An under-appreciated property of arrays of many types is the fact that double (and more complex) thin-wire elements can serve as substitutes for impractical fat elements. As we reduce the diameter of an element, the mutual coupling between elements within arrays--both phased and parasitic--decreases, with a consequent decrease in array gain and an offset in frequency of other array properties. Much, but not all, of an array's properties can be fully restored by the substitution for the impractical fat element and the single thin wire element of a double-wire element. For most standard arrays, only the gain will suffer. The double-wire element will restore a good portion of the gain. However, the higher losses of the double wire elements relative to the original fat element will limit the degree of restoration. The larger the number of 1/2 wavelength elements or their equivalents, the lower the percentage of gain restoration. Nonetheless, the use of double-wire elements to preserve such operating parameters as the pattern shape, the front-to-back ratio, and the feedpoint impedance often suffices to make the use of double-wire elements preferable to single thin-wire elements.

The key question for these notes is how effectively to model double-wire elements so that we meet two criteria:

- 1. The substitute element is an effective substitute for the original fat element; and
- 2. The substitute double-wire element is adequately modeled to assure accurate modeling results.

The first step in the process is sketched in **Fig. 1**. We take one representative
element from the original array and find its self-resonant frequency. Then we
construct a double-thin-wire element model of the same length and place it at the
same frequency. We next vary the spacing between the wires each side of a center
line until the new element is resonant. Resonance technically means having a
purely resistive source impedance with no reactance. There are no task-
independent standards for what counts as resonance, but my experience suggests
that resonating an antenna within +/-j1 Ohm of reactance is not a difficult task.

To be exact, we should perform the same exercise with each element within any array. However, unless there are oddities to the array, modeling a single representative element normally suffices to provide a usable uniform wire spacing for double-wire elements throughout the substitute array.

Critical to the first step and to rebuilding the subject array with double-wire
elements is figuring out how to model the substitute elements. **Fig. 2** provides
one useful technique.

Assuming a driven element, we need a source wire. For most modeling, the segment with the source should be the same length as the immediately adjacent segments. Hence, the first task is to model a source wire (designated W6 in the sketch) at the element center (assuming for simplicity a symmetrical set of elements in the array). How long we should make the wire and its segments is a function of the wires labeled W4, W5, W7, and W8. The length of each of these wires is one-half the spacing of the double wires in the ultimate substitute element. Hence, the length of the source wire should be close to 1.5 times the spacing of the wires. Arriving at the final number will, of course, require some trial-and-revision modeling in pursuit of the double wire spacing figure in the first step in the process.

The end wires (W1 and W11) ideally should be composed of 2 segments each. Equally ideal would be the case of keeping all of the segments in the model the very same length. This practice is the most accurate, but can result in large models even of single elements. The minimum requirement is that the segmentation for each of the 4 long wires (W2, W3, W9, and W10) should be identical to ensure that the segment junctions in parallel wires align with each other. Otherwise, NEC may show some inaccuracies. The level of segmentation along these wires can be determined by experimenting with levels of segmentation on a single element. As the segmentation is reduced from the ideal level, the element reactance will increase. The user must determine at what level the reactance is too high for us to use the element as modeled as a substitute for the original fat-wire element. A few Ohms reactance relative to a resistive value of 70 Ohms is normally acceptable for most design exercises. However, the lower the resistive source impedance values encountered in the array, the higher the need to use more adequate segmentation on the substitute elements.

These practices are sufficient for modeling double wire substitute elements for common parasitic arrays, such as Yagis. A 3-element 80-meter Yagi modeled with 4" diameter elements will show considerably more gain than a version made from #12 or #14 AWG wire, even when both are optimized for their wire sizes. Most of the gain--within a few tenths of a dB--can be restored using double-wire equivalents for the 4" elements.

**A More Complex Case: the Quad Loop**

Quad beams show relatively narrow operating bandwidth with respect to some parameters, largely because we conventionally construct them from thin wire elements. For most installations, the use of fatter aluminum tubing is impractical. Therefore, the double-wire substitute for a better-performing fat-element version of the quad becomes a desirable alternative.

**Fig. 3** shows two ways of creating double-wire elements for quads. On the left
is the Tee assembly, which places one wire ahead of the original element position
and one wire behind the original element position, using a cross or Tee bar
attached to the normal quad arm support to hold the wires at a constant distance.
Note the use of a shorting wire between the elements at each corner. An
alternative to the Tee assembly is the planar arrangement of loops in a double-
wire substitute element. The planar assembly places both loops at the same
distance from other elements in the array, but one loop must be larger than the
other. A good starting point in developing such a loop is to place each loop the
same amount larger and smaller than the original element. Extensive modeling
with each type of double wire element has shown that in every normal parasitic
array tested, there is no performance difference between the two double-loop
arrangements so long as the wire spacing is the same within each element.

