble to maintain its environment. In the real world, it is commonplace for owners of masts to allow other antenna growths to sprout without informing the proprietors of existing antennas on the mast.

THE PROBLEM
One of the problems with so-called "high-performance" antennas can be that the tuning is critical. For example, this is the case with physically short inductively loaded monopoles, such as are used commonly in mobile and hand-held applications. Such antennas have a high Q-factor and the reactive part of the radiation impedance is large and sensitive to objects in the near field. This results in a very variable mismatch to the feed, resulting in large variations in accepted power as the antenna is transported through the environment. In general, we can expect behavior problems with most compact antenna designs.

Well-behaved (to Jefferies) means that, like the satellite Yagi on his lab roof, it could tolerate birds landing on it with little change in its characteristics. Or it resists de-tuning by near-by objects such as trees, towers, the antenna tower/boom, or buildings. Dean Straw's (N6BV) criterion is how it would behave in the winter with a one-inch thick layer of ice on it. So, reliability and the ability to maintain all the design parameters are, we all agree, an important characteristic of what we describe as a "well-behaved antenna". Also, a well-behaved antenna should not be sensitive to small dimensional changes: we have all seen sorry-looking antenna structures, bent almost beyond recognition by the buffeting of storms and the collective weight of bouncing birds.

SCOPE OF REVIEW
In this article we propose to present the design and the theory behind one such well-behaved antenna. The antenna is a parasitic two-element array of rectangular loops made of 1-inch diameter aluminum alloy tubing and is designed for 2 meters. It has high gain, a very high f/b ratio and a remarkably wide SWR bandwidth. The lessons learned from designing this antenna are applicable to all loop arrays on all bands, including the common Quad antenna.

Some of the ideas and concepts that we will be dealing with came from "anomalies" or unusual behavior in Handelsman's modeling data that were explained by Jefferies. Others came from theoretical predictions by Jefferies that were confirmed by Handelsman.

In order to get started, and before going into parasitic arrays, we should examine single loops and see how they behave. We will start with square or quad loops and then stretch them into rectangles and see what happens to their characteristics.

Single Loops:
The quad has been known for almost 60 years. It was the invention of Clarence Moore, W9LZX in 1942 and was designed specifically to eliminate arcing which occurred at the ends of a Yagi at a high-power short-wave station at high altitude in Quito, Ecuador. Changing the dipole elements of the antenna to full-wave-perimeter loops eliminated this problem. Later, the loop was found to have a higher gain than the dipole and has become the basis for many amateur antennas, including quads from 80 meters to UHF, triangles and rectangles.

The major problem with loop antennas is that no one knows what their size should be. They are all of slightly greater than one wavelength in circumference. Many formulas have been given for the loop circumference of a quad loop - all of them wrong. L.B. Cebik, in the February 2000 issue of antenneX addressed and answered the question of what the loop size should be, depending on the frequency and the wire diameter. Handelsman has determined the formulas for the loop size of any rectangular antenna (see footnote).

To summarize the problem and the solution, the loop size of any 4-sided loop depends on two factors: the ratio of the wire diameter to the operating wavelength and the distance between the two radiating wires. The thicker the wire the greater the loop circumference at any frequency.

We begin with Figures 1 and 2 which show the quad and the rectangle. These antennas, like all of the ones we shall discuss, will be fed at the center of the bottom wire and will be horizontally polarized. The top wire also radiates, and the distance between the top and bottom radiators determines the phasing of the currents in the two radiators, and to a lesser extent the magnitude of the currents in the two radiators, and therefore the "array-gain" of the antenna. A square quad loop can be thought of as being equivalent to two dipoles, separated by 0.25 wavelengths and fed in phase.

Figure 1

Figure 2

In addition to the effect on the loop size, thicker wires have other benefits. For thicker wires there is smaller local stored magnetic energy, so they contribute less inductive reactance to the structure. So there is less change in reactance with change in frequency and their SWR bandwidth is greater. Because of the increase in size of the sides of the loop with wire thickness, the gain is greater as is the radiation resistance.

Table 1 summarizes the gain, feed-point resistance and SWR-2 bandwidth of three quad antennas designed for 146 MHz. They are different only in the wire sizes used: from 0.01-1 inches. You might ask why use 0.01 inch wire? The answer is that this is approximately the wire diameter to wavelength ratio of a loop on 20 meters (14 MHz) made of #10 AWG wire (approx. 0.1 inch - 2.5mm). This way you will be able to see how a 20-meter quad will play. The antennas were modeled with lossless wire to eliminate the widening of the SWR bandwidth due to resistive losses. Any practical antenna will therefore have wider bandwidth than the results of this model.

