Phase array question
 

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

Phase array question
 

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

Phase array question
 

"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.

Phase array question
 

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

Phase array question
 

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

Phase array question
 

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

Phase array question
 

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

Phase array question
 

"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

Phase array question
 

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

Phase array question
 

"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

Phase array question
 

Joel Koltner wrote:
"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.")

Not directly. What counts (considering the simple case of two elements)
is the magnitude and phase of the current in the other element, and
their spacing, orientation, and lengths. A good way to look at the
effect of mutual coupling is as "mutual impedance", i.e., the amount of
impedance change caused by mutual coupling. (Johnson/Jasik covers this
concept well.) If you were to feed two elements with constant current
sources (as in the Cardioid.EZ EZNEC example), mutual coupling doesn't
change the element currents, but only the feedpoint impedances. With any
other kind of feed system, the impedance change causes the currents to
change, which in turn affects the impedances. So the feed method
certainly does have an effect on the currents you get, which affects
both mutual coupling and pattern.

There's a lot more about this, and how to design feed systems which will
effect the desired currents, in the _ARRL Antenna Book_.

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.)

I just recently purchased Miron's book but haven't yet looked at it in
any depth. It appears to be most interesting to anyone wanting a better
understanding of method of moments numerical methods. If you can read
German, you might be interested in _Kurze Antennen_ by Gerd Janzen. But
your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two". They'll be
inefficient, which will hurt you both receiving and transmitting at VHF
and above. The book I'd go to for researching the possibilities would be
Lo & Lee's _Antenna Handbook_. You might also get some ideas from
Bailey, _TV and Other Receiving Antennas_, since TV antennas have to be
pretty broadband.

Roy Lewallen, W7EL

Phase array question
 

Cecil Moore wrote:
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."

Fields superpose, numbers add, and power is the rate of change in energy.

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.

It's almost as if you think that if you don't always point it out,
energy won't be conserved! :-)

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)

According to fig. 7.1 in Born and Wolf, that's useful for showing how
light intensity varies as a function of phase, and hence position. It's
just that there's no valid way to multiply by the cosine of the angle
between two scalars.

Maybe wave problems are best solved using waves.

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.

(A+B)*(A+B) = A^2 + B^2 + 2AB

Must the first order term (2AB) in such equations always be referred to
as "The Interference Term", Cecil? Doing so seems to impart a greater
level of importance to it than to the other, unnamed terms in the
equation. The factored form must then be least important of all.

Beats, interference, and modulation are fundamentally the same
phenomenon. There's no need to get all worked up about one of them in
deference to the others, just as there's no need to worry about there
being a node for every antinode.

73, ac6xg

Phase array question
 

Jim Kelley wrote:
Cecil Moore wrote:
Ptotal = P1 + P2 + 2*SQRT(P1*P2)*cos(A)

According to fig. 7.1 in Born and Wolf, that's useful for showing how
light intensity varies as a function of phase, and hence position. It's
just that there's no valid way to multiply by the cosine of the angle
between two scalars.

You are pretty confused. The angle is between the electric
field intensities of the two waves being superposed.

Must the first order term (2AB) in such equations always be referred to
as "The Interference Term", Cecil?

From "Optics", by Hecht, 4th Edition, page 387 & 388:

"I12 = 2E1*E2 ... and is known as the interference term."
E1 and E2 are electric field intensities. * is the dot product.
indicates a time average value.

"The interference term becomes I12 = 2*SQRT(I1*I2)cos(delta)"

Hecht calls it the "interference term" and I am only quoting
him.
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:
Jim Kelley wrote:
Cecil Moore wrote:
Ptotal = P1 + P2 + 2*SQRT(P1*P2)*cos(A)

According to fig. 7.1 in Born and Wolf, that's useful for showing how
light intensity varies as a function of phase, and hence position.
It's just that there's no valid way to multiply by the cosine of the
angle between two scalars.

You are pretty confused.

I think you know that I'm just pointing out the problem inherent in
using a valid equation in the way you describe without considering the
many assumptions being made. It led you, for example, to write that
there is a 4th mechanism of reflection - even in violation of Maxwell's
equations! Do you still believe that interference actually moves power
from one place to another? It is that kind of nonsense that amateur
radio would be better off without.

Hecht calls it the "interference term" and I am only quoting
him.

