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 dot kirkby at onetel dot net

Oops, I made a couple of mistakes the


Dave wrote:
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.

Dave wrote:
Oops, I made a couple of mistakes the


Dave wrote:
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.

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
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

Dave wrote:
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.
. . .

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.


. . .

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.

. . .

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).

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.

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).

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

Roy Lewallen wrote:

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.


. . .

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.


. . .

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).

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.

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).

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.

Roy Lewallen, W7EL

Dave wrote:
. . .

OK. I'm just a bit suspicious of computer programs some times, as
someone will have to choose an algorithm of some sort.

.. . .

I might be able to attack it with a computer
algebra system - maths never was my strongest subject.

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
are talking of something like NEC which breaks antennas into segments.

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.

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).

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.

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.

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.

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

Roy Lewallen wrote:

But I assume you
are talking of something like NEC which breaks antennas into segments.

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.

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:

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

which has some source. I downloaded one

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++ [-iinput-file-name] [-ooutput-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.

Dave wrote:

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:

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

which has some source. I downloaded one

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

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

Dave wrote:
Roy Lewallen wrote:


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.

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.

Dave writes:

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:

NEC-2 can be built for Linux, and binaries are available for some
distributions. I am sure NEC-4 also can, if you get a license.

However, I have seen strange results come out of NEC-2 on Debian
Linux. Looked like numerical instability, and could be caused by a
problem in the toolchain. I don't think these binaries are used much,
so they don't receive the amount of tender loving care that they
deserve.

73
LA4RT Jon, Trondheim, Norway