Optimum beamwidth for a 3 sector cell using a 2-element array
This applet permits the optimum beamwidth for
a mobile cell that has three separate antennas spaced 120° apart. In addition, it permits the optimum beamwidth
to be determined for any angle of rotation of the three antennas, rotated in
unison, up to an angle of 30° from their initial position. The cell shape follows the classic
hexagon shape. The rationale behind the design,
mathematics involving hexagon geometry , some basic
information on CDMA and some information on the programme design is given separately in the
hyperlinked documents. Below is an explanation of the applet and how to use it.
Explanation of the scrollbars
Figure 1 shows a screen-shot of
the scrollbars used in the applet.
Figure 1 A
screen-shot of the scrollbars used in the applet.
Next to each scrollbar
description is the initialization of the applet value and the range of values
this scrollbar takes.
Number of
Subscribers in cell (199) [ 5 - 9999]
This scrollbar is used to insert
into the hexagon, the number of subscribers that the user wishes to accommodate
in a single cell. This number does not mean that there will be this number of
mobiles in the cell, because of the way the subscribers are packed. For a
uniform distribution throughout the hexagon, a square has been used for each
subscriber. This square will vary in the value of its side according to the
number of subscribers entered using the scrollbar. For a low value of
“Number of Subscribers in cell”, the squares will not exactly fit
into a hexagon, so the “Total subs in cell” readout value will be
less than that entered by the scrollbar. The “% of subs in cell/scrollbar
value” will indicate the percentage value. As the number of subscribers
is increased, the packing of squares into the hexagon improves. For large
values of subscribers, the percentage of “Total subs in cell” to
the “Number of Subscribers in cell” can become 98%. Should you wish
to have a particular number of subscribers in the cell, then the scrollbar “Number
of Subscribers in cell”, should be adjusted to give a value of
“Total subs in cell” read from the printout on the orange canvas, equal
to your requirements. The subscribers in the cell are represented by small yellow
and blue dots. A yellow dot represents a subscriber who is receiving the
base-station signal at the correct level and can communicate with the
base-station. If the dots are blue, then the subscribers are out of range of
the base-station.
On the orange canvas, the readout
“Subs receiving OK” provides an indication of the number of yellow
dots in the hexagon. This number will not equal the “Total subs in
cell” as there are subscribers in the cell that are out of range of the
base-station. The ratio of the “Subs receiving OK” to the
“Total subs in cell” is given as a percentage by the readout
“% of Subs Recv OK/Cell subs”.
Offset from
top of cell (0.986) [0 to 1.0]
As the number of subscribers in
the cell changes, the packing will vary. To equalize the packing distribution
of subscribers, this scrollbar allows the adjustment of the offset of the top
row of subscribers from the top of the hexagon to balance distance of the
bottom row of subscribers from the bottom of the hexagon and also to ensure
that the distribution is properly centered. It is important that this adjustment is made
prior to any curves being drawn, etc. in order to provide consistency in the
results. The reason for the existence of this scrollbar is discussed in the hexagon geometry document.
Reception
cut-off radius (0.707) [0 to1.0]
This is normally set to 0.707r to
agree with the beamwidth circle and to indicate that a subscriber on the
perimeter of the polar circle when moving away from bore-sight will find a radial
where the antenna polar pattern has fallen to 0.7071 of the bore-sight value
and which will just allow him to receive the base-station. Any further around,
away from bore-sight will cause the subscriber to be out of range of the
base-station. However, there may be occasions where a reception cut-off radius
is desired to be different from the 0.707 value. This scrollbar provides this
flexibility. Additional information may be found in the rationale behind the
design document. In addition, the reception cut-off radius of 0.707, allows
the beamwidth (degrees), which is the angle subtended by a polar pattern
between the intersections of the polar pattern and the 0.707r circle, to be
directly read from the polar plots.
Element
spacing (wavelengths) (0.5) [0.303 to 0.5]
This is the same as “beamwidth”. The
scrollbar has not been relabeled to beamwidth as it gives information on the
spacing of a 2-element array operating as a broadside array. More information
on linear arrays and applets demonstrating the principles can be found by
inspecting antennas and array scanning. The beamwidth adjustment
is one of the more important parameters that the applet uses. This is because
the number of available subscribers is a function of the beamwidth in this
applet. The readout “Antenna beamwidth” provides the value of the
antenna beamwidth. This is determined from the point where the main polar
pattern under consideration intersects the white 0.7071 radius circle shown on
the applet. From the position of one of the two intersection points to the
boresight radial is half the beamwidth of the polar pattern. From this, the
programme determines the antenna beamwidth. The range of the element spacing
scrollbar is designed to provide a beamwidth range from 60° to 111° which is
sufficient to observe the optimum beamwidth and lie within the valid range of
the fixed distance between the main and interfering polar patterns, as
discussed below under the Adjacent interference level scrollbar.
