Multi-loop Rectangles
and Compact Arrays
By Dan Handelsman, N2DT
and David J. Jefferies

Introduction
e propose to examine a VHF array that packs a lot of gain into a small volume. In order to do so, we shall begin with the simplest element of the array—a unique antenna which is composed of four rectangular loops. We shall then assemble various arrangements of these elements and show how they work and what can be achieved with them.

Background
One of the co-authors, Dan Handelsman, N2DT began to investigate a class of antennas which is based on the simple rectangle. It has been long known in the amateur radio literature that you could attach two simple rectangles together to create a double-loop antenna. The question then arose as to what happens when you attach even more rectangular loops to each other.

Idle modeling was followed by the construction of a 2-meter quadruple loop which will be the basis for the antennas in this study. The antenna, shown in Figure 1, is composed of four rectangles. But, before studying this antenna and its use as an array component, we should start at the beginning and see how it came about.

The Rectangle
The rectangle is simply one manifestation of the one-wavelength-circumference (1 wl)  loop. A more commonly known type of such a loop is the quad loop which is simply a square. These loops can be fed in the center of either a vertical or a horizontal side - leading to vertical or horizontal polarization. For our purposes we shall only discuss antennas that will be horizontally polarized.

At one extreme, if you stretch a loop so that its radiators are long and the distance between them is small you get the folded dipole. If, on the other hand you take the loop and stretch it so that the non-radiating wires are longer than the radiators, you end up with the rectangle. The other extreme of such an antenna is the “limit” rectangle—an antenna that approaches 0.5 wl per long side and whose small (“point-source”) radiators approach zero length. It is essentially a 0.5 wl transmission line which is shorted at both ends. The loop circumference is exactly 1 wl, while its feedpoint impedance approaches zero and the currents approach infinity.

As a convention, to eliminate any confusion about dimensions, we will refer to the radiator length as the “width” of the antenna and the inter-radiator distance as the “height”. To maintain the approximate 1 wl+ of loop circumference at resonance, you must decrease the width as you increase the height.

You might ask: “Why bother with the rectangle?” Well, the quad loop has a gain in free space of 3.1 dBi. If you stretch it to near the limit size, a lossless rectangle has a gain that approaches 6 dBi. This almost doubles the gain of the quad loop. However, two problems arise with constructing such a near-limit rectangle. The first is that the radiation resistance decreases as the radiators shrink and this results in radiator currents and losses which increase exponentially. Beyond a certain height or distance between the radiators the losses more than offset the gains—a phenomenon best seen on the low HF bands when one uses relatively thin wires. The second is the radiation resistance itself—at some point the antenna becomes impossible to feed. In our case, since we will be modeling antennas at 146 MHz with 1" 6061-T6 aluminum tubing, the losses do not enter into the picture. We are limited only by the feed-point resistance as the radiators get shorter. Nevertheless, useable rectangles can be constructed with a gain of over 5 dBi.

How Does the Rectangle Work?
The rectangle can be considered as an array of two dipoles, separated by the rectangle’s inter-radiator distance or “height” in our terminology, fed in phase and having broadside gain. Figure 2, which compares the gains of lossless rectangles and of their analogous two-dipole arrays, shows us that the gains are virtually identical up to the limit of 0.5 wl of radiator separation. Figure 2 is drawn from an earlier article on 80-meter rectangles but the data is valid in our case.

Figure 2

All arrays based on the simple rectangle can be thought of as such stacks of dipoles. The double loop, which we will discuss next, is analogous to three stacked dipoles. The general approximation for the gain of an n-multi-loop array is

Gain (numerical) = gain of dipole (1.65) times number of radiators (n+1)

or

Gain (dBi) = 2.2 + 10log(Array Factor).

More Loops
Out of curiousity, Handelsman then decided to model multi-rectangles with three, four, five, and six loops. The gains of antennas of 1 - 6 loops are shown in Figure 4^[v]. As expected from our discussion of the rectangle earlier, the longer loops (the ones with the greatest inter-radiator separation) have the higher gain. It seems that there is no limit to the number of contiguous loops, except of course the diminishing returns in the gain as you tack more of them on.

Figure 4

After an extensive search of the literature, no previous designs based on such variants of the rectangle could be unearthed. Various “grid arrays” have been described, some of which are commercial products used in microwave/radar, but all are composed of much larger loops of 3 wl circumference and in different configurations.

The rectangular array chosen for all further discussion in this article is the 4x or quadruple rectangle loop antenna with five radiating rod sections. It could also be called the SQR or symmetrical quadruple rectangle but 4x is a suitable abbreviation. The reason for limiting ourselves to 4 loops is strictly practical—they exhaust the modeling capacity of the version of NEC-2 used and exhaust the patience of the user because of the long run times. There is no reason why arrays based on a 6x or larger rectangle can’t be used. All the driven elements of the arrays discussed here have the feed point at the center of the center wire - wire number 3 of 5 radiators.

