In the preceding episode, we examined the modeling of series feedline connections at the source of a model. We looked at a few examples of arrays that used both 2-line and 3-line combinations to familiarize ourselves with both the modeling techniques that we need and with the differences between parallel and series connections of feedlines.

The arrays that we examined had something in
common, regardless of the connection. Each individual transmission-line
termination had the same impedance. Therefore, we were able to use a
simplified set of calculations, outlined in **Fig. 1**. For a
parallel connection of individual termination impedances at each
feedline, the net impedance was 1/N (Z), where N is the number of lines
connected and Z is the impedance of the individual connection. Because
we used a series representation of the impedance (R +/- jX Ohms), we
could arrive at the net impedance by handling the resistance and
reactance values individually.

The series connection presented us with no more difficult an arithmetic task than the parallel connection. For three identical impedance values in series, the net impedance is the sum of the individual impedances, that is, N (Z). With the impedance values shown as R +/- jX Ohms, we could simply multiply R and X by the number (N) of lines being joined in serial fashion.

The two forms in **Fig. 1** contain a
reminder that series and parallel connections doe have a difference. A
parallel connection of feedlines, relative to the source, shows a
constant voltage across each line but divides the total source current
equally among the individual lines. In contrast, a series connection of
feedlines shows the same current at each line, but there is a voltage
drop across each line equal to the source voltage divided by the number
of equal loads presented by the lines at the junction. These notes, of
course, assume a lossless situation, which is consistent with the
lossless lines created by the TL facility within NEC. We also assume,
in accord with NEC, that the losses associated with the structure and
loading of the elements within the array are equal and therefore do not
disturb the basic calculations that rest on the impedance values that
appear at the source-end of each transmission line.

With respect to modeling parallel and series
connections, we do encounter a difference, as suggested by **Fig. 2**.
On the left is the very simple scheme required within NEC for modeling
a parallel set of transmission lines in conjunction with a source. We
need a single 1-segment very short, very thin wire--normally at a
remote location relative to the radiating elements of the model--in
order to join the lines and the source in parallel. The wire can be as
short as about 0.001-wavelength. 1-mm (about AWG #20) is a good
diameter, and if the program allows it, the wire can be lossless. Some
programs, such as EZNEC, allow the specification of a virtual wire that
automatically meets these criteria and does not appear in the graphic
view is the antenna.

Creating a series connection among the
transmission lines and the source requires a more complex structure, as
suggested by the remaining two outlines in **Fig. 2**. For each
transmission line and for the source, we require a separate very thin,
very short wire. These wires connect in series forming a complete
circuit. Hence, two lines plus a source requires a triangle, while 3
lines plus a source requires a square. The construct creates the
required series connection among the element. However, the structure
has a finite dimension that forms a loop. The consequences are twofold.
First, we need to check the average gain test (AGT) score to determine
that the construct has not significantly changed the AGT value relative
to the value obtained from a parallel connection. With very thin and
very short wires in the construction, the AGT value at HF will normally
change by no more than 0.001, an acceptable value under virtually all
circumstances. Second, the loop formed by the construct will often add
inductive reactance to the net impedance as related to the simple sum
of the individual impedance values without the construct. The amount is
normally small and should not be surprising. However, it may require
noting relative to any physical implementation of the model under
development.

The models that we have so far examined presented equal loads to the junction of lines with the source. At the end of the preceding episode, I indicated that there are many situations in which the lines will not present equal loads. A question arises about how, under these circumstances, we may move from a parallel to a series connection. The modeling technique will remain the same. We shall switch from a single-wire junction to a more complex series construct of wires. However, to understand what is occurring, we may need to look at modeling practices and at the analysis of behavior of voltage and current under parallel and series connections. In this episode, we shall examine a single example using both types of connections. Our goal is to understand how we can model both ways to obtain a reasonably full analysis and understanding of how and why the two arrays differ.