Adjacent
interference level (dB) (-8.0) [-
The adjacent sector interference
level in dB is the level of interference below bore-sight for each of the
antenna polar patterns. It is represented by a circle on the applet diagram. If
the radius of the main polar circle is r, the radius of this circle is
determined from 
, so for -8dB setting of the scrollbar, the interference
level radius would be 0.3981r. As the radius where the subscriber becomes
unavailable is 0.7071r, the difference in these two radii provides the fixed
difference between the main lobe and the overlapping interference lobe that
needs to be maintained if a subscriber is to remain available. As this
difference is to be kept constant, on the main bore-sight side of this fixed
difference line the subscribers remain available, whereas on the side away from
bore-sight, the subscribers become unavailable for reception. As the beamwidth
of the antenna increases, the interference pattern also increases in level,
thus, to maintain the fixed difference there is a movement to a different
radial towards bore-sight of this constant difference. The result is to reduce
the number of available subscribers, counteracting the increase in the number
of available subscribers resulting from the increase in beamwidth. This is
described in the rationale behind the design document. The applet
provides a screen print out of the “Interference level difference”
which is the fixed difference mentioned above. As this fixed difference is a
decimal number, in the programming of the applet where a difference between two
levels cannot be determined exactly, the difference is valid if it lies within
a small range of possible values (difference +/- 0.00141r). The actual value of
the difference used and which is used to determine the pertinent radial, is
provided in the readout from the applet as the “Actual level
difference”
Normally, in this applet, the adjacent
interference level parameter is set to a value and kept at that value for the
duration of the experiment. With this parameter fixed, the beamwidth is varied
and the change in the number of available subscribers is observed.
Beam orientation
(degrees) (0) [ 0 to 30]
On initialization of the applet,
the beam orientation is set to 0°. This represents the bore-sights of the
three beams pointing to corners of the hexagon. As the scrollbar is changed
from 0° to 30°, the three polar patterns of the three
antennas rotate clockwise, so that in the 30° position the bore-sights of the three
beams point to sides of the hexagon. At any angle of rotation of the beams all
other scrollbars can be changed to provide valid results. One of the main
reasons for this applet is to compare the value of available subscribers for
different beamwidths at 0° and 30° beam orientation. This particular
experiment permits the optimum beamwidth for the different rotations to be
obtained and also permits the number of available subscribers to be compared
for the different beam orientations.
Explanation of the canvases
The antenna
polar pattern canvas
This black background canvas
presents the three polar patterns in the colours green, yellow and orange. By
varying the “Element spacing (wavelengths)” scrollbar, all three
patterns will change their width in unison. By changing the “Beam
Orientation (degrees)” scrollbar, from the initial position of 0° up to 30°, the three
polar patterns will rotate clockwise. The angle of 0° indicates
that the polar pattern beam bore-sight is pointing to a corner of the hexagon,
whereas, the angle of 30° shows the beam pointing to a side of the
hexagon. Overlaid on this canvas, in the colour magenta, are the main polar
circle and the radials. The radials are spaced at 10°. The 0.7071r
circle is coloured white. In addition, there are two other circles. The
reception cut-off radius that is initialized to 0.7071r can be varied using the
“Reception cut-off radius” scrollbar and the inner magenta coloured
circle that can be varied by the “Adjacent interference level (dB)”
scrollbar.
Superimposed within the hexagon
boundary are yellow dots indicating subscribers who can communicate with the
base-station at the centre of the hexagon and blue dots that indicate
subscribers who are out of range of the base-station. The base-station is
indicated by a dark grey triangle at the centre of the hexagon. The edges of
the triangle indicate the sides of the ground plane behind each 2 element array
that suppress the back lobe. The vertical positioning of the rows of dots can
be varied, for proper dot or subscriber distribution, by varying the
“Offset from top of cell” scrollbar. The total number of coloured
dots, representing available and unavailable subscribers, contained within the
hexagon can be varied using the “Number of subscribers in cell”
scrollbar. By varying the “Element spacing (wavelengths)”
scrollbar, the number of unavailable subscribers (blue dots) can be seen to
vary with the beamwidth.
