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#1
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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 dot kirkby at onetel dot net |
#2
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Does reactance of dipole depend on diameter ??
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. |
#3
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Does reactance of dipole depend on diameter ??
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 |
#4
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Does reactance of dipole depend on diameter ??
On Sun, 27 Jul 2008 19:46:40 +0100, 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. 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 |
#5
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Does reactance of dipole depend on diameter ??
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 |
#6
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Does reactance of dipole depend on diameter ??
Richard Clark wrote:
On Sun, 27 Jul 2008 19:46:40 +0100, 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. 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. 73's Richard Clark, KB7QHC |
#7
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Does reactance of dipole depend on diameter ??
Roy Lewallen wrote:
... 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 |
#8
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Does reactance of dipole depend on diameter ??
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 |
#9
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Does reactance of dipole depend on diameter ??
On Sun, 27 Jul 2008 23:04:02 +0100, Dave wrote:
As someone else said, this stuff can get very complex very quickly. 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: http://home.comcast.net/~kb7qhc/ante.../Cage/cage.htm 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. http://www.qsl.net/kb7qhc/antenna/Discone/discone.htm 73's Richard Clark, KB7QHC |
#10
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Does reactance of dipole depend on diameter ??
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 |
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