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.

Thanks,
---Joel

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

"Joel Koltner" wrote in message
...
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.

Thanks,
---Joel



yes, all true. and that is where many of the arguments on here begin,
trying to add powers instead of fields, voltages, or currents. and yes,
theoretically you can keep making the beamwidth narrower and get more and
more gain, that is one reason lasers are so intense with such low power,
they have extremely narrow beamwidths.

It's also 'true' that to get more gain in one direction you
typically have less gain in some other direction. You can't get
something for nothing, so you are only redirecting what you've already
got, so to speak (how directional antennas work).
- 'Doc

Joel Koltner wrote:
"Powers don't add, field strengths do"

"Add" is a rather loosely defined term. A more technically
precise statement would be: "Powers don't superpose, field
strengths do." When fields superpose, they still must obey
the conservation of energy principle, i.e. the total energy
before the superposition must equal the total energy after
the superposition.

Given two RF waves in a transmission line and the phase angle,
A, between the two electric fields, the following Power equation,
published in QEX, gives us a valid method of "adding" two powers.

Ptotal = P1 + P2 + 2*SQRT(P1*P2)*cos(A)

Reference: "Wave Mechanics of Transmission Lines, Part 3",
by Steven R. Best, VE9SRB, "QEX", Nov/Dec 2001, (Eq 13),
page 4.

The last term is known in optics as the "interference"
term, positive for constructive interference and negative
for destructive interference. Angle A, the phase angle
between the two electric fields, determines the sign
of the last term and thus whether interference is
destructive or constructive.

Reference: "Optics", by Hecht, 4th Edition:
Chapter 7: The Superposition of Waves
Chapter 9: Interference
--
73, Cecil http://www.w5dxp.com

On Thu, 7 Aug 2008 17:55:29 -0700, "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.

Thanks,
---Joel

What Roy did not tell you is that his program has a free demo version
(http://eznec.com/) that will will provide quick answers. The learning
curve for EZNEC is pretty sharp for about 10 minutes and then it
shallows out.

John Ferrell W8CCW

Hi Roy,

"Roy Lewallen" wrote in message
treetonline...
There are two errors in your analysis. The first is that you've neglected
mutual coupling between the elements.

Yes, I assumed it was negligible. When we analyzed arrays in class some years
ago, the starting point was always, "assume each antenna has the pattern of a
single dipole in isolation, is matched to the transmission lines, with input
current 1A @ angle whatever." We did analyze the patterns for a couple of
local Oregon TV & radio stations, BTW, including whichever radio? station it
is up in Portland (very roughly) near you off of I-205 as you drive past
Oregon City.

But that happens only in special cases and not by any means all cases.

The phased arrays I had in mind were those that were usually separated by a
"significant" fraction of a wavelength, e.g., lambda/8 or more. That's
probably not far enough apart to neglect coupling?

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).

6dB vs. 3dB is a rather embarassing outright brain fart on my part. :-) (The
usual case of confusing "twice the power = 3dB" with "twice the voltage =
6dB").

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.

Thanks Roy, I'll take a look!

---Joel

"Cecil Moore" wrote in message
...
Ptotal = P1 + P2 + 2*SQRT(P1*P2)*cos(A)

So... let's see... my two 1/2W antennas now, in the "preferred" location, get
you... 0.5+0.5+2*sqrt(0.5*0.5)*cos(0) = 2W... yep, same as the field strength
analysis. Cool!

Presumably you could demonstrate all this with a "ripple tank" (the kind with
water used back in high school physics) -- set things up so that, in a
preferred direction, the wave height is 1.414 even though the wave height made
by each "radiator" in isolation is 0.707.

Thanks Cecil,
---Joel

Joel Koltner wrote:
Hi Roy,

"Roy Lewallen" wrote in message
treetonline...
There are two errors in your analysis. The first is that you've neglected
mutual coupling between the elements.

Yes, I assumed it was negligible. When we analyzed arrays in class some years
ago, the starting point was always, "assume each antenna has the pattern of a
single dipole in isolation, is matched to the transmission lines, with input
current 1A @ angle whatever." We did analyze the patterns for a couple of
local Oregon TV & radio stations, BTW, including whichever radio? station it
is up in Portland (very roughly) near you off of I-205 as you drive past
Oregon City.

But that happens only in special cases and not by any means all cases.

The phased arrays I had in mind were those that were usually separated by a
"significant" fraction of a wavelength, e.g., lambda/8 or more. That's
probably not far enough apart to neglect coupling?

Not by a long shot! Here's a simple example from the EZNEC demo program,
using example file Cardioid.EZ. It's a two element array of quarter
wavelength vertical elements spaced a quarter wavelength apart and fed
with equal currents in quadrature to produce a cardioid pattern. The
impedance of a single isolated element is 36.7 + j1.2 ohms. In the
array, the impedances are 21.0 - j18.7 and 51.6 + j20.9 ohms, and the
elements require 29 and 71 percent of the applied power respectively in
order to produce equal fields. The deviation is due to mutual coupling.
This particular array is a special case of another kind -- there is no
net effect of the mutual coupling on the array gain, so it has 3.0 dB
gain over a single element. This isn't true in the general case, however.

Neglecting the mutual coupling is convenient for the professors because
it simplifies the problem and allows them to illustrate the simple
addition of fields. The problem is that it leads some students to think
they have the whole story.

In very large arrays such as those used for radar, the vast majority of
elements are in essentially the same environment relative to each other
so the mutual coupling has the same effect on all except the outer few
elements. But it simply can't be ignored in arrays of a few elements.

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).

6dB vs. 3dB is a rather embarassing outright brain fart on my part. :-) (The
usual case of confusing "twice the power = 3dB" with "twice the voltage =
6dB").

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.

Thanks Roy, I'll take a look!

Another source which has a good discussion of the topic is Johnson's
_Antenna Engineering Handbook_, or earlier editions edited by Jasik. Be
wary of amateur and hobbyist publications (other than the _ARRL Antenna
Book_ -- very few authors understand the topic, and pass along their
misconceptions.

Roy Lewallen, W7EL

"Roy Lewallen" wrote in message
treetonline...
Not by a long shot! Here's a simple example from the EZNEC demo program,
using example file Cardioid.EZ. It's a two element array of quarter
wavelength vertical elements spaced a quarter wavelength apart and fed with
equal currents in quadrature to produce a cardioid pattern. The impedance of
a single isolated element is 36.7 + j1.2 ohms. In the array, the impedances
are 21.0 - j18.7 and 51.6 + j20.9 ohms, and the elements require 29 and 71
percent of the applied power respectively in order to produce equal fields.
The deviation is due to mutual coupling.

That's a much, much greater difference than I would have guessed. Wow...

Isn't the input impedance of one element affected not only by the relative
position of the other element, but also how it's driven? I.e., element #1
"sees" element #2 and couples to it, but how much coupling occurs depends on
whether the input of element #2 is coming from a 50 ohm generator vs. a 1 ohm
power amplifier (close to a voltage source), etc.? (Essentially viewing the
antennas as loosely coupled transformers, where the transformer terminations
get reflected back to the "primary.")

Thanks for the book links. Do you happen to have a copy of "Small Antenna
Design" by Douglas Miron? And have an opinion about it? Or some other book
on electrically small antennas? (Not phased arrays, though :-) -- more like
octave bandwidth VHF or UHF antennas that are typically 1/10-1/40 lambda in
physical size.)

---Joel