I'll bet he'd prefer that you didn't. :-)

73, ac6xg

Phase array question
 

Jim Kelley wrote:
I think you know that I'm just pointing out the problem inherent in
using a valid equation in the way you describe without considering the
many assumptions being made. It led you, for example, to write that
there is a 4th mechanism of reflection -

Here's a quote from my energy article:

"Note that the author previously used the word "reflection" for both
actions involving a single wave and the interaction between two waves.
Now the word "reflected" is being used only for single waves and the
word "redistributed" is being used for the two wave interference scenario."

Nowhere in my present article do I say there is a 4th mechanism
of reflection. Why do you continue to incessantly harp on past
semantic blunders that were corrected years ago?

Do you still believe that interference actually moves power
from one place to another?

Do you ever stop beating dead horses? :-)

Since I stated in my article that power doesn't flow, you are just
once more bearing false witness. Maybe you should have that burr
under your blanket looked at by a competent veterinarian. :-)

I said that the redistribution of energy, which necessarily obeys
the conservation of energy principle, is associated with a wave
cancellation interference event. I never uttered your false
statement that "interference moves power". Here's what I said:

"The term "power flow" has been avoided in favor of "energy flow".
Power is a measure of that energy flow per unit time through a
plane. Likewise, the EM fields in the waves do the interfering.
Powers, treated as scalars, are incapable of interference."

Yet, a couple of times a year just like clockwork, you accuse
me of saying that power moves (which I have never said). One
wonders what drives your never-ending vendetta obsession.

Here is the definition that I am using for RF "interference"
adopted from "Optics", by Hecht:

RF wave interference corresponds to the interaction of two
(or more) RF waves yielding a resultant power density for
the total wave that deviates from the sum of the two power
densities in the superposed component waves.

It is simple physics to realize that (V1+V2)^2 is not usually
equal to (V1^2 + V2^2). When they are not equal, interference
has occurred. Why do you have such a problem with such a simple
concept?

In a transmission line, the power equation indicates exactly
by how much the resultant power deviates from the sum of the
component powers. The magnitude of that deviation from the
sum of the component powers is called the "interference term"
according to Hecht.

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

'A' is the angle between the V1 and V2 voltage phasors.
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:
Jim Kelley wrote:

Do you still believe that interference actually moves power
from one place to another?

Since I stated in my article that power doesn't flow, you are just
once more bearing false witness.

Let me try this then: Do you still think that interference is what
moves ENERGY from one place to another?

"The term "power flow" has been avoided in favor of "energy flow".

Yet, a couple of times a year just like clockwork, you accuse
me of saying that power moves (which I have never said). One
wonders what drives your never-ending vendetta obsession.

Note that the reason the author included the disclaimer about "power
flow" was because the term "power flow" had not been avoided by said
author in this newsgroup, in an argument which must have gone on for 6
weeks. In fact, it was a point that was never actually conceded.
Rather, thusly, he "avoided" conceding it. (Reminder: Now you come back
by mentioning how Poynting vectors show how much and in which direction
the power is flowing.)

It is simple physics to realize that (V1+V2)^2 is not usually
equal to (V1^2 + V2^2). When they are not equal, interference
has occurred.

Well, 8th grade algebra is supposed to help us realize that (V1+V2)^2 is
not equal to (V1^2 + V2^2). But the fact that (V1+V2)^2 is equal to
V1^2 + V2^2 + 2V1*V2 doesn't depend in the least on whether
"interference has occurred", Cecil. That was the whole point of my
comment about it.

In a transmission line, the power equation indicates exactly
by how much the resultant power deviates from the sum of the
component powers. The magnitude of that deviation from the
sum of the component powers is called the "interference term"
according to Hecht.

You'd think it would be enough for someone to throw out the whole idea
of "the sum of the powers" once and for all. But no. The inclination
instead is apparently to keep refining one's epicycle formulary. Hecht
makes no such connection between 'power' and 'interference', Cecil. And
why would he? There isn't one.....except in certain amateur radio
articles and newsgroup postings.

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

'A' is the angle between the V1 and V2 voltage phasors.

....and NOT between the two 'powers'. Still, it's a very useful
expression for finding a quick, albeit simplified solution. That is
after all its intended purpose.

73, ac6xg

Phase array question
 

Jim Kelley wrote:
Let me try this then: Do you still think that interference is what
moves ENERGY from one place to another?

To the best of my knowledge, I addressed all of your objections
in a revision to my energy article which was done many months
ago. The reasons for your objections don't even exist any more.

Note that the reason the author included the disclaimer about "power
flow" was because the term "power flow" had not been avoided by said
author in this newsgroup, in an argument which must have gone on for 6
weeks.

