[TowerTalk] Antenna Formulas

[Jim Lux]

At 07:39 AM 5/9/2005, Daniel Hileman wrote:
>Hi Everyone,
>Now I'm gonna feel real stupid asking this...buuuuut,
>I really would like to know...so, here goes. How come
>the formula for a half wave dipole is 468/Freq but the
>formula for a Full wave loop is 1005/Freq. I guess My
>real question is, why is the loop formula 1005/Freq.,
>and not just 936/Freq ??  I've always wondered that,
>and just never asked, so NOW I am, lol. Like for
>instance, if I wanted to make a 3/8 wave inverted "L"
>would I use 377/freq or 351/freq??
>
>Thanks and 73,
>Daniel N9WX


Not a stupid question at all.  It's one that they spend quite a bit of time 
on in Antennas classes, because, at first glance you'd think that it should 
be exactly a half wavelength long ( half wavelength (feet) = 
491.7/Frequency(MHz)).  It has to do with where the antenna is "resonant" 
(i.e. the feed point is purely resistive) versus where there's exactly a 
halfwavelength long.

A center fed infinitely thin wire half a wavelength long in free space 
would have a feedpoint impedance that's somewhat inductive 
(73.08+42.5j).  Since, by and large, antennas that are purely resitive are 
preferred, you can slightly detune the antenna from an ideal half wave by 
shortening it, essentially cancelling out the inductive component. Note 
that a reactive feedpoint impedance isn't necessarily bad: It just means 
that some of the energy in the antenna is stored in the reactance connected 
to the antenna, as well as in the magnetic field around the antenna itself. 
(leaving aside issues like feedline loss, matching network losses, or 
whether your box at the feedpoint can tolerate a reactive load)

This trick of lengthening or shortening an antenna to change the feedpoint 
reactance (it doesn't change the pattern very much.. certainly not 
measureably, for small few percent changes) is used in things like quad 
helixes and turnstiles, where you want the two antennas to be out of phase 
with each other.  Make one a bit long, and the other a bit short, so the 
parallel combination of the two is just right. The current in the long one 
will lag the current in the short one.

Furthermore, since most people don't build infinitely thin antennas in free 
space, there's ALWAYS something around the antenna that will interact with 
the currents in the antenna, which will always have some effect on the 
impedance.  So, the advice to "prune to resonance", or, given that dipoles 
are a pretty broad band device anyway, you can come up with a "rule of 
thumb" (like 468/Freq) that is "close enough", and you won't be that far off.
(interestingly, an infinitely thin doublet, 468/F long (47.6% of free space 
wavelength), would have a feedpoint impedance of 63.4-29j.  It doesn't get 
to resonance until you get a bit thicker: somewhere around 0.0015 
wavelength in radius (which would be 3cm at 20m), which is quite fat.)


Now, as to "why is a half wave antenna not resonant".  There are a lot of 
ways to look at the problem, of varying degrees of fidelity to 
electromagnetic theory. You'll see names like Schelkunoff, Pocklington, 
King, and Hallen turning up in these derivations, where they start with 
some mathematically tractable shape (like a transmission line that is 
gradually opened out, or a pair of cones) and then manipulate from 
there.  A very common analysis for "thin wire antennas" is to assume that 
there's a wave propagated from the feedpoint, which is reflected back from 
the end, forming a standing wave.  All very nice if radiation didn't exist, 
because you could solve it with the classical "wave equation" used for 
things like vibrating strings, etc. (But, if this WERE the case, then a 
resonant antenna would be exactly a free space half wave, and it's not..)


Suffice it to say that the reality is somewhat complex, in that antennas 
look like a lossy(i.e. radiating) transmission line, where you have an 
interesting interplay between charge moving back and forth in the wire and 
the electromagnetic fields around the wire.  The charge moving (i.e. 
current) in one section of the wire induces a voltage in all the other 
sections of wire, which, in turn results in a current.  Not only that, but 
when charges accelerate, some of the energy is radiated away.  Some of the 
energy radiates to the far field, but a goodly portion actually flows back 
into the physical antenna.

Putting a dielectric around the wire (or immersing it in a dielectric, as 
in a ground radial) changes how fast those travelling waves propagate, 
which changes the resonance.  Partly it's a propagation speed 
effect  (1/sqrt(epsilon)) and partly it's a charge storage in the 
dielectric effect (think of it as distributed  capacitive loading); But, 
really, they're the same thing, when you get right down to it, because the 
reason there's increased capacitance is because the EM wave propagates 
slower in the dielectric.



If you're really interested, check out the chapter on "Self and Mutual 
Impedances" in the textbook Antennas, by John Krauss (W8JK,SK).  The 
textbook by Balanis also covers this, but isn't nearly as accessible (in my 
opinion).  There's also an online electromagnetics textbook by a EE 
professor at Rugets, S.J.Orfanidis: 
http://www.ece.rutgers.edu/~orfanidi/ewa , but I didn't find his 
explanation particularly revelatory (his book is quite handy in other ways, 
with lots of useful equations and stuff, and even better, a bunch of 
downloadable Matlab routines (which will also work with Octave, as 
mentioned in the lastest QEX))


Summing up... all those equations like 468/F or 1005/F essentially embody 
empirical experience, and almost never do you need "gnat's eyelash" 
precision for simple low gain antennas.  Use the rule of thumb, throw up 
the wire, and be done with it.  Any differences between equation and 
reality can be taken care of in the tuner, and if you're only a few percent 
off, the losses won't be all that great.

In my opinion, trying to rationalize or conceptualize why it's so in terms 
of "dielectric loading" or "end effects" or "isotropic self capacitance 
from large diameter tubes", etc., just gets in the way in the long 
run.  Those sorts of approximations work for a limited number of cases, 
and, most troubling, using such approximations doesn't give you any insight 
into where or why the approximation breaks down.  Developing that insight 
is not easy, if even possible in the general case. Presumably, if you make 
your living as an antenna engineer, you MIGHT have that insight, because if 
you didn't, you probably wouldn't have chosen that career. Some people are 
really good at it, some aren't. (I'm in the latter category... my brain 
does not "see" electromagnetic fields, like really gifted antenna folk do.)

{DANGER WILL ROBINSON!!! THIS IS NOT A TROLL!!} Look at all the discussions 
about current distributions in antennas with loading coils, most of which 
(in my opinion) are due to trying to understand a really complex system 
with too-simple models.


These days, with fast computers, if you need sub 1% accuracy, you might as 
well just numerically model it, and all the gory EM details are buried in 
the code.  Nobody  (well, very few people) in the professional antenna 
world agonize about analytical models these days. You use the approximation 
to get your design started, then let the fine details be ground out on the 
computer.


Jim, W6RMK