5. Multiple Tower Arrays

All of our samples in the 4 preceding episodes
used a single tower centered on the coordinate system center (X=0,
Y=0). Typical of those one-tower models was the near resonant 234'
tower at 1 MHz, with an 18" face of a triangular tower. The model that
we used earlier looked almost like the one that we shall show here. **
**

CM near-resonant monopole, perfect ground

CM NAB substitute single-wire monopole

CE

GW 1 41 0 0 0 0 0 234 0.555

GW 30901 1 9901.0000 9901.0000 9901.0000 9901.0001 9901.0001 9901.0001 .00001

GS 0 0 .3048

GE 1 0 0

GN 1

EX 0 30901 1 0 0.0 7.4897

NT 30901 1 1 1 0 0 0 1 0 0

FR 0 1 0 0 1 1

RP 0 181 1 1000 -90 0 1.00000 1.00000

RP 0 1 361 1000 90 0 1.00000 1.00000

RP 1 1 1 0000 0 0 1.00000 1.00000 1609.344

RP 1 1 1 0000 0 0 1.00000 1.00000 3218.688

EN

The only difference between past models and this
one is that the new version adds a second RP1 command at a distance of
2 miles to the original that uses a distance of 1 mile. Both commands
use ground level as the observation height for the command. The basic
data collection is in the following lines. **
**

Near Resonant (234') 18" Face Triangular Single-Wire Monopole Model Data

Impedance (Ohms) Current (Apk) Gain (dBi) AGT AGT-dB F-S @ 1 mile F-S @ 2 miles

35.65 - j 1.29 7.4897 5.14 1.999 0.00 275.1 mV/m @ -45.6 deg 137.5 mV/m @ -178.1 deg

If the only field-strength value in which we have any interest is the magnitude (in peak mv/m as shown or adjusted to RMS), then we need not add the second RP1 request. For perfect ground, field-strength magnitude values decrease linearly with distance from the antenna. However, if we have any interest in the phase angle, the second request is necessary to obtain the additional figures.

In this episode, the tower that we have just modeled will play a significant role, but not solo. In this episode, we shall look at some very basic cases that employ two towers with considerations of the current magnitude and phase angle at each source--remembering that we are using the standard method in NEC to provide current sources. The task will sometimes involve more than simply adding a second tower to the GW portion of the list.

**Two Towers Fed In-Phase for a Broadside Pattern**

Suppose that we need a pattern like the one shown
in **Fig. 1** to fulfill broadcast needs and restrictions. The
simplest way to obtain it is with two towers, in this case, using
broadside array techniques. In the present sample, we shall use only
simple arrays to illustrate the modeling aspects. Actual arrays may be
considerably more complex, and the resultant patterns may be equally
complex. The pattern is laid out according to the compass-rose azimuth
conventions favored by some agencies and many field engineers.

The desired coverage calls for a moderate
increase in gain along the N-S axis with lesser gain in the E-W
directions. One way to obtain such coverage is to arrange two towers
about 1/4-wavelength apart in the E-W plane (+Y and -Y) and to feed
them in phase. Initially, this feed requirement will use two sources,
each supplied with the same current magnitude and phase angle, with the
magnitude determined by our standard 1-kW power level. We shall use our
single-wire substitute for a 234' tower in each case. A wavelength at 1
MHz is 983.571 feet, so the separation between towers is 245.893'. **
**

CM 2 near-resonant monopoles, perfect ground

CM NAB substitute single-wire monopole

CM in-phase feeding--1/-wl spacing

CE

GW 1 41 0 122.946 0 0 122.946 234 0.555

GW 2 41 0 -122.946 0 0 -122.946 234 0.555

GW 30901 1 9901.0000 9901.0000 9901.0000 9901.0001 9901.0001 9901.0001 .00001

GW 30902 1 9902.0000 9902.0000 9902.0000 9902.0001 9902.0001 9902.0001 .00001

GS 0 0 .3048

GE 1 0 0

GN 1

EX 0 30901 1 0 0.0 4.3339

EX 0 30902 1 0 0.0 4.3339

NT 30901 1 1 1 0 0 0 1 0 0 0

NT 30902 1 2 1 0 0 0 1 0 0 0

FR 0 1 0 0 1 1

RP 0 181 1 1000 -90 0 1.00000 1.00000

RP 0 1 361 1000 90 0 1.00000 1.00000

RP 1 1 37 0000 0 0 1.00000 10.00000 1609.344

EN

The simple data collection, as we can see from
the following lines, does not tell the full story, as it did for the
single tower models. The collection also omits the 2-mile
field-strength report. **
**

