Gentlemen:
(I tried to send this with the three attachments, but got the bounce from Yahoo
for too big a file. (7.8 MB, 680 KB and !.5 MB, respectively)
If you want the attachments and they didn't get through to you, please contact
me at AG6CX1@gmail.com.
I also got a bounceback for Larry Banks alum,mit.edu email. Larry, if you want
the attachments, and they don't get through, let me know where to send them.
Original message text follows below.)
You have posed a most interesting question that has gone dormant in ham ran=
ks ever since ARRL and others started trying to explain the difference
between the "468" and the "492" in the canned formula for dipole length
calculation. The Antenna Book gave up on detail on the matter so long ago I
can't find it either.
But, first of all, David is quite right:
"That whole "self capacitance end effect" is a hand-wavey thing that is a
conceptual explanation that isn't particularly accurate, but does seem
to work."
For a variety of reasons, I have been looking at the "end-effect", accumulated
charge at the end of the wire that increase capacitance and affects
length for some time.
In fact, in the early 1970s, while studying under Lan Jen Chu at 77 Mass
Avenue, I asked him that very question. His response was that if I were
really interested in the physics of the matter I should take the bus down to
Harvard Square and ask for Professor (RWJ) King! It was Physics there and EE
here.
We had a good laugh and went to lunch.
In recent years, I unearthed every QST back to Noah;s Ark Maritime Mobile
Edition, only to find the hand-wavey solutions referred to by David.
ARRL Technical Desk could offer nothing more than a few comments.
The actual investigation of this topic goes back to the day when there were
only a few closed form solutions, largely based on the geometry.
In 1898. Abraham (the German Physicist, not the prophet!) calculated the free
period of an infinitely extended but rather narrow metallic ellipsoid
of revolution when excited by an electrical impulse. Cutting to the chase, he=
found to a god approximation that the fundamental natural period was
related to the major axis length by the expression lambda/L =3D 2. (See
Abraham,=
Ann. der Phys., 66, 435, 1898, Die electrischen Schwingungen um einen stab=
formigen Leiter, behandelt nach der Maxwell'schen Theorie)
In 1902 Macdonald, a Scot, was awarded a prestigious prize when he solved
a similar math problem, but came up with the answer lamba/L = 2.53. (See
MacDonald, Electric Waves, page 111-112)
A pissing contest rage for years as to which was correct, lambda/2 =L or
lambda/2.53=L
until Lord Rayleigh himself in a three page piece in Philosophical Magazine,
VIII, page 105-107, 1904, On the Electrical Vibrations Associated With Thin
Terminated Conducting Rods.
He offered that while he had not followed Abraham's thesis in detail, he saw
no reason to distrust it. He goes on to quote from Abraham that in an
elongated ellipsoid of an infinitely thin rod, taken to limit (which , if,
using
your imagination, you stretch the ellipsoid far enough, you get a linear
wire) , you get a second approximation including a term lambda=2L(1 + 5.6 x
epsilon**2), where epsilon
lies in the expression 1/epsilon =4 log (2L/d), where d = diameter of the
conductor.
For a bunch of values, the correction factor ends up being on the order of 4-5%.
(To this figure the hand wavers sometimes add another five percent for
insulated vs non-insulated wire, but that's another story for another day.)
Aren't you glad you asked?
Remember, these guys were trying to solve Maxwell's Equations in elliptical or
spherical coordinates,
with whole scale integration that must have taken a host of
school boys to figure out. And it was for an ellipsoid of perfectly
conducting wire in infinite system, that became a thin rod, (sort of) at its
limit.
If truncated, and terminated (in say an insulator!) the correction factor
might account
for the unexpected shortening (or lengthening) due to
accumulating charge (which you can read as an increase in capacitance.)
I once asked Kirk McDonald, of Princeton, why he didn't have a crew of
freshmen solving the problem as an extension to his great work in
electrostatics and electromagnetics. He responded that he had better things to
do.
I concluded (accurately) I fear, that no one really gave a fig, and the 5% =
for "end effect offered by ARRL led to the 468 vs 492, oh well, its only
4.87% difference. That meant cutting the dipole for 133.71 feet instead of
140.57 feet, running her up the pole, and checking the SWR.
A good pal, Rick DJ0IP. a master of the OCFD multiband antenna, , common
mode chokes, and baluns required to keep the OCFDs radiating without too much
RF in the shack, would concur that my fear was probably well-founded, and urge
me to move on.
