[TowerTalk] Dipole length vs diameter -- a theory question

[jimlux]

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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