[Towertalk] limitations of stacking

[Michael Keane, K1MK]

+3 dB is the maximum array factor for a two element array: the max gain 
over a single array element that can be realized for an array of two 
identical, non-interacting radiators having uniform excitation (equal 
currents).

For vertical stacking over ground, it would be unusual to achieve an array 
factor of +3 dB because the ground reflections from antennas placed at two 
different heights do not coincide at low angles and so the radiated fields 
don't add up to give the full factor of two.

Now, the stacking gain for a pair of real antennas differs even further 
from the idealized array factor because the antennas are going to interact 
with each other. And that mutual coupling may result in a stacking gain of 
either more or less than +3 dB depending on the details of the interaction.

Under certain simple circumstances the interaction between antennas in an 
array (stack) can be constructive. This in the case for a pair of half-wave 
dipoles where for separations in the vicinity of 0.7 waves the mutual 
impedance is real and negative. The negative impedance lowers the radiation 
resistance of the elements in the array resulting in additional gain. A 
pair stacked dipoles can be described analytically with a closed-form 
solution and the theoretical maximum gain for a pair of stacked dipoles in 
free space is +4.9 dB over a single dipole. One can describe that net gain 
as being composed of +3 dB from the array factor plus an additional +1.9 dB 
from the mutual coupling.

In an array built from more "complicated" antennas, such as Yagis whose 
performance as individual antennas results from the summation of fields 
generated by currents flowing in multiple driven and parasitic elements, 
any change to the amplitudes or phases of these currents from mutual 
coupling is far more likely to result in a loss of gain than an increase. 
Still, for 2 or 3 element Yagis, it would be unusual but not theoretically 
impossible to find a situation in which the effects of mutual coupling due 
to stacking result in a slight increase in gain rather than a slight decrease.

Because of mutual coupling, element lengths and spacings for Yagis used in 
a stack may benefit from re-optimization for the specific stack in which 
the antenna will be used. And when modelling how a single antenna in a 
stack performs, e.g. when driving the top antenna only or the bottom 
antenna only, it may be necessary to retain the un-excited antennas (and 
their feed lines & terminations) in the model because there is mutual 
coupling with the other antennas in the stack even when the other antennas 
are not being directly driven.

73,
Mike K1MK

At 09:51 AM 3/22/02, Pete Smith wrote:
>Is there a theoretical reason why the gain resulting from stacking two HF
>yagis over real ground is limited to 3 dB?  I have a model of two C-3Es
>that shows 3.5 dB increase over a single C-3E in the top position.  I'm
>trying to figure out if this is a modeling anomaly of some sort...
>
>73, Pete N4ZR


Michael Keane, K1MK
k1mk@alum.mit.edu