**A 2-Element Horizontal Phased Array with
Parallel Phaseline Connections with the Source**

Let's consider a 2-element phased array for 28.5 MHz. The design that we shall use emerges from an old design that I once developed to convert a driver-reflector Yagi of good performance of its type to better performance as a phased array, all without altering the elements. The driver and reflector used relatively thin stepped diameter elements so that the whole antenna broke into parts that stored within a PVC boom. The inner section consisted of 0.25" diameter rod, and the total length of the inner sections was 108" (54" each side of the centerline). The outer sections consisted of 0.1875" (3/16") diameter rod. The total length of the driver was 198" (45" per section), while the total length of the reflector was 211.2" (51.6" per section). I spaced the element 57.6" apart. As a Yagi, the array yielded a free-space gain of 6.24 dBi with a front-to-back ratio of 10.88 dB. The design frequency feedpoint impedance was about 40 + j8 Ohms, which provided a 50-Ohm SWR of less than 2:1 from 28 to 29 MHz.

Out of curiosity, I developed a phasing harness
for the Yagi that improved performance, as shown in **Fig. 3**.
Using phasing techniques, the gain increased slightly, but the
front-to-back ratio jumped by 10 dB. The Yagi configuration is limited
to using only the geometry of the antenna elements to obtain its
results. Adding a phase lines allows a broader control of the current
magnitude and phase angle values on the individual elements to increase
at least some of the performance values.

As the outline shows, the phasing system consists of two lines, one from each element, to a main feedline junction. Essentially, we may analyze all 2-element phased arrays on this model, even thought some--like the well-known ZL-Special--used a zero length line from the forward element to the feedline junction. In this design, the forward phasing line consists of 6" of 50-Ohm, 0.78 VF cable, while the rearward line uses the same cable, but with a 64" length. The electrical lengths of these cables, of course, are the physical lengths divided by the velocity factor (VF).

We may develop a fuller understanding of the
phased array in the present (and, by extension, any other) design by
using two models of the antenna and doing some external calculations. **Fig.
4** shows the outlines of the two models. The upper model establishes
the relative current magnitude and phase angle on the two elements
required to produce the required performance (as listed in **Fig. 3**).
Normally, we assign a current magnitude of 1.0 and a phase angle of
zero on the forward element and then find the current magnitude and
phase angle required for the rear element. The arrangement of lines
used to establish phasing conditions must meet these requirements to
obtain the listed performance. (At this point, we shall not concern
ourselves with the resulting impedance at the junction of the phase
lines in the lower portion of **Fig. 4**, although we shall
eventually work with that detail.) Our fundamental question is the
conditions that must exist at the junction of the two phase lines to
obtain correct element phasing when we use the standard parallel
connection of the lines and the source. In this and many other cases,
the rearward line (TL2) has a half twist to effect a 180-degree phase
shift relative to the junction or to the element phasing that would
obtain with a normal line.

Every transmission line (including those in phase-line service) transforms the impedance at the load end to another value at the source end. If the line happens to match the load, the impedance is constant along the line, but any difference between the line characteristic impedance (Zo) and the load impedance (Zl) results in a different impedance values at the source end except for lines that are exactly a multiple of 1/2-wavelength electrically. What amateurs often forget is that the current and voltage also undergo transformation along the line, and they are more critical to the phased array's performance than the impedance. Voltage and current undergo only one transformation per 360 degrees of electrical change.

The required conditions for proper performance of a phased array include establishing the desired relative current magnitude levels and the required phase angle difference at the two elements. The junction end of the line also has conditions. A parallel connection divides the source current between the two lines. Each line's share of the current must result in the desired ratio of current magnitude and the desired difference in phase angle at the element ends of the line. In addition, the transformations along the line must result in identical voltage magnitudes and phase angles at the junction of the two lines with the source. Very often, we can obtain these conditions, but the combination of parallel voltage and current magnitudes and phase angles results in an unusable or at least a highly undesirable feedpoint impedance. Therefore, with available lines, the designer's options are limited. For example, a combination calling for a total physical length that is less than the spacing between elements would be unusable. As well, we cannot simply change the spacing, since a phased array is a combination of parasitic and directly fed energy at each element. Therefore, for the required performance, changing the element spacing would require a different ratio of current magnitude and an altered phase angle difference to arrive at the specified performance.