Two-Tower Broadside Array 18" Face Triangular Single-Wire Monopole Model Data

Impedance (Ohms) Current (Apk) Gain (dBi) AGT AGT-dB F-S @ 1 mile

53.24 - j17.40 x2 4.3339/tower 6.23 max. 1.999 0.00 311.5 mV/m @ -47.2 deg

The impedance reports are for each tower, as is
the current magnitude. The gain and the field-strength values are
maximum values, taken in the northern direction (0 degrees, which
corresponds to the +X direction on the geometry coordinate system).
However, we shall be interested in both the gain and the field strength
in various directions around the pattern. Because the field strength is
likely to be the more important figure, we may wish to examine a table
of figures taken at suitable intervals. The following sample from the
model traces 1/4 of the pattern (because it is symmetrical) at
10-degree intervals. **
**

**** Electric Field: Phi Pattern ****

Z=0, Freq=1, File=fcc51.NOU

---E (Theta)--- --- E (Phi) ---

Phi Magnitude Phase Magnitude Phase

Degrees Volts/m Degrees Volts/m Degrees

0.00 3.1157E-001 -47.23 5.3620E-022 -108.51

10.00 3.0868E-001 -47.23 4.3494E-022 105.73

20.00 3.0040E-001 -47.23 3.2047E-022 -167.59

30.00 2.8786E-001 -47.23 1.9626E-022 137.96

40.00 2.7271E-001 -47.23 6.6090E-023 -66.91

50.00 2.5687E-001 -47.23 6.6090E-023 113.09

60.00 2.4224E-001 -47.23 1.9626E-022 -42.04

70.00 2.3051E-001 -47.23 3.2047E-022 12.41

80.00 2.2293E-001 -47.23 4.3494E-022 -74.27

90.00 2.2032E-001 -47.23 5.3620E-022 71.49

The E(theta) columns represent the vertical
component of the field-strength calculations. The horizontal component
(E(phi) is too small to be significant. To better visualize the changes
in field strength as we move around the overall pattern, we may also
graph the values, as shown in **Fig. 2**.

The combination of data allows significant evaluation of the likely performance of the 2-tower broadside array. Of course, the sample selects a spacing between towers that yields less than the full broadside bi-directional gain of such towers. Wider spacing will yield more gain in the N-S direction with less gain in the E-W direction. As a certain point as we increase spacing, the oval pattern will gradually evolve into a figure-8.

Our interest does not lie in what we can do with
towers so much as it lies in what we can include in and show by
appropriate modeling. For example, we may wish to include in the model
a composite feed system so that we have only a single source. The
normal form of feeding the system would be to bring transmission lines
from each tower to a central point so that each line is equal in length
(and characteristic impedance) to the other. We may set up such lines
by selecting the junction point and placing a short, thin wire to serve
as the source as well as the junction between lines. Let's arbitrarily
set up two 600-Ohm transmission lines, one from each tower. The
terminal points for the lines and the source will be a position exactly
centered between the towers (Y=0) and 245' (1/4-wavelength) away from
the towers. The general outline of the model will have the appearance
of the set-up in **Fig. 3**.

