# Does reactance of dipole depend on diameter ??

D

#### Dave

Jan 1, 1970
0
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]));

(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 <dot> kirkby <at> onetel <dot> net

D

#### Dave

Jan 1, 1970
0
Oops, I made a couple of mistakes there:

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

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.
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.

Sorry, I mean the reactance does not depend on radius when the dipole is
0.5 wavelengths in length.

D

#### Dave

Jan 1, 1970
0
Richard said:
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).

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

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

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

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.
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.

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.

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.

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

J

#### John Smith

Jan 1, 1970
0
Roy said:
...
Roy Lewallen, W7EL

This is just one more example of an anomaly which reminds one of "The
Emperor's New Clothes--Hans Christian Anderson."

That, for some strange reason, science cannot explain the relationship
of conductor diameter to length in an ABSOLUTELY predictable manner
smacks of "charlatanism." This is proof, in my humble opinion, that
gross errors exist on a very basic level of our
understanding/formulas/equations of RF and light ...

Einstein suspected "the answers" in a "small equation", perhaps as short
as an inch and a half long which would encompass "the theory of
everything." After pages of complex calculus/algerba/geometric
equations--we end up little better than "a guess." :-(

Surely there are some out there knowledgeable of a "hand-job!" <grin>

Obviously, something is amiss, and until that is corrected we do better
than "guess."

Regards,
JS

D

#### Dave

Jan 1, 1970
0
Roy said:
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]));

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.

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.
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.

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.
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.

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.
But again, you can get extremely accurate results from readily available
antenna modeling programs.

OK, thank you for that.

D

#### Dave

Jan 1, 1970
0
Roy said:
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.

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.

http://www.si-list.net/swindex.html

http://www.si-list.net/NEC_Archives/necpp-1.1.1.tar.gz

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<input-file-name>] [-o<output-file-name>]
-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.

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.

OK, I understand that.
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.

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.

D

#### Dave

Jan 1, 1970
0
Roy said:
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

Thank you.
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

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

J

#### Jim Lux

Jan 1, 1970
0
Dave said:
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.

Yes, it does.
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.

yes, it does vary
K X05 X045
-------------------------

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]));

(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
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).

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.

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).

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
http://www.ece.rutgers.edu/~orfanidi/ewa/

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.

J

#### Jim Lux

Jan 1, 1970
0
Dave said:
Roy Lewallen wrote:

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.

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

There should be versions out there that don't link with the matrix math
packages.
Anyway, if you have any recommendations for the best freely available
Unix/Linux code, I would be interested.

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.

D

#### Dave

Jan 1, 1970
0
Jim said:
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)

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

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.
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.

Thank you for that.

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