Rectangles:
If you stretch a square loop by increasing the distance between the radiators and then shrink the radiators to maintain the approximately 1 wavelength perimeter loop size, you can theoretically increase the gain of the antenna to nearly 6 dBi. Of course, this is impossible to attain since, when such an antenna becomes 0.5 wavelengths long and its radiators have shrunk to nearly point sources, its feed-point resistance approaches zero ohms and its currents approach infinity. Long before you get to this point the losses overcome the antenna gain. The rectangles modeled here, made of 1" tubing, have no appreciable losses to speak of - less than 0.03 dB.

We also see in Table 1 what happens to these rectangles as you stretch their height (inter-radiator distance) from 26-32 inches. First, their width - radiator size - narrows as their height increases. This is associated with an increase in gain, a decrease in their radiation resistance and a decrease in their SWR bandwidths. The 32-inch rectangle is a nice antenna on 2 meters - it has a gain of more than 1.2 dB over the quad loop and its feed-point impedance is a more easily matched 40 ohms. Its 9 MHz bandwidth is not a limiting factor. Taller and narrower loops, although having more gain, have much lower feed-point resistance and narrower bandwidth.

Figure 3 shows you the SWR curves of six antennas; three quads of different wire sizes and three rectangles made from 1-inch diameter tube, whose heights are varied from 28-32 inches. Thicker is better from the wire point of view and the more you depart from the square loop the narrower the bandwidth.

Figure 3

Two Element Parasitic Rectangular Arrays:
We have just seen what happens with single loops. Now let us see what happens when you arrange them in the typical quad array; a driven loop and reflector. The loop elements we shall use will be various squares and rectangles. A rectangle parasitic array is pictured in Figure 4.

Figure 4

You can see what happens to the gains of such arrays in Figure 5. First, look at the three antennas with the open symbols. These are quads (squares) composed of three different wires: 1 inch diameter aluminum, 0.1 inch copper and 0.0025 inch wire with its resistivity corrected to approximate #10 wire on 80 meters. The 0.1-inch copper was included since it forms the basis of many loop antennas on 2 meters, including Quagis which are loop-rod hybrids.

Figure 5

There is not much difference in gain between the 0.1 and 1 inch wires but there is a considerable difference in gain-bandwidth product. This might be taken as a performance measure. But the bandwidth of the 0.1-inch wire antenna is more than sufficient to cover the 2-meter band. If you look at the bandwidth of the antenna with 0.0025-inch wire you can see how an 80-meter quad array would perform - there is definitely a loss of gain and a narrower bandwidth.

It is important now to examine why the gain is so low with this thin-wire antenna. It has a lot of bearing on what we will be discussing later about quad arrays that are "well-behaved". In Figure 6 we see the gain curve of the extremely thin-wire (0.0025 inch) array. First, it is narrow-banded as we have already discussed. This is in large part due to the very small wire diameter/wavelength and the tremendous changes in reactance with movement off resonance in frequency. The problem will be even more acute with rectangular loops as they are made taller and narrower since their Q increases rapidly and results in narrower bandwidths.

Figure 6

The loss curve is very important. The losses should increase with the frequency as the antenna becomes longer in terms of wavelength. This is exactly what happens far below the design frequency of 146 MHz. As we approach 5 MHz below the design frequency the losses of the 2 element quad array begin to drop. Something is obviously happening in a two-element quad which lowers the losses as we approach and exceed the design frequency where the antenna gain begins to approach the lossless model. Further on, very far above the design frequency, the losses increase again as expected.

We are now going to get to the heart of this article - what is happening to an array of loop elements which results in beneficial changes.

The "Coupling" or mutual interaction between loop elements:
Across the entire spectrum of rectangle lengths, from square to extreme 36-inch (0.44 wavelength) tall loops, arrays using wires of 0.01 to 1" can attain extremely high f/b ratios. With careful adjustment of the loop dimensions of the driven and reflector elements and the loop spacing between them, f/b ratios can exceed 60 dB.

Although "careful adjustment" might smack of "trying to balance a pencil on its point", negating the objective of a well-behaved antenna, Handelsman found well-behaved antennas with front-to-back ratios over 40 dB.