To model this situation, without altering the
tower positions or other attributes, we need a model that resembles the
following lines. **
**

CM 2 near-resonant monopoles perfect ground

CM NAB substitute single-wire monopole

CM in-phase common feeding--1/4-wl spacing

CE

GW 1 41 0 122.946 0 0 122.946 234 0.555

GW 2 41 0 -122.946 0 0 -122.946 234 0.555

GW 3 1 -245 0 1 -245 0 2 0.0001

GW 30901 1 9901.0000 9901.0000 9901.0000 9901.0001 9901.0001 9901.0001 .00001

GS 0 0 .3048

GE 1

GN 1

EX 0 30901 1 0 0.0 1.5138

NT 30901 1 3 1 0 0 0 1 0 0

TL 1 1 3 1 600 0 ! User Defined VF

TL 2 1 3 1 600 0 ! User Defined VF

FR 0 1 0 0 1 1

RP 0 181 1 1000 -90 0 1 1

RP 0 1 361 1000 90 0 1 1

RP 1 1 37 0000 0 0 1.00000 10.00000 1609.344

EN

Form this model we may obtain the usual data
collection. **
**

Two-Tower Broadside Array, Common Source, 18" Face Triangular Single-Wire Monopole Model Data

Impedance (Ohms) Current (Apk) Gain (dBi) AGT AGT-dB F-S @ 1 mile

872.8 - j1457.8 1.5138 6.23 max. 1.999 0.00 311.5 mV/m @ -47.2 deg

Only the impedance and the required current for 1kW differ from the dual-source model. To confirm the high source impedance, we may independently calculate the current transformation down each 600-Ohm line, with a length of 274.12' based on the separate source impedance values of 53.24 - j17.40 Ohms. The result will be separate impedances of about 1754 - j2920 Ohms, which combine in parallel to 877 - j1460 Ohms, very close to the modeled values, considering the rapid change in value for each small increment of length.

If the impedance is inconvenience due to the
100.3-degree lines required, we may always change the position of the
junction. The shortest lines occur when we place the junction in line
with the towers at Y=0, as suggested by **Fig. 4**. The lines have
shrunk to 45 degrees.

The only change to the model is in the placement
of GW3, as the following partial model file shows. **
**

GW 1 41 0 122.946 0 0 122.946 234 0.555

GW 2 41 0 -122.946 0 0 -122.946 234 0.555

GW 3 1 0 0 1 0 0 2 0.0001

We do not required changes in the TL command
entries because we have used zeros (after the characteristic impedance
entry of 600) to specify that the line length is the actual distance
between the terminal points as defined by the wire entries. **
**

TL 1 1 3 1 600 0 ! User Defined VF

TL 2 1 3 1 600 0 ! User Defined VF

In the data collection, we find that the only
resultant differences occur in the entries for the composite source
impedance and the required peak current level needed at this impedance
to achieve a 1-kW power level. **
**

Two-Tower Broadside Array, Common Source, 18" Face Triangular Single-Wire Monopole Model Data

Impedance (Ohms) Current (Apk) Gain (dBi) AGT AGT-dB F-S @ 1 mile

49.96 + j 278.9 6.3272 6.23 max. 1.999 0.00 311.5 mV/m @ -47.2 deg

NEC employs lossless transmission lines for its calculations. At 1 MHz for virtually any line less than 1/2-wavelength long, the values for lossless line calculations will not differ significantly from calculations including line losses. The model set-ups also presume a velocity factor of 1.0. If the velocity factor of a line departs significantly from that value, one may always insert the electrical line length in place of our use of zero to force the program to use the actual distance between terminal points on the line.

Not all arrays require patterns with maximum
field-strength values going north and south. Suppose that we require
that the pattern have its gain maximum point aligned along an axis
defined by compass heading of 60 and 240 degrees. In general, there are
two major ways to achieve this goal. One is to set up each tower so
that the broadside direction is automatically along the desired axis.
The other method, shown here, is to set up the model in the simple
manner shown earlier and then to turn the entire array around the
Z-axis by the required 60 degrees. Note in the following model lines,
that to turn the axis clockwise--as the present situation requires, we
specify -60 degrees in the GM line. (+60 degrees turns the pattern
counterclockwise.) **
**