But that argument happened many, many years ago. You convinced
me that power doesn't flow. I respect the fact that it is a
commonly accepted concept defined in "The IEEE Dictionary",
rampant within every power company, and accepted by many
members of this newsgroup.

If you will check back over the years, you will find a posting
of mine where I said the dimensions of power flow would be
joules/sec/sec which doesn't make any physical sense.

But the fact that (V1+V2)^2 is equal to
V1^2 + V2^2 + 2V1*V2 doesn't depend in the least on whether
"interference has occurred", Cecil. That was the whole point of my
comment about it.

Yes, and you are still wrong according to Hecht. If the
interference term in the power equation is not zero,
(V1^2+V2^2) does not equal (V1+V2)^2. In the special case
where (V1^2+V2^2) = (V1+V2)^2, the interference
term is zero, i.e. zero interference.

Please reference page 388 in "Optics", by Hecht, 4th Edition.

Hecht
makes no such connection between 'power' and 'interference', Cecil.

But Hecht certainly makes a connection between 'power density' and
'interference'. It is a trivial matter to convert the power density
irradiance equation to the power equation by multiplying by the
cross-sectional area of a transmission line.

The units of irradiance (power density) are joules/sec/unit-area.

Multiply the irradiance equation by the unit-area of the coax,
e.g. 1 in^2, and you get joules/sec = power which is what a
Bird wattmeter indicates.

If you want, you can convert the Bird wattmeter reading to
irradiance by dividing by the cross-sectional area of the
coax.

'A' is the angle between the V1 and V2 voltage phasors.

...and NOT between the two 'powers'.

*Nobody* has ever said there is a phase angle between two powers
yet you persist in that false strawman implication.
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:
Jim Kelley wrote:
Let me try this then: Do you still think that interference is what
moves ENERGY from one place to another?

To the best of my knowledge, I addressed all of your objections
in a revision to my energy article which was done many months
ago. The reasons for your objections don't even exist any more.

May we assume that the term "4th mechanism of reflection" will be
avoided in future publications?

You convinced me that power doesn't flow.

Thank you, Jesus.

But the fact that (V1+V2)^2 is equal to V1^2 + V2^2 + 2V1*V2 doesn't
depend in the least on whether "interference has occurred", Cecil.
That was the whole point of my comment about it.

Yes, and you are still wrong according to Hecht.

Well, Hecht and I both understand the 8th grade algebra, and I don't
disagree with him. How then could he disagree with me? I think that's
the symmetric equality property. :-)

One possible way to resolve the apparent dichotomy would be to suppose
that you misunderstand what we are each saying.

Hecht makes no such connection between 'power' and 'interference', Cecil.

But Hecht certainly makes a connection between 'power density' and
'interference'. It is a trivial matter to convert the power density
irradiance equation to the power equation by multiplying by the
cross-sectional area of a transmission line.

The units of irradiance (power density) are joules/sec/unit-area.

Multiply the irradiance equation by the unit-area of the coax,
e.g. 1 in^2, and you get joules/sec = power which is what a
Bird wattmeter indicates.

Which of course explains how it is that your answers come out correctly.
I believe I already mentioned that it does (obviously) produce correct
answers, given all the underlying assumptions are correct.

By the way, what assumptions are you making?

If you want, you can convert the Bird wattmeter reading to
irradiance by dividing by the cross-sectional area of the
coax.

Wouldn't I first have to buy into the idea that power is flowing through
it? :-)

'A' is the angle between the V1 and V2 voltage phasors.

...and NOT between the two 'powers'.

*Nobody* has ever said there is a phase angle between two powers
yet you persist in that false strawman implication.

It's hard to imagine how a simple, mutually agreed upon statement of
fact could be construed as a "false strawman implication", but there it is.

73, ac6xg

Phase array question
 

Jim Kelley wrote:
May we assume that the term "4th mechanism of reflection" will be
avoided in future publications?

Yes, exactly as it has been for the past six months since
I revised my article. In less time than it takes to condemn
me for saying something in the far distant past, you could
have just read my article. Here's the half-year old footnote
from my article.

[10] Revision 1.1, Feb. 20, 2008 - In the original version, the
redistribution of energy due to wave cancellation was dubbed a
"reflection". W5DXP has dropped that designation in favor of a
"redistribution" as described by the FSU web page. The word "reflection"
is reserved for describing something that happens to a single wave when
it encounters an impedance discontinuity. The word "redistribution" of
energy is adopted for describing what happens to the energy in canceled
waves. In like manner, since interference can occur with or without wave
cancellation, any reference to interference as the cause of the
redistribution of energy has been removed.

