## Loops in Arrays

In the previous two articles in this series we have examined antennas which were based on variations of the simple full wavelength loop and how these antennas perform in free-space and over real ground. In this article we shall examine what happens when these loops are arrayed parasitically. We shall be looking at 2 element arrays in order to show how they function. This is important since I have found that the optimized 2 element -driver/reflector – “cell” is the best starting point for multi-element arrays of up to 5 elements.

To this end we shall begin, as we have before, with the basic square or quad loop. We shall then examine what variables go into array performance and then look into all of these when we examine the performance of the other loop variants.

In the end, a reader will be able to balance all of the conflicting factors to arrive at a configuration which is best suited for his needs. The main impetus for these studies is antenna gain—the more the better. Unfortunately, gain must be traded off for almost everything else that is important in antenna performance; namely three kinds of bandwidth (BW) related to gain, SWR and front-to-back ratio

#### Background

The quad was originally developed as a 2-element square array to be used as a substitute for a 3-element Yagi which was being literally consumed by corona discharge. This was high in the Andes in Quito, Ecuador at radio station HCJB. The original array has been called the “Cubical Quad” for obvious reasons. It was only later that arrays of more than 2 elements and single element loops were popularized.

Over the years many myths have arisen about the performance of quad arrays. The most popular one is that a quad with “n” elements performs equally to a Yagi with “n+1″ elements.

On the other hand, actual measurements by **Wayne Overbeck, N6NB**** ^{ 1}** in the 1970s determined that multi-element quad arrays fell considerably short of their expected gain and were inferior to Yagis from this point of view and also because of their complexity and vulnerability to the elements. What we know for certain about quad performance is that a 2 element quad will have a far higher front-to-back (f/b) ratio than a 2-element Yagi.

At the time of N6NB’s article, “science” had come to the Yagi design, beginning with the NBS studies, the work of the late **Jim Lawson, W2PV**** ^{ 2}** and others and NEC/MININEC

**computer modeling. The design of multi-element loop arrays has, until recently, languished in the Middle Ages. It is easy to get a quad array to work – it is difficult to get it to work optimally based on all the published dimensions.**

^{3}The major problem with 2-element quad arrays at the present time is that, while we can make the gain competitive with Yagis and get far higher f/b ratios than 2-el Yagis, we cannot sustain a decent f/b over a large BW. ** L.B. Cebik, W4RNL** has undertaken a systematic study to determine the optimum dimensions of the loops in quad arrays and the optimum spacing for highest gain and f/b ratio compatible with decent BW. Since most of the ills of the quad are related to the thin wires used in its construction, LB has developed a multi-wire substitute for the single wire in each loop – somewhat like the “cage” dipole. These studies will be published starting with the October 2000 issue of

**.**

*antenneX*My studies parallel his but in a different way – by comparing different kinds of basic loop elements. All of the antennas I shall be discussing are composed of 2 elements – one driven and the other a parasitic reflector. First off, I shall show you how well these arrays can perform when not hindered by thin wires. This I shall do with VHF/UHF arrays where we can use aluminum tubing elements. Then we shall see how the other kinds of elements alter the performance of the quad array – for better and for worse. The other array elements we will study are the simple rectangular loop and two variations on the SDR or symmetrical double rectangle; one composed of two squares and the other of two rectangles. **Figures 1- 4** show what the square/quad loop, the rectangle, the double-square symmetrical double rectangle (SDR) and the double-rectangle SDR look like.

#### Array Basics

**David Jefferies, G6GPR**** ^{ 4}** and I have examined the nature of the coupling

**between loop elements. Due to the greater area available for mutual interaction, loop elements couple better than simple rod or dipole elements. In the case of a 2-element loop array, we can easily find an optimum distance between the elements which leads to equality of currents and all the good things that derive from this state. These are extremely high f/b ratios, wide SWR BWs and decreased losses.**

^{ 5}Since the greater part of the coupling is via the non-radiating or transmission line wires, it stands to reason that rectangles with narrower radiators and longer inter-radiator transmission lines will couple better. As we shall see later on this has some detrimental effects on array gain.

