I did something similar to this as part of my Master's research, except I
used a 'brute force' algorithm written in Fortran.

As the other poster suggested, you have to consider the mutual coupling.
This can really mess up the pattern if ignored, except in one special case.
That special case is when the array is completely symmetrical. That is each
dipole 'sees' the same surroundings as all other dipoles. Since you have
them in a circular arrangement, the dipoles will have to be placed radially
on the circle. That is, one end of each should point towards the center of
the circle.

What is happening is that the elements are coupling to each other. This
introduces an additional complex impedance in addition to the self impedance
of the dipoles themselves. You then have to 'dither' the phase of the
driving current (for an all driven array) to get the desired pattern. In
the case of an array with parasitic elements (that is, the driving current
is 0.0), what is typically done is to dither the self impedance of the
parasitic elements by either placing a lumped constant (inductance or
capacitance) in the parasitic element, or changing the length of the
parasitic element slightly. The more elements you have, the more
complicated the problem becomes.

Note that the pattern for receiving will be the same as for transmitting
because of reciprocity. It is usually easier to calculate the pattern for
the transmit case.


Two dipoles, as you found, are always symmetrical to each other. Get
yourself a copy of J. D. Kraus' "Antennas" and study the section on mutual
coupling and reciprocity. The newer editions also have a section on the
"Method of Moments", which is what NEC and similar programs use.

For my thesis project, I used a brute force method. Kraus gives canned
formulas for the self and mutual impedances of simple antenna
elements---dipoles, monopoles, etc. (actually, he only gives part of the
formulas and refers you to other papers). I then wrote some complex matrix
math routines in Fortran, and let it crank away. For N elements, you need
an NxN matrix of complex numbers to describe the relationship between all of
the elements. The self impedances will be on the diagonal, and the mutual
impedances for each pairing of elements will be elsewhere in the matrix.
What you do then is solve the matrix equation I=V/Z, with V being a unit
matrix (for a uniformly driven array). This will give you the currents in
the elements from which the pattern can be calculated. To 'dither' the
currents, change the complex values in the V diagonal. For a parasitic
array (I never got far enough to actually try this) all but the feed element
in V is 0.0.

Fortunately, NEC does all of this for you.

By the way, a herd of bugs have 'antennae', a herd of radios have
'antennas'. (As in Kraus' Antennas).

Jim
N8EE