You convinced me that power doesn't flow.

Thank you, Jesus.

That was at least eight years ago, Jim. What is wrong with
you? I have no doubt that six months from now, you will again
be accusing me of believing that power flows. You seem to be
suffering from dementia.

Which of course explains how it is that your answers come out correctly.
I believe I already mentioned that it does (obviously) produce correct
answers, given all the underlying assumptions are correct.

However, correct answers don't seem to be enough for you.
You seem to be looking for a stone tablet handed down from God.

If you want, you can convert the Bird wattmeter reading to
irradiance by dividing by the cross-sectional area of the
coax.

Wouldn't I first have to buy into the idea that power is flowing through
it? :-)

No, irradiance, like power, doesn't flow. If you don't know
that, that's probably the source of your confusion.

It's hard to imagine how a simple, mutually agreed upon statement of
fact could be construed as a "false strawman implication", but there it is.

Your multiple unethical attempts to imply that I said power
has a phase angle is more than obvious to everyone.
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:

What is wrong with
you?

Glutton for punishment I guess. Thanks for asking.

73, ac6xg

Phase array question
 

Jim Kelley wrote:
Cecil Moore wrote:
What is wrong with you?

Glutton for punishment I guess. Thanks for asking.

Many years ago, five or more, I agreed with you that power
doesn't flow. Yet, more than a dozen times in the ensuing
years, you have accused me of believing that power flows.
I denied it every time, yet you did it again this very month.

Many months ago, I revised my energy article based on your
inputs. Yet, you continue to accuse me of promoting a "4th
mechanism for reflection caused by interference".

Incidentally, whether interference causes reflections or
not certainly depends upon the definitions of "interference"
and "reflection" which seem ill-defined to start with.

Whether your rraa behavior is deliberate or not, it points to
a personality disorder of some kind. I cannot find "glutton
for punishment" in the "Diagnostic and Statistical Manual of
Mental Disorders" but maybe you can plead 302.83 or 302.84.
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:
Many months ago, I revised my energy article based on your
inputs. Yet, you continue to accuse me of promoting a "4th
mechanism for reflection caused by interference".

Not everything is an accusation, Cecil. Please stop with the
melodramatics act. You wrote that bit about the 4th Mechanism of
Reflection in your paper against every bit of advice I had given you for
months. Buck up and take responsibility for your yourself for once.

73, ac6xg

Phase array question
 

Jim Kelley wrote:
You wrote that bit about the 4th Mechanism of
Reflection in your paper against every bit of advice I had given you for
months. Buck up and take responsibility for your yourself for once.

I bucked up and took responsibility 6 months ago.
When are you going act like a man and let it go?
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:
Jim Kelley wrote:
You wrote that bit about the 4th Mechanism of Reflection in your paper
against every bit of advice I had given you for months. Buck up and
take responsibility for your yourself for once.

I bucked up and took responsibility 6 months ago.

I missed your retractions. If you wouldn't mind, please repost them.
Thanks very much.

73 ac6xg

Phase array question
 

Jim Kelley wrote:
Cecil Moore wrote:
I bucked up and took responsibility 6 months ago.

I missed your retractions. If you wouldn't mind, please repost them.
Thanks very much.

One more time: "[10] Revision 1.1, Feb. 20, 2008 - In the
original version, the redistribution of energy due to wave
cancellation was dubbed a "reflection". W5DXP has dropped
that designation in favor of a "redistribution" as described
by the FSU web page. The word "reflection" is reserved for
describing something that happens to a single wave when it
encounters an impedance discontinuity. The word "redistribution"
of energy is adopted for describing what happens to the energy
in canceled waves. In like manner, since interference can occur
with or without wave cancellation, any reference to interference
as the cause of the redistribution of energy has been removed."
--
73, Cecil http://www.w5dxp.com

Phase array question
 

"Roy Lewallen" wrote in message
treetonline...
Not directly. What counts (considering the simple case of two elements) is
the magnitude and phase of the current in the other element, and their
spacing, orientation, and lengths.

Understood, thanks.

But your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two".