The gain of a quad loop is due to the “stacking” gain of its two radiators which are separated by a nominal .25 wl. Jim Lawson, in analyzing this loop, calculated that due to the incomplete reinforcement based on the overlap of the vertical apertures of the two radiators the added gain over that of a dipole is about 1.1 dB. So, the predicted gain of a quad loop of infinitely thin wire, derived from field theory, is about 3.2 dBi. This is in concordance with the modeled data.

From the studies of Lawson and from NEC modeling we can also predict that the gain of a 1 wl perimeter rectangular loop will increase as we narrow the radiators and separate them by greater distance. The gain limit is about 6.1 dBi with the hypothetical lossless rectangle composed of two isotropic radiators separated by .5 wl. As we can see in the earlier article on the “Well-behaved Antenna” the practical limit on the rectangle’s gain is about 5 dBi.

The question is how will this increase in element gain be translated into performance when we create 2-element arrays with these loops? As we shall see further along, the major performance tradeoff as we increase the gain of the component rectangles is a loss of f/b bandwidth. The BW turns out to be directly related to radiator size. The narrower the radiators, the narrower the BW – and vice versa.

Now, if we were to use a double loop or SDR as the array element we can get additional gain if we are willing to pay the price of making the antenna twice as tall. For a single-element double square loop this translates into a gain of about 4.5 dBi and, for the “limit” rectangular SDR, the ceiling on the gain is about 7.2 dBi. The practical limit for SDR gain – from the point of view of BW and Rin – is about 6.2 dBi. We shall also look into how much of this increase in gain is recovered in array gain.

#### The Two-Element Quad Array

Let us begin by looking at antennas having relatively thick elements (0.5″ or 12.7 mm) of aluminum tubing and designed for a center frequency of 146 MHz – for both resonance and peak f/b. The minimum f/b that I will be targeting is 20 dB. **Table 1** shows what happens to gain, f/b ratio and BW when we narrow the element separation from that yielding the best f/b ratio (14″ or .17 wl).

**Table 2** shows the dimensions of these antennas. I am not showing what happens when we separate the elements even more since both the f/b and the gain decrease markedly. The point to be made here is that the gain is greater with narrower element spacing but the price to be paid for it is in both SWR and f/b BW.

Another point to be made with quad arrays and all of the other rectangular arrays we shall be studying is that the frequency of peak gain is well below the frequency of peak f/b. By the time we get to the range of f/b that we are targeting the gain is always on the way down. This is shown in **Figure 5**. We will always be comparing antenna gain in the targeted range of 144-148 MHz – not at the frequency of peak gain.

The most important point to note in array design with these elements is that the limiting factor in BW is the f/b ratio and not the SWR. The minimum f/b target will always be reached before the SWR becomes the limiting factor. And, this is even more so when we deal with HF antennas with thinner wires relative to wl. Nevertheless, until the f/b BW decreases to the design minimum limit you can “push” the gain by narrowing the element separation. Note that the ideal element separation is 14″ which is about .17 wl on 2 meters. The minimal separation, consistent with minimum f/b, of 12″ is about .15 wl. These numbers are far greater than those found in the common quad construction articles. **Even with these very thick elements (6.25E-03 wl) the widest possible BW consistent with a f/b ratio of >20 is 4%.**

Based on the “ideal” 2-element quad design, the reference standard for all of the other antennas being discussed is, over 144-148 MHz, a range in gain of about 7.5 – 7.1 dBi consistent with a f/b > 20 dB. Rin is not a limiting factor – it will always be over 50 ohms. The average gain from 144-146 MHz is 7.26 dBi.

**The Two-Element Rectangular Array**

The dimensions for a resonant single square loop composed of .5″ (12.7 mm) wire are 20.94″ per side. Let us now look at what happens when we change the loops into taller and narrower rectangles. As we increase the inter-radiator distance to 28” and 30″ the widths of the elements must narrow commensurately. The feed point resistance drops – but it is still high enough for an easy match to a feedline. These antennas were modeled only at one separation – that giving peak f/b and maximum f/b BW.

We can note the following: the gain increases appreciably, the SWR BW is still very wide but the f/b becomes the limiting factor. The 28″ rectangular array’s f/b at 144 MHz is slightly below our design criterion. The average gains for the 28” and 30″ rectangular arrays are 7.49 and 7.57 dBi respectively. Compare the gain patterns over 144-148 MHz of the quad and the two rectangles (marked square, R-28 and R-30) in **Figure 6**. The decline in gain is steeper the taller and narrower you make the rectangles.

