Does reactance of dipole depend on diameter ??

I wish to know if the reactance of a dipole that is physically 0.5000 wavelengths in length depends on the diameter of the wire or not.

I know a dipole 0.5 wavelength long is not resonate, but inductive so you need to shorten it a few percent to bring it to resonance. I know the length at resonance depends on wire diameter.

But I'm interested if the reactance does very with wire diameter when the antenna is physically 0.5 wavelengths long, which means it will be somewhat inductive.

A book published by the ARRL by the late Dr. Laswon (W2PV)

Lawson J. L., ?Yagi Antenna Design?, (1986), The American Radio Relay League. ISBN 0-87259-041-0

has a table of reactance vs the ratio K (K=lambda/a, where a is the radius) for antennas of 0.45 and 0.50 wavelengths in length. I've reproduced that table below.

The first column (K) is lambda/a

The second column (X05) is the reactance of a dipole 0.5 wavelengths in length.

The third column X045 is the reactance for a dipole 0.45 wavelengths in length.

K X05 X045

------------------------- 10 34.2 23.1 30 36.7 6.4 100 38.2 -14.1 300 39 -33.6 1000 39.6 -55.5 3000 40 -75.7 10000 40.4 -98.1 30000 40.6 -118.6 100000 40.8 -141.1 300000 41.0 -161.8

1000000 41.1 -184.4

What one notices is:

1) Reactance for 0.45 lambda is very sensitive to radius, varying by more than 200 Ohms as K changes from 10 (fat elements) to 1000000 (thin elements).

2) The value for a dipole 0.5 lambda in length changes much less (only

6 Ohms), but it *does* change.

3) For infinitely thin elements (K very large), the reactance of a dipole 0.5 lambda in length looks as though it is never going to go much above 41.2 Ohms. Certainly not as high as 42 Ohms.

Now I compare that to a professional book I have:

Balanis C. A., ?Antenna Theory ? Analysis and Design?, (1982), Harper and Row. ISBN 0-06-0404458-2

There is a formula in Balanis' book for reactance of a dipole of arbitrary radius and length, in terms of sine and cosine integrals. It's hard to write out, but the best I can do gives:

Define:

eta=120 Pi k=2/lambda

reactance = (eta/(4*Pi)) (2 SinIntegral[k l] + Cos[k l]*(2 SinIntegral[k l] - SinIntegral[2 k l]) - Sin[k l]*(2 CosIntegral[k l] - CosIntegral[2 k l] - CosIntegral[(2 k a^2)/l]));

where 'a' is the radius.

(It's in Mathematica notation)

What is interesting about that is that for a length of 0.5 lambda, the reactance does not depend on wavelength at all - it is fixed at 42.5445 Ohms. So two different books give two quite different results.

Numerically evaluating the above formula gives this data.

K X05 X045

------------------------- 10 42.5 35.7183 30 42.5 15.5269 100 42.5 -6.79382 300 42.5 -27.1632 1000 42.5 -49.4861 3000 42.5 -69.8555 10000 42.5 -92.1784 30000 42.5 -112.548 100000 42.5 -134.871 300000 42.5 -155.24

1000000 42.5 -177.563

Does anyone have any comments? Any idea if Balanis's work is more accurate? It is more up to date, but perhaps its an approximation and the amateur radio book is more accurate. (The ham book seems quite well researched, and is not full of the voodoo that appears in a lot of ham books).

BTW, I'm also looking for an exact formula for input resistance of a dipole of arbitrary length. I know is 73.13 Ohms when 0.5 wavelengths long, but I'm not sure exactly how much it varies when the length changes (I believe it is not a lot).

Dave

david kirkby onetel net

Reply to
Dave
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Oops, I made a couple of mistakes there:

Dave wrote:

k = 2 Pi / lambda,

not 2 / lambda.

You can possibly see that when the length is 0.5 lambda, the sine term in there is always zero, so the radius 'a' has no effect on the reactance.

Sorry, I mean the reactance does not depend on radius when the dipole is

0.5 wavelengths in length.
Reply to
Dave

Yes, it does.