To illustrate the required conditions, we can use the standard equations for impedance, voltage, and current transformation down a transmission line from load to source, that is, the junction end of each line. We shall let Zo be the characteristic impedance of the line at the design frequency, while Zl is the load impedance and Zs is the source end impedance. The script "l" is the electrical length of the line in either degrees or radians, according to the calculator's preferred measurement method.

To calculate the voltage at the source end of the line, where El is the load voltage and Es is the source voltage, we can use a comparable standard equation.

The current calculation, where Il is the load current and Is is the source current, also uses a standard equation.

All of these equations are the versions for lossless lines, the type that appear in the NEC TL system. Therefore, any external calculations based on these equations should yield results that closely coincide with the NEC reports, especially if we are using NEC data for the input values. The calculations require separating and recombining the real and imaginary portions of the equations, a task well-suited to a spreadsheet or a utility program. TLD and TLW are suitable programs, although they include loss factors. However, the short length of the lines should make any differences inconsequential.

The result of applying these equations to the
situation of a modeled phased array allows us to examine how the arrays
do their work. **Table 1** provides relevant calculation results
for the parallel junction version of our phased array. Note that the
input voltage and current for the rearward line have been phase shifted
by 180 degrees to account for the half-twist in that line.

The boldface entry for the calculated voltage values at the junction of lines shows the identity (within a very close approximation) of voltage, with the current being split between the two lines. The net impedance is simply the parallel combination of the two impedance values. The calculation shows a very close coincidence with the reported feedpoint impedance from the NEC model. This should come as no surprise, since NEC makes calculations very similar to these in the course of a core run for the model with phase lines as TL command functions.

The resulting impedance may seem troublesome, since it is neither resonant nor convenient to common matching systems. Had the impedance been close to resonance at about 25 Ohms, we might have applied a common equation to construct a 1/4-wavelength matching section. A 35.5-Ohm line (composed of RG-83 or of two parallel sections of 70-Ohm line) would have yielded a 50-Ohm final impedance value.

Most amateurs forget that the special formula for exact 1/4-wavelength sections is only a single point along a spectrum of slowly changing impedance values. Therefore, we can create sections that are longer or shorter than 1/4-wavelength to approximate the exact match. A quarter-wavelength at the design frequency is about 103" with a physical length that is the electrical length value times the lines velocity factor. In this case, we need a much shorter length of line. For a VF of 0.66, 34" will do, while for a VF of 0.78, 40" will provide a resonant feedpoint impedance of about 58 Ohms.

The array, then, is quite usable, although I am not here recommending its use. Rather, the phased array with its parallel connections of feedlines and the feedpoint serves as a good example of its type.

**A 2-Element Horizontal Phased Array with
Series Phaseline Connections with the Source**

For every parallel solution to a design
challenge, there should also be a series solution. Therefore, let's use
the same set of elements and see if we can develop a series phaseline
and feedpoint connection. My purpose is not to claim that one or the
other array is superior, but only to show what a series set of
connections will entail. The final design for the revised array appears
in outline form in **Fig. 5**, along with a free-space E-plane
pattern. The gain and front-to-back data on the graphic show the near
identity of performance between the new array and its parallel cousin.

The basic outline shows that the feedpoint is not located in the same position as in the case of the parallel junction. The earlier system used 70" of total 50-Ohm, VF 0.78 line divided into a 6" forward section and a 64" rearward section (with its half twist). The new system uses a total of 79" of the same line, with a 25" forward section and a 54" rearward length (again, with a half twist). The new phasing lines are not identical in any way to the old ones, nor are they a mirror image of the old system. Series connections answer to a different set of conditions relative to parallel connections.