Yeah, some years ago I read Wheeler's 1947 paper on the fundamental limits
there (although one of my old professors, James McLean, was fond of mentioning
how many peoples' interpretation of that paper wasn't quite right... and he
wrote his own take on it back in 1996). In my case, I intend to use the usual
folding techniques to obtain significantly greater overall length than what
I'm allowed in any one dimension. When I look at such "meander line"
antennas, though, it's often not clear to me if the designer took coupling
between the folds into account or not, or instead just used some rules of
thumbs to avoid "significant" coupling and then empirically trimmed the
antenna to get the input impedance they were after.

The book I'd go to for researching the possibilities would be Lo & Lee's
_Antenna Handbook_.

Four volumes... wow!

You might also get some ideas from Bailey, _TV and Other Receiving
Antennas_, since TV antennas have to be pretty broadband.

That I have a copy of... I think I got it a few years ago when the topic of
how "rabbit ear" TV antennas actually "work" was being discussed. (These days
I think of them as similar to a tapered slot antenna, which is wideband.)

Thanks,
---Joel

Phase array question
 

Interesting paper on how the fundamental limits on the Q of antennas applies
to real-world devices: http://www.centurion.com/home/pdf/wp...ion_limits.pdf --
some antennas perform better than one might initially predict based on
coupling to adjacent ground planes (etc.) that effectively make the antenna
electrically larger than initially intended!

Phased array question
 

Ahhhhhhhh. That's better.

Ed, NM2K

Phase array question
 

Joel Koltner wrote:
"Roy Lewallen" wrote in message
treetonline...
Not directly. What counts (considering the simple case of two elements) is
the magnitude and phase of the current in the other element, and their
spacing, orientation, and lengths.

Understood, thanks.

But your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two".

But actually, that's not the principle.. The actual limits have to do
with the ratio of stored energy vs radiated power in the antenna (Q, in
the energy storage sense, not in the "resonant circuit" sense). For
example, a lot of the efficiency issues in practical antenna systems are
more to deal with the reactive component of the feedpoint impedance, and
the reactive/lossy network used to make it look like the feedline or
transmitter output impedance.

And the classical formulation also makes some not necessarily always
valid assumptions: linearity and reciprocity of components being the
notable one (Foster vs non-Foster terminations, for instance).

Phase array question
 

On Tue, 19 Aug 2008 09:17:01 -0700, Jim Lux
wrote:

But your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two".

But actually, that's not the principle.. The actual limits have to do
with the ratio of stored energy vs radiated power in the antenna (Q, in
the energy storage sense, not in the "resonant circuit" sense).

Hi Jim,

Different meanings of Q? The measure of Q may vary according to
arbitrary usage: the choice of SWR limits to define bandwidth which
infers Q. Some choose 2:1, classic Q would go further. Either way,
and for either quantitative result, the meaning of Q remains
essentially the same.

For
example, a lot of the efficiency issues in practical antenna systems are
more to deal with the reactive component of the feedpoint impedance, and
the reactive/lossy network used to make it look like the feedline or
transmitter output impedance.

That isn't Q, that is matching considerations. Certainly there is a Q
for the system and Q can be extremely high in detriment to getting
transmitting power out of the antenna. This is, albeit, a largely
unattainable situation, but try sending voice communications through a
very efficient 1M loop at 160M.

73's
Richard Clark, KB7QHC

Phase array question
 

Richard Clark wrote:
On Tue, 19 Aug 2008 09:17:01 -0700, Jim Lux
wrote:

But your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two".
But actually, that's not the principle.. The actual limits have to do
with the ratio of stored energy vs radiated power in the antenna (Q, in
the energy storage sense, not in the "resonant circuit" sense).

Hi Jim,

Different meanings of Q? The measure of Q may vary according to
arbitrary usage: the choice of SWR limits to define bandwidth which
infers Q. Some choose 2:1, classic Q would go further. Either way,
and for either quantitative result, the meaning of Q remains
essentially the same.


Actually, the Q you use is not the actual definition, which is the ratio
of the energy stored in the system vs the energy lost per cycle (or
possibly per radian, so there's factor of 2pi in there)

The articles that talk about size, efficiency, and Q, use this
definition (Energy stored in near field vs energy radiated away).
(e.g. the papers by Chu, Harrington, etc.)


For a single LCR tuned circuit with reasonably high Q, it just happens
that the BW/CF works out to the same thing (because it's a quadratic
equation that determines both..)