However, the thread you guys put on TowerTalk revived my hope that optimism
springs forward in the human heart.
Hence, you have awakened the sleeping dragon, and I offer attachments of th=
e two papers I mentioned above, plus the most recent on the topic, from
C.R. Englund, in the Bell Systems Technical Journal ( BSTJ, Vol7, 1928,
The Natural Period of Linear Conductors)
You will be pleased to have at your fingertips a world-class bibliography o=
n the subject, should you have enough bourbon and firewood to read through =
them all.
Thanks for dredging up such an interesting subject.
By the way, my schoolboy German has long since vanished, but I'm happy to
put $50 in the pot if anyone has an interest and a source of someone to
create a reasonably-priced translation of Abraham's fine paper. My flexibility
(if ever such existed!) with Cosine and Sine integrals have also slipped in=
to the fog, and it is so easy to look towards Livermore Labs or wherever
the finite element solution EZNEC and NEC lives. Let me know if there is suff=
icient interest.
73 to you and your inquiring minds!
Ed McCannAG6CXSausalito CA
On Tuesday, January 28, 2020, 2:14:08 PM PST, jimlux <jimlux@earthlink.net>
wrote:
On 1/28/20 12:19 PM, David Gilbert wrote:
>
> As you point out, the resonance of a conductor is determined by length
> (inductance) and diameter (distributed capacitance to itself). I don't
> know the formula for that either, but I'm pretty sure that whatever you
> get by reply to your question will be for a straight conductor. A bent
> conductor like your halo will have somewhat more capacitance to itself
> than a straight one.
It's not exactly accurate to relate length to inductance and diameter to
capacitance for determining antenna resonant frequency. The dominant
factor is the length. Changes in diameter will change the impedance
bandwidth but not the resonant frequency (very much).
The K-factor graph can be derived semi-analytically - there are several
analytical expressions for the complex impedance of an antenna (and you
can solve for where X is zero) over a restricted range. Or, you can
numerically integrate the field equations - which is what people have
been doing since the late 1800s.
That whole "self capacitance end effect" is a hand-wavey thing that is a
conceptual explanation that isn't particularly accurate, but does seem
to work.
As Dave says - the way you solve this is to use a method of moments code
(like NEC and its ilk) which numerically integrates the electric field
equation.
EZNEC (and NEC) do not model "capacitance" per-se. What they model is
the current induced in a small piece of the antenna by the currents
flowing in all the other pieces of the antenna, subject to the
constraint that the voltages on the ends of connected pieces are the same.
It basically sets up a huge set of simultaneous equations (the
admittance matrix) and then solves it.
>
> Also, proximity has more effect at high voltage positions than at low
> voltage positions ... which is how top hats work.
>
> All that is why I usually just generate an EZNEC+ model, which at least
> tries to geometrically take into account distributed capacitance. As a
> general rule, almost every model I've ever done says that as I increase
> the width (as long as it's an appreciable percent of a wavelength) the
> resonant frequency goes down and the bandwidth increases ... but
> configuration has a large effect.
>
> 73,
> Dave AB7E
>
>
>
> On 1/28/2020 11:19 AM, Larry Banks via TowerTalk wrote:
>> Hi TTers,
>>
>> A friend of mine asked me what first appeared to be a simple
>> question. Paraphrasing:
>>
>> “How do I calculate the length of my HB 2M halo, based on
>> the diameter of the aluminum rod. Is it like propagation
>> velocity with coax?”
>>
>> My quick answer was: “No, propagation velocity only relates to
>> transmission lines. Use the graph in the literature for your design
>> to start. Modeling will help. But let me do some research.”
>>
>> I had realized that I really didn’t know the answer. I have looked in
>> my two usual places: the ARRL Antenna Book and Wikipedia and found
>> lots of hand-waving and the usual references to the “K-factor” graph,
>> which appears to be derived experimentally. BUT NO THEORY, other than
>> vague references to the capacitance and inductance of the rod changing
>> with dimensional changes which, in fact, is similar to transmission
>> lines.
>>
>> Do any of you have a reference to some real theory and an equation
>> that allows me to calculate this based on length, diameter, and
>> material characteristics? (Ignoring environment effects of course.
>> This would be for free space.)
>>
>> 73 -- Larry -- W1DYJ
>>
>> _______________________________________________
>>
>>
>>
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