This is the formulation by S.A. Schelkunoff, which is one of many approximations to the general problem of finding the reactance of a simple cylindrical dipole of arbitrary length and diameter. The general problem was attacked for decades by some very skilled mathematicians and engineers including R.W.P. King, David Middleton, Charles Harrison, G.H. Brown, D. D. King, F. G. Blake, M.C. Gray, and others. You'll find their works scattered about the IRE (now IEEE), British IEE, and various physics journals. The problem can't be solved in closed form, so all these people proposed various approximations, some of which work better in some situations and others in others. A good overview can be found in "The Thin Cylindrical Antenna: A Comparison of Theories, by David Middleton and Rolond King, in _J. of Applied Physics_, Vol. 17, April 1946.

As I mentioned above, some approximations are better in some circumstances (e.g., dipoles of moderate diameter near a half wave in length) and some in others (e.g. fat dipoles or ones near multiples of a half wave in length). I don't know which is better for your particular question. The easy way to find out is to get one of the readily available antenna modeling programs, any of which is capable of calculating the answer to very high accuracy, and compare this correct answer with the various approximations you find published.

There is no exact formula for that, either. Calculating an exact answer requires knowledge of the current distribution, which is a function of wire diameter. Assuming a sinusoidal distribution gets you very close for thin dipoles, but it's not exact. You'll find calculations based on this assumption in just about any antenna text such as Balanis or Kraus. But again, you can get extremely accurate results from readily available antenna modeling programs.

Roy Lewallen, W7EL

Reply to
Roy Lewallen

Hi Dave,

Yes, it does.

You are working with source material with conflicting agendas. One is simply interested in what is called a dipole for the sake of field studies and the characteristics of that dipole are a good first order approximation. This means thin-wire by and large. The other source is examining the antenna itself (or so it seems by both accounts).

The fatter the wire, the lower the inductance. Naturally the reactance must follow. The fatter the wire, the more wavelength it encompasses for a given length, hence the length can be shorter for resonance. This shorthand hardly matters for conventional wire antennas as "fat" is in the extreme, and wire is hardly the proper nomenclature when we get into these gross dimensions.

Approximations of "fat" come with cage structures that attempt to mimic a solid of revolution.

If you want to find the author who developed the first principles of thin vs. fat, that is Dr. Sergei Alexander Schelkunoff (with Friis).

In what has been decried in this forum as the failed metaphor of an antenna as transmission line, the antenna formulas from Schelkunoff were derived from (beat) a transmission line, albeit a special one.

To attempt to draw parallels between transmission lines and antennas is fraught with failures, true. Specifically, the traditional dipole in its thin-wire implementation has no linear Impedance relationship along its length. The wire separation is always growing with distance from the feed point and thus the Z varies with distance. This failure was anticipated by Schelkunoff, and folded into field theory through using conic sections for the dipole arms. Hence the biconical dipole, the conical monopole, and the discone. The transmission line analogy survives through this legacy.

All formulas that you have probably recited are the degenerative forms for his based on the conic sections.

Now as to that degeneration of the conic section into "thick" wire to "thin" wire. The conic section is certainly thick at the distal end, no doubt there. It is also thin at the feed point. The advantage is lowered capacitance bridging the feedpoint compared to that if the thickness were constant from the distal end - for a given thickness/length/resonance. Also the conic sections most nearly approach the shape of the emerging wave's initial spherical front.

Well, the long and short of it is to seek: "Antennas: Theory and Practice," Sergei A. Schelkunoff and Harald T. Friis, Bell Telephone Laboratories, New York : John Wiley & Sons, 1952.

73's Richard Clark, KB7QHC
Reply to
Richard Clark

First, as you point out one book is using an approximation where the other may be using calculated data. I believe the approximations start by assuming a perfectly sinusoidal current distribution, which may not be entirely correct, but does make the math easier. The ham radio book would be more likely to use output from an antenna modeler, since antennas are an area where theory gets damn complicated damn quick, but the modelers can do a pretty good job if you treat them right.

