**Multi-loop Rectangles
and Compact Arrays**

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)**.

The Array Factor of a
rectangular array of “n” loops is “n+1" - which is the number of
radiating elements.

**The Double Loop or SDR**

This antenna has been known since the 1950s and is composed of two equal-sized rectangles.
It was first called the “skeleton slot”^{[i]}. Later, when described
as a vertically polarized antenna for 80 meters, it was named the DMS or “double
magnetic slot”^{[ii]}.
A 30-meter incarnation was called the “H-Double Bay”^{[iii]}.
In an earlier article on the double loops, Handelsman, at the advice of L.B. Cebik, W4RNL,
decided to call the entire genre by a more basic descriptive term; the symmetrical double
rectangle or SDR^{[iv]}.
This antenna is pictured in **Figure 3**. The gain of a lossless SDR of close
to 1 wl overall height - composed of two almost 0.5 wl height rectangles - is over 7 dBi.

**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.

The first array in Table 2
has a slightly lower gain but has a significantly higher f/b ratio and feed-point
impedance. Within the range of heights of these two arrays are a host of others which may
be optimized for whatever parameter one wants; gain, f/b ratio and feed-point impedance.
The other dimensions that might be optimized are the spacing of the parasitic elements and
the separation of the collinear arrays. The separations for the two arrays discussed here
are good for maximum gain but too wide from the point of view of the cleanliness of the
azimuth pattern. The sidelobes can be cleaned up, at a slight sacrifice in gain, by
narrowing the collinear separation.

**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}**

In the case of the 1.48 by
0.45 by 0.69 wavelength antenna, optimised for f/b and having an acceptable input
impedance, this numerical gain is estimated to be 35 or 15.4 dBi which compares to the
15.9 dBi of the simulation. Sticking our necks out, if we had an approximately cubical
antenna about 2 wavelengths on a side we might expect a maximum gain of about 22 dBi.

Yagis of the equivalent gain
(16 dBi) have tremendously long booms. This Yagi would have to have 16-17 elements and a
boom length of approximately 4 wl. On 146 MHz this translates into 27' (8.2m). In
comparison the antenna we have discussed is “compact”.

**Conclusion**

The authors would like to bring this new antenna design to the attention of experimenters.
It packs a lot of gain into a small volume and should be useable anywhere in the VHF/UHF
spectrum. N2DT is going to try a single loop version of the 4x on 10 meters when the
weather permits.

The two described here are
merely starting points for experimentation. More refined designs should be even better,
with higher gains, f/b ratios or both. **-30-**

Dan Handelsman, N2DTI was first licensed as WA2BCG in 1957 and have been N2DT since 1977. I am a DX'er and contester but have been inactive since I reached the top of the Honor Roll in 1990. I took up the challenge again last year and began to play QRP. I've always been interested in antennas and, with the help of LB Cebik. W4RNL, got into antenna designs based on the rectangle. My profession is that of a Pediatric Endocrinologist and I hold an M.D. and a J.D. degree. I am presently a Clinical Professor of Pediatrics at the New York Medical College and also consult in litigation as a medical expert witness. It is clear that, with respect to antennas, I am an "amateur" in the true sense of the word. |

David Jefferies PhD CEng CPhysDepartment of Electrical Engineering University of Surrey Guildford GU2 5XH UK Click here for David Jefferies' Biography |

**NOTES:**

[i]. Peter Dodd, G3LDO, The HF Skeleton Slot Antenna, The
ARRL Antenna Compendium, Vol. 6, p. 70, ARRL, 1999. This is a multi-band 10-30 meter
version of a SDR. According to Dodd, the antenna was first published in: B. Sykes, G2HCG,
Skeleton Slot Aerials, RSGB Bulletin, January 1953.

[ii]. Lew Gordon, K4VX, “The Double Magnetic Slot
Antenna For 80 Meters”, ARRL Antenna Compendium, Vol. 4, p. 18, ARRL, 1995.

[iii]. Paul Carr, N4PC, “The H-Double Bay Antenna”,
p. 28, CQ, September, 1995. This is an SDR for 30 meters which is horizontally polarized
and fed at the lower end-wire.

[iv]. Dan Handelsman, N2DT, The Double Rectangle, Three
variations on a rectangular theme, Communications Quarterly, Spring 1999, p. 67.

[v]. The antennas described in Figure 4 are 0.3 - 0.45 wl in
height or inter-radiator spacing. These were fed at an end-wire but the gains there and at
the center are similar.

[vi]. See reference 1
above. In Figure 2, Dodd shows a picture of a TV antenna marketed in England in the 50's
and 60's. This was the SDR or skeleton slot with two bays of parasitic elements,
consisting of a reflector and a group of directors, each at the level of the two radiating
elements.

[vii] The ARRL Antenna Book, Figure 29, p. 18-19, 18^{th}
Edition, ARRL Newington Ct., 1997

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