Joel Koltner wrote:
I've taken college classes in antennas and hence have a pretty good feel for
some of the mathematics behind it all, but I've found that at times I don't
have good, intuitive explanations for various antenna behaviors -- and I'm not
at all good at being able to look at some fancy antenna and start rattling off
estimates of the directivity, front to back ratio, etc. -- so I wanted to ask
a simple question on a two-element phased array:

First, start with one antenna. Feed it 1W, and assume that in some
"preferred" direction at some particular location the (electric) field
strength is 1mV/m.

Now, take two antennas, and space them and/or phase their feeds such that in
the same preferred direction the individual antenna patterns add. I.e., we're
expecting a 6dB gain over the single antenna (but only at that location).
Since we start off by splitting the power to each antenna (1/2W to each), that
initially seems impossible, since 1/2W+1/2W = 1W -- should imply the same
1mV/m field strength. But this is an incorrect analysis, in that powers don't
add directly. Instead, the fields add... hence, each antenna alone will now
produce 707uV/m (at the one particular location in question), so the two
together produce 1.414mV/m which is the same as if the single antenna had been
fed with 2W. Hence the 6dB gain we're after! (This analysis also implies
there must be other locations that now receive 1mV/m in order to conserve
energy.)

Is that correct? "Powers don't add, field strengths do" is obvious enough,
but definitely leads to some slightly non-intuitvely-obvious (to me) results.
By extension of the above, though, it becomes obvious that (in theory) one can
build an array with any desired amount of gain, the beamwidth just has to
become narrower and narrower, of course.

There are two errors in your analysis. The first is that you've
neglected mutual coupling between the elements. In some special cases,
this will result in equal feedpoint impedances, so that equal powers
will result in equal currents, which in turn result in equal field
strengths. But that happens only in special cases and not by any means
all cases. In the general case, splitting the power equally between
elements won't result in equal fields from them.

Moving on, let's assume that you've got a special case where the equal
power split results in equal field strength. Your analysis is then
correct up until you calculate the dB gain. Your correct value of 1.414
mV/m is correct, but it represents a 3, not 6, dB gain relative to a
single element (which produced 1 mV/m). In the absence of mutual
coupling, and if the elements are spaced and phased such that there's
some direction in which the fields can completely reinforce, then the
maximum pattern gain relative to a single element is 10 * log(N) where N
is the number of elements, e.g., 3 dB for two elements, 6 dB for four
elements, etc.

For a much more detailed explanation of these phenomena, I recommend
reading the treatment of phased arrays in Chapter 8 of the _ARRL Antenna
Book_. I'm admittedly a bit partial to this particular treatment, since
I wrote it.

Roy Lewallen, W7EL