However a general trend emerged. Figure 7 shows that, as the height of the rectangle elements was increased, the array element spacing needed to maximize the f/b ratio also increased.

Figure 7

The peak f/b is reached with loop arrays in the same way as with Yagis - it is a maximum at the point where radiator currents are equal, or nearly so. Only in this way can the radiated fields from the parasitic element cancel the fields from the driven element (in the backward direction). The increase in spacing lead to the conjecture that inter-element coupling increased with the lengthening of the loops.

Why does coupling increase as the rectangles become taller and narrower?

Even though coupling seemed to relate to height or the distance between the radiators, Jefferies, early on, thought that coupling between quad, and to a lesser extent rectangle elements, was due to electrostatic coupling between the CORNERS of the driven and parasitic elements. Handelsman's modeling showed the currents at the corners approached the feed-point currents more closely as the radiators became narrower and the loops taller. This we expect, as the current is maximum at the feed and falls off cosinusoidally with distance along the loop, to reach zero at a point a quarter of a wavelength away from the feed along the loop perimeter.

Two questions arose from this finding. First, why is this so? And, second, how do the corners result in the mutual interaction between array elements known as "coupling"?

The currents in two adjacent rods, for example in a Yagi array, are known to be oppositely directed. If we connect the ends of the rod elements to make a folded dipole or rectangular loop, the currents in the rods are co-directed. An experimental study and a description of why this is so is to be found on Jefferies' web site:
http://www.ee.surrey.ac.uk/Personal/D.Jefferies/dipimp.html

From this we conclude that the coupling between two unconnected rods, or unconnected loops, is by electrostatic fields that are strongest at the current minima, for it is at these points that the voltages are maximum. The coupling is not by magnetic fields, which, if they encircled both the driven and parasitic elements in the same sense, would set up co-directed currents. We have seen from the folded dipole example that the currents are opposed, and therefore the coupling is by positive charge on the driven element inducing negative charge on the parasitic element at the corresponding lateral point, and so the induced currents are in opposition. In the case of the wider spacing (in our antenna) between the driven and parasitic elements, an additional phase shift (delay) occurs between the currents due to the transit time electromagnetic radiation takes to get from one to the other at the speed of light, 30 cms per nanosecond.

In the case of coupled quad elements, this occurs at the corners. In the case of the coupled rectangles the coupling is by way of electrostatic fields from the non-radiating sides, the long sides in the case of our stretched rectangles, as well as the corners. Thus Jefferies conjectured that longer rectangles should couple more strongly, and that thickening the conductors at the corners would also help significantly. This would provide a greater capacitance between the driven and coupled elements and tie the charges on the elements to be more nearly equal and opposite.

"Good Behavior":
Jefferies also conjectured that, at the specific dimensions that lead to the highest f/b ratios, the mutual interaction between elements would be such that the array would have the greatest SWR bandwidth. Handelsman then compared three loop arrays using 26", 28" and 30" rectangles. Because of his new found desire to make the antennas "well-behaved", Handelsman decided to model arrays with element spacing increasing by 1 inch increments. As we mentioned above, playing with spacing and loop dimensions fractionally would always yield a high f/b ratio. But using only integer inch spacing resulted in f/b ratios of low-40s for the 26 and 30-inch antennas but a really nice almost-60 dB for the 28-inch.

Figure 8 is the result of looking at what happens to the SWR over a frequency range of 146 MHz (the design frequency at which the f/b peak was normalized for all antennas in this article) -8 to +26 MHz. Over a BW of 34 MHz (23%), all 3 arrays have SWRs of 2 or less. The 28" array, which fortuitously hit the highest f/b peak had the best SWR curve over that range.

Figure 8

Individual loops show a decrease in SWR bandwidth as they are made taller. To see this, look at Figure 3 again. Here we have an antenna - 28" - which should show a narrower SWR curve than the 26" but, instead improves upon it when in an array. The next question is why? But, first let us look at two more curves, Figures 9 and 10.

Handelsman took a 28" rectangle array and resonated and adjusted the elements to that the reactance was zero and the f/b peaked at exactly 146 MHz. Figure 9 is the SWR curve, with the now-familiar flattening above the design frequency. Figure 10 gives us a better idea of what is happening to the resistance and reactance above and below the design frequency.