The graph
plot canvas
This orange canvas contains a
grid that permits the number of subscribers that can communicate with the
base-station (Subs receiving OK) along the y-axis to be plotted against the beamwidth
from 60° to 111° along the x-axis. The y-axis is self-scaling
for the number of subscribers contained in the hexagon. That is, as the
“Number of subscribers in cell” scrollbar is increased, there will
be an increase in the “Subs receiving OK”. This change is reflected
by the self-scaling property of the “Subs receiving OK” axis.
Plotted on this grid is a black
dot that moves as the beamwidth is changed. It is this black dot that indicates
for a particular value of beamwidth, the number of subscribers that are
receiving the base-station. A family of black dot plots can be produced for
changes in parameters brought about by changing any or all of the scrollbars.
It is this black dot that is the main feature of the applet and which indicates
the optimum beamwidth for the various parameters such as interference level and
beam rotation.
In addition to the plot, are
given the more important parameters and calculations. These are indicated by
screen printouts. One pair is the
antenna beamwidth and the effective beamwidth. The effective beamwidth is the
movement of the beamwidth radial to accommodate the effect of adjacent sector
interference. The difference between the antenna and effective beamwidths can
be observed from these screen printouts.
To enhance the reading of the
position of the black dot on the plot, the “Subs receiving OK” is
given and this value expressed as a percentage of the number of subscribers in
the cell by “% of subs Recv OK/cell subs”.
The final pair of screen
printouts is the “Interference level difference” and the
“Actual level difference”. The “Interference level
difference” is the difference between 0.7071r and the radius of the
scrollbar value of the “Adjacent interference level (dB)”. In
practice, this difference value does not always exist due to opposite points on
the main lobe and interfering lobe that lie on a radial not providing an exact decimal
value of the required difference. A window (+/- 0.00141r) is made in the
program that allows values to vary around the required difference value. These
values that lie within the window are the actual values of the interference
level difference that is used in the applet. The printout “Actual level
difference” gives the actual values used. By observing these two
different differences it can be observed how far the actual difference is away
from the required difference. In some cases, the difference between the
opposite points on the main lobe and the interfering lobe lie outside of the
window. When this occurs, there will be a discontinuity in the plot. This is
because, the effect of the interference has not been taken into account and the
number of subscribers receiving the base-station defaults to a calculation made
without interference. The black dot on the plot, in this case, jumps to a
higher value. This sudden jump upwards and then back can be observed after the
optimum beamwidth has been passed. If the window is made wider, then a number
of values for differences can be found. The problem that occurs in this case is
that a hysterisis effect is brought into play. When increasing the beamwidth the
first difference value found that lies in this window is obtained and plotted.
When decreasing the beamwidth back to a position already passed, the second or
maybe third difference value now becomes the first and this is then plotted. As
it is not the same, there will be a different hysterisis value of “subs receiving OK” plotted.
Using the applet
When first initializing the
applet, the default initialization values can be used. All that needs to be
done is to slowly vary the “Element spacing (wavelengths)” from 0.5
down to 0.303 and observe the variation of the number of subscribers receiving
the base-station by the variation of the black dot on the orange canvas. At the
same time, watch the change in beamwidth on the black canvas. At a value of 0.375
of the element spacing scrollbar, corresponding to a beamwidth of 83.4°, a yellow
fixed difference line appears on the black canvas at about 140°, between the
main beam at 180° and the adjacent sector beam tail. It
will be noticed that as the beamwidth increases, the yellow fixed difference
line moves anti-clockwise to the end of the interference pattern tail at 150° on the
canvas. This occurs at a beamwidth about 111°. As the yellow fixed difference line can
go no further along the interfering pattern as it has reached the limit of the
pattern, it show that increasing the beamwidth to a larger value would be
meaningless in terms of the effect of adjacent sector interference.
On a second time around, using
the same initialization values, note the antenna beamwidth on the black canvas
by estimating the angle between the cuttings of the polar pattern of a single
antenna by the 0.7071 circle. Compare this with the antenna beamwidth printout
on the yellow canvas. Also, as the fixed difference line appears when the
element spacing is the same or slightly greater 0.303 and moves anti-clockwise as
the element spacing (beamwidth) is increased; note the difference between the
antenna beamwidth and the effective beamwidth on the orange canvas printouts. The
subscribers receiving OK printout can be seen to reduce and the number of blue
dots increase as the beamwidth is increased further. The plot shows a maximum value
of the subscribers receiving OK is reached and then a reduction that follows in
accordance with the screen printout on the orange canvas.
For your comments and suggestions, mail the author: A.A.R.Townsend
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