CM 2 near-resonant monopoles perfect ground

CM NAB substitute single-wire monopole

CM in-phase common feeding--1.4-wl spacing

CM rotated for 60/240-deg AZ axis

CE

GW 1 41 0 122.946 0 0 122.946 234 0.555

GW 2 41 0 -122.946 0 0 -122.946 234 0.555

GW 3 1 -245 0 1 -245 0 2 0.0001

GM 0 0 0 0 -60 0 0 0

GW 30901 1 9901.0000 9901.0000 9901.0000 9901.0001 9901.0001 9901.0001 .00001

GS 0 0 .3048

GE 1

GN 1

EX 0 30901 1 0 0.0 1.5138

NT 30901 1 3 1 0 0 0 1 0 0

TL 1 1 3 1 600 0 ! User Defined VF

TL 2 1 3 1 600 0 ! User Defined VF

FR 0 1 0 0 1 1

RP 0 181 1 1000 -90 0 1 1

RP 0 1 361 1000 90 0 1 1

RP 1 1 37 0000 0 0 1.00000 10.00000 1609.344

EN

We need not show the data collection, since it
has not changed. What has changed is the field-strength table. The
magnitudes will be the same as the earlier sample shown, but the
headings on which they occur will differ. Remember that for tabular
information, NEC uses the phi or counterclockwise convention.
Therefore, the 60-degree compass-rose azimuth bearing coincides with
the phi 300-degree bearing in the following table. **
**

**** Electric Field: Phi Pattern ****

Z=0, Freq=1, File=fcc53.NOU

---E (Theta)--- --- E (Phi) ---

Phi Magnitude Phase Magnitude Phase

Degrees Volts/m Degrees Volts/m Degrees

0.00 2.4224E-001 162.75 9.3641E-023 -108.70

10.00 2.3051E-001 162.75 7.5958E-023 105.52

20.00 2.2294E-001 162.75 5.5967E-023 -167.82

30.00 2.2032E-001 162.75 3.4275E-023 137.71

40.00 2.2294E-001 162.75 1.1542E-023 -67.17

50.00 2.3051E-001 162.75 1.1542E-023 112.83

60.00 2.4224E-001 162.75 3.4275E-023 -42.29

70.00 2.5687E-001 162.75 5.5967E-023 12.18

80.00 2.7271E-001 162.75 7.5958E-023 -74.48

90.00 2.8786E-001 162.75 9.3641E-023 71.30

100.00 3.0041E-001 162.75 1.0848E-022 106.98

110.00 3.0869E-001 162.75 1.2002E-022 53.35

120.00 3.1158E-001 162.75 1.2792E-022 -66.10

130.00 3.0869E-001 162.75 1.3192E-022 134.16

140.00 3.0041E-001 162.75 1.3192E-022 -39.16

150.00 2.8786E-001 162.75 1.2792E-022 161.09

160.00 2.7271E-001 162.75 1.2002E-022 41.65

170.00 2.5687E-001 162.75 1.0848E-022 -11.99

180.00 2.4224E-001 162.75 9.3641E-023 23.69

190.00 2.3051E-001 162.75 7.5958E-023 169.48

200.00 2.2294E-001 162.75 5.5967E-023 82.82

210.00 2.2032E-001 162.75 3.4275E-023 137.29

220.00 2.2294E-001 162.75 1.1542E-023 -17.83

230.00 2.3051E-001 162.75 1.1542E-023 162.17

240.00 2.4224E-001 162.75 3.4275E-023 -42.71

250.00 2.5687E-001 162.75 5.5967E-023 -97.18

260.00 2.7271E-001 162.75 7.5958E-023 -10.52

270.00 2.8786E-001 162.75 9.3641E-023 -156.31

280.00 3.0041E-001 162.75 1.0848E-022 168.01

290.00 3.0869E-001 162.75 1.2002E-022 -138.35

300.00 3.1158E-001 162.75 1.2792E-022 -18.91

310.00 3.0869E-001 162.75 1.3192E-022 140.84

320.00 3.0041E-001 162.75 1.3192E-022 -45.84

330.00 2.8786E-001 162.75 1.2792E-022 113.90

340.00 2.7271E-001 162.75 1.2002E-022 -126.65

350.00 2.5687E-001 162.75 1.0848E-022 -73.02

360.00 2.4224E-001 162.75 9.3641E-023 -108.70

**Fig. 6** re-confirms the successful rotation
by showing the far-field pattern for the revised model. The lines on
either side of the main axis lines indicate the half-power beamwidth,
suggesting that the gain is about 3 dB weaker at right angles to the
main axis. You may correlate this to the ratio of the relevant
field-strength reports by the usual equation in which PdB = 20 log_{(10)}(E1/E2).

