[Amps] Thoughts on half-wave dipole

[Steve Cook]

By the time you figure this one out mathematically, you could already have had the answer with an antennalizer and a pair of wire cutters.

-S
  ----- Original Message ----- 
  From: Dr. David Kirkby 
  To: 'AMPS' 
  Sent: Sunday, July 27, 2008 12:10 PM
  Subject: [Amps] Thoughts on half-wave dipole


  I know this is a bit off-topic. but possibly someone here has some 
  thoughts. I've posted it to rec.radio.amateur.antenna and 
  sci.electronics.design, but someone here might know. I've corrected a 
  couple of typos that appeared on the newsgroup post.


  --------

  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 Pi/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 a2)/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 diameter 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 G8WRB.






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