On 1/28/20 4:42 PM, Edward Mccann wrote:
Gentlemen:
You have posed a most interesting question that has gone dormant in ham
ranks 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
There was an article about this a while ago - I think it's someone who
miscopied it from handwritten notes of empirical data, and it got
captured. Ward Silver wrote it up.
Also at: https://www.kb6nu.com/468-ham-radios-magic-number/
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",
accumulation of charge at the end of the wore that increase capacitance
and affects length for some time.
In fact, in the early 1970s, while studying under Lan Jen Chu at 77
Massachusetts Avenue, I asked him that 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!
We had a good laugh and went to lunch.
Yup. and the usual quick numerical approximation is King's.
Prof. Orfanidis has a free to download text book and accompanying matlab
code:
https://www.ece.rutgers.edu/~orfanidi/ewa/ Probably chapter 24 or 25 is
what you want.
If you're hardcore, Pocklington's paper is online, as is a paper by John
Strutt (Lord Rayleigh) with some corrections. and then the paper by
Rayleigh you reference below.
Kraus Antennas 2nd Ed also has a nice writeup on the various equations
(self and mutual Z)
In recent years, I unearthed every QST back to Noah;s Ark Maritime
Mobile station, only to find the hand-wavey solutions referred to by
David. ARRL Technical Guest 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 of
the situation.
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 = 2. (See Abraham, Ann. der Phys., 66, 435, 1898, /Die
electrischen Schwingungen um einen stabformigen 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 as to which was correct lambda/2 =l or
lambda/2.53=L for a number of years, until Lord Rayleigh himself in a
three page note 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 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 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 limits.
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. He had better thing 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.
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
the 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 on the subject, should you have enough bourbon and firewood
to read through them all.
This is an incredible rabbit hole to dive down. Not necessarily useful,
but interesting.
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 into 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 sufficient interest.
Orfanidis's book with the Matlab has been my go-to source for quick
computer software answers if I don't want to fool with NEC. I've
converted a number of his routines to Python (since I'm moving to
SciPy/NumPy instead of Octave/Matlab)
He has sine and cosine integral functions.
73 to you inquiring minds!
Ed McCann
AG6CX
Sausalito
*********************************************************
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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