The notes so far have dealt with the simple case in which the sources for each broadside element are identical with respect to current magnitude and phase angle. Not all arrays of towers have such an easy requirement.

**An Endfire Array of Two Towers**

For directional patterns, that is, patterns with a dominant lobe in only one direction, array designers generally use end-fire techniques so that the pattern is in line with the towers rather than broadside to them. We shall employ only a very basic two-tower array to note the key modeling points of interest. However, some installations have used up to 4 towers to obtain specific pattern shapes. As well, in some instances, designs have combined broadside with end-fire techniques for truly large arrays. Since there are texts devoted to the design of such arrays, we may focus on translating endfire arrays into models over perfect ground. We shall retain our 234' tower with the single-wire equivalent of an 18" face on a triangular structure. As was clear in the broadside array, mutual coupling between towers in relatively close proximity alters the source impedance so that each tower in the array is no longer self-resonant. (Compare the source impedance values for the initial 2-source broadside model with the source impedance of the reference single-tower model at the beginning of these notes.) Our present exercise will require even closer attention to the impedances reported for each tower.

Our sample will use two towers separated by
1/4-wavelength. To set the main-lobe direction at north (0-degrees
azimuth), we align the towers along the X-axis. To ensure that we place
the array center at the coordinate center, each tower is 1/8-wavelength
from X=0. The resulting geometry is simply our broadside array turned
90 degrees. In fact, if we were to feed the two sources in phase, we
would obtain the earlier broadside pattern with the stronger
field-strength reading east and west. **
**

CM 2 near-resonant monopoles, perfect ground

CM NAB substitute single-wire monopole

CM end-fire two-tower array

CM 90-degree feeding--1/4wl spacing

CE

GW 1 41 122.946 0 0 122.946 0 234 0.555

GW 2 41 -122.946 0 0 -122.946 0 234 0.555

GW 30901 1 9901.0000 9901.0000 9901.0000 9901.0001 9901.0001 9901.0001 .00001

GW 30902 1 9902.0000 9902.0000 9902.0000 9902.0001 9902.0001 9902.0001 .00001

GS 0 0 .3048

GE 1 0 0

GN 1

EX 0 30901 1 0 5.3579 0

EX 0 30902 1 0 0.0 5.3579

NT 30901 1 1 1 0 0 0 1 0 0 0

NT 30902 1 2 1 0 0 0 1 0 0 0

FR 0 1 0 0 1 1

RP 0 181 1 1000 -90 0 1.00000 1.00000

RP 0 1 361 1000 90 0 1.00000 1.00000

RP 1 1 37 0000 0 0 1.00000 10.00000 1609.344

EN

In the model, GW 1 is the forward tower, that is, the tower in the direction of the main lobe, with GW 2 to the rear. Basic array theory tells us that we shall obtain a highly directional pattern if we feed the towers so that the rear tower has the same current magnitude as the forward tower. However, the phase angle of the rear tower current should be +90-degree relative to the forward tower (or the forward tower phase angle should be -90 degrees relative to the rearward tower). Some software allows the modeler to enter the desired values directly into the input interface screens. However, we shall do it the "old-fashioned" way by manipulating the currents on the remote EX entries for our current-fed array.

The problem at hand is simplified by the use of
equal current magnitudes. However, the EX entry in NEC lists the
excitation voltage in terms of real and imaginary components of the
voltage that we shall transform into a current via the NT entries. **Fig.
8** shows the help screens (a composite of 2 screens, one for each
source) to assist us in sorting out the entries. The screens list both
the components and the magnitude and phase angle, and we may set up the
line by placing the values in either format. As the screen shows, the
forward tower (1) is 90 degrees behind the rearward tower (2) with
respect to the phase angle. Compare these entries to the EX commands in
the model.