The array composed of the tallest rectangle has a gain advantage (average) over that of a quad array of only .3 dB. Yet the gain of a 30″ rectangle (single loop) is 4.5 dBi and this is about 1.1 dB greater than the quad’s. The question that now arises is why is the gain increase of the individual rectangle over that of the square loop not recoverable when the loops are arrayed?

The answer is “coupling”. The element separations for the 28” and 30″ rectangles, at maximal f/b and current equality between elements, are 18” (.22 wl) and 20″ (.25 wl) respectively. As we discussed earlier, the mutual interaction between rectangular elements is directly related to the length of the transmission wires (vertical in this case) connecting the radiators. The taller they are the greater the coupling. Because of the narrow f/b BWs we must aim for ideal coupling and current equality and this happens at such a large inter-element separation that the gain falls down markedly. To put it another way, we could get much higher gains with narrower spacing but the f/b BW deteriorates rapidly. So the increased coupling of the taller antennas is a disadvantage due to the greater element separation needed at the point of maximum f/b. The result is a lower gain than one would expect simply from comparing the performance of individual loops.

Lastly, we can note that even the most extreme array, composed of the 30″ rectangles, has a SWR BW four times as wide as the f/b BW. The conclusion is again that the most limiting factor in array design is the f/b BW. SWR, gain and Rin are never issues.

**Conclusion:** there is little to be “gained” by altering the shape of a square loops into rectangles when one uses them in 2 element arrays.

**Double-Loop SDR Arrays**

As with all the other antennas we have discussed, these were composed of .5″ (12.7 mm) aluminum tubing. All of the SDRs were fed in the center of the center radiator.

#### Double Squares

**F/B ratio:**

Let us look at **Table 4**. Here we are modeling 2-element SDR arrays composed of square loops – or “double squares”. From our prior experience we can predict that the squares will have a greater BW than SDRs which are composed of narrower rectangles. **Table 4** shows what happens to f/b as we vary the element spacing from 14-20″. Maximum f/b is reached at an element separation of 20″. Note however, that we meet our f/b minimum criterion across the range of 144-148 MHz at all spacings, but the narrowest.

**Gain:**

We now examine what happens to the gain with spacing. We again see what we saw with the single loop arrays – the gain increases as we narrow the element spacing. **Table 4** summarizes the findings. We can get useable antennas from the point of view of gain and f/b ratio at spacings of 16-18″. The average gains over the desired range are 8.27 and 8.15 dBi respectively. The antenna with 16″ (40 cm) spacing shows a gain increase over that of the quad array of 1 dB.

We see the same phenomenon as with the simple rectangle arrays; the element spacing increases due to the coupling of the taller antennas and we cannot fully recover the gain increase of the single loop over that of the reference square loop. We would expect 1.5 dB but we only recover 1 dB.

**Bandwidth:**

Compare the SWR BWs in **Table 4** with those of the quad arrays’ in **Table 1**. The BWs are astonishing. We have more than doubled the BWs of the quads. There are clearly compensatory changes in R and X with frequency which serve to stabilize the overall feed point impedance. **Figure 7** shows the variation in SWR with frequency for an array with 18″ element spacing as we sweep from 130 – 200 MHz.

**Figure 8** illustrates the changes in R and X over the same range.

We can look at the changes in the feed point impedance over a narrower range – 137-163 MHz – in **Figure 9**. The Zin is incredibly stable and shows a “transformer effect” due to the second loop. What I mean by this term is that the reactance – over part of the range – varies such that it becomes more negative with higher frequency.

#### Double Rectangles

**F/B ratio:**

The SDR rectangles exhibit peak f/b at a certain element spacing and this is related to their overall heights. We again see the same phenomenon as we did with the simple rectangle arrays; the taller and narrower you make the component loops the narrower the f/b BW. This is shown in **Table 5** and **Figure 10**. By the time we get to a SDR element of 50″ – composed of two 25″ tall loops – the f/b barely meets our minimum criterion at 144 and 148 MHz. These are not “extreme” loops. The starting square loop in the antennas we discussed immediately above was just under 22″/side. **Table 5** does not show the spacing at peak f/b ratio only the minimum spacing compatible with maximizing our gain while still retaining our minimum f/b goals. The element spacings at peak f/b are greater by about 3″.