Chu, 1948, defines Q = 2*omega*mean electric energy stored beyond input
terminals/(power dissipated in radiation) (page 1170). he goes on to
say,"We have computed the Q of an antenna from the energy stored in the
equivalent circuit and the power radiated, and *interpreted it freely*
as the reciprocal of the fractional bandwidth." (my emphasis added) " To
be more accurate, one must define the bandwidth in terms of allowable
impedance variation or the tolerable reflection coefficient over the
band. For a given antenna, the bandwidth can be increased by choosing a
proper matching network. The theoretical aspect of this problem has
been dealt with by R.M. Fano."

Harrington, 1965, considering directional antennas (Chu dealt with
omnis) defines Q as = 2*omega*W/Pin [eq 54], which is slightly different
than Chu. W is either We or Wm (the energy stored in the E or H field
respectively), and Pin is the input power to the array. He goes on to
say,"If the Q is large, it is related to the frequency bandwidth of the
array as follows. Consider the array to be resonated by a suitable
reactance network at the frequency of interest, omega/sub/r. Define the
ferquency bandwidth of the array in the usual manner to be the
fractional frequency increment between the 0.707 points on the
normalized input |Z|,
beta = deltaOmega/Omega/sub/r. [eq 56]

If the Q is high (say Q10) then we have the relationship:
Q approx= 1/beta [eq 57]"


Note well the "approximately equal" and the "resonated by a suitable
reactance network"


SWR bandwidth is something totally different of course..

But again, if you pick the right SWR value for your bandwidth
measurement, then a dipole antenna is modeled pretty well by a single
pole LCR resonant, so the math works out conveniently the same.

Once you start straying away from antennas that can be modeled as a
single LCR, the "bandwidth" vs "Q" relationship goes away.

A good example would be some forms of phased arrays with non-reciprocal
devices. Antennas with multiple resonances would be another case
(except if you are only working over a restricted range, where the
single resonance in your range is approximated well by a single LCR)

Phase array question
 

On Tue, 19 Aug 2008 12:36:30 -0700, Jim Lux
wrote:

Richard Clark wrote:
On Tue, 19 Aug 2008 09:17:01 -0700, Jim Lux
wrote:

But your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two".
But actually, that's not the principle.. The actual limits have to do
with the ratio of stored energy vs radiated power in the antenna (Q, in
the energy storage sense, not in the "resonant circuit" sense).

Hi Jim,

Different meanings of Q? The measure of Q may vary according to
arbitrary usage: the choice of SWR limits to define bandwidth which
infers Q. Some choose 2:1, classic Q would go further. Either way,
and for either quantitative result, the meaning of Q remains
essentially the same.


Actually, the Q you use is not the actual definition, which is the ratio
of the energy stored in the system vs the energy lost per cycle (or
possibly per radian, so there's factor of 2pi in there)

Hi Jim,

If you will note, my example was a measure, not a definition of Q.
The half power points bandwidth compared to the center frequency is a
classic computation of Q. As I point out, the choice of 2:1 SWR, not
being half power points, is arbitrary but in no way diverges from the
sense of Q (and could be extrapolated anyway).

The articles that talk about size, efficiency, and Q, use this
definition (Energy stored in near field vs energy radiated away).
(e.g. the papers by Chu, Harrington, etc.)

Use "which" definition? You offer what appear to be two, and that is
one too many.

For a single LCR tuned circuit with reasonably high Q, it just happens
that the BW/CF works out to the same thing (because it's a quadratic
equation that determines both..)


Chu, 1948, defines Q = 2*omega*mean electric energy stored beyond input
terminals/(power dissipated in radiation) (page 1170). he goes on to
say,"We have computed the Q of an antenna from the energy stored in the
equivalent circuit and the power radiated, and *interpreted it freely*
as the reciprocal of the fractional bandwidth." (my emphasis added) " To
be more accurate, one must define the bandwidth in terms of allowable
impedance variation or the tolerable reflection coefficient over the
band. For a given antenna, the bandwidth can be increased by choosing a
proper matching network. The theoretical aspect of this problem has
been dealt with by R.M. Fano."

Harrington, 1965, considering directional antennas (Chu dealt with
omnis) defines Q as = 2*omega*W/Pin [eq 54], which is slightly different
than Chu. W is either We or Wm (the energy stored in the E or H field
respectively), and Pin is the input power to the array. He goes on to
say,"If the Q is large, it is related to the frequency bandwidth of the
array as follows. Consider the array to be resonated by a suitable
reactance network at the frequency of interest, omega/sub/r. Define the
ferquency bandwidth of the array in the usual manner to be the
fractional frequency increment between the 0.707 points on the
normalized input |Z|,
beta = deltaOmega/Omega/sub/r. [eq 56]

If the Q is high (say Q10) then we have the relationship:
Q approx= 1/beta [eq 57]"

Hence the two definitions? What I see are computational models for
different systems which arbitrarily restrain loss to inhabit their
model or exclude it. Q can deteriorate considerable if you open the
definition to include more components going back to the power supplies
to the finals.