Second, check to see if they're both using the same equivalent circuit

-- if one is looking at parallel equivalent reactance and the other serial, that would account for the difference.

Third, check to see if the ham book is giving you figures for a dipole way out in free space, or one that's mounted some known wavelength above some known ground, or that has some real resistivity in the wire, or some other 'real world' assumption that a pure theory book may not bother with.

--
Tim Wescott
Wescott Design Services
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Reply to
Tim Wescott

I can't say I understand what you mean here.

True.

OK

I've probably got some stuff on him here. I've got quite a few technical books - including Krass, Balanis and a few more.

As someone else said, this stuff can get very complex very quickly.

I'm not sure if the stuff in Lawsons book might be experimentally measured. It references some stuff by Uda et al, but it was published in a Tokyo University book - not exactly easy to trace, and I very much doubt in English.

That's not one I have. If I get involved in this work again, I might buy a copy.

Reply to
Dave

Thank you for that. If by chance you have that as a PDF, perhaps you can mail it to me. But if not, I'll try to get it for interest sake. I needed this for a piece of work, but the work will have finished by the time I get much more done. But at least I have a better understanding now.

OK. I'm just a bit suspicious of computer programs some times, as someone will have to choose an algorithm of some sort. But I assume you are talking of something like NEC which breaks antennas into segments.

Balanis has it, but leaves it as an integral, without simplifying like he does for the real part. Yet the formuals for hte real and imaginary parts look very similar. I might be able to attack it with a computer algebra system - maths never was my strongest subject.

I thought I'd looked in Krauss and not found it, but perhaps it is there. I think there is a relatively new version of Kraus, but my copy is quite old.

OK, thank you for that.

Reply to
Dave

Hi Dave,

Your kick-off was already complex. A thick-wire can be monstrously thick and not do much about the overall length at the principle resonance:

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It does have its advantages higher in frequency.

When we look at cage conicals, the flare angle of the conical shows interesting relationships - not so much at resonance as for the continuum of reactance and resistance. What I describe as optimal bears upon an arbitrary 50 Ohm relationship, but others might mine significance from the steeper skirts of the discone.

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73's Richard Clark, KB7QHC
Reply to
Richard Clark

. . .

I suppose it's natural to be more suspicious of others' work than your own. I've personally found the opposite to often be more appropriate.

But I assume you

Yes. You can find a good description of the method of moments in the second and later editions of Kraus. The fundamental equation can only be solved numerically, and the method of moments, used by NEC and MININEC, is an efficient way to do it.

Hallen's integral equation is exact, but it's not a formula, since you can't plug numbers into one side and get a result on the other. Nor can it be solved in closed form at all. That's why so much work was done on approximate solutions and on developing numerical solution methods. Feel free to write your own program to solve it, but such programs have existed for decades and have been verified countless times as well as being highly optimized.

Getting the resistance is pretty straightforward once you assume the shape of the current distribution. Assume some arbitrary current at the feedpoint which, along with the assumed current distribution, gives you the field strength in any direction. With the impedance of free space, this directly gives the power density. Integrate the power density over all space to get the total radiated power. Then you know how much power is radiated per ampere of current at the feedpoint, from which you can calculate the feedpoint resistance.

This calculation is done in all editions of Kraus, I'm sure; I have only the first and second, but I can't imagine it was deleted in later ones. Be careful when reading Kraus, however. Unlike many authors, he uses a uniform, rather than triangular, current distribution for his short elemental dipole examples. This is equivalent to a very short dipole with huge end hats, not just a plain short dipole. The half wavelength and other dipoles in his text are conventional.

Roy Lewallen, W7EL

Reply to
Roy Lewallen

Since you clearly know more about this stuff than me, do you know of the best freely available software for this which works under Unix? (I use Sun's Solaris for 99% of the things I do, including sending this message. I use Solaris on my laptop too, rather than Windows).

Hence I'm almost certainly looking for source code in either C, C++ or Fortran. Anything that works under Linux would almost certainly be able to be compiled for Solaris without too much effort. I found this page:

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which has some source. I downloaded one

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It would not compile immediately on my Sun. gcc 4.3.1 complained about some ambiguous code. gcc 3.4.1 did not, so I got past that bit.