How we then handle the double-wire elements follows the same procedure as for
linear elements. See **Fig. 4** for the outline of a 2-element quad beam using the
Tee arrangement. Of course, the sketch does not show the support structure,
which we shall assume is invisible to RF.

Especially notable in **Fig. 4** is the feedpoint arrangement, which follows the same
rules as for the linear elements we examined. The source is on a 3-segment wire
at the center of the lower portion of the driven element. The double wire
arrangement begins and ends as did the linear element. The only difference is
that the elements do not have ends, but form loops. The corner shorting wires
are necessary to ensure that each loop in an element has virtually the same
current at each corresponding point of each wire. In practice, adding further
shorting wires at mid-side points would likely be good building practice.

The planar loop structure appears in **Fig. 5**, a version of the very same quad
using the alternative form of loop construction. For many installations, planar
loops would be simpler to construct, since they require no addition attachments
to the support arms except as necessary to hold the loop corners in place.
Likewise, the source treatment for the planar loop driver is the same as for the
Tee-assembly. With the planar loop model, it is important to use a level of
segmentation that is dictated by the wire spacing in order to keep the segment
junctions well aligned. Note at the corners that the outer loop has exactly 2
more segments than the inner loop, which is a function of setting the segment
length equal to half the wire spacing. That value is initially directed by the
length of the wires from the source wire to each of the longer loop wires.

Although linear double-wire elements are quite straightforward to model, loop structures can be sufficiently complex to make it difficult for the modeler to keep his place. Hence, utilizing every available modeling aid, including a good plan on paper before model construction, is always sound advice. However, access to a modeling-by-equation facility can go a long ways toward making the process very easy.

**Fig. 6** shows the equations page of a NEC-Win Plus model of a 2-element quad. In
this simplified version, each of the loop half-sides is defined by a simple
equation referenced to the length of a wave at the design frequency (variables
A and B). D defines the driver-reflector spacing. H is the user input wire
size, corresponding here to #14 AWG wire. We can change the design for other
wire sizes by entering the new size and changing the constants in the equations
for A, B, and D. We can also change the design frequency in variable G.

The essential loop-creation variable is E, which specifies half the distance between #14 wires. The total spacing will be 5"--the selected substitute for the original 0.5" elements upon which the model is based. The rest of the task is simply to set up the wires for the model to make use of these variables.

**Fig. 7** shows the wire page in symbolic form for the 2-element quad. Wires 1-19
represent the driven element, with the source wire shown in the top place in the
listing. Wires 20-33 list the reflector loop, which lacks the need for a source
wire and hence uses fewer entries on the wires page. Out of view below the
bottom of the figure are the last 4 wires (30-33) which form the corner
connectors for the reflector double loop.

The length of the source wire is defined in numerical terms so that it is 1.5 times the spacing between wires. Otherwise, the entire structure is set up in terms of variables. The inner and outer loops of each element are set by using the baseline dimensional variable (A or B) and adding to it or subtracting from it half the spacing distance, as represented by variable E. Although the wire page may look complex to newer modelers, consider the ease of introducing errors--if only by transposing digits here and there--should every wire spreadsheet cell need a numeric entry. For example, in the present model, the values of A and B are 51.90357 and 57.3246, which are in fact not used directly on the wire page. Instead, for each entry, we add 2.5 or subtract 2.5 to obtain each of the loop corners.

Still, there must be a somewhat easier way to model double-wire elements to arrive at models with fewer segments and even fewer wire entries.

**Some Simplifications and Cautions**

We can significantly reduce the level of segmentation if we can do away with the
source wire as a separate single wire. There are two ways to accomplish this,
as shown in **Fig. 8**.

Below the "standard" treatment of a double-wire assembly, we see an element having two wires and two sources. This wire set might be the center of a linear element or the feedpoint area of a quad loop. By using two sources, we not only eliminate the separate source wire assembly, but as well, we increase the ideal segment length to the actual spacing between the two wires. Once more, we might judiciously reduce segmentation further by sampling a single element as a means of discovering how using few and long segments affects the self-resonant impedance of the element. Once more, the limits of allowable variation depend on the task at hand and are a user-responsibility. For loops, corner shorting wires are required to ensure similar current patterns on the two wires.