Now let's perform one more comparison: the EX
entries with the currents that appear on the source segments of the two
towers. We may glean this information from the NEC output file. **
**

**** Segment Current versus Frequency ****

FREQUENCY SEG. TAG COORD. OF SEG. CENTER SEG. - - - CURRENT (AMPS) - - -

(MHz) NO. NO. X Y Z LENGTH REAL IMAG. MAG. PHASE

1.000000 1 1 0.1250 0.0000 0.0029 0.00580 -9.6451E-16 -4.3339E+00 4.3339E+00 -90.000

1.000000 42 2 -0.1250 0.0000 0.0029 0.00580 4.3339E+00 -1.4008E-16 4.3339E+00 0.000

Although we entered the source voltages with phase angles of 0 and 90 degrees for towers 1 and 2, respectively, the currents on the sources have phase angles of -90 and 0 degrees, respectively. We now understand two things. First, the voltage entries for the EX line have preserved their phase difference in the conversion to current values on the source segments. Second, the NT command responsible for the conversion shifts the entered phase angle by -90 degrees relative to the final current reports on the affected segments. If we forget this second fact, it shows up quite rapidly, since the pattern for entering the phase angles backwards will also be backwards.

**Fig. 9** shows the resulting far-field
patterns that merges from the model that we have constructed. If we
truly needed to reduce the rearward radiation further, we may juggle
both the magnitude and the phase angle of the EX entries until
satisfied. However, once we have established the desired pattern, we
would need to re-adjust the current magnitudes with respect to the
total power supplied to the array as indicated by the power budget
portion of the NEC output report, using the technique shown in the
first of these episodes. The values shown are for our pre-set power
level of 1 kW.

The methods for obtaining a main-lobe direction
other than north are the same as for the broadside array. We may
perform pre-modeling calculations so as to place the towers in the
correct positions to yield a pattern with the desired heading, or we
may construct the tower using the X-axis as the main line and then
rotate the tower wires using the GM command. Let's rotate the array so
that the main lobe has a heading of 315 degrees on the compass-rose
azimuth scale. We need to inform the GM command to rotate the structure
+45 degrees to effect the counterclockwise rotation, as shown in the
following model. **
**

CM 2 near-resonant monopoles, perfect ground

CM NAB substitute single-wire monopole

CM end-fire two-tower array

CM 90-degree feeding--1/4wl spacing

CM 315-deg AZ heading via GM

CE

GW 1 41 122.946 0 0 122.946 0 234 0.555

GW 2 41 -122.946 0 0 -122.946 0 234 0.555

GM 0 0 0 0 45 0 0 0

GW 30901 1 9901.0000 9901.0000 9901.0000 9901.0001 9901.0001 9901.0001 .00001

GW 30902 1 9902.0000 9902.0000 9902.0000 9902.0001 9902.0001 9902.0001 .00001

GS 0 0 .3048

GE 1 0 0

GN 1

EX 0 30901 1 0 5.3579 0

EX 0 30902 1 0 0.0 5.3579

NT 30901 1 1 1 0 0 0 1 0 0 0

NT 30902 1 2 1 0 0 0 1 0 0 0

FR 0 1 0 0 1 1

RP 0 181 1 1000 -90 0 1.00000 1.00000

RP 0 1 361 1000 90 0 1.00000 1.00000

RP 1 1 37 0000 0 0 1.00000 10.00000 1609.344

EN

**Fig. 10** shows the resulting pattern.

The data collection for both of our sample
endfire arrays is the same. **
**

Two-Tower Endfire Array, Source at 90-Degree Phasing, 18" Face Triangular Single-Wire Monopole Model Data

Tower Impedance (Ohms) Current (Apk) Gain (dBi) AGT AGT-dB F-S @ 1 mile

1 50.74 + j16.91 5.3579 @ -90 8.25 max. 1.999 0.00 393.5 mV/m @ -90.7 deg

2 18.93 - j19.90 5.3579 @ 0 deg

Obtaining the desired phase shift and power division with a single ultimate source is subject to many techniques that we shall leave to external calculations. However, it is possible to construct a fairly complex model with a combination of TL and NT entries to incorporate the desired technique into the model. However, for most purposes, obtaining the individual source impedance values and the source-segment current magnitudes and ratios allow these calculations to proceed most efficiently externally to the model.