It then tries to link with the 'blas', 'atlas' and 'lapack_atlas' libraries, none of which my Sun has.

I then swapped to the Sun C/C++ and Fortran compilers, removed references to 'blas', 'atlas' and 'lapack_atlas' , and replaced them with 'sublibperf' which is the optimised library on Solaris. That worked ok, and I had an executable:

$ ./nec2++ usage: nec2++ [-i] [-o] -g: print maximum gain to stdout. -b: Perform NEC++ Benchmark. -h: print this usage information and exit. -v: print nec2++ version number and exit.

I've not done any more than that at this point, but proved it will compile on Solaris with little effort.

Anyway, if you have any recommendations for the best freely available Unix/Linux code, I would be interested.

OK, I understand that.

I think I found what I was looking for in either Kraus or Balanis last night. The book is beside the bed, and as my wife is still asleep I'm not going to look for it.

Reply to
Dave

Sorry, my knowledge doesn't extend to that of programs suitable for Unix or Linux. The "Unofficial NEC archives" site is the best I know of for various compilations. Hopefully some of the other readers can help you out.

My program, EZNEC, has been reported to run under Linux using some versions of the wine Windows emulator, but not with others. You'll have a lot more to choose from if you can emulate Windows.

Roy Lewallen, W7EL

Reply to
Roy Lewallen

Thank you.

Thanks a lot. I'm just a bit anti-windows myself. Fed up with all the hassle of viruses etc.

The CPUs in this machine are not even capable or running Windows, as they are not AMD/Intel compatible.

I do of course have access to Windows machines, and might well check out your code later. I assume analysing a simple dipole will be trivual and not require a huge amount of work.

But you have helped me understand the differences in impedance between the different sets of data I have seen, which was my main concern. For the immediate future at least, I am not going to do any numerical analysis, but I might well look at that for a later date.

Dave

Reply to
Dave

Yes, it does.

yes, it does vary

Balanis is giving the usual closed form expression for self Z.. I think the original is from Schelkunoff or King.. I don't have my copy of Kraus in front of me so I can't check.

Perhaps Lawson is using a different approximation?

Some formulas make the assumption of a sinusoidal current distribution, others are more refined.

73.1 +j42.5 to be more accurate..

"exact" as in analytical expression with no error? Or good to less than a percent? Accounting for resistance of the element?

As a practical matter, I use NEC for this kind of thing (which does take into account resistance, etc.) You can set it up to zap out a table that you can then interpolate into, for instance.

However, there are a variety of formulas that one can use. I suggest taking a look at Orfanidis's book

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Chapter 16 is probably the one you want. Figure 16.3.1 for instance. As you noted, X varies a lot more slowly for fat elements (which is to be expected.. ). Chapter 22 is also quite handy. Equation 22.2.10 is the expression for Z11, which is an integration of F(z), given in

22.2.11. The author makes the point "In evaluating the self impedance of an antenna with a small radius, the integrand F(z) varies rapidly around z = 0. To maintain accuracy in the integration, we split the integration interval into three subintervals, as we mentioned in Sec. 21.10"

He has matlab procedures and functions for most of them.. imped.m is probably the one you want.

Reply to
Jim Lux

FORTRAN would be the language of choice (since that's what NEC was written *and validated* in.. one would be concerned about a C translation, although I'm sure there are C versions out there which have been validated)

There should be versions out there that don't link with the matrix math packages.

What you've got is probably as good as anything else, especially if you're just looking for a table of Z vs length and diameter.

Reply to
Jim Lux

Good point.

The Sun library 'libsunperf' has all the functions of blas, atlas and lapack_atlas (well at least alls those used by NEC). I simply needed to link against that one library, rather than the other 3, and I soon had a n executable. I've not used it yet, as I have more pressing things to do.

That sun library should be highly optimised for the UltraSPARC processors in my workstation.

Thank you for that.

Reply to
Dave

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