Note well the "approximately equal" and the "resonated by a suitable
reactance network"


SWR bandwidth is something totally different of course..

But again, if you pick the right SWR value for your bandwidth
measurement, then a dipole antenna is modeled pretty well by a single
pole LCR resonant, so the math works out conveniently the same.

Once you start straying away from antennas that can be modeled as a
single LCR, the "bandwidth" vs "Q" relationship goes away.

A good example would be some forms of phased arrays with non-reciprocal
devices.

This example needs a sub-example: Non-reciprocal devices? I don't see
how this will alter the concept of Q.

Antennas with multiple resonances would be another case
(except if you are only working over a restricted range, where the
single resonance in your range is approximated well by a single LCR)

All antennas have multiple resonances and it is a classic
differentiator between themselves and LCR circuits. I see no example
here that is not already offered by my original post.

Let's simply return to the quote I responded to:
On Tue, 19 Aug 2008 09:17:01 -0700, Jim Lux
wrote:
(Q, in the energy storage sense, not in the "resonant circuit" sense)
embodies two explicit definitions of Q:
1. Q for energy storage, and
2. Q for "resonant circuit."
The measurement of Q might deviate by the result offered, but
conceptually energy storage and resonance are inextricably congruent.
If by resonance you are suggesting ONLY the peak frequency, then it
follows there is(are) some other frequency(ies) that are not resonant
(a tautology), and Q follows by exactly the same relation and degree
as it does for energy storage.

If there is any suitable distinction of Q in an antenna, then it is
that the degradation of Q is a benefit to the antenna, IFF the
substantial portion of R is radiation resistance. If the benefit of
filtering (phasing) due to resonance were not an issue, then a Q of 1
would be the Holy Grail of antenna design. Even in the design of the
finals stage in a tube amplifier, Terman teaches us that the final's
tank should NOT have an excessive working Q (beyond 10-15); hence a
high Q is NOT beneficial. Q and efficiency are a slippery topic when
you try to tie them together.

73's
Richard Clark, KB7QHC

Phase array question
 

On Aug 8, 1:13 pm, Roy Lewallen wrote:
Joel Koltner wrote:
"Roy Lewallen" wrote in message
streetonline...
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.")

Not directly. What counts (considering the simple case of two elements)
is the magnitude and phase of the current in the other element, and
their spacing, orientation, and lengths. A good way to look at the
effect of mutual coupling is as "mutual impedance", i.e., the amount of
impedance change caused by mutual coupling. (Johnson/Jasik covers this
concept well.) If you were to feed two elements with constant current
sources (as in the Cardioid.EZ EZNEC example), mutual coupling doesn't
change the element currents, but only the feedpoint impedances. With any
other kind of feed system, the impedance change causes the currents to
change, which in turn affects the impedances. So the feed method
certainly does have an effect on the currents you get, which affects
both mutual coupling and pattern.

There's a lot more about this, and how to design feed systems which will
effect the desired currents, in the _ARRL Antenna Book_.

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.)

I just recently purchased Miron's book but haven't yet looked at it in
any depth. It appears to be most interesting to anyone wanting a better
understanding of method of moments numerical methods. If you can read
German, you might be interested in _Kurze Antennen_ by Gerd Janzen. But
your search for small, broadband antennas puts you bump-up against the
principle "small - broadband - efficient, choose any two". They'll be
inefficient, which will hurt you both receiving and transmitting at VHF
and above. The book I'd go to for researching the possibilities would be
Lo & Lee's _Antenna Handbook_. You might also get some ideas from
Bailey, _TV and Other Receiving Antennas_, since TV antennas have to be
pretty broadband.

