[TowerTalk] Dipole length vs diameter -- a theory question
jimlux
jimlux at earthlink.net
Tue Jan 28 22:24:40 EST 2020
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 at 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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