The Big Arrays
Co-author Jefferies decided to tease Handelsman, the other co-author, by asking him to model a “three dimensional array”. Apparently this has not been done before and involves a 3-D lattice array of elements that have more “internal” radiators than “surface” radiators. This goal has not been attained here. On the other hand, what has been found is an array that takes up little cubic space, packs a lot of gain and has an excellent front-to-back (f/b) ratio.

We shall attempt to outline the thought processes that went into two versions of the “monster” we have created. One class of antennas was chosen to show what might be done with the f/b ratio and another was chosen to maximize gain.

Table 1 - dimensions (inches) Multiply by 2.54 for CM

Table 2 - operating parameters of various 4x loop configurations

Here, the gain column units are dBi, the f/b column units are dB, the Rin and Xin column units are ohms, and the numerical factor column is the total number of radiators in the array. For the basic array element, a 4x loop, this is n+1 or 5 where n is the number of loops.

Table 1 gives you the dimensions of the arrays and their individual elements. Table 2 shows the evolution of two sets of arrays based on the above criteria. The calculated gain is derived from the second of the formulas in the text. Please understand that only cursory attempts were made to optimize the gains and f/b ratios of these antennas. These are impressive already but can certainly be improved on.

One 4x loop (driven) plus one 4x loop (reflector) parasitic array.

Handelsman had noted that rectangles want to couple. In fact, the worst antenna from the point of view of parasitic coupling is the square quad loop. Two element quads have f/b ratios that are never higher than the mid-20s in dB. Over a specific range of inter-radiator distances - a topic for another article - parasitic rectangles can achieve the same f/b ratios as phased arrays, in the 50+ dB range.

The 4x loop, which is the array element of all of the antennas here, was chosen because it yielded good f/b ratios in a parasitic reflector/driven configuration. This element was combined in various parasitic/collinear configurations to show what can be done with them. The results are in Table 2.

One might ask; “why not use a dipole reflector a la a Yagi?” Jefferies suggested that in order to couple dipole parasitic elements to a loop that has “n” radiators, one has to use a rack of “n” rods, one behind each of the loop radiators. Wayne Overbeck, N6NB, the inventor of the “Quagi” in 1976 - an antenna with a loop reflector and driven element and rod directors - mentioned that he chose the loop reflector because it gave higher gain than a rod reflector. In fact, he found that a trigonal reflector was best but too complicated to construct. This may have been an early clue as to the need for bays of stacked parasitics^[vi].

The “Three Element Parasitic Loop 4X Array”: one driven 4x loop, one reflector 4x loop and one director 4x loop.

Handelsman, in modeling the above antenna using a 5-rod bay for a director, found only a minuscule gain increase of about 0.5 dB. Also the f/b ratio deteriorated significantly. Attempts were then made to use a smaller 4x loop, tuned to a higher frequency, as a director.

There are two possible ways to tune such a director. One way is to shorten all the wires proportionately and this yields an antenna that is both lower and narrower. The results of this were quite poor from both the gain and f/b points of view. The other way, suggested by the studies of Jefferies, was to maintain the overall heights of the individual rectangles and to shorten only the radiators. Jefferies thought that the coupling of rectangular loops is via the longer non-radiating or phasing wires. This configuration then yielded the best 3 element parasitic 4x arrays. Again, for the dimensions you can review Table 1 and for the results see Table 2.

Now, Handelsman, tantalized by the prospects of a “3-D” array decided to stack two of these three element 4x loop parasitic arrays to form a collinear array having two identical elements. So two such elements, each a 3 element “parasitic array”, were modeled side-by-side, fed in phase and separated to the optimum spacing to yield maximum gain. This resulted in the antenna pictured in Figure 5. This is the end product, a 2x3 array composed of two 3-element parasitic arrays fed collinearly. The two feeds each see an impedance with the resistive part close to 100 ohms (for one of the designs); if the elements are fed in parallel there is a natural match to a 50 ohm coaxial cable.

Six Element Parasitic/collinear Array of 4x Loops
The approximation for the expected gain of such an array, using the gain formulas discussed earlier in the text, is about 16.9 dBi. This is based on 30 elements - six arrays of 5 radiators each. The second antenna has gain of 16.6 dBi (see Table 2) and so approaches this figure. This antenna was not optimized extensively but it isn’t bad for a start.

Why This Antenna
The first antenna, which is best for f/b and impedance, measures 1.48 x .45 x .69 wl and occupies a volume of .46 cubic wavelengths.  The second, with the higher gain, measures 1.74 x .45 x .84 wl or .66 cubic wavelengths.  These antennas are compact since their long dimension is quite small. There is a suggestion that for a cubical antenna structure such as this, the maximum effective area Ae of the antenna is approximately the square of the maximum linear dimension, and that the gain (numerical) is then given by:

Gain = (4 pi Ae)/lambda ²