Figure 9

Figure 10

Bandwidth and Losses - why is there an improvement?
What is it about the coupling of two elements that, at the point of highest f/b - and near-equality of currents - widens the SWR BW so much?

Well, the stored magnetic energy in the near field is a minimum at the point where the oppositely-directed currents are most nearly equal. Look at Figure 4 again at the arrows which indicate the current phase direction. The radiated power for the coupled loops is more than for the single loop as there is double the length of radiating current element. Thus the Q factor, defined as the ratio of stored energy to radiated energy per radian of oscillation, is smaller and minimizes at the peak f/b ratio for the array. The bandwidth is proportional to 1/Q, so we expect maximum bandwidth at maximum f/b ratio. The losses are also less for a structure consisting of coupled elements than they are for a single loop; a discussion of why this is so may be found at Jefferies' web site: http://www.ee.surrey.ac.uk/Personal/D.Jefferies/topics.html

The Well-behaved array:
We now focus on the 28-inch rectangle array and see why it behaves so well by looking at what happens to the gain, f/b and SWR as we change the element spacing. This is important because the findings in Figures 11-13 are generically applicable to all loop arrays. Here we take the spacing that yields the highest f/b ratio - 17 inches (0.21 wavelengths) and look at what happens when we increase it to 21" (0.26 wavelengths) and decrease it to 13 inches (0.16 wavelengths). These curves are important in that in a chapter in a well-known antenna book the author, speaking of 2-meter quads, says that the spacing doesn't make much difference.

Figure 11

Figure 12

Figure 13

Lets look at gain in Figure 11. In the range covered here, the closer the element spacing the higher the array gain. The f/b, Figure 12, as expected, shows a peak at a specific spacing. You also see the typical shift downward in the frequency of peak f/b when you narrow the spacing and upward as you widen it. Figure 13, showing the SWR pattern, indicates that the BW narrows with narrower spacing and vice-versa.

Figure 13 also shows us how well behaved this antenna is. Remember, in these curves we have been looking over a frequency range of 36 MHz or fractional bandwidth of 25 percent. The SWR change is paltry with changes in spacing from 0.16 to 0.26 wavelengths. The bandwidth far exceeds the 2-meter band coverage requirement. Additionally, and not shown here, changes in element dimensions may move the frequencies of peak gain and peak f/b around, and affect the feed-point impedance to some extent. The overall performance doesn't change very much. The bandwidth stays the same.

Lastly, at the prodding of Jefferies, Handelsman modeled various structures in the near-field region of the antenna. A tree was out of the reach of NEC, as were birds perching on the elements. A coating of ice was also out of the question. The best that could be done was in modeling with a 4 inch metal boom and various thick metal objects in the near and far-fields such as masts/towers. The antenna was rock-solid in all its parameters.

Jefferies has a saying for his students that "antenna designs can be made which are surprisingly uncritical. It is for this reason that such a plethora of functionally useful designs exist in the literature; almost anything approaching lambda/2 in dimensions can be made to radiate efficiently, and this holds for many larger designs as well."

The corollary of this observation is that many such antennas are difficult to "tweak" experimentally; quite large alterations in geometry and disposition may seem to have little measurable effect on the performance.

Summary:
The 28-inch rectangular loop 2 element parasitic array is an extremely well-behaved antenna. Although its individual loops have relatively narrow bandwidths, the parasitic combination shows an extremely wide bandwidth which is resistant to changes in the element spacing and to changes in dimensions (unless really gross).

The studies that went into creating this array have yielded practical design figures, and, more importantly, a theoretical insight into the behavior of parasitic loop arrays. We have shown that rectangular antennas, squares included, couple mainly via their corners.

Parasitic arrays of these elements have a) much wider bandwidths and b) lower losses due to:

1. Lower stored local magnetic energy, as the fields due to the oppositely-directed currents in driven element and parasitic element tend to cancel, and

2. For a given radiated power, the currents are lower as the total radiating length of rod is larger.

These issues are important because the simple rectangle loops are the basis for more complex arrays involving multi-rectangles of 2-6 attached loops - in single and multi-element parasitic and fed arrangements. An understanding of how loop arrays couple has practical consequences in the design of other loop antennas be they multi-element quads and quagis. -30-

Footnote:
The Rectangle Family of Antennas, Part 1 - The Simple Rectangle by Dan Hanselsman, is scheduled to be published in an upcoming issue in QEX/Communications Quarterly.

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