Roy Lewallen, W7EL

Simply in an effort to provide a bit more insight, or perhaps I should
better say to suggest a math tool that may lead you to more insight,
consider what "mutual impedance" means. In a simple circuit where
there's a current through an impedance, the voltage drop across it is
given by V = Z * I. If you expand this idea to include mutual
impedances, the Z becomes a matrix, and I and V are vectors. So in an
antenna system with, say, four feedpoints, V becomes a vector of four
voltages, one for each feedpoint, and I similarly is a vector of four
currents, one for each feedpoint. Z is then a four-by-four matrix
with self-impedances along the diagonal and mutual impedances off the
diagonal. It is clear that if you know the four currents, you can
find the voltages. Further, if you can invert the Z matrix, then you
can calculate the currents if you know the voltages. That also
suggests how to find the mutual impedances: if you excite one
feedpoint with a known current and leave all the rest open, you can
measure the voltages at each (including phase) and that immediately
gives you the mutual impedance from the excited feedpoint to each of
the others: your I vector has only one non-zero component. Repeat
for each feedpoint.

You can use the same sort of analysis with other systems which have
interaction among components. For example, a system of inductors
which share magnetic fields can easily be characterized by a matrix of
self-inductances and mutual inductances. For a single inductor, V =
L*di/dt; if you have two coupled inductors, V1 = L1 * di1/dt + M12 *
di2/dt, and V2 = M21 * di1/dt + L2 * di2/dt -- or in matrix notation,
V = L * d/dt(I). That can expand to as many inductors as you care to
consider.

(If this is confusing, it's probably best to just ignore this
suggestion...)

Cheers,
Tom

Phase array question
 

"K7ITM" wrote in message
...
(If this is confusing, it's probably best to just ignore this
suggestion...)

Not at all, thanks for the additional details, Tom.

Somewhere I have copies of IEEE tutorial articles on this sort of generalized
circuit theory... they ended up as references for work I did in college. (We
spent the bulk of our time worrying about scattering parameter matrices,
however. I did write some Matlab code to do n-port conversions between S, Y,
and Z parameters, though -- but based off of formulas from papers: I question
if I could have correctly derived the conversions between, e.g., S and Y
myself in a reasonable period of time, since the port terminations for the
scattering parameters were allowed to be arbitrary at each port and the math
for this requires a decent background in linear algebra.)

---Joel

Phase array question
 

Jim Kelley wrote:
Cecil Moore wrote:
Ptotal = P1 + P2 + 2*SQRT(P1*P2)*cos(A)

According to fig. 7.1 in Born and Wolf, that's useful for showing how
light intensity varies as a function of phase, and hence position. It's
just that there's no valid way to multiply by the cosine of the angle
between two scalars.

I suspect you knew if I ever found my Born and Wolf after my
move, you would be in trouble - and you are.

You previously said that Born and Wolf did not agree with
Hecht, but they do, contrary to your assertions.

Their equation for irradiance (intensity) agrees with Hecht.

Itot = I1 + I2 + J12

where J12 = 2E1*E2 = 2*SQRT(I1*I2)*cos(A)

On page 258 of "Principles of Optics", by Born and Wolf,
4th edition, they label J12 as the *interference term*,
contrary to your assertions. (Hecht labels that term I12)
--
73, Cecil http://www.w5dxp.com

Phase array question
 

Cecil Moore wrote:
Jim Kelley wrote:
Cecil Moore wrote:
Ptotal = P1 + P2 + 2*SQRT(P1*P2)*cos(A)

According to fig. 7.1 in Born and Wolf, that's useful for showing how
light intensity varies as a function of phase, and hence position.
It's just that there's no valid way to multiply by the cosine of the
angle between two scalars.

I suspect you knew if I ever found my Born and Wolf after my
move, you would be in trouble - and you are.

No. I assumed you could find fig 7.1 and see it for yourself. The plot
has phase on the X-axis and intensity on the Y-axis. The caption reads
Interference of two beams of equal intensity; variation of intensity
with phase difference. " At the top of the page: "Let us consider the
distribution of intensity resulting from the superposition of two waves
which are propagated in the z-direction...."

The relation is useful for precisely the reason I indicated. That is
why Heckt also includes it in his book.

I deleted the rhetorical blithering from your post. And again, please
quote my remarks directly whenever you wish to discuss them.

73, ac6xg

Phase array question
 

Jim Kelley wrote:
The relation is useful for precisely the reason I indicated. That is
why Heckt also includes it in his book.

Your Freudian slip concerning your feelings about Hecht
is more than apparent, i.e. "To Heck with Hecht"! :-)

One wonders why you said something to the effect that
Hecht had been discredited in favor of Born and Wolf.
When I recommended the 57 page chapter on interference
in "Optics", by Hecht, you said something to the effect
that interference is unimportant, yet Born and Wolf's
chapter on interference is 113 pages long and mostly
agrees with Hecht's writings.
--
73, Cecil http://www.w5dxp.com