Calculating the actual source impedance from this virtual parallel feed system requires only a bit of hand-calculator work. Suppose that a quad loop returned values of 165 + j2 Ohms for Source 1 and 138 - j3 Ohms for Source 2. We need not do any fancy vector work to arrive at the final single source value. Instead, use the hand calculator to add the inverses of the two resistances and then take the inverse of the result (75.1). Do the same for the reactance values, taking into account their signs (+j6).

Perhaps the only thing that the hand calculation robs us of is using program facilities to determine the SWR and to plot such values over a sweep of frequencies. Only if we can reduce the parallel impedance to a single value within the program can we use these conveniences.

The lower portion of **Fig. 8** shows one technique that works with good accuracy.
We select one of the two wires in the driven double-wire element to be the source
wire. From the source segment to the corresponding segment on the other wire
within the element, we create a transmission line using the TL facility within
NEC. Since the TL line is strictly mathematical, we may choose for it any value
of characteristic impedance and any length. The characteristic impedance should
be close to the median resistive value between those that would appear on each
of the two lines in a parallel source model. Using the figures that we just
examined, a characteristic impedance of 150 Ohms would be quite reasonable.
Precision is not too critical, since we shall make the line almost impossibly
short. Any transmission line effects an impedance transformation continuously
down its length. Thus, an extremely short line is needed so that the impedance
placed in parallel with the source is as close as possible to that occurring on
the second wire of the pair. I have used lines as short as 0.001' with success,
although that practice may be a bit fussy. The sample problem return a source
impedance of 75.2 + j5.6 Ohms, which is certainly as close as one needs to the
calculated parallel values.

There are some cautions that must be observed if the TL substitute for a double source is to provide reasonable results--not only in terms of the source impedance, but as well, in terms of the reported far-field pattern. The orientation of the TL line--that is, whether it is "Normal" or "Reversed"--will be a function of how we construct the double wire elements. For example, the portions of the planar quad loops that we connected with a TL line for a single feed both proceeded from -X to +X. Hence, the current direction was the same for both wires. Therefore, we employed the TL as Normal.

One convention for constructing continuous loop double wire linear elements is to go around the horn, that is to let the bottom wire move from -X to +X, then to create the end wire, next to make the other long wire move from +X to -X, and finally to close the loop with the other end wire. In this situation, the current direction on the two wires is in opposition. For a single wire element or even such a wire within a parasitic array, there are no negative consequences for the far-field pattern. However, if we had applied the TL line to achieve a simulated parallel source, we would need to use the "Reverse" option, which in fact places a half twist on the line and reorients the sources in parallel.

Observing this requirements becomes especially important if we choose to model certain types of driven arrays using double-wire elements as substitutes for single fat elements. The LPDA makes a fine example. It consists of a sequence of linear elements, each of which is connected to the next, both fore and aft, by a phasing line. The line is reversed between each pair of elements.

**Fig. 9** sketches the forward-most 2 elements of an LPDA array. Assume for the
moment that we model each substitute double-wire element as a continuous loop,
so that the direction of modeling is reversed for each long wire of each element
loop. In order to capture the action of the LPDA, we must parallel the two wires
into a virtual single feed point for the phasing line and then we must have a
phase reversal between the first element and the second.

Using the modeling convention chosen, each element contains within it a phase reversal relative to a parallel feed. Hence, the very short TL we create to connect the two wires within an element must be reversed.

We shall connect the "rear" wire of one element to the "forward" wire of the next element. How do we effect the required phase reversal from one element to the next? We do so by making use of the fact that the connected wires already have a reversed phase relative to each other. Hence, we use a phase-line section in its normal mode.

It is possible to use an alternative convention in creating the double-wire elements. We can let each wire in the element pair have the same modeling direction, say, from -X to +X. In this case, we would use a normal very short phase line between wires within an element and a reversed phase line to connect one element to the next.

Either system will return the same results in terms of array source impedance and far-field pattern values. With careful model construction, both are capable of very useful and accurate results. However, mixing systems tends to yield a bewildering array of meaningless results.

Properly and carefully used, the techniques we have explored can allow the modeler to create full models that once seemed too complex to tackle. Often, models use simplified fat-element models for double-wire elements and simply presume that they are "accurate enough." That presumption is wholly unnecessary, since there are a host of techniques, not only to fully model double-wire elements, but as well to do so in reasonably compact models. Therefore, even those having segment-limited NEC modeling programs should be able to handle most of the cases that arise. Moreover, the results obtained--when compared to both fat-wire and single thin-wire models--can be edifying. In fact, they can go a long way toward helping to make design and construction decisions.