**Fig. 6** shows the two steps through which
we shall proceed in hopes of obtaining a better understanding of the
series connections. The lower portion of the graphic shows the
feedpoint junction area. I could have avoided the double crossing of
wires in the schematic representation, but the cross in TL2 is a
reminder that the rear line receives a half twist. The junction area of
the diagram shows the required connections to obtain a set of series
junctions among the lines and the source.

The key to analyzing the series situation is the
understanding the each line creates a voltage drop across its
transformed version of the element load. Therefore, the current through
each of the source ends of the lines is the same. The actual element
currents may differ in magnitude and phase angle, but the lines must be
of the right length for the characteristic impedance so that the
current magnitudes and phase angles at the junction are the same. We
may go through the same set of calculations that we used for the
parallel connection array to check our work and establish the correct
condition at the feedpoint. **Table 2** provides the results of
those external calculations. Once more, the initial rear element
voltage and current phase angles have been adjusted for the half-twist
and its 180-degree alteration of the phase angle between the source and
load ends of TL2.

The boldface entry is for the current, which shows a very close coincidence of values. The net impedance at the feedpoint is simply the sum of the individual impedance values. The resistive component shows a tight alignment with the value reported by the NEC model. The model shows a higher reactance than we obtain from the calculations. However, the calculations do not include the triangular loop that creates the series connections. If we remodel the array for a separate source for each phase line, we obtain from NEC impedance values of 64.3 + j55.6 and 84.3 + j7.1 Ohms, for a net series impedance of 148.6 + j62.7 Ohms, very close to the calculated value. Although the connection triangular loop may seem small, it is significant.

Neither of the two versions of the phased array is directly suited to the use of a 50-Ohm main feedline. We raised the impedance of the parallel connected version by the use of a section of 35-Ohm cable, usually constructed from two parallel sections of 70-Ohm cable. Even though the matching section cable impedance is not the geometric mean between the array impedance and the main line impedance, we found a length that provided a satisfactory 50-Ohm SWR. Similarly, we may use a 93-Ohm cable (such as RG-62 with a VF of 0.84) to serve as a matching section for the series connected version of the array. The array impedance is not perfect for use with a 1/4-wavelength of this cable, but a slightly longer cable (108" physically, 128.5" electrically) provides a satisfactory design-frequency resonant impedance (40.9 Ohms).

In fact, both versions of the array, with the proper matching section, provide complete coverage of the first MHz of 10 meters with well under a 2:1 SWR value. Since both arrays also provide essentially the same performance in all vital categories, performance cannot be the deciding factor in selecting which version to use. Of course, these notes are not necessarily recommending either version for actual use.

Careful exploration of both series and parallel connected models can uncover other differences that might impact a selection. Both forms of the array undergo a considerable range of front-to-back values across the defined 1-MHz passband. However, the parallel-connected version shows a much higher change of gain from one end of the passband to the other: 0.88 dB. In contrast, the series connected version varies by only 0.17 dB across the same spread. Despite the gain stability advantage of the series connected version, the parallel-connected form may be simpler to construct. The phaseline system may use available coaxial cable connectors to effect all junctions, both with the elements and at the main junction of the phase lines.

**Conclusion**

The goal of these notes has been to show the modeling techniques necessary to replicate in NEC a series connection of feedlines and a source in cases more complex than those shown in the preceding episode. We selected a 2-element phased array that required two phase lines constructed using the TL facility. Since the two elements required different terminal current magnitude and phase angle values to obtain the desire performance, the parallel and the series pairs of phase lines proved to be quite different. The differences stemmed in large part from the fact that parallel connections divide current, while series connections divide voltage. Although NEC provides output data that is both reliable and useful in both cases, we resorted to external calculations to show that both systems established the conditions appropriate to each type.

The external analysis was largely post-facto, that is, applied after finding the correct phase line values, including the characteristic impedance and the length. In this case, the element geometry was given, but any number of other geometries is possible. Since the process depends upon setting realistic performance goals in terms of the gain and the front-to-back ratio for any given geometry, I am not aware of any system for automatically calculating such phase lines. A systematic search among available line Zo values and length combinations within NEC remains one of the fastest routes to a reasonable design.