The data collection shows the maximum values for
gain and field-strength (the latter still in peak form and needing
conversion to RMS). Since most installations will need values in many
directions to correlate with field measurements, the modeler should
attend to the RP1 tabular output. The sample that follows shows the
values for the rotated example. Once more, remember that NEC output
reports employ the phi or counterclockwise convention for listing
azimuth angles. Therefore, the value applicable to a compass-rose
heading of 315 degrees occurs between the phi entries for 40 and 50
degrees. **
**

**** Electric Field: Phi Pattern ****

Z=0, Freq=1, File=fcc55.NOU

---E (Theta)--- --- E (Phi) ---

Phi Magnitude Phase Magnitude Phase

Degrees Volts/m Degrees Volts/m Degrees

0.00 3.8121E-001 -90.93 4.6724E-022 -153.51

10.00 3.8834E-001 -90.77 3.7881E-022 60.73

20.00 3.9181E-001 -90.66 2.7900E-022 147.41

30.00 3.9313E-001 -90.58 1.7082E-022 92.96

40.00 3.9347E-001 -90.54 5.7515E-023 -111.91

50.00 3.9347E-001 -90.54 5.7515E-023 68.09

60.00 3.9313E-001 -90.58 1.7082E-022 -87.04

70.00 3.9181E-001 -90.66 2.7900E-022 -32.59

80.00 3.8834E-001 -90.77 3.7881E-022 -119.27

90.00 3.8121E-001 -90.93 4.6724E-022 26.49

100.00 3.6887E-001 -91.12 5.4160E-022 62.13

110.00 3.5005E-001 -91.35 5.9964E-022 8.46

120.00 3.2409E-001 -91.64 6.3956E-022 -111.03

130.00 2.9115E-001 -92.01 6.6011E-022 89.18

140.00 2.5232E-001 -92.49 6.6063E-022 -84.18

150.00 2.0949E-001 -93.14 6.4106E-022 116.03

160.00 1.6509E-001 -94.10 6.0193E-022 -3.46

170.00 1.2182E-001 -95.61 5.4442E-022 -57.13

180.00 8.2234E-002 -98.30 4.7023E-022 -21.49

190.00 4.8653E-002 -103.96 3.8163E-022 124.27

200.00 2.3437E-002 -119.59 2.8130E-022 37.59

210.00 1.1594E-002 -170.26 1.7232E-022 92.04

220.00 1.3540E-002 146.90 5.8035E-023 -63.09

230.00 1.3540E-002 146.90 5.8035E-023 116.91

240.00 1.1594E-002 -170.26 1.7232E-022 -87.96

250.00 2.3437E-002 -119.59 2.8130E-022 -142.41

260.00 4.8653E-002 -103.96 3.8163E-022 -55.73

270.00 8.2234E-002 -98.30 4.7023E-022 158.51

280.00 1.2182E-001 -95.61 5.4442E-022 122.87

290.00 1.6509E-001 -94.10 6.0193E-022 176.54

300.00 2.0949E-001 -93.14 6.4106E-022 -63.97

310.00 2.5232E-001 -92.49 6.6063E-022 95.82

320.00 2.9115E-001 -92.01 6.6011E-022 -90.82

330.00 3.2409E-001 -91.64 6.3956E-022 68.97

340.00 3.5005E-001 -91.35 5.9964E-022 -171.54

350.00 3.6887E-001 -91.12 5.4160E-022 -117.87

360.00 3.8121E-001 -90.93 4.6724E-022 -153.51

**Conclusion**

The notes in this episode have focused on the modeling convention, methods, and cautions applicable to multi-tower installations. I have used very simple arrays in order to set the modeling aspects of the situation in bold relief. Far more complex arrays are possible--and with them come far more complex models.

Some implementations of NEC are set up to ease the process of modeling arrays. For example, EZNEC provides RMS input and output values of voltages and currents. As well, the use of current sources is completely hidden, allowing the user simply to set in place the desired source values for current magnitude and phase. Our use of a more generic form of NEC has had the goal of showing some of what may go on "behind the scenes" in such interfaces.

A five-episode run of notes on a single topic--however broad--might seem to answer most of the beginning level questions one might have about tower modeling. Unfortunately, there is at least one major category of question left over at the interface between AM BC tower modeling and tower modeling in general.