**Gain:****Table 5** and **Figure 11** relate the gains of SDR rectangles of overall heights of 44-50″. These are the maximum gains compatible with a f/b minimum of 20 dB. You can see that you get to a point of diminishing returns as you make the antennas taller since the gain of the tallest drops off the most precipitously over the modeled range. The best SDR from the gain point of view is one of 48″ (two 24″ loops). The average gain is 8.5 dBi or about 1.25 dB greater than that of a quad array. An individual double rectangle 48″ antenna has a gain of 5.21 dBi or almost 2 dB higher than that of the quad loop. Again, we only recover part of that gain increment.

**BW:**

The SWR BWs of the 2 element SDR rectangular arrays are nowhere as wide as those of the square arrays. But they are wide enough. **Table 5** shows the SWR at the 144 and 148 MHz limits for all of the antennas. They are all very low with the highest being only 1.17. I did a frequency sweep on the 48″ SDR and the SWR-2 BW was from 131-172 MHz or 41 MHz.

#### HF: 10 meters

The square looped SDR has the widest BW of all – as we have seen earlier. The question is how it would fare on the widest higher HF band which is 10 meters and which covers a band segment of about 6%. Let’s look at **Table 6**.

Please excuse me but I decided to make the dimensions more difficult for the Americans and easier for the rest of the world. Everything is metric. At the request of David Jefferies I will be “metric” in all future articles. (** Ed: Plus, there is always this handy conversion chart just a Click Away.**)

**The equivalent AWG wire diameter is #10. This corresponds to a diameter/wl ratio of 2.5E-04.**

I tuned this antenna for maximum f/b ratio to see how wide of a bandwidth it would cover. Even so, a f/b of >20 is maintained only over a span of 28.5 – 29.3 MHz (0.8 MHz). If we lower our f/b criterion to 15 dB the BW is still only from 28.3-29.6 MHz (1.3 MHz). The question now is whether we can do better.

#### ADR Arrays

The asymmetrical double rectangle is composed of two loops, one nominally 1 wl in perimeter and the other which is smaller. The introduction of such small secondary loops makes the antenna highly negatively reactive. If we tune it to resonance by widening the radiators we may effectively increase the f/b BW. We could possibly cover all of 10 meters. This will be the subject of the next article in this series. The final article will cover the multi-element loop arrays.

**Summary**

We can substitute other types of loops, namely the rectangle and the two variants of the SDR, for the simple square loops of the 2-element quad. The simple rectangle is not useful since it has little to offer from the gain point of view. The others will give an appreciable gain increase over that of the “cubical quad”.

The major problem which arises when these other loops are arrayed is that their effective area for mutual interaction or coupling is greater and that necessitates more of an element separation to attain the highest f/b ratios. This increase in spacing is detrimental to the gain so that the gain increase of the individual loops over that of the reference quad loop cannot be fully recovered. Nevertheless the gain is there.

The other, generic, problem with any loop array is that the f/b BW is limited by the size of its radiators. This holds for all the variants and is a problem, fortunately, only in covering one band – 10 meters. In the next article we shall pursue this 10-meter problem.

From the point of view of SWR or impedance BW, the double square array is a particularly attractive antenna on VHF and UHF where feedline losses enter the picture. Its feed point resistance is very high and the changes in this and in its reactance are very low. When matched, the SWR is virtually flat over a far greater range than where it would be limited by gain or f/b ratio. This holds true for the SDR rectangles as well but to a slightly lesser degree.

#### References

1- Overbeck, Wayne N6NB, Measuring Antenna Gain with Amateur Methods, Oct 1977 QST p.11.

2 – Lawson, James, Yagi Antenna Design, Chapter 4, ARRL, 1986.

3 – The program used for antenna modeling is NecWin Plus by Nittany Scientific.

4 – Jefferies, David and Handelsman, Dan, The Well-behaved Antenna, Antennex April 2000

5 – The program that I used for calculating mutual impedances is Grant Bingeman’s RF Designer.

Sorry but this is the only document that i could find – MD0MDI

Originally posted on the AntennaX Online Magazine by Dan Handelsman, N2DT